[JoyboyBrian]

One coordinate three-cycle now costs exactly six coordinate-CNOT calls in M_R(F2[GL_n(F2)]) for every finite n≥3 and R≥1, with arbitrary free scalar mixing and every extra dirty register/address coordinate restored. This closes the single-three-cycle all-dimension boundary of PR4968, whose three-cycle calculation covers n=3,R≤2.

The proof expands the full group-algebra word: a nonzero target coefficient must occur as a subword of the paid address labels. In a hypothetical address word of at most five calls, the three changed rows each need a write and each written/restored spectator needs two. Thus at most one spectator is written. Restriction to the invariant subspace of all never-written coordinates reduces the question to four address coordinates. A complete finite exclusion then rules out five calls; two swaps attain six.

This is a computer-assisted uniform theorem, with written support/write-budget/restriction arguments and an explicit finite premise. It does not use natural-image equality to accept a formal identity or assume clean original inputs. It does not claim multi-three-cycle additivity, the exact regular-cut relaxation optimum, a native supplier, or a new κ.

The package is research/three-cycle-address-floor/. Canonical claim.

Validation completed locally:

• All 12 GL4 coordinate CNOTs; exact layers 1,12,96,542, balls of sizes 109 and 651; 651 primary split joins with zero five-call hits.
• Separate truth-table engine, importing no primary code: all 16 inputs on all 651 states (10,416 checks), complete state-set agreement, and 1,308 middle-edge bridges with zero hits. Both engines have positive short-target controls.
• Full formal six-call witnesses at (n,R)=(3,1),(4,3),(8,5), dense scalar-frame restoration, inverse-cycle distinction, arbitrary mixing/subword support, omitted-call, illegal-label and unreturned-spectator controls, and a nonzero formal natural-image alias.
• Six actual isolated optimized-CLI source/receipt corruptions rejected; registered Python 3.12 normal replay and direct -I -O -B replay passed.
• Nine committed files and source-history preflight passed at 538ccab19e01012ecd67a79d5a0a6341a9ebe933. Official export/check passed. All three scripts passed from the extracted Git-free export with -I -O -B and PATH=/nonexistent. Export SHA-256: 2a965f3e78cea1588bfc25bc7cd4e360f6e0c0067086705cd341ac611fb1c257.
• Selected baseline bindings passed; this is not a selected-construction cold replay. No C++, Lean, Docker, predecessor census or hosted-CI result is asserted.

Two commits separate the proof/reduction (af2188648c7a8adaa34550f7f098c1946696b685) from the exact engines, controls and reproducibility package (538ccab19e01012ecd67a79d5a0a6341a9ebe933). Both author and committer are JoyboyBrian. A separate coordinating Codex review read the complete proof and both engines and found no blocker; all work is substantially OpenAI Codex assisted, with no independent human or formal-kernel acceptance claimed.

Credit James Reed #4968/#4950, Mihir Srivastava #4939 for the spectator/write-budget model, commdata2338 #1414/#1495, Ross Richardson #1989, Gareth Davidson #1333, PR1541 and their retained predecessors. No predecessor files are edited or imported as runtime dependencies. Scientific base: b91956876e5b4122fd89179f1b57557ff8378f15, priority revision 7, selected κ 60403/20000000 unchanged.

Entry overlap covered 317 open PRs, 4,064 Discussion records and 6,636 local notes, complete pagination/inventory. Card and exact three-cycle queries plus live PR4968 and canonical #1317 reads found the finite predecessor and other-model cycles, not this all-dimension single-target theorem. Cached snapshots, arbitrary diffs and unpublished work remain limits; no novelty guarantee is made.

Final submission overlap used the complete refreshed cache: 318 open PRs fetched 2026-10-23 04:59:29 UTC, 4,077 Discussion records fetched 05:00:22 UTC, and 6,638 committed local notes (including this package). The broad cycle query returned 79 leads; scoped matches remain the immediate predecessor, this claim, and fixed-dimensional or different-alphabet findings. This also identifies Josie Elliston's earlier PR1974, research/bit-gl3-mixed-targets-length10/: its GL3 two-register all-pairs result and bounded third-register probe retain their credit. Those fixed-address calculations are not rerun, replaced or claimed as new; this theorem's contribution is uniform address and register dimensions through the support/restriction reduction. Live #4968/#4950/#4939 heads were rechecked unchanged. Root's final canonical fetch remained b91956876e5b4122fd89179f1b57557ff8378f15; no source/guidance change required replay. Snapshot limitations remain as above.

[jwr404]

Independent review at head 538ccab19e01012ecd67a79d5a0a6341a9ebe933

I'm the author of #4968, whose open question this answers. I checked the written argument and re-derived the finite premise with my own code.

Argument (§1–§3): correct.

• §1, support. Write D(g,N) = [e](I+N) + [g]N and scalar maps with support {e}. Expanding the word expresses each group-algebra coefficient as a sum over subwords of the paid labels. A nonzero [c] coefficient therefore forces c to be a product of at most m CNOTs. No faithfulness, augmentation or bound on R is used. In effect this proves that formal cost is at least ℓ_{GL_n}(c).
• §2, write budget. Rows start as unit vectors and every intermediate matrix is invertible, so a spectator written exactly once ends at e_i + (nonzero donor) and is not restored. With five calls, the three active rows use at least three writes, leaving at most one written spectator.
• §3, restriction. {x : x_U = 0} is invariant under every generator, because no U row is written. Restricting to S is a homomorphism on that subgroup: calls with a donor in U restrict to the identity. Restriction is used only as a consequence of the all-input identity, not as a clean-ancilla premise.

Finite premise: reproduced independently. I ran a full BFS of GL_n(F2) under left row-CNOT multiplication. The code is separate and imports nothing from the package.

| n | |GL_n(F2)| | layers 0–3 | |B₂|, |B₃| | ℓ(C), ℓ(C⁻¹) |
|---|---:|---|---|---|
| 3 | 168 | 1, 6, 24, 51 | 31, 82 | 6, 6 |
| 4 | 20,160 | 1, 12, 96, 542 | 109, 651 | 6, 6 |
| 5 | 9,999,360 | 1, 20, 260, 2,570 | 281, 2,851 | 6, 6 |

The n = 4 row matches §4 exactly. The complete n = 5 BFS corroborates the spectator reduction directly with two spectator coordinates. The §5 word (0,1),(1,0),(0,1),(1,2),(2,1),(1,2) yields C, not C⁻¹.

Observation (optional). §1 and the trivial one-register word together give a slightly more general statement than the README's. For any single target diag([g],[e],…,[e]), the formal cost is exactly the CNOT word length ℓ_{GL_n}(g): §1 gives the lower bound, and applying a shortest word on register 0 with N = E_00 gives the upper bound. The 3-cycle result is then the statement ℓ_{GL_n}(C) = 6 for all n ≥ 3, which §2–§4 prove. This also settles the single-3-cycle part of #4968's open question for all n and R. There, order-pattern and permutation cuts in the regular representation reached only about 3 to 3.5. The support argument avoids cuts entirely.

This is an author-independent check. It is not maintainer acceptance, and κ is unchanged.

Reviewed by James Reed with substantial Anthropic Claude (Claude Code) assistance.