DONE: Two-write spectator projection in the residual five-coordinate two-cycle problem

[JoyboyBrian]

DONE: the scoped two-write projector lemma and its independent checks are published in PR #6090.

PR6012 leaves a two-write spectator lemma open in its residual five-coordinate, ten-call two-cycle problem. This package derives the exact row-stabilizer projection while retaining the middle labels selected by e_j^T b=e_l^T and their required relabelling g b h.

For an endpoint [c]I in the two-role binary formal model, a coordinate written exactly twice has equal rank-one scalar projectors when c restores that row, and complementary projectors when c moves it. The moved-row result uses the projections of both the word and its inverse; the forward equation alone admits additional pairs. In PR6012's residual (6,4) case, these restrictions apply to both writer and reader pairs.

A standard five-call commutator gives an exact identity with a spectator written and read twice. It disproves an unrestricted three-write spectator rule; inserting it before the known twelve-call endpoint supplies a seventeen-call positive control. It is not a ten-call counterexample. The full length-ten and length-eleven exclusions remain open.

Validation passes on exact head 188cee674895aa8c1e63aafa91003e34c237e420: normal and optimized registered checks; 3,600 formal projection contexts; all 576 augmented matrix cases; six commutator identities; a separate complete-permutation group-label engine checking 288 return filters, 72 outer contexts and six seventeen-call endpoints; source pins/history and committed bytes; official export integrity; and read-only Git-free empty-PATH replays with source mutation rejection. A separate read-only agent reviewed the algebra and independently checked the matrix and group fixtures. This is not independent human or proof-kernel acceptance.

Reproduce with python3.12 -B scripts/research_package.py --package research/two-write-spectator-projection.

Credit Nick Petalas/#6012 for the projection/reduction question, Mihir Srivastava/#4939 for the formal model, James Reed/jwr404 #5005/#5379 for the cited small-dimension results, and JoyboyBrian/#4974 for the earlier single-cycle reduction. The commutator is standard. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Advance claim and canonical discussion.

No native supplier, tariff, selected-frontier change or new kappa is claimed. Hosted planning failed the same two current-status navigation tests as the canonical base; scientific replay was skipped. Existing PR #6050 repairs those documentation mismatches. Local validation above is separate from hosted CI.

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[JoyboyBrian]

HANDOFF / scoped lemma established; not packaged; separate root argument review and a fresh connected/color fixture check found no blocking issue. No independent human or proof-kernel acceptance.

Claim: #6075 (comment)
Source: PR6090 188cee674895aa8c1e63aafa91003e34c237e420, research/two-write-spectator-projection/README.md; PR6012 3d9d5eb2ae784356497771661960ad0455b137db, research/two-three-cycles-every-dimension/PROOF.md.

New reduction: a residual five-coordinate ten-call two-cycle word cannot have a connected read/write incidence graph. More generally, it cannot have all its call projectors in one complementary pair {N,I+N}. This is a structural elimination, not a complete length-ten exclusion.

Proof. Make one vertex for each call position. For each coordinate, join its two writer positions and its two reader positions. The source lemma says the two projectors are equal for a spectator coordinate and complementary for a target coordinate. Each vertex has exactly one writer edge and one reader edge, so this is a union of alternating even cycles (parallel writer/reader edges allowed). Following a cycle must give an even number of complement edges, else N=I+N. In a connected graph every projector is N or I+N for one rank-one idempotent N. A single scalar basis diagonalizes them all, separating the formal word into two ordinary coordinate-CNOT group words. Their lengths sum to ten and each must implement the same coordinate 3-cycle c.

Each ordinary CNOT word implementing c with any restored spectator coordinates needs at least six gates. The general connectivity/diagonal argument gives L >= n-v+z, with z=3. If any spectator is connected to the target support component, the final interaction graph requires at least three joins, so L>=6. Otherwise the operations on the three target coordinates form a word in GL3(F2); the complete 168-state shortest-path graph has distance six to each of the two coordinate 3-cycles. Thus both scalar lanes need at least six calls, contradicting ten total.

A concrete connected configuration passes every pairwise projector constraint, showing that the new condition is stronger than parity alone: chronological address labels are (i,i+1 mod5) for i=0..4 followed by (i,i+2 mod5) for i=0..4. Take targets0,1,2 and spectators3,4. The single alternating10-cycle has a consistent projector coloring, in position order, [0,0,1,0,1,1,1,0,0,1]. The two scalar lanes each have five calls, so the complete target is impossible. This does not claim that this literal word realizes the target.

Checks actually run: exact construction of this pairing graph, propagation of every equality/complement constraint, verification that it is connected with lane counts5/5; an independent stdlib breadth-first traversal of GL3(F2), all168 states, returning both cycle distances6. No full formal-word enumeration, mathematical kernel, export or hosted CI ran. These are exploratory local checks; the written universal argument above is the actual scoped result.

Remaining obligation: disconnected alternating cycles can use different complementary projector pairs, which may not commute. Determine which such component patterns survive the full off-subgroup equations, retaining selected middle labels and their g b h relabelling. Do not infer a full ten-call exclusion from the incidence constraints alone. A package would need to retain exact predecessor lemma bytes and add literal formal controls, source pins and replay/export checks.

Credit Nick Petalas/PR6012, JoyboyBrian/PR6090, Mihir Srivastava/PR4939 and James Reed's coordinate-cycle lineage; Bu–Fan–Joo for the general connectivity/diagonal bound. New incidence reduction prepared by JoyboyBrian with substantial OpenAI Codex assistance. No new kappa, supplier or complete multiplication theorem.

This claim is released. No continuing reservation, selected-frontier change or supplier admission is implied.

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