Negative result: twisted helper exits have no target-to-helper feedback gain at any fixed 3+3 or five-register exit, plus an every-word contamination-charge lemma (no κ claim)

[thakrarsagar]

Statement. Scoped proof plus finite certificates. No new supplier, no κ effect, no selected-frontier change.

1. Every-word contamination-charge lemma (written proof in PROOF.md, constructive implementation in lemma.py). Model: H helpers, Y targets, dirty inputs, unit-cost directed XORs, fixed labels (PR1306's model). P = P_{H,Y} is the restricted group (no target→helper XOR); it contains every exit [[T,0],[K,I]]. For any word w of length L whose product M lies in P, an explicit tracking construction outputs a restricted word computing exactly M, of length L - f_v + c(w). Here f_v counts absorbed feedback gates, and c(w) is an explicit charge: a nonsingular contaminated read costs |Z_a| + |Z[:,i]|, a singular read costs |Z_a| + |Z[:,i]| - 2, and each final in-block relabel swap costs 3. No step assumes an invertible target block; the singular step exchanges one helper and one target.
   • Corollary 1: if every read is clean, deleting all feedback gives a restricted word, so d_P <= L - f.
   • Corollary 2: a strict feedback advantage needs every minimum unrestricted word to have c(w) >= f_v + 1 >= 2, so it must contain a non-clean (contaminated or singular) read. This is only a necessary condition.
   • Positive control: in the five-register model exactly 120 non-exit elements of P (60 in P_{2,3}, 60 in P_{3,2}, all with target block D != I) have d_full = d_P - 1. So the exit condition D = I is load-bearing. Corollary 2 is asserted on a minimum word of each.
2. 3+3 decision (finite certificate). For all 85,504 exits [[T,0],[K,I_3]] with T != I, d_full = d_P: no strict feedback advantage. Common minimum histogram 1:6, 2:96, 3:744, 4:3,594, 5:11,463, 6:23,948, 7:29,259, 8:15,438, 9:956 (max 9), over 3,478 S3×S3 classes. The 512 T = I shears are PR1306's census; they are cited, not re-claimed (the run agrees: 0 advantage).
3. Five-register shapes (finite certificate). No strict advantage at any of the 333,552 twisted exits: 2+3 (320, max 6), 3+2 (10,688, max 8), 4+1 (322,544, max 10). The run also covers 160 already-settled T = I exits (64 + 64 PR1306, 16 PR2567 in 4+1, 16 in 1+4), for 333,712 in total.

Milestone and decision informed. Local module: a scoped exact interface lemma plus finite certificates. PR1868 question 3 asks whether target-to-helper feedback helps a replacement helper contract at fixed twisted exits.

• Closed here: the fixed-exit question in PR1306's model for 3+3 (the 3x3 twisted case PR2494 left open) and the twisted five-register shapes 2+3, 3+2, 4+1. With PR2494's H + Y <= 4 census, this covers every twisted shape with at most five registers.
• Not closed: twisted 2+4, 4+2, 5+1 and larger; the actual seed02 D boundary (at least 3 helpers and 99 targets), where only Corollary 2's necessary condition applies; C-absorption pricing; twisted-versus-restored exits.
• No κ effect either way: the bit side does not bind (research/r439-twisted-star-absorption-seed02).

Base and pins. Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90 (priority revision 6; canonical main unchanged at submission). Git blob and SHA-256 pins in SOURCES.json. The diff adds only research/bit-twisted-feedback-contamination-lemma/ (9 files).

Reproduce.

python3.12 scripts/research_package.py --package research/bit-twisted-feedback-contamination-lemma --install-deps

Expected:

PASS_TWISTED_FEEDBACK_CONTAMINATION_LEMMA exits33=86016 twisted=85504 classes=3478 strict_feedback_advantage=0 max_restricted_min=9 lemma_words=15028 controls=5
PASS_TWISTED_FEEDBACK_NUMPY_CROSSCHECK twisted33=85504 classes=3478 five_register_exits=333712 twisted5=333552 strict_feedback_advantage=0 lemma_words=333712 nonexit_advantage_controls=120
PASS_REGISTERED_RESEARCH_PACKAGE research/bit-twisted-feedback-contamination-lemma outcome=negative-result

Local run: about 75 s, under 0.5 GB RSS (uv CPython 3.12.15 + numpy 2.5.4). verify.py is stdlib only.

Certificates and independence.

• 3+3 primary (independent33.py, stdlib): column-major encoding, restricted MITM B^P_6 × B^P_3 (exact to 9), full B_5 × B_3 in GL6 (exact to 8, so d_full >= 9 for the 956 exits with d_P = 9); asserts its restricted balls stay in P_{3,3}.
• 3+3 second method (numpy_crosscheck.py, numpy==2.5.4): row-major, dense BFS over all 14,450,688 elements of P_{3,3}, left-multiplication B_4 × B_4 MITM, own orbit canonicalisation. Same class count, histograms and 0 advantage. Per-class results are not cross-joined inside the package.
• Five registers: dense BFS over all 2^25 matrices and each parabolic.
• Lemma corroboration: invariant asserted after every gate on 6,628 words (two per element of every P_{H,Y} with n <= 4), 8,400 random P-words (n = 2..8) and 2,800 free words; plus one greedy minimum word per five-register exit (333,712). 35 of the 4+1 greedy words have a non-clean read with c = f_v + 1, so the lemma is not tight there (outputs length L + 1); those exits are decided by the census.

Negative controls (verify.py): h^=y; y^=h zeroes the target block yet the construction covers the double-swap word via two singular reads; naive feedback deletion on h^=y0; y1^=h; h^=y0 gives the wrong map; the charge is necessary there (L - f = 1 < d_P = 2); a forged gate fails replay; a self-XOR is rejected.

Scope, assumptions, unresolved obligations. PR1306's model (fixed labels, dirty inputs, unit-cost XORs; no free permutations, COPY/ERASE, frames or extra registers). Fixed exits are compared; twisted and restored exits are not interchangeable. Open: physical continuation/absorption of T, R439 substitution, chronology across nonzero frames, hosting, precision/tape, all-size compilation, native supplier replay. No general no-feedback theorem beyond the lemma; no Lean proof. A correction to the original hypothesis is recorded: c = 0 does not imply every read is clean (a singular read can have surcharge 0); only the converse holds.

Search record. 3+3: all 86,016 exits, exhaustive. Five registers: exhaustive dense BFS. Random lemma words from seed 20261019. BFS runs to exhaustion; MITM is exact up to a + b. No sampling in the census, no pivot.

Credit and concurrent work. Predecessors (authors and heads checked with gh): ywz65 (#2494, four-register twisted census, H + Y <= 4); Sagar Thakrar (#1868, twisted-helper contract and question 3); @commdata2338 (#1306, model, substitution lemma, MITM, 3x3 T = I census); Hanson Sun (@bbMisk) (#1557, 3x4 census); rtyu193-rgb (#1706, 4x4 census; #2567, to-8 package with H = 1 / Y = 1 lemma). What this adds: the every-word charge lemma (including singular steps), the twisted 3+3 domain, and the twisted five-register shapes. Open #2721 (geckods, framed-transfer no-target-feedback clause) is related context only; this package is distinct and does not depend on it. An overlap refresh at submission (open PRs, PRs updated since 2026-10-19T11:20Z, Discussions, scripts/find_overlap.py) found no other work on twisted-exit feedback.

Claim. No Discussion claim or post was made for this work; this PR body carries the result.

Review provenance. Reviewed by 7 independent AI review passes in the same loop (two rounds: reproducibility/controls, mathematical audit, novelty/provenance, referee), not by a human. The AI math audits re-implemented the construction from PROOF.md and recomputed the 3+3 and five-register censuses with separate encodings; all review issues were addressed.

Attribution. Sagar Thakrar (@thakrarsagar), with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5) via Claude Code in an automated research loop.

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