[apoursh]

Change and mathematical scope

This proves N(c) = Θ(c log c) for the unchanged unsigned immutable-source/one-dirty-connector interface of #4231/#4248. It resolves #6035 open question 3 and #6061 open question 2. Only research/connector-gray-growth/ is added.

For every k ≥ 1 and c ≥ max(2, 2^(k−1)), a binary reflected Gray path yields a depth-two k-source block with exactly c calls per source. Concatenating floor(c/2) blocks and the inherited depth-one pair when c is odd gives

N(c) ≥ (1 + floor(log₂ c)) floor(c/2) + 2(c mod 2).

The new construction splits any zero-start one-bit cube-covering walk into its upward-label subsequence A and downward-label subsequence B. If coordinate j flips f_j ≤ c times, the unsigned word

A · ∏_j j^(c−f_j) · reverse(B)

has two chronological prefixes whose integer count vectors sum to c·1 plus each visited binary mask. Reflected Gray paths have max f_j = 2^(k−1). Padding readouts to c makes the arbitrary dirty contribution vanish; every operator attains its c-call floor. The README proves the all-parameter lemma, composition, and a self-contained version of the inherited counting upper bound.

At c=256 this gives 1,152 sources versus 1,024 from #6061's eight-source block; at c=512, 2,560 versus 2,048. Earlier finite templates remain stronger at small c.

Scope: arbitrary compatible linear maps, including singular/noncommuting maps, over any nonzero unital ring of characteristic c≥2. No extra operation, clean row or connector is added. This is a scoped proof and constructive verifier, not a new κ. Full word-dependent admission/layout, coefficient preparation, schedule/dispatch, complete primitive prices and a native supplier remain unpriced; the README retains the complete conditional invoice. Large corollary words have exponentially many outputs and are proved by decomposition rather than materialized.

Task and overlap

Base 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8; canonical main was pulled and re-fetched before submission with no change. Task connector-alphabet-change, scope changed to the explicitly released general-family handoff. The original full card and reason are retained in TASK.json. This resolves an asymptotic local interface question; it does not pass a supplier budget gate.

Canonical collaboration claim.

Nearest work:

• Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231/Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248: interface, floor, prefix criterion, concatenation/deletion, pair block and counting upper bound.
• Prove N(c) ≥ ⌊5c/2⌋ for the unsigned connector via five-source depth-two blocks in every c ≥ 5 (no new κ) #6035 at 119de1d7fca5124fa8d01ea49d258560aca618e6: five-source construction and unbounded-ratio question.
• Depth-two connector blocks on six, seven and eight sources: N(c) ≥ 6⌊c/2⌋ for c ≥ 8, up to 8⌊c/2⌋ for c ≥ 27 (no new κ) #6061 at babc5e99421403284fa530e9f135b618f31805bb: six/seven/eight-source blocks and explicit all-k handoff.
• Unsigned connectors reach their call floor on 3c−1 sources for every c≥8 #6057 at f92bd41f60564175bd68f944e9d6450da8965734: parallel six-source construction.
• Prove and execute the paid clean-row Gray optimum #5372 uses a clean helper; Research: six-source universal dirty-connector boundary, f(6)=16 (no new kappa) #1185 has characteristic-two optima and Gray upper bounds without c calls per source. Those contracts differ.

The three canonical predecessor READMEs are pinned. Pending PRs are motivation/comparison, not executable dependencies. Live handoffs and all-state PR searches were checked; sparse local coverage (11/8,348 notes) is explicitly incomplete and is not a novelty certificate.

Validation

CPython 3.12.14, macOS arm64; standard library only.

python3.12 -I -W error -B research/connector-gray-growth/verify.py
python3.12 -I -O -W error -B research/connector-gray-growth/verify.py
python3.12 -B scripts/research_package.py --package research/connector-gray-growth
python3.12 -B scripts/check_research_commit.py --package research/connector-gray-growth --check-source-history --check-modes

Normal and optimized registered replay pass (about 0.9 s per direct run):

• 36 production blocks through k=12, 24,534 masks; 14 repeated/backtracking covering walks.
• Five full composed words for c=2..6: 847 outputs and 4,778 calls, executing the production call iterator on all source/dirty input columns with noncommuting 2×2 matrices.
• A separate recursive construction/grouped-boundary interpreter: 27 blocks, five compositions, 52,682 calls.
• Counting thresholds for c=2..128; 27 combined guard/semantic controls rejected, including omitted calls, wrong padding, wrong prefixes, unreversed downward suffix and exceeded depth.
• Exact source pins, committed bytes/history/modes, official ten-file export/integrity and normal/optimized read-only Git-free replay all pass at a0f08278dd755e0bf228855e45c5efaa3b8ad97a. Exported bytes remain unchanged.

The card's 945-pairing positive control and selected-baseline binding check passed as orientation. The selected cold replay, Lean, Docker and native physical supplier were not run; none is claimed. Finite tests support the written proof rather than establish its infinite quantifiers.

Credit

Prepared by apoursh with substantial OpenAI Codex assistance, including separate construction, execution and proof-audit agents. Credit Nick Petalas (#4231/#4248/#6035), JosieElliston (#6061), zxv (#6057), Ross Richardson and retained connector lineage. Classical Gray codes are not new. No independent human or proof-kernel review claimed. Apache-2.0; predecessor files unchanged.

Open next questions

See the README/TASK.json for relevance, first steps and blockers: sharpen the leading constant of N(c)/(c log₂ c), or bind a concrete characteristic-c consumer to a fully paid invoice. Neither is implied by the source-count theorem.