Research: high-arity connector toggle cost floor

docs/ci-verification.md

@@ -2,6 +2,18 @@
 
 ## Current selected-family replay
 
+Resolve the [selected record](../certificates/selected-result.json) before
+running a historical checkpoint. The current cold replay is
+`make frontier4359-verify`, using the
+[proof review](research/pr-intake-evidence/pr4359/PROOF-REVIEW.md),
+[validation](research/pr-intake-evidence/pr4359/VALIDATION.json) and
+[source manifest](../certificates/frontier4359-source.json).
+The [frontier4359 workflow](../.github/workflows/frontier4359.yml) provides
+hosted replay evidence. Metadata and source-pin checks are separate from
+operator replay and do not discharge the inherited theorem assumptions.
+
+## Historical frontier1094 replay
+
 Start with the [selected record](../certificates/selected-result.json). Its
 `reproduce`, `review`, `source_manifest` and `validation` fields identify the
 replay command, scoped assessment, source bindings and integration evidence.

docs/research/current-status.md

@@ -1,12 +1,14 @@
 # Current reviewed status
 
-Selected conditional κ: **0.0023 = 23/10000**, zxv’s #4359 productive-helper construction, using Jacob Alan Sussman’s mixed-target recursion and the credited cumulative native/outer lineage.
+Selected conditional κ: **0.0023** (= 23/10000), zxv’s #4359 productive-helper construction, using Jacob Alan Sussman’s mixed-target recursion and the credited cumulative native/outer lineage.
 
 [Selected record](../../certificates/selected-result.json) · [Proof review](pr-intake-evidence/pr4359/PROOF-REVIEW.md) · [Integration receipt](pr-intake-evidence/pr4359/INTEGRATION.json) · [Contributors](community-productive4359-contributors.md).
 
 Cold replay: `make frontier4359-verify`. The fresh 23-stage replay and 59 normalized artifacts pass; the changed interfaces have an AI-assisted proof review under retained analytic/native/fixed-tape hypotheses. Winning uniform templates are not materialized. This is not a complete formal or independent human verification. #4404 independently checks a narrower arithmetic/flow scope.
 
-## Historical selected status through #1094
+All sections below are a historical snapshot; the selected-result.json record above is authoritative.
+
+## Historical checkpoint through #1094
 
 Historical selected conditional κ: **0.001672660197076918**, Rohan Arun's #1094 on Jeremy Kothe's #1087 packing and the cumulative k8/slot/T16 construction. The bit supplier remains #589 on #555.
 

research/d8-mixed-target-seeda97c8e/SOURCES.json

@@ -4,25 +4,25 @@
   "files": [
     {
       "path": "autoresearch.md",
-      "usage": "input",
+      "usage": "provenance",
       "git_blob": "8d5403e0f10f68a61d0cb9294fa2e83e872d1d82",
       "sha256": "215b3222f6a943edc69422a28ac21ea1f397ddf54770f1d87ac4195385d73699"
     },
     {
       "path": "certificates/selected-result.json",
-      "usage": "input",
+      "usage": "provenance",
       "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
       "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
     },
     {
       "path": "docs/research/agent-task-catalog.json",
-      "usage": "input",
+      "usage": "provenance",
       "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
       "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
     },
     {
       "path": "docs/research/current-status.md",
-      "usage": "input",
+      "usage": "provenance",
       "git_blob": "d73b4ff9da41f044e793422db4318be17e234a6c",
       "sha256": "adb54af005eb11edaf84d782fe729338680ff908f6d4fbf39ba1062d2e7d8a77"
     },

research/d8-mixed-target-seeda97c8e/VERIFY.json

@@ -4,10 +4,6 @@
   "python": "3.12",
   "requirements": [],
   "inputs": [
-    "autoresearch.md",
-    "certificates/selected-result.json",
-    "docs/research/agent-task-catalog.json",
-    "docs/research/current-status.md",
     "research/bit-three-register-lower-bound/README.md",
     "research/d8-mixed-target-seeda97c8e/verify.py"
   ],

research/historical-1042-link/RESULT.json

@@ -2,8 +2,9 @@
   "historical_href": "../../certificates/selected-result.json",
   "checkpoint": "certificates/checkpoints/frontier1042.json",
   "historical_kappa": "1672188119205421/1000000000000000000",
-  "live_kappa": "836330098538459/500000000000000000",
+  "live_kappa": "23/10000",
   "same_href": true,
   "kappas_differ": true,
-  "no_new_kappa": true
+  "no_new_kappa": true,
+  "intermediate_checkpoint": "certificates/checkpoints/frontier1094.json"
 }

research/historical-1042-link/SOURCES.json

@@ -11,20 +11,29 @@
     {
       "path": "docs/research/current-status.md",
       "usage": "input",
-      "git_blob": "4e0a8b7cdacc1d9680f7f81d03624118bb20416d",
-      "sha256": "00a08925c51b26d168b8099ddeed659ac4381f7e04f3c7fd0e9de8e71f206823"
+      "git_blob": "532af84312cbfeb39a1e4659d8398deb560f1e4a",
+      "sha256": "54c843968d0b178369a7e56161be44c4692935c3f46ce261678aeedadb309891",
+      "pin": "head:41beb2e687da0ad1e06e2fcec00eb377560ca2db:docs/research/current-status.md"
     },
     {
       "path": "certificates/selected-result.json",
       "usage": "input",
-      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
-      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9",
+      "pin": "head:41beb2e687da0ad1e06e2fcec00eb377560ca2db:certificates/selected-result.json"
     },
     {
       "path": "certificates/checkpoints/frontier1042.json",
       "usage": "input",
       "git_blob": "f451ced6626a1f03aba6915ce23a136b01ebe51d",
       "sha256": "207d1f49f113c585024b9d3c17eadbb8b1be17c1321e3d190aabe2f18ce1c5b7"
+    },
+    {
+      "path": "certificates/checkpoints/frontier1094.json",
+      "usage": "input",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+      "pin": "head:41beb2e687da0ad1e06e2fcec00eb377560ca2db:certificates/checkpoints/frontier1094.json"
     }
   ]
 }

research/historical-1042-link/VERIFY.json

@@ -6,16 +6,25 @@
   "inputs": [
     "docs/research/current-status.md",
     "certificates/selected-result.json",
-    "certificates/checkpoints/frontier1042.json"
+    "certificates/checkpoints/frontier1042.json",
+    "certificates/checkpoints/frontier1094.json"
   ],
   "commands": [
     {
-      "argv": ["{python}", "-B", "{package}/verify.py"],
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
       "timeout_seconds": 30,
       "checks": "Compare the #1042 status link with the live record and the frontier1042 checkpoint."
     },
     {
-      "argv": ["{python}", "-B", "{package}/test_verify.py"],
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/test_verify.py"
+      ],
       "timeout_seconds": 30,
       "checks": "Check the printed decimal and the historical section cut."
     }

research/historical-1042-link/verify.py

@@ -2,7 +2,8 @@
 """Separate the #1042 status link from the live selected record.
 
 The historical section states the #1042 kappa and links to the live record.
-The matching checkpoint is the selected record's previous_record field.
+The matching checkpoint is reached through the selected record's historical
+previous_record chain (currently via frontier1094).
 """
 
 import hashlib
@@ -16,6 +17,7 @@
 MARKER = "PASS_HISTORICAL_1042_LINK kappas_differ=true no_new_kappa"
 HREF = "../../certificates/selected-result.json"
 CHECKPOINT = "certificates/checkpoints/frontier1042.json"
+INTERMEDIATE = "certificates/checkpoints/frontier1094.json"
 SHOWN = "0.001672188119205421"
 
 
@@ -43,6 +45,7 @@ def authenticate():
         "docs/research/current-status.md",
         "certificates/selected-result.json",
         "certificates/checkpoints/frontier1042.json",
+        "certificates/checkpoints/frontier1094.json",
     }
     require(set(pinned) == expected, "input inventory changed")
     for path, row in pinned.items():
@@ -64,12 +67,15 @@ def evaluate():
     require(HREF in section, "the historical section lost the live-record link")
     require(SHOWN in section, "the historical kappa sentence moved")
     require(Fraction(SHOWN) == Fraction(old["kappa"]), "the checkpoint kappa left the printed decimal")
-    require(live["previous_record"] == CHECKPOINT, "the previous-record pointer moved")
+    require(live["previous_record"] == INTERMEDIATE, "the live previous-record pointer moved")
+    intermediate = json.loads((ROOT / INTERMEDIATE).read_text(encoding="utf-8"))
+    require(intermediate["previous_record"] == CHECKPOINT, "the checkpoint ancestry link moved")
     require(Fraction(live["kappa"]) != Fraction(old["kappa"]), "the live and historical kappas agree")
     require(text.index(HREF) < text.index("# Historical status through #1042"), "the live section lost its record link")
     return {
         "historical_href": HREF,
         "checkpoint": CHECKPOINT,
+        "intermediate_checkpoint": INTERMEDIATE,
         "historical_kappa": old["kappa"],
         "live_kappa": live["kappa"],
         "same_href": True,

research/six-source-higher-arity-toggle-floor-seedd45c17a9/EXTERNAL-PINS.json

@@ -0,0 +1,32 @@
+{
+  "academic_source": {
+    "author": "Volodymyr Gavrylkiv",
+    "title": "Bases in Dihedral and Boolean Groups",
+    "publication": "Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.1",
+    "url": "https://cs.uwaterloo.ca/journals/JIS/VOL20/Gavrylkiv/gav4.pdf",
+    "locator": "Introduction definition of basis; Table 3, page 7: r[(C2)^6]=14",
+    "downloaded_sha256": "3418224e0e4427bba18ef907f14000069de9aed7400dfc240dafba1cf29f6584"
+  },
+  "duo_witness": {
+    "pr": 2128,
+    "head": "697369a971eed02c0fbcd75ebefec9f59945f08e",
+    "path": "research/connector-six-source-duo/CERTIFICATE.json",
+    "sha256": "e611d16e100eb9fd9861d9c872c21ae7d1ac280880efe783d2407151dd1e6bb2"
+  },
+  "closest_predecessors": [
+    {
+      "pr": 2156,
+      "head": "05b4672fa4450ef50b06315e3a0893d4ba43d42d",
+      "scope": "DUO-domain six-source full-batch optimum and exact 142-add floor"
+    },
+    {
+      "pr": 2146,
+      "head": "302b0e87d9328fabc2a670248afb357bbb44041b",
+      "scope": "one fixed six-source single-TRIO call profile"
+    },
+    {
+      "pr": 1518,
+      "scope": "atomic triple-sum connector budget result"
+    }
+  ]
+}

research/six-source-higher-arity-toggle-floor-seedd45c17a9/README.md

@@ -0,0 +1,65 @@
+# Six-source additive-cost floor for high-arity connector toggles
+
+**Scoped negative result; no supplier or new κ.** Extending the six-source universal dirty-connector alphabet with one source-hyperedge operation for every subset of size four through six cannot improve its full 63-mask batch when every source tap is charged. The exact optimistic full-batch Pareto point remains 140 total queries and 142 total additions. The known attaining word uses 14 source calls and 16 source additions. The argument closes the entire arity-4..6 family, including schedules that use more than two output readouts, without enumerating its words.
+
+## Assignment and changed scope
+
+- Cycle 43 seed: `d45c17a9abef1858e6b314d2bb66be28`; track `recursive-interfaces`; task `connector-alphabet-change`; priority revision 4.
+- Base: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+- Selected record SHA-256: `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- Catalogue SHA-256: `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`.
+- The broad card is refined to a new parametrized operation family, `H_S: z += sum(B_j x_j)` for `4 <= |S| <= 6`, on the fixed six-source, all-63-output instance. It keeps read-only sources, arbitrary dirty connector contents, complete restoration, and separate output readouts. Its most favorable full price is one dispatch plus `|S|` charged source additions; any routing, frame, or implementation cost adds to this bill.
+
+This is a changed connector contract relative to the selector's broader starting suggestions. Earlier work covers the pair-source DUO witness (#2128), a fixed one-TRIO profile (#2146), and the DUO full-batch optimum (#2156). This result derives an all-arity floor for source subsets of size four through six. It neither claims those earlier constructions nor independently replays their searches.
+
+## Exact lower bound
+
+A source word is a closed walk `q_0,...,q_k` in `F2^6`; a hyperedge call on `S` changes `q` by the indicator vector of `S`. Let `P` be the distinct prefix set and let `D(P)={a+b:a,b in P}`. A mask in `D(P)` can be realized with two output readouts. The exact group basis result `r[(C2)^6]=14` (Volodymyr Gavrylkiv, *Bases in Dihedral and Boolean Groups*, Journal of Integer Sequences 20 (2017), Article 17.8.1, Table 3) implies that a prefix set of size at most 13 cannot cover every mask.
+
+The lower bound must also allow outputs with more than two readouts. Each output needs an even number of `A_i` reads to cancel its arbitrary dirty-connector contribution. A nonzero output needs at least two reads; if its mask is missing from `D(P)`, two reads cannot realize it, so it needs at least four. If `M` of the 63 nonzero masks are missing, the total output-read count is `R >= 126 + 2M`.
+
+Now split by the number `k` of source-call dispatches. If `k >= 14`, then `k+R >= 140`. If `k` is 12 or 13, `|P| <= k <= 13`, so the basis-size result forces `M >= 1`; hence `k+R >= k+128 >= 140`. If `k <= 11`, at most `binom(k,2)` nonzero pair differences occur, so `M >= 63-binomial(k,2)`. Thus
+
+```text
+k + R >= k + 126 + 2(63 - binomial(k,2)) >= 153   for 0 <= k <= 11.
+```
+
+Therefore **every schedule, including those with four or more readouts on some outputs, has at least 140 total queries**. This case split depends on source-prefix geometry, not on whether one source call toggles one, two, or six sources.
+
+For the source-add charge, expand every hyperedge step `S` into `|S|` singleton toggles in any fixed order. The expanded path retains every original boundary, so all original output-readout placements remain valid. It is a singleton source word of length `W=sum |S|`; the pinned exact six-source result `f(6)=16` in `research/universal-connector-six-source` gives `W >= 16`. Also `R >= 126`, hence total additions `W+R >= 142`.
+
+The most favorable complete price is therefore
+
+```text
+queries Q >= 140
+source additions W >= 16
+output readouts R >= 126
+total additions W+R >= 142
+price >= 140 o + 142 p, for o,p >= 0
+```
+
+Here `o` is the fully charged dispatch/entry-exit cost and `p` the per-add cost. Any routing, frame, or implementation surcharge only increases the bill. The 14-step DUO witness from PR #2128 attains `Q=140`, `W=16`, and `R=126`, so it reaches `(140,142)`. Its witness is attributed to that PR and pinned in [EXTERNAL-PINS.json](EXTERNAL-PINS.json).
+
+## Verification
+
+The local verifier checks the cited six-source census receipt, the exact pinned 14-step witness and its all-mask difference coverage, expansion of every six-bit source subset, the complete 63-row dirty-connector schedule, reverse restoration, the all-k query-floor case split, and three failure controls. The six-source census prerequisite was also run locally under CPython 3.14.7; its output is recorded in `RESULT.json`.
+
+From repository root:
+
+```sh
+python3 -I -O -B research/six-source-higher-arity-toggle-floor-seedd45c17a9/verify.py
+python3 scripts/research_package.py --package research/six-source-higher-arity-toggle-floor-seedd45c17a9 --check-only
+python3.12 scripts/research_package.py --package research/six-source-higher-arity-toggle-floor-seedd45c17a9
+```
+
+## Limits
+
+This excludes an improvement only for source-hyperedge calls whose fully charged source-add price is at least the sum of their constituent source taps, with the six-source universal dirty-connector contract and 63 separate outputs fixed. It does not cover subadditive physical fusion, mutable sources, new output operations, different target families, supplier placement, or a multiplication exponent. The result is an abstract interface bound, not physical admission into the selected supplier.
+
+## Open next questions
+
+**Question:** Can a concrete selected-supplier interface realize any source-hyperedge of arity at least four with a total cost strictly below its constituent source reads while preserving arbitrary dirty-state restoration? First step: name one existing supplier frame, compile one such hyperedge into literal native operations, and charge its complete route/frame/precision bill. It matters because only a physically realizable subadditive charge escapes the floor proved here. The present six-source abstract connector has no admitted supplier realization, which is the blocker.
+
+## Credit
+
+The exact `f(6)=16` source-word result is predecessor work in `research/universal-connector-six-source`. The 14-call/16-add upper witness is Aaron Wang/Navinor's PR #2128. PR #2156 provides the closely related DUO-domain global optimum and credits Gavrylkiv for the group basis value. PR #2146 tests one fixed six-source TRIO profile. Prepared by Cinna Davis with substantial OpenAI Codex assistance; this package adds the all-arity source-expansion lower bound and full charged interpretation. No independent human review is claimed.

research/six-source-higher-arity-toggle-floor-seedd45c17a9/RESULT.json

@@ -0,0 +1,57 @@
+{
+  "status": "PASS_HIGH_ARITY_CONNECTOR_FLOOR",
+  "scope": "F2^6, full 63-row batch, source hyperedges |S|=4..6, sources read-only, arbitrary dirty connector, separate readouts, per-dispatch o and per-source-add p",
+  "lower_bounds": {
+    "total_queries": 140,
+    "source_adds": 16,
+    "separate_readouts": 126,
+    "total_additions": 142
+  },
+  "priced_lower_bound": "140 o + 142 p, for o,p >= 0, before nonnegative implementation surcharge",
+  "attaining_predecessor": {
+    "pr": 2128,
+    "head": "697369a971eed02c0fbcd75ebefec9f59945f08e",
+    "steps": [
+      2,
+      16,
+      8,
+      16,
+      1,
+      4,
+      48,
+      4,
+      3,
+      4,
+      32,
+      8,
+      4,
+      16
+    ],
+    "source_calls": 14,
+    "source_adds": 16,
+    "full_difference_coverage": true
+  },
+  "proof_inputs": {
+    "singleton_f6": "research/universal-connector-six-source/EXPECTED.json; source pins in SOURCES.json",
+    "difference_basis": "Gavrylkiv, JIS 20 (2017), Article 17.8.1, Table 3; exact PDF hash in EXTERNAL-PINS.json"
+  },
+  "dirty_replay_cases": 8064,
+  "limitations": [
+    "abstract connector model only",
+    "no physical supplier or selected-bound admission",
+    "no kappa change claimed",
+    "does not cover subadditive physical fusion or changed output alphabet"
+  ],
+  "predecessor_check": {
+    "command": "python3 -I -O -B research/universal-connector-six-source/verify.py",
+    "expected": "PASS_SIX_SOURCE_CONNECTOR f(6)=16 max14=59/64 max12=49/64 batch63=142 f(7) in [18,24]",
+    "observed": "PASS_SIX_SOURCE_CONNECTOR f(6)=16 max14=59/64 max12=49/64 batch63=142 f(7) in [18,24]",
+    "environment": "CPython 3.14.7"
+  },
+  "query_bound_case_split": {
+    "k_ge_14": "Q>=k+126>=140",
+    "k_12_or_13": "r((C2)^6)=14 implies at least one missing pair mask; Q>=k+128",
+    "k_le_11": "M>=63-binomial(k,2); Q>=k+126+2M>=153",
+    "allows_more_than_two_readouts": true
+  }
+}