Research: q<=4 barrier for a noncentral UT4 root pair

research/ut4-noncentral-commutator-seedbfe59/BASELINE.json

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+{
+  "command": "python3 -B scripts/research_baseline.py",
+  "status": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "selected_kappa": "836330098538459/500000000000000000",
+  "review": "docs/research/community-batch1099-review.md",
+  "certificate": "research/community-frontier1094-audit/ASSEMBLY.json",
+  "full_replay": "make frontier1094-verify",
+  "bit_saving": "1733427039589844234749/1000000000000000000000000",
+  "complex_saving": "67018507072324140917/40000000000000000000000",
+  "binding_primitive": "complex",
+  "ideal_balanced_cap": "67018507072324140917/40067018507072324140917",
+  "target_kappa": "1/100",
+  "target_primitive_under_retained_transfer": "1/99",
+  "required_bit_multiplier": 5.827190801985042,
+  "required_complex_multiplier": 6.028788489788009,
+  "scope": "Selected source bindings and arithmetic only. The a/(1+a) transfer is a planning model under retained hypotheses, not a universal requirement or supplier verification."
+}

research/ut4-noncentral-commutator-seedbfe59/README.md

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+# Four-call barrier for a noncentral UT4 root pair
+
+**Exact scoped negative result; no new supplier or kappa.** In the full formal
+group algebra F2[UT4(F2)], with paid labels restricted to the three adjacent
+transvections, the endpoint
+
+~~~text
+diag([xyxy], [yzyz], [1], [1])
+~~~
+
+has no realization in at most four calls
+D_N(g) = I + ([g]+[1])N, even when every binary rank-one idempotent
+N in M4(F2) is allowed on every call. The exact search covers all ordered
+words through length four. A direct eight-call word realizes the endpoint, so
+the minimum in this fixed formal model is between five and eight. Lengths
+five through seven were not searched.
+
+## Model and cost screen
+
+Encode a group element by its six strict-upper entries in the order
+(12,13,14,23,24,34). Let x=I+E12, y=I+E23, and z=I+E34; these involutions
+generate the 64-element group UT4(F2). Their commutators
+u=xyxy=I+E13 and v=yzyz=I+E24 are noncentral. The target is
+diag([u],[v],[1],[1]) over the formal group algebra.
+Indeed, u and z have commutator I+E14, as do x and v; these nonidentity
+commutators are checked in the executable structural validation.
+
+There are 120 rank-one idempotents: each is ab^T for nonzero a,b in F2^4
+with b^T a=1. This gives 360 allowed call matrices. Because each paid label
+is an involution and N^2=N, every call is itself an involution:
+(I+([g]+[1])N)^2=I.
+
+The direct construction uses x,y,x,y on register one and y,z,y,z on
+register two, for eight unit-priced formal calls. In the natural
+16-dimensional representation of the four-register group-algebra matrix, the
+target displacement has rank two. Each allowed call has displacement rank
+one, since it is a tensor product of a rank-one idempotent and a rank-one
+generator displacement. Thus one call is excluded by rank, while the
+optimistic rank floor is two. This formal count assigns one unit per call.
+Source, helper, recursive, hosting, restoration, precision, and interface
+charges do not exist in this abstract identity model; treating them as zero
+would only make a hypothetical short word cheaper. A formal identity alone
+would still need a realized supplier and complete interface bill before it
+could affect the selected conditional bound.
+
+## Exact search
+
+For each of the 360 allowed factors, form the full 4-by-4 matrix over
+F2[G], where a group-algebra coefficient vector is a 64-bit mask. The exact
+ordered two-factor table contains 129,600 words and 87,016 distinct matrix
+products (42,584 ordered collisions).
+
+The exhaustive split is:
+
+1. Length two: test the target against the complete pair table.
+2. Length three: for each of 360 final calls C, test target*C against the
+   pair table, using C^-1=C.
+3. Length four: for each ordered suffix (C,D) among all 129,600 pairs, test
+   target*D*C against the pair table, using (CD)^-1=DC.
+
+The diagonal target makes each right-side multiplication an exact row
+translation in the group algebra. The script nevertheless stores and compares
+all 16 matrix-entry coefficient masks; it does not project to a representation
+or quotient. The certificate records the full pair-table digest and the
+complete candidate counts. It also checks the direct eight-call construction,
+an omitted-call corruption, and a wrong-label corruption.
+
+## Reproduce
+
+Python 3.10+ and the standard library suffice:
+
+~~~sh
+python3 -I -B research/ut4-noncentral-commutator-seedbfe59/verify.py
+~~~
+
+Expected output:
+
+~~~text
+PASS_UT4_NONCENTRAL_Q4 factors=360 pair_words=129600 distinct_pairs=87016 q3=360 q4=129600 outcome=no_word_of_length_at_most_4 baseline=complex
+~~~
+
+To regenerate certificate.json explicitly:
+
+~~~sh
+python3 -I -B research/ut4-noncentral-commutator-seedbfe59/search.py --write-certificate
+~~~
+
+search.py constructs the group table, all legal factors, full group-algebra
+matrix products, and the exhaustive meet-in-the-middle search.
+verify.py recomputes the result and requires exact certificate equality.
+certificate.json pins the seed, base, selected-record digest, table counts,
+rank screen, outcome, and direct/corruption controls. Before searching, the
+script cross-checks every one of the 4,096 ordered UT4 group products against
+ordinary 4-by-4 binary matrix multiplication. The verifier also reruns the
+selected baseline binding control and requires exact equality with BASELINE.json.
+
+## Scope, overlap, and attribution
+
+This refines catalogue task bit-new-register-model to one four-register
+nonabelian group, one fixed three-label alphabet, one noncentral two-root
+target, and the predeclared q<=4 search. It does not settle q=5..8, arbitrary
+nonabelian groups, arbitrary target tuples, or a physical supplier. Closed
+PR #1333 established a rank-sum minimum with the full rank-one affine-label
+alphabet; this result restricts to three fixed generators and uses a
+different noncentral target. Open UT4 PRs #1662, #1680, and #1690 concern
+central targets. Active Discussion #1738 concerns a three-register UT3
+central pair; #1740 concerns an S4 pair; #1743 concerns helper-star costs.
+These are nearby search families, not the target and endpoint proved here.
+
+The live overlap refresh paginated all 434 open PRs and all 392 Discussions
+with no warnings. Direct checks of active claims #1738, #1740, and #1743 found
+different groups, targets, and mechanisms. The claim was kept local during
+the initial GraphQL rate limit, then posted after reset as
+[Discussion #1759](https://github.com/CrocSwap/integer-mult-bounds/discussions/1759).
+The finding and handoff comment links this package's [PR #1761](https://github.com/CrocSwap/integer-mult-bounds/pull/1761).
+TASK.json records exact refresh timestamps, active heads, and limits.
+
+The current selected record is a maintainer-reviewed conditional finite
+construction, with kappa 836330098538459/500000000000000000. Its review is
+docs/research/community-batch1099-review.md, its selected source manifest is
+research/community-frontier1094-audit/SELECTED-SOURCE.json, and its assembly
+certificate is research/community-frontier1094-audit/ASSEMBLY.json (SHA-256
+1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045). The
+selected record names the integration receipt and reproduces with
+make frontier1094-verify. Exact base-commit Git blob and SHA-256 pins for
+these artifacts and the selected record are in SOURCES.json. No selected
+result, upstream source, or pre-existing certificate was changed; this
+package adds its own certificate.json.
+
+The task's baseline positive control returned
+PASS_RESEARCH_BASELINE_BINDINGS_ONLY: the complex primitive remains binding.
+Under the retained planning transfer, the baseline arithmetic reports a
+required bit multiplier of 5.827190801985042 for the target kappa 1/100.
+This is source-binding and arithmetic evidence only, not a full supplier
+replay or a universal transfer bound. The exact output is preserved in
+BASELINE.json, and the baseline script is pinned in SOURCES.json. Consequently
+this finite bit-model result does not change the selected frontier.
+
+Prepared by Cinna Davis (@1337isnot1337) with substantial OpenAI Codex
+assistance in mathematics, implementation, verification, and writing.
+
+## Open next questions
+
+Does this fixed endpoint have a word of length five, six, or seven in the same
+full group-algebra alphabet? Start from the q<=4 pair table and use a
+three-factor split, with a cost screen before retaining any new construction.
+Even a positive formal word would need a complete supplier, recursive-moment,
+and assembly argument before any kappa claim.

research/ut4-noncentral-commutator-seedbfe59/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "research/bit-three-register-lower-bound/README.md",
+      "usage": "input",
+      "git_blob": "25e41c370c101a33a6deb978de08b03ed7539b79",
+      "sha256": "c12b33ef388deece0cbf891505a2ce2c0779a0c3a6cf1f0f7f7039bab59cbebd"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "provenance",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "provenance",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "provenance",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "docs/research/community-batch1099-integration.json",
+      "usage": "provenance",
+      "git_blob": "9d60e86c0e494d17bec5e28eda41b275c9ede64e",
+      "sha256": "684c50d1ac69863e1fcc0a34bf76c54bac576f6bbde79b3c67d8432a495c8c1c"
+    },
+    {
+      "path": "Makefile",
+      "usage": "provenance",
+      "git_blob": "448f13b58fbe3492e2ea6b2005c8822aefe2765c",
+      "sha256": "029076ba6558f4c97d59d508baeb47847e0fad3545ed1fd22943ab9d6c84c2b9"
+    },
+    {
+      "path": "scripts/research_baseline.py",
+      "usage": "input",
+      "git_blob": "6b607b36eefd65718120ade3c8187681f11bf879",
+      "sha256": "59d9b04dc33e1bb3f81506d187243435de4902138a038b935e36ebaa5f37db67"
+    }
+  ]
+}