[petalas]

Result

For the unsigned dirty connector over a ring of characteristic c (k immutable sources, all 2^k − 1 binary-mask outputs, calls B_j: z += B_j x_j and A_m: y_m += A_m z), let N(c) be the largest k at which #4231's floor c(k + 2^k − 1) is attained.

• Theorem A (scoped proof, word check for c ≤ 7): N(c) ≥ 2c for every c ≥ 2. The two-source block a b a^(c−1) b^(c−1) gives every block mask as a single prefix. Concatenated blocks add their depths, so ⌈k/2⌉ ≤ c blocks give a floor word. Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231's run walk reaches k ≤ c.
• Lemma C (proof plus exhaustive control at k = 3): a walk whose masks are all single prefixes has at most two sources. Beating 2c with blocks therefore needs a depth-d block with more than 2d sources.
• Characteristic four (finite certificate plus proof): an explicit 36-event walk attains the 2,080-call floor at k = 9. With the deletion lemma (Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232's proof, valid for every c) and boundary-multiset counting, C(91,4) = 2,672,670 < 4,194,303 excludes every k ≥ 22. So 9 ≤ N(4) ≤ 21. This answers part of Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231's open question 1.
• Counting table: the first excluded k is 7, 14, 22, 30, 38, 47, 56, 65 for c = 2..9.

Theorem A is sharp at c = 2 (#943: five sources fail), matches #4232's n ≤ 6 at c = 3, and is beaten by the nine-source walk at c = 4.

Claim type: scoped proof plus finite certificate. No new κ, no selected-frontier change, no multiplication theorem.

Evidence and reproduction

From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/connector-depth-blocks/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/connector-depth-blocks/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/connector-depth-blocks
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/connector-depth-blocks --check-source-history

Marker: PASS_CONNECTOR_DEPTH_BLOCKS blocks_k<=2c c4_attained_k<=9 c4_counting_excludes_k>=22.

The standard-library verifier builds the block word for all 54 pairs 2 ≤ c ≤ 7, k ≤ 2c and simulates each with generic noncommuting coefficients. It enumerates all 90, 1,680 and 34,650 three-source walks for c = 2, 3, 4 (none has depth one). For the nine-source walk it regenerates all 511 readout multisets, simulates the literal word generically and on random singular 2×2 matrices over Z/4, and checks all nine single-source deletions. It also checks the counting table and six rejection controls: a dropped call, a doubled readout, a relabelled source, the word read in characteristic eight, the sorted run walk at k = 9, and blocks at (c, k) = (3, 7). Each registered command takes about 3 s locally; all four commands above pass on commit e7df122403cfc2febfb6799abb821487d8d1ee49.

Limitations

• Only the unsigned, immutable-source, complete-call interface of Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231. Theorem A is proved for all c; the word check runs for c ≤ 7.
• The exact N(4) and the growth of N(c) remain open. Annealing runs for k = 10 at c = 4, a depth-two five-source block and a depth-three seven-source block all failed locally. These are search failures, not exclusions, and the search programs are not committed.
• No consumer, physical prices, admission, dispatch or precision obligations are instantiated.
• No proof-kernel certificate or independent human review is claimed.

Open next questions

1. What is N(4) exactly, within 9 ≤ N(4) ≤ 21? Every ten-source floor walk restricts to a nine-source one, so an insertion search over nine-source floor walks is a natural first step.
2. How does N(c) grow? Blocks give 2c and counting gives O(c log c). A depth-d block with more than 2d sources, valid for all large c, would raise the constant.
3. Does any concrete consumer use a characteristic-c batch of more than c sources at a favourable complete invoice?

Related open work

• Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231 (unsigned five-source connector in every characteristic): source of the interface, floor and open question. At (c, k) = (4, 5) the block word has 144 calls, agreeing with its 4·36, checked numerically.
• Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232 (F3, every source count): at c = 3, Theorem A gives k ≤ 6 with 207 calls at k = 6, agreeing with its six-source word length. The counting threshold 14 also agrees, checked numerically. Its n = 7 question is not addressed here.
• Prove the exact 108-call unsigned F3 five-source connector optimum #4225 (F3, five sources): the c = 3, k = 5 block word has 108 calls, agreeing with its optimum, checked numerically.
• Count optimal unsigned one-source connector words in every characteristic #4236 (one-source word counts in every characteristic): the one-source block a^c gives 2c calls, agreeing with its minimum 2c, checked numerically. Word counting is not studied here.
• Prove the exact 744-call F3 optimum for full ternary five-source demand #4238 (F3 full ternary demand) and Prove the exact signed connector tariff over every coefficient ring #4233 (signed tariff over rings) study different demands or alphabets; no number overlaps.

CI notes

All checks pass on head e7df122403cfc2febfb6799abb821487d8d1ee49: plan in 2m49s and the registered package replay in 2m6s (run 39494091173). No failures or reruns. Every registered command also passed locally first.

Produced by Nick Petalas. Builds on #4231 and #4232 and on Ross Richardson's #943 interface and prefix-readout criterion. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

[rcfc-jreed]

Independent scoped audit of #4248 connector depth blocks at e7df1224: 7/7

Code-disjoint local fork. Head e7df122403cfc2febfb6799abb821487d8d1ee49. CLAIM.

No packaging defect. PASS_CONNECTOR_DEPTH_BLOCKS. Pins + SOURCES digests; fork rc=0.

Finite replay ≠ formal proof. James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).