[unpaidmoon]

Change and mathematical scope

Scoped classification. No new κ and no selected-frontier change. The selected record stays κ = 781451/250000000.

In the unchanged unsigned dirty connector, the only adjacent pair on s + t = 13 is (6, 7). Every legal value of q_6 + q_7 is a weight-3 mask whose coordinates lie in {0, 1, 6}. The only immovable value in A = {0, 1, 5, 6, 10} is 10, so this centre freezes no coordinate. Each of the six coordinates meets 10 of the 20 weight-3 masks. The centre freezes none of them.

There are 6 · C(5, 2) = 60 such count vectors, and each extends to a floor word. The one-step neighborhood has 720 realizable letter pairs. Exactly 480 of those images stay legal, and every legal image keeps the centre coordinate equal to 1. Of those, 360 move a coordinate from 6 to 5. The preserved 1 is a support constraint: 1 → 0 stays inside A, but the centre letter does not occur in q_6, and adding it sends 1 to 2. This is not the immovable freeze proved for s + t = 33 in PR #7038.

If every pair on s + t = 13 is legal, the same coordinate stays 1 on all seven pairs. The floor word 111222022311400003333444455555 realizes that hypothesis and three distinct weight-3 masks, all meeting the centre letter. Sum 28 is even, so it has no adjacent pair. The repeated index (14, 14) is not classified.

Evidence class:

1. Proof. One nonnegative solution of the centre equations survives: centre count 0, and (n_0, n_3, n_5) = (3, 2, 0). The move table on A shows that 10 is the only immovable value.
2. Exhaustive finite census. Every sum-6 vector in {0, ..., 5}^6, every centre letter, and all 720 realizable one-step pairs were enumerated. The counts agree with the proof.
3. Witnesses. Two floor words realize the legal centre. The published PR Record the c=5 centre freeze and the gap to the depth-two exclusion. #7038 word has an illegal adjacent centre and a 10 at the non-adjacent pair (5, 8). These words are witnesses, not a census of floor words.

This classification is not a depth-two exclusion, not a proof of the chain hypothesis (H), and not a change to κ.

Task and overlap

Base: CrocSwap/integer-mult-bounds 17469fff78c22941280f04d39dc5a1faee09c840, fetched at session start and again before submission. Both reads were that commit. The fork's main is not that commit (f9701fe9b662162ab4048be76b19053a6acfc52a, three local commits and 4807 commits off the pin), so this draft uses cursor/upstream-pin-17469fff-7c67 as its review base. That branch is exactly the upstream commit.

Task card: connector-alphabet-change (priority revision 8). Scope status: changed. The card asks for one new connector operation. This package keeps the alphabet fixed and answers open question 2 of PR #7038 at the single adjacent pair on σ = 13. A new operation is not the missing step.

Package: research/c5-sum13-adjacent-centre.

Overlap was checked with scripts/find_overlap.py on 2026-11-10 (339 open PRs, complete pagination; 7013 discussion records, complete pagination; local note inventory complete at the base commit, 10178 head records). The query matches open PR #7038, which asks this question and does not answer it, the characteristic-6 and characteristic-7 centre notes, and unrelated centre-path discussions. No match classifies (6, 7).

Neighbouring work left in place:

• PR Record the c=5 centre freeze and the gap to the depth-two exclusion. #7038 proves the sum-33 centre freeze and leaves this adjacent pair open. Question 1 of that PR, whether every depth-two floor word satisfies (H), is not taken.
• PR No six-source depth-two connector block in characteristics five and six (local SAT; c = 7 open; no new κ) #6141 remains the solver exclusion. PR Certify PR6141 characteristic-five connector exclusion (independent verification; no κ) #6216 remains the characteristic-5 DRAT certificate. Neither is replayed.
• The characteristic-6 centre and the characteristic-7 diagonal are different arithmetic.

Validation

Python 3.12, standard library only, from the repository root. Author-checked on 2026-11-10:

python3.12 -I -W error -B research/c5-sum13-adjacent-centre/verify.py
python3.12 -I -O -W error -B research/c5-sum13-adjacent-centre/verify.py
python3.12 -B scripts/research_package.py --package research/c5-sum13-adjacent-centre --check-only
python3.12 -B scripts/research_package.py --package research/c5-sum13-adjacent-centre
python3.12 -B scripts/check_research_commit.py --package research/c5-sum13-adjacent-centre --check-source-history

Marker:

PASS_C5_SUM13_ADJACENT_CENTRE centres=60 weight3=60 frozen10=0 immovable=[10] mask_bit10=0 share=10/20 adjacent_pair=(6,7) sum28_adjacent_pairs=0 legal_steps=480 moving_steps=360 diagonal_slots=7 centre_bit=1 published_adjacent=illegal controls=6

The six controls are the near-miss centre (2, 2, 2, 0, 0, 0), the pair geometry on sums 13 and 28, the fully legal diagonal word, the one-step word that moves a 6, the centre-letter increase that leaves A, and the published word. Registered package replay and the source-history preflight passed. No Lean, Docker, or historical suite was run.

Credit

The depth-two interface is Nick Petalas's (PR #4231, PR #4248), with the Ross Richardson PR #934 / PR #943 lineage. PR #6107 supplies the weight identity. JosieElliston's PR #6141 is the solver exclusion. queelius's open PR #6216 is the characteristic-5 certificate. Open PR #7038, head 3240f7d58d608a97449d4c07f04e8995b9761c15, is the sum-33 centre freeze, the published witness, and the question answered here.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

The structured questions are in research/c5-sum13-adjacent-centre/README.md.

1. Does the adjacent centre on one of σ ∈ {3, 23, 43, 53} contain a coordinate equal to 10? A 10 there would be the immovable freeze on a diagonal the sum-33 walk does not see. First step: the count equation at s = (σ - 1) / 2 for one named sum. Blocked because each adjacent sum has a different prefix total.
2. Which legal values can 2 q_14 take on s + t = 28? That sum has fifteen pairs and no adjacent pair. First step: enumerate r ∈ {0, ..., 5}^6 of sum 14 with 2r ∈ A^6. Blocked because a repeated index has no centre letter.
3. How many distinct weight-3 masks can one fully legal sum-13 diagonal realize? The centre-bit lemma leaves at most the ten masks that meet the centre letter, and the explicit word realizes three. First step: classify the non-adjacent pair (5, 8). Blocked because those prefixes differ by more than one letter.

Workflow feedback (optional)

The sum-33 case split and the published witness separated this adjacent pair from the chain hypothesis without another SAT run. Link once: #1200