Research: exact two-role GL3 coordinate-CNOT minimum for one target

research/gl3-coordinate-cnot-pair-seed1385e243/CLAIM.md

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+# Coordination claim
+
+Posted as [Discussion #1798](https://github.com/CrocSwap/integer-mult-bounds/discussions/1798). The exact claim is preserved below.
+
+```text
+CLAIM
+Question: In the exact formal model over F2[GL_3(F2)] with paid address labels restricted to the six coordinate CNOTs, can arbitrary free scalar GL_2(F2) mixing implement diag([A],[A]) in at most five paid calls, where A=T12 T13 T21? Here Tij=I+Eij, acting by row i <- row i + row j. The direct bill is six calls; rank(A-I)=2 gives the optimistic two-role floor four.
+Base commit: 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
+Track/seed/task: bit-mechanisms / 1385e243b2eaf19e9860b06a32a8c515 / bit-new-register-model; priority revision 3; scope key 7084092dd726e7ba1f2f.
+Predecessors: PR1333's rank-priced binary-affine model charges this A by displacement rank rather than restricted coordinate-CNOT word length. research/coordinate-cnot-register-bound proves the d=2 swap case but leaves a d=3 gap target open. This scope uses a three-dimensional target and all six coordinate CNOT labels; it does not use its D8 target or third-dirty-register claim #1738. PR720 is the rational abelian three-register context.
+Method/domain: First certify A's shortest coordinate-CNOT word length in the complete 168-element GL_3(F2) Cayley graph. Then exactly enumerate all products of at most five factors D_N([g]) in the full group algebra, for g among the six coordinate CNOTs and N among the six rank-one idempotents on exactly two scalar registers. A witness must equal diag([A],[A]) formally; if none exists, the bounded conclusion is min=6 in this two-register model, since the direct six-call word is explicit.
+Expected artifact: reproducible exact search receipt, full formal witness if a five-call construction exists, otherwise an exact bounded no-shorter certificate. No transfer to the selected rational-address bit supplier and no new kappa are claimed.
+Claim expires: 2026-10-14 11:15 UTC.
+```
+
+## Finding
+
+Posted the bounded result in [Discussion #1798, comment 18921618](https://github.com/CrocSwap/integer-mult-bounds/discussions/1798#discussioncomment-18921618): exact minimum six for this target with exactly two scalar registers; no word through five in the full formal `F2[GL_3(F2)]` model. The six-call direct word and exact search receipt are in this package. No supplier adoption or new `κ` is claimed.
+
+Focused review package: [PR #1804](https://github.com/CrocSwap/integer-mult-bounds/pull/1804).

research/gl3-coordinate-cnot-pair-seed1385e243/README.md

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+# A fixed 3D coordinate-CNOT target cannot share calls across two roles
+
+**Exact scoped negative result; no new κ.** In the formal group algebra
+`F2[GL_3(F2)]`, the target `diag([A],[A])` below needs exactly six paid
+coordinate-CNOT calls when there are exactly two scalar registers and every
+rank-one scalar idempotent is available. The direct word uses six calls. An
+exact meet-in-the-middle search found no word of length at most five in the
+complete declared alphabet. This is one fixed target and register count; it
+does not settle other three-dimensional targets or extra dirty registers.
+
+The experiment tests a concrete higher-dimensional branch left open by the
+two-dimensional [coordinate-CNOT register result](../coordinate-cnot-register-bound/README.md).
+It changes the target and address dimension, not just the search seed. The
+result is an abstract paid-call fact. The selected bit supplier uses a different
+rational-address model, and the complex supplier still binds, so this gives no
+immediate κ change.
+
+## Exact model and target
+
+Let `G = GL_3(F2)`. For each ordered pair `i != j`, the only paid address label
+is the coordinate CNOT
+
+```text
+Tij = I + Eij, acting by row i <- row i + row j.
+```
+
+There are six labels. A paid call has unit cost and is
+`D_N([g]) = I + ([g] + 1)N` in characteristic two, where `N = xy^T` is a
+rank-one idempotent on the two scalar registers (`x,y ∈ F2^2`, `y^T x = 1`).
+There are six such idempotents. Free scalar `GL_2(F2)` basis changes normalize
+to this full set of six projectors, as in the cited predecessor work. No
+address operation outside the six coordinate CNOTs, extra register, copy,
+erasure or free address transformation is allowed.
+
+Fix
+
+```text
+A = T12 T13 T21
+```
+
+with the product order shown. In row-bit encoding its matrix is `(6,3,4)`.
+Exact binary row reduction gives `rank(A-I)=2`. A complete Cayley BFS over all
+168 matrices in `G` gives coordinate-CNOT distance 3, with layer counts
+`[1,6,24,51,60,24,2]`. The direct two-role word uses the three factors for
+`A` with projector `E11`, followed by the same three factors with projector
+`E22`; it realizes `diag([A],[A])` at cost six.
+
+The broader [binary-affine rank-priced model](../binary-affine-register-bound/README.md)
+supplies an optimistic four-call floor for two copies because their total
+displacement rank is four. The test
+therefore asks whether register mixing can use five or fewer coordinate-CNOT
+calls, saving at least one from the direct bill of six. All costs inside this
+abstract model are included: every label application is one paid call, and
+there are no auxiliary roles or hidden cleanup calls. No physical bit-supplier
+cost is inferred.
+
+## Exact bounded result
+
+The formal address algebra is the full group algebra of the finite group `G`.
+Each of the four scalar-matrix entries is stored as an exact 168-bit coefficient
+vector. The verifier enumerates the complete 36-factor alphabet: six address
+labels times six rank-one idempotents.
+
+Every coordinate-CNOT label is an involution, so each paid factor is
+involutive. Any word of length at most five splits as a prefix of length at
+most two and a suffix of length at most three. The verifier enumerates the
+exact formal-matrix state balls at those radii, retains one shortest word for
+each state, and checks for every prefix `P` whether `P^-1 diag([A],[A])` is in
+the suffix ball. No match exists. Since the explicit direct word has length
+six, the exact minimum in this fixed two-register model is six.
+
+The replay records these deterministic receipts:
+
+| State set | Layer sizes, including radius zero | Distinct states | SHA-256 |
+|---|---:|---:|---|
+| Formal-matrix prefix ball, radius 2 | `1, 36, 924` | 961 | `aedf570c8cd33c107fad63ce967cb06618e8d171081007e4eea851f16659cd48` |
+| Formal-matrix suffix ball, radius 3 | `1, 36, 924, 21112` | 22,073 | `8455d502835c5a4bed62f8521061065e787248651462f374b444caa178e4ea96` |
+
+The zero intersection is an exhaustive statement for the declared finite
+formal alphabet and the five-call cutoff, not a natural-representation or
+finite-column test. The verifier also compares 252 one-factor updates against
+a separate sparse generic matrix-product implementation. The direct two-copy
+word and the one-copy/one-identity three-call word are positive controls. All
+six one-call omissions from the direct word and a wrong-label mutation are
+rejected. Admitting `A` itself as an extra paid address label gives the
+expected two-call endpoint and confirms that the six-label boundary matters.
+
+## Scope, overlap and limits
+
+The nearest result is the exact two-dimensional two-SWAP theorem in
+[coordinate-cnot-register-bound](../coordinate-cnot-register-bound/README.md).
+That package leaves a genuinely three-dimensional target with a rank-versus-
+word-length gap as an open question. This note tests one such fixed target
+using all six `GL_3(F2)` coordinate CNOTs, rather than its `GL_2(F2)` target
+and two-label algebra. PR #1333 prices all binary affine labels by displacement
+rank, so its alphabet is broader and its direct cost for `A` is different.
+The active Discussion #1738 claim concerns a third dirty register for a D8
+central target; it does not cover this target, dimension or register count.
+Immediately before submission, the overlap helper refreshed 465 open PR records
+and 444 Discussion records with complete pagination and no warnings. The exact
+scope search returned this claim and this local package only; the helper does
+not inspect code diffs or arbitrary nested files. Other pinned predecessor and
+selected-record context is listed in [SOURCES.json](SOURCES.json). The claim
+and its expiry are recorded in
+[CLAIM.md](CLAIM.md) and [Discussion #1798](https://github.com/CrocSwap/integer-mult-bounds/discussions/1798).
+The focused package is submitted as [PR #1804](https://github.com/CrocSwap/integer-mult-bounds/pull/1804).
+
+The finite search does not address `r > 2`, other target matrices, translation
+labels, general affine labels, arbitrary free address maps, rational frames,
+physical scalar or hosting costs, compiler realization or selected-supplier
+admission. It is not a multiplication construction and does not improve the
+selected bound. The selected
+`κ=836330098538459/500000000000000000` remains unchanged; the selected complex
+side still binds. No new κ is claimed.
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3 -I -B research/gl3-coordinate-cnot-pair-seed1385e243/verify.py
+python3 -I -O -B research/gl3-coordinate-cnot-pair-seed1385e243/verify.py
+python3 scripts/research_package.py --package research/gl3-coordinate-cnot-pair-seed1385e243 --check-only
+```
+
+The exact expected summary begins:
+
+```text
+PASS_GL3_COORDINATE_CNOT_TWO_ROLE gl3=168 address_layers=[1, 6, 24, 51, 60, 24, 2] target_rank=2 target_length=3
+```
+
+Python 3.12 is registered in `VERIFY.json`. The local replay interpreter and
+version are recorded in `VALIDATION.md` after final checks.
+
+## Open next questions
+
+Does adding one restored dirty register permit a five-call realization of this
+same `GL_3(F2)` target? This isolates whether the only missed sharing resource
+is an extra scalar role. First derive a pruning invariant for three-register
+formal matrices, then use an exact search only if its precomputed state bound
+fits a declared memory budget. The 168 scalar rank-one idempotents make a
+direct radius-three expansion too large for this cycle.
+
+## Credit
+
+Credit commdata2338 for the preceding two-dimensional coordinate-CNOT target
+and its explicit three-dimensional open question; credit Gareth Davidson's
+PR #1333 for the rank-priced binary-affine model and augmentation normalization.
+This package does not claim either operation alphabet or general register
+mixing as new. Prepared by Cinna Davis with substantial OpenAI Codex
+assistance. No independent human review or supplier adoption is claimed.
+Apache-2.0.

research/gl3-coordinate-cnot-pair-seed1385e243/SOURCES.json

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