diff --git a/research/staged544-refined-endpoint/.gitattributes b/research/staged544-refined-endpoint/.gitattributes
new file mode 100644
index 0000000000..ec67521fb7
--- /dev/null
+++ b/research/staged544-refined-endpoint/.gitattributes
@@ -0,0 +1,3 @@
+* text eol=lf
+*.gz binary
+vendor/** -text
diff --git a/research/staged544-refined-endpoint/CLAIM.json b/research/staged544-refined-endpoint/CLAIM.json
new file mode 100644
index 0000000000..f33933b11e
--- /dev/null
+++ b/research/staged544-refined-endpoint/CLAIM.json
@@ -0,0 +1,13 @@
+{
+  "complex_lower": "173057843726624681874071321804985755655644807397156243331276259008002643975681/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "complex_upper": "86528921863312340937035660902492877827822403698578121665638129504001321987841/50000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "endpoint_digits": 80,
+  "fresh_native_replay": false,
+  "gain": "762590080026439756810099999999999999999999000099999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "kappa": "1730578437266246818740713218049857556556448073971562433312762590080026439756809999999999999999999998999999999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "physical": 323069,
+  "previous": "17305784372662468187407132180498575565564480739715624333119999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "private_zero_buffers": 91,
+  "replicas": 360,
+  "selected_frontier_changed": false
+}
diff --git a/research/staged544-refined-endpoint/FORMAL-CHANGE-AUDIT.json b/research/staged544-refined-endpoint/FORMAL-CHANGE-AUDIT.json
new file mode 100644
index 0000000000..ded0828307
--- /dev/null
+++ b/research/staged544-refined-endpoint/FORMAL-CHANGE-AUDIT.json
@@ -0,0 +1,30 @@
+{
+  "status": "PROOF_BODIES_UNCHANGED",
+  "changed_lines": {
+    "CapCertificate.lean": [
+      {
+        "line": 48,
+        "before_prefix": "def upper : Fin 7 \u2192 \u211a := ![(135077310945238073835664751237162262628210",
+        "after_prefix": "def upper : Fin 7 \u2192 \u211a := ![(135077310945238073835664751237162262628210"
+      }
+    ],
+    "AnalyticCertificate.lean": [
+      {
+        "line": 38,
+        "before_prefix": "def endpoint : \u211a := 54080576164570213085647288064058048642389002311611",
+        "after_prefix": "def endpoint : \u211a := 17305784372662468187407132180498575565564480739715"
+      },
+      {
+        "line": 41,
+        "before_prefix": "def upper : Fin 7 \u2192 \u211a := ![(100745261008386462509737099312640790310316",
+        "after_prefix": "def upper : Fin 7 \u2192 \u211a := ![(503726305041932312548685496563203951551581"
+      },
+      {
+        "line": 92,
+        "before_prefix": "/-- The actual real exponential moment at the published 60-digit endpo",
+        "after_prefix": "/-- The actual real exponential moment at the proposed 80-digit endpoi"
+      }
+    ]
+  },
+  "scope": "Textual change audit, not Lean elaboration/kernel check"
+}
diff --git a/research/staged544-refined-endpoint/FORMAL-DATA.json b/research/staged544-refined-endpoint/FORMAL-DATA.json
new file mode 100644
index 0000000000..4bdae6ee3d
--- /dev/null
+++ b/research/staged544-refined-endpoint/FORMAL-DATA.json
@@ -0,0 +1,32 @@
+{
+  "endpoint": "173057843726624681874071321804985755655644807397156243331276259008002643975681/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "log_upper": {
+    "1": "42904594411483911290921088574385425709047528448715976645956988571617899759205973/10000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "2": "179865613029422290983743836799018300141462635525567120523750942383419817697545129/50000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "3": "319184715248028143769686362051601686625726228704884821286100452352429568270198833/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "4": "290416508002850051025764461452218943475375257615108715635433883817500273198120787/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "5": "268102152871429075449134952421235493137915149060307994268305096568761098821293953/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "8": "44220357989371104016808249861280257333565048835816638044673176573632182200230263/20000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "48": "41925843024050020002793513468331059395526174960782719637828448555072119482320947/100000000000000000000000000000000000000000000000000000000000000000000000000000000"
+  },
+  "weight_upper": {
+    "1": "5037263050419323125486854965632039515515813123979119739049327448446247437415523839971594780304879053/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "2": "1006244849300832117227334850803589471353259962367000112587337836237263822185857853193329141685299883/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "3": "10055390258478318702735196789299380974598979106244048870901713891769435640370131535754404057635525197/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "4": "2512596341033269936571327050488918900797498810133573533611463569761893864457578011954826530420192591/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "5": "2511626245512671220465894009973910441561771954881139752642633015326322249610212835151571034808264101/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "8": "2509584174213792103224997148477336282910856045665342057938515539919371590220915369255384660757605463/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "48": "10007258228810539199340477838604803218049186436146540908000107607859989792524228358661467016084073803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+  },
+  "log_degree": 60,
+  "exp_degree": 26,
+  "root_upper": {
+    "1": "1709682071524341872977914446059068945687158662089760087588298513648466431140986964131832379/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "2": "78389268609117970295183841420183634117602514499308989820003491013074224898274587103932377/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "3": "59612205786190783200497858238480871280010105534840193468536084961687039700052165204929563/40000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "4": "718832762584181935863410006114861039343341562058796885073492442373725223855973513855130509/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "5": "349529725287554381854657482572542092721239471300292172144051890309678870642203459561245573/250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "8": "1318345099355356699940061975994465396292039242917024224119936446107390011400428568787269981/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "48": "3293140230695581904239101463758503566499283368816211814999286028419201556213646303933641/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+  }
+}
diff --git a/research/staged544-refined-endpoint/FRONTIER-COMPARISON.json b/research/staged544-refined-endpoint/FRONTIER-COMPARISON.json
new file mode 100644
index 0000000000..24905a43d9
--- /dev/null
+++ b/research/staged544-refined-endpoint/FRONTIER-COMPARISON.json
@@ -0,0 +1,9 @@
+{
+  "gain": "762590080026439756810099999999999999999999000099999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "head": "ef4315b8caa605e47a2515004845eb2d2ed71a86",
+  "new_kappa": "1730578437266246818740713218049857556556448073971562433312762590080026439756809999999999999999999998999999999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "physical_changed": false,
+  "previous_kappa": "17305784372662468187407132180498575565564480739715624333119999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "previous_pr": 3857,
+  "profile_changed": false
+}
diff --git a/research/staged544-refined-endpoint/LEAN-AXIOMS.txt b/research/staged544-refined-endpoint/LEAN-AXIOMS.txt
new file mode 100644
index 0000000000..40d2d83080
--- /dev/null
+++ b/research/staged544-refined-endpoint/LEAN-AXIOMS.txt
@@ -0,0 +1,6 @@
+info: CapCertificate.lean:69:0: 'Staged544Certificate.exact_certificate' depends on axioms: [propext, Classical.choice, Quot.sound]
+info: CapCertificate.lean:70:0: 'Staged544Certificate.admission' depends on axioms: [propext, Classical.choice, Quot.sound]
+info: AnalyticCertificate.lean:115:0: 'Staged544.actual_moment_lt_one' depends on axioms: [propext, Classical.choice, Quot.sound]
+info: AnalyticCertificate.lean:117:0: 'Staged544.actual_rpow_moment_lt_one' depends on axioms: [propext, Classical.choice, Quot.sound]
+Target module kernel replay: lake env leanchecker AnalyticCertificate, exit0.
+All-imports fresh kernel replay: NOT_RUN.
diff --git a/research/staged544-refined-endpoint/NOTICE b/research/staged544-refined-endpoint/NOTICE
new file mode 100644
index 0000000000..f53b314d9c
--- /dev/null
+++ b/research/staged544-refined-endpoint/NOTICE
@@ -0,0 +1,43 @@
+Endpoint refinement and packaging by hexapode with Codex assistance.
+Direct predecessor: Dugongue PR3857, head ef4315b8caa605e47a2515004845eb2d2ed71a86.
+Retained proof bodies, engines and provenance follow below without removed credit.
+
+Dugongue, with substantial OpenAI Codex assistance. New work: Apache-2.0.
+
+Direct current predecessors:
+- ChaloupeMusic / PR3815, head 77ffbf287b6ca198b4a31e3835db3bae60718b93:
+  changed T360 clock/fill, exact fixed-inventory packing cap and complete release.
+- hexapode / PR3802, head de2cd5c58009205210e3bf328b0ef5cadf3903af:
+  first publication of the same prior endpoint via T385; PR3795 is the T488 prior.
+- dataisfire / PR3734, head fa041037a3374eaa08feeca0b26a5d6893194147:
+  staged helper 544 word, current native release and retained outer arithmetic.
+- zxv / PR3638 and PR3682: chronological T360 integration and retimed suppliers.
+- Ross Richardson / PR3605: staged-entry mechanism and source supplier.
+- roger-sato / PR3642 and PR3693: commuting-release retiming and whole-word adapter.
+- Sagar Thakrar / PR3687: arbitrary-dirty independent interpreter.
+- Arham Tahir: retained exact free-operator/frame audit lineage.
+- SovereignSteak / PR3531: positive-binomial precision precedent; dataisfire /
+  PR3361: source-bound precision integration precedent. No numerical-method priority.
+
+The original native, rank-seven, geometry, owner, hosting, COPY, prefix and outer
+contributors retain their credit: Jeremy Diviney, dcharb78, Nick Petalas,
+Marius-Paul Dumitrean, Jeremy Kothe, Mason Hale, Yiyang Lu, Tekalign, Philippe
+Van Velzen, Sagar Thakrar and all authors named in retained source notices.
+Vlado Vojdanovski supplies the outer construction; Chafik Boukhalfa's fine-prime
+padding uses Tim Trudgian's cited prime result; gauge-free/union contributors,
+eumemic's 47-row assembly and the PR589 BIT lineage remain credited.
+
+The extracted MathProof sufficient-transfer proof has Dugongue/Codex provenance.
+Mathlib supplies the sufficient real exponential/Taylor/rpow lemmas. These are
+standard methods; no claim that nobody has previously applied them is made.
+
+The vendored native receipt came from PR3734's optional source archive:
+https://raw.githubusercontent.com/CrocSwap/integer-mult-bounds/fa041037a3374eaa08feeca0b26a5d6893194147/research/staged544-producer-return/REPLAY-INPUTS.tar.gz
+Its archive and member SHA-256 hashes are documented by the retained upstream
+REPLAY-INPUTS manifest. Bounded replay authenticates the vendored receipt and
+actual words/native frames, not omitted full cache bytes. The archive is an
+explicit optional full-review source, not a hidden runtime dependency.
+
+All exact retained files keep their source-specific attribution and Apache-2.0
+notices. SOURCE-CREDITS.md under vendor/staged544-producer-return preserves
+that package's own predecessor list.
diff --git a/research/staged544-refined-endpoint/OVERLAP.md b/research/staged544-refined-endpoint/OVERLAP.md
new file mode 100644
index 0000000000..942352c4ad
--- /dev/null
+++ b/research/staged544-refined-endpoint/OVERLAP.md
@@ -0,0 +1,7 @@
+# Scope and prior work
+
+This is a precision refinement of PR3857, head ef4315b8caa605e47a2515004845eb2d2ed71a86. It uses the unchanged PR3815 T360 owner allocation and PR3734 staged544 native supplier, with the same normalized profile as PR3802. No new circuit, physical packing, independent proof method, or method priority is claimed.
+
+The team refreshes open contributions during work; publication must recheck the then-current supported frontier. This draft itself does not assert exhaustive Discussion coverage or absence of overlap. Historical predecessor intake is preserved at predecessor/OVERLAP.md and is not a current freshness claim.
+
+The proposed improvement is only the strictly larger certified conditional endpoint in CLAIM.json. All physical, compiler, hosting, precision, exact recovery, analytic transfer and BIT premises remain as in PR3857. The unrelated fiber-norm proposal is excluded. No PR or selected-record change has been made for this draft.
diff --git a/research/staged544-refined-endpoint/PACKAGE-PINS.json b/research/staged544-refined-endpoint/PACKAGE-PINS.json
new file mode 100644
index 0000000000..246ae77b6c
--- /dev/null
+++ b/research/staged544-refined-endpoint/PACKAGE-PINS.json
@@ -0,0 +1,79 @@
+{
+  "excluded_self": "PACKAGE-PINS.json",
+  "files": {
+    ".gitattributes": "e7f3a9dced277ff518316d4450e6e51d79fdbf06056d1d234ee945fbb658c22e",
+    "CLAIM.json": "320de617fe7362d6c5ad01f454adbd1954c5365b203678bc73dc57e4bd895803",
+    "FORMAL-CHANGE-AUDIT.json": "e5bd4698065b66dcb71be0367dfd9815ec1655815b9b5b5c27683a61c0c9fa0f",
+    "FORMAL-DATA.json": "2727f051d8cafbb1a3f13e7efd8e094c93ee62dd704b11e2b34f7fbec08ce830",
+    "FRONTIER-COMPARISON.json": "d661aca9fac45babe27b35d0410c94277c9d518ffb630f112b07e427b1a19299",
+    "LEAN-AXIOMS.txt": "96eb521eb3d110ad7d20bd2ed38cd604be47c280b2524a3163f03800e58010e4",
+    "NOTICE": "6d619ff1d1ba1fcee39d3d466cb277ff3d94bcfdeeb45f5324c0aa7b9f64aca8",
+    "OVERLAP.md": "fa82bc5dd0a485c795f3c430d1ca79d79dd7cabb4c786372aa19650a83e41d0a",
+    "PRECEDENTS.md": "93ab87a7153a327f6814ac4231de670b31ddcc97f3ee811826bc7b5fa317fa0e",
+    "PREDECESSOR-CLAIM.json": "2b3a4ede0d77e4d9908f50f756559fa4a5f371f9d04a185a308641b601c8857a",
+    "PROOF.md": "10d0883086e4b3e55dee7880155742c118972183ff821fec5f3a5de001678962",
+    "README.md": "f4d5dcf711cc08f6a28d9b7e7bd4474dce33d2329909142787257e06d6e7f77d",
+    "RESULT.json": "6cbacc0c0e6a2c55bb7aa03b25963382354ce7d1cf53d5e1315b3b113349fe9b",
+    "SEARCH.md": "d7c806a984faf043ad6f9429d88f2e5a83bbd0313a790b52e97b27a2f25bad18",
+    "SOURCES.json": "0d0bf357f50639eff9a68c735ad57c1b748cc43c137a7cfd1fd7ba6c28c4ed24",
+    "TASK.json": "553e08c1cbab11652febb3cbbd803a35891f5737809ea3b4bc862e5d265b8018",
+    "VALIDATION.json": "a48ee96ba3e1f738c2f675089ae60169935fc2f8f57417c764d59ea5de3ff005",
+    "VENDOR.json": "30ee97d3076de3b117c544496758d02609259f9543cf5f1e698faf70d8f89443",
+    "VERIFY.json": "de52ae06af66d5664c75bfc2281354846be43c4c5f5de5d217a1c562c3c70435",
+    "dyadic_moment.py": "8dc2233a5c6138e26ac239a63c78d547f234097e8d05b8831dcf279ad4debbe0",
+    "evidence/LEAN-BUILD.log": "adb98c6ba7f47d1249710fd996267d33a728b8b37dad8a80ad95c50e5cdf5a81",
+    "evidence/LEAN-KERNEL.log": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855",
+    "lean/AnalyticCertificate.lean": "44b6c75086d95163e876563621676289f3c6348c584358106c301b6c8195ffaf",
+    "lean/CapCertificate.lean": "b1ba8ad1b65eeecaddf34168456d2f9a89eed5f36212bfcfa24107aa8fbe4833",
+    "lean/lake-manifest.json": "c6e31911303dc9025795b078e1be70b2639f6a54185c894d0d2c796bf15c49d0",
+    "lean/lakefile.toml": "3834d8881ac1275b9bc6ce7fee85d69941f5a3af130cffa3aa9b7fa06d9693ca",
+    "lean/lean-toolchain": "d5edba4e4b8faad9c1baeadb265716d20d03be4d1a2647dc5e35b0c0325bea7b",
+    "predecessor/CLAIM.json": "2b3a4ede0d77e4d9908f50f756559fa4a5f371f9d04a185a308641b601c8857a",
+    "predecessor/LEAN-AXIOMS.txt": "e85b210e44d543ab8267e2364eaffd17353e43c2ff3569d30731d18f8542e2b3",
+    "predecessor/OVERLAP.md": "6e94fd6c4aca5c5d7bffd5fa826b744a664f41831f69f49baccd02739c57249a",
+    "predecessor/PRECEDENTS.md": "d411509009449c377f21ecb81de23f8cc81f3c3b107012e0730ab608e4424d1c",
+    "predecessor/PROOF.md": "5817158d8a52188ff9c224625e269c33f22d7da16cedbeb2d648c2d9f527113a",
+    "predecessor/README.md": "a10b7b08a81b237f2e5fd5091d92e2b2bd5b52f61064c717559494fd58671c27",
+    "predecessor/SEARCH-HASHES.json": "659dab80d2ac83d547382d59cfc3822eede0e0f5b27c17e27e43cf5d7599fc9d",
+    "predecessor/SEARCH.md": "12acdc42cd62b274ed5160a0fe4e06001e81d505d09f9376df45a9d7340e79dc",
+    "predecessor/TASK.json": "71861df3b4ac868a05f9aa5eb06227acb35c9714435d76c213df83c82d82658b",
+    "predecessor/VALIDATION.json": "695a8162562e3f2efae80795b2a24da625fb6673236feaf949da92ba091865c1",
+    "test_arithmetic.py": "0d846c9673c2bb2ab750074f2de2f3718ff63a412d9b49893349cb996114cfab",
+    "test_cli.py": "3ef153d3b85299ecaa835cac343daccc00cc27cf31365bc578412f52de10a89e",
+    "vendor/bootstrap/assemble.py": "f25e1e870df5d5a16a1e367b3df8843cb01644ab4b75895d7939af81e4fcb833",
+    "vendor/bootstrap/outer.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+    "vendor/native/NATIVE-RECEIPT.json": "7ddf3f80b2fa61d1ab12035da9ef1c95d43f76077df819d7683144d7862a3f93",
+    "vendor/staged544-clock-ceiling/PRICE.json": "f989112add9b51d3a9d8be458defafadc85d84add9a4f476ec0f81d7a72f1be0",
+    "vendor/staged544-clock-ceiling/README.md": "0c52ebccffc4fbd7b95512506851157a553bd15b65cfb8ee0ca11247bffb3f41",
+    "vendor/staged544-clock-ceiling/ceiling_certificate.py": "2276f1c5d341db63dcf511f6398733b1efbe9d09e2e9aa86b50d7a160a0f7d03",
+    "vendor/staged544-clock-ceiling/clock.py": "b29fcc9386c60ed4e7aa6c35f7e1403beaac82c6cd2ed472436ef4f1c4a39aaa",
+    "vendor/staged544-clock-ceiling/evidence/RELEASE.json": "9e61d0c0e1d59a6719234a64f558d4e5918bb69555019a7276ecafb1b34dfc1c",
+    "vendor/staged544-clock-ceiling/inputs/FILL.json.gz": "50e4635123c0551a1a3c8f4cab870bcccc3e579c24447abe9d81336fde91f883",
+    "vendor/staged544-clock-ceiling/inputs/ORDER.json": "743e74b538432e2b2045e754b0de9c18f36fc5039fdb8eb542177a898ec78e44",
+    "vendor/staged544-clock-ceiling/inputs/PROFILE.json": "9686df0d8e710c2e94aef3887f4bf130a05b64215faeae702cccb59e95a469d5",
+    "vendor/staged544-clock-ceiling/verify.py": "df925b428dded034edc316518661e15c46d952f7071488a22ac64c14d43f1ac7",
+    "vendor/staged544-producer-return/PRICE.json": "eb89e98609f7c6b3c4281feededbde11b30ae56696b2756a88b318447d1c92ca",
+    "vendor/staged544-producer-return/PROOF.md": "be5fa8e381da957d52d02b154e2420f4459d7c8b30e23e82e245c11c978e3b62",
+    "vendor/staged544-producer-return/REPLAY-INPUTS.json": "727691274be6f91059c763dfa8206f7bca4516c5acf4703da403789a49b5e1e7",
+    "vendor/staged544-producer-return/SOURCE-CREDITS.md": "d4a13d8b8ed9cdb594f00efb3d1a336e09d934423330f02a1514308a720c278b",
+    "vendor/staged544-producer-return/audit_core.py": "888679ca8502cdcb100420fe5382a052529a494bf76cb332928b86e62612b951",
+    "vendor/staged544-producer-return/audit_verify.py": "99971d3cc143cc894600e08c8cfbad3e8f537a544212f21dfbe4271cfa97354d",
+    "vendor/staged544-producer-return/evidence/NATIVE-base.json": "d12ca528547af356176837b83e0de08a33fe5e59cc851238f321388dfbd81383",
+    "vendor/staged544-producer-return/evidence/NATIVE-extra.json": "dd3c7d52685a056415c7a5d0582cf83b4b1248b7e06913c288efd5b05725eeb6",
+    "vendor/staged544-producer-return/inputs/DISPATCH.json": "5d6fa711d704a1b498e5fe066ea821c4fa0217463702e82a7ce5be9da4023e8a",
+    "vendor/staged544-producer-return/inputs/NATIVE-base.json.gz": "bfe0cc0badac6c6d18461ceea0b299162c0f280c05f81b63f11fcc06cf9180e6",
+    "vendor/staged544-producer-return/inputs/NATIVE-extra.json.gz": "c5eec94a64cecfd6b83de05619628ad4868e889d62a55b49973f09705cc60448",
+    "vendor/staged544-producer-return/inputs/ORDER.json": "a2f308937e66c354a1c612556a8ad34c45bf685401238f5358a7bb9d65965ed4",
+    "vendor/staged544-producer-return/inputs/PLAN.json.gz": "d566cfb87b71639d4138e1f89b543e2dd6a7772070226a2a14c799a6835bf21e",
+    "vendor/staged544-producer-return/inputs/TEMPLATE.json.gz": "21297b20f1f2dc76eb9dd8124416068c3cb096bab9df17fa4da19bc68dfe082e",
+    "vendor/staged544-producer-return/outer-replay/outer/check_ledger.py": "1dd56009986eb14dd6603c9193c7ad71dd45c7d005bf71c9cd3f916e66f54d98",
+    "vendor/staged544-producer-return/outer-replay/outer/independent_check.py": "a5a0361539ec7cae9eb6c0371f857919335a1720040f22af291aa3e34fc864c7",
+    "vendor/staged544-producer-return/owner_audit.py": "56405eacc7c2dd0f7834ed5d62a4db8303c71657300eb4e4bbe020319552cde1",
+    "vendor/staged544-producer-return/rational_price.py": "ee38ccbc94411b2c9c7ab88b1f6d1b34debdafa896bd5f4b7214f20168f90dab",
+    "vendor/staged544-producer-return/words/BORROW-WORD-43base.json.gz": "5ca4be99851f8878cbfe0fd6c454df926b11218f17feefa1664721cf03156699",
+    "vendor/staged544-producer-return/words/BORROW-WORD-43extra.json.gz": "965268a9be73d3f62e7ef680d18f108cf064bfe1f44b82157baa61443232904e",
+    "verify.py": "a2efb04f43915bb55292c7c66312cc156410d17d753f9484c93d74f9bd3f998a"
+  },
+  "scope": "Exact private release file inventory; repository registration and publication performed separately.",
+  "status": "FROZEN_PRIVATE_RELEASE"
+}
diff --git a/research/staged544-refined-endpoint/PRECEDENTS.md b/research/staged544-refined-endpoint/PRECEDENTS.md
new file mode 100644
index 0000000000..5ab4633329
--- /dev/null
+++ b/research/staged544-refined-endpoint/PRECEDENTS.md
@@ -0,0 +1,5 @@
+# Retained provenance
+
+PR3857 by Dugongue supplies the dyadic evaluator, generic Lean proof bodies, formal logarithm data and portable verification structure reused here. The changed endpoint and concrete rational upper weights are prepared by hexapode with Codex assistance. Earlier physical and analytic sources retain their in-file licenses, VENDOR.json pins, NOTICE, and predecessor/PRECEDENTS.md credits.
+
+Historical validation belongs only to the predecessor sources. VALIDATION.json records whether the changed concrete Lean certificates have actually been checked.
diff --git a/research/staged544-refined-endpoint/PREDECESSOR-CLAIM.json b/research/staged544-refined-endpoint/PREDECESSOR-CLAIM.json
new file mode 100644
index 0000000000..9d92276c83
--- /dev/null
+++ b/research/staged544-refined-endpoint/PREDECESSOR-CLAIM.json
@@ -0,0 +1,13 @@
+{
+  "complex_lower": "54080576164570213085647288064058048642389002311611326041/31250000000000000000000000000000000000000000000000000000000",
+  "complex_upper": "1730578437266246818740713218049857556556448073971562433313/1000000000000000000000000000000000000000000000000000000000000",
+  "endpoint_digits": 60,
+  "fresh_native_replay": false,
+  "gain": "7132180498575565564481739715624333120000000000999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "kappa": "17305784372662468187407132180498575565564480739715624333119999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "physical": 323069,
+  "previous": "17305784372662468187399999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+  "private_zero_buffers": 91,
+  "replicas": 360,
+  "selected_frontier_changed": false
+}
diff --git a/research/staged544-refined-endpoint/PROOF.md b/research/staged544-refined-endpoint/PROOF.md
new file mode 100644
index 0000000000..77b1016f44
--- /dev/null
+++ b/research/staged544-refined-endpoint/PROOF.md
@@ -0,0 +1,9 @@
+# Proof scope and changed constants
+
+All generic proof bodies and analytic enclosure arguments are retained from PR3857 (predecessor/PROOF.md). The changed concrete endpoint and upper weights are regenerated at80 digits. The unchanged dyadic evaluator certifies both adjacent signs; the retained independent alternating rational engine also certifies them. The complete physical profile is unchanged.
+
+The Lean analytic certificate retains the same rational eighth-root and logarithm bounds,60-term logarithm Taylor check and26-term exponential bound. Each new rational exponential upper bound is rounded outward to a100-digit grid. Exact rational preflight verifies the weighted bound strictly below73*(896*360). Both changed concrete Lean modules were freshly compiled; target-module kernel replay of AnalyticCertificate passed. The printed theorem dependencies are only propext, Classical.choice and Quot.sound. A fresh replay of all imported Mathlib modules was not run; historical predecessor success is not used as a pass for changed files.
+
+Set C to CLAIM.json complex_lower, DIG=80. The unchanged outer ledger uses R0=10^-160, epsilon=1−(1025/1024)R0, floor grid10^126, and backoff10^-102. It derives kappa=C−10^-126−10^-102, with63 strict rows and17 rejection controls. The separate retained2048-payload carried bootstrap checks47 strict rows and4 controls. No fiber-prefix proposal or degree3 change is included.
+
+Native evidence is authenticated reuse, exactly as PR3857; actual dirty words, owner allocation, local bill and full seven-rank profile are freshly checked by the portable verifier. No new native-chain replay or unconditional multiplication theorem is asserted.
diff --git a/research/staged544-refined-endpoint/README.md b/research/staged544-refined-endpoint/README.md
new file mode 100644
index 0000000000..bede59f6ac
--- /dev/null
+++ b/research/staged544-refined-endpoint/README.md
@@ -0,0 +1,38 @@
+# Certified endpoint refinement of PR3857
+
+This package refines the same conditional multiplication construction to
+
+```
+kappa = 0.00173057843726624681874071321804985755655644807397156243331276259008002643975681
+      - 10^-102 - 10^-126.
+```
+
+The strict gain over PR3857 is approximately 7.6259e-61. This is a numerical refinement, not a new geometry or additional verification coverage merely from more digits. The seven-rank profile, 323,069 physical families, 91 private zero buffers, and 2,048-bit payload allowance are unchanged.
+
+The regenerated concrete Lean certificates compile and target-module kernel replay passes with Lean 4.34.1 and pinned Mathlib. Python 3.12 scientific and corruption checks pass. Full native caches, all-size compiler/hosting, source, precision/tape/recovery, analytic and BIT assumptions remain inherited. No fresh complete native-chain run, all-imports fresh kernel replay, independent human review, or unconditional multiplication theorem is claimed.
+
+## Reproduction
+
+From the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/staged544-refined-endpoint
+python3.12 -B scripts/check_research_commit.py --package research/staged544-refined-endpoint --check-source-history
+```
+
+The portable verifier can also run directly as `python3.12 -B verify.py` inside this directory. It regenerates finite scientific evidence and checks proof-source bindings; it does not invoke Lean. To reproduce the separate formal checks, enter `lean/`, install the pinned toolchain, then run:
+
+```sh
+lake exe cache get
+lake build
+lake env leanchecker AnalyticCertificate
+```
+
+See VALIDATION.json and evidence/ for the completed checks. Tiny sign summaries inherited from the 60-decimal display may round to zero; exact rational comparisons and the dyadic/Lean margins retain their strict signs.
+
+## Open next questions
+
+- Can the exact finite moment certificate be composed with a formal all-size supplier and outer-transfer theorem? Start by connecting the strict moment hypothesis to the pinned profile. The implementation and compiler assumptions remain the blocker.
+- Which changed supplier can improve the attained staged544 profile? Screen an exact operator change against the complete inventory cost before native admission. A material gain requires such a construction rather than further endpoint digits.
+
+Prepared by hexapode with substantial OpenAI Codex assistance. Dugongue's PR3857 dyadic evaluator, generic Lean proofs, formal logarithm bounds, and inherited physical/outer engines retain their attribution and licenses. The construction retains ChaloupeMusic/PR3815, hexapode/PR3802, dataisfire/PR3734 and all source ancestry credited in NOTICE and predecessor/PROOF.md.
diff --git a/research/staged544-refined-endpoint/RESULT.json b/research/staged544-refined-endpoint/RESULT.json
new file mode 100644
index 0000000000..06bf07a72b
--- /dev/null
+++ b/research/staged544-refined-endpoint/RESULT.json
@@ -0,0 +1,979 @@
+{
+  "arithmetic": {
+    "alternating_signs": {
+      "accepted_upper": "0",
+      "rejected_lower": "0"
+    },
+    "certificate": {
+      "bits": 512,
+      "digits": 80,
+      "evaluations": 261,
+      "exp_terms": 96,
+      "log_terms": 256,
+      "lower": "173057843726624681874071321804985755655644807397156243331276259008002643975681/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "margin_lower": "8365237875737848726635318629890738259139943616589179184110613667149466124665907/548111188176053369430586141926654989691356474745817041281339191819345713549406600411660621309063012123968502362663549758901458729352375779410207571368717844480",
+      "margin_upper": "28719308285112189142614235368182726006072454394191073626743058500867677154045077/2192444752704213477722344567706619958765425898983268165125356767277382854197626401646642485236252048495874009450654199035605834917409503117640830285474871377920",
+      "upper": "86528921863312340937035660902492877827822403698578121665638129504001321987841/50000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "weights": {
+        "1": {
+          "lower": "13507731094523807383566475123716226262821080735013659937896950134837434477991620304205855756237792292311750260853556038594859174301966598591383088728416814",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13507731094523807383566475123716226262821080735013659937896950134837434477991620304205855756237792292311750260853556038594859174301966598591383088728416915"
+        },
+        "2": {
+          "lower": "13491537669919590444070783937941464752554609528295312364120075762630530816279756695957039929900401848255443706168261710262397613766253017942706209006837101",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13491537669919590444070783937941464752554609528295312364120075762630530816279756695957039929900401848255443706168261710262397613766253017942706209006837203"
+        },
+        "3": {
+          "lower": "13482074124629314266984853423762132196619008555772618820195884044083409682596544977661357694011530838439984600059835445921466208152729089541843752149002481",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13482074124629314266984853423762132196619008555772618820195884044083409682596544977661357694011530838439984600059835445921466208152729089541843752149002581"
+        },
+        "4": {
+          "lower": "13475363658420252292726653566578878147646435678341933346136213519761447932248379861556076157777577929289644834201241414284156073003775118317955380101246396",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13475363658420252292726653566578878147646435678341933346136213519761447932248379861556076157777577929289644834201241414284156073003775118317955380101246497"
+        },
+        "48": {
+          "lower": "13417539623722925629684767919185678866775428178768770102551675998974478591678481675509425432160414596693798883228869483043188943937259390146824381395561826",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13417539623722925629684767919185678866775428178768770102551675998974478591678481675509425432160414596693798883228869483043188943937259390146824381395561924"
+        },
+        "5": {
+          "lower": "13470160916654698189586785076891163906547046740939775262388900537080309894800792109484478840209598056770772511656870622915958070296415813340584899035271169",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13470160916654698189586785076891163906547046740939775262388900537080309894800792109484478840209598056770772511656870622915958070296415813340584899035271269"
+        },
+        "8": {
+          "lower": "13459209036752850346611399332698520957888451347969204958854153028399849634185585362065661352009007298720371270575038487256688169510878983253236696889686538",
+          "scale": "13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096",
+          "upper": "13459209036752850346611399332698520957888451347969204958854153028399849634185585362065661352009007298720371270575038487256688169510878983253236696889686637"
+        }
+      }
+    },
+    "complex_lower": "173057843726624681874071321804985755655644807397156243331276259008002643975681/100000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "complex_upper": "86528921863312340937035660902492877827822403698578121665638129504001321987841/50000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "gain": "762590080026439756810099999999999999999999000099999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "kappa": "1730578437266246818740713218049857556556448073971562433312762590080026439756809999999999999999999998999999999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "previous": "17305784372662468187407132180498575565564480739715624333119999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "profile": {
+      "1": 7989840,
+      "2": 2349000,
+      "3": 749088,
+      "4": 330480,
+      "5": 69120,
+      "8": 25920,
+      "48": 137952
+    }
+  },
+  "bootstrap": {
+    "controls": [
+      "next_grid",
+      "zero_eta",
+      "zero_beta",
+      "unit_beta"
+    ],
+    "cutoffs": [
+      {
+        "carry_bits_bit_length": 36,
+        "cutoff_log2_power_of_two_exponent": 45,
+        "exponent_numerator_sha256": "374c8fe18aa79a341e421b1b8db32b6ddbd737b5c5307a005f192ca989b476f5",
+        "gap_numerator_sha256": "d7708ffc90f82426fa11c5a7334a267e63ebd4b17603f7f1a435562de261e627",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "1a1cf797fabe7f95836fabeca626907c77b3e6c9aff7c2290b396a238c69362e",
+        "step": 0
+      },
+      {
+        "carry_bits_bit_length": 48,
+        "cutoff_log2_power_of_two_exponent": 66,
+        "exponent_numerator_sha256": "55777245a25fd911a79a70a1b3209330230a470858e7ba91027f14b8508dc699",
+        "gap_numerator_sha256": "2915e0eae11ffe78b387a1a908e995320ec287b43ec05ffd8d5aad562289b99d",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "4c8082d12f373fef16d09a213fe9edbeb95dc0ce17025aaa4b4f5e6ed1ea0067",
+        "step": 1
+      },
+      {
+        "carry_bits_bit_length": 60,
+        "cutoff_log2_power_of_two_exponent": 87,
+        "exponent_numerator_sha256": "ca20ed235f8b9ac5089f6011b5b7848b77ef77474a1552925a4b8a7f4441f40c",
+        "gap_numerator_sha256": "6d22fbc58e1abb4aa06ed4e211e724d153e6d1ae030d63202d7dd8afc07542e3",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "38e413929983db6b0d2656b56c36a0bd1902d0f2e7bbf8c0a152c62dc342b399",
+        "step": 2
+      },
+      {
+        "carry_bits_bit_length": 72,
+        "cutoff_log2_power_of_two_exponent": 108,
+        "exponent_numerator_sha256": "3ca5c580502a8ec8909a21e3119c676973684f92659786025f01044141479903",
+        "gap_numerator_sha256": "f53b87b775a098a74a252de0b3f780cbea96459ac0adbc1ed4db3861f12fcee3",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "119699ac1ada3de61face2c8217cce0cc21aa0dbad31259b7e67c15418756b49",
+        "step": 3
+      },
+      {
+        "carry_bits_bit_length": 85,
+        "cutoff_log2_power_of_two_exponent": 130,
+        "exponent_numerator_sha256": "ee2ccaba410aa74d8b1374800d11bbb91394f98288b6667e70f8a5483dba984b",
+        "gap_numerator_sha256": "a3ea719f3bcdb00ba0ab5b4783d0efe710ad30d57969da5a3c0895a0d82c65b7",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "12ac9d4cc858a49f571b8f7fe55b2e9763e69ba26b39f4c1eb44cfd59b60f3c5",
+        "step": 4
+      },
+      {
+        "carry_bits_bit_length": 97,
+        "cutoff_log2_power_of_two_exponent": 152,
+        "exponent_numerator_sha256": "54df1c803fad145ab7eff23124feed466e3ee89bdf2f838d454969984fb1ad6b",
+        "gap_numerator_sha256": "d2a5b61408028cc6c79c3e60b09e390381b74117e7d50edf85886e06a60bcc24",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "56b93d36d348411e843244865ee133fd19dd7822a387b0fad9622b8a69c6a4c0",
+        "step": 5
+      },
+      {
+        "carry_bits_bit_length": 109,
+        "cutoff_log2_power_of_two_exponent": 173,
+        "exponent_numerator_sha256": "80d8a2499a63b3c8b89ce0df3f6150d41cc7af52253c13100afd752013362bc5",
+        "gap_numerator_sha256": "c004dfed547910e4bbe3d472d0e7e66fc1fcf1b1514abe29e2b2024a61ce2a54",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "fb9c5bcb016bc82ac97b54403bb567af60d3be422a6f966c9ef0dcfe59f9b0f8",
+        "step": 6
+      },
+      {
+        "carry_bits_bit_length": 121,
+        "cutoff_log2_power_of_two_exponent": 194,
+        "exponent_numerator_sha256": "17fbc2e2407579c1b44cb8ab8dd6547c3a4efa0c981cda2eda07046e592670bd",
+        "gap_numerator_sha256": "39ab746f47a7ef2d686c1fb2d726e4c8b0c9cda39c8c10e350fa479cab0c64c0",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "7b83afcf22052370a31748146fac3d66c1d5a30971d6cedcc8a8940343994fee",
+        "step": 7
+      },
+      {
+        "carry_bits_bit_length": 133,
+        "cutoff_log2_power_of_two_exponent": 215,
+        "exponent_numerator_sha256": "02d2e7429453ebd89dff9657c9598dcfc551f8eb9fc5cee23a081fbf65cefcca",
+        "gap_numerator_sha256": "da047ac2aef0e13757a97c17934912f8a2a1b7992e0ad91af02f980a5f57ac33",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "e48f014be7244ab246d92419e0d2316251b8cfb7936063d1b1619611e9819ac7",
+        "step": 8
+      },
+      {
+        "carry_bits_bit_length": 145,
+        "cutoff_log2_power_of_two_exponent": 236,
+        "exponent_numerator_sha256": "777414108e06c705551360142621d66fa930730133f486702330db86e9b3b4bb",
+        "gap_numerator_sha256": "db8c72606e038eb06ab997930383466ca7d7676a7fd9acfc0bfbb14d7c533e82",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "3324e60bfc1d6ccf32d87a3e5f853884f223f7fcd3080b5cbc35aa58a54b92fe",
+        "step": 9
+      },
+      {
+        "carry_bits_bit_length": 158,
+        "cutoff_log2_power_of_two_exponent": 258,
+        "exponent_numerator_sha256": "488558bbf61dd6cf87ce69e916d42299fabd2b34d770b1d5cd8b63006628d40f",
+        "gap_numerator_sha256": "8b91f412275194ba3d948633ea22bce8d11aa48038858031abdae8d7188604f0",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "502e9d7d978ae2371a16049ea2c0f55939ad01b3429834196df22212a1f5b3aa",
+        "step": 10
+      },
+      {
+        "carry_bits_bit_length": 170,
+        "cutoff_log2_power_of_two_exponent": 280,
+        "exponent_numerator_sha256": "8e63979c55fc1ad9c294d3ad63e9734af2474b6f57f08d9b134e751d014a07bb",
+        "gap_numerator_sha256": "ef80c8bb5c26d3f2617bcafcbc93ca2a4a43ecbc00f128a8a4a6568f842e17a2",
+        "grid_bits": 240,
+        "log2C_bound_sha256": "a1078cc38627491719de86e8a63d18a109757fa311c81035a19a574257d6b124",
+        "step": 11
+      },
+      {
+        "all_rejected": true,
+        "control": "uncarried_cutoffs_fail_carry_rule",
+        "stages_checked": [
+          2,
+          3,
+          4,
+          5,
+          6,
+          7
+        ]
+      }
+    ],
+    "legacy_balanced_kappa": "863794354748563234780516486717720277/500000000000000000000000000000000000000",
+    "strict_constraints": {
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+              "decimal": "1E-160",
+              "rational": "1/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
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+            "q_below_effective": {
+              "decimal": "5E-161",
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+            "stop_positive": {
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+            },
+            "strassen_integer_power_gap": {
+              "decimal": "1951441519231",
+              "rational": "1951441519231"
+            },
+            "support_router_span": {
+              "decimal": "3.0029296875E-160",
+              "rational": "123/409600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+            },
+            "theta_positive_polynomial_power": {
+              "decimal": "2",
+              "rational": "2"
+            },
+            "union_header_fits_record": {
+              "decimal": "3.0029296875E-160",
+              "rational": "123/409600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+            },
+            "union_tag_work_per_visit_fits_record": {
+              "decimal": "3.0029296875E-160",
+              "rational": "123/409600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+            }
+          },
+          "xi": {
+            "decimal": "0.9079805784372662468187407132180498575565564480739715624333127625900800264397568099999999999999999999999999999999999999999999999999999999999999999999999999999997997291169044728610362100360304198140629423477269706199712748455415203098534423114990234375000000000000000000000000000000",
+            "rational": "1859544224639521273484780980670566108275827605655493759863424537784483894148621946879999999999999999999999999999999999999999999999999999999999999999999999999999589845231420360419402158153790299779200905928144835829701170883669033594579849853950000000000000000000000000000000000000000000000000000000000000000000000000000041/2048000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+          },
+          "zeta": {
+            "decimal": "1E-162",
+            "rational": "1/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+          }
+        }
+      },
+      "scope": "Conditional fine-prime padding composition. Fresh native admission is required by verify.py. The cited prime theorem, written gauge-free theorem and inherited all-size interfaces are not proved by this arithmetic. No independent human review is claimed.",
+      "source_sha256": {
+        "CLAIM.json": "24ca6b3c4a72a378cb5627fa0d0412c49d4296611dcdcf70170160a2790d943b",
+        "new/SPEC.json": "e69c6d934a8caefa137da1068ecf1b1c82ea553471a9ba40198775f0aeeda1c3"
+      },
+      "status": "PASS_FINE_PRIME_PADDING_EXACT_ARITHMETIC"
+    }
+  },
+  "scope": "Conditional unchanged PR3815/PR3802 supplier; all inherited general realization, compiler, hosting, precision, analytic and BIT assumptions remain",
+  "status": "PASS_STAGED544_CURRENT_CONDITIONAL_ENDPOINT"
+}
diff --git a/research/staged544-refined-endpoint/SEARCH.md b/research/staged544-refined-endpoint/SEARCH.md
new file mode 100644
index 0000000000..c6fc3fcd25
--- /dev/null
+++ b/research/staged544-refined-endpoint/SEARCH.md
@@ -0,0 +1 @@
+No new circuit search. Exact adjacent-grid evaluation at65,70,75,80digits;80 selected after two independent enclosures and formal numeric checks. This is a small endpoint refinement of unchanged geometry. Structural searches remain private and are not part of this package.
diff --git a/research/staged544-refined-endpoint/SOURCES.json b/research/staged544-refined-endpoint/SOURCES.json
new file mode 100644
index 0000000000..0593d8fbd1
--- /dev/null
+++ b/research/staged544-refined-endpoint/SOURCES.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    }
+  ]
+}
diff --git a/research/staged544-refined-endpoint/TASK.json b/research/staged544-refined-endpoint/TASK.json
new file mode 100644
index 0000000000..c52eb44605
--- /dev/null
+++ b/research/staged544-refined-endpoint/TASK.json
@@ -0,0 +1,73 @@
+{
+  "actual_scope": "Strict conditional endpoint refinement of PR3857 on exactly unchanged physical supplier/profile, with regenerated concrete finite analytic certificates.",
+  "ai_assistance": "Codex assistance; retained predecessor proof and engines credited to Dugongue and their source authors.",
+  "author": "hexapode",
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "bottleneck_link": "Small certified frontier increase; structural research remains separate.",
+  "budget_minutes": 60,
+  "evidence": "Changed concrete finite and analytic Lean modules compile; target-module kernel replay passes with standard axioms. Python3.12 portable scientific and semantic-corruption controls pass. All-imports fresh kernel replay not run; no fresh nativechain.",
+  "open_questions": [
+    {
+      "blocked_by": "The inherited all-size/compiler, hosting, source, precision, physical and BIT assumptions remain outside this finite certificate.",
+      "first_step": "Freeze one existing outer transfer statement and connect its strict moment hypothesis to the exact current profile and endpoint.",
+      "question": "Can the exact finite moment certificate be composed with a formal sufficient all-size supplier/outer theorem?",
+      "why_it_matters": "This would turn finite analytic admission into a stronger multiplication theorem under explicit implementation hypotheses."
+    },
+    {
+      "blocked_by": "The bounded retiming, staged-entry and commutator searches in this session yielded no beating paid-profile witness.",
+      "first_step": "Screen a source-bound exact operator change against the full fixed-inventory dual before complete native admission.",
+      "question": "Which changed word raises the attained staged544 inventory ceiling?",
+      "why_it_matters": "The numerical refinement here is tiny; a material improvement requires changing the paid construction."
+    }
+  ],
+  "original_card": {
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "first_experiment": "Prioritize a bounded independent audit of #2979: reconstruct one joint group and its single-lowering control, check the declared frames/chronology, then trace changed children, idle widths and cleanup buffers into the complete invoice and both supplier caps. Choose one still-missing implication after checking existing audits. Stop with exact unresolved dependencies if full admission exceeds the session; do not launch a new lowering census on an unvalidated premise or substitute extra root digits for mechanism checking.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #2979 joint rank-7 end-frame proposal and the revision-6 evidence status. Its claimed kappa is not selected. Compare coordinated changes with the losing single-change controls and locate the complete physical ledger; an aggregate moment alone does not validate the mechanism.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "id": "independent-moment",
+    "methods": [
+      "Independently reconstruct a coordinated frame/circuit change and its single-change control, then derive the complete rank and physical ledger.",
+      "Implement certified moment inequalities through a different exact enclosure method."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering"
+    ],
+    "question": "Does one changed supplier or assembly claim lack a source-bound independent check of a specific new implication?",
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Promoting #2979 from merged source archive to validated supplier solely from author PASS receipts or aggregate pricing."
+    ],
+    "starting_points": [
+      "certificates/selected-result.json",
+      "research/rank7-joint-endframes/README.md",
+      "docs/research/research-family-status.md"
+    ]
+  },
+  "outcome": "completed",
+  "priority_revision": 6,
+  "publication_status": "Locally validated private release; no PR or selected-frontier change yet.",
+  "schema_version": 1,
+  "scope_change_reason": "User explicitly requests publishing every proven kappa improvement; this is extra certified endpoint precision, not new mechanism or new verification coverage.",
+  "scope_status": "changed",
+  "stop_rule": "Bound new-word searches and preserve failures. Keep the final quarter for proof/kernel, registered replay, source history and publication checks; do not extend the attained fixed-inventory packing family.",
+  "task_id": "independent-moment"
+}
diff --git a/research/staged544-refined-endpoint/VALIDATION.json b/research/staged544-refined-endpoint/VALIDATION.json
new file mode 100644
index 0000000000..ee93d541b3
--- /dev/null
+++ b/research/staged544-refined-endpoint/VALIDATION.json
@@ -0,0 +1,65 @@
+{
+  "inherited_scope": "All compiler/hosting/precision/allsize/analytic/BIT assumptions unchanged. Payload2048 retained.",
+  "lean": {
+    "actual_theorems": [
+      "Staged544.actual_moment_lt_one",
+      "Staged544.actual_rpow_moment_lt_one"
+    ],
+    "all_imports_fresh_replay": "NOT_RUN; no complete fresh replay of Mathlib dependencies claimed.",
+    "axioms": [
+      "propext",
+      "Classical.choice",
+      "Quot.sound"
+    ],
+    "build_log_sha256": "adb98c6ba7f47d1249710fd996267d33a728b8b37dad8a80ad95c50e5cdf5a81",
+    "compiled_source_sha256": {
+      "AnalyticCertificate.lean": "44b6c75086d95163e876563621676289f3c6348c584358106c301b6c8195ffaf",
+      "CapCertificate.lean": "b1ba8ad1b65eeecaddf34168456d2f9a89eed5f36212bfcfa24107aa8fbe4833"
+    },
+    "elaboration_exit_code": 0,
+    "mathlib_commit": "d13f23b723b8a846827a245b89c10fc7d3f11612",
+    "platform": "macOS arm64",
+    "status": "PASS_CHANGED_CONCRETE_CERTIFICATES",
+    "target_module_kernel_replay": {
+      "command": [
+        "lake",
+        "env",
+        "leanchecker",
+        "AnalyticCertificate"
+      ],
+      "exit_code": 0,
+      "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855"
+    },
+    "version": "4.34.1"
+  },
+  "native_reuse": "Unchanged PR3734 native evidence and PR3815 allocation, exactly as PR3857. No fresh nativechain.",
+  "predecessor_validation": "predecessor/VALIDATION.json is historical only and does not certify new files.",
+  "python": "CPython3.12.1 macOS",
+  "registered_checks": [
+    {
+      "exit_code": 0,
+      "name": "portable_regenerated_full_result",
+      "python": "3.11.6",
+      "seconds": 25.758022785186768
+    },
+    {
+      "exit_code": 0,
+      "name": "actual_CLI_readonly_and_corruption_controls",
+      "python": "3.12.1",
+      "seconds": 32.752
+    },
+    {
+      "exit_code": 0,
+      "name": "arithmetic",
+      "python": "3.12.1",
+      "seconds": 1.107
+    },
+    {
+      "exit_code": 0,
+      "name": "arithmetic_optimized",
+      "python": "3.12.1",
+      "seconds": 1.233
+    }
+  ],
+  "status": "PASS_PORTABLE_PYTHON_AND_CHANGED_LEAN_TARGET_KERNEL"
+}
diff --git a/research/staged544-refined-endpoint/VENDOR.json b/research/staged544-refined-endpoint/VENDOR.json
new file mode 100644
index 0000000000..d3204d1009
--- /dev/null
+++ b/research/staged544-refined-endpoint/VENDOR.json
@@ -0,0 +1,205 @@
+{
+  "files": [
+    {
+      "path": "vendor/staged544-clock-ceiling/clock.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/clock.py",
+      "sha256": "b29fcc9386c60ed4e7aa6c35f7e1403beaac82c6cd2ed472436ef4f1c4a39aaa"
+    },
+    {
+      "path": "vendor/staged544-clock-ceiling/ceiling_certificate.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/ceiling_certificate.py",
+      "sha256": "2276f1c5d341db63dcf511f6398733b1efbe9d09e2e9aa86b50d7a160a0f7d03"
+    },
+    {
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+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/verify.py",
+      "sha256": "df925b428dded034edc316518661e15c46d952f7071488a22ac64c14d43f1ac7"
+    },
+    {
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+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/PRICE.json",
+      "sha256": "f989112add9b51d3a9d8be458defafadc85d84add9a4f476ec0f81d7a72f1be0"
+    },
+    {
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+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/inputs/FILL.json.gz",
+      "sha256": "50e4635123c0551a1a3c8f4cab870bcccc3e579c24447abe9d81336fde91f883"
+    },
+    {
+      "path": "vendor/staged544-clock-ceiling/inputs/ORDER.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/inputs/ORDER.json",
+      "sha256": "743e74b538432e2b2045e754b0de9c18f36fc5039fdb8eb542177a898ec78e44"
+    },
+    {
+      "path": "vendor/staged544-clock-ceiling/inputs/PROFILE.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/inputs/PROFILE.json",
+      "sha256": "9686df0d8e710c2e94aef3887f4bf130a05b64215faeae702cccb59e95a469d5"
+    },
+    {
+      "path": "vendor/staged544-clock-ceiling/evidence/RELEASE.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/evidence/RELEASE.json",
+      "sha256": "9e61d0c0e1d59a6719234a64f558d4e5918bb69555019a7276ecafb1b34dfc1c"
+    },
+    {
+      "path": "vendor/staged544-clock-ceiling/README.md",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-clock-ceiling/README.md",
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+    },
+    {
+      "path": "vendor/staged544-producer-return/PROOF.md",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/PROOF.md",
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+    },
+    {
+      "path": "vendor/staged544-producer-return/PRICE.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/PRICE.json",
+      "sha256": "eb89e98609f7c6b3c4281feededbde11b30ae56696b2756a88b318447d1c92ca"
+    },
+    {
+      "path": "vendor/staged544-producer-return/owner_audit.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/owner_audit.py",
+      "sha256": "56405eacc7c2dd0f7834ed5d62a4db8303c71657300eb4e4bbe020319552cde1"
+    },
+    {
+      "path": "vendor/staged544-producer-return/rational_price.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/rational_price.py",
+      "sha256": "ee38ccbc94411b2c9c7ab88b1f6d1b34debdafa896bd5f4b7214f20168f90dab"
+    },
+    {
+      "path": "vendor/staged544-producer-return/inputs/DISPATCH.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/inputs/DISPATCH.json",
+      "sha256": "5d6fa711d704a1b498e5fe066ea821c4fa0217463702e82a7ce5be9da4023e8a"
+    },
+    {
+      "path": "vendor/staged544-producer-return/inputs/NATIVE-base.json.gz",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/inputs/NATIVE-base.json.gz",
+      "sha256": "bfe0cc0badac6c6d18461ceea0b299162c0f280c05f81b63f11fcc06cf9180e6"
+    },
+    {
+      "path": "vendor/staged544-producer-return/inputs/NATIVE-extra.json.gz",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/inputs/NATIVE-extra.json.gz",
+      "sha256": "c5eec94a64cecfd6b83de05619628ad4868e889d62a55b49973f09705cc60448"
+    },
+    {
+      "path": "vendor/staged544-producer-return/inputs/PLAN.json.gz",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/inputs/PLAN.json.gz",
+      "sha256": "d566cfb87b71639d4138e1f89b543e2dd6a7772070226a2a14c799a6835bf21e"
+    },
+    {
+      "path": "vendor/staged544-producer-return/inputs/TEMPLATE.json.gz",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/inputs/TEMPLATE.json.gz",
+      "sha256": "21297b20f1f2dc76eb9dd8124416068c3cb096bab9df17fa4da19bc68dfe082e"
+    },
+    {
+      "path": "vendor/staged544-producer-return/inputs/ORDER.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/inputs/ORDER.json",
+      "sha256": "a2f308937e66c354a1c612556a8ad34c45bf685401238f5358a7bb9d65965ed4"
+    },
+    {
+      "path": "vendor/staged544-producer-return/audit_core.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/audit_core.py",
+      "sha256": "888679ca8502cdcb100420fe5382a052529a494bf76cb332928b86e62612b951"
+    },
+    {
+      "path": "vendor/staged544-producer-return/audit_verify.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/audit_verify.py",
+      "sha256": "99971d3cc143cc894600e08c8cfbad3e8f537a544212f21dfbe4271cfa97354d"
+    },
+    {
+      "path": "vendor/staged544-producer-return/words/BORROW-WORD-43base.json.gz",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/words/BORROW-WORD-43base.json.gz",
+      "sha256": "5ca4be99851f8878cbfe0fd6c454df926b11218f17feefa1664721cf03156699"
+    },
+    {
+      "path": "vendor/staged544-producer-return/words/BORROW-WORD-43extra.json.gz",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/words/BORROW-WORD-43extra.json.gz",
+      "sha256": "965268a9be73d3f62e7ef680d18f108cf064bfe1f44b82157baa61443232904e"
+    },
+    {
+      "path": "vendor/staged544-producer-return/SOURCE-CREDITS.md",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/SOURCE-CREDITS.md",
+      "sha256": "d4a13d8b8ed9cdb594f00efb3d1a336e09d934423330f02a1514308a720c278b"
+    },
+    {
+      "path": "vendor/staged544-producer-return/evidence/NATIVE-base.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/evidence/NATIVE-base.json",
+      "sha256": "d12ca528547af356176837b83e0de08a33fe5e59cc851238f321388dfbd81383"
+    },
+    {
+      "path": "vendor/staged544-producer-return/evidence/NATIVE-extra.json",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/evidence/NATIVE-extra.json",
+      "sha256": "dd3c7d52685a056415c7a5d0582cf83b4b1248b7e06913c288efd5b05725eeb6"
+    },
+    {
+      "path": "vendor/staged544-producer-return/outer-replay/outer/check_ledger.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/outer-replay/outer/check_ledger.py",
+      "sha256": "1dd56009986eb14dd6603c9193c7ad71dd45c7d005bf71c9cd3f916e66f54d98"
+    },
+    {
+      "path": "vendor/staged544-producer-return/outer-replay/outer/independent_check.py",
+      "origin_commit": "77ffbf287b6ca198b4a31e3835db3bae60718b93",
+      "origin_path": "research/staged544-producer-return/outer-replay/outer/independent_check.py",
+      "sha256": "a5a0361539ec7cae9eb6c0371f857919335a1720040f22af291aa3e34fc864c7"
+    },
+    {
+      "path": "vendor/native/NATIVE-RECEIPT.json",
+      "origin_archive": "research/staged544-producer-return/REPLAY-INPUTS.tar.gz",
+      "origin_commit": "fa041037a3374eaa08feeca0b26a5d6893194147",
+      "archive_member": "NATIVE-RECEIPT.json",
+      "sha256": "7ddf3f80b2fa61d1ab12035da9ef1c95d43f76077df819d7683144d7862a3f93"
+    },
+    {
+      "path": "vendor/bootstrap/assemble.py",
+      "origin_archive": "research/staged544-producer-return/REPLAY-INPUTS.tar.gz",
+      "origin_commit": "fa041037a3374eaa08feeca0b26a5d6893194147",
+      "archive_member": "source/code/assemble.py",
+      "sha256": "f25e1e870df5d5a16a1e367b3df8843cb01644ab4b75895d7939af81e4fcb833"
+    },
+    {
+      "path": "vendor/bootstrap/outer.py",
+      "origin_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+      "origin_path": "research/adaptive-kernel-demotion/vendor/engines/outer.py",
+      "sha256": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c"
+    },
+    {
+      "path": "vendor/staged544-producer-return/REPLAY-INPUTS.json",
+      "origin_commit": "fa041037a3374eaa08feeca0b26a5d6893194147",
+      "origin_path": "research/staged544-producer-return/REPLAY-INPUTS.json",
+      "sha256": "727691274be6f91059c763dfa8206f7bca4516c5acf4703da403789a49b5e1e7"
+    }
+  ],
+  "optional_native_archive": {
+    "required_for_bounded_replay": false,
+    "origin_commit": "fa041037a3374eaa08feeca0b26a5d6893194147",
+    "path": "research/staged544-producer-return/REPLAY-INPUTS.tar.gz",
+    "sha256": "e74d545c18f77b7a9b8f0cc3d4901a87bb686888edaf3e41b30e5b2748751041",
+    "coverage": "Not fetched or authenticated by bounded replay; local preparation authenticated original bytes before extracting the retained receipt and bootstrap source."
+  }
+}
diff --git a/research/staged544-refined-endpoint/VERIFY.json b/research/staged544-refined-endpoint/VERIFY.json
new file mode 100644
index 0000000000..3f9febc66f
--- /dev/null
+++ b/research/staged544-refined-endpoint/VERIFY.json
@@ -0,0 +1,46 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Complete fresh scoped result: current literal owners/ceiling, both dirty words, independent interval signs, carried bootstrap, modern outer and data binding."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/test_arithmetic.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Seven standalone arithmetic/domain groups including independent exact-series root signs and rank-mass-preserving false profile."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-O",
+        "-B",
+        "{package}/test_arithmetic.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Standalone arithmetic remains guarded with assertions disabled."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/test_cli.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Actual read-only Git-free verifier and semantic source/claim/proof-data/result/optimization corruption controls."
+    }
+  ]
+}
diff --git a/research/staged544-refined-endpoint/dyadic_moment.py b/research/staged544-refined-endpoint/dyadic_moment.py
new file mode 100644
index 0000000000..76b9c1a2e5
--- /dev/null
+++ b/research/staged544-refined-endpoint/dyadic_moment.py
@@ -0,0 +1,131 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Independent dyadic interval evaluator for finite positive rank moments.
+
+All runtime arithmetic is integer/Fraction; no floating point, Decimal, NumPy,
+or imports from predecessor moment engines. The analytic enclosures and outward
+rounding argument are proved in PROOF.md. This code is not formally verified.
+Dugongue, with substantial OpenAI Codex assistance.
+"""
+from fractions import Fraction
+
+
+def need(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def ceil_div(a, b):
+    need(b > 0, 'positive denominator')
+    return -((-a) // b)
+
+
+class DyadicMoment:
+    def __init__(self, profile, dimension, live, bits=512, log_terms=256, exp_terms=96):
+        need(type(dimension) is int and dimension > 1, 'dimension')
+        need(type(live) is int and live > 0, 'live count')
+        need(type(bits) is int and bits >= 32, 'precision bits')
+        need(type(log_terms) is int and log_terms > 0, 'log terms')
+        need(type(exp_terms) is int and exp_terms > 0, 'exp terms')
+        need(bool(profile), 'nonempty profile')
+        need(all(type(r) is int and 0 < r < dimension and
+                 type(n) is int and n > 0 for r,n in profile.items()), 'positive strict child ranks')
+        self.profile = dict(profile)
+        self.dimension, self.live = dimension, live
+        self.scale = 1 << bits
+        self.bits, self.log_terms, self.exp_terms = bits, log_terms, exp_terms
+        self.logs = {r:self.log_ratio(dimension,r) for r in profile}
+
+    def log_unit(self, num, den):
+        """Enclose log(num/den) for 1 <= num/den <= 2."""
+        need(0 < den <= num <= 2*den, 'log reduction')
+        s = self.scale
+        lo = s*(num-den)//(num+den)
+        hi = ceil_div(s*(num-den), num+den)
+        sqlo, sqhi = lo*lo//s, ceil_div(hi*hi,s)
+        plo, phi = lo, hi
+        total_lo = total_hi = 0
+        for j in range(self.log_terms):
+            total_lo += (2*plo)//(2*j+1)
+            total_hi += ceil_div(2*phi,2*j+1)
+            plo, phi = plo*sqlo//s, ceil_div(phi*sqhi,s)
+        # phi / s bounds t^(2N+1); 2/(1-t^2) <= 9/4.
+        total_hi += ceil_div(9*phi,4*(2*self.log_terms+1))
+        return total_lo,total_hi
+
+    def log_ratio(self,num,den):
+        need(num >= den > 0, 'positive ratio >= 1')
+        k = 0
+        while num >= 2*den:
+            den *= 2
+            k += 1
+        a,b = self.log_unit(num,den)
+        c,d = self.log_unit(2,1)
+        return a+k*c,b+k*d
+
+    def exp_interval(self,lo,hi):
+        """Enclose exp(x) whenever lo/S <= x <= hi/S <= 1/2."""
+        s = self.scale
+        need(0 <= lo <= hi <= s//2, 'exponential small-argument domain')
+        tlo = thi = s
+        alo = ahi = s
+        for j in range(1,self.exp_terms+1):
+            tlo = tlo*lo//(s*j)
+            thi = ceil_div(thi*hi,s*j)
+            alo += tlo
+            ahi += thi
+        # The first omitted term bounds subsequent ratios by <= 1/2.
+        first = ceil_div(thi*hi,s*(self.exp_terms+1))
+        return alo,ahi+2*first
+
+    def weights(self,b):
+        need(type(b) is Fraction and b >= 0, 'nonnegative exact rational endpoint')
+        out = {}
+        for r,(lo,hi) in self.logs.items():
+            a = lo*b.numerator//b.denominator
+            z = ceil_div(hi*b.numerator,b.denominator)
+            out[r] = self.exp_interval(a,z)
+        return out
+
+    def moment(self,b):
+        weights = self.weights(b)
+        a = sum(n*r*weights[r][0] for r,n in self.profile.items())
+        z = sum(n*r*weights[r][1] for r,n in self.profile.items())
+        den = self.scale*self.dimension*self.live
+        return Fraction(a,den),Fraction(z,den)
+
+    def bracket(self,digits=60):
+        need(type(digits) is int and 1 <= digits <= 100, 'requested grid')
+        grid = 10**digits
+        lo,hi = 0,grid//100
+        need(self.moment(Fraction(lo,grid))[1] < 1,'lower bracket')
+        need(self.moment(Fraction(hi,grid))[0] > 1,'upper bracket')
+        evaluations = 2
+        while hi-lo > 1:
+            mid = (lo+hi)//2
+            a,z = self.moment(Fraction(mid,grid))
+            evaluations += 1
+            if z < 1:lo = mid
+            elif a > 1:hi = mid
+            else:raise ValueError('precision insufficient at trial point')
+        return Fraction(lo,grid),Fraction(hi,grid),evaluations
+
+
+def certify(profile,dimension,live,digits=60):
+    evaluator = DyadicMoment(profile,dimension,live)
+    lo,hi,count = evaluator.bracket(digits)
+    lower = evaluator.moment(lo)
+    upper = evaluator.moment(hi)
+    need(lower[1] < 1 < upper[0],'strict adjacent grid certificate')
+    return {'lower':str(lo),'upper':str(hi),'digits':digits,'bits':evaluator.bits,
+            'log_terms':evaluator.log_terms,'exp_terms':evaluator.exp_terms,
+            'evaluations':count,'margin_lower':str(1-lower[1]),
+            'margin_upper':str(upper[0]-1),
+            'weights':{str(r):{'lower':str(a),'upper':str(b),'scale':str(evaluator.scale)}
+                       for r,(a,b) in evaluator.weights(lo).items()}}
+
+
+if __name__ == '__main__':
+    import json
+    profile={1:7989840,2:2349000,3:749088,4:330480,5:69120,8:25920,48:137952}
+    print(json.dumps(certify(profile,73,896*360),indent=2))
diff --git a/research/staged544-refined-endpoint/evidence/LEAN-BUILD.log b/research/staged544-refined-endpoint/evidence/LEAN-BUILD.log
new file mode 100644
index 0000000000..96ec911029
--- /dev/null
+++ b/research/staged544-refined-endpoint/evidence/LEAN-BUILD.log
@@ -0,0 +1,7 @@
+ℹ [2038/2040] Built CapCertificate (6.2s)
+info: CapCertificate.lean:69:0: 'Staged544Certificate.exact_certificate' depends on axioms: [propext, Classical.choice, Quot.sound]
+info: CapCertificate.lean:70:0: 'Staged544Certificate.admission' depends on axioms: [propext, Classical.choice, Quot.sound]
+ℹ [2039/2040] Built AnalyticCertificate (6.0s)
+info: AnalyticCertificate.lean:115:0: 'Staged544.actual_moment_lt_one' depends on axioms: [propext, Classical.choice, Quot.sound]
+info: AnalyticCertificate.lean:117:0: 'Staged544.actual_rpow_moment_lt_one' depends on axioms: [propext, Classical.choice, Quot.sound]
+Build completed successfully (2040 jobs).
diff --git a/research/staged544-refined-endpoint/evidence/LEAN-KERNEL.log b/research/staged544-refined-endpoint/evidence/LEAN-KERNEL.log
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/research/staged544-refined-endpoint/lean/AnalyticCertificate.lean b/research/staged544-refined-endpoint/lean/AnalyticCertificate.lean
new file mode 100644
index 0000000000..463d501247
--- /dev/null
+++ b/research/staged544-refined-endpoint/lean/AnalyticCertificate.lean
@@ -0,0 +1,117 @@
+import Mathlib.Analysis.SpecialFunctions.Pow.Real
+import CapCertificate
+import Mathlib.Tactic.Linarith
+import Mathlib.Analysis.SpecialFunctions.Log.Basic
+import Mathlib.Analysis.Complex.Exponential
+import Mathlib.Data.Rat.BigOperators
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.FinCases
+import Mathlib.Data.Fin.VecNotation
+
+open scoped BigOperators
+namespace Staged544
+
+def taylorQ (x : ℚ) (n : ℕ) : ℚ := ∑ i ∈ Finset.range n, x ^ i / (i.factorial : ℚ)
+
+lemma taylor_cast (x : ℚ) (n : ℕ) :
+    ((taylorQ x n : ℚ) : ℝ) = ∑ i ∈ Finset.range n, (x : ℝ) ^ i / (i.factorial : ℝ) := by
+  simp [taylorQ]
+
+lemma log_upper_of_taylor (y x : ℚ) (n : ℕ) (hy : 0 < y) (hx : 0 ≤ x)
+    (hpoly : y ≤ taylorQ x n) : Real.log (y : ℝ) ≤ (x : ℝ) := by
+  apply (Real.log_le_iff_le_exp (by exact_mod_cast hy)).2
+  have hp : (y : ℝ) ≤ ((taylorQ x n : ℚ) : ℝ) := by exact_mod_cast hpoly
+  rw [taylor_cast] at hp
+  exact hp.trans (Real.sum_le_exp_of_nonneg (by exact_mod_cast hx) n)
+
+lemma exp_upper_of_taylor (x u : ℚ) (n : ℕ) (hn : 0 < n)
+    (hx : 0 ≤ x) (hx1 : x ≤ 1)
+    (hpoly : taylorQ x n + x ^ n * (n + 1) / ((n.factorial : ℚ) * n) ≤ u) :
+    Real.exp (x : ℝ) ≤ (u : ℝ) := by
+  have h := Real.exp_bound' (by exact_mod_cast hx : 0 ≤ (x : ℝ))
+    (by exact_mod_cast hx1 : (x : ℝ) ≤ 1) hn
+  have hp : ((taylorQ x n : ℚ) : ℝ) + (x : ℝ) ^ n * (n + 1) /
+      ((n.factorial : ℝ) * n) ≤ (u : ℝ) := by exact_mod_cast hpoly
+  rw [taylor_cast] at hp
+  exact h.trans hp
+
+def endpoint : ℚ := 173057843726624681874071321804985755655644807397156243331276259008002643975681 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000
+def rank : Fin 7 → ℚ := ![1, 2, 3, 4, 5, 8, 48]
+def logUpper : Fin 7 → ℚ := ![(42904594411483911290921088574385425709047528448715976645956988571617899759205973 / 10000000000000000000000000000000000000000000000000000000000000000000000000000000), (179865613029422290983743836799018300141462635525567120523750942383419817697545129 / 50000000000000000000000000000000000000000000000000000000000000000000000000000000), (319184715248028143769686362051601686625726228704884821286100452352429568270198833 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000), (290416508002850051025764461452218943475375257615108715635433883817500273198120787 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000), (268102152871429075449134952421235493137915149060307994268305096568761098821293953 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000), (44220357989371104016808249861280257333565048835816638044673176573632182200230263 / 20000000000000000000000000000000000000000000000000000000000000000000000000000000), (41925843024050020002793513468331059395526174960782719637828448555072119482320947 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000)]
+def upper : Fin 7 → ℚ := ![(5037263050419323125486854965632039515515813123979119739049327448446247437415523839971594780304879053 / 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (1006244849300832117227334850803589471353259962367000112587337836237263822185857853193329141685299883 / 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (10055390258478318702735196789299380974598979106244048870901713891769435640370131535754404057635525197 / 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (2512596341033269936571327050488918900797498810133573533611463569761893864457578011954826530420192591 / 2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (2511626245512671220465894009973910441561771954881139752642633015326322249610212835151571034808264101 / 2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (2509584174213792103224997148477336282910856045665342057938515539919371590220915369255384660757605463 / 2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (10007258228810539199340477838604803218049186436146540908000107607859989792524228358661467016084073803 / 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)]
+
+set_option maxRecDepth 100000
+set_option maxHeartbeats 20000000
+
+def rootUpper : Fin 7 → ℚ := ![(1709682071524341872977914446059068945687158662089760087588298513648466431140986964131832379 / 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (78389268609117970295183841420183634117602514499308989820003491013074224898274587103932377 / 50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (59612205786190783200497858238480871280010105534840193468536084961687039700052165204929563 / 40000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (718832762584181935863410006114861039343341562058796885073492442373725223855973513855130509 / 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (349529725287554381854657482572542092721239471300292172144051890309678870642203459561245573 / 250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (1318345099355356699940061975994465396292039242917024224119936446107390011400428568787269981 / 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), (3293140230695581904239101463758503566499283368816211814999286028419201556213646303933641 / 3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000)]
+
+
+lemma root_power (i : Fin 7) : 73 / rank i ≤ (rootUpper i) ^ 8 := by
+  fin_cases i <;> norm_num [rank, rootUpper]
+
+lemma log_numeric (i : Fin 7) : rootUpper i ≤ taylorQ (logUpper i / 8) 60 := by
+  fin_cases i <;> norm_num [taylorQ, logUpper, rootUpper, Finset.sum_range_succ]
+
+lemma logUpper_valid (i : Fin 7) :
+    Real.log ((73 / rank i : ℚ) : ℝ) ≤ (logUpper i : ℝ) := by
+  have hroot : (0 : ℚ) < rootUpper i := by fin_cases i <;> norm_num [rootUpper]
+  have hlog : (0 : ℚ) ≤ logUpper i / 8 := by fin_cases i <;> norm_num [logUpper]
+  have hy : (0 : ℚ) < 73 / rank i := by fin_cases i <;> norm_num [rank]
+  have hp : ((73 / rank i : ℚ) : ℝ) ≤ (rootUpper i : ℝ) ^ 8 := by
+    exact_mod_cast root_power i
+  have h := log_upper_of_taylor (rootUpper i) (logUpper i / 8) 60 hroot hlog (log_numeric i)
+  have hl := Real.log_le_log (by exact_mod_cast hy : (0 : ℝ) < ((73 / rank i : ℚ) : ℝ)) hp
+  rw [Real.log_pow] at hl
+  push_cast at h
+  norm_num at hl
+  push_cast
+  linarith
+
+lemma exp_numeric (i : Fin 7) :
+    taylorQ (endpoint * logUpper i) 26 +
+    (endpoint * logUpper i) ^ 26 * (26 + 1) / ((26 : ℕ).factorial * 26 : ℚ) ≤ upper i := by
+  fin_cases i <;> norm_num [taylorQ, endpoint, logUpper, upper, Finset.sum_range_succ]
+
+noncomputable def weight (i : Fin 7) : ℝ :=
+  Real.exp ((endpoint : ℝ) * Real.log ((73 / rank i : ℚ) : ℝ))
+
+lemma weight_le_upper (i : Fin 7) : weight i ≤ (upper i : ℝ) := by
+  have h0 : (0 : ℚ) ≤ endpoint * logUpper i := by fin_cases i <;> norm_num [endpoint, logUpper]
+  have h1 : endpoint * logUpper i ≤ (1 : ℚ) := by fin_cases i <;> norm_num [endpoint, logUpper]
+  have hx : (0 : ℝ) ≤ (endpoint : ℝ) := by norm_num [endpoint]
+  have hm := mul_le_mul_of_nonneg_left (logUpper_valid i) hx
+  have he := Real.exp_le_exp.mpr hm
+  have hb := exp_upper_of_taylor (endpoint * logUpper i) (upper i) 26 (by decide) h0 h1 (exp_numeric i)
+  simpa [weight] using he.trans (by simpa using hb)
+
+def coefficient : Fin 7 → ℚ := ![7989840, 4698000, 2247264, 1321920, 345600, 207360, 6621696]
+
+lemma exact_upper_certificate : (∑ i : Fin 7, coefficient i * upper i) < (73 * (896 * 360) : ℚ) := by
+  norm_num [Fin.sum_univ_succ, coefficient, upper]
+
+/-- The actual real exponential moment at the proposed 80-digit endpoint is below 1. -/
+theorem actual_moment_lt_one :
+    (∑ i : Fin 7, (coefficient i : ℝ) * weight i) / (73 * (896 * 360)) < 1 := by
+  apply (IntegerMultBounds.MomentTransfer.normalized_admission _ _ (by norm_num)).2
+  have h := IntegerMultBounds.MomentTransfer.rational_upper_certificate Finset.univ coefficient upper weight
+    (73 * (896 * 360)) (by intro i _; fin_cases i <;> norm_num [coefficient])
+    (fun i _ => weight_le_upper i) exact_upper_certificate
+  simpa [IntegerMultBounds.MomentTransfer.weighted] using h
+
+/-- The same certificate in the repository's positive real-power moment notation. -/
+theorem actual_rpow_moment_lt_one :
+    (∑ i : Fin 7, (coefficient i : ℝ) *
+      (((73 / rank i : ℚ) : ℝ) ^ (endpoint : ℝ))) / (73 * (896 * 360)) < 1 := by
+  have h : ∀ i : Fin 7, (((73 / rank i : ℚ) : ℝ) ^ (endpoint : ℝ)) = weight i := by
+    intro i
+    have hp : (0 : ℚ) < 73 / rank i := by fin_cases i <;> norm_num [rank]
+    rw [Real.rpow_def_of_pos (by exact_mod_cast hp : (0 : ℝ) < ((73 / rank i : ℚ) : ℝ))]
+    simp [weight, mul_comm]
+  simp_rw [h]
+  exact actual_moment_lt_one
+
+end Staged544
+
+#print axioms Staged544.actual_moment_lt_one
+
+#print axioms Staged544.actual_rpow_moment_lt_one
diff --git a/research/staged544-refined-endpoint/lean/CapCertificate.lean b/research/staged544-refined-endpoint/lean/CapCertificate.lean
new file mode 100644
index 0000000000..1ca4871a95
--- /dev/null
+++ b/research/staged544-refined-endpoint/lean/CapCertificate.lean
@@ -0,0 +1,70 @@
+import Mathlib.Data.Rat.BigOperators
+import Mathlib.Basic.Real.Basic
+import Mathlib.Algebra.BigOperators.Fin
+import Mathlib.Data.Fin.VecNotation
+import Mathlib.Algebra.Order.BigOperators.Group.Finset
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.FinCases
+
+/-! Certificate transfer extracted, with unchanged proof bodies, from Dugongue's
+MathProof IntegerKappaMomentTransfer20261015.lean. Substantial OpenAI Codex
+assistance. Apache-2.0. This proves only the stated conditional real-weight
+admission, not the logarithm/exponential bounds or the multiplication theorem. -/
+
+namespace IntegerMultBounds.MomentTransfer
+open scoped BigOperators
+variable {ι : Type*}
+noncomputable def weighted (s : Finset ι) (c w : ι → ℝ) : ℝ :=
+  ∑ i ∈ s, c i * w i
+
+theorem weighted_mono_weights (s : Finset ι) (c u v : ι → ℝ)
+    (hc : ∀ i ∈ s, 0 ≤ c i) (huv : ∀ i ∈ s, u i ≤ v i) :
+    weighted s c u ≤ weighted s c v := by
+  exact Finset.sum_le_sum fun i hi => mul_le_mul_of_nonneg_left (huv i hi) (hc i hi)
+
+theorem rational_upper_certificate (s : Finset ι) (c upper : ι → ℚ) (w : ι → ℝ)
+    (budget : ℚ) (hc : ∀ i ∈ s, 0 ≤ c i)
+    (hupper : ∀ i ∈ s, w i ≤ (upper i : ℝ))
+    (hcertificate : (∑ i ∈ s, c i * upper i) < budget) :
+    weighted s (fun i => (c i : ℝ)) w < (budget : ℝ) := by
+  have h := weighted_mono_weights s (fun i => (c i : ℝ)) w (fun i => (upper i : ℝ))
+    (fun i hi => by exact_mod_cast hc i hi) hupper
+  have hcast : weighted s (fun i => (c i : ℝ)) (fun i => (upper i : ℝ)) < (budget : ℝ) := by
+    change (∑ i ∈ s, (c i : ℝ) * (upper i : ℝ)) < (budget : ℝ)
+    exact_mod_cast hcertificate
+  exact h.trans_lt hcast
+
+/-- Dividing by a positive budget connects paid sums to moment < 1. -/
+theorem normalized_admission (cost budget : ℝ) (hbudget : 0 < budget) :
+    cost / budget < 1 ↔ cost < budget := by
+  rw [div_lt_one hbudget]
+
+end IntegerMultBounds.MomentTransfer
+
+namespace Staged544Certificate
+open scoped BigOperators
+open IntegerMultBounds.MomentTransfer
+def coefficient : Fin 7 → ℚ := ![7989840, 4698000, 2247264, 1321920, 345600, 207360, 6621696]
+def upper : Fin 7 → ℚ := ![(13507731094523807383566475123716226262821080735013659937896950134837434477991620304205855756237792292311750260853556038594859174301966598591383088728416915 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096), (13491537669919590444070783937941464752554609528295312364120075762630530816279756695957039929900401848255443706168261710262397613766253017942706209006837203 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096), (13482074124629314266984853423762132196619008555772618820195884044083409682596544977661357694011530838439984600059835445921466208152729089541843752149002581 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096), (13475363658420252292726653566578878147646435678341933346136213519761447932248379861556076157777577929289644834201241414284156073003775118317955380101246497 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096), (13470160916654698189586785076891163906547046740939775262388900537080309894800792109484478840209598056770772511656870622915958070296415813340584899035271269 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096), (13459209036752850346611399332698520957888451347969204958854153028399849634185585362065661352009007298720371270575038487256688169510878983253236696889686637 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096), (13417539623722925629684767919185678866775428178768770102551675998974478591678481675509425432160414596693798883228869483043188943937259390146824381395561924 / 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096)]
+
+theorem coefficient_nonnegative (i : Fin 7) : 0 ≤ coefficient i := by
+  fin_cases i <;> norm_num [coefficient]
+
+set_option maxRecDepth 4096 in
+set_option maxHeartbeats 2000000 in
+theorem exact_certificate : (∑ i : Fin 7, coefficient i * upper i) < (73 * (896 * 360) : ℚ) := by
+  norm_num [Fin.sum_univ_succ, coefficient, upper]
+
+/-- All seven actual real weights below the published rational bounds suffice. -/
+theorem admission (w : Fin 7 → ℝ) (hw : ∀ i, w i ≤ (upper i : ℝ)) :
+    (∑ i : Fin 7, (coefficient i : ℝ) * w i) / (73 * (896 * 360)) < 1 := by
+  apply (normalized_admission _ _ (by norm_num)).2
+  have h := rational_upper_certificate Finset.univ coefficient upper w
+    (73 * (896 * 360)) (fun i _ => coefficient_nonnegative i)
+    (fun i _ => hw i) exact_certificate
+  simpa [weighted] using h
+
+end Staged544Certificate
+
+#print axioms Staged544Certificate.exact_certificate
+#print axioms Staged544Certificate.admission
diff --git a/research/staged544-refined-endpoint/lean/lake-manifest.json b/research/staged544-refined-endpoint/lean/lake-manifest.json
new file mode 100644
index 0000000000..eb1123cf72
--- /dev/null
+++ b/research/staged544-refined-endpoint/lean/lake-manifest.json
@@ -0,0 +1,117 @@
+{
+  "version": "1.2.0",
+  "packagesDir": ".lake/packages",
+  "packages": [
+    {
+      "url": "https://github.com/leanprover-community/mathlib4.git",
+      "type": "git",
+      "subDir": null,
+      "scope": "",
+      "rev": "d13f23b723b8a846827a245b89c10fc7d3f11612",
+      "name": "mathlib",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "d13f23b723b8a846827a245b89c10fc7d3f11612",
+      "inherited": false,
+      "configFile": "lakefile.lean"
+    },
+    {
+      "url": "https://github.com/leanprover-community/plausible",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "118aa17ee84656b8bd727fef7c458ee8c833385c",
+      "name": "plausible",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "main",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    },
+    {
+      "url": "https://github.com/leanprover-community/LeanSearchClient",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "ddf04cf3949fa556442341e87d47f9f6e6074707",
+      "name": "LeanSearchClient",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "main",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    },
+    {
+      "url": "https://github.com/leanprover-community/import-graph",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "e928b72544873815af278d38681b31c0293588e3",
+      "name": "importGraph",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "main",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    },
+    {
+      "url": "https://github.com/leanprover-community/ProofWidgets4",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "106ff4fafc74ef4ac99d81dbf3ab399118f497a5",
+      "name": "proofwidgets",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "main",
+      "inherited": true,
+      "configFile": "lakefile.lean"
+    },
+    {
+      "url": "https://github.com/leanprover-community/aesop",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "355695d523e41d0554926416cba2a2b3544fbbc9",
+      "name": "aesop",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "master",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    },
+    {
+      "url": "https://github.com/leanprover-community/quote4",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "6a489d9af5d0c47e5b259e2e8bcdfc1811b5a259",
+      "name": "Qq",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "master",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    },
+    {
+      "url": "https://github.com/leanprover-community/batteries",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover-community",
+      "rev": "f2effa3d803fda822b1f97b806c47cf2adfbcbc2",
+      "name": "batteries",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "main",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    },
+    {
+      "url": "https://github.com/leanprover/lean4-cli",
+      "type": "git",
+      "subDir": null,
+      "scope": "leanprover",
+      "rev": "e92c9f15fdfacc8536f31cfb3b7ad26c3c8cd204",
+      "name": "Cli",
+      "manifestFile": "lake-manifest.json",
+      "inputRev": "v4.34.0",
+      "inherited": true,
+      "configFile": "lakefile.toml"
+    }
+  ],
+  "name": "staged544_certificate",
+  "lakeDir": ".lake",
+  "fixedToolchain": false
+}
diff --git a/research/staged544-refined-endpoint/lean/lakefile.toml b/research/staged544-refined-endpoint/lean/lakefile.toml
new file mode 100644
index 0000000000..af51027ef7
--- /dev/null
+++ b/research/staged544-refined-endpoint/lean/lakefile.toml
@@ -0,0 +1,14 @@
+name = "staged544_certificate"
+version = "0.1.0"
+defaultTargets = ["AnalyticCertificate"]
+
+[[require]]
+name = "mathlib"
+git = "https://github.com/leanprover-community/mathlib4.git"
+rev = "d13f23b723b8a846827a245b89c10fc7d3f11612"
+
+[[lean_lib]]
+name = "CapCertificate"
+
+[[lean_lib]]
+name = "AnalyticCertificate"
diff --git a/research/staged544-refined-endpoint/lean/lean-toolchain b/research/staged544-refined-endpoint/lean/lean-toolchain
new file mode 100644
index 0000000000..ba8ebf2dba
--- /dev/null
+++ b/research/staged544-refined-endpoint/lean/lean-toolchain
@@ -0,0 +1 @@
+leanprover/lean4:v4.34.1
diff --git a/research/staged544-refined-endpoint/predecessor/CLAIM.json b/research/staged544-refined-endpoint/predecessor/CLAIM.json
new file mode 100644
index 0000000000..9d92276c83
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/CLAIM.json
@@ -0,0 +1,13 @@
+{
+  "complex_lower": "54080576164570213085647288064058048642389002311611326041/31250000000000000000000000000000000000000000000000000000000",
+  "complex_upper": "1730578437266246818740713218049857556556448073971562433313/1000000000000000000000000000000000000000000000000000000000000",
+  "endpoint_digits": 60,
+  "fresh_native_replay": false,
+  "gain": "7132180498575565564481739715624333120000000000999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "kappa": "17305784372662468187407132180498575565564480739715624333119999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "physical": 323069,
+  "previous": "17305784372662468187399999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+  "private_zero_buffers": 91,
+  "replicas": 360,
+  "selected_frontier_changed": false
+}
diff --git a/research/staged544-refined-endpoint/predecessor/LEAN-AXIOMS.txt b/research/staged544-refined-endpoint/predecessor/LEAN-AXIOMS.txt
new file mode 100644
index 0000000000..c7bddb366a
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/LEAN-AXIOMS.txt
@@ -0,0 +1,2 @@
+'Staged544.actual_moment_lt_one' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Staged544.actual_rpow_moment_lt_one' depends on axioms: [propext, Classical.choice, Quot.sound]
diff --git a/research/staged544-refined-endpoint/predecessor/OVERLAP.md b/research/staged544-refined-endpoint/predecessor/OVERLAP.md
new file mode 100644
index 0000000000..28df7fb32a
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/OVERLAP.md
@@ -0,0 +1,64 @@
+# Current source and overlap checkpoint
+
+The session read current-main AGENTS.md, autoresearch.md and CONTRIBUTING.md
+first, pinned `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, and fetched canonical
+main again before verification. Priority revision remains 6 and the selected
+result remains `836330098538459/500000000000000000`. The first working baseline
+check passed after materializing the sparse checkout's exact selected closure;
+1,626 files were byte-checked against the fresh main tree. This was a bindings
+check, not a cold scientific replay of the selected construction.
+
+The donor explicitly required best-known κ and rejected the stale T740 work.
+That earlier draft was preserved outside this package. The relevant current
+construction is the cap-attaining staged-helper line, not a descendant of our
+unpublished T740 endpoint. The current-main original `independent-moment` card
+is retained verbatim in TASK.json; its scope change is explicit. A picker draw
+was not substituted for the donor's assigned target.
+
+Entry intake fully paginated all 539 open PR bodies and read the 100 most
+recently updated closed PR bodies. An incremental public-API refresh at
+2026-10-18 22:58 UTC paginated updates until an older-than-entry row, covered
+all 16 changed records since entry through #3846, and updated the open snapshot
+to 552 records. The repository overlap helper searched the resulting cache,
+including current closed/reopened states and local package notes. Closed-history
+coverage is partial. This is title/body coverage, not an exhaustive code-diff
+or unpublished-work search; snapshots are not atomic.
+
+Visible Discussion start/handoff pages #1163/#1166 and the relevant #2612 thread
+were read, but all Discussion comments/replies were not paginated. The helper
+reports no full Discussion cache and explicit offline freshness warnings; no
+absence-of-overlap or method-priority guarantee follows. Relevant direct PR
+comments were separately fetched: #3802 credits #3815's equal endpoint/cap;
+#3734 points to that solution of its changed-clock question; #3815 had no
+comments in that check.
+
+Exact current comparisons:
+
+| PR | Head | Relevant contribution |
+|---|---|---|
+| [#3815](https://github.com/CrocSwap/integer-mult-bounds/pull/3815) | `77ffbf287b6ca198b4a31e3835db3bae60718b93` | T360 clock/fill, exact packing ceiling and full release |
+| [#3802](https://github.com/CrocSwap/integer-mult-bounds/pull/3802) | `de2cd5c58009205210e3bf328b0ef5cadf3903af` | Earlier publication of the same prior κ at T385 |
+| [#3734](https://github.com/CrocSwap/integer-mult-bounds/pull/3734) | `fa041037a3374eaa08feeca0b26a5d6893194147` | Staged helper 544 word, current native inputs and release |
+| [#3682](https://github.com/CrocSwap/integer-mult-bounds/pull/3682) | `431e6c6023ef352c41696469de7fb614cd4ded8c` | Retimed T360 integration |
+| [#3605](https://github.com/CrocSwap/integer-mult-bounds/pull/3605) | `3efbb5de06a49ffeea6684482f2d3fe48cdca3fb` | Staged-entry mechanism |
+| [#3642](https://github.com/CrocSwap/integer-mult-bounds/pull/3642) | `6568db75adbb61437190545f6fef1534c70e91e5` | Retiming and corrected owner-control coverage |
+| [#3531](https://github.com/CrocSwap/integer-mult-bounds/pull/3531) | `550a660a97308d3aeb3620ee1f4218f77e2c2742` | Earlier binomial/T740 precision and reuse precedent |
+
+New #3836 fixes the nine-stage bridge idle alphabet at its scoped boundary;
+#3828 develops centre-kernel theory. Neither updated body supplies a higher
+complete conditional κ. Earlier donation headlines #3184/#3190/#3198/#3200/
+#3251 are now closed and have published operator objections; their numbers
+are not treated as current admissible suppliers. No claim that all changed
+architectures or repairs are impossible is made.
+
+The new bound strictly improves #3802/#3815, the highest complete
+source-backed conditional endpoint located in this search. The mathematical
+addition is a closed finite real-power moment proof plus independently replayed
+exact arithmetic, not merely extra digits or a new physical word. A global
+unconditional best-known multiplication theorem is not established. No selected
+record, shared scientific source, maintainer decision, or other PR is changed.
+
+No public claim or PR was posted from this execution context because the donor's
+GitHub credential is available in their normal PowerShell session, not this
+sandbox. The prepared publication command uses that existing session. Routine
+publication is authorized; no additional permission step is required.
diff --git a/research/staged544-refined-endpoint/predecessor/PRECEDENTS.md b/research/staged544-refined-endpoint/predecessor/PRECEDENTS.md
new file mode 100644
index 0000000000..4dbdde3bf6
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/PRECEDENTS.md
@@ -0,0 +1,27 @@
+# Existing MathProof proof material used
+
+The registered source catalog and canonical Lean tree were read without changing
+the authoritative DAG or importing a staging copy as a competing active module.
+The catalog's canonical
+`Mathproof/Computability/IntegerKappaMomentTransfer20261015.lean` has SHA-256
+`68e89786008dd3f222632920a00eafd6fc64661e08548c5325a3daa4ba964522` and
+status `NO_LOCAL_ARTIFACT`. That is source-only provenance, not an old kernel
+receipt. This package checks its extracted sufficient argument freshly.
+
+Its `weighted_mono_weights`, `rational_upper_certificate` and
+`normalized_admission` supply the finite rational-to-real admission assembly.
+Their proof bodies are retained unchanged in CapCertificate.lean; the new
+instance is the actual current seven-rank bill. The public package imports
+only Mathlib and needs no private MathProof directory or prior compiled module.
+
+The existing `IntegerKappaProviderExchange20261015.lean` and
+`IntegerSuperiorKappa20261015.lean` were also consulted during route comparison.
+They support concavity and replica-monotonicity arguments at their abstract
+boundaries. They do not create a new physical word or beat the current attained
+packing cap. No numerical saving is attributed to an unconstructed use of them.
+
+The session reused our earlier exact commuting-retiming implementation as a
+search template on the newly pinned current word. It did not reuse a stale κ.
+The current formal analytic route adds sufficient eighth-root/Taylor inequalities
+and composes them through the existing admission theorem. Proof status is bound
+to final source bytes in VALIDATION.json; retrieval alone is never proof evidence.
diff --git a/research/staged544-refined-endpoint/predecessor/PROOF.md b/research/staged544-refined-endpoint/predecessor/PROOF.md
new file mode 100644
index 0000000000..752ebecd37
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/PROOF.md
@@ -0,0 +1,132 @@
+# Exact finite admission and conditional endpoint transfer
+
+Let m = 73, T = 360, W = 896T, and let nᵣ be the seven positive counts in
+CLAIM.json. The paid moment is
+
+    M(b) = Σᵣ nᵣ r (73/r)^b / (73W).
+
+Its rank mass is 65,088T. Thus M(0) < 1, and M is continuous and strictly
+increasing because every paid rank is positive and below 73. The exact
+interval witnesses prove M(C) < 1 < M(C + 10⁻⁶⁰). Consequently its unique
+root lies between these adjacent grid points. Strict positivity of the lower
+and upper margins is recorded in RESULT.json, not inferred from printed digits.
+
+## Independently implemented dyadic enclosures
+
+An integer interval (a,z) at scale S = 2⁵¹² means [a/S,z/S]. For nonnegative
+intervals, floor(ab/S) and ceil(zw/S) bound products. Addition and multiplication
+by a positive exact rational preserve enclosures under outward rounding.
+
+For 1 ≤ y ≤ 2, put t = (y−1)/(y+1), so 0 ≤ t ≤ 1/3. The identity
+
+    log y = 2 Σⱼ≥0 t^(2j+1)/(2j+1)
+
+has a positive tail after N terms bounded above by
+
+    2 t^(2N+1) / ((2N+1)(1−t²)) ≤ 9 t^(2N+1)/(4(2N+1)).
+
+The program propagates intervals for t, t² and each odd power, rounding every
+term outward. At loop exit its upper power bounds t^(2N+1); the stated tail
+is rounded upward and added. Here N = 256. Exact binary range reduction
+x = 2ᵏy gives log x = k log 2 + log y without a floating-point operation.
+
+Multiplying the logarithm interval by exact b bounds each exponential argument.
+For 0 ≤ x ≤ 1/2, the exponential Taylor terms are positive. After degree 96,
+the first omitted upper term T bounds the whole omitted tail by 2T because
+all subsequent term ratios are at most 1/2. Directed recurrence rounding
+therefore encloses exp x. Every actual argument is checked in this domain.
+Positive weighted addition gives the moment interval. An ambiguous sign in
+bisection fails; it is never accepted by a heuristic.
+
+The retained current-source engine uses log(r), outward decimal quantization
+and alternating partial sums for exp(−b log r), with the signed r = 73 live
+term. Its representation and exponential-tail argument differ from the new
+positive ratio/exponential evaluator. It independently proves the new signs
+without changing its precision parameters. The separate test oracle sums
+exact Fractions with geometric atanh and exponential remainders and rounds
+only at the logarithm/final reporting boundaries.
+
+## Sufficient formal assembly from MathProof
+
+The existing MathProof theorem `rational_upper_certificate` proves:
+nonnegative rational coefficients cᵢ, real weights wᵢ ≤ uᵢ, and an exact
+rational inequality Σ cᵢuᵢ < B imply Σ cᵢwᵢ < B. Its unchanged proof bodies
+are extracted in CapCertificate.lean and specialized to cᵢ = nᵣr and
+B = 73 × 896 × 360. The exact 512-bit rational certificate is checked with
+ordinary `norm_num`, without `native_decide`.
+
+The stronger analytic route uses only sufficient Taylor inequalities from
+Mathlib. FORMAL-DATA.json contains outward 80-digit logarithm bounds Lᵣ,
+90-digit eighth-root bounds zᵣ and 70-digit upper weights uᵣ. The sufficient
+checks are
+
+    73/r ≤ zᵣ⁸,
+    zᵣ ≤ Σⱼ<60 (Lᵣ/8)^j/j!,
+    Σⱼ<26 (CLᵣ)^j/j! + (CLᵣ)^26·27/(26!·26) ≤ uᵣ,
+    Σᵣ nᵣr uᵣ < 73W.
+
+All are finite rational inequalities. Positivity and Mathlib's
+`Real.sum_le_exp_of_nonneg` give zᵣ ≤ exp(Lᵣ/8), hence log(73/r) ≤ Lᵣ.
+Mathlib's `Real.exp_bound'` gives exp(CLᵣ) ≤ uᵣ because 0 ≤ CLᵣ ≤ 1.
+Monotonicity then bounds the actual exp(C log(73/r)). The MathProof
+certificate-transfer theorem proves M(C) < 1. The real-power form follows
+from `Real.rpow_def_of_pos`; it does not assume any real-weight upper bound
+as an unresolved hypothesis. The complete analytic route passed source compilation, the separate
+target-module kernel replay and the transitive-axiom audit, with only
+propext, Classical.choice and Quot.sound. The separate all-imports fresh replay
+exceeded its 300-second budget and is not counted as a pass; VALIDATION.json
+retains that limit.
+These are standard analytic methods, not a priority claim for Taylor bounds.
+
+## Actual finite source and unchanged physical invoice
+
+PR #3815's exact current clock and fill are replayed with PR #3734's unchanged
+owner auditor. Every helper identity, strict lower/closed upper source window,
+same-call physical family, compact chronological prefix and bin capacity is
+checked. The complete idle bill, zero data/helper alias count and rank mass
+are rederived, with actual corrupted inputs rejected by production gates.
+The exact dual checks all 2,551 patterns and the original six semantic controls.
+
+Both current words are rechecked by the retained arbitrary-dirty interpreter:
+632 live columns and 91 independent buffer columns, both orientations, reverse
+pull-back and cleanup boundary. Widths and paid local frames are recomputed;
+no new local interpreter or native-chain rebuild is claimed. Retained native
+receipts bind these exact word hashes and canonical native-frame hashes.
+Full native-cache bytes are outside this bounded check; the optional immutable
+source archive is explicitly identified. Its inherited correctness/admission
+premises remain conditional.
+
+The root grid changes a proof parameter, not any instruction or coefficient.
+No gate, owner, COPY, restoration, routing or physical stock charge disappears.
+Physical stock remains 323,069, including 91 private buffers, and the charged
+2,048-bit reserve is retained. All-size and implementation obligations do not
+follow merely from a source/receipt hash.
+
+## Fresh complete outer arithmetic
+
+The unchanged carried-bootstrap function is extracted from its authenticated
+assembly source. The same exact 47-row engine runs 12 carried stages, including
+the carried-constant cutoff rule and four original controls, at the new C.
+Its legacy balanced endpoint is auxiliary, not the final modern κ.
+
+The current admitted modern checker and independent balance checker are copied
+unchanged into fresh scratch space. At endpoint_digits = 60 their existing
+formulas give
+
+    r₀ = 10⁻¹⁴⁰, δ = (1025/1024)r₀,
+    ε = 1−δ, β = 1−3δ,
+    b = C−r₀, q = C−2r₀,
+    FLOOR = 10¹⁰⁶, BACKOFF = 10⁻⁸².
+
+The checker derives 0 < C−εq < 1/FLOOR and that C·FLOOR is integral. Therefore
+floor(FLOOR·εq)/FLOOR = C−1/FLOOR, yielding
+
+    κ = C − 10⁻¹⁰⁶ − 10⁻⁸².
+
+The candidate is derived by the checker; it is not supplied as its comparison
+point. The comparison field is the previous PR #3815 endpoint. All 63 strict
+rows and 17 original rejection controls run, and the independent derivation
+agrees. Exact rational subtraction proves the gain in CLAIM.json. The retained
+BIT endpoint remains above C. The general mathematical/physical premises of
+this outer transfer are inherited; this finite proof does not admit an
+unconditional integer-multiplication theorem or change the selected frontier.
diff --git a/research/staged544-refined-endpoint/predecessor/README.md b/research/staged544-refined-endpoint/predecessor/README.md
new file mode 100644
index 0000000000..2e7bd785d2
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/README.md
@@ -0,0 +1,126 @@
+# A certificate for the current staged-helper moment
+
+For the unchanged cap-attaining staged-helper construction of
+[PR #3802](https://github.com/CrocSwap/integer-mult-bounds/pull/3802) and
+[PR #3815](https://github.com/CrocSwap/integer-mult-bounds/pull/3815), the new
+conditional endpoint is
+
+```
+κ = 0.001730578437266246818740713218049857556556448073971562433312
+    − 10⁻⁸² − 10⁻¹⁰⁶.
+```
+
+This is approximately **7.1321804986 × 10⁻²⁵ above their published endpoint**.
+The numerical gain is tiny. The useful proof target is the actual seven-term
+real moment, supported by an independent dyadic evaluator and our existing
+MathProof sufficient certificate-transfer argument. [PROOF.md](PROOF.md)
+states the mathematics and [VALIDATION.json](VALIDATION.json) records the exact
+completed proof and replay scope. No changed physical mechanism is claimed.
+
+This supersedes this session's abandoned T740 direction: the comparison is
+fresh main `20ecf2891d80b5acf18d980b2164fb44ea7faa90` and current PRs, not
+an old root. [OVERLAP.md](OVERLAP.md) records coverage and exact heads. The
+selected maintainer-reviewed result remains κ = 0.001672660197076918.
+
+## Concrete construction and evidence
+
+The T360 realization uses 3,240 calls, 1,321,920 literal helper roles,
+323,069 physical families and 91 private zero buffers. The complete profile is
+`{1:7989840, 2:2349000, 3:749088, 4:330480, 5:69120, 8:25920, 48:137952}`.
+PR #3802's T385 realization has the same normalized profile; it reached the
+24-digit endpoint first. PR #3815 supplies the smaller T360 realization and
+the packing-cap proof used here.
+
+The direct verifier regenerates the following:
+
+* The exact clock, all literal owner roles and their three actual corruption
+  controls, the zero helper/data alias count and all 2,551 cap patterns, using
+  the retained source implementations and their six additional controls.
+* Both changed supplier words on all 632 live inputs plus 91 arbitrary dirty
+  buffers, forward, dual and reverse pull-back, including the cleanup boundary.
+* Independent 512-bit integer interval signs at C and C + 10⁻⁶⁰. The unchanged
+  alternating-series engine separately certifies both signs. A third exact
+  Fraction-series implementation supplies an additional cross-check in tests.
+* The retained 12-stage carried bootstrap with 47 strict rows and four controls;
+  the actual modern outer derivation with 63 strict rows, 17 controls and the
+  independent balance derivation.
+
+Lean proves `Staged544.actual_rpow_moment_lt_one` for the actual real-power
+moment, with all seven analytic upper bounds discharged. Its transitive axioms
+are only `propext`, `Classical.choice` and `Quot.sound`. The portable proof files
+and exact rational data are under `lean/` and `FORMAL-DATA.json`; the Python
+replay binds their checked bytes and does not silently invoke Lean.
+
+## Reproduce
+
+The bounded scientific replay is Python 3.12, standard library only:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/staged544-certified-endpoint
+```
+
+A Git-free export needs only this package:
+
+```sh
+python3 -B research/staged544-certified-endpoint/verify.py
+```
+
+Verification is read-only. Optional `--output NEW.json` refuses to overwrite an
+existing file. The complete fresh result must equal RESULT.json. Inherited
+assertion-based checkers require normal Python; integrated `-O` execution is
+refused explicitly. Standalone arithmetic tests pass under `-O`. All child
+imports use a fresh bytecode namespace and retained input bytes are authenticated.
+
+For the separate proof, use Lean 4.34.1 and the pinned Mathlib commit
+`d13f23b723b8a846827a245b89c10fc7d3f11612`:
+
+```sh
+cd research/staged544-certified-endpoint/lean
+lake exe cache get
+lake build
+lake env lean AnalyticCertificate.lean
+lake env leanchecker AnalyticCertificate
+```
+
+## Conditional scope and provenance
+
+The full multiplication endpoint remains conditional on inherited all-size
+compilation and recursive rounding, physical/common-label and mutable-source
+realization, whole-call hosting, SCALAR/COPY/CLEAR and owned-zero contracts,
+uniform coefficient/stock, bootstrap, precision/tape/recovery, analytic outer
+transfer and BIT H1–H5 assumptions. A proved finite real moment does not prove
+those implementation or transfer hypotheses.
+
+Native-chain evidence is retained and bound to the actual words and canonical
+native frames. The full native cache and archive bytes are not authenticated
+or regenerated by this bounded replay. Their immutable optional source archive
+is identified in [NOTICE](NOTICE) for a full review; it is not a hidden runtime
+input. The original full-release records are reused at their stated conditional
+scope. This is not a fresh complete cold/native run or independent human review.
+
+All runtime source bytes are vendored exactly with origins and SHA-256 hashes
+in VENDOR.json. No unmerged predecessor Git objects, network, source-tree
+aliases or local MathProof paths are required for bounded replay. Retained
+scientific source files are unchanged.
+
+## Open next questions
+
+A material improvement requires a changed paid word, inventory or interface
+outside the attained cap. First require a source-bound exact operator and
+complete optimistic bill before native work. The bounded new-word searches in
+this session found no beating witness; [SEARCH.md](SEARCH.md) gives their scope
+without claiming a universal obstruction. A separate next proof is to compose
+the finite moment certificate with an explicit sufficient all-size supplier
+and outer theorem. [TASK.json](TASK.json) records first steps and blockers.
+
+## Credit
+
+ChaloupeMusic/PR #3815 supplies the changed T360 clock, literal fill and exact
+cap; hexapode/PR #3802 reached the same prior endpoint first at T385.
+dataisfire/PR #3734 supplies the staged helper 544 word and native release.
+Ross Richardson/PR #3605, zxv/PR #3638 and #3682, roger-sato/PR #3642,
+Sagar Thakrar/PR #3687 and #3693, and all retained owners/native/outer authors
+keep their credit. Our earlier MathProof moment-transfer work supplies the
+sufficient formal assembly. See [PRECEDENTS.md](PRECEDENTS.md), [NOTICE](NOTICE)
+and the retained source notices for exact origins. Dugongue, with substantial
+OpenAI Codex assistance. New work is Apache-2.0.
diff --git a/research/staged544-refined-endpoint/predecessor/SEARCH-HASHES.json b/research/staged544-refined-endpoint/predecessor/SEARCH-HASHES.json
new file mode 100644
index 0000000000..077c5b8722
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/SEARCH-HASHES.json
@@ -0,0 +1,17 @@
+{
+  "scope": "Private bounded-search artifact identities; not runtime proof inputs or claimed public replay",
+  "files": {
+    "search_retime_fast.py": "7ebd105542fc228bfb7f701b8556af5d5b2847734ff98ecf5567b3d299f855b0",
+    "fast-retime-result.json": "09c519b860c8bc29f89349f4f6df08ba638ec1a57b6085b83edecbe03786cdb8",
+    "search_commutators.py": "bb3598d9a4624381b23e62a049a6d63447d9c0c229a5d5bf5132ca519bf76de4",
+    "commutator-screen.json": "f43e35d15f7b660e4c1a24d786c7fa5730c266eb60cd458a40fecd9839811e86",
+    "search_prefix_full.py": "ead487150c40629c381d59440c860dd288a605ebe6f9190054dbd52297f85fdf",
+    "prefix-search-full.json": "fe78dbd9a5bfabf7525947dfc4dd5f993dadffcf0a51706f94c50c99dd2589ea",
+    "screen_global_staging.py": "f65bdee14fc4ae42170d3b688d579714ca179fa8e368d5c5dae9599e177e9265",
+    "global-staging-screen.json": "7d35b5cbc23252c41abe022f7e1204974b86ec7808d562286dbdc5050b41c66e",
+    "staging-candidate.json": "b5a02e7e49af18db0c0989dc1ab400fc2b59d4a41ee233bde85f66e844fc41d1",
+    "staging-rejection.json": "611798c537e2f09ce93d63e1686acf45213a9b7dd893aa6e84d7350c80518e0c",
+    "candidate-fast-retime-result.json": "408e230393ab0453399bef6548066a74d9da6cd6fb0f77459bd7c164143df66d",
+    "candidate-commutator-screen.json": "97d0256a2a16b8552500b4274626f1d419710622ae07b42ca5c1f4cf6d4b3c46"
+  }
+}
diff --git a/research/staged544-refined-endpoint/predecessor/SEARCH.md b/research/staged544-refined-endpoint/predecessor/SEARCH.md
new file mode 100644
index 0000000000..b60b9ec798
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/SEARCH.md
@@ -0,0 +1,38 @@
+# Bounded new-word searches and scope changes
+
+These exploratory searches ran on the exact current PR #3734 words before
+settling on the finite analytic certificate. They are not the research result,
+exhaustive architectural exclusions, or κ gains. Private artifact hashes in
+SEARCH-HASHES.json preserve the local checkpoints without pretending they
+are independently replayable public certificates.
+
+* Exact commuting B/restore retiming used our prior MathProof implementation
+  as a template. The accelerated current-word scan evaluated 4,982 moves,
+  with minimal moved frames, retained other frames and convergence back to
+  the unchanged suffix; no favorable paid-profile delta was found.
+* An adjacent compensated-commutator screen evaluated 101 eligible three-register
+  rewrites, each with its explicitly paid extra ADD; none improved the bill.
+* A sparse rational prefix-kernel search used fixed-prefix identities and
+  actual scalar/warp constraints over the 408 live helper columns. It found
+  666 two-donor relations plus selected sparse basis solutions at the stated
+  first-producer cuts. Those are sufficient algebraic candidates, not valid
+  physical words by themselves. Other cut choices, more general non-scalar
+  eigenvectors and changed native interfaces are outside the search.
+* One changed helper 472 word passed all 723-column forward/dual/reverse and
+  cleanup-boundary checks for the base provider. It changed the local bill by
+  `15H1 − 9H2 − 2H3 + 2H4` and the inventory by one width-8 helper becoming
+  width 7. Its new local profile was `{1:2455, 2:716, 3:222, 4:104, 5:20, 8:8}`.
+  A fresh exact dual over all 2,551 patterns, using that actual new inventory,
+  is feasible and strictly rejects the current target even before chronology.
+  Thus this one candidate cannot beat the current endpoint in that model;
+  its extra provider and full native admission were not pursued.
+* Coordinate and adjacent-node-pair frame changes, with endpoints and native
+  warp contracts fixed, produced no lower bill on that candidate. A subsequent
+  4,973-move retiming screen and 102 compensated-commutator cases also found no
+  favorable delta. These do not exhaust simultaneous graph/contract changes.
+
+The discriminating decision was to stop investing in a new allocation or
+native admission of that losing candidate. The submitted positive result is
+instead the actual current-profile real moment certificate and its tiny strict
+conditional endpoint refinement. No failed search is relabelled as a universal
+obstruction or a new mechanism.
diff --git a/research/staged544-refined-endpoint/predecessor/TASK.json b/research/staged544-refined-endpoint/predecessor/TASK.json
new file mode 100644
index 0000000000..3963e9915f
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/TASK.json
@@ -0,0 +1,73 @@
+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does one changed supplier or assembly claim lack a source-bound independent check of a specific new implication?",
+    "starting_points": [
+      "certificates/selected-result.json",
+      "research/rank7-joint-endframes/README.md",
+      "docs/research/research-family-status.md"
+    ],
+    "methods": [
+      "Independently reconstruct a coordinated frame/circuit change and its single-change control, then derive the complete rank and physical ledger.",
+      "Implement certified moment inequalities through a different exact enclosure method."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #2979 joint rank-7 end-frame proposal and the revision-6 evidence status. Its claimed kappa is not selected. Compare coordinated changes with the losing single-change controls and locate the complete physical ledger; an aggregate moment alone does not validate the mechanism.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Prioritize a bounded independent audit of #2979: reconstruct one joint group and its single-lowering control, check the declared frames/chronology, then trace changed children, idle widths and cleanup buffers into the complete invoice and both supplier caps. Choose one still-missing implication after checking existing audits. Stop with exact unresolved dependencies if full admission exceeds the session; do not launch a new lowering census on an unvalidated premise or substitute extra root digits for mechanism checking.",
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Promoting #2979 from merged source archive to validated supplier solely from author PASS receipts or aggregate pricing."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering"
+    ]
+  },
+  "scope_status": "changed",
+  "scope_change_reason": "The donor explicitly requested the best-known kappa and then rejected the stale T740 scope. Fresh current-main/539-open-PR intake selected the staged544 cap-attaining T360/T385 frontier. The new scope checks its actual real moment and refines its exact conditional endpoint; no claim of a new physical mechanism.",
+  "actual_scope": "Independent real-moment certificate and strict conditional endpoint refinement of PR3815/PR3802, reusing MathProof sufficient moment-transfer work and exact current finite source evidence.",
+  "bottleneck_link": "Closes the concrete numerical admission implication for the complete current seven-rank bill. Extra digits alone are not the claimed independent verification coverage.",
+  "budget_minutes": 60,
+  "stop_rule": "Bound new-word searches and preserve failures. Keep the final quarter for proof/kernel, registered replay, source history and publication checks; do not extend the attained fixed-inventory packing family.",
+  "outcome": "completed",
+  "evidence": "Closed Lean proof of the actual real-power moment, with all seven analytic upper bounds discharged and only standard axioms; separate target-module kernel replay passes. Registered Python 3.12 scientific/control replay and actual Git-free CLI pass. Full imported-library --fresh replay timed out at 300 seconds; no fresh native chains.",
+  "author": "Dugongue",
+  "ai_assistance": "Substantial OpenAI Codex assistance in research, proof, implementation, verification and packaging; no independent human review claimed.",
+  "open_questions": [
+    {
+      "question": "Can the exact finite moment certificate be composed with a formal sufficient all-size supplier/outer theorem?",
+      "why_it_matters": "This would turn finite analytic admission into a stronger multiplication theorem under explicit implementation hypotheses.",
+      "first_step": "Freeze one existing outer transfer statement and connect its strict moment hypothesis to the exact current profile and endpoint.",
+      "blocked_by": "The inherited all-size/compiler, hosting, source, precision, physical and BIT assumptions remain outside this finite certificate."
+    },
+    {
+      "question": "Which changed word raises the attained staged544 inventory ceiling?",
+      "why_it_matters": "The numerical refinement here is tiny; a material improvement requires changing the paid construction.",
+      "first_step": "Screen a source-bound exact operator change against the full fixed-inventory dual before complete native admission.",
+      "blocked_by": "The bounded retiming, staged-entry and commutator searches in this session yielded no beating paid-profile witness."
+    }
+  ],
+  "publication_status": "Local result; publication script uses the donor authenticated PowerShell session. No automatic merge or selected-frontier change."
+}
diff --git a/research/staged544-refined-endpoint/predecessor/VALIDATION.json b/research/staged544-refined-endpoint/predecessor/VALIDATION.json
new file mode 100644
index 0000000000..748df89647
--- /dev/null
+++ b/research/staged544-refined-endpoint/predecessor/VALIDATION.json
@@ -0,0 +1,92 @@
+{
+  "schema_version": 1,
+  "source_base": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "scientific_scope": "Unconditional finite real-power moment certificate; conditional full multiplication endpoint under the stated inherited premises.",
+  "python": "CPython 3.12.10, Windows x86_64, standard library only",
+  "registered_checks": [
+    {
+      "name": "metadata",
+      "exit_code": 0,
+      "seconds": 0.362
+    },
+    {
+      "name": "registered",
+      "exit_code": 0,
+      "seconds": 79.358
+    }
+  ],
+  "baseline_bindings": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY on current-main selected inputs; not a cold replay",
+  "lean": {
+    "version": "4.34.1",
+    "mathlib_commit": "d13f23b723b8a846827a245b89c10fc7d3f11612",
+    "compiled_source_sha256": {
+      "AnalyticCertificate.lean": "1edd0cdcdc5873be01aa085ecc2da46b97cf3949598a3a691608627f966bbe6e",
+      "CapCertificate.lean": "3992defbc43fd2fda494fffed2ccc1ce3667e16e2a003aeb1492ffe0e721ee50"
+    },
+    "actual_theorems": [
+      "Staged544.actual_moment_lt_one",
+      "Staged544.actual_rpow_moment_lt_one"
+    ],
+    "actual_analytic_upper_bounds_discharged": true,
+    "elaboration_exit_code": 0,
+    "axioms": [
+      "propext",
+      "Classical.choice",
+      "Quot.sound"
+    ],
+    "module_kernel_replay": {
+      "exit_code": 0,
+      "seconds": 36.07,
+      "command": [
+        "lake",
+        "env",
+        "leanchecker",
+        "AnalyticCertificate"
+      ]
+    },
+    "all_imports_fresh_replay": {
+      "exit_code": 124,
+      "seconds": 300.056,
+      "command": [
+        "lake",
+        "env",
+        "leanchecker",
+        "--fresh",
+        "AnalyticCertificate"
+      ]
+    },
+    "fresh_replay_limit": "The all-imports --fresh attempt timed out at 300 seconds and is not counted as a pass. Source compilation, theorem transitive-axiom audit and separate target-module kernel replay passed."
+  },
+  "finite_coverage": {
+    "clock_owner_roles": 1321920,
+    "calls": 3240,
+    "live_inputs_per_word": 632,
+    "independent_dirty_buffers_per_word": 91,
+    "orientations": [
+      "forward",
+      "dual",
+      "reverse pull-back"
+    ],
+    "packing_patterns": 2551,
+    "bootstrap_strict_rows": 47,
+    "bootstrap_original_controls": 4,
+    "modern_outer_strict_rows": 63,
+    "modern_outer_original_controls": 17,
+    "owner_original_controls": 3,
+    "clock_cap_extra_controls": 6
+  },
+  "reuse": "Retained native receipts and source metadata; no fresh native chains or omitted cache-byte authentication. Exact retained interpreter and outer engines keep their credit.",
+  "remaining_obligations": [
+    "Independent scientific admission of all inherited all-size/compiler, physical hosting/source and precision/tape/recovery contracts",
+    "Full all-size outer transfer and BIT H1-H5 premises",
+    "Independent human review and hosted CI; no selected-frontier update"
+  ],
+  "log_sha256": {
+    "analytic-lean.log": "e85b210e44d543ab8267e2364eaffd17353e43c2ff3569d30731d18f8542e2b3",
+    "kernel-replay.log": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855",
+    "kernel-fresh.log": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855",
+    "registered-current.log": "33c6531080f7d0821f10360b6500022ceb1bb91da38823f9beee9af6a735e0bf",
+    "metadata-current.log": "689a43f8278e023fb84fc0cc029f6f20b7c3777f852bd6f83904e80f5d843c99",
+    "baseline-current.log": "767acbd9da247c78bcbc5758c05cfad26c5fe7c162a1be37d2528bc3bb3e5ee0"
+  }
+}
diff --git a/research/staged544-refined-endpoint/test_arithmetic.py b/research/staged544-refined-endpoint/test_arithmetic.py
new file mode 100644
index 0000000000..7387f65c7e
--- /dev/null
+++ b/research/staged544-refined-endpoint/test_arithmetic.py
@@ -0,0 +1,104 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Independent exact-series oracle and behavioral checks, including -O.
+
+The oracle follows the classical atanh/Taylor identities also used by the
+credited predecessor root.py; it sums exact Fractions and rounds only at the
+logarithm boundary, unlike the dyadic evaluator's repeated interval rounding.
+This is a second arithmetic implementation, not a new transcendental method.
+Dugongue, with substantial OpenAI Codex assistance.
+"""
+from fractions import Fraction as Q
+from functools import lru_cache
+from pathlib import Path
+import hashlib
+import json
+import types
+import unittest
+
+P=Path(__file__).resolve().parent
+module=types.ModuleType('dyadic_source')
+exec(compile((P/'dyadic_moment.py').read_bytes(),'dyadic_moment.py','exec'),module.__dict__)
+D=module.DyadicMoment
+PROFILE={1:7989840,2:2349000,3:749088,4:330480,5:69120,8:25920,48:137952}
+
+@lru_cache(None)
+def log_oracle(x):
+    k=0
+    while x>=2:x/=2;k+=1
+    def unit(y):
+        t=(y-1)/(y+1);power=t;total=Q(0)
+        for j in range(128):
+            total+=2*power/(2*j+1);power*=t*t
+        return total,total+2*power/(257*(1-t*t))
+    lo,hi=unit(x);a,b=unit(Q(2));lo+=k*a;hi+=k*b
+    scale=10**100
+    return Q((lo*scale).__floor__(),scale),Q((hi*scale).__ceil__(),scale)
+
+def exp_oracle(x):
+    if not 0<=x<=Q(1,2):raise ValueError('oracle exponential domain')
+    term=total=Q(1)
+    for j in range(1,65):term*=x/j;total+=term
+    return total,total+term*x/65/(1-x/66)
+
+def oracle_moment(b):
+    low=high=Q(0)
+    for r,n in PROFILE.items():
+        a,z=log_oracle(Q(73,r));weight=Q(n*r,73*896*360)
+        low+=weight*exp_oracle(b*a)[0]
+        high+=weight*exp_oracle(b*z)[1]
+    # Final directed decimal rounding keeps the public receipt compact.
+    scale=10**90
+    return Q((low*scale).__floor__(),scale),Q((high*scale).__ceil__(),scale)
+
+class ArithmeticTests(unittest.TestCase):
+    def test_exact_zero(self):
+        e=D(PROFILE,73,896*360)
+        self.assertEqual(e.log_unit(1,1),(0,0))
+        self.assertEqual(e.exp_interval(0,0),(e.scale,e.scale))
+        self.assertEqual(e.moment(Q(0)),(Q(65088,73*896),)*2)
+
+    def test_rational_oracle_root(self):
+        a=json.loads((P/'CLAIM.json').read_bytes());c=Q(a['complex_lower'])
+        self.assertLess(oracle_moment(c)[1],1)
+        self.assertGreater(oracle_moment(c+Q(1,10**80))[0],1)
+
+    def test_independent_precision_replay(self):
+        c=Q(json.loads((P/'CLAIM.json').read_bytes())['complex_lower'])
+        for bits in (384,512,640):
+            e=D(PROFILE,73,896*360,bits=bits)
+            self.assertLess(e.moment(c)[1],1)
+            self.assertGreater(e.moment(c+Q(1,10**80))[0],1)
+
+    def test_log_intervals_against_series(self):
+        e=D(PROFILE,73,896*360,bits=256)
+        for numerator,denominator in ((1,1),(2,1),(3,2),(73,48),(73,1),(1024,1)):
+            a,b=e.log_ratio(numerator,denominator);lo,hi=log_oracle(Q(numerator,denominator))
+            self.assertLessEqual(Q(a,e.scale),hi)
+            self.assertLessEqual(lo,Q(b,e.scale))
+
+    def test_exp_half_boundary(self):
+        e=D(PROFILE,73,896*360,bits=256)
+        for x in (Q(0),Q(1,1000),Q(1,2)):
+            a,b=e.exp_interval((x*e.scale).__floor__(),(x*e.scale).__ceil__())
+            lo,hi=exp_oracle(x)
+            self.assertLessEqual(Q(a,e.scale),hi)
+            self.assertLessEqual(lo,Q(b,e.scale))
+
+    def test_rank_mass_insufficient(self):
+        # Preserve weighted rank mass, but concentrate it on rank 1.
+        bad={1:65088*360};c=Q(json.loads((P/'CLAIM.json').read_bytes())['complex_lower'])
+        self.assertEqual(sum(r*n for r,n in bad.items()),65088*360)
+        self.assertGreater(D(bad,73,896*360).moment(c)[0],1)
+
+    def test_domain_guards(self):
+        for profile,m,w in (({0:1},73,1),({73:1},73,1),({1:-1},73,1),({},73,1),({1:1},73,0),({True:1},73,1)):
+            with self.assertRaises(ValueError):D(profile,m,w)
+        e=D(PROFILE,73,896*360)
+        for lo,hi in ((-1,0),(2,1),(0,e.scale//2+1)):
+            with self.assertRaises(ValueError):e.exp_interval(lo,hi)
+        with self.assertRaises(ValueError):e.weights(0.001)
+        with self.assertRaises(ValueError):e.weights(Q(-1))
+        with self.assertRaises(ValueError):D(PROFILE,73,896*360,bits=32).bracket(60)
+
+if __name__=='__main__':unittest.main(verbosity=2)
diff --git a/research/staged544-refined-endpoint/test_cli.py b/research/staged544-refined-endpoint/test_cli.py
new file mode 100644
index 0000000000..5238177521
--- /dev/null
+++ b/research/staged544-refined-endpoint/test_cli.py
@@ -0,0 +1,32 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Actual standalone CLI acceptance and fail-closed source/claim/proof controls.
+Dugongue, with substantial OpenAI Codex assistance.
+"""
+from pathlib import Path
+import hashlib,json,os,shutil,subprocess,sys,tempfile,unittest
+P=Path(__file__).resolve().parent
+
+class CliTests(unittest.TestCase):
+ def test_actual_export_and_corruptions(self):
+  with tempfile.TemporaryDirectory(prefix='staged544-cli-') as folder:
+   package=Path(folder)/'package';shutil.copytree(P,package,ignore=shutil.ignore_patterns('__pycache__','.lake'))
+   def hashes():return {p.relative_to(package).as_posix():hashlib.sha256(p.read_bytes()).hexdigest() for p in package.rglob('*') if p.is_file()}
+   def run(*options,extra=()):
+    env=dict(os.environ,PYTHONPYCACHEPREFIX=str(Path(folder)/'bytecode'));env.pop('PYTHONOPTIMIZE',None)
+    return subprocess.run([sys.executable,*options,'-B',str(package/'verify.py'),*extra],env=env,stdout=subprocess.PIPE,stderr=subprocess.STDOUT,timeout=120)
+   original=hashes();result=run();self.assertEqual(result.returncode,0,result.stdout.decode(errors='replace'));self.assertIn(b'PASS_STAGED544_CURRENT_CONDITIONAL_ENDPOINT',result.stdout);self.assertEqual(hashes(),original)
+   for name,mutator in [('CLAIM.json',lambda x:x.__setitem__('kappa','0')),('CLAIM.json',lambda x:x.__setitem__('fresh_native_replay',True)),('CLAIM.json',lambda x:x.__setitem__('physical',323068)),('RESULT.json',lambda x:x.__setitem__('scope','false scope')),('FORMAL-DATA.json',lambda x:x['log_upper'].__setitem__('1','0'))]:
+    with self.subTest(file=name):
+     file=package/name;before=file.read_bytes();data=json.loads(before);mutator(data)
+     try:file.write_text(json.dumps(data),encoding='utf-8');self.assertNotEqual(run().returncode,0)
+     finally:file.write_bytes(before)
+   for file in [package/'vendor/staged544-clock-ceiling/clock.py',package/'lean/AnalyticCertificate.lean']:
+    before=file.read_bytes()
+    try:file.write_bytes(before+b'\n# source mutation\n');self.assertNotEqual(run().returncode,0)
+    finally:file.write_bytes(before)
+   self.assertNotEqual(run('-O').returncode,0,'inherited assertions must remain enabled')
+   existing=Path(folder)/'existing.json';existing.write_bytes(b'preserve this')
+   self.assertNotEqual(run(extra=('--output',str(existing))).returncode,0);self.assertEqual(existing.read_bytes(),b'preserve this')
+   self.assertEqual(hashes(),original,'read-only source and all corruption preimages restored')
+if __name__=='__main__':unittest.main(verbosity=2)
diff --git a/research/staged544-refined-endpoint/vendor/bootstrap/assemble.py b/research/staged544-refined-endpoint/vendor/bootstrap/assemble.py
new file mode 100644
index 0000000000..b317e9b35d
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/bootstrap/assemble.py
@@ -0,0 +1,292 @@
+"""Gauge44 complete global admission with the retained native exact-complex moment.
+
+Prepared with OpenAI Codex assistance. The rational engines and outer
+constraints are retained from PR547/PR548 and their preserved ancestry.
+The bit operator is the cited unchanged PR589 supplier (PR555's punctured
+R439 word, v = 111, decoder constant 63, two A-section transports), shipped
+as inputs/PR589-EXPECTED.json and pinned by git blob id. Its rank-mass
+deficit is derived from the word's parameters, T(2Kv - L*cst) at nine
+stages (PR545/PR555 rank-mass identity), not hard-coded. The complex owner
+and invoice inputs must be freshly emitted by the packaged physical replay.
+All conservative admission guards and conservative moment/outer witnesses run
+before the PR559 native-complex correction. The bit surcharge is retained.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from decimal import Decimal, localcontext
+import argparse,json,hashlib,importlib.util,os,sys
+from collections import Counter
+assert __debug__, 'Assertions required'
+P=Path(sys.argv[1] if __name__=='__main__' else os.environ['GAIN_PR564_SOURCE']).resolve()
+read=lambda p:json.loads(p.read_text())
+sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest()
+hash_ok=lambda x:isinstance(x,str) and len(x)==64 and all(t in '0123456789abcdef'for t in x)
+digest=lambda x:hashlib.sha256(json.dumps(x,separators=(',',':')).encode()).hexdigest()
+def load(n):
+ s=importlib.util.spec_from_file_location(n,P/'engines'/(n+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,B,A=[load(n)for n in ('moment','base_two_moment','outer')]
+import hashlib
+try:
+ from gmpy2 import mpz   # optional accelerator, identical integer results
+except ImportError:
+ mpz=int
+def dec(x):
+ with localcontext()as ctx:ctx.prec=3120;return str(Decimal(x.numerator)/Decimal(x.denominator))
+import sys as _sys;_sys.path.insert(0,str(Path(__file__).resolve().parent));import fine_moment as BF,fine_mercator as MF,fine_cache as FC;from exact_coarse import moment as exact_coarse_moment
+from witness import SPEC,DISPATCH,CALL_ORDER,COUNTS,PIDS,X,DISCOVER,pin,SPOT,spot_widths,GEO,REPLICAS,CALLS,DATA,DEFICIT,PAYLOAD,PARENT_LIVE,PARENT_PHYSICAL,PARENT_IDLE,PARENT_HELPERS
+# Cited bit supplier: PR589's tune/EXPECTED.json (merged on main as research/e8-bit-p111-tuned/tune/EXPECTED.json).
+BIT_PATH=Path(__file__).resolve().parents[1]/'inputs/PR589-EXPECTED.json'
+BIT_BLOB='e293ddc557742637f9f679f083a1a282d87b1b13'
+# Nine-stage bit template (PR463/PR510 chain, 32 replicas): L stages, T replicas, K=(L-1)//2 pairs, moment width M=1+7L.
+BIT_L,BIT_T=9,32;BIT_K=(BIT_L-1)//2;BIT_M=1+7*BIT_L
+def git_blob(data):return hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest()
+def load_bit(data):
+ """The pinned supplier record: its full record (word parameters) and its nine-stage price."""
+ assert git_blob(data)==BIT_BLOB,'bit supplier blob pin'
+ return json.loads(data)
+def bit_deficit(word,live):
+ """Rank-mass deficit M*live - mass = T(2Kv - L*cst) of a v-port bit word with decoder constant cst (PR545/PR555).
+ cst is the sum of the retained totals' COPY ranks and live = 2KvT data wires; both are cross-checked here, and the
+ caller checks the result against the priced profile, so neither v nor cst is free."""
+ v,cst,ranks=word['v'],word['decoder_constant'],word['copy_ranks']
+ assert type(v)is int and type(cst)is int and all(type(r)is int and 0<r<=8 for r in ranks)
+ assert len(ranks)==word['retained_totals']==BIT_L and cst==sum(ranks)
+ assert live==2*BIT_K*v*BIT_T
+ d=BIT_T*(2*BIT_K*v-BIT_L*cst);assert d>0;return d
+
+
+def assemble(owner,invoice,owner_sha,invoice_sha):
+ # These are admission receipts, not optional collections of successful rows.
+ # Require both orientations and every finite check before pricing anything.
+ assert owner['status']=='PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE'
+ assert invoice['status']=='PASS_INDEPENDENT_MIXED_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE'
+ assert all(hash_ok(x)for x in (owner_sha,invoice_sha,owner['template_sha256'],owner['controller_audit_sha256'],invoice['parent_template_sha256']))
+ checks={'local_primitive_bound','logical','router','literal','mass','precision','tape','row','halving'}
+ assert set(invoice['strict_checks'])==checks
+ assert all(invoice['strict_checks'][name] is True for name in checks)
+ assert set(invoice['providers'])==set(PIDS)
+ assert invoice['provider_counts']==COUNTS
+ assert owner['all_live_owners_per_call_distinct'] is True
+ # R (helpers per provider word) is read from the invoice and must be shared by every provider; n=2v+R.
+ R=invoice['helpers'];assert type(R) is int and R>0 and all(p['helpers']==R for p in invoice['providers'].values())
+ assert owner['owners_per_call']==2*112+R and owner['actual_helper_owner_charts']==owner['hosted_residuals']==CALLS*R==pin('expected.hosted_residuals',owner['hosted_residuals'])
+ pin('expected.rank_mass',owner['rank_mass'])
+ pin('expected.actual_73column_sigmas',owner['actual_73column_sigmas']);pin('expected.cross_replica_helper_roles',owner['cross_replica_helper_roles'])
+ pin('expected.idle_remainders',owner['explicit_idle_remainder_children'])
+ assert invoice['complete_invocation_sweeps']==invoice['bridge_sweeps']==CALLS
+ assert invoice['live']==owner['live'] and invoice['physical']==owner['physical']
+ assert invoice['centres']==8
+ assert invoice['physical_stock_identity']==dict(data_and_owner_families=DATA,local_helpers=R,private_centres=8,private_zero_buffers=invoice['private_zero_buffers'],coefficient_work=2)
+ assert invoice['private_zero_buffers']==max(p['zero_buffers']for p in invoice['providers'].values())
+ for p in invoice['providers'].values():
+  for z in p['local'].values():
+   assert z['private_zero_buffer_peak']==p['zero_buffers'] and z['private_zero_buffer_clears']==2*p['zero_buffers']
+   assert z['macros']==z['literal_scalar_macros']+z['private_zero_buffer_clears']
+   assert z['real_component_primitives']==z['literal_real_component_primitives']+2*z['private_zero_buffer_clears']
+ assert sum(invoice['physical_stock_identity'].values())==invoice['physical']
+ assert invoice['controller_audit_sha256']==owner['controller_audit_sha256']
+ for pid,p in invoice['providers'].items():
+  assert set(p['local'])=={'forward','backward'}
+  assert p['helpers']==R and p['centres']==8 and p['n']==2*112+R and p['v']==112 and p['h']==9
+  K='providers.'+pid+'.';pin(K+'entering',p['entering_helpers']);pin(K+'entrance',p['entrance_rank'])
+  pin(K+'inventory',p['width_inventory'])
+  assert all(type(w) is int and 0<w<=9 for w in p['widths'])
+  assert Counter(p['widths'])==Counter({int(r):n for r,n in p['width_inventory'].items()})
+  assert sum(p['widths'])==9*R-p['entrance_rank']-p['terminal_codimension'] and len(p['widths'])==R   # HPE: widths |final|-|start|
+  pin(K+'width_digest',digest(p['widths']));assert tuple(p['widths'][h]for h in SPOT)==spot_widths()
+  pin(K+'expected_profile',p['paid_profile'])
+  assert p['children']==sum(p['paid_profile'].values())
+  pin(K+'program_hashes',p['program_hashes'])
+  for name in ('CANONICAL-WORD.json.gz','LOCAL-PROGRAMS.json.gz','LOCAL-CHECKS.json','INDEPENDENT-LOCAL-REPLAY.json','CLIFFORD-RECIPES.json','CLIFFORD-REVIEW.json','RESIDUAL-ADAPTERS.json'):
+   assert hash_ok(p['artifacts'][name]) and p['artifacts'][name]==owner['source_artifacts']['providers/'+pid+'/'+name]
+   if name in ('LOCAL-CHECKS.json','CLIFFORD-RECIPES.json','RESIDUAL-ADAPTERS.json'):pin(K+'artifacts.'+name,p['artifacts'][name])
+  pin(K+'artifact_semantic_sha256',p['semantic_artifact_sha256'])
+  pin(K+'complete_contract',{k:v for k,v in p.items()if k!='artifacts'})
+  assert owner['provider_artifact_semantic_sha256'][pid]==p['semantic_artifact_sha256']
+  pin(K+'semantic_phase_sha256',p['semantic_artifact_sha256']['CLIFFORD-REVIEW.json'])
+  assert owner['provider_phase_semantic_sha256'][pid]==invoice['provider_phase_semantic_sha256'][pid]==p['semantic_artifact_sha256']['CLIFFORD-REVIEW.json']
+  for name in ('forward','backward'):
+   z=p['local'][name];pinned=pin(K+'expected_local.'+name,{k:z[k]for k in('events','macros','real_component_primitives','prefix_l1','coefficient_work_primitive_count')});primitives=pinned['real_component_primitives'];coefficient_work=pinned['coefficient_work_primitive_count']
+   assert z['all_dirty_helper_columns_restored'] is True and z['private_centres_and_coefficient_work_cleared'] is True
+   assert z['all_exact_columns']==2*112+R and z['dirty_helper_columns']==R
+   assert z['events']==pinned['events'] and z['macros']==pinned['macros']
+   assert z['real_component_primitives']==primitives and z['coefficient_work_primitive_count']==coefficient_work
+   assert z['pre_wrappers']==z['post_wrappers']==p['entering_helpers']+p.get('cold_hyperplane_ended_helpers',0);pin(K+'wrappers',z['pre_wrappers'])   # COLD-HPE
+   assert 0<z['prefix_bits']<=192 and 0<Q(z['prefix_l1'])<2**192
+   peak=Q(z['prefix_l1']);assert z['prefix_bits']==peak.numerator.bit_length()-peak.denominator.bit_length()+1
+   assert peak==Q(pinned['prefix_l1'])
+  assert p['compiler']==64*74**3
+  assert p['local_adapters']==16*(max(z['macros']for z in p['local'].values())+max(z['warp_events']for z in p['local'].values())+p['children']+2*112+R+8+p['zero_buffers']+2)+64*73
+  assert p['local_primitive_bound']==max(z['real_component_primitives']for z in p['local'].values())+p['local_adapters']*p['compiler']
+ calls=invoice['selected_calls'];assert len(calls)==CALLS
+ assert [(c['stage'],c['replica'])for c in calls]==CALL_ORDER
+ assert {(c['stage'],c['replica']) for c in calls}=={(s,r) for s in range(9) for r in range(REPLICAS)}
+ for c in calls:
+  pid=DISPATCH[c['stage'],c['replica']]
+  assert c['provider_id']==pid
+  # The retained nine-stage recipe alternates its two reflected shears.
+  assert c['orientation']==('forward' if c['stage']%2==0 else 'backward')
+  p=invoice['providers'][pid];z=p['local'][c['orientation']]
+  assert c['program_sha256']==p['program_hashes'][c['orientation']]
+  for name in ('events','macros','real_component_primitives','coefficient_work_primitive_count','pre_wrappers','post_wrappers'):
+   assert c[name]==z[name]
+  assert c['entering_helpers']==p['entering_helpers'] and c['entrance_rank']==p['entrance_rank']
+ assert invoice['selected_real_component_primitives']==sum(c['real_component_primitives'] for c in calls)
+ assert invoice['selected_coefficient_work_primitive_count']==sum(c['coefficient_work_primitive_count'] for c in calls)
+ assert invoice['selected_scalar_macros']==sum(c['macros'] for c in calls)==sum(invoice['providers'][c['provider_id']]['local'][c['orientation']]['macros'] for c in calls)
+ assert invoice['factors_per_wrapper']==144
+ assert invoice['wrapper_elementary_factors_charged']==sum(c['pre_wrappers']+c['post_wrappers'] for c in calls)*144
+ paid=Counter()
+ for c in calls:paid.update({int(r):n for r,n in invoice['providers'][c['provider_id']]['paid_profile'].items()})
+ assert dict(paid)=={int(r):n for r,n in invoice['selected_local_paid_profile'].items()}
+ idle=Counter({int(r):n for r,n in owner['paid_profile'].items()});idle.subtract(paid);assert all(n>=0 for n in idle.values())
+ idle=+idle;assert sum(idle.values())==owner['explicit_idle_remainder_children']
+ pin('expected.idle_remainder_profile',{str(r):n for r,n in sorted(idle.items())})
+ paid.update(idle)
+ assert dict(paid)=={int(r):n for r,n in owner['paid_profile'].items()}
+ assert invoice['compiler']==64*74**3
+ assert invoice['local_adapters']==max(p['local_adapters']for p in invoice['providers'].values())
+ assert invoice['local_primitive_bound']==max(p['local_primitive_bound']for p in invoice['providers'].values())
+ # Recount the retained dedicated-bank envelope independently. Its geometry
+ # stays fixed, while every local primitive and wrapper comes from the actual
+ # selected provider/orientation sums above.
+ parent_live,parent_physical,parent_idle=PARENT_LIVE,PARENT_PHYSICAL,PARENT_IDLE
+ parent_banks,rem=divmod(parent_live-2*112*4*REPLICAS,9);assert rem==0 and parent_physical==parent_live+PARENT_HELPERS+8+2   # five bank groups with 8,8,8,8,5 active lanes and R0=504 helpers; shared private work is restored
+ assert owner['source_idle_children_charged']==parent_idle
+ bridges,shifts=2*112*CALLS,4*112*REPLICAS;moves=2*shifts
+ nonlocal_items=(4*112*REPLICAS*9+2*112*REPLICAS*9+2*CALLS*R+parent_idle+9*parent_banks+bridges+shifts+moves+parent_live+parent_physical+invoice['wrapper_elementary_factors_charged'])
+ outside=8*nonlocal_items+64*73
+ pervertex=(invoice['selected_real_component_primitives']+bridges+shifts+moves+parent_live+(CALLS*invoice['local_adapters']+outside)*invoice['compiler']+64*parent_physical)
+ rebased=pervertex+65*(invoice['physical']-parent_physical)+(invoice['live']-parent_live)
+ assert invoice['unhosted_pervertex']==pervertex and invoice['hosted_rebased_pervertex']==rebased
+ used=max(pervertex,rebased)
+ prior=invoice['retained_prior_overcharge']
+ if prior is not None:
+  assert hash_ok(prior['sha256']) and type(prior['previous_pervertex_allowance']) is int and prior['previous_pervertex_allowance']>0
+  assert prior['added_complete_new_pervertex']==pervertex
+  assert prior['policy']=='Retain old allowance and add entire fresh mixed recount.'
+  used=max(used,prior['previous_pervertex_allowance']+pervertex)
+ assert invoice['conservative_pervertex_used']==used
+ finite=invoice['finite']
+ assert 0<finite['physical_row_coefficient']<finite['retained_row_coefficient']==20161
+ assert 0<finite['required_C0_bits']<finite['C0_exponent']==210000
+ assert finite['G_exponent']==30000 and finite['E_exponent']==100000
+ assert finite['halving_integer_inequality']==[5329,4608]
+ assert 0<invoice['global_prefix_bits']<=invoice['charged_prefix_bits']==PAYLOAD
+ assert invoice['row_prefix_extra']==2*PAYLOAD
+ assert invoice['global_prefix_bits']==max(z['prefix_bits']for p in invoice['providers'].values()for z in p['local'].values())+37+CALLS+CALLS+4
+ group=2**(36*36)
+ for j in range(1,37):group*=2**(2*j)-1
+ stock=group*invoice['physical'];streamed_mass=group*owner['rank_mass']
+ logical=group*used+64;router=invoice['compiler']*(logical+1)*(stock+1)**2
+ literal=2*2**30000*stock**2+8*streamed_mass+4*stock+4+32*73
+ total_mass=streamed_mass+2**100000
+ assert finite['physical_stock_bits']==stock.bit_length() and finite['router_bits']==router.bit_length() and finite['literal_bits']==literal.bit_length()
+ assert finite['physical_row_coefficient']==2*stock.bit_length()+9909+252+2*PAYLOAD
+ assert finite['required_C0_bits']==(32*73*total_mass*total_mass).bit_length()
+ actual_checks=dict(local_primitive_bound=invoice['local_primitive_bound']<2**48,logical=logical<2**6050,router=router<2**30000,literal=literal<2**100000,mass=total_mass<2**100001,precision=32*73*total_mass*total_mass<2**210000,tape=2*total_mass+18<2**210000,row=finite['physical_row_coefficient']<20161,halving=73*73>2*48*48)
+ assert actual_checks==invoice['strict_checks'] and all(actual_checks.values())
+ bit_record=load_bit(BIT_PATH.read_bytes())
+ bit=bit_record['nine']
+ assert 'nine-stage' in bit['pipeline'] and '32 replicas' in bit['pipeline']
+ c=Q(bit['bit_lower'])
+ bit_D=bit_deficit(bit_record['word'],bit['bit']['live'])
+ H={int(r):n for r,n in owner['paid_profile'].items()};pin('expected.global_profile',owner['paid_profile'])
+ # Claimed lower endpoints come from the spec; discovery derives them with the same certified moment engine.
+ # Either way they are verified below (accepted at b, rejected at b+1e-24).
+ if DISCOVER:b=Q(M.certify(H,73,owner['live'],False)['lower']);conservative_b=Q(M.certify(H,73,owner['live'],True)['lower'])
+ else:b=Q(X['complex_lower']);conservative_b=Q(X['conservative_complex_lower'])
+ assert owner['template_sha256']==invoice['template_sha256'] and owner_sha==invoice['owner_audit_sha256']
+ pin('expected.live',owner['live']);pin('expected.physical',owner['physical']);assert owner['deficit']==DEFICIT
+ assert owner['banked_residuals']==0
+ def check_moment(side,h,m,W,lower,deficit,bad,digits=24):
+  assert m*W-sum(r*n for r,n in h.items())==deficit and m*m>2*max(h)**2
+  rows=[]
+  def certify(z,accept,eM,eB):
+   if eM is MF:   # finer engines: memoised across stages and prefetched concurrently (fine_cache.py); results are the engines' own outputs
+    l,u=FC.moment('fine_mercator',h,m,W,z,bad);ll,uu=FC.moment('fine_moment',None,m,W,z,False,list(h.items()))
+    if bad:
+     fl,fu=FC.moment('fine_moment',None,m,W,z,False,[(1,32*m*m*sum(h.values()))]);ll+=fl/10**16;uu+=fu/10**16
+   else:
+    l,u=eM.moment(h,m,W,z,bad);ll,uu=eB.moment(m,W,list(h.items()),z)
+    if bad:
+     fl,fu=eB.moment(m,W,[(1,32*m*m*sum(h.values()))],z);ll+=fl/10**16;uu+=fu/10**16
+   assert (u<1 and uu<1)if accept else(l>1 and ll>1)
+   rows.append(dict(side=side,surcharge=bad,endpoint=str(z),accept=accept,mercator=[str(l-1),str(u-1)],base_two=[str(ll-1),str(uu-1)]))
+  coarse=Q((lower*10**24).__floor__(),10**24)   # retained engines certify the inherited 1e-24 bracket containing the endpoint
+  certify(coarse,True,M,B);certify(coarse+Q(1,10**24),False,M,B)
+  if digits>24:   # finer-constant copies of both engines certify the 1e-digits bracket (accept at the endpoint, reject at endpoint+1e-digits)
+   FC.prefetch([dict(name=nm,H=hh,m=m,W=W,z=zz,bad=bb,entries=ee) for zz in (lower,lower+Q(1,10**digits)) for nm,hh,bb,ee in [('fine_mercator',h,bad,None),('fine_moment',None,False,list(h.items()))]+([('fine_moment',None,False,[(1,32*m*m*sum(h.values()))])] if bad else [])])
+   certify(lower,True,MF,BF);certify(lower+Q(1,10**digits),False,MF,BF)
+  return rows
+ bitH={int(r):n for r,n in bit['profile'].items()}
+ assert BIT_M==64 and sum(r*n for r,n in bitH.items())==bit['bit']['rank_mass']
+ bit_moments=check_moment('bit',bitH,BIT_M,bit['bit']['live'],c,bit_D,True)
+ conservative_moments=check_moment('conservative_complex',H,73,owner['live'],conservative_b,DEFICIT,True,SPEC['outer_refinement']['endpoint_digits'])
+ assert bit['bit']['two_level_coefficient_bits']<bit['bit']['charged_two_level_coefficient_bits']==512 and bit['bit']['row_coefficient']<20161
+ def _outer_witness_raw(bb):
+  options=SPEC["outer_refinement"]
+  assert set(options)=={"grid_digits","margin_digits","bootstrap_steps","endpoint_digits"}
+  assert all(type(v)is int and v>0 for v in options.values())
+  grid=10**options["grid_digits"]; margin_grid=10**options["margin_digits"]
+  eta=beta=Q(1,margin_grid);s=min(c,(1-beta)*bb)-Q(1,1000*margin_grid);assert 0<s<c and s<(1-beta)*bb
+  # Bootstrap chain on the 2^-KB grid in exact integers (no rational normalisation). The internal parameter s is replaced by its floor s' = S/2^KB on the grid
+  # (s' <= s, so every exponent computed below is also valid for s: a smaller s only weakens the bootstrap; s - s' < 2^-KB is far below the kappa grid), and every
+  # exponent is the recursion value (1-s')s'+s'*old rounded DOWN. All strict conditions are checked exactly on the rounded values.
+  # Cutoff lemma instantiation (PR387 PROOF; FINITE_BIT_ACCOUNTING.md; repair (a) of PR1956, Josie Elliston): stage j needs L_j with L_j*gap_j^2 >= 36 and
+  # L_j*gap_j >= 2*(4+ell_j), where ell_j >= log2 C_j and C_j dominates the stage's fresh constants (< 2^PAYLOAD) PLUS THE CARRIED previous constant: the next constant
+  # is C_large + C_old*2^(L_j*(a_{j+1}-a_j)), so ell_{j+1} = max(PAYLOAD, ell_j + ceil(L_j*(a_{j+1}-a_j))) + 1. L_j is the least power of two (up to one extra doubling)
+  # meeting both inequalities; a larger cutoff satisfies the same hypotheses, and kappa is computed from the chain end only.
+  KB=int(3.6*options["margin_digits"])+64;ONE=mpz(1)<<KB;S=(mpz(s.numerator)<<KB)//s.denominator;assert 0<S<ONE and Q(int(S),int(ONE))<=s
+  O=384599*ONE//10**10   # chain[0] = 384599/10^10 rounded down onto the grid
+  chain0=Q(int(O),int(ONE));cuts=[];N1=36*ONE**4;ONE2=ONE*ONE;ell=PAYLOAD;unc_prev=None;unc_failed=[]
+  for step in range(options["bootstrap_steps"]):
+   A_=(S*(ONE-S)+S*O)>>KB
+   g=min((S-A_)*ONE,(ONE-A_-S)*ONE,ONE*ONE-A_*ONE-S*(ONE-O),(ONE-S)*ONE)   # gap = g/ONE^2: the same four terms as min(s-a,1-a-s,1-a-s*(1-old),1-s)
+   assert O<A_ and A_<S and S<ONE-A_ and g>0
+   N2=2*(4+ell)*ONE2
+   e=max(N1.bit_length()-(2*g.bit_length()-1)+2,N2.bit_length()-g.bit_length()+2,0);cut=1<<e
+   assert cut*g*g>=N1 and cut*g>=N2   # both lemma inequalities with the carried log2 C bound ell
+   carry=-((-(cut*(A_-O)))>>KB)      # ceil(L*(a_{j+1}-a_j)): log2 of the factor 2^(L*Delta a) carried into the next stage's constant
+   if step<7:   # control: the UNCARRIED least cutoff of the old formula cannot pay the carry from stage 2 on (shortfall ratio about 2(1-s))
+    Lu=max(-(-N1//(g*g)),-(-2*(4+PAYLOAD)*ONE2//g))
+    if unc_prev is not None:
+     cu=-((-(unc_prev[0]*unc_prev[1]))>>KB);unc_failed.append(Lu*g<2*(4+cu)*ONE2);assert unc_failed[-1],step
+    unc_prev=(Lu,A_-O)
+   h=lambda v:hashlib.sha256(('%x'%int(v)).encode()).hexdigest()
+   cuts.append(dict(step=step,grid_bits=KB,exponent_numerator_sha256=h(A_),gap_numerator_sha256=h(g),log2C_bound_sha256=h(ell),cutoff_log2_power_of_two_exponent=e,carry_bits_bit_length=int(carry).bit_length()))
+   ell=max(PAYLOAD,ell+carry)+1;O=A_
+  cuts.append(dict(control='uncarried_cutoffs_fail_carry_rule',stages_checked=list(range(2,2+len(unc_failed))),all_rejected=all(unc_failed)))
+  chain=[chain0,Q(int(O),int(ONE))]
+  q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin*grid;k=Q((ticks.numerator-1)//ticks.denominator,grid)
+  bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}}
+  outer=A.assembly(chain[-1],bb,bridge,k,eta=eta,beta=beta);assert len(outer['strict_constraints'])==47 and all(v>0 for v in outer['strict_constraints'].values())
+  controls=[]
+  for name,kk,ee,be in [('next_grid',k+Q(1,grid),eta,beta),('zero_eta',k,Q(0),beta),('zero_beta',k,eta,Q(0)),('unit_beta',k,eta,Q(1))]:
+   try:A.assembly(chain[-1],bb,bridge,kk,eta=ee,beta=be)
+   except AssertionError:controls.append(name)
+   else:raise AssertionError(name)
+  return k,cuts,outer,controls
+ def outer_witness(bb):   # memoised across stages (fine_cache.memo): the exact outer witness is a pure function of these inputs and the two source files
+  return FC.memo('outer_witness',(str(bb),str(c),json.dumps(SPEC['outer_refinement'],sort_keys=True),PAYLOAD,sha(Path(__file__)),sha(Path(A.__file__))),lambda:_outer_witness_raw(bb))
+ conservative_k,conservative_cuts,conservative_outer,conservative_controls=outer_witness(conservative_b)
+ pin('expected.conservative_kappa_decimal',dec(conservative_k))
+ # Only after every conservative supplier/admission/outer check succeeds is
+ # the reviewed native exact-complex recurrence selected. Bit charge stays.
+ native_moments=check_moment('complex',H,73,owner['live'],b,DEFICIT,False,SPEC['outer_refinement']['endpoint_digits'])
+ k,cuts,outer,controls=outer_witness(b)
+ pin('expected.kappa_decimal',dec(k))
+ assert exact_coarse_moment(M,H,73,owner['live'],b,True)[0]>1
+ assert M.moment(bitH,64,bit['bit']['live'],c+Q(1,10**24),False)[1]<1
+ model_controls=['native_complex_endpoint_rejected_with_old_surcharge','omitting_bit_surcharge_would_fail_next_endpoint_rejection']
+ moments=bit_moments+native_moments
+ input_hashes={'owner_audit':owner_sha,'fresh_invoice':invoice_sha,'package:inputs/PR589-EXPECTED.json':sha(BIT_PATH)}
+ input_hashes.update({'engines/'+name+'.py':sha(P/'engines'/(name+'.py'))for name in ('moment','base_two_moment','outer')})
+ return dict(status='PASS_GAUGE44_NATIVE_EXACT_GLOBAL_ASSEMBLY',kappa=str(k),kappa_decimal=dec(k),complex_lower=str(b),complex_decimal=dec(b),bit_lower=str(c),bit_decimal=dec(c),binding='complex',gain_over_pr635_percent=dec(100*(k/Q(X['gain_baseline_kappa'])-1)),complex_profile=H,provider_counts=invoice['provider_counts'],surcharge_model={'complex':False,'bit':True},conservative_complex_lower=str(conservative_b),conservative_complex_decimal=dec(conservative_b),conservative_kappa=str(conservative_k),conservative_kappa_decimal=dec(conservative_k),conservative_moments=conservative_moments,conservative_bootstrap=conservative_cuts,conservative_outer={n:str(v)for n,v in conservative_outer['strict_constraints'].items()},conservative_controls=conservative_controls,model_controls=model_controls,native_correction_gain=dec(k-conservative_k),template_sha256=owner['template_sha256'],input_sha256=input_hashes,bit_supplier=dict(pr=589,repository='CrocSwap/integer-mult-bounds',commit='a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46',merged_as='b3eb785ac5b4b1ca2beea0f80db078a505626a90',path='research/e8-bit-p111-tuned/tune/EXPECTED.json',git_blob_sha1=BIT_BLOB,word=dict(v=bit_record['word']['v'],decoder_constant=bit_record['word']['decoder_constant'],copy_ranks=bit_record['word']['copy_ranks']),derived_deficit=bit_D,deficit_formula='T(2Kv-L*cst), L=%d T=%d K=%d M=%d'%(BIT_L,BIT_T,BIT_K,BIT_M),replay_scope='Cited unchanged supplier; this assembly reprices its full profile and does not rebuild its bit circuit.'),moments=moments,bootstrap=cuts,outer={n:str(v)for n,v in outer['strict_constraints'].items()},controls=controls,scope='Conditional R480 seven-slot k8 construction with a source-bound replica count. The owner map attains the fixed-inventory bin bound. The full conservative invoice, native moment correction, PR589 bit surcharge and 47 outer inequalities are checked. Local admission and all-size/compiler, mutable-source, whole-call hosting, precision/tape, exact-recovery, analytic-transfer and bit H1-H5 assumptions remain.')
+if __name__=='__main__':
+ import sys
+ package,C,out=map(Path,sys.argv[1:4]);assert package.resolve()==P and not out.exists()
+ owner_path=C/'audit/OWNERS-AND-PROFILE.json';invoice_path=C/'audit/PREFIX-AND-INVOICE.json'
+ owner,invoice=read(owner_path),read(invoice_path)
+ assert owner['template_sha256']==read(C/'hosted/HOSTED.json')['template_sha256']
+ r=assemble(owner,invoice,sha(owner_path),sha(invoice_path));out.write_text(json.dumps(r,indent=2)+'\n');print(r['status'],r['kappa_decimal'])
diff --git a/research/staged544-refined-endpoint/vendor/bootstrap/outer.py b/research/staged544-refined-endpoint/vendor/bootstrap/outer.py
new file mode 100644
index 0000000000..43007488ad
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/bootstrap/outer.py
@@ -0,0 +1,58 @@
+#!/usr/bin/env python3
+"""Exact semantic assembly with the balanced prefix, for the paired-cube network.
+
+Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The 47 strict constraints, the seven margins and
+the finite bridge are unchanged from icekylinx's scripts/structured_bulk_assembly.py (PR144, adapted from Zhihao
+Chen's PR23). The prefix is the RaD project's (hipotures) balanced positional transform, reviewed in
+references/semantic-bulk/rad20/reports/review-balanced-transform.md: the prefix margin becomes 1-eps instead of
+1-eps(1+c), and eps(1+c)<1 remains a strict geometric condition (K_geometry) whose gap need not exceed kappa.
+The parameters c=q+eta/4 and eps=(1-eta)/(1+q) are those of the reviewed balanced assembly in
+research/copied-fixed/balanced_assembly.py, with its backoff h equal to eta. Nothing else differs from
+structured_bulk_assembly.assembly.
+"""
+from fractions import Fraction as Q
+
+
+def assembly(bit_saving,complex_saving,bridge,kappa,eta=Q(1,10**8),beta=Q(1,4)):
+    a,b=bit_saving,complex_saving
+    tau,sigma=1-a,1-b
+    q=a*(1-2*eta); lp=1-q; lam=(tau+lp)/2
+    c=q+eta/4; eps=(1-eta)/(1+q)
+    minimum=eps*q; r=(minimum+1-eps)/2; delta=eta/8
+    internal=tau+(1-beta)*max(sigma-tau,Q(0));leaf=sigma+beta*(1-sigma)
+    margins=dict(balanced_prefix=1-eps,coordinate_movement=a,
+        compact_phase_layer=minimum,bulk_exposure=a,
+        Gaussian_arithmetic=min(1-eps-delta,r-delta),
+        scalar_work=1-eps-delta,dimension=eps)
+    slacks=dict(bit_positive=a,complex_above_bit=b-a,
+        complex_below_one_over32=Q(1,32)-b,beta_positive=beta,
+        beta_below_one=1-beta,leaf_saving_above_bit=(1-beta)*b-a,
+        q_positive=q,q_below_internal=1-internal-q,q_below_leaf=1-leaf-q,
+        c_positive=c,c_below_one=1-c,q_below_reservations=c-q,
+        lambda_above_tau=lam-tau,lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal,lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf,compact_reservations=lp-(1-c),
+        lambda_prime_below_one=q,epsilon_positive=eps,epsilon_below_one=1-eps,
+        guard_width=1-eps,K_geometry=1-eps*(1+c),K_dominates_log=eps*c,
+        record_suffix=1-eps,phase_local=1-eps-delta,phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r,cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps,alpha_positive=r,alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r,delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta,short_record_fallback=eps-a,
+        small_field_exposure=1-eps-minimum,
+        artificial_boundary=8-eps+r-delta-minimum,
+        literal_scalar_guard=Q(bridge['semantic']['strict_literal_gap']),
+        row_product_gap=bridge['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()})
+    assert len(slacks)==47 and len(margins)==7
+    assert all(v>0 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0}
+    assert min(margins.values())==minimum
+    assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2
+    assert 1-eps*(1+c)==eta-eps*eta/4
+    return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma,
+            eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam,
+            lambda_prime=lp,alpha_squared_power=r,delta=delta,
+            C0=bridge['semantic']['C0'],C1=1,kappa=kappa),
+        strict_constraints=slacks,margins=margins,minimum_margin=minimum,
+        absorption_gap=minimum-kappa,
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c))
diff --git a/research/staged544-refined-endpoint/vendor/native/NATIVE-RECEIPT.json b/research/staged544-refined-endpoint/vendor/native/NATIVE-RECEIPT.json
new file mode 100644
index 0000000000..09a6b8bfda
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/native/NATIVE-RECEIPT.json
@@ -0,0 +1,139 @@
+{
+  "cache_files": {
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+    "_receipts/e0eefc96567fbb45e19e6de0ca295381dbe0e686c08ddc783639b065c7c791d4/RECEIPT.json": "bc1bc13637846b3807119ae55dd006c8076815f355c93e0a1f57c1dce3ebeb3e",
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+    "_receipts/e0eefc96567fbb45e19e6de0ca295381dbe0e686c08ddc783639b065c7c791d4/STAGES.json": "b0d0263a87e95da1dc03e552c878ba4da4494184d1560ecc66e4dd93d7129a0b",
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+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local-replay.log": "dd643a58258431c3b07d3b06d9060c49677fa23b4fafc9f10988ed1f6867bd9a",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/CANONICAL-WORD.json.gz": "f69c0ca2c217eab87a231b67e548cd6491286314b1215e19f0b1d7463afd5e91",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/CLIFFORD-RECIPES.json": "f1b5a509e39293f5d7ffea7b269fd9e5443e625a9286a022caa1fd4b62af8063",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/CLIFFORD-REVIEW.json": "1eac718473827728f905380793bc91b8f1c89521fcf1fb978bd9091170d3e8ee",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/FULL-WORD.json.gz": "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/INDEPENDENT-LOCAL-REPLAY.json": "70b4b787a5a4529452a58e9ac48826e848748a35b95a9b20c5358f4355705058",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/LOCAL-CHECKS.json": "e52e7feece9cb22e83c1d687c0f98731f8d50b7e8c863732ec662c89453fbcd4",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/LOCAL-PROGRAMS.json.gz": "81d4aefacaa27f2f215e7d35559d5a079ca521f4ca2eb9a9c6a6bd7062179bff",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/RESIDUAL-ADAPTERS.json": "89e5779ba4fcc1b1a76bed05c9b36600bc1dc3c6efd7e623865692ec715c30c5",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/SCALAR-FRAME-AUDIT.json": "4be1b04a3504e228e3fc95ef8cb4805272dacd6bd0616b8b4aef649c80a901ae",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/local/WRAPPER-CHECKS.json": "fc7e24778302326fdacfd08dd3d5e63483cf4539cd5022a6c77f016752fac1a3",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/native.log": "9bb4169a54cf20008b1bffcb08e2bacd59b4aeab4faa7bbae6c462c5083cb07a",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/phases.log": "da5230122cb781ac3fa5c98b08356c0af2050cf70e13f2f52ebec19ffb39aea1",
+    "e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e/wrappers.log": "74b13cf8570ca2f32ce0018575b1b73e95af77a28e59d09556fec59260bdbba5"
+  },
+  "declared_source_reconstruction_changed_only": true,
+  "fresh_current_provider_count": 2,
+  "manifest_sha256": "4c4f7067b1b5404b59213f480e9d91c96ffc79b5458e2736b159f5a522ed2dc3",
+  "native_scientific_engines_unchanged_from_manifest": "105d03bbe5e1f9ca0a476043b37c5456bbfc70fac1201dfc3a4084de0b0e7c67",
+  "providers": {
+    "base": {
+      "complete_native_files": {
+        "STEPS.json": "9aa0d30ffa17375b77b1a9cc745057d90fdb3a240c276069d0f41d3c40af32aa",
+        "local/CANONICAL-WORD.json.gz": "bfe0cc0badac6c6d18461ceea0b299162c0f280c05f81b63f11fcc06cf9180e6",
+        "local/CLIFFORD-RECIPES.json": "4890935bbf7a525161316ff9f00ca0fee83d531b3552d7f3317ed1594defc47f",
+        "local/CLIFFORD-REVIEW.json": "a53b4e7531d8b6ab6c25658d0d066ba5f63f381698106b82853d8e9b63a07630",
+        "local/FULL-WORD.json.gz": "0630efbce2c051de1bf56a18dfce182cfc56095e9fa53e7c3f34e8a24d70d1b7",
+        "local/INDEPENDENT-LOCAL-REPLAY.json": "e9f29854974884eb163b64a15ecb628a5bec932c02b48b036f41819dac82cba7",
+        "local/LOCAL-CHECKS.json": "93289eb9c8858a828c07040433c0e0a04c3f402ad37eb39aa82973e909e69ff5",
+        "local/LOCAL-PROGRAMS.json.gz": "441693817d93306d7a81aa463e98638ba69a7787e76bd4d9e8d2b5698248de35",
+        "local/RESIDUAL-ADAPTERS.json": "9352dc86d0161f64d987b4134c27533ba93753e09f29cfb8a53fd517baa5c39b",
+        "local/SCALAR-FRAME-AUDIT.json": "1713279c8716815c40dbc6485352c2d6c52c78a5352a06a09607a5b3a412ba7a",
+        "local/WARP-RECIPE-CHECKS.json": "1dff01dd5e3660c5f7e5e34730eab6d98c57aa9885af7de1963b4635c89f5f8a",
+        "local/WRAPPER-CHECKS.json": "74fd56850634a46ede861f97b2e7670a63423cfdc2d46786ccd39b03423db7fc"
+      },
+      "word_sha256": "5ca4be99851f8878cbfe0fd6c454df926b11218f17feefa1664721cf03156699"
+    },
+    "extra": {
+      "complete_native_files": {
+        "STEPS.json": "b897c5db7c439ad97bd3835db1fa7cda602417dcecfa65c1f6908ead33410b60",
+        "local/CANONICAL-WORD.json.gz": "c5eec94a64cecfd6b83de05619628ad4868e889d62a55b49973f09705cc60448",
+        "local/CLIFFORD-RECIPES.json": "b7965e69d748a642da92a63e877e1c9642b2aded0a4fbe070422b77502710621",
+        "local/CLIFFORD-REVIEW.json": "22dbfb53f7ccb135a2c74fbf0ee5ea34e7edc92b7dd2917597fbe9a5f12c4097",
+        "local/FULL-WORD.json.gz": "30927c26cdcc675f2905ec8db6b6662a304f5dc85180ee8273d1016171362d6e",
+        "local/INDEPENDENT-LOCAL-REPLAY.json": "d7a743d38b7f9ff7bfa183cdb1f5c7a6f33bbe0adec249e400fbc9c44e6f483d",
+        "local/LOCAL-CHECKS.json": "93289eb9c8858a828c07040433c0e0a04c3f402ad37eb39aa82973e909e69ff5",
+        "local/LOCAL-PROGRAMS.json.gz": "441693817d93306d7a81aa463e98638ba69a7787e76bd4d9e8d2b5698248de35",
+        "local/RESIDUAL-ADAPTERS.json": "67003353a7fbd614b23b69848210fbfb709e298cb3ce73a0db0109ab608826fe",
+        "local/SCALAR-FRAME-AUDIT.json": "494b514a66d3b663d7c1975a53091957ea3cb25443cce21a180b6540a9f43d4a",
+        "local/WARP-RECIPE-CHECKS.json": "33bf797268214c3ada7611ee9a0ee772111883c6aa58cec3e42d8ad376f677fa",
+        "local/WRAPPER-CHECKS.json": "74fd56850634a46ede861f97b2e7670a63423cfdc2d46786ccd39b03423db7fc"
+      },
+      "word_sha256": "965268a9be73d3f62e7ef680d18f108cf064bfe1f44b82157baa61443232904e"
+    }
+  },
+  "retained_parent_receipt_sha256": "bc1bc13637846b3807119ae55dd006c8076815f355c93e0a1f57c1dce3ebeb3e",
+  "retained_reuse_receipt_sha256": "063ca4d8c1486786088a2fbf40101bd9ed0d32e393139edbfdaa290255a48a00",
+  "reused_retained_native_count": 2,
+  "reused_retained_parent": true,
+  "source_commit": "staged544-literal-v1",
+  "status": "PASS_COMPOSED_FRESH_NATIVE_PROVIDERS_AND_AUTHENTICATED_CACHE"
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/PRICE.json b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/PRICE.json
new file mode 100644
index 0000000000..b7dbe2496d
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/PRICE.json
@@ -0,0 +1,21 @@
+{
+  "status": "PASS_EXACT_ACTUAL_CHRONOLOGICAL_PROFILE_PRICE",
+  "replicas": 360,
+  "profile": {
+    "1": 7989840,
+    "2": 2349000,
+    "3": 749088,
+    "4": 330480,
+    "5": 69120,
+    "8": 25920,
+    "48": 137952
+  },
+  "root_lower": "86528921863312340937/50000000000000000000000",
+  "root_upper": "1730578437266246818741/1000000000000000000000000",
+  "lower_decimal": "0.001730578437266246818740",
+  "upper_decimal": "0.001730578437266246818741",
+  "predecessor_pr": 3734,
+  "predecessor_root_lower": "86522418010381773087/50000000000000000000000",
+  "gain_lower_over_predecessor": "130077058611357/1000000000000000000000",
+  "scope": "Exact rational enclosure of the paid-profile root for the changed-clock literal owner fill, with the predecessor rational_price.py. The headline outer kappa needs the complete admission recorded separately."
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/README.md b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/README.md
new file mode 100644
index 0000000000..f3c0bbfbe5
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/README.md
@@ -0,0 +1,210 @@
+# staged544-clock-ceiling
+
+**The staged544 packing ceiling is a certified cap, and PR3734's own T360 clock reaches it
+with one rule. No new κ: the value equals PR3802's, which was published first.**
+
+Answers the first open next question of
+[PR3734](https://github.com/CrocSwap/integer-mult-bounds/pull/3734)
+(`research/staged544-producer-return`): *"Can a changed source clock attain more of the
+ideal544 packing gain?"* Stacked on PR3734's head
+`fa041037a3374eaa08feeca0b26a5d6893194147`; no file of that package changes.
+
+## What this adds next to PR3802
+
+[PR3802](https://github.com/CrocSwap/integer-mult-bounds/pull/3802) (hexapode, head
+`de2cd5c58009205210e3bf328b0ef5cadf3903af`) reached the same conditional
+κ = `0.001730578437266246818740 − 10^-46 − 10^-70` with a T385 combined clock while this
+work was in progress; its idle histogram is exactly 385/360 times the one here, rank by
+rank. This package does not claim priority on that value. It adds:
+
+1. **A certificate that this κ cannot be beaten by any owner fill of this inventory**,
+   under any clock and any replica count (`ceiling_certificate.py`, exact rationals,
+   standard library). PR3734 and PR3802 make no packing-optimality claim. Further gains
+   for the staged544 supplier therefore need a changed inventory, template or word, not
+   another allocation.
+2. **The cause of PR3734's loss**: its own clock, at the start, measured by an exact
+   per-tick LP (`construction/hall_lp.py`) that rates any clock in about 20 s.
+3. **The ceiling on PR3734's own clock and stock**: one deterministic rule (`clock.py`,
+   727 of 3240 calls moved), T360, 323,069 physical families (PR3802: T385, 345,469; both
+   need 19 stock bits, so no margin changes).
+4. **PR3734's complete admission run end to end** through its `full_replay.py` wrapper on
+   the changed inputs (PR3734 and PR3802 each report that their wrapper was not).
+
+## Result
+
+| | PR3734 | this package |
+|---|---|---|
+| source clock | `inputs/ORDER.json` | one rule applied to it, 727 of 3240 calls moved |
+| idle histogram | `{1:84732, 3:23211, 5:4679, 32:1, 48:137911}` | `{1:84240, 3:23328, 5:4320, 48:137952}` |
+| paid profile root, lower end | `0.001730448360207635461740` | `0.001730578437266246818740` |
+| conditional κ (complete admission) | `0.00173044836020763546173999…99` | `0.00173057843726624681873999…99` |
+
+The new idle histogram is the fixed-inventory optimum, and its root is the "ideal
+fixed-inventory packing root" quoted in PR3734's PROOF.md as infeasible on the fixed clock.
+
+## The ceiling certificate
+
+Per owner replica the inventory is fixed by PR3734's supplier and template, whatever the
+clock or replica count: nine calls of 408 helpers `{4:26, 5:2, 6:18, 7:53, 8:96, 9:213}`,
+idle bins 672 × rank 8, 448 × rank 16, 448 × rank 32, 448 × rank 48, the local profile
+`{1:2440, 2:725, 3:224, 4:102, 5:20, 8:8}` per call and 896 live families of rank 73. Each
+helper takes its width in one idle bin and each remainder l > 0 is paid as one rank-l child,
+as in `owner_audit.py`.
+
+- The root is unique. `F(a) = Σ n_r r^(1−a) − 896 T 73^(1−a)` has the sign of
+  `G(a) = Σ n_r r (73/r)^a − 65408 T`, strictly increasing since every paid rank is below
+  73. So "root < a" is "F(a) > 0", and every term scales with T.
+- F(a) is linear in the counts of bin patterns. Any dual `(π_w, μ_c)` with
+  `π·p + μ_c ≤ h(l_p)` for every pattern p of every rank c bounds, by weak duality, the
+  idle part of F for every packing: integer or fractional, with or without time windows
+  and the one-helper-per-call-per-bin rule.
+- The dual is solved exactly on the ten patterns of the ideal, with outward rational
+  enclosures of `r^(1−a)` from PR3734's `rational_price.py` at `a = root_upper =
+  0.001730578437266246818741`. All 2551 patterns of ranks 8, 16, 32 and 48 are dual
+  feasible (minimum reduced cost exactly 0) and the bound on min F is positive
+  (≈ 5.3e-20 per replica).
+
+Hence every owner fill of this inventory has root below `0.001730578437266246818741`,
+while this package's fill has root above `0.001730578437266246818740`. Controls: at
+`root_lower` the same certificate gives a non-positive bound, and shifting the width-9
+dual by 10^-30 makes it infeasible.
+
+## Why PR3734's clock lost 8.65 bill units
+
+A clock fixes for each owner replica nine stage times τ0 < … < τ8. The nine idle child
+pairs (224 bins each) are open on fixed stage windows:
+
+| children | rank | window |
+|---|---|---|
+| 0, 2 | 16 | (−1, τ1] |
+| 1, 3 | 8 | (τ1, τ2] |
+| 4, 6 | 32 | (−1, τ3] |
+| 5, 7 | 8 | (τ3, τ4] |
+| 8, 10 | 48 | (−1, τ5] |
+| 9, 11 | 8 | (τ5, τ6] |
+| 12, 13 | 48 | (τ2, τ8] |
+| 14, 15 | 32 | (τ4, τ8] |
+| 16, 17 | 16 | (τ6, τ8] |
+
+Both providers have the same native widths in the same helper order, so every call
+supplies the same 408 helpers and the clock only shapes the windows. For one width,
+assigning helpers to interval windows is feasible exactly when every tick interval [A, B]
+supplies at least the demand of the windows it contains (Hall). `construction/hall_lp.py`
+solves the resulting LP at single-tick resolution, generating these rows and the patterns
+on demand. It relaxes integrality and the per-call rule, so it bounds a given clock from
+above; it accepts PR3734's own fill (every Hall row holds) and returns the ideal when the
+Hall rows are dropped.
+
+On PR3734's clock it stops at `0.0017304575193851`, only 9.2e-9 above the actual fill:
+the clock, not the solver, cost 8.0 of the 8.65 units. The binding rows were a start cut
+for width 7 (b = 359) and end cuts for widths 4, 5, 8 (b = 2655). Each stage-1 call closes
+its owner's rank-16 pair (−1, τ1], whose ideal content is 224 × (7, 9), while every call
+supplies 53 width-7 helpers, and 8 replicas reach τ1 between ticks 40 and 47. Once the
+start is throttled the LP reaches the ideal and the end cuts stop binding, with the tail
+of the clock unchanged.
+
+## The rule
+
+`clock.py`: greedy merge of PR3734's call list. At tick t the next call is the first
+listed call whose replica has placed every earlier stage, except that a stage-1 call
+waits while it would raise the number of placed stage-1 calls above `53/224 · (t + 1)`
+(integer test `224 (placed + 1) > 53 (t + 1)`). Stage-1 calls move later by at most 114
+ticks and other calls earlier by at most 36. Coverage and per-replica stage order are
+kept, which is all `owner_audit.py`, `admit544.py` and the block-norm context require of a
+clock; the block-norm helper/data alias premise holds because τs never lies in a window of
+its own replica's pair `stage_pair[s]`, and `verify.py` checks it on every role.
+
+Tighter ratios (0.9 and 0.8 × 53/224) create one-tick windows and lose again. Moving every
+stage-8 call into a final block on top of the throttle also reaches the ideal (2591 calls
+moved, late windows of at least 376 ticks); the smaller change is kept.
+
+## Construction (provenance, not rerun by the verifier)
+
+1. `construction/hall_lp.py` on the changed clock, with the per-call rows of step 2: the
+   LP optimum is integral (3061 classes with one pattern, 179 with two or three), an
+   alternative vertex of the ideal with the same idle histogram.
+2. `construction/build_fill.py` places helpers tick by tick **in reverse time**, serving
+   first the open class that opens latest, so that the residual flood falls at the start
+   of the clock, where the 1080 classes (−1, τ] are all open; there bins are filled
+   emptiest first. Forward deadline order fails: the classes that close last are then
+   filled by the last few calls, one helper per bin. For the last 40 ticks the LP also
+   carries the per-call matching condition (for each set S of widths, at least
+   Σ_{w∈S} s_w open bins with a slot in S); 12 ticks were not enough on this clock because
+   its tail stays compressed.
+3. `owner_audit.py` replayed unchanged on the result.
+
+Environment: CPython 3.12.14, numpy 2.5.4, scipy 1.18.1, highspy 1.15.1 (macOS arm64).
+The solver comments are in French.
+
+## Reproduce
+
+```sh
+python3 scripts/research_package.py --package research/staged544-clock-ceiling --check-only
+python3.12 scripts/research_package.py --package research/staged544-clock-ceiling
+```
+
+`verify.py` (standard library, about 15 s) regenerates the clock from its rule, runs
+PR3734's byte-pinned `owner_audit.py` on the changed `ORDER.json`, `FILL.json.gz` and
+`PROFILE.json` beside PR3734's unchanged DISPATCH, NATIVE, PLAN and TEMPLATE inputs,
+checks the helper/data alias premise, encloses the root with PR3734's `rational_price.py`,
+checks the ceiling certificate, binds the recorded complete admission to these inputs, and
+rejects six semantic controls: PR3734's clock with the new fill, a same-replica stage swap,
+a bracket shifted by 10^-24, one idle rank-48 child split into 47 + 1, the certificate at
+`root_lower`, and a dual perturbed by 10^-30.
+
+## Complete admission
+
+PR3734's own `full_replay.py --reuse-current-native` ran end to end from a copy of
+`research/staged544-producer-return` whose `inputs/ORDER.json`, `inputs/FILL.json.gz`,
+`inputs/PROFILE.json` and `PRICE.json` were replaced by this package's files (nothing else
+changed; the pinned replay archive was reused as is). Recorded in
+[evidence/RELEASE.json](evidence/RELEASE.json):
+
+- `PASS_STAGED544_COMPLETE_CONDITIONAL_RELEASE`, admission
+  `PASS_REPLICA_ORDER_FILL_DERIVATIVE_COMPLETE_RELEASE`;
+- conditional κ = `0.0017305784372662468187399999999999999999999998999999999999999999999999`
+  (root lower end − 10^-46 − 10^-70, as for PR3734);
+- stages owners 17 s, parent-lift 39 s, rebind 3 s, discovery 352 s, both outer ledgers,
+  release 553 s (CPython 3.12.14, numpy 2.5.4, scipy 1.18.1, sympy 1.14.0, mpmath 1.3.0,
+  macOS arm64).
+
+`verify.py` does not rerun it; it binds the record to this package (input hashes, profile,
+headline = root lower end − 10^-46 − 10^-70, strict gain over PR3734 and PR3682). Native
+chains and retained ancestry were reused under their pinned receipts; no fresh native replay
+or third-party review is claimed.
+
+## Scope and remaining obligations
+
+Conditional finite construction in exactly PR3734's sense: every inherited all-size
+compiler, realization, hosting, precision/recovery, analytic and BIT assumption of PR3734
+and its chain remains. The ceiling certificate is scoped to this supplier, idle-child
+template and paid-remainder rule; it says nothing about changed words or templates. No
+unconditional theorem, formal verification or selected-frontier change is claimed. If
+PR3734 changes head, this package must be rebased and rerun.
+
+## Open next questions
+
+The structured list is in [TASK.json](TASK.json).
+
+- **Raise the ceiling.** The packing of this supplier is now provably at its cap, so a
+  gain needs a changed T360 inventory (child ranks or counts) or word. First step: price
+  candidate inventories with the fixed-inventory LP (`construction/hall_lp.py` without
+  Hall rows, or the dual of `ceiling_certificate.py`), then with a throttled clock.
+  Blocker: physical stock, coverage law and template checks of the complete admission.
+- **Other chains.** Do other chronological owner fills of this model (for example
+  PR3682's T360 incumbent) fall short of their own fixed-inventory ceiling because of the
+  clock? First step: run `construction/hall_lp.py` on the chain's ORDER.json and act only
+  if its exact per-tick LP is below the ideal. Blocker: the 53/224 ratio belongs to this
+  template's rank-16 (7, 9) closures; PR3694 reports that PR3633's T488 allocation already
+  attains its ceiling.
+
+## Credit
+
+Prepared by Erwan Secousses (ChaloupeMusic), at human direction, with substantial
+assistance from Claude (Anthropic, Claude Code). Builds directly on dataisfire's PR3734
+(staged544 producer return, owner audit, rational price, fixed-inventory ideal and the
+open question answered here) and, through it, PR3682 (zxv, retiming and T360 framework),
+PR3638 (chronological allocation and integral network construction), PR3605 (staged
+entry), PR3687/3693 (retained independent interpreter) and the authors credited in
+PR3734's SOURCE-CREDITS.md. PR3802 and PR3795 (hexapode) are concurrent work; PR3802
+reached the same κ first. Apache-2.0.
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/ceiling_certificate.py b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/ceiling_certificate.py
new file mode 100644
index 0000000000..e447b15b8b
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/ceiling_certificate.py
@@ -0,0 +1,90 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Exact dual certificate: no owner fill of the staged544 inventory beats the ideal root.
+
+Inventory per owner replica (PR3734 template and native supplier, any clock, any replica
+count T): nine calls of 408 helpers with widths {4:26, 5:2, 6:18, 7:53, 8:96, 9:213}; idle
+bins 672 of rank 8, 448 of rank 16, 448 of rank 32, 448 of rank 48; local paid profile
+{1:2440, 2:725, 3:224, 4:102, 5:20, 8:8} per call; 896 live families of rank 73. Every
+helper occupies its width in one idle bin and each bin's remainder l > 0 is paid as one
+rank-l child (owner_audit.py).
+
+The root is unique: F(a) = sum n_r r^(1-a) - 896 T 73^(1-a) has the sign of
+G(a) = sum n_r r (73/r)^a - 65408 T, strictly increasing because every paid rank is < 73.
+So "root < a_hi" is "F(a_hi) > 0". F(a_hi) is linear in the bin pattern counts y, so for
+any dual (pi, mu) with pi . p + mu_c <= h(l_p) for every pattern p of every rank c, weak
+duality gives sum y h(l) >= sum_c mu_c B_c + sum_w pi_w 9 s_w for every packing, integer or
+not, with or without time windows or the one-helper-per-call-per-bin rule. The dual is the
+exact solution on the ten patterns of the fixed-inventory ideal, with outward rational
+enclosures of r^(1-a) from PR3734 rational_price.py. Standard library only.
+"""
+from fractions import Fraction as Q
+
+W = (4, 5, 6, 7, 8, 9)
+SUPPLY = (26, 2, 18, 53, 96, 213)                  # helpers per call
+BINS = {8: 672, 16: 448, 32: 448, 48: 448}         # idle bins per owner replica
+LOCAL = {1: 2440, 2: 725, 3: 224, 4: 102, 5: 20, 8: 8}
+CAPS = (8, 16, 32, 48)
+# the ten patterns of the fixed-inventory ideal (counts of widths 4..9)
+BASIS = [(8, (0, 0, 0, 0, 1, 0)), (16, (0, 0, 0, 1, 0, 1)), (32, (1, 0, 0, 0, 0, 3)),
+         (32, (0, 0, 1, 0, 1, 2)), (32, (0, 1, 0, 0, 0, 3)), (32, (0, 0, 0, 1, 2, 1)),
+         (32, (0, 0, 0, 0, 0, 3)), (32, (0, 0, 0, 2, 0, 2)), (48, (0, 0, 0, 0, 0, 0)),
+         (48, (0, 0, 0, 0, 0, 5))]
+
+
+def patterns(cap):
+    out = []
+
+    def rec(i, rest, cur):
+        if i == len(W):
+            out.append(tuple(cur))
+            return
+        for k in range(rest // W[i] + 1):
+            rec(i + 1, rest - k * W[i], cur + [k])
+    rec(0, cap, [])
+    return out
+
+
+def left(cap, p):
+    return cap - sum(k * w for k, w in zip(p, W))
+
+
+def solve(A, b):
+    """Exact Gaussian elimination over the rationals."""
+    n = len(A)
+    M = [[Q(x) for x in row] + [Q(y)] for row, y in zip(A, b)]
+    for c in range(n):
+        piv = next(r for r in range(c, n) if M[r][c] != 0)
+        M[c], M[piv] = M[piv], M[c]
+        for r in range(n):
+            if r != c and M[r][c] != 0:
+                f = M[r][c] / M[c][c]
+                M[r] = [x - f * y for x, y in zip(M[r], M[c])]
+    return [M[i][n] / M[i][i] for i in range(n)]
+
+
+def certify(R, a, perturb=Q(0)):
+    """Lower bound on min over all packings of F(a) per owner replica, with its dual.
+
+    R is PR3734 rational_price module; lower enclosures of h keep the bound valid.
+    perturb shifts the width-9 dual after solving (a control: feasibility must then fail)."""
+    a = Q(a)
+    h = {l: R.h_bounds(l, a)[0] for l in range(1, 49)}
+    cost = lambda cap, p: h[left(cap, p)] if left(cap, p) else Q(0)  # noqa: E731
+    rows, rhs = [], []
+    for cap, p in BASIS:
+        rows.append(list(p) + [1 if cap == c else 0 for c in CAPS])
+        rhs.append(cost(cap, p))
+    sol = solve(rows, rhs)
+    pi, mu = sol[:6], dict(zip(CAPS, sol[6:]))
+    pi[5] += perturb
+    worst = None
+    for cap in CAPS:
+        for p in patterns(cap):
+            rc = cost(cap, p) - sum(x * k for x, k in zip(pi, p)) - mu[cap]
+            if worst is None or rc < worst[0]:
+                worst = (rc, cap, p)
+    dual = sum(mu[c] * BINS[c] for c in CAPS) + sum(x * 9 * s for x, s in zip(pi, SUPPLY))
+    local = sum(9 * n * R.h_bounds(r, a)[0] for r, n in LOCAL.items())
+    live = 896 * R.h_bounds(73, a)[1]
+    return dict(min_reduced_cost=worst[0], worst_pattern=(worst[1], worst[2]), dual_feasible=worst[0] >= 0,
+                lower_bound=local + dual - live, patterns=sum(len(patterns(c)) for c in CAPS))
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/clock.py b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/clock.py
new file mode 100644
index 0000000000..6411de359a
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/clock.py
@@ -0,0 +1,43 @@
+# SPDX-License-Identifier: Apache-2.0
+"""The changed T360 source clock: one deterministic rule applied to PR3734's ORDER.json.
+
+Greedy merge of the predecessor call list. At tick t the next call is the first listed
+call whose replica has already placed every earlier stage, except that a stage-1 call
+is skipped while it would raise the number of placed stage-1 calls above 53/224 (t + 1).
+
+Why 53/224: a stage-1 call closes its owner's two rank-16 children (-1, tau1]. In the
+fixed-inventory ideal every rank-16 slot holds (7, 9), so each closing pair of children
+needs 224 width-7 helpers, while every call supplies 53. Integer arithmetic only.
+"""
+
+
+def throttle(order, stage=1, num=53, den=224):
+    pending = [tuple(x) for x in order]
+    replicas = 1 + max(r for _, r in pending)
+    nxt = [0] * replicas                       # next stage expected per replica
+    out, placed = [], 0
+    while pending:
+        t = len(out)
+        for i, (s, r) in enumerate(pending):
+            if nxt[r] != s:
+                continue
+            if s == stage and den * (placed + 1) > num * (t + 1):
+                continue
+            break
+        else:
+            raise ValueError('no admissible call')
+        s, r = pending.pop(i)
+        out.append([s, r])
+        nxt[r] += 1
+        placed += s == stage
+    return out
+
+
+def check_order(order, replicas=360, stages=9):
+    if len(order) != stages * replicas or {tuple(x) for x in order} != {(s, r) for s in range(stages) for r in range(replicas)}:
+        raise ValueError('call coverage')
+    seen = [0] * replicas
+    for s, r in order:
+        if s != seen[r]:
+            raise ValueError('per-replica stage order')
+        seen[r] += 1
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/evidence/RELEASE.json b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/evidence/RELEASE.json
new file mode 100644
index 0000000000..82b4bcf27e
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/evidence/RELEASE.json
@@ -0,0 +1,93 @@
+{
+  "admission_kappa_decimal": "0.0017305784372662468187399999999999999999999998999999999999999999999999",
+  "admission_stages": [
+    {
+      "exit_code": 0,
+      "log_sha256": "b1f015112123bdd23ce17fb6f250a73db83ecdaf9ac43790afdbef4401e52011",
+      "name": "owners",
+      "script_sha256": "5472811c24aef3852dc7e189dff5060076a6c0b25bfde928da50dcfe9b5bdc15",
+      "seconds": 17.09042025002418
+    },
+    {
+      "exit_code": 0,
+      "log_sha256": "2d738a7322c4303d36deb694e6d79b6b51bfe098642bf5cda8abc319f03d66ab",
+      "name": "parent-lift",
+      "script_sha256": "9a1edd7d4ab4a2f2ac8d15b42b7e40f93653d65026bac59e4c1ce80b486bd60e",
+      "seconds": 39.15233549999539
+    },
+    {
+      "exit_code": 0,
+      "log_sha256": "a008477700970daae2738d7499964a797470cb3b3748d296bf9fb510e71dc6fb",
+      "name": "rebind",
+      "script_sha256": "52f8b4cec2dab69cbe590e1fd0f7a68e1b35de5a540bd22fd2adfce07a92b46b",
+      "seconds": 3.0021492910163943
+    },
+    {
+      "exit_code": 0,
+      "log_sha256": "0698d7bbe8ffa568e2c0cea6bdd2fea81cb7fe17df8a4ed58e0f10e6bf14a421",
+      "name": "discovery",
+      "script_sha256": "4f5c0926b70edb5aef3af0f8a8d9fa8119fa7af6603df1721c0dc31760b08254",
+      "seconds": 352.38219449997996
+    },
+    {
+      "exit_code": 0,
+      "log_sha256": "b66b80965a59ac5aec812a16e8cbcddf6c97c02b3e6dd35b6be8ce02ff9174c4",
+      "name": "outer-check_ledger.py",
+      "script_sha256": "1dd56009986eb14dd6603c9193c7ad71dd45c7d005bf71c9cd3f916e66f54d98",
+      "seconds": 0.03600720799295232
+    },
+    {
+      "exit_code": 0,
+      "log_sha256": "9d162a1a0b56ad0acfa2727e226138805c9f388d600a56d5229baf5d49c23def",
+      "name": "outer-independent_check.py",
+      "script_sha256": "a5a0361539ec7cae9eb6c0371f857919335a1720040f22af291aa3e34fc864c7",
+      "seconds": 0.02200825000181794
+    },
+    {
+      "exit_code": 0,
+      "log_sha256": "acdf066ba832393ac990976f6705748d72a1cc0d521b394711516c874e7a4796",
+      "name": "release",
+      "script_sha256": "6cfefe80c02a6c0f243f50ca9e146e1ab1440b84a033d8a71d42be5a5fa27f6b",
+      "seconds": 553.10043804202
+    }
+  ],
+  "admission_status": "PASS_REPLICA_ORDER_FILL_DERIVATIVE_COMPLETE_RELEASE",
+  "command": "python -B full_replay.py --work WORK --reuse-current-native, run from a copy of research/staged544-producer-return whose inputs/ORDER.json, inputs/FILL.json.gz, inputs/PROFILE.json and PRICE.json were replaced by this package files",
+  "comparison_kappa": "17304188997660711638039999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+  "comparison_pr": 3682,
+  "complex_lower": "86528921863312340937/50000000000000000000000",
+  "dispatch_sha256": "5d6fa711d704a1b498e5fe066ea821c4fa0217463702e82a7ce5be9da4023e8a",
+  "environment": {
+    "mpmath": "1.3.0",
+    "numpy": "2.5.4",
+    "platform": "macOS-26.6.2-arm64-arm-64bit",
+    "python": "3.12.14",
+    "scipy": "1.18.1",
+    "sympy": "1.14.0"
+  },
+  "fill_sha256": "50e4635123c0551a1a3c8f4cab870bcccc3e579c24447abe9d81336fde91f883",
+  "kappa": "17305784372662468187399999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+  "order_sha256": "743e74b538432e2b2045e754b0de9c18f36fc5039fdb8eb542177a898ec78e44",
+  "profile": {
+    "1": 7989840,
+    "2": 2349000,
+    "3": 749088,
+    "4": 330480,
+    "48": 137952,
+    "5": 69120,
+    "8": 25920
+  },
+  "release_receipt_sha256": "ca48aec4d7bac2c7412edad85e38cb9035f6321c8229308c3b4137e24725112f",
+  "replay_archive_sha256": "e74d545c18f77b7a9b8f0cc3d4901a87bb686888edaf3e41b30e5b2748751041",
+  "retained_ancestry_reused": true,
+  "reuse_current_native": true,
+  "scope": "Complete PR3734 conditional release on the changed clock and fill, local run, 2026-10-18. Native chains and retained ancestry reused under their pinned receipts (--reuse-current-native); no fresh native replay and no third-party review.",
+  "status": "PASS_STAGED544_COMPLETE_CONDITIONAL_RELEASE",
+  "wrapper_stages": [
+    {
+      "exit_code": 0,
+      "log_sha256": "3181655a0ebbf5266a1f329d913742b4b2ba6f27f5b949589c1a234a4c5dd4a8",
+      "name": "complete-admission"
+    }
+  ]
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/FILL.json.gz b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/FILL.json.gz
new file mode 100644
index 0000000000..6ac0a2c528
Binary files /dev/null and b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/FILL.json.gz differ
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/ORDER.json b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/ORDER.json
new file mode 100644
index 0000000000..58d4b4c341
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/ORDER.json
@@ -0,0 +1 @@
+[[0, 0], [0, 45], [0, 90], [0, 135], [0, 180], [0, 225], [0, 270], [0, 315], [0, 1], [0, 46], [0, 91], [0, 136], [0, 181], [0, 226], [0, 271], [0, 316], [0, 2], [0, 47], [0, 92], [0, 137], [0, 182], [0, 227], [0, 272], [0, 317], [0, 3], [0, 48], [0, 93], [0, 138], [0, 183], [0, 228], [0, 273], [0, 318], [0, 4], [0, 49], [0, 94], [0, 139], [0, 184], [0, 229], [0, 274], [0, 319], [1, 0], [1, 45], [1, 90], [1, 135], [1, 180], [1, 225], [1, 270], [1, 315], [0, 5], [0, 50], [0, 95], [0, 140], [0, 185], [0, 230], [0, 275], [0, 320], [1, 1], [1, 46], [1, 91], [1, 136], [1, 181], [1, 226], [0, 6], [1, 271], [0, 51], [0, 96], [0, 141], [1, 316], [0, 186], [0, 231], [0, 276], [0, 321], [1, 2], [0, 7], [0, 52], [0, 97], [1, 47], [0, 142], [0, 187], [0, 232], [1, 92], [0, 277], [0, 322], [0, 8], [1, 137], [0, 53], [0, 98], [0, 143], [1, 182], [0, 188], [0, 233], [0, 278], [1, 227], [0, 323], [0, 9], [0, 54], [0, 99], [1, 272], [0, 144], [0, 189], [0, 234], [1, 317], [0, 279], [0, 324], [2, 0], [1, 3], [2, 45], [2, 90], [2, 135], [1, 48], [2, 180], [2, 225], [2, 270], [2, 315], [1, 93], [0, 10], [0, 55], [0, 100], [1, 138], [0, 145], [0, 190], [0, 235], [1, 183], [0, 280], [0, 325], [2, 1], [1, 228], [2, 46], [2, 91], [2, 136], [2, 181], [1, 273], [2, 226], [2, 271], [2, 316], [1, 318], [0, 11], [0, 56], [0, 101], [1, 4], [0, 146], [0, 191], [0, 236], [1, 49], [0, 281], [0, 326], [2, 2], [1, 94], [2, 47], [2, 92], [2, 137], [2, 182], [1, 139], [2, 227], [2, 272], [2, 317], [1, 184], [0, 12], [0, 57], [0, 102], [1, 229], [0, 147], [0, 192], [0, 237], [1, 274], [0, 282], [0, 327], [2, 3], [2, 48], [1, 319], [2, 93], [2, 138], [2, 183], [1, 5], [2, 228], [2, 273], [2, 318], [1, 50], [0, 13], [0, 58], [0, 103], [1, 95], [0, 148], [0, 193], [0, 238], [1, 140], [0, 283], [0, 328], [2, 4], [2, 49], [1, 185], [2, 94], [2, 139], [2, 184], [1, 230], [2, 229], [2, 274], [2, 319], [1, 275], [3, 0], [3, 45], [3, 90], [1, 320], [3, 135], [3, 180], [3, 225], [3, 270], [1, 6], [3, 315], [0, 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177], [5, 222], [5, 267], [5, 312], [5, 357], [6, 41], [6, 86], [6, 131], [6, 176], [6, 221], [6, 266], [6, 311], [6, 356], [7, 40], [7, 85], [7, 130], [7, 175], [7, 220], [7, 265], [7, 310], [7, 355], [8, 39], [8, 84], [8, 129], [8, 174], [8, 219], [8, 264], [8, 309], [8, 354], [4, 44], [4, 89], [4, 134], [4, 179], [4, 224], [4, 269], [4, 314], [4, 359], [5, 43], [5, 88], [5, 133], [5, 178], [5, 223], [5, 268], [5, 313], [5, 358], [6, 42], [6, 87], [6, 132], [6, 177], [6, 222], [6, 267], [6, 312], [6, 357], [7, 41], [7, 86], [7, 131], [7, 176], [7, 221], [7, 266], [7, 311], [7, 356], [8, 40], [8, 85], [8, 130], [8, 175], [8, 220], [8, 265], [8, 310], [8, 355], [5, 44], [5, 89], [5, 134], [5, 179], [5, 224], [5, 269], [5, 314], [5, 359], [6, 43], [6, 88], [6, 133], [6, 178], [6, 223], [6, 268], [6, 313], [6, 358], [7, 42], [7, 87], [7, 132], [7, 177], [7, 222], [7, 267], [7, 312], [7, 357], [8, 41], [8, 86], [8, 131], [8, 176], [8, 221], [8, 266], [8, 311], [8, 356], [6, 44], [6, 89], [6, 134], [6, 179], [6, 224], [6, 269], [6, 314], [6, 359], [7, 43], [7, 88], [7, 133], [7, 178], [7, 223], [7, 268], [7, 313], [7, 358], [8, 42], [8, 87], [8, 132], [8, 177], [8, 222], [8, 267], [8, 312], [8, 357], [7, 44], [7, 89], [7, 134], [7, 179], [7, 224], [7, 269], [7, 314], [7, 359], [8, 43], [8, 88], [8, 133], [8, 178], [8, 223], [8, 268], [8, 313], [8, 358], [8, 44], [8, 89], [8, 134], [8, 179], [8, 224], [8, 269], [8, 314], [8, 359]]
\ No newline at end of file
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/PROFILE.json b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/PROFILE.json
new file mode 100644
index 0000000000..9f91bdd265
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/inputs/PROFILE.json
@@ -0,0 +1 @@
+{"1": 7989840, "2": 2349000, "3": 749088, "4": 330480, "5": 69120, "8": 25920, "48": 137952}
\ No newline at end of file
diff --git a/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/verify.py b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/verify.py
new file mode 100644
index 0000000000..d84e715661
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-clock-ceiling/verify.py
@@ -0,0 +1,203 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Changed T360 clock for PR3734: literal owner fill at the fixed-inventory packing ceiling,
+and an exact certificate that the ceiling caps every owner fill of this inventory.
+
+Standard library only. Checks, in order:
+1. clock: inputs/ORDER.json is exactly clock.throttle(PR3734 ORDER.json), covers every call
+   and keeps every replica's stage order 0..8;
+2. owners: PR3734's unmodified owner_audit.py (byte-pinned) passes on the changed ORDER,
+   FILL and PROFILE beside PR3734's unchanged DISPATCH, NATIVE, PLAN and TEMPLATE inputs,
+   and rejects its three mutants;
+3. blocknorm premise: no helper role of call (s, r) lands in owner r's pair stage_pair[s];
+4. price: PR3734's rational_price.py encloses the root of the audited profile in
+   PRICE.json's [root_lower, root_upper]; the root equals the ideal fixed-inventory root
+   quoted by PR3734 PROOF.md and strictly exceeds PR3734's root_lower;
+   ceiling_certificate.py: an exact dual over all 2551 bin patterns shows that every owner
+   fill of this inventory, under any clock and any replica count, has root < root_upper;
+   the recorded complete admission is bound to these inputs;
+5. controls: PR3734's own clock with this FILL is rejected by the owner audit; a stage
+   swap is rejected by the clock check; a shifted bracket and a moved idle child are
+   rejected by the price check; the certificate does not clear root_lower and a dual
+   perturbed by 1e-30 is infeasible.
+"""
+import gzip
+import hashlib
+import importlib.util
+import json
+import shutil
+import subprocess
+import sys
+import tempfile
+from fractions import Fraction as Q
+from pathlib import Path
+
+P = Path(__file__).resolve().parent
+S = P.parent / 'staged544-producer-return'
+sys.path.insert(0, str(P))
+import ceiling_certificate  # noqa: E402
+import clock  # noqa: E402
+
+UNCHANGED = ['DISPATCH.json', 'NATIVE-base.json.gz', 'NATIVE-extra.json.gz', 'PLAN.json.gz', 'TEMPLATE.json.gz']
+STAGE_PAIR = [0, 0, 0, 1, 1, 2, 2, 3, 3]
+IDEAL_DECIMAL = '0.001730578437266246818740'
+
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+
+def read(p):
+    raw = p.read_bytes()
+    return json.loads(gzip.decompress(raw) if p.suffix == '.gz' else raw)
+
+
+def owner_audit(order_path):
+    """Run PR3734's owner_audit.py, byte for byte, in a fresh directory."""
+    tmp = Path(tempfile.mkdtemp(prefix='staged544-clock-'))
+    try:
+        (tmp / 'inputs').mkdir()
+        shutil.copyfile(S / 'owner_audit.py', tmp / 'owner_audit.py')
+        need((tmp / 'owner_audit.py').read_bytes() == (S / 'owner_audit.py').read_bytes(), 'audit bytes')
+        for name in UNCHANGED:
+            shutil.copyfile(S / 'inputs' / name, tmp / 'inputs' / name)
+        shutil.copyfile(order_path, tmp / 'inputs/ORDER.json')
+        for name in ('FILL.json.gz', 'PROFILE.json'):
+            shutil.copyfile(P / 'inputs' / name, tmp / 'inputs' / name)
+        r = subprocess.run([sys.executable, '-B', str(tmp / 'owner_audit.py')], capture_output=True, text=True)
+        return r.returncode, r.stdout, r.stderr
+    finally:
+        shutil.rmtree(tmp)
+
+
+def load_price_module():
+    spec = importlib.util.spec_from_file_location('rational_price_3734', S / 'rational_price.py')
+    mod = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    return mod
+
+
+def root_signs(R, profile, low, high):
+    signed = dict(profile)
+    signed[73] = signed.get(73, 0) - 896 * 360
+    return R.interval(signed, low)[1] < 0 < R.interval(signed, high)[0]
+
+
+def decimal24(x):
+    return '0.' + str((x.numerator * 10**24) // x.denominator).rjust(24, '0')
+
+
+def main():
+    need(__debug__, 'assertions enabled')
+    # 1. clock
+    old_order = read(S / 'inputs/ORDER.json')
+    order = read(P / 'inputs/ORDER.json')
+    clock.check_order(old_order)
+    clock.check_order(order)
+    need(clock.throttle(old_order) == order, 'changed clock is the stated rule')
+    moved = sum(a != b for a, b in zip(order, old_order))
+    need(moved == 727, 'moved call count')
+
+    # 2. owners
+    code, out, err = owner_audit(P / 'inputs/ORDER.json')
+    need(code == 0, 'owner audit failed: ' + err[-800:])
+    audit = json.loads(out)
+    need(audit['positive']['status'] == 'PASS_STANDARD_LIBRARY_INDEPENDENT_LITERAL_AUDIT', 'owner audit status')
+    need(audit['controls'] == ['same_call_family', 'duplicate_helper', 'unpaid_hole'], 'owner audit controls')
+    profile = {int(r): n for r, n in audit['positive']['profile'].items()}
+    need(profile == {int(r): n for r, n in read(P / 'inputs/PROFILE.json').items()}, 'audited profile file')
+
+    # 3. blocknorm helper/data alias premise
+    fill = read(P / 'inputs/FILL.json.gz')
+    plan = read(S / 'inputs/PLAN.json.gz')
+    pairs = {int(c): v for c, v in plan['pairs'].items()}
+    sides = {int(c): v for c, v in plan['sides'].items()}
+    template = read(S / 'inputs/TEMPLATE.json.gz')
+    recipe = {x['stage']: x['pair'] for x in template['recipe'] if x['kind'] == 'local'}
+    need([recipe[s] for s in range(9)] == STAGE_PAIR, 'recipe stage pairs')
+    alias = 0
+    for t, h, o, c, p, z in fill['rows']:
+        s, r = order[t]
+        alias += (896 * o + (2 * pairs[c] + sides[c]) * 112 + p) // 224 == 4 * r + STAGE_PAIR[s]
+    need(alias == 0, 'helper/data alias')
+
+    # 4. price
+    R = load_price_module()
+    price = read(P / 'PRICE.json')
+    need({int(r): n for r, n in price['profile'].items()} == profile, 'priced profile is the audited profile')
+    low, high = Q(price['root_lower']), Q(price['root_upper'])
+    need(high - low == Q(1, 10**24) and root_signs(R, profile, low, high), 'exact root enclosure')
+    need(decimal24(low) == price['lower_decimal'] == IDEAL_DECIMAL, 'root decimal')
+    need(('ideal fixed-inventory packing root' + IDEAL_DECIMAL) in (S / 'PROOF.md').read_text(), 'PR3734 ideal root quote')
+    old = Q(read(S / 'PRICE.json')['root_lower'])
+    need(low > old and low - old == Q(price['gain_lower_over_predecessor']), 'strict gain over PR3734 root_lower')
+
+    # 4a. ceiling: every owner fill of this inventory (any clock, any replica count) has root < root_upper
+    cert = ceiling_certificate.certify(R, high)
+    need(cert['patterns'] == 2551 and cert['dual_feasible'] and cert['lower_bound'] > 0, 'fixed-inventory ceiling certificate')
+    idle_ideal = {1: 234, 3: Q(324, 5), 5: 12, 48: Q(1916, 5)}    # per owner replica
+    need({int(r): Q(n, 360) for r, n in audit['positive']['idle'].items()} == idle_ideal, 'ideal idle histogram')
+
+    # 4b. recorded complete admission is bound to these exact inputs (not rerun here)
+    rel = read(P / 'evidence/RELEASE.json')
+    sha = lambda p: hashlib.sha256(p.read_bytes()).hexdigest()  # noqa: E731
+    headline = low - Q(1, 10**46) - Q(1, 10**70)
+    need(rel['status'] == 'PASS_STAGED544_COMPLETE_CONDITIONAL_RELEASE'
+         and rel['admission_status'] == 'PASS_REPLICA_ORDER_FILL_DERIVATIVE_COMPLETE_RELEASE', 'release status')
+    need(Q(rel['kappa']) == headline and Q(rel['complex_lower']) == low, 'release headline is root_lower - 1e-46 - 1e-70')
+    need({int(r): n for r, n in rel['profile'].items()} == profile, 'release profile')
+    need(rel['order_sha256'] == sha(P / 'inputs/ORDER.json') and rel['fill_sha256'] == sha(P / 'inputs/FILL.json.gz'), 'release input binding')
+    need(rel['dispatch_sha256'] == sha(S / 'inputs/DISPATCH.json'), 'release dispatch binding')
+    need(all(s['exit_code'] == 0 for s in rel['admission_stages']), 'release stages')
+    old_headline = old - Q(1, 10**46) - Q(1, 10**70)
+    need(headline > old_headline > Q(rel['comparison_kappa']), 'strict headline gain over PR3734 and PR3682')
+
+    # 5. controls
+    rejected = []
+    tmp = Path(tempfile.mkdtemp(prefix='staged544-clock-order-'))
+    try:
+        (tmp / 'ORDER.json').write_bytes((S / 'inputs/ORDER.json').read_bytes())
+        code, out, err = owner_audit(tmp / 'ORDER.json')
+        need(code != 0 and 'strict lower and closed upper window' in err, 'old clock must reject the new fill')
+        rejected.append('predecessor_clock_with_new_fill')
+    finally:
+        shutil.rmtree(tmp)
+    bad = [list(x) for x in order]
+    r0 = bad[0][1]
+    i, j = bad.index([0, r0]), bad.index([1, r0])
+    bad[i], bad[j] = bad[j], bad[i]
+    try:
+        clock.check_order(bad)
+    except ValueError as e:
+        need('stage order' in str(e), 'wrong clock rejection')
+        rejected.append('same_replica_stage_swap')
+    else:
+        raise ValueError('stage swap accepted')
+    step = Q(1, 10**24)
+    need(not root_signs(R, profile, low + step, high + step), 'shifted bracket accepted')
+    rejected.append('bracket_shifted_up')
+    moved_child = dict(profile)
+    moved_child[48] -= 1
+    moved_child[47] = moved_child.get(47, 0) + 1
+    moved_child[1] += 1
+    need(not root_signs(R, moved_child, low, high), 'moved idle child accepted')
+    rejected.append('one_idle_rank48_split_47_1')
+    need(ceiling_certificate.certify(R, low)['lower_bound'] <= 0, 'certificate must not clear root_lower')
+    rejected.append('ceiling_certificate_at_root_lower')
+    need(not ceiling_certificate.certify(R, high, perturb=Q(1, 10**30))['dual_feasible'], 'perturbed dual accepted')
+    rejected.append('ceiling_dual_perturbed_1e-30')
+
+    result = dict(status='PASS_STAGED544_CHANGED_CLOCK_CEILING_OWNERS_AND_PRICE', moved_calls=moved,
+                  roles=audit['positive']['roles'], physical=audit['positive']['physical'],
+                  idle=audit['positive']['idle'], profile={str(r): n for r, n in sorted(profile.items())},
+                  root_lower=price['root_lower'], lower_decimal=price['lower_decimal'],
+                  recorded_release_kappa=rel['kappa'],
+                  ceiling_certificate=dict(at=price['root_upper'], patterns=cert['patterns'], dual_feasible=True,
+                                           min_F_per_replica_lower_bound_positive=True),
+                  gain_lower_over_pr3734=price['gain_lower_over_predecessor'], rejected_controls=rejected,
+                  fill_payload_sha256=hashlib.sha256(gzip.decompress((P / 'inputs/FILL.json.gz').read_bytes())).hexdigest())
+    print(json.dumps(result, sort_keys=True), flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/PRICE.json b/research/staged544-refined-endpoint/vendor/staged544-producer-return/PRICE.json
new file mode 100644
index 0000000000..1131efad13
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/PRICE.json
@@ -0,0 +1,21 @@
+{
+  "status": "PASS_EXACT_ACTUAL_CHRONOLOGICAL_PROFILE_PRICE",
+  "replicas": 360,
+  "profile": {
+    "1": 7990332,
+    "2": 2349000,
+    "3": 748971,
+    "4": 330480,
+    "5": 69479,
+    "8": 25920,
+    "32": 1,
+    "48": 137911
+  },
+  "root_lower": "86522418010381773087/50000000000000000000000",
+  "root_upper": "1730448360207635461741/1000000000000000000000000",
+  "lower_decimal": "0.001730448360207635461740",
+  "upper_decimal": "0.001730448360207635461741",
+  "gain_lower": "1841277597768621/62500000000000000000000",
+  "beats_baseline": true,
+  "scope": "Exact integer aggregate chronological allocation and full paid-profile pricing; literal physical realization and full outer admission remain required."
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/PROOF.md b/research/staged544-refined-endpoint/vendor/staged544-producer-return/PROOF.md
new file mode 100644
index 0000000000..d5eba79e80
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/PROOF.md
@@ -0,0 +1,15 @@
+# Staged helper544 after a returning producer excursion
+
+This construction changes the retimed PR3682 word. It raises ordinary incoming helper544 from span(51) to span(414,51), deletes its early B43 target208 +=544/2 read, and adds 672 +=544/2 and676 -=544/2 immediately before the old B693 producer. Helper544's paid residual width decreases8 to 7.
+
+The exact fixed-prefix relation uses v=e544+e672/2-e676/2. Prefix P through B246 satisfies P v=v on all704 register components; the intervening temporary donor676/566 producer excursion is included. The old first-producer census excluded this case. Because544 is unchanged before B693, the two releases commute from cut247 to that cut. The new prefix is the old prefix times the corresponding input shear. Literally rederived inverse cleanup and the independent arbitrary-dirty interpreter check the complete program and its inverse transpose on632 live inputs plus91 arbitrary buffer inputs. No zero-dirt assumption is used.
+
+The actual minimizing frame assignment is retained, not just an optimum vector. Its per-call profile is {1:2440,2:725,3:224,4:102,5:20,8:8}, and its inventory is {4:26,5:2,6:18,7:53,8:96,9:213}. Both providers have the same bill. Native operator identities and maps are preserved; exact phases, wrappers, both orientations and rejection controls were freshly replayed for both changed words. The new span(414,51) to FULL residual also has a separate512-column Gaussian check including global phase and three rejecting controls.
+
+The ideal fixed-inventory packing root0.001730578437266246818740 is NOT the claimed bound: its tight patterns are infeasible on the fixed T360 clock. The actual finite integer packing, 1,321,920 literal roles and323,069 physical families, gives profile {1:7990332,2:2349000,3:748971,4:330480,5:69479,8:25920,32:1,48:137911}. Independent physical replay reconstructs source windows, exact widths, same-call physical-family uniqueness, chronological prefixes and all idle remainders. The integer solver reached its time limit with a feasible incumbent; no optimality claim is made. Every incumbent equation was checked with integers and its literal expansion checked independently.
+
+The exact finite profile root lies in [0.001730448360207635461740,0.001730448360207635461741]. The proposed conditional endpoint is the lower decimal minus10^-46 minus10^-70, strictly above PR3682's exact endpoint by at least2.9460441564297936e-8. Full release evidence is required separately and is checked by verify.py.
+
+This is a conditional finite construction. Uniform/all-size compilation, source/mutable-label realization, whole-call hosting, precision/tape/recovery, analytic transfer, recurrence and BIT assumptions remain inherited. The selected repository certificate is unchanged. No unconditional theorem, independent human review, formal proof or global architectural/packing optimum is claimed.
+
+Credit: PR3682 retiming/T360 framework; PR3638 chronological allocation and exact integral network construction; PR3605 staged-entry mechanism and cleanup; PR3687/3693 retained independent interpreter; PR3133/2979 coupled-word lineage and the native, owner, prefix, BIT and outer authors named in retained sources. New investigation and adapters: dataisfire with substantial OpenAI Codex assistance. Apache-2.0. Retained independent code is reused; this is not a newly independent third-party review.
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/REPLAY-INPUTS.json b/research/staged544-refined-endpoint/vendor/staged544-producer-return/REPLAY-INPUTS.json
new file mode 100644
index 0000000000..1c96d298a5
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/REPLAY-INPUTS.json
@@ -0,0 +1,761 @@
+{
+  "archive_sha256": "e74d545c18f77b7a9b8f0cc3d4901a87bb686888edaf3e41b30e5b2748751041",
+  "files": {
+    "NATIVE-RECEIPT.json": "7ddf3f80b2fa61d1ab12035da9ef1c95d43f76077df819d7683144d7862a3f93",
+    "admit544.py": "76f1b45793abb32a902a8d3cd353b39a018a35a277bb4e223e7508226b71eb6e",
+    "extension_blocknorm.py": "6e6b8683cc87a84e5410c399b1128eb68694f852a3927cb124067f503ed7d7c7",
+    "helpers/lift_parent.py": "9a1edd7d4ab4a2f2ac8d15b42b7e40f93653d65026bac59e4c1ce80b486bd60e",
+    "helpers/reconstruct_replica.py": "5472811c24aef3852dc7e189dff5060076a6c0b25bfde928da50dcfe9b5bdc15",
+    "helpers/replica_parent.py": "48bdbf28f59d80edb266803ec61bf5e22e8fb5e5a0e2b88004b6039147589bd4",
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+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/CAPPED-SIGNOFF.md": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json": "c98559147bb498ac78af998eed63ab6661af934fa1a0df6844176b1811753fb9",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/PARAMETERS.json": "9f72cfffa34844764ca65a6b4431cc5c779141751df746019db9d3ad8f1dee92",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/Q-EMBEDDING-CONTROLS.json": "24ead3534482397c00c23ad3dbbd4e930e02290e00c97765353d8783c6575c7d",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/README.md": "0ec721f76dfa943e808afd3c2cc39ee69f1a6e5f4f77ba3c7c1373ad60db25af",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/UNION-CONTROLS.json": "6c5262144215cc603bbc3bee93a9a6d9d2318d0177a59115fe560762998498f9",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_capped_layout.py": "267d93ab5114ec160f88a6e65fd34c9f0696a686c36ba2dd194d6b70fa32a6bd",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_q_embedding.py": "ac858a14bbaf37e0e412a7ada282cd1586632f52b8a979733e775f50130969b6",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_union.py": "74608380d72567edcd4470c8d226b15848f9d3a4e5d77d652bdd8963480ab9ce",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/CONTROLS.json": "12d9207e887fda73159a9e2e4776003ca6f905fc3dc45d77ced588a1c9f80cb2",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/ERASURE-ROUNDING-CONTROL.json": "cceb2ade8ce5e18be3871f771955e38258f5eb4e39790c016aab96d8cf0488ef",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/FIXED-A-LDU-CONTROLS.json": "d6bda457fff01e04b6565d84c7bdbac870569a2ba7ef18a1fc8db11b738faf38",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/FIXED-A-PERIODIC-NORM-AUDIT.md": "21027a569588922c491b2c5b4f31e01ddb784f1f20cdf48b7d482d0e85b66bdb",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md": "c6a032ed495325806ddf491c5c85bc822a7fdd613aff906a7be553960d0f85a3",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-CONTROLS.json": "3f7e98ce524d5fb2ba2f911a350081340215e806ae945abf15a2fa07988ca2f8",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/LEDGER.json": "87eab0eb334ddc16e50cedbcab56320d2ff056c3dc87d7a5191171bb5a4c206e",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/NEAREST-NEIGHBOR-CONTROLS.json": "b85eca6098847e7b2f77b49a6c8c35e33cf790bc5dd01b95e1334daef9af47f4",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/NEAREST-NEIGHBOR-NORM.md": "7955383d50fc59b5c4c4d839c7e4d630add6208a3a233d7372a80309085ac6b2",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md": "811a3a0c88a90708add5b12fb0a05c25623f0af155010d5dcacd56021c55cba5",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/POSITIVE-TREE-CONTROLS.json": "7b31811402ddda4668fe8375f6c90a6047511d50836dbb7cfc7ffcf24fe19a11",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/Q-LDU-CONTROLS.json": "370a07aa1a56c656ec1f44d7651dfbe999ad0c19dede9036257cc2f29dc39452",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/README.md": "4d65ea3c8b6a0d5faef819a58c8b8277681d11bed54686e9a7f78de29736e68e",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md": "b4367eac012624f803b3d2e3a72339e050a0071bbec37e6d33b2cf322bd70bc5",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md": "ec2f8a1c4273b81217545d2710bcffdd09b5c9ba9349e887b5f7bf6f5bddfdc7",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/SYLVESTER-CONTROLS.json": "198af9f58c2950f17d8b1861ba4566bb2f6a0108a889a56db328fe67f8faa1a2",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/TASK.md": "5eef77fb23f26f5bfb5dc8e376ed25e769924f2b6420ebfc8671e3b2a2f1f611",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_constructions.py": "91387c9bd763a5c55c22a9907947bd07b6002f739aa6feedf6d3a02adea1e316",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py": "486efd4692665a548b94b79877abb76ad13157268f05c270c4e75d8f97700795",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py": "15134147317a7d789c9ea0ebd71db5ff11e59a20ef140f681d87478977734cd0",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_holes.py": "99c5dc4601ea88b553f52e235d36fba0aeb867e7fcd54130cdb3eb115a9bb0d8",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_ledger.py": "254342c05949e5cefe4f507a35487e0a1ffb7137a60d2b97831d5ad55d7d7ce5",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py": "21bf2be683cefa7a760bf1e1bdaa647f57bf47ab86b13fd7a74e8d650f492913",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py": "46db2ce9122ff57bb4eeaa723db1f351891e671a7eaa7bf4fb61f85dafee9e37",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py": "ec2589ca9abe71d674ba9ccf81f289b1526f56f3001c48d22977accb1be3e4dc",
+    "source/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_sylvester_factor.py": "60b21fda4ff451582087d30222a072eb1a214d531277e92883eaf28221aafced",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json": "82656563e7c8725f95c4aea7e273805a3d3faf34c495a330f1c91a036765414a",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/INDEPENDENT-SUBSTITUTION-RECEIPT.json": "49a8450289f1c1730c7d11d7c1506ad4af26fe9a1f76341f808a3025123e9519",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/INDEPENDENT-SUBSTITUTION-REVIEW.md": "b48d3e14083829db81336b639a6cfc698ac0dccc76fa2606d11b4be7a968ddeb",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json": "ec4e89e0513958c1d553736cd06eedbc5f2aca61b40f92aeade3eb882bf588bc",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/REPRODUCE.txt": "76808b685f1623813658f5b596a4b8191a07e7c0810550f6fada7fb4a4992913",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex": "d053a6b69de2ff4005ee52eb3b673de606326aa0f0e290354675f9f08f915b99",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/bind_native_admission.py": "166541ca2b7f3066f4c2421e5a9f1886d58e1fdd7677904976261ab2c82b32cb",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py": "02f7f9ebf29487331171954c7ceb37ff5315e2f28a9d56096ebbf9fb865c70ab",
+    "source/vendor/pr1673/capped-phase-fusion/frontier_fusion/independent_substitution_check.py": "579bbde41e90792818c2726b38dcab26cf5a2695d3348731e814175146b0a737",
+    "source/vendor/pr1673/capped-phase-fusion/gaussian_fixed_width_note.tex": "6f3edb7b238f2b6af4f3e2d98dee90fe6f4ba063a5ccd120422e45c941712aab",
+    "source/vendor/pr1673/capped-phase-fusion/make_native_capsule.py": "8f342ca212b8d601a11f704042c577c1b088e0079bda04b3065467c2f3c36ea0",
+    "source/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex": "b5957f38c98a68f2584a42cedf91d753180dab438129fdf749b441074f00d4e1",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/full-1661-verification/RESULT.json": "11c57817a4d43a489010a13af7621d37eeedc398308c036e63a2a97119443861",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/full-1661-verification/STAGES.json": "c2edb01a61c03e04bf48e605422ff83c1d304e4d586268959d4ea24b743640fb",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/pr1661/research/gmx-t74-p566-ep400/CLAIM.json": "70aacd45f9e6fac8675687cf50da32e0ad4e479d9a5c28f2afd1550311ae8849",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/pr1661/research/gmx-t74-p566-ep400/MANIFEST.json": "d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/pr1661/research/gmx-t74-p566-ep400/NOTICE": "3918a42010a435782de0fe8c38fab8f7df5d48a375ecf603c2bad1b818a16d6f",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/pr1661/research/gmx-t74-p566-ep400/inputs/PR589-EXPECTED.json": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4",
+    "source/vendor/pr1673/capped-phase-fusion/target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json": "d6b6a2140f3a61ae8cfdd8b51a1131934d63d2dd8573c1b13eb2b9528f3c3db8",
+    "source/vendor/pr1673/capped-phase-fusion/verify.py": "93ef0f46e132485da6194587199b8f039e25b1b9f099f93bbdaeac5c0d66de7f",
+    "source/vendor/pr1786/FINE-PRIME-PADDING.md": "47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce",
+    "source/vendor/pr1786/NOTICE.md": "dcc1adb2461aa93ca62b6a912dbd8b215599e6af8053dfbacb7423b544dd659a",
+    "source/verify.py": "196cf2d1cda601b7d88929cb17cbebbe3d4d06bc66d9e0d4b5edca750861d3ee",
+    "source/verify_manifest.py": "723d726dbbe5b65a49caecae0f1aab664a1413fb639abfd3e826a5a538e14e5b",
+    "source/verify_release.py": "c43680a2da390f79925547a5ee1919d8a76637cccf98e2c8164b6f667fbc3183"
+  }
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/SOURCE-CREDITS.md b/research/staged544-refined-endpoint/vendor/staged544-producer-return/SOURCE-CREDITS.md
new file mode 100644
index 0000000000..c054fad87e
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/SOURCE-CREDITS.md
@@ -0,0 +1,14 @@
+# Source credit and independence
+
+New work: dataisfire with substantial OpenAI Codex assistance, Apache-2.0. No independent human review, fresh third-party AI review or proof-kernel verification is claimed.
+
+The construction extends [PR3682](https://github.com/CrocSwap/integer-mult-bounds/pull/3682), head431e6c6023ef352c41696469de7fb614cd4ded8c. Its retimed word, actual T360 source clock and immutable native/parent framework are retained. The complete new source/receipt snapshot is bound by REPLAY-INPUTS.json. The new ordinary source stage is an explicit pinned record applied after the unchanged predecessor reconstruction.
+
+- [PR3605](https://github.com/CrocSwap/integer-mult-bounds/pull/3605), Ross Richardson, supplies the staged-entrance mechanism, literal cleanup and exact Gaussian residual-check pattern. staged_pair.py is retained; the new544 shear and its proof are separate.
+- [PR3638](https://github.com/CrocSwap/integer-mult-bounds/pull/3638), head d62e1acd7518a231440383f0d0dddf54d6edf8c1, supplies the T360 source-window clock, chronological aggregate model and integral-network/literal construction, with its retained PR3605 and earlier credits. The new inventory, integer incumbent and literal fill are recomputed. No predecessor owner receipt is used as proof of the new fill.
+- [PR3687](https://github.com/CrocSwap/integer-mult-bounds/pull/3687) and [PR3693](https://github.com/CrocSwap/integer-mult-bounds/pull/3693), Sagar Thakrar and roger-sato with their credited AI assistance, supply audit_core.py and the whole-word semantics adapter audit_verify.py. These bytes are retained and reused. They are independent of the current constructor in the stated interpreter scope, not a newly written independent audit or new reviewer approval.
+- The coupled-word source lineage includes PR3133, PR2979, Marius-Paul Dumitrean, dcharb78, Jeremy Kothe and the retained native/owner/block-norm/outer authors. All notices and scientific engines inside the source snapshot remain intact.
+
+owner_audit.py is a standalone standard-library adaptation of the source-window/family/prefix/invoice predicates in PR3638's independent_literal.py. It imports no producer, optimizer or retained owner checker. Its source template, plan and actual native canonical words are authenticated inputs. Three controls mutate actual literal roles and must fail for the intended reason. It shares the credited contract interpretation; it is not evidence of independent human agreement.
+
+The rational finite-price enclosure is retained from the PR3682 sources. New script adapters bind it to the actual finite histogram. Native forward/backward scalar replay, wrapper/phase lowering and all-column warp recipes were freshly run for both new words; unchanged native and parent ancestry is explicitly reused. The separate new544 span(414,51) residual check includes global Gaussian phase.
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/audit_core.py b/research/staged544-refined-endpoint/vendor/staged544-producer-return/audit_core.py
new file mode 100644
index 0000000000..8b5e1d5f2a
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/audit_core.py
@@ -0,0 +1,480 @@
+"""Independent exact semantics and binary-frame bill for the BORROW-WORD format.
+
+Standard library only. Nothing is imported from the audited authors' code
+(PR3133/PR3605 producers, cleanup generator, evaluator or frame solver); the word
+format and its semantic contract were read from those pinned sources and are
+credited there.  The frame-bill and free-word structure (point-set spans, warp
+frame and lifted-frame checks, target levels keyed by register, dual symbol
+naming) re-implements, without importing, Arham Tahir's
+research/coupled-carrier-independent-audit/verify.py (Apache-2.0) and shares its
+reading of the contract, so this module is not independent of that audit.
+
+Sagar Thakrar (thakrarsagar), with substantial AI assistance from Claude
+(Anthropic, Claude Opus 5.5). Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+
+V, N, H = 112, 704, 9          # sources 0..111, targets 112..223, helpers 224..703
+DUAL = 10 ** 6                  # offset naming the independent dual symbols D_k = K_k^(-T)
+
+
+class AuditFailure(Exception):
+    """Explicit failure; never disabled by python -O."""
+
+
+def need(ok, *why):
+    if not ok:
+        raise AuditFailure(' '.join(map(str, why)))
+
+
+def is_target(r):
+    return V <= r < 2 * V
+
+
+def touched(g):
+    return g[1:3] if g[0] == 'add' else g[1:2]
+
+
+# ----------------------------------------------------------------------------
+# Binary frames: a subspace of F_2^9 is stored as a 512-bit point-set mask
+# (bit x set iff x lies in the subspace).  Containment is mask inclusion.
+# ----------------------------------------------------------------------------
+_SPAN = {}
+
+
+def span(vectors):
+    key = tuple(vectors)
+    mask = _SPAN.get(key)
+    if mask is None:
+        points = {0}
+        for u in key:
+            need(type(u) is int and 0 <= u < 512, 'frame vector out of range', u)
+            if u not in points:
+                points |= {p ^ u for p in points}
+        mask = sum(1 << p for p in points)
+        _SPAN[key] = mask
+    return mask
+
+
+def points(mask):
+    return [x for x in range(512) if mask >> x & 1]
+
+
+def dim(mask):
+    size = bin(mask).count('1')
+    need(size and size & (size - 1) == 0, 'not a subspace point set')
+    return size.bit_length() - 1
+
+
+FULL = (1 << 512) - 1
+ZERO = 1
+
+
+def hyperplane(port):
+    return sum(1 << x for x in range(512) if bin(x & port).count('1') % 2 == 0)
+
+
+def warp_images(columns):
+    """Images of all 512 vectors under the F_2-linear map with columns G e_i."""
+    need(len(columns) == H and all(type(c) is int and 0 <= c < 512 for c in columns), 'warp columns')
+    images = [0] * 512
+    for x in range(1, 512):
+        low = x & -x
+        images[x] = images[x ^ low] ^ columns[low.bit_length() - 1]
+    need(len(set(images)) == 512, 'singular warp')
+    return images
+
+
+def image_mask(images, mask):
+    return sum(1 << images[x] for x in points(mask))
+
+
+def lagrangian_action_ok(images, U, E):
+    """K_G(x,z)=(Gx, Gx+G^{-T}(x+z)) is symplectic and maps L_U to L_E, L_0 and L_FULL to themselves.
+
+    L_U={(x,x+p): x in U, p in U^perp}.  This is the binary frame action only; it
+    is not a Clifford phase or Gaussian-product certificate.  For invertible G with
+    G(U)=E it is implied by the frame-map check (<G^-T p, Gu> = <p, u>), so it is a
+    consistency check of the implementation, not extra evidence.
+    """
+    inverse = [0] * 512
+    for x, y in enumerate(images):
+        inverse[y] = x
+    # G^{-T} z: bit j of the result is <G^{-1} e_j, z>.
+    git = lambda z: sum((bin(inverse[1 << j] & z).count('1') & 1) << j for j in range(H))
+    lift = lambda x, z: (images[x], images[x] ^ git(x ^ z))
+    pair = lambda a, b: (bin(a[0] & b[1]).count('1') + bin(a[1] & b[0]).count('1')) & 1
+    basis = [(1 << i, 0) for i in range(H)] + [(0, 1 << i) for i in range(H)]
+    lifted = [lift(*b) for b in basis]
+    if any(pair(lifted[i], lifted[j]) != pair(basis[i], basis[j]) for i in range(18) for j in range(i + 1, 18)):
+        return False
+
+    def frame(mask):
+        us = points(mask)
+        perp = [p for p in range(512) if all(bin(p & u).count('1') % 2 == 0 for u in us)]
+        return {(u, u ^ p) for u in us for p in perp}
+    return all({lift(*pt) for pt in frame(a)} == frame(b) for a, b in ((ZERO, ZERO), (FULL, FULL), (U, E)))
+
+
+# ----------------------------------------------------------------------------
+# Chronological binary-frame bill (BORROW-WORD contract)
+# ----------------------------------------------------------------------------
+def frame_bill(w, lagrangian=False):
+    rep = w['report']
+    need((rep['v'], 2 * rep['v'] + rep['R'], rep['h']) == (V, N, H), 'word geometry')
+    removed = set(w['removed'])
+    need(len(removed) == len(w['removed']) and all(2 * V <= r < N for r in removed), 'removed helpers')
+    live = [r for r in range(N) if r not in removed]
+    liveset = set(live)
+    B, warps = w['B'], w['warps']
+    for k, (pos, lender, cols, U, E) in enumerate(warps, 1):
+        need(0 <= pos <= len(B), 'warp position', k)
+        need(lender in liveset and not is_target(lender), 'warp lender', k)
+        images = warp_images(cols)
+        su, se = span(U), span(E)
+        need(dim(su) == len(U) == len(E) == dim(se), 'warp frame bases', k)
+        need(image_mask(images, su) == se, 'warp does not map its frame', k)
+        if lagrangian:
+            need(lagrangian_action_ok(images, su, se), 'lifted frame action', k)
+    for name in ('A', 'B', 'restore'):
+        for g in w[name]:
+            if g[0] != 'warp':
+                need(all(r in liveset for r in touched(g)), 'removed/out-of-range register', name, g)
+                if g[0] == 'add' and not is_target(g[1]):
+                    need(not is_target(g[2]), 'non-target reads a target', name, g)
+    need(all(g[-1] is None and is_target(g[1]) and g[0] == 'add' for g in w['cut']), 'cut gates')
+    need(all(g[-1] == [] and is_target(g[1]) for g in w['precondition'] + w['compensation']), 'zero-frame wrapper writes')
+    need(all(not is_target(g[1]) for g in w['restore']), 'restore writes a target')
+
+    ports = w['ports']
+    start, end = {}, {}
+    for r in live:
+        if r < V:
+            start[r], end[r] = span((ports[r],)), FULL
+        elif r < 2 * V:
+            start[r], end[r] = ZERO, hyperplane(ports[r - V])
+        else:
+            start[r], end[r] = ZERO, FULL
+    for q, U in w['start_extra'].items():
+        need(int(q) in liveset and int(q) >= 2 * V, 'entrance names a live helper', q)
+        start[int(q)] = span(U)
+    for q, U in w['helper_final'].items():
+        need(int(q) in liveset and int(q) >= 2 * V, 'terminal names a live helper', q)
+        end[int(q)] = span(U)
+
+    current, bill, chain = dict(start), Counter(), {}
+
+    def climb(r, mask, where):
+        need(current[r] & ~mask == 0, 'frame containment fails', where, r)
+        d = dim(mask) - dim(current[r])
+        if d:
+            bill[d] += 1
+        chain.setdefault(r, []).append((where, d, mask))
+        current[r] = mask
+
+    npre = w['prelude_events']
+    for i, g in enumerate(w['A']):
+        if g[0] == 'warp':
+            need(i < npre and len(g) == 5, 'warp outside the prelude', i)
+            _, r, op, fr, to = g
+            need(op != 0 and abs(op) <= len(warps) and warps[abs(op) - 1][1] == r, 'prelude warp lender', i)
+            images = warp_images(warps[abs(op) - 1][2])
+            if op < 0:
+                inverse = [0] * 512
+                for x, y in enumerate(images):
+                    inverse[y] = x
+                images = inverse
+            need(image_mask(images, span(fr)) == span(to), 'prelude warp frame action', i)
+            need(current[r] == span(fr), 'prelude warp entry frame', i, r)
+            current[r] = span(to)
+            chain.setdefault(r, []).append((('A', i), 'warp', span(to)))
+        else:
+            if i < npre:
+                need(is_target(g[1]), 'prelude scalar writes a non-target', i)
+            for r in touched(g):
+                climb(r, span(g[-1]), ('A', i))
+    levels = {V + int(t): span(U) for t, U in w['target_levels'].items()}
+    need(all(current[t] == levels.get(t, ZERO) for t in range(V, 2 * V)), 'target cut levels')
+    for q, U in zip(w['centers'], w['center_frames']):
+        U = span(U)
+        need(current[q] == U, 'centre entry frame', q)
+        served = {levels.get(g[1], ZERO) for g in w['cut'] if g[2] == q}
+        for T in sorted(served, key=lambda m: -dim(m)):
+            need(T & ~U == 0, 'centre descent not nested', q)
+            d = dim(U) - dim(T)
+            if d:
+                bill[d] += 1
+                chain.setdefault(q, []).append((('centre',), d, T))
+            U = T
+    at = {}
+    for k, rec in enumerate(warps, 1):
+        at.setdefault(rec[0], []).append(k)
+    for p in range(len(B) + 1):
+        for k in at.get(p, ()):
+            _, lender, _, U, E = warps[k - 1]
+            need(current[lender] == span(U), 'native warp entry frame', k, lender)
+            current[lender] = span(E)
+            chain.setdefault(lender, []).append((('warp', k), 'warp', span(E)))
+        if p < len(B):
+            for r in touched(B[p]):
+                climb(r, span(B[p][-1]), ('B', p))
+    for i, g in enumerate(w['restore']):
+        for r in touched(g):
+            climb(r, span(g[-1]), ('restore', i))
+    for r in live:
+        climb(r, end[r], ('end',))
+    restored = set(map(int, w['helper_final']))
+    need(all(end[r] == FULL for r in live if r not in restored and not is_target(r)), 'a FULL-cleanup register ends low')
+    widths = Counter(dim(end[r]) - dim(start[r]) for r in live if r >= 2 * V)
+    return dict(sorted(bill.items())), dict(sorted(widths.items())), chain, start, end
+
+
+def hygiene(w):
+    """Inherited structural contract: dyadic scalars, power-of-two scales, wrapper and correction access."""
+    entering = set(map(int, w['start_extra']))
+    restored = set(map(int, w['helper_final']))
+    live = set(range(N)) - set(w['removed'])
+    for name in ('precondition', 'compensation', 'A', 'cut', 'B', 'restore', 'cleanup'):
+        for g in w[name]:
+            if g[0] == 'warp':
+                continue
+            q = Q(g[3], g[4]) if g[0] == 'add' else Q(g[2], g[3])
+            need(q != 0 and q.denominator & (q.denominator - 1) == 0, 'non-dyadic scalar', name, g)
+            if g[0] == 'scale':
+                need(abs(q.numerator) & (abs(q.numerator) - 1) == 0, 'non power-of-two scale', name, g)
+    for g in w['precondition'] + w['compensation']:
+        need(not any(r in entering | restored for r in touched(g)), 'wrapper touches an entering or restored helper', g)
+    for c in w['initial_corrections']:
+        word = c['operator_word']
+        need(is_target(c['target']) and c['source'] in live and c['source'] >= 2 * V
+             and c['source'] not in entering, 'correction access', c)
+        need(word and all(w['warps'][abs(o) - 1][1] == c['source'] for o in word), 'correction operator lender', c)
+        q = Q(c['coefficient'])
+        need(q != 0 and q.denominator & (q.denominator - 1) == 0, 'correction coefficient', c)
+    return True
+
+
+# ----------------------------------------------------------------------------
+# Literal inverse cleanup, derived from the contract
+# ----------------------------------------------------------------------------
+def ordinary_forward(w):
+    """Chronological non-wrapper actions whose inverse forms cleanup."""
+    out = [g for g in w['A'][w['prelude_events']:]]
+    at = {}
+    for k, rec in enumerate(w['warps'], 1):
+        at.setdefault(rec[0], []).append(['warp', rec[1], k])
+    for p in range(len(w['B']) + 1):
+        out += at.get(p, [])
+        if p < len(w['B']):
+            out.append(w['B'][p])
+    return out + list(w['restore'])
+
+
+def derive_cleanup(w):
+    restored = set(map(int, w['helper_final']))
+    buf = lambda r: r + N if r in restored else r
+    full = [256, 128, 64, 32, 16, 8, 4, 2, 1]
+    out = []
+    for g in reversed(ordinary_forward(w)):
+        if g[0] == 'warp':
+            out.append(['warp', buf(g[1]), -g[2], full])
+        elif is_target(g[1]):
+            continue
+        elif g[0] == 'add':
+            out.append(['add', buf(g[1]), buf(g[2]), -g[3], g[4], full])
+        else:
+            need(g[0] == 'scale', 'unknown primitive', g)
+            out.append(['scale', buf(g[1]), g[3], g[2], full])
+    return out
+
+
+def same_cleanup(a, b):
+    norm = lambda c: [(g[:-1], span(g[-1])) for g in c]
+    return len(a) == len(b) and norm(a) == norm(b)
+
+
+# ----------------------------------------------------------------------------
+# Exact free-operator semantics.  A value is a dict {(input, word): Fraction};
+# word is a reduced tuple of signed operator ids (leftmost = applied last).
+# ----------------------------------------------------------------------------
+def program(w, cleanup=None):
+    """Literal executed sequence: corrections, wrapper, A, cut, B+warps, restore | cleanup."""
+    items = [('corr', c['target'], c['source'], tuple(c['operator_word']), Q(c['coefficient']))
+             for c in w['initial_corrections']]
+    items += w['precondition'] + w['compensation'] + w['A'] + w['cut']
+    at = {}
+    for k, rec in enumerate(w['warps'], 1):
+        at.setdefault(rec[0], []).append(['warp', rec[1], k])
+    for p in range(len(w['B']) + 1):
+        items += at.get(p, [])
+        if p < len(w['B']):
+            items.append(w['B'][p])
+    items += w['restore']
+    return items, list(w['cleanup'] if cleanup is None else cleanup)
+
+
+def dual_op(op):
+    return (1 if op > 0 else -1) * (abs(op) + DUAL)
+
+
+def _left(op, word):
+    return word[1:] if word and word[0] == -op else (op,) + word
+
+
+def _right(word, op):
+    return word[:-1] if word and word[-1] == -op else word + (op,)
+
+
+def _axpy(dst, src, q):
+    for key, x in src.items():
+        y = dst.get(key, 0) + q * x
+        if y:
+            dst[key] = y
+        else:
+            dst.pop(key, None)
+
+
+def forward_rows(w, cleanup=None, dual=False, symbolic_buffers=True, snapshot=None):
+    """Schrodinger pass: every register's value as a combination of initial inputs.
+
+    Live inputs are independent symbols.  Private buffers are independent symbols
+    when symbolic_buffers is set (dirty), otherwise zero.  The optional snapshot
+    callback sees the rows at the cleanup boundary.
+    """
+    removed = set(w['removed'])
+    live = [r for r in range(N) if r not in removed]
+    restored = sorted(map(int, w['helper_final']))
+    rows = {r: {(r, ()): Q(1)} for r in live}
+    for c in restored:
+        rows[c + N] = {(c + N, ()): Q(1)} if symbolic_buffers else {}
+    prefix, tail = program(w, cleanup)
+
+    def apply(g):
+        kind = g[0]
+        if kind == 'add':
+            t, s, q = g[1], g[2], Q(g[3], g[4])
+            need(t != s and q, 'add gate', g)
+            if dual:
+                t, s, q = s, t, -q
+            _axpy(rows[t], rows[s], q)
+        elif kind == 'scale':
+            q = Q(g[2], g[3])
+            need(q != 0, 'zero scale', g)
+            if dual:
+                q = 1 / q
+            rows[g[1]] = {k: q * x for k, x in rows[g[1]].items()}
+        elif kind == 'warp':
+            op = dual_op(g[2]) if dual else g[2]
+            rows[g[1]] = {(var, _left(op, word)): x for (var, word), x in rows[g[1]].items()}
+        elif kind == 'corr':
+            _, t, s, word, c = g
+            if dual:
+                src, ops, dst, c = rows[t], [-dual_op(o) for o in word], s, -c
+            else:
+                src, ops, dst = rows[s], list(reversed(word)), t
+            moved = {}
+            for (var, wd), x in src.items():
+                for o in ops:
+                    wd = _left(o, wd)
+                moved[var, wd] = x
+            _axpy(rows[dst], moved, c)
+        else:
+            raise AuditFailure('unknown primitive ' + repr(g))
+
+    for g in prefix:
+        apply(g)
+    if snapshot is not None:
+        snapshot(rows)
+    for g in tail:
+        apply(g)
+    return rows
+
+
+def reverse_columns(w, cleanup=None, upto_cleanup=False):
+    """Heisenberg pull-back: process the forward program backwards.
+
+    cols[s] = {(output, word): coefficient}: the final output register 'output'
+    contains coefficient*word*x_s.  Words grow on the right.  Every live input and
+    every private buffer is an independent symbol.
+    """
+    removed = set(w['removed'])
+    live = [r for r in range(N) if r not in removed]
+    restored = sorted(map(int, w['helper_final']))
+    regs = live + [c + N for c in restored]
+    cols = {r: {(r, ()): Q(1)} for r in regs}
+    prefix, tail = program(w, cleanup)
+    seq = prefix if upto_cleanup else prefix + tail
+    for g in reversed(seq):
+        kind = g[0]
+        if kind == 'add':
+            t, s, q = g[1], g[2], Q(g[3], g[4])
+            need(t != s and q, 'add gate', g)
+            _axpy(cols[s], cols[t], q)
+        elif kind == 'scale':
+            q = Q(g[2], g[3])
+            need(q != 0, 'zero scale', g)
+            cols[g[1]] = {k: q * x for k, x in cols[g[1]].items()}
+        elif kind == 'warp':
+            op = g[2]
+            cols[g[1]] = {(out, _right(word, op)): x for (out, word), x in cols[g[1]].items()}
+        elif kind == 'corr':
+            _, t, s, word, c = g
+            moved = {}
+            for (out, wd), x in cols[t].items():
+                for o in word:
+                    wd = _right(wd, o)
+                moved[out, wd] = x
+            _axpy(cols[s], moved, c)
+        else:
+            raise AuditFailure('unknown primitive ' + repr(g))
+    return cols
+
+
+def transpose(cols):
+    rows = {}
+    for s, col in cols.items():
+        for (r, word), x in col.items():
+            rows.setdefault(r, {})[s, word] = x
+    return rows
+
+
+def endpoint_defects(w, rows, dual=False):
+    """Compare final rows with the contract on the zero-buffer (live-input) domain.
+
+    Returns (bad live registers, dirty buffers, buffer->live leaks, buffer self-map identity).
+    """
+    removed = set(w['removed'])
+    live = [r for r in range(N) if r not in removed]
+    restored = sorted(map(int, w['helper_final']))
+    bad = []
+    for r in live:
+        expect = {(r, ()): Q(1)}
+        if dual and r < V:
+            expect[r + V, ()] = Q(-1)
+        if not dual and is_target(r):
+            expect[r - V, ()] = Q(1)
+        got = {k: x for k, x in rows.get(r, {}).items() if k[0] < N}
+        if got != expect:
+            bad.append(r)
+    dirty = [c for c in restored if any(k[0] < N for k in rows.get(c + N, {}))]
+    leaks = sorted({r for r in live for k in rows.get(r, {}) if k[0] >= N})
+    buffer_identity = all({k: x for k, x in rows.get(c + N, {}).items() if k[0] >= N} == {(c + N, ()): Q(1)}
+                          for c in restored)
+    return bad, dirty, leaks, buffer_identity
+
+
+def boundary_defects(w, rows):
+    """Contract at the cleanup boundary (forward): targets sheared, restored helpers clean and isolated."""
+    restored = set(map(int, w['helper_final']))
+    out = []
+    for t in range(V, 2 * V):
+        if {k: x for k, x in rows[t].items() if k[0] < N} != {(t, ()): Q(1), (t - V, ()): Q(1)}:
+            out.append(('target', t))
+    for c in restored:
+        if rows[c] != {(c, ()): Q(1)}:
+            out.append(('restored', c))
+    for r, row in rows.items():
+        if r < N and r not in restored and any(k[0] in restored for k in row):
+            out.append(('restored input read', r))
+    return out
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/audit_verify.py b/research/staged544-refined-endpoint/vendor/staged544-producer-return/audit_verify.py
new file mode 100644
index 0000000000..7f9fc1df42
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/audit_verify.py
@@ -0,0 +1,254 @@
+#!/usr/bin/env python3
+"""Whole-word cross-check of published RETIMED and DELAYED supplier words.
+
+Reuses Sagar Thakrar's exact PR3687 audit_core, unchanged and authenticated
+before execution. This is not a new interpreter. No construction/native checker
+is imported. roger-sato with substantial OpenAI Codex assistance; Apache-2.0.
+"""
+from collections import Counter
+from copy import deepcopy
+import gzip
+import hashlib
+import json
+from pathlib import Path
+import sys
+import types
+
+HERE = Path(__file__).resolve().parent
+STAGED = {1: 2439, 2: 726, 3: 224, 4: 102, 5: 20, 8: 8}
+RETIMED = {1: 2437, 2: 724, 3: 226, 4: 102, 5: 20, 8: 8}
+WIDTHS = {4: 26, 5: 2, 6: 18, 7: 52, 8: 97, 9: 213}
+MOVES = ((591, 1045, 640, 644, 2, [423, 65, 20, 10]),
+         (641, 1085, 656, 660, -1, [427, 96, 24, 6]))
+SOURCE_COMMITS = {
+    3605: '3efbb5de06a49ffeea6684482f2d3fe48cdca3fb',
+    3642: '044d24f12e74f228dd462c07ae2d7f3ea35e2be9',
+    3653: '607a09ff441c79b215a8127a81f4cc70021d0d9b',
+    3682: '431e6c6023ef352c41696469de7fb614cd4ded8c',
+    3687: 'f798838eb8ede4178e94f240e7b7e54b3f80c82d',
+}
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def authenticate():
+    rows = json.loads((HERE / 'SOURCES.json').read_text())['snapshots']
+    expected = {(3687, 'vendor/audit_core.py')}
+    for provider in ('base', 'extra'):
+        expected.update((number, f'inputs/{kind}-{provider}.json.gz') for number, kind in
+                        ((3605, 'STAGED'), (3642, 'RETIMED'), (3653, 'RETIMED'), (3682, 'DELAYED')))
+    keys = [(row['source_pr'], row['local']) for row in rows]
+    need(len(keys) == len(expected) and set(keys) == expected, 'source mapping inventory differs')
+    for row in rows:
+        need(row['source_commit'] == SOURCE_COMMITS[row['source_pr']], 'source commit differs')
+        raw = (HERE / row['local']).read_bytes()
+        need(len(raw) == row['bytes'], 'source byte length: ' + row['local'])
+        need(hashlib.sha256(raw).hexdigest() == row['sha256'], 'source SHA256: ' + row['local'])
+        blob = hashlib.sha1(b'blob ' + str(len(raw)).encode() + b'\0' + raw).hexdigest()
+        need(blob == row['git_blob'], 'source Git blob: ' + row['local'])
+    core = types.ModuleType('retained_audit_core')
+    path = HERE / 'vendor/audit_core.py'
+    # Execute the authenticated source bytes, without accepting a .pyc cache.
+    exec(compile(path.read_bytes(), str(path), 'exec'), core.__dict__)
+    return core
+
+
+def load(kind, provider):
+    return json.loads(gzip.decompress((HERE / 'inputs' / f'{kind}-{provider}.json.gz').read_bytes()))
+
+
+def events(w):
+    result = []
+    for pos in range(len(w['B']) + 1):
+        result += [('warp', k, deepcopy(rec)) for k, rec in enumerate(w['warps']) if rec[0] == pos]
+        if pos < len(w['B']):
+            result.append(('gate', pos, deepcopy(w['B'][pos])))
+    return result
+
+
+def replace_events(w, seq):
+    result = deepcopy(w)
+    body, warps = [], [None] * len(w['warps'])
+    for kind, label, gate in seq:
+        gate = deepcopy(gate)
+        if kind == 'gate':
+            body.append(gate)
+        else:
+            gate[0] = len(body)
+            warps[label] = gate
+    need(all(rec is not None for rec in warps), 'lost warp in reconstruction')
+    result['B'], result['warps'] = body, warps
+    return result
+
+
+def reconstruct(C, staged, kind):
+    """Literal prose recipes: adjacent shear swap, or move before first producer.
+
+    This repeats the credited local structural arguments as an input control;
+    the new evidence is the full word calculation below.
+    """
+    seq = events(staged)
+    for start, stop, donor, pivot, coefficient, frame in MOVES:
+        source = next(x for x in seq if x[:2] == ('gate', start))
+        need(source[2] == ['add', donor, pivot, coefficient, 1, [429, 30]], 'source release')
+        if kind == 'RETIMED':
+            target = next(x for x in seq if x[:2] == ('gate', start + 1))
+            need(target[2][0] == 'add' and target[2][2] == pivot, 'adjacent same-pivot shear')
+            need(not any(rec[0] == start + 1 for rec in staged['warps']), 'unexpected crossed warp')
+            where = seq.index(target) + 1
+        else:
+            target = next(x for x in seq if x[:2] == ('gate', stop))
+            writes = [i for i, g in enumerate(staged['B']) if i > start and g[1] == pivot]
+            need(writes[0] == stop, 'first producer cut')
+            where = seq.index(target)
+        changed = deepcopy(source[2])
+        changed[-1] = frame
+        seq.insert(where, ('gate', -pivot, changed))
+        seq.remove(source)
+    result = replace_events(staged, seq)
+    result['cleanup'] = C.derive_cleanup(result)
+    result['report']['local_profile'] = {str(k): v for k, v in RETIMED.items()}
+    return result
+
+
+def semantics(C, w):
+    """All live and buffer inputs are symbols, with no zero-buffer assumption."""
+    def endpoint(name, result):
+        bad_live, dirty_buffers, leaks, buffer_identity = result
+        need(not bad_live and not dirty_buffers and not leaks and buffer_identity,
+             name + ' endpoint: ' + repr(result))
+    boundary = {}
+    rows = C.forward_rows(w, symbolic_buffers=True,
+                          snapshot=lambda values: boundary.setdefault('defects', C.boundary_defects(w, values)))
+    endpoint('forward', C.endpoint_defects(w, rows))
+    dual_rows = C.forward_rows(w, dual=True, symbolic_buffers=True)
+    endpoint('dual', C.endpoint_defects(w, dual_rows, dual=True))
+    need(not boundary['defects'], 'cleanup boundary: ' + repr(boundary['defects']))
+    pulled = C.transpose(C.reverse_columns(w))
+    need(pulled == {r: row for r, row in rows.items() if row}, 'reverse pull-back differs from forward')
+    pre = C.transpose(C.reverse_columns(w, upto_cleanup=True))
+    pre = {r: pre.get(r, {}) for r in set(pre) | set(range(C.N))}
+    need(not C.boundary_defects(w, pre), 'reverse cleanup-boundary contract')
+    live = C.N - len(w['removed'])
+    buffers = len(w['helper_final'])
+    need((live, buffers, len(rows)) == (632, 91, 723), 'complete symbolic domain')
+    # Endpoint expectations are derived by the retained checker: live x,y,h
+    # map to x,y+x,h; all private buffer symbols return unchanged.
+    return {'live_inputs': live, 'dirty_buffers': buffers, 'matrix_dimension': len(rows),
+            'forward': 'PASS', 'dual': 'PASS', 'reverse_pullback_equals_forward': True,
+            'cleanup_boundary_forward_and_reverse': 'PASS',
+            'buffer_to_live_leaks': 0, 'buffer_self_map': 'identity',
+            'nonzero_endpoint_terms': sum(len(row) for row in rows.values())}
+
+
+def inspect(C, w, profile):
+    C.hygiene(w)
+    bill, widths, chain, _, _ = C.frame_bill(w)
+    need(bill == profile, 'computed profile differs')
+    need(w['report']['local_profile'] == {str(k): v for k, v in bill.items()}, 'declared profile differs')
+    need(widths == WIDTHS, 'helper widths differ')
+    need(w['cleanup'] == C.derive_cleanup(w), 'literal cleanup differs')
+    return {'semantics': semantics(C, w), 'profile': bill, 'widths': widths,
+            'rank_mass': sum(r * n for r, n in bill.items())}, chain
+
+
+def expect_rejection(C, fn, phrase):
+    try:
+        fn()
+    except (ValueError, C.AuditFailure) as exc:
+        need(phrase in str(exc), 'wrong rejection: ' + str(exc))
+        return phrase
+    raise ValueError('mutant unexpectedly passed: ' + phrase)
+
+
+def negative_controls(C, staged, w, kind):
+    controls = []
+    for _, _, donor, pivot, _, _ in MOVES:
+        seq = events(w)
+        release = next(x for x in seq if x[0] == 'gate' and x[2][0] == 'add' and x[2][1:3] == [donor, pivot])
+        for defect in ('omit', 'flip', 'cross_first_producer'):
+            changed = deepcopy(seq)
+            index = seq.index(release)
+            if defect == 'omit':
+                changed.pop(index)
+            elif defect == 'flip':
+                changed[index][2][3] *= -1
+            else:
+                producer = next(x for x in changed[index + 1:] if x[0] == 'gate' and x[2][1] == pivot)
+                moved = changed.pop(index)
+                changed.insert(changed.index(producer) + 1, moved)
+            bad = replace_events(w, changed)
+            bad['cleanup'] = C.derive_cleanup(bad)
+            # Repair the literal inverse so stale cleanup is not the rejection.
+            reason = expect_rejection(C, lambda: semantics(C, bad), 'forward endpoint')
+            controls.append({'defect': defect, 'pivot': pivot, 'rejected_by': reason})
+    old_cleanup = deepcopy(w)
+    old_cleanup['cleanup'] = staged['cleanup']
+    # The inverse shears commute: this is a format failure, not a wrong operator.
+    semantic_control = semantics(C, old_cleanup)
+    need(old_cleanup['cleanup'] != C.derive_cleanup(old_cleanup), 'old cleanup unexpectedly literal')
+    controls.append({'defect': 'old_cleanup_order', 'semantics': semantic_control,
+                     'rejected_by': expect_rejection(C, lambda: inspect(C, old_cleanup, RETIMED), 'literal cleanup differs')})
+    old_frames = deepcopy(w)
+    for gate in old_frames['B']:
+        if gate[0] == 'add' and gate[1:3] in ([640, 644], [656, 660]):
+            gate[-1] = [429, 30]
+    need(semantics(C, old_frames) == semantic_control, 'frame-only mutation changed operator')
+    if kind == 'RETIMED':
+        need(C.frame_bill(old_frames)[0] == STAGED, 'old frame control does not restore staged bill')
+        controls.append({'defect': 'old_release_frames', 'semantic_operator': 'PASS',
+                         'profile': STAGED, 'rejected_by': expect_rejection(C, lambda: inspect(C, old_frames, RETIMED), 'computed profile differs')})
+    else:
+        # The later release encounters a donor that has already climbed, so
+        # the original rank-2 frame is now illegal, despite the same operator.
+        controls.append({'defect': 'old_release_frames', 'semantic_operator': 'PASS',
+                         'rejected_by': expect_rejection(C, lambda: inspect(C, old_frames, RETIMED), 'frame containment fails')})
+    fake_report = deepcopy(w)
+    fake_report['report']['local_profile']['1'] += 1
+    controls.append({'defect': 'wrong_declared_profile', 'rejected_by':
+                     expect_rejection(C, lambda: inspect(C, fake_report, RETIMED), 'declared profile differs')})
+    return controls
+
+
+def run(C):
+    results = {}
+    for provider in ('base', 'extra'):
+        staged = load('STAGED', provider)
+        control, old_chain = inspect(C, staged, STAGED)
+        records = {'STAGED': control}
+        for kind in ('RETIMED', 'DELAYED'):
+            w = load(kind, provider)
+            need(reconstruct(C, staged, kind) == w, 'prose reconstruction differs: ' + kind)
+            record, chain = inspect(C, w, RETIMED)
+            deltas = {}
+            for register in set(chain) | set(old_chain):
+                before = Counter(d for _, d, _ in old_chain.get(register, []) if type(d) is int and d > 0)
+                after = Counter(d for _, d, _ in chain.get(register, []) if type(d) is int and d > 0)
+                delta = {r: after[r] - before[r] for r in set(after) | set(before) if after[r] != before[r]}
+                if delta:
+                    deltas[register] = delta
+            need(deltas == {640: {1: -1, 2: -1, 3: 1}, 656: {1: -1, 2: -1, 3: 1}}, 'per-register invoice differs')
+            record['prose_reconstruction'] = 'exact complete JSON equality'
+            record['changed_register_bills'] = deltas
+            record['controls'] = negative_controls(C, staged, w, kind)
+            records[kind] = record
+            print('CHECKED', provider, kind, '723-register forward/dual/pull-back and 9 controls', flush=True)
+        results[provider] = records
+    return {'status': 'PASS_RETIMED_AND_DELAYED_WHOLE_WORD_AUDIT',
+            'scope': 'Reused PR3687 interpreter; finite exact operator/frame audit only. No native Gaussian, owner, root, or complete supplier claim.',
+            'providers': results}
+
+
+if __name__ == '__main__':
+    need(sys.argv[1:] in ([], ['--write']), 'usage: verify.py [--write]')
+    C = authenticate()
+    result = run(C)
+    output = json.dumps(result, sort_keys=True, indent=2) + '\n'
+    if sys.argv[1:] == ['--write']:
+        (HERE / 'RESULT.json').write_text(output)
+    else:
+        need((HERE / 'RESULT.json').read_text() == output, 'committed result differs')
+    print(result['status'])
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/evidence/NATIVE-base.json b/research/staged544-refined-endpoint/vendor/staged544-producer-return/evidence/NATIVE-base.json
new file mode 100644
index 0000000000..7fb1c27441
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/evidence/NATIVE-base.json
@@ -0,0 +1,55 @@
+{
+  "status": "PASS_COMPLETE_T74_NATIVE_REPLAY",
+  "engine": "sparse",
+  "word_sha256": "5ca4be99851f8878cbfe0fd6c454df926b11218f17feefa1664721cf03156699",
+  "source_sha256": {
+    "accelerators/SPARSE-NOTICE.md": "3da80a5155c625bff907a42a82c65bfcf92854d2c960bcea4537c19c99efd4f2",
+    "accelerators/SPARSE-PROOF.md": "d73e4522602404caba8e7f247005a45c8a8cd26ffd2a70d65f4f9556a391d053",
+    "accelerators/sparse_state.py": "e91210ecd5085cf9add872f40bb434806838da488e12152c63c86e51fb5a7a37",
+    "accelerators/tables.py": "2fe087ba4558bd21ca49945dbedaa4047e4170e8fd4e2280701bb2f0481811b5",
+    "native/new/code/borrow/NOTICE-PR2226.md": "6b6730303818f6edc2a253eb4d05471eee58f28b5815fb250c32ae376f774c8a",
+    "native/new/code/borrow/bw_adapters.py": "c5907050a9a7fa36d4fcba3ad4fe440d1bec2063360c095013891785ba65c3ce",
+    "native/new/code/borrow/bw_audit_local.py": "08d5864d9b77902fc5ab5a062a8ea843fe1e572243b8c580abb0a421b2e46d07",
+    "native/new/code/borrow/bw_check_recipes.py": "d600b959a0ac462dc28deb8c0d5d8f9e0c76d7f7d7b6f92a296c5ef9e0a1de19",
+    "native/new/code/borrow/bw_emit.py": "99b59a6fa3feda86785e4225157dcb1a6169d256cec4f5e0acc9cea7a701df76",
+    "native/new/code/borrow/bw_gauss.py": "0884b14d6d77cde419a3c773ad9b21d0d18de1e6e20e6b6d5add458ed32ba4cb",
+    "native/new/code/borrow/bw_phases.py": "9d1a620e26715f4c42fd0347ecc14ef5a4a9031867869b64e8bae4d8526a821e",
+    "native/new/code/borrow/bw_tamper.py": "eab4b026965b9796eee92239660e1fb1fa629f6d9db44bc27b13911a018a6ee9",
+    "native/new/code/borrow/bw_transport.py": "9818e08b819310b40f1441b89be8c4369584a5b3fa5dfe9b86b1df9a7f5d6a76",
+    "native/new/code/borrow/vendor/PR2226-NOTICE.md": "f1d45d886926915531e3a59968abf0a4c974d7038f869848e71bd3a21b834422",
+    "native/new/code/borrow/vendor/clifford/clifford_exact.py": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+    "native/new/code/borrow/vendor/pr2226_operators.py": "b37b67a04681c776dafcdc6c9af0e8a2dbfa9893b02e0fa2d77ccc9182a87988",
+    "native/new/code/borrow/vendor/pr2226_replay.py": "0f9a47cdaeed0e22938c63251c232d6f23aeee2b62b0b1223cd4473c5428bcf7",
+    "native/new/run_native_borrow.py": "8ec7db430f4f4771c06bdcfc31736193614f6ec209d57b63d4e2b75f17fb3ac8"
+  },
+  "steps": [
+    {
+      "step": "native",
+      "exit_code": 0,
+      "seconds": 0.7970000000000255
+    },
+    {
+      "step": "ledgers",
+      "exit_code": 0,
+      "seconds": 0.8429999999998472
+    },
+    {
+      "step": "local-replay",
+      "exit_code": 0,
+      "seconds": 29.71900000000005
+    },
+    {
+      "step": "wrappers",
+      "exit_code": 0,
+      "seconds": 1.3589999999999236
+    },
+    {
+      "step": "phases",
+      "exit_code": 0,
+      "seconds": 40.95400000000018
+    }
+  ],
+  "seconds": 73.6880000000001,
+  "python": "3.12.6",
+  "scope": "Full five-stage native replay including independent all-column warp recipes and original rejection controls; supplier/outer admission separate."
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/evidence/NATIVE-extra.json b/research/staged544-refined-endpoint/vendor/staged544-producer-return/evidence/NATIVE-extra.json
new file mode 100644
index 0000000000..1901edc093
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/evidence/NATIVE-extra.json
@@ -0,0 +1,55 @@
+{
+  "status": "PASS_COMPLETE_T74_NATIVE_REPLAY",
+  "engine": "sparse",
+  "word_sha256": "965268a9be73d3f62e7ef680d18f108cf064bfe1f44b82157baa61443232904e",
+  "source_sha256": {
+    "accelerators/SPARSE-NOTICE.md": "3da80a5155c625bff907a42a82c65bfcf92854d2c960bcea4537c19c99efd4f2",
+    "accelerators/SPARSE-PROOF.md": "d73e4522602404caba8e7f247005a45c8a8cd26ffd2a70d65f4f9556a391d053",
+    "accelerators/sparse_state.py": "e91210ecd5085cf9add872f40bb434806838da488e12152c63c86e51fb5a7a37",
+    "accelerators/tables.py": "2fe087ba4558bd21ca49945dbedaa4047e4170e8fd4e2280701bb2f0481811b5",
+    "native/new/code/borrow/NOTICE-PR2226.md": "6b6730303818f6edc2a253eb4d05471eee58f28b5815fb250c32ae376f774c8a",
+    "native/new/code/borrow/bw_adapters.py": "c5907050a9a7fa36d4fcba3ad4fe440d1bec2063360c095013891785ba65c3ce",
+    "native/new/code/borrow/bw_audit_local.py": "08d5864d9b77902fc5ab5a062a8ea843fe1e572243b8c580abb0a421b2e46d07",
+    "native/new/code/borrow/bw_check_recipes.py": "d600b959a0ac462dc28deb8c0d5d8f9e0c76d7f7d7b6f92a296c5ef9e0a1de19",
+    "native/new/code/borrow/bw_emit.py": "99b59a6fa3feda86785e4225157dcb1a6169d256cec4f5e0acc9cea7a701df76",
+    "native/new/code/borrow/bw_gauss.py": "0884b14d6d77cde419a3c773ad9b21d0d18de1e6e20e6b6d5add458ed32ba4cb",
+    "native/new/code/borrow/bw_phases.py": "9d1a620e26715f4c42fd0347ecc14ef5a4a9031867869b64e8bae4d8526a821e",
+    "native/new/code/borrow/bw_tamper.py": "eab4b026965b9796eee92239660e1fb1fa629f6d9db44bc27b13911a018a6ee9",
+    "native/new/code/borrow/bw_transport.py": "9818e08b819310b40f1441b89be8c4369584a5b3fa5dfe9b86b1df9a7f5d6a76",
+    "native/new/code/borrow/vendor/PR2226-NOTICE.md": "f1d45d886926915531e3a59968abf0a4c974d7038f869848e71bd3a21b834422",
+    "native/new/code/borrow/vendor/clifford/clifford_exact.py": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+    "native/new/code/borrow/vendor/pr2226_operators.py": "b37b67a04681c776dafcdc6c9af0e8a2dbfa9893b02e0fa2d77ccc9182a87988",
+    "native/new/code/borrow/vendor/pr2226_replay.py": "0f9a47cdaeed0e22938c63251c232d6f23aeee2b62b0b1223cd4473c5428bcf7",
+    "native/new/run_native_borrow.py": "8ec7db430f4f4771c06bdcfc31736193614f6ec209d57b63d4e2b75f17fb3ac8"
+  },
+  "steps": [
+    {
+      "step": "native",
+      "exit_code": 0,
+      "seconds": 0.7970000000000255
+    },
+    {
+      "step": "ledgers",
+      "exit_code": 0,
+      "seconds": 0.8589999999999236
+    },
+    {
+      "step": "local-replay",
+      "exit_code": 0,
+      "seconds": 28.71900000000005
+    },
+    {
+      "step": "wrappers",
+      "exit_code": 0,
+      "seconds": 1.3589999999999236
+    },
+    {
+      "step": "phases",
+      "exit_code": 0,
+      "seconds": 41.077999999999975
+    }
+  ],
+  "seconds": 72.84400000000005,
+  "python": "3.12.6",
+  "scope": "Full five-stage native replay including independent all-column warp recipes and original rejection controls; supplier/outer admission separate."
+}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/DISPATCH.json b/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/DISPATCH.json
new file mode 100644
index 0000000000..9e93daeaf8
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/DISPATCH.json
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diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/PLAN.json.gz b/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/PLAN.json.gz
new file mode 100644
index 0000000000..7a1db55a4d
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diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/TEMPLATE.json.gz b/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/TEMPLATE.json.gz
new file mode 100644
index 0000000000..114808c32b
Binary files /dev/null and b/research/staged544-refined-endpoint/vendor/staged544-producer-return/inputs/TEMPLATE.json.gz differ
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/outer-replay/outer/check_ledger.py b/research/staged544-refined-endpoint/vendor/staged544-producer-return/outer-replay/outer/check_ledger.py
new file mode 100644
index 0000000000..d3b6149c5c
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/outer-replay/outer/check_ledger.py
@@ -0,0 +1,260 @@
+#!/usr/bin/env python3
+"""Exact ledger for fine-prime padding (PR1786, chafreaky / Chafik Boukhalfa, with Tim Trudgian's prime theorem) at the free point epsilon = 1 - (1025/1024)*10^-(DIG+80), explicit Strassen boundary, retuned to the fixed-grid maximum kappa = C - 1/FLOOR - BACKOFF.
+
+Adapted from PR1786's ledger, itself adapted from PR1731's cost ledger, credited to Jeremy Kothe et al. and
+Vlado Vojdanovski (PR1673); uses PR1758's gauge-free chirp and per-visit tag
+accounting. New source-prime, coefficient-setup and halo lemmas are in
+FINE-PRIME-PADDING.md. Finite arithmetic does not prove the all-size theorem.
+"""
+from fractions import Fraction as Q
+from decimal import Decimal, localcontext
+from pathlib import Path
+from math import factorial
+import hashlib, json, sys
+sys.set_int_max_str_digits(0)
+
+if not __debug__: raise RuntimeError('Assertions must be enabled')
+HERE=Path(__file__).resolve().parent
+WORK=HERE.parent
+SPEC=WORK/'new/SPEC.json'
+CLAIM=WORK/'CLAIM.json'
+BJ=json.loads((HERE/'B.json').read_text())
+spec=json.loads(SPEC.read_text())
+# Free strict slacks of the PR1673 proof (retuned in this package, see RETUNED-SLACKS.md); every one only needs to be > 0.
+DIG=spec['outer_refinement']['endpoint_digits']   # the supplier endpoint is certified to 10^-DIG (accept at the endpoint, reject at endpoint + 10^-DIG); every slack sits 10 digits below
+# Fixed-grid maximum (PR2180 / PR2161, dataisfire; proof PROOF-GRID-MAX.md): the free gaps shrink together to R0 = 10^-(DIG+80); grid, backoff and certified endpoint are unchanged.
+R0=Q(1,10**(DIG+80)); ZETA=R0/100; SLACK=R0; GAP=R0; STOP=R0/10**6
+FLOOR=10**(DIG+46); BACKOFF=Q(1,10**(DIG+22)); B_GAP=R0
+# Fine prime padding: theta is between 1/(4p^2) and 1/(2p^2-1).
+# Online chirp guard beta+2*xi-2<1; coefficient setup uses O(p^2 log p) bits.
+# epsilon = 1 - DELTA_E is a FREE parameter (outer/FINE-PADDING-COMPOSITION.md sec. 2): every row is strictly satisfied for every DELTA_E > r0 = 10^-(DIG+80); beta = 1 - 3 DELTA_E lies in the admissible window
+# [1 - DELTA_E (1 + 1/q), 1 - 2 DELTA_E). DELTA_E = (1025/1024) r0 keeps the gamma spare power 1-eps-r0 = r0/1024 > 0 and puts C - eps*q below 1/FLOOR (PROOF-GRID-MAX.md).
+DELTA_E=Q(1025,1024)*R0
+EPS=1-DELTA_E
+BETA=1-3*DELTA_E
+THETA_POWER=Q(2)
+HALO_POWER=48
+CORE_POWER=128
+STRASSEN_TAU=Q(45,16)
+KAPPA_1828=Q('0.0016797549561255445213356709939207025769983114385025802954603275201627999999999999999999999999979966404900867489109573286580121585')  # exact kappa of rohanarun's PR1828 (slack refinement of PR1800 at 1e-96), from its body
+KAPPA_1839=Q('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263049999999997996640490086748910957328658012158594')  # exact kappa of PR1839 (previous package: 1e-90 endpoint)
+KAPPA_1875=Q('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328446322744513169085862209720704481201654121427713006366929567157349625273906229277148454180995537563371124755588139767841405530386745578819565618798006892541712197674874836946756930569827889227573192555649999999997996640490086748910957328658012158594')  # exact kappa of rohanarun's PR1875 (1e-300 endpoint), from its body
+KAPPA_1876=Q('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310799999999999999999997996640490086748910957328658012158594846')  # exact kappa of rohanarun's PR1876 (1e-400 endpoint), from its body
+KAPPA_1897=Q('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310881185746917580802513640545967852810801680966856133978576162943792322682482152988532923811998938861379689883167389728443314312230815059559906903773756886872476136837071871203030423643505345473735135533')  # exact kappa of M6LI's PR1897 (1e-500 grid), from its body
+KAPPA_1786=Q('0.0016797547881500489087805478664999999999999999999999999999999998000000199999')  # exact kappa of chafreaky's PR1786 ledger (EXPECTED-BALANCE.json), headline 0.001679754788150048
+KAPPA_PKG5=Q('0.001677992607205282520732088774830808372222877897034109604209719298829805473412451530867883424937597')  # previous package (rectangular k=2, tau_eff=2.2500351)
+KAPPA_PKG4=Q('0.0016778199023144364336310551358776001013829401215770658599799349282716974076737897457922714197448847')  # previous package (radix union schedule, square boundary tau=2.372864)
+KAPPA_1712=Q('0.0016776417313433545642517275958639984231708545722735432722117400060454625429077941422798526440584702')  # pkg of PR1712 (Le Gall tau=12/5, old union row)
+KAPPA_1699=Q('0.001677641731343354564251058287063644749065568493')  # exact kappa of PR1699's ledger (Chafik Boukhalfa), same supplier profile
+certified_complex=Q(spec['expected']['complex_lower'])
+bit=Q('0.001733427039589844234749')
+# Keep short independently checkable decimals strictly inside both endpoints.
+b=certified_complex-B_GAP   # strictly inside the certified supplier endpoint; B.json carries the same exact value
+assert b==Q(BJ['b'])
+q=b-GAP
+a=(q+b)/2
+online=Q(spec['expected']['kappa'])
+old=Q('0.001673360105405778')
+assert b<certified_complex<bit
+profile={int(r):int(n) for r,n in spec['expected']['global_profile'].items()}
+m=spec['geometry']['dimension']; W=spec['expected']['live']
+T=spec['geometry']['replicas']
+assert type(T) is int and T>0
+assert m==73 and W==896*T and W==BJ['live']
+EXPECTED_PROFILE={int(k):v for k,v in BJ['profile'].items()}
+assert profile==EXPECTED_PROFILE
+
+GB=int((DIG+130)*3.3220)+16   # enclosure grid 2^-GB, about 50 digits below the slacks
+ONE=1<<GB
+try:
+    from gmpy2 import mpz   # optional accelerator, identical integer results
+except ImportError:
+    mpz=int
+def _cd(a,b):return -((-a)//b)
+def down(x):return Q((x*ONE).__floor__(),ONE)
+def up(x):return Q(_cd(x.numerator*ONE,x.denominator),ONE)
+LN=GB//3+12   # atanh terms: z<=1/3, so the tail is below 9^-LN < 2^-GB
+EN=GB//6+12   # exponential terms: x<0.01, so x^EN/EN! < 2^-GB
+def unit_log(y):
+    # 2*atanh((y-1)/(y+1)) in directed-rounding integer fixed point: floor every term for the lower sum, ceil every term for the upper sum, geometric tail
+    z=(y-1)/(y+1);p,q=mpz(z.numerator),mpz(z.denominator);p2,q2=p*p,q*q
+    pl=ONE*p//q;ph=_cd(ONE*p,q);sl=sh=mpz(0)
+    for j in range(LN):
+        sl+=pl//(2*j+1);sh+=_cd(ph,2*j+1);pl=pl*p2//q2;ph=_cd(ph*p2,q2)
+    tail=_cd(2*ph*q2,(2*LN+1)*(q2-p2))
+    return Q(int(2*sl),ONE),Q(int(2*sh+tail),ONE)
+ln2=unit_log(Q(2))
+def log(x):
+    k=0
+    while x>=2:x/=2;k+=1
+    lo,hi=unit_log(x)
+    return down(lo+k*ln2[0]),up(hi+k*ln2[1])
+def exp(x):
+    assert 0<=x<Q(1,100)
+    xl=mpz(x.numerator)*ONE//x.denominator;xu=_cd(mpz(x.numerator)*ONE,x.denominator)
+    tl=th=sl=sh=mpz(ONE)
+    for j in range(1,EN):
+        tl=((tl*xl)>>GB)//j;th=_cd(-((-th*xu)>>GB),j);sl+=tl;sh+=th
+    tn=_cd(-((-th*xu)>>GB),EN)    # x^EN/EN! (rounded up); the tail is at most twice this
+    return Q(int(sl),ONE),Q(int(sh+2*tn+2),ONE)
+def moment(endpoint):
+    lo=hi=Q(0)
+    for r,n in profile.items():
+        l,h=log(Q(m,r)); weight=Q(n*r,m*W)
+        lo+=weight*exp(endpoint*l)[0]
+        hi+=weight*exp(endpoint*h)[1]
+    return down(lo),up(hi)
+
+native=moment(b)
+assert native[1]<1
+b_up=Q(BJ['b_above_endpoint'])
+assert b_up>certified_complex and moment(b_up)[0]>1   # a larger test endpoint is rejected
+
+def ledger(kind, *, q_=q, zeta=ZETA, slack=SLACK,
+           old_scan=False, full_gauge=False, full_key=False, old_union=False, old_theta=False, old_linear=False, old_tag=False,
+           eps_=EPS, beta_=BETA, tau_eff=STRASSEN_TAU, halo_power=HALO_POWER, core_power=CORE_POWER):
+    r=R0; stop=STOP
+    # Existing explicit Strassen format, 7^16<2^45; c=3/32.
+    # Union: radix-distributed schedule (outer/UNION-SCHEDULE.md, new in this package): row exponent eps+1/2-xi. The boundary row binds with main; the Gaussian row has slack at this explicit point.
+    c=(3-tau_eff)/2
+    eps=eps_; beta=beta_
+    xi=eps*(1+q_)-c+slack
+    saving=eps*q_
+    # Deliberate strict gap (BACKOFF) after decimal truncation to the FLOOR grid.
+    kappa=Q((saving*FLOOR).__floor__(),FLOOR)-BACKOFF
+    # kappa is derived, not silently asserted to equal an expected decimal.
+    tau=1-a; sigma=1-b
+    internal=tau+(1-stop)*max(sigma-tau,Q(0))
+    leaf=sigma+stop*(1-sigma)
+    lam=1-(q_+a)/2
+    key=beta+1-eps
+    margins={
+        'main_transform':saving,
+        'global_coordinate_router':a,
+        'fused_gaussian':1-eps+q_*beta,
+        'group_bit_router':1-eps+a*beta,
+        'group_scans':1-eps+beta,
+        'boundary_algebraic_matrix':c+xi-eps,
+        'sparse_union_movement':xi+Q(1,2)-eps,
+        'polynomial_products':eps,
+        'microbox_face_corrections':1-(eps+beta+halo_power-core_power+2),
+    }
+    if old_union:margins['old_g_factor_union_row']=1-eps-beta-Q(1,2)+xi
+    if old_scan:margins['old_per_axis_scan']=1-eps
+    if full_key:margins['unnecessary_full_key_sort']=margins['sparse_union_movement']-key
+    checks={
+        'actual_bit_above_effective':bit-a,
+        'complex_above_effective':b-a,
+        'q_positive':q_, 'complex_endpoint':b-q_,
+        'q_below_effective':a-q_, 'stop_positive':stop,
+        'stop_below_one':1-stop,
+        'leaf_saving_above_effective':(1-stop)*b-a,
+        'q_below_internal':1-internal-q_, 'q_below_leaf':1-leaf-q_,
+        'lambda_above_tau':lam-tau, 'lambda_above_sigma':lam-sigma,
+        'lambda_above_internal':lam-internal,
+        'lambda_prime_above_lambda':1-q_-lam,
+        'lambda_prime_above_leaf':1-q_-leaf,
+        'epsilon_positive':eps,'epsilon_below_one':1-eps,
+        'cap_above_band':xi-Q(1,2),'cap_below_cell':eps-xi,
+        'group_positive':beta,'group_below_dimension':eps-beta,
+        'support_router_span':1-beta,
+        'group_gauge_power_slack':1-(2*xi-THETA_POWER)-r-beta,
+        'actual_logarithmic_group_guard':1-(2*xi-THETA_POWER)-beta,
+        'global_gamma_power_slack':1-eps-r,
+        'actual_logarithmic_gamma_guard':1-eps,
+        'alpha_logarithm_absorption':r,
+        'group_key_fits_record':1-key,
+        'membership_arithmetic_fits_record':1-beta-2*(1-eps),
+        'microbox_halo_volume':core_power-halo_power-beta,
+        'positive_layout_slack':slack,
+        'short_record_fallback':eps-a,
+        'improvement_over_online_final':kappa-online,
+        'improvement_over_previous_local_final':kappa-old,
+        'complex_below_one_over_32':Q(1,32)-b,
+    }
+    checks['strassen_integer_power_gap']=Q(2**tau_eff.numerator-7**tau_eff.denominator)
+    checks['theta_positive_polynomial_power']=THETA_POWER
+    checks['cap_below_phase_cell_power']=THETA_POWER-xi
+    checks['halo_covers_generic_inverse_radius']=Q(halo_power-43)
+    checks['union_tag_work_per_visit_fits_record']=1-beta
+    checks['union_header_fits_record']=1-beta
+    checks['source_axis_superpolynomial']=1-eps
+    checks['improvement_over_pr1786']=kappa-KAPPA_1786
+    checks['improvement_over_pr1828']=kappa-KAPPA_1828
+    checks['improvement_over_pr1839']=kappa-KAPPA_1839
+    checks['improvement_over_pr1875']=kappa-KAPPA_1875
+    checks['improvement_over_pr1876']=kappa-KAPPA_1876
+    checks['improvement_over_pr1897']=kappa-KAPPA_1897
+    checks['improvement_over_pr1758']=kappa-Q('0.001679745508802299051546533685199973672313473419790800679752457380932784242972585600630181672224983')
+    checks['improvement_over_pr1699']=kappa-KAPPA_1699
+    checks['improvement_over_pr1712']=kappa-KAPPA_1712
+    checks['improvement_over_previous_package']=kappa-KAPPA_PKG4
+    checks['improvement_over_pkg_k2']=kappa-KAPPA_PKG5
+    if full_gauge:checks['unrecentered_group_gauge']=1-THETA_POWER-r-beta
+    if old_theta:checks['old_theta_chirp_guard']=1-(2*xi-eps)-r-beta
+    if old_linear:checks['old_linear_guard']=1-xi-r-beta
+    if old_tag:checks['old_union_tag_2beta']=1-2*beta
+    # Main-binding family invariant (PR2180): the Gaussian saving stays above the main saving. The fixed kappa grid (FLOOR) is coarser than the free gaps,
+    # so the *_above_kappa rows alone would no longer exclude a beta outside the main-binding window; this row is delta*(1-2q) > 0.
+    checks['gaussian_above_main']=margins['fused_gaussian']-saving
+    checks.update({n+'_above_kappa':v-kappa for n,v in margins.items()})
+    assert margins['boundary_algebraic_matrix']==saving+slack
+    assert margins['sparse_union_movement']==saving+slack+(tau_eff-2)/2
+    return {'kind':kind,'kappa':kappa,'epsilon':eps,'xi':xi,'beta':beta,
+            'q':q_,'a':a,'r0':r,'zeta':zeta,'slack':slack,
+            'margins':margins,'strict_checks':checks}
+
+def encode(x):
+    if isinstance(x,Q):
+        with localcontext() as c:
+            c.prec=DIG+200
+            return {'rational':str(x),'decimal':str(Decimal(x.numerator)/Decimal(x.denominator))}
+    if isinstance(x,dict):return {str(k):encode(v) for k,v in x.items()}
+    if isinstance(x,(list,tuple)):return [encode(v) for v in x]
+    return x
+
+rows={};controls={}
+for kind in ['struct_boundary_radix_union']:
+    row=ledger(kind)
+    assert all(v>0 for v in row['strict_checks'].values())
+    assert min(row['margins'].values())==row['margins']['main_transform']
+    # fixed-grid maximum: C*FLOOR is an integer and 0 < C - eps*q < 1/FLOOR, so floor(FLOOR*eps*q)/FLOOR = C - 1/FLOOR exactly and kappa attains the grid/backoff ceiling
+    assert (certified_complex*FLOOR).denominator==1 and 0<certified_complex-EPS*q<Q(1,FLOOR)
+    assert row['kappa']==certified_complex-Q(1,FLOOR)-BACKOFF
+    assert len(row['strict_checks'])==63
+    rows[kind]=row
+    variants=[('endpoint_equality',{'q_':b}),('zero_layout_slack',{'slack':Q(0)}),
+              ('old_per_axis_scan',{'old_scan':True}),('full_cell_gauge',{'full_gauge':True}),
+              ('full_key_sort',{'full_key':True}),('old_g_factor_union_row',{'old_union':True}),
+              ('old_theta_chirp_guard',{'old_theta':True}),('old_linear_guard',{'old_linear':True}),
+              ('old_union_tag_2beta',{'old_tag':True}),('matrix_below_strassen',{'tau_eff':Q(14,5)}),
+              ('halo_too_short',{'halo_power':40}),('core_too_small',{'core_power':48}),
+              ('group_equals_dimension',{'beta_':EPS}),('epsilon_equals_one',{'eps_':Q(1)}),('epsilon_not_above_r0_slack',{'eps_':1-R0}),('beta_below_gaussian_window',{'beta_':1-700*DELTA_E}),('beta_above_membership_window',{'beta_':1-2*DELTA_E})]
+    controls[kind]={}
+    for name,kwargs in variants:
+        bad=ledger(kind,**kwargs)
+        failed={n:v for n,v in bad['strict_checks'].items() if v<=0}
+        assert failed,(kind,name)
+        controls[kind][name]=failed
+    assert 'old_g_factor_union_row_above_kappa' in controls[kind]['old_g_factor_union_row'] and 'unnecessary_full_key_sort_above_kappa' in controls[kind]['full_key_sort']
+# PR1786 p^10 umbrella (vendor/pr1786/FINE-PRIME-PADDING.md, "a conservative common eventual bound p^10", and check_fine_padding.py norm_power = 10) tied to THETA_POWER:
+# the inverse-norm bound ||K^-1|| = O(alpha^5 rho^2 theta^-2 exp(a rho/3)) has theta^-2 <= 16 p^(2*THETA_POWER) (theta >= 1/(4 p^THETA_POWER)), an exponential factor <= p^2 eventually,
+# alpha^5 polylogarithmic; a further half power is allowed for the polylogarithmic and bounded-constant factors of the augmented matrices. The umbrella must exceed that budget.
+UMBRELLA_POWER=10
+def umbrella_margin(theta_power):return UMBRELLA_POWER-(2*theta_power+2+Q(1,2))
+assert umbrella_margin(THETA_POWER)>0 and umbrella_margin(Q(15,4))<=0   # control: the umbrella would fail silently for THETA_POWER >= 3.75
+HALO_UMBRELLA={'umbrella_power':UMBRELLA_POWER,'theta_power':THETA_POWER,'exponent_budget':2*THETA_POWER+2+Q(1,2),'margin':umbrella_margin(THETA_POWER),'fails_from_theta_power':Q(15,4)}
+out={'status':'PASS_FINE_PRIME_PADDING_EXACT_ARITHMETIC',
+     'scope':'Conditional fine-prime padding composition. Fresh native admission is required by verify.py. The cited prime theorem, written gauge-free theorem and inherited all-size interfaces are not proved by this arithmetic. No independent human review is claimed.',
+     'geometry_changes':{'theta_power':THETA_POWER,'halo_power':HALO_POWER,'core_power':CORE_POWER,'matrix_exponent':STRASSEN_TAU},
+     'halo_umbrella':HALO_UMBRELLA,'complex_lower_short':b,'complex_lower_pinned':certified_complex,
+     'native_moment':native,'profile':profile,'m':m,'W':W,
+     'online_final_kappa':online,'rows':rows,'rejecting_controls':controls,
+     'pr1673_kappa':Q('0.001677160151714381'),'improvement_over_pr1673':rows['struct_boundary_radix_union']['kappa']-Q('0.001677160151714381'),
+     'source_sha256':{p.relative_to(WORK).as_posix():hashlib.sha256(p.read_bytes()).hexdigest() for p in [SPEC,CLAIM]}}
+(HERE/'LEDGER.json').write_text(json.dumps(encode(out),indent=2)+'\n')
+print(out['status'])
+for name,row in rows.items():
+    print(name,'FINAL KAPPA',encode(row['kappa'])['decimal'],
+          'positive checks',len(row['strict_checks']),
+          'gain over online',encode(row['kappa']-online)['decimal'])
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/outer-replay/outer/independent_check.py b/research/staged544-refined-endpoint/vendor/staged544-producer-return/outer-replay/outer/independent_check.py
new file mode 100644
index 0000000000..b838c430ef
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/outer-replay/outer/independent_check.py
@@ -0,0 +1,60 @@
+"""Independent balance-equation check (PR1673, adapted: this package's supplier b and SPEC), not an execution of the ledger in check_ledger.py."""
+from fractions import Fraction as F
+from pathlib import Path
+from decimal import Decimal,localcontext
+import json,hashlib,sys
+sys.set_int_max_str_digits(0)
+D=Path(__file__).resolve().parent;W=D.parent
+src=W/'new/SPEC.json'
+BJ=json.loads((D/'B.json').read_text())
+spec=json.loads(src.read_text());pinned=F(spec['expected']['complex_lower'])
+DIG=spec['outer_refinement']['endpoint_digits'];T=10**(DIG+80)   # all free slacks are 10^-(DIG+80) or below (fixed-grid-maximum point, PR2180 / PR2161 dataisfire)
+b=F(BJ['b']);q=b-F(1,T);eff=(q+b)/2;actual_bit=F('0.001733427039589844234749')
+r=F(1,T);z=F(1,T*100);s=F(1,T)
+assert 0<q<eff<b<pinned<actual_bit
+# Exact inequalities needed by the stopped support-local supplier.
+stop=F(1,T*10**6);assert (1-stop)*b>eff
+rows={}
+# FINE-PRIME PADDING (outer/FINE-PADDING-COMPOSITION.md): explicit point epsilon=1-delta, beta=1-3 delta, delta=(1025/1024)*10^-(DIG+80), Strassen boundary tau=45/16 (7^16<2^45), c=3/32,
+# theta in (1/(4p^2),1/(2p^2-1)], chirp guard beta+2xi-2<1, halo power 48, core power 128. Derived here WITHOUT the ledger code; the structured boundary is not used.
+delta=F(1025,1024*T);E=1-delta;B=1-3*delta
+assert 7**16<2**45;tau=F(45,16);c=(3-tau)/2;assert c==F(3,32)
+nu=2;halo=48;core=128
+X=E+E*q+s-c                      # boundary saving c+X-E = E q + s
+assert c+X-E==E*q+s
+# the Gaussian row has slack: 1-E+qB-Eq = delta(1-2q)
+assert 1-E+q*B-E*q==delta*(1-2*q)>0
+# admissible beta window for this epsilon: Gaussian row needs beta >= 1-delta(1+1/q); membership needs beta < 1-2 delta
+assert 1-delta*(1+1/q)<=B<1-2*delta
+assert F(1,2)<X<E<1 and 0<B<E and X<nu
+assert 1-(2*X-nu)-r-B>z>0 and 1-E-r>0 and core-halo-B>0 and halo>43
+assert 1-B>0 and 1-B-2*(1-E)==delta>0
+costs={'transform':1-E*q,'global_router':1-eff,'gaussian':E-q*B,'group_router':E-eff*B,'group_scans':E-B,'boundary':E-X+(tau-1)/2,'sparse_union':E+F(1,2)-X,'polynomials':1-E,'microfaces':E+B+halo-core+2}
+K=F((E*q*10**(DIG+46)).__floor__(),10**(DIG+46))-F(1,10**(DIG+22))
+assert max(costs.values())==costs['transform'] and all(v<1-K for v in costs.values())
+# earlier packages' guards fail at this point (they are rejecting controls in the ledger): linear guard 1-X-r-B<0, old theta chirp guard 1-(2X-E)-r-B<0, 2B<1 test fails
+assert 1-X-r-B<0 and 1-(2*X-E)-r-B<0 and 1-2*B<0
+assert K>F('0.0016797547881500489087805478664999999999999999999999999999999998000000199999')   # PR1786 exact kappa
+assert K>F('0.0016797549561255445213356709939207025769983114385025802954603275201627999999999999999999999999979966404900867489109573286580121585')  # PR1828 exact kappa
+assert K>F('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263049999999997996640490086748910957328658012158594')  # PR1839 exact kappa
+assert K>F('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328446322744513169085862209720704481201654121427713006366929567157349625273906229277148454180995537563371124755588139767841405530386745578819565618798006892541712197674874836946756930569827889227573192555649999999997996640490086748910957328658012158594')  # PR1875 exact kappa
+assert K>F('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310799999999999999999997996640490086748910957328658012158594846')  # PR1876 exact kappa
+assert K>F('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310881185746917580802513640545967852810801680966856133978576162943792322682482152988532923811998938861379689883167389728443314312230815059559906903773756886872476136837071871203030423643505345473735135533')  # PR1897 exact kappa
+assert K>F('0.001679745508802299051546533685199973672313473419790800679752457380932784242972585600630181672224983')  # PR1758
+assert K<q
+# epsilon is free: every row is strictly satisfied for every delta > r0 (beta = 1-3 delta); only kappa moves, kappa = (1-delta) q - O(1e-76)
+# fixed-grid maximum: saving < C and C*FLOOR integral give K <= C-1/FLOOR-BACKOFF; this point attains it (0 < C-E*q < 1/FLOOR)
+FL=10**(DIG+46);assert (pinned*FL).denominator==1 and 0<pinned-E*q<F(1,FL) and K==pinned-F(1,FL)-F(1,10**(DIG+22))
+sweep=[]
+for dd in [F(1,10**7),F(1,10**12),F(1,10**30),F(1,10**60),F(1025,1024*T),F(11,T*10)]:
+    Ed=1-dd;Bd=1-3*dd;Xd=Ed+Ed*q+s-c
+    assert 1-Ed-r>0 and 1-dd*(1+1/q)<=Bd<1-2*dd and 1-Ed+q*Bd-Ed*q>0 and Ed-Bd>0 and 1-Bd-2*(1-Ed)>0
+    assert F(1,2)<Xd<Ed and 1-(2*Xd-nu)-r-Bd>0 and core-halo-Bd>0
+    assert 1-Ed+eff*Bd>Ed*q and eff>Ed*q and 1-Ed+Bd>Ed*q and Ed>Ed*q and 1-(Ed+Bd+halo-core+2)>Ed*q
+    sweep.append((str(dd),str(float(q-Ed*q))))
+assert 1-F(1,T)<=1-r   # delta=r0 itself is rejected by 1-eps-r0>0
+assert not (1-(1-r)-r>0)
+rows['struct_boundary_radix_union']={'epsilon':str(E),'cap':str(X),'beta':str(B),'kappa':str(K),'margin_above_kappa':str(E*q-K),'checked_cost_powers':len(costs),'gap_to_q':str(q-K),'epsilon_free_sweep_delta_gap_to_q':sweep}
+report={'status':'PASS_INDEPENDENT_BALANCE_DERIVATION','source_sha256':hashlib.sha256(src.read_bytes()).hexdigest(),'b':str(b),'rows':rows,'scope':'Independent balance derivation and margin comparisons for this package\'s supplier endpoint; the supplier native admission is bound by verify_release.py; inherited all-size assumptions remain.'}
+(D/'INDEPENDENT.json').write_text(json.dumps(report,indent=2)+'\n')
+print(json.dumps(report,indent=2))
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/owner_audit.py b/research/staged544-refined-endpoint/vendor/staged544-producer-return/owner_audit.py
new file mode 100644
index 0000000000..3ab635aafe
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/owner_audit.py
@@ -0,0 +1,87 @@
+# Standalone standard-library physical owner audit, adapted from credited PR3638 predicates.
+from pathlib import Path
+from collections import Counter,defaultdict
+import gzip,json,hashlib
+P=Path(__file__).resolve().parent
+
+def need(ok,why):
+ if not ok:raise ValueError(why)
+def read(p):
+ raw=p.read_bytes();return json.loads(gzip.decompress(raw) if p.suffix=='.gz' else raw)
+def rank(U):
+ piv={}
+ for x in U:
+  while x:
+   b=x.bit_length()
+   if b in piv:x^=piv[b]
+   else:piv[b]=x;break
+ return len(piv)
+def audit(mutator=None):
+ D=P/'inputs';T=360;order=read(D/'ORDER.json');dispatch=read(D/'DISPATCH.json');N=9*T
+ need(len(order)==N and set(map(tuple,order))=={(s,r) for s in range(9) for r in range(T)},'call coverage')
+ clock={tuple(v):i for i,v in enumerate(order)}
+ need(all([clock[s,r] for s in range(9)]==sorted(clock[s,r] for s in range(9)) for r in range(T)),'source order')
+ providers={(s,r):pid for s,r,pid in dispatch};need(len(providers)==len(dispatch)==N and set(providers)==set(clock),'dispatch')
+ plan=read(D/'PLAN.json.gz');template=read(D/'TEMPLATE.json.gz');children={c['child']:c['rank'] for c in template['idle_children']};K=max(children)+1
+ windows={}
+ for owner in range(T):
+  touched={};t=None
+  for ev in template['execution_schedule']:
+   kind=ev['kind']
+   if kind in ('bank_padding_cover_sweep','translate_every_live_family'):continue
+   if kind=='local_cover_sweep':t=clock[ev['stage'],owner];touched[next(x['pair'] for x in template['recipe'] if x['kind']=='local' and x['stage']==ev['stage'])]=t
+   elif kind=='bridge':
+    need(t is not None,'bridge source');touched[ev['src']]=t;touched[ev['dst']]=t
+   elif kind=='idle_cover_sweep':
+    need(t is not None,'idle source')
+    for c in ev['child_indices']:
+     if c in children:
+      need((owner,c) not in windows,'unique child');windows[owner,c]=(children[c],touched.get(ev['pair'],-1),t)
+   else:need(kind in ('terminal_Y_translations','terminal'),'source event')
+  need(all((owner,c) in windows for c in children),'complete windows')
+ families={c:2*plan['pairs'][str(c)]+plan['sides'][str(c)] for c in children}
+ fwin=defaultdict(list)
+ for (o,c),(cap,lo,hi) in windows.items():fwin[o,families[c]].append((lo,hi))
+ for rows in fwin.values():need(all(max(a,c)>=min(b,d) for j,(a,b) in enumerate(rows) for c,d in rows[j+1:]),'source family windows disjoint')
+ widths={};local=None
+ for tag,pid in [('base','gauge44'),('extra','extra')]:
+  w=read(D/f'NATIVE-{tag}.json.gz');need(w['R']==408 and len(w['zero_buffers'])==91,'native stock')
+  widths[pid]=[rank(w['frames'][w['final'][r]])-rank(w['frames'][w['start'][r]]) for r in range(224,632)]
+  need(Counter(widths[pid])=={4:26,5:2,6:18,7:53,8:96,9:213},'native widths')
+  loc={int(r):n for r,n in w['local_profile'].items()};need(local is None or local==loc,'provider paid profiles');local=loc
+ need(local=={1:2440,2:725,3:224,4:102,5:20,8:8},'changed local bill')
+ fill=read(D/'FILL.json.gz')
+ if mutator:mutator(fill['rows'])
+ need(fill['columns']==['time','helper','owner_replica','child','port','offset'] and fill['dispatch_overrides']==[],'literal schema')
+ need(len(fill['rows'])==N*408,'role count');used=Counter();callfamilies=set()
+ for j,row in enumerate(fill['rows']):
+  need(len(row)==6 and all(type(x)is int for x in row),'integer role')
+  t,h,o,c,p,z=row;need(t==j//408 and h==j%408,'ordered complete helper identities')
+  if h==0:callfamilies=set()
+  need((o,c) in windows and 0<=p<112,'owner child port')
+  cap,lo,hi=windows[o,c];need(lo<t<=hi,'strict lower and closed upper window')
+  family=o*896+families[c]*112+p;need(family not in callfamilies,'same-call physical family uniqueness');callfamilies.add(family)
+  w=widths[providers[tuple(order[t])]][h];key=(o*K+c)*112+p
+  need(z==used[key] and z+w<=cap,'compact chronological prefix and capacity');used[key]+=w
+ idle=Counter()
+ for (o,c),(cap,lo,hi) in windows.items():
+  for p in range(112):
+   left=cap-used[(o*K+c)*112+p];need(left>=0,'bin overflow')
+   if left:idle[left]+=1
+ profile=Counter({r:N*n for r,n in local.items()});profile.update(idle)
+ expected={int(r):n for r,n in read(D/'PROFILE.json').items()};need(dict(profile)==expected,'complete paid histogram')
+ need(sum(r*n for r,n in profile.items())==65088*T,'mass deficit')
+ return dict(status='PASS_STANDARD_LIBRARY_INDEPENDENT_LITERAL_AUDIT',replicas=T,calls=N,roles=N*408,physical=896*T+408+8+91+2,idle=dict(idle),profile=dict(profile))
+def controls():
+ def reused_family(rows):
+  rows[1][2:5]=rows[0][2:5]
+ def duplicate_helper(rows):rows[1][1]=rows[0][1]
+ def unpaid_hole(rows):rows[0][5]+=1
+ results=[]
+ for name,mutator,phrase in [('same_call_family',reused_family,'same-call physical family uniqueness'),('duplicate_helper',duplicate_helper,'ordered complete helper identities'),('unpaid_hole',unpaid_hole,'compact chronological prefix')]:
+  try:audit(mutator)
+  except ValueError as ex:
+   need(phrase in str(ex),'wrong owner rejection: '+str(ex));results.append(name)
+  else:raise ValueError('owner mutant accepted '+name)
+ return results
+if __name__=='__main__':print(json.dumps(dict(positive=audit(),controls=controls()),sort_keys=True),flush=True)
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/rational_price.py b/research/staged544-refined-endpoint/vendor/staged544-producer-return/rational_price.py
new file mode 100644
index 0000000000..cc53a4ef64
--- /dev/null
+++ b/research/staged544-refined-endpoint/vendor/staged544-producer-return/rational_price.py
@@ -0,0 +1,77 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Rational enclosures for the finite paid-profile comparison.
+
+For 1<=x<2, log(x)=2*sum(z^(2j+1)/(2j+1)), z=(x-1)/(x+1).
+The positive remainder is at most 2*z^(2n+1)/((2n+1)*(1-z*z)).
+Range reduction gives z<=1/3. Alternating exponential partial sums then
+bound exp(-a*log(r)); no floating arithmetic or root solver is used.
+"""
+from fractions import Fraction as Q
+from functools import lru_cache
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def log_series(x, n=100):
+    z = (x - 1) / (x + 1)
+    total = Q(0)
+    power = z
+    for j in range(n):
+        total += 2 * power / (2*j + 1)
+        power *= z*z
+    tail = 2 * power / ((2*n + 1) * (1-z*z))
+    return total, total + tail
+
+
+@lru_cache(None)
+def log_bounds(rank):
+    need(type(rank) is int and 1 <= rank <= 73, 'priced rank domain')
+    exponent = rank.bit_length() - 1
+    a, b = log_series(Q(rank, 1 << exponent))
+    c, d = log_series(Q(2))
+    lo, hi = a + exponent*c, b + exponent*d
+    # Outward rational quantization keeps subsequent denominators bounded.
+    scale = 10**90
+    low = (lo.numerator * scale) // lo.denominator
+    high = -((-hi.numerator * scale) // hi.denominator)
+    return Q(low, scale), Q(high, scale)
+
+
+def exp_negative(x, degree):
+    need(0 <= x < 1, 'alternating exponential range')
+    term = Q(1)
+    total = term
+    for j in range(1, degree + 1):
+        term *= -x / j
+        total += term
+    return total
+
+
+@lru_cache(None)
+def h_bounds(rank, a):
+    need(Q(0) < a < Q(1, 100), 'exponent domain')
+    lo, hi = log_bounds(rank)
+    return rank * exp_negative(a*hi, 31), rank * exp_negative(a*lo, 30)
+
+
+def interval(profile, a):
+    a = Q(a)
+    lower = upper = Q(0)
+    for r, count in profile.items():
+        need(type(r) is int and type(count) is int, 'integer signed profile')
+        lo, hi = h_bounds(r, a)
+        lower += count * (lo if count >= 0 else hi)
+        upper += count * (hi if count >= 0 else lo)
+    return lower, upper
+
+
+def compact(bounds):
+    # Report outward intervals at 60 decimal places, retaining rational signs.
+    lo, hi = bounds
+    scale = 10**60
+    low = (lo.numerator*scale)//lo.denominator
+    high = -((-hi.numerator*scale)//hi.denominator)
+    return {'lower': str(Q(low, scale)), 'upper': str(Q(high, scale))}
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/words/BORROW-WORD-43base.json.gz b/research/staged544-refined-endpoint/vendor/staged544-producer-return/words/BORROW-WORD-43base.json.gz
new file mode 100644
index 0000000000..674ea18ade
Binary files /dev/null and b/research/staged544-refined-endpoint/vendor/staged544-producer-return/words/BORROW-WORD-43base.json.gz differ
diff --git a/research/staged544-refined-endpoint/vendor/staged544-producer-return/words/BORROW-WORD-43extra.json.gz b/research/staged544-refined-endpoint/vendor/staged544-producer-return/words/BORROW-WORD-43extra.json.gz
new file mode 100644
index 0000000000..b17e2be67e
Binary files /dev/null and b/research/staged544-refined-endpoint/vendor/staged544-producer-return/words/BORROW-WORD-43extra.json.gz differ
diff --git a/research/staged544-refined-endpoint/verify.py b/research/staged544-refined-endpoint/verify.py
new file mode 100644
index 0000000000..12aaa763dc
--- /dev/null
+++ b/research/staged544-refined-endpoint/verify.py
@@ -0,0 +1,144 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Independent endpoint certificate on the current PR3815/PR3802 construction.
+Endpoint refinement by hexapode with Codex assistance. Retained engines and generic proof bodies by Dugongue, with substantial OpenAI Codex assistance. Outer integration follows
+dataisfire PR3361; exact retained scientific engines keep their attribution.
+"""
+import argparse,ast,gzip,hashlib,json,os,subprocess,sys,tempfile,textwrap,types
+from pathlib import Path
+from fractions import Fraction as Q
+P=Path(__file__).resolve().parent
+S=P/'vendor/staged544-producer-return'
+B=P/'vendor/staged544-clock-ceiling'
+VENDOR_SHA='30ee97d3076de3b117c544496758d02609259f9543cf5f1e698faf70d8f89443'
+PROFILE={1:7989840,2:2349000,3:749088,4:330480,5:69120,8:25920,48:137952}
+DIGITS=80
+PROOF_PINS={'CapCertificate.lean': 'b1ba8ad1b65eeecaddf34168456d2f9a89eed5f36212bfcfa24107aa8fbe4833', 'AnalyticCertificate.lean': '44b6c75086d95163e876563621676289f3c6348c584358106c301b6c8195ffaf'}
+
+def need(condition,message):
+ if not condition:raise ValueError(message)
+def read(path):return json.loads(path.read_bytes())
+def sha(path):return hashlib.sha256(path.read_bytes()).hexdigest()
+def save(path,value):path.write_bytes((json.dumps(value,indent=2,sort_keys=True)+'\n').encode())
+def load(path,name):
+ module=types.ModuleType(name);module.__file__=str(path)
+ exec(compile(path.read_bytes(),str(path),'exec'),module.__dict__)
+ return module
+
+def authenticate():
+ need(sha(P/'VENDOR.json')==VENDOR_SHA,'vendor manifest and usage classes')
+ records=read(P/'VENDOR.json')['files']
+ expected={x['path']:x['sha256'] for x in records}
+ actual={f.relative_to(P).as_posix():sha(f) for f in (P/'vendor').rglob('*') if f.is_file()}
+ need(actual==expected,'exact complete vendor source inventory')
+ need(not any(f.is_symlink() for f in (P/'vendor').rglob('*')),'regular retained inputs')
+
+def arithmetic():
+ authenticate()
+ profile={int(r):n for r,n in read(B/'inputs/PROFILE.json').items()}
+ need(profile==PROFILE and sum(r*n for r,n in profile.items())==65088*360,'complete fixed profile and rank mass')
+ D=load(P/'dyadic_moment.py','new_dyadic');certificate=D.certify(profile,73,896*360,DIGITS)
+ C=Q(certificate['lower']);high=Q(certificate['upper']);K=C-Q(1,10**102)-Q(1,10**126)
+ need(sha(P/'PREDECESSOR-CLAIM.json')=='2b3a4ede0d77e4d9908f50f756559fa4a5f371f9d04a185a308641b601c8857a','pinned PR3857 claim')
+ old=Q(read(P/'PREDECESSOR-CLAIM.json')['kappa'])
+ need(old<K<C<high,'strict positive conditional improvement')
+ R=load(S/'rational_price.py','credited_alternating_price');signed=dict(profile);signed[73]=-896*360
+ below=R.interval(signed,C);above=R.interval(signed,high)
+ need(below[1]<0<above[0],'unchanged alternating engine independently certifies both new signs')
+ for r in (1,48):
+  bad=dict(profile);bad[r]+=1
+  need(D.DyadicMoment(bad,73,896*360).moment(C)[0]>1,'unpaid child must reject')
+ return dict(profile=profile,complex_lower=str(C),complex_upper=str(high),kappa=str(K),previous=str(old),
+             gain=str(K-old),certificate=certificate,alternating_signs=dict(accepted_upper=R.compact(below)['upper'],rejected_lower=R.compact(above)['lower']))
+
+def finite(work):
+ env=dict(os.environ,PYTHONPYCACHEPREFIX=str(work/'bytecode'));env.pop('PYTHONOPTIMIZE',None)
+ run=subprocess.run([sys.executable,'-B',str(B/'verify.py')],stdout=subprocess.PIPE,stderr=subprocess.STDOUT,env=env,timeout=180)
+ need(run.returncode==0,'retained literal construction: '+run.stdout.decode('utf-8',errors='replace')[-1200:])
+ baseline=json.loads(run.stdout)
+ need(baseline['status']=='PASS_STAGED544_CHANGED_CLOCK_CEILING_OWNERS_AND_PRICE','actual predecessor finite status')
+ need({int(r):n for r,n in baseline['profile'].items()}==PROFILE,'fresh actual owner profile')
+ C=load(S/'audit_core.py','credited_arbitrary_dirty_interpreter');A=load(S/'audit_verify.py','credited_semantics_adapter')
+ receipt=read(P/'vendor/native/NATIVE-RECEIPT.json');words={}
+ for tag in ('base','extra'):
+  path=S/'words'/('BORROW-WORD-43'+tag+'.json.gz');word=json.loads(gzip.decompress(path.read_bytes()))
+  C.hygiene(word);local,widths,*_=C.frame_bill(word)
+  need(local=={1:2440,2:725,3:224,4:102,5:20,8:8},'actual local paid bill')
+  need(widths=={4:26,5:2,6:18,7:53,8:96,9:213} and word['cleanup']==C.derive_cleanup(word),'actual widths and literal cleanup')
+  native=read(S/'evidence'/('NATIVE-'+tag+'.json'))
+  need(native['status']=='PASS_COMPLETE_T74_NATIVE_REPLAY' and native['word_sha256']==sha(path),'retained native evidence binds exact word')
+  need([x['step'] for x in native['steps']]==['native','ledgers','local-replay','wrappers','phases'] and all(x['exit_code']==0 for x in native['steps']),'retained native stages')
+  binding=receipt['providers'][tag]
+  need(binding['word_sha256']==sha(path) and binding['complete_native_files']['local/CANONICAL-WORD.json.gz']==sha(S/'inputs'/('NATIVE-'+tag+'.json.gz')),'retained archive receipt binds actual canonical native frames')
+  words[tag]=dict(semantics=A.semantics(C,word),local=local,widths=widths,word_sha256=sha(path))
+ return dict(actual_clock_owners_and_ceiling=baseline,actual_dirty_words=words,
+             native_scope='Authenticated retained PR3734 native evidence; no fresh native-chain rebuild or omitted cache-byte authentication')
+
+def bootstrap(C):
+ source=(P/'vendor/bootstrap/assemble.py').read_text(encoding='utf-8')
+ node=next(n for n in ast.walk(ast.parse(source)) if isinstance(n,ast.FunctionDef) and n.name=='_outer_witness_raw')
+ body=textwrap.dedent('\n'.join(source.splitlines()[node.lineno-1:node.end_lineno])+'\n')
+ env=dict(Q=Q,mpz=int,hashlib=hashlib,PAYLOAD=2048,c=Q('0.001733427039589844234749'),
+  SPEC=dict(outer_refinement=dict(endpoint_digits=80,grid_digits=39,margin_digits=49,bootstrap_steps=12)),
+  A=load(P/'vendor/bootstrap/outer.py','credited_47_row_engine'))
+ exec(compile(body,'<unchanged carried bootstrap>','exec'),env)
+ k,cuts,outer,controls=env['_outer_witness_raw'](C)
+ need(len(outer['strict_constraints'])==47 and all(v>0 for v in outer['strict_constraints'].values()),'47 strict carried-bootstrap rows')
+ need(len(cuts)==13 and len(controls)==4,'12 carried stages and four controls')
+ return dict(legacy_balanced_kappa=str(k),cutoffs=cuts,controls=controls,strict_constraints={n:str(v) for n,v in outer['strict_constraints'].items()})
+
+def outer(a,work):
+ outerdir=work/'outer';outerdir.mkdir();(work/'new').mkdir()
+ for name in ('check_ledger.py','independent_check.py'):
+  (outerdir/name).write_bytes((S/'outer-replay/outer'/name).read_bytes())
+ C=Q(a['complex_lower']);K=Q(a['kappa'])
+ save(work/'new/SPEC.json',dict(geometry=dict(dimension=73,replicas=360),expected=dict(complex_lower=str(C),kappa=a['previous'],global_profile=a['profile'],live=896*360),outer_refinement=dict(endpoint_digits=80)))
+ save(work/'CLAIM.json',dict(kappa=str(K),scope='Conditional fixed-supplier endpoint refinement'))
+ save(outerdir/'B.json',dict(b=str(C-Q(1,10**160)),b_above_endpoint=a['complex_upper'],profile=a['profile'],live=896*360))
+ for name in ('check_ledger.py','independent_check.py'):
+  with (work/(name+'.log')).open('wb') as log:
+   subprocess.run([sys.executable,'-B','-X','pycache_prefix='+str(work/'outer-bytecode'),str(outerdir/name)],stdout=log,stderr=subprocess.STDOUT,check=True,timeout=60)
+ ledger=read(outerdir/'LEDGER.json');row=ledger['rows']['struct_boundary_radix_union'];balance=read(outerdir/'INDEPENDENT.json')
+ need(Q(row['kappa']['rational'])==K,'derived modern endpoint')
+ need(len(row['strict_checks'])==63 and all(Q(v['rational'])>0 for v in row['strict_checks'].values()),'63 strict modern rows')
+ controls=ledger['rejecting_controls']['struct_boundary_radix_union']
+ need(len(controls)==17 and all(v and all(Q(w['rational'])<=0 for w in v.values()) for v in controls.values()),'17 actual rejection controls')
+ need(balance['status']=='PASS_INDEPENDENT_BALANCE_DERIVATION' and Q(balance['rows']['struct_boundary_radix_union']['kappa'])==K,'independent balance derivation')
+ return dict(ledger=ledger,independent=balance)
+
+def check_lean_data(a):
+ for name,digest in PROOF_PINS.items():need(sha(P/'lean'/name)==digest,'checked proof bytes '+name)
+ source=(P/'lean/CapCertificate.lean').read_text(encoding='utf-8');weights=a['certificate']['weights'];ranks=sorted(a['profile'])
+ counts='def coefficient : Fin 7 → ℚ := !['+', '.join(str(r*a['profile'][r]) for r in ranks)+']'
+ bounds='def upper : Fin 7 → ℚ := !['+', '.join('('+weights[str(r)]['upper']+' / '+weights[str(r)]['scale']+')' for r in ranks)+']'
+ need(source.count(counts)==1 and source.count(bounds)==1,'concrete Lean numbers match regenerated certificate')
+ need('sorry' not in source and 'native_decide' not in source,'proof placeholders or native trust shortcut')
+ analytic=(P/'lean/AnalyticCertificate.lean').read_text(encoding='utf-8');endpoint=Q(a['complex_lower'])
+ need(('def endpoint : ℚ := '+str(endpoint.numerator)+' / '+str(endpoint.denominator)) in analytic,'formal actual endpoint')
+ data=read(P/'FORMAL-DATA.json');need(data['endpoint']==a['complex_lower'] and data['log_degree']==60 and data['exp_degree']==26,'formal data scope')
+ need('sorry' not in analytic and 'native_decide' not in analytic,'analytic proof placeholder or native trust shortcut')
+ for field,name in [('log_upper','logUpper'),('weight_upper','upper'),('root_upper','rootUpper')]:
+  values=[Q(data[field][str(r)]) for r in ranks]
+  declaration='def '+name+' : Fin 7 → ℚ := !['+', '.join('('+str(v.numerator)+' / '+str(v.denominator)+')' for v in values)+']'
+  need(analytic.count(declaration)==1,'formal data declaration '+name)
+
+def reproduce(a,work):
+ return dict(status='PASS_STAGED544_CURRENT_CONDITIONAL_ENDPOINT',arithmetic=a,
+  finite=finite(work),bootstrap=bootstrap(Q(a['complex_lower'])),outer=outer(a,work),
+  scope='Conditional unchanged PR3815/PR3802 supplier; all inherited general realization, compiler, hosting, precision, analytic and BIT assumptions remain')
+
+def main():
+ ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--output',type=Path);args=ap.parse_args()
+ need(not sys.flags.optimize,'inherited checkers require enabled assertions')
+ need(not args.output or not args.output.exists(),'output must be fresh')
+ a=arithmetic();check_lean_data(a);claim=read(P/'CLAIM.json')
+ for key in ('complex_lower','complex_upper','kappa','previous','gain'):need(claim[key]==a[key],'public exact claim '+key)
+ need(claim['endpoint_digits']==80 and claim['fresh_native_replay'] is False and claim['selected_frontier_changed'] is False,'public scope flags')
+ need(claim['replicas']==360 and claim['physical']==323069 and claim['private_zero_buffers']==91,'complete fixed stock')
+ with tempfile.TemporaryDirectory(prefix='staged544-endpoint-') as folder:
+  result=reproduce(a,Path(folder));authenticate()
+  encode=lambda x:json.dumps(json.loads(json.dumps(x)),sort_keys=True,separators=(',',':'))
+  need(encode(result)==encode(read(P/'RESULT.json')),'complete regenerated scoped result')
+  if args.output:save(args.output,result)
+ print(result['status'],'gain',a['gain'])
+if __name__=='__main__':main()