Mister-Meeseeks · Oct 17, 2026 · Oct 16, 2026
diff --git a/research/rank7-r1-t740-cap/CLAIM.json b/research/rank7-r1-t740-cap/CLAIM.json
@@ -0,0 +1,76 @@
+{
+ "allocation": {
+  "payload_sha256": {
+   "DISPATCH.json": "90edc935320dd6264e98dc735d868df3c95f5901c70d7d96171001fcb295cce0",
+   "FILL.json.gz": "b56bfed2c542e63f4139e8e3ebb9fe0c979a3bab1ba6a934e2113f40f5c7e0f0",
+   "ORDER.json": "927226d702bf3cada076f4c459da9063c0df7e7154a62955cf75c40cf30215b9",
+   "PROFILE.json": "83130e5231b3693c2af871c430833601de83fdd8851c024df4f78efc76c8deef"
+  },
+  "relocations": 332,
+  "source": "PR2979 allocation/ (T74 ORDER/DISPATCH/FILL, used unchanged by PR3015), ten lockstep copies (PR2969 convention)",
+  "touched_bins": 263
+ },
+ "baseline_head": "e244296debb45fe6f4da84d29e4d64fbab55fda0",
+ "baseline_kappa": "17291509735342244286949999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "baseline_pr": 3015,
+ "best_open_claim_head": "217edac2ebfa6250536bd6df153de14901a69395",
+ "best_open_claim_kappa": "0.0017291512039830184429979999999999999999999998999999999999999999999999",
+ "best_open_claim_pr": 3021,
+ "calls": 6660,
+ "complex_lower": "345830551200866186907/200000000000000000000000",
+ "fixed_inventory_cap": {
+  "cap_lower": "345830551200866186907/200000000000000000000000",
+  "cap_upper": "17291527560043309345351/10000000000000000000000000",
+  "realized": "attained: the literal idle histogram is exactly 740 x the optimal basic idle profile (verify.py)"
+ },
+ "gain_over_pr3015": "22280876331323/12500000000000000000000",
+ "gain_over_pr3015_decimal": "1.78247010650584E-9",
+ "gain_over_pr3021": "1552021312491537/1000000000000000000000000",
+ "gain_over_pr3021_float": 1.552021312491537e-09,
+ "headline_convention": "floor of the exact root lower end at 24 digits, minus 10^-46 and 10^-70, as PR2979/PR3015/PR3021 (tools/audit_allocation_moment.py)",
+ "helpers": 408,
+ "idle": {
+  "1": 173160,
+  "3": 49617,
+  "48": 281903,
+  "5": 17205
+ },
+ "kappa": "17291527560043309345349999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "kappa_decimal": "0.0017291527560043309345349999999999999999999998999999999999999999999999",
+ "live_families": 663040,
+ "local_profile": {
+  "1": 2443,
+  "2": 707,
+  "3": 232,
+  "4": 108,
+  "5": 18,
+  "8": 8
+ },
+ "meaning": "Moment nomination: PR3015's R1 words (PR2979's words with Jeremy Kothe's declared restore edit R1) with a literal T740 allocation that attains their fixed-capacity packing cap. PR3015's complete T74 release covers the words; the T740 allocation's complete admission/outer release is separate.",
+ "physical_families": 663549,
+ "profile": {
+  "1": 16443540,
+  "2": 4708620,
+  "3": 1594737,
+  "4": 719280,
+  "48": 281903,
+  "5": 137085,
+  "8": 53280
+ },
+ "replicas": 740,
+ "roles": 2717280,
+ "root_lower": "345830551200866186907/200000000000000000000000",
+ "root_upper": "17291527560043309345351/10000000000000000000000000",
+ "scope": "Packing composed with PR3015's declared R1 edit; no other word, gate, frame, scalar or cleanup change. Inherited all-size/compiler/theorem hypotheses remain. No selected-frontier change and no independent human review is claimed.",
+ "status": "FINITE_OWNER_AND_MOMENT_CANDIDATE_OUTER_PENDING",
+ "width_inventory": {
+  "4": 26,
+  "5": 2,
+  "6": 16,
+  "7": 52,
+  "8": 99,
+  "9": 213
+ },
+ "words": "PR3015 edits/FINAL-WORD-base.json.gz and FINAL-WORD-extra.json.gz (vendored as data/R1-WORD-*.json.gz), unchanged",
+ "zero_buffers": 91
+}
diff --git a/research/rank7-r1-t740-cap/README.md b/research/rank7-r1-t740-cap/README.md
@@ -0,0 +1,177 @@
+# R1 words at their packing cap (T740): conditional κ = 0.001729152756004330934535 − 10⁻⁴⁶ − 10⁻⁷⁰
+
+```text
+Status / UTC update / priority revision: DONE (moment nomination; see "Scope") · 2026-10-16 · priority revision 6 · base main d8e0f83f
+Concrete question and changed premise: PR3015's R1 restore edit keeps every helper width, window and the 91-buffer
+    namespace of PR2979, and its fixed-capacity cap certificate has the same optimal basis as PR2979's. So an allocation
+    that attains PR2979's cap also attains R1's. Can that be done literally?
+Established result and evidence status: yes. A literal T740 allocation is ten lockstep copies of PR2979's T74 fill
+    (which PR3015 uses unchanged) plus 332 whole-role relocations. Its idle histogram is exactly 740 x the optimal basic
+    idle profile, and with the R1 words its exact root equals the R1 cap. Gains: about 1.5520e-9 over PR3021's released
+    headline (R1 + PR3018 packing) and 1.7825e-9 over PR3015. Packing only; author-checked (AI-assisted); not
+    maintainer-reviewed.
+Public artifact URL + exact commit: this package (see the PR).
+Next useful task: the complete T740 admission/outer release. Further gains need more word edits (R2, ...).
+Active claim / expiry: released.
+```
+
+## Result
+
+| | κ (headline; each − 10⁻⁴⁶ − 10⁻⁷⁰) | words | T | packing |
+|---|---|---|---:|---|
+| PR #3015 (Jeremy Kothe) | 0.001729150973534224428695 | PR2979 + R1 | 74 | PR2979's fill, 1.78e-9 below the R1 cap |
+| PR #3021 (Nick Petalas) | 0.001729151203983018442998 | PR2979 + R1 | 74 | PR3018's journal, 1.55e-9 below |
+| **this package** | **0.001729152756004330934535** | **PR2979 + R1** | **740** | **attains the R1 cap** |
+
+- Exact gains are `1552021312491537/10^24` over PR3021 and `178247010650584/10^23` over PR3015 (CLAIM.json).
+- The words are PR3015's two R1 words, used unchanged (`data/R1-WORD-*.json.gz`; provenance in VENDOR.json).
+  - Local profile per call: {1:2443, 2:707, 3:232, 4:108, 5:18, 8:8}.
+  - Width inventory: {4:26, 5:2, 6:16, 7:52, 8:99, 9:213}.
+  - 91 private buffers.
+- The allocation has 740 replicas, 6,660 calls and 2,717,280 helper roles.
+- Complete paid profile: {1:16441320, 2:4708620, 3:1594737, 4:719280, 5:137085, 8:53280, 48:281903}.
+- Physical stock: 663,040 live + 408 helpers + 8 centres + 91 buffers + 2 guards = **663,549**.
+- The exact root bracket [0.001729152756004330934535, 0.0017291527560043309345351] is exactly PR3015's certified R1
+  cap.
+
+**Concurrent work.** Nick Petalas' [PR3024](https://github.com/CrocSwap/integer-mult-bounds/pull/3024) attains the same
+kind of cap at T740 for PR2979's words without R1. It uses a 550-row journal on PR3018's fill. This package's 332
+relocations were found independently from PR2979's own fill; they reach the same idle histogram. PR3024 records the R1
+composition as a separate obligation, and that composition is what this package adds.
+
+## Why T740 and why it composes with R1
+
+**Why T740.** On the optimal face (12 tight patterns), exact row multipliers fix three counts per replica:
+
+| pattern | count per replica |
+|---|---|
+| [9,9,9]@32 | 93/4 |
+| [9,9,9,9,9]@48 | 1341/20 |
+| []@48 | 7619/20 |
+
+- Since 1341 = 3²·149 is prime to 20, an integral optimum needs 20 | T.
+- T740 is the first multiple of PR2917's T74 order with that property.
+- At T74 the best integral mix is 1.55e-9 below the cap, which is where PR3021 sits.
+
+**Why it composes with R1.** R1 changes the local profile by −2H₁ + H₂ per call and nothing else that the allocation
+reads: helper widths, entrances, endpoints, windows, order, dispatch and buffers are all unchanged. PR3015's R1 cap
+certificate has the same 10-pattern basis and the same idle profile as PR2979's. `verify.py` re-derives:
+
+- the R1 providers from the words;
+- the face counts and the optimality of every literal bin pattern;
+- the equality of the root and the R1 cap.
+
+## Construction
+
+`lift.py` defines the allocation completely, in two steps.
+
+1. **Lift.** PR2979's literal T74 ORDER, DISPATCH and FILL (PR3015 uses them unchanged) are copied ten times in
+   lockstep, in Dugongue's PR2969 convention. With j = 0, …, 9:
+   - call (s, r) at time t → call (s, r + 74j) at time 10t + j;
+   - role (t, h, o, c, p, z) → role (10t + j, h, o + 74j, c, p, z).
+2. **Relocate.** The roles in `allocation/RELOCATIONS.json` move to their listed bins, and the 263 touched bins are
+   repacked as compact prefixes.
+
+### What the relocations change
+
+    60 x [4,8,9,9,9,9]@48  ->  57 x [9,9,9,9,9]@48 + 3 x []@48
+    75 x [9,9,9]@32        ->  60 x [4,9,9,9]@32 + 15 x [8,8,8,8]@32
+
+The displaced 4s and 8s are late, while almost all [9,9,9] bins close early. So each one travels along a chain of
+same-width exchanges (`construction/plan_t740.py`, seeded greedy; `construction/CHAINS.json`):
+
+1. It enters an unlocked bin whose window contains its call.
+2. It evicts a same-width role of an earlier call, which moves on in turn.
+
+Every intermediate bin keeps its pattern. In all, 332 roles move: 150 in step A, 45 in B, 80 in C, 12 in D and 45 in E.
+The search reads only widths, windows and families, so the same relocations serve PR2979's words and R1's.
+
+## Verification
+
+Both registered commands use the standard library only, under Python 3.12.
+
+**`verify.py` (about 70 seconds)** checks:
+
+- **Pins.**
+  - Live pins of the PR2979 inputs on main.
+  - Vendored-file pins (VENDOR.json): exact PR heads, Git blob IDs and SHA-256.
+- **Providers.** The R1 words re-derive the providers:
+  - widths identical to PR2979's registry;
+  - local profile = R1;
+  - 91 buffers.
+- **Allocation.** The deterministic T740 rebuild has pinned canonical payload SHA-256.
+- **Owner engine.** PR2979's vendored engine checks all 2,717,280 roles:
+  - windows, same-call families, capacities, masks and compact prefixes;
+  - the complete paid profile and the rank mass.
+- **Physical stock.** 663,549.
+- **Cap and optimal face.**
+  - PR3015's R1 cap certificate is re-verified with PR2979's `relax_exact.py`.
+  - Exact fixed face counts.
+  - The idle histogram is 740 × the basic optimum.
+  - Every bin pattern is tight.
+  - The root bracket equals the cap bracket.
+- **Moment.** The dyadic moment auditor is re-pinned to PR3015's headline and shows a strict gain. An exact
+  comparison also shows a strict gain over PR3021.
+- **Claim.** CLAIM.json is bound to the results.
+
+**`controls.py` (about four minutes)** checks seven controls:
+
+| control | result |
+|---|---|
+| bare lift | valid, but its root is PR3015's (below the cap) |
+| dropped chain link | rejected by the engine |
+| window violation | rejected by the engine |
+| family collision | rejected by the engine |
+| stale profile | rejected by the engine |
+| tampered relocations | the payload pin fails |
+| wrong moment baseline | rejected |
+
+```sh
+python3 scripts/research_package.py --package research/rank7-r1-t740-cap --check-only
+python3.12 scripts/research_package.py --package research/rank7-r1-t740-cap
+python3 -B research/rank7-r1-t740-cap/construction/plan_t740.py --out /tmp/t740-plan   # provenance: regenerates RELOCATIONS.json
+```
+
+## Scope
+
+- **Packing composed with R1.** No word, gate, frame, scalar or cleanup changes are made beyond PR3015's declared R1
+  edit.
+- **Inherited claims.** PR3015's (and PR2979's) word-level, native and release claims are inherited unchanged. So are
+  all of PR2979's inherited all-size, compiler, hosting, precision and analytic hypotheses.
+- **Review status.** PR2979 is an archived source package and PR3015/PR3021 are open. None is scientifically admitted,
+  and the independent audits they need apply here too.
+- **Open finite obligation.** The complete fresh-native admission and modern outer release have not yet been rerun for
+  the T740 allocation. PR3015's and PR3021's releases are at T74. This is a moment nomination with literal T740 owners,
+  the same status as PR3024 and PR2969.
+- **What is not claimed.** No selected-frontier change, no new mechanism, and no independent human review.
+- **Optimality.** The R1 cap is the optimum of the fixed-capacity relaxation, so no allocation of the R1 words in this
+  model can do better.
+
+## Open next questions
+
+1. **Complete T740 release.**
+   - First step: rerun PR3015's reproduce.py chain with the T740 ORDER/DISPATCH/FILL that `verify.py --work DIR` writes.
+     T-dependent physical and prefix constants come from the order.
+   - Blocked by: replay time and memory at ten times PR3015's 271,728 roles.
+2. **More declared edits.**
+   - Why it matters: packing is exhausted for these words, so only a changed local profile raises κ further.
+   - First step: price candidate edits with PR3015's exhaustive 2-4-gate restore-block search. Each new local profile
+     with the same basis inherits this allocation's cap attainment.
+
+These are recorded as structured `open_questions` in TASK.json.
+
+## Credit
+
+- **Jeremy Kothe (PR #3015):** the declared R1 edit, R1 words and R1 cap certificate.
+- **Marius-Paul Dumitrean (PR #2979):** the words under R1, the literal T74 fill lifted here and the checkers.
+- **Nick Petalas (PR #3018, #3021, #3022, #3024):** the integrality analysis lineage and the concurrent T740 cap for
+  PR2979's words.
+- **Dugongue (PR #2969):** the lockstep T-lift convention and cross-copy moves.
+- **themilkman216:** the observation about the T at which these caps become integral.
+- **Mason Hale (PR #2917):** chronology and owner/moment checks.
+- **Tekalign:** the independent owner checker.
+- **Jeremy Diviney (PR #2908):** the moment auditor arithmetic.
+- **Yiyang Lu:** the flow model.
+- **icekylinx:** the exact auditor.
+
+New work by Rohan Arun, prepared with substantial AI assistance (Anthropic Claude). Apache-2.0. Retained notices apply.
diff --git a/research/rank7-r1-t740-cap/SOURCES.json b/research/rank7-r1-t740-cap/SOURCES.json
@@ -0,0 +1,96 @@
+{
+  "schema_version": 1,
+  "base_commit": "d8e0f83f00191df62174b641e9fff055c357f641",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/rank7-joint-endframes/CLAIM.json",
+      "usage": "input",
+      "git_blob": "8ec2ce4e440c0b584ee982c551e526a011c9324c",
+      "sha256": "436e0d297cbdf195445d3697d496ac675e95210b79e70ce602a6a6667a4c6ce9"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/DISPATCH.json",
+      "usage": "input",
+      "git_blob": "9544fee4e6ee1f202cee22bfaeaf229ee3b2515e",
+      "sha256": "65e435bd68e97243c003677271b88485a29d4d0fae2dff1354100404a145e74c"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/FILL.json.gz",
+      "usage": "input",
+      "git_blob": "8d4fd644d60ba2c98106dc770803b647afdf0963",
+      "sha256": "c2136be9434cda511a1a8a44d98f9f72ea5376830a8426674253b4ee351c4c39"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/ORDER.json",
+      "usage": "input",
+      "git_blob": "7409c4df1092241729739596c7648bda504e6aad",
+      "sha256": "519d1d5dbb997a219fcfcad34c0b03e7838bef24199784d7e964fa10090f66e1"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/PROVIDERS.json",
+      "usage": "input",
+      "git_blob": "e1d99e12cfa3ac8375984d85c0b040dc36ea1312",
+      "sha256": "f2edb7922f3029a5d607437e737775ff3371bca49ebab62e5e2495d16d3bf573"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/vendor/PLAN.json.gz",
+      "usage": "input",
+      "git_blob": "7a1db55a4d53ee99280be4f3ab7d29f7b2033adf",
+      "sha256": "d566cfb87b71639d4138e1f89b543e2dd6a7772070226a2a14c799a6835bf21e"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/vendor/TEMPLATE.json.gz",
+      "usage": "input",
+      "git_blob": "114808c32b1799da53524bdc25cea67efb60009e",
+      "sha256": "21297b20f1f2dc76eb9dd8124416068c3cb096bab9df17fa4da19bc68dfe082e"
+    },
+    {
+      "path": "research/rank7-joint-endframes/allocation/vendor/independent_owners.py",
+      "usage": "input",
+      "git_blob": "16013201b19054f7e45398ecf10af74c9c6a9114",
+      "sha256": "77a848729159e271ba64f403dfd59dddb13044b17b2950fa56700f442e2fe461"
+    },
+    {
+      "path": "research/rank7-joint-endframes/tools/audit_allocation_moment.py",
+      "usage": "input",
+      "git_blob": "623ae14fbeaacf2f1b2302ae23b40c52586b7342",
+      "sha256": "c632527ecf092f013862660dbd7f52b599a50b321a8aa88ddc45af0547e79cb5"
+    },
+    {
+      "path": "research/rank7-joint-endframes/tools/relax.py",
+      "usage": "input",
+      "git_blob": "97dfe35c434283ca6e83818e61aed65740522f50",
+      "sha256": "0fed1c56b50a2e6d44deeeb6db0f1c7e1f71b86d1747130e1a9202216249a728"
+    },
+    {
+      "path": "research/rank7-joint-endframes/tools/relax_exact.py",
+      "usage": "input",
+      "git_blob": "0be97756bb9d8b072c962c147d4b4d3b3122f884",
+      "sha256": "baf1855d6095c951a90d19a0d8094bd33290519940b732e2b758c5647830f044"
+    },
+    {
+      "path": "research/rank7-joint-endframes/tools/root.py",
+      "usage": "input",
+      "git_blob": "bd6dc2eb4e5013625d17cc08461b67622bcfed69",
+      "sha256": "55f5aca39b5ee2fe4687ac8863e6c24569010f27a332883c069b8e14b8fe09e3"
+    },
+    {
+      "path": "research/rank7-joint-endframes/vendor/preload_move.py",
+      "usage": "input",
+      "git_blob": "55b435035bbc439a9d55e8abd125993208cef8d9",
+      "sha256": "c8fda4674644fcd5cbf39b6645ce148a21daa90a998b678cbe51f970b78b0044"
+    }
+  ]
+}