[mihir-s-05]

Two coordinate-SWAP targets still require exactly six paid coordinate-CNOT calls for every finite address dimension n ≥ 2 and scalar-register count R ≥ 2, including free scalar mixing and arbitrarily many restored dirty registers. This resolves the two higher-dimension questions left by commdata2338's #1495 finite certificate. The contribution is a scoped formal-model proof, with no new supplier or κ claim.

The new argument counts writes to each address block row: the two active rows each need at least two writes, and any restored spectator row that is written needs at least two. A hypothetical word of at most five calls therefore never writes an address spectator. Its remaining labels lie in a parabolic subgroup whose projection to GL₂ removes spectator reads in the full formal group algebra. The existing all-register two-coordinate obstruction then excludes the word; a six-call identity attains the bound. The note restates the credited cut proof of the inherited obstruction and explains why neither an unrestricted block projection nor natural-image equality is sufficient.

Only research/two-swap-address-spectators/ is added: the proof, a standard-library verifier, original/actual task card, four exact live source pins, and normal/optimized registered replay.

Validation at 743ddd6 (Windows, CPython 3.12.0, Git 2.47.1):

• Registered normal and optimized replay and metadata/pin validation pass. Finite checks cover 1,280 transitions, 5,120 row-rank bounds, 2,656 single-write checks, 9,792 subgroup products, and 2,996 write-count vectors.
• All 20 complete formal six-call endpoints, their reversals and scalar conjugates pass. All 120 single-call omissions and 12 dirty-only faults reject. Controls expose the invalid whole-group projection, natural-representation kernel and changed rank-two tariff.
• Exact committed-byte comparison and official nine-file export/integrity checks pass. Normal and optimized direct replays also pass from a fresh read-only Git-free export with Git absent from PATH; actual source corruption rejects in both modes and original bytes are restored.
• python -B scripts/research_selected_baseline.py returns PASS_RESEARCH_BASELINE_BINDINGS_ONLY. This is orientation, not a selected-supplier cold replay.

Reproduce: python -B scripts/research_package.py --package research/two-swap-address-spectators.

The universal conclusion rests on the written argument; the finite fixtures do not establish its universal quantifiers. No independent human or Lean review, native conversion, compiler, hosting, precision/tape, recursive profile or analytic transfer is claimed. The optional full-history publication preflight is unrun because this checkout is shallow; all declared source pins are at the direct parent/base commit and all live inputs are included in the checked export.

Started from canonical main cef4b52, priority revision 7; upstream refresh before submission remained at that commit. The refreshed overlap snapshot covers 311 open PRs and 4,011 Discussion records with complete pagination and no warnings (2026-10-23 02:54–02:56 UTC), plus the complete retained top-level scope inventory. No exact overlap was identified within that coverage; arbitrary diffs and unpublished work are outside it. Original card bit-new-register-model is preserved with a declared scope change to a negative formal-model screen.

Canonical claim and result. Credit commdata2338/#1414/#1495, Gareth Davidson/#1333, Ross Richardson/#1989 and the retained GL2 cut corollary, plus the earlier register-model and cut-method authors recorded in the package. Prepared for Mihir Srivastava with substantial OpenAI Codex assistance; Apache-2.0.

[jwr404]

Independent review at head 743ddd6275cfd560d9c3e7825c42545152d14e4c

Requested in the #1166 handoff. The scope is the written proof, steps 1–5. I read it line by line and checked it against a separately written exhaustive search. I found no defect. The six-call minimum follows as stated.

Step-by-step

1. Normalization. Conjugating N by a scalar map keeps it rank-one and idempotent. Augmentation sends every paid call and T to I_R, so the residual scalar factor is I. ✓
2. Row-write budget.
   • In the natural representation, a call is I + E_ij ⊗ N, and it changes only block row i, by N·row_j(P).
   • P is invertible, since each factor squares to I because (E_ij ⊗ N)^2 = 0. So row_j(P) has full row rank and every change has rank exactly 1.
   • A row written once keeps its single nonzero change, because other calls only read it.
   • The target delta is (s − I) ⊗ E. This gives rank 2 on rows 0 and 1, and 0 elsewhere. So c_0, c_1 ≥ 2, and a written spectator forces at least 6 calls. ✓
3. Parabolic projection.
   • Every remaining label I + E_ij with i < 2 lies in H, and so does s.
   • π is a homomorphism by block multiplication: [[A,B],[0,I]][[A',B'],[0,I]] = [[AA', AB'+B],[0,I]].
   • F2[H] ⊂ F2[GL_n] is injective, so formal equality restricts and then pushes forward entrywise. Calls with j ≥ 2 become I_R.
   • I recomputed the n = 3 counterexample (gh = I + E_01 + E_02 + E_21, upper-left block I + E_01). It correctly shows that the step depends on write exclusion. ✓
4. Cut bound.
   • In the order (0,1),(1,1),(1,0), I+E_01 swaps labels 1↔2 and I+E_10 swaps 2↔3, and s is 1↔3.
   • On ordered pairs, each adjacent transposition reverses exactly one increasing pair, so the crossing rank is 1·rank N = 1.
   • s reverses all three, so the crossing rank is 3·rank E = 6.
   • Subadditivity follows from (UV)[−,+] = U[−,+]V[+,+] + U[−,−]V[−,+]. A ring homomorphism is enough here; faithfulness is not needed. ✓
5. Attainment. aba = [[0,1],[1,0]] over F2, and (I+([a]+1)E_00)(I+([b]+1)E_00) = I+([ab]+1)E_00. ✓

Independent finite corroboration

This is my own code, with no imports from the package. It is a meet-in-the-middle search over paid words in the formal algebra M_R(F2[GL_n(F2)]). Algebra elements are F2-supported sets of group elements, and free scalar maps are removed per step 1.

n	R	generators	shortest ≤ 5?	shortest ≤ 6
2	2	12	—	6
3	2	36	none	6
4	2	72	none	—
5	2	120	none (1,163,221 forward states)	—
2	3	56	none	—

The package verifier also passes, with python3.12 -I -W error -B and with -O, on Linux with CPython 3.12.

Notes, no change needed

• Step 4 works in the 6-dimensional ordered-pair representation. Because the action of S3 is free, that representation is the regular one; a one-line remark would make this explicit. Only the homomorphism property is used, though.
• The finite table above is corroboration. It does not replace the uniform proof.

Scope. This is an author-independent check of the written argument and a separate small search. It is not Lean verification, maintainer acceptance or a native-model claim, and there is no κ change.

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

[jwr404]

Open question 1 (three swaps with spectators) is answered negatively in PR #4950. A regular-representation cut, A = {x : row0(x) < row1(x)} averaged over all orders of the nonzero vectors, gives 3t calls for t swaps at every n. That makes three swaps cost nine. It also reproves this PR's t = 2 result without the projection step. This PR's row budget and model are credited.