[unpaidmoon]

Change and mathematical scope

Scoped classification. No new κ and no selected-frontier change. The selected record stays κ = 363005817/100000000000.

Chosen sum, fixed before the argument: σ = 43. In the unchanged unsigned dirty connector, the adjacent centre is the pair (21, 22). Every legal value of q_21 + q_22 is a weight-3 mask with either two or three coordinates equal to 10. There are two shapes, sixty centres each:

• centre count 0, other counts two 3s and three 5s: one 1, two 6s, three 10s;
• centre count 2, other counts three 3s and two 5s: one 5, three 6s, two 10s.

The only immovable value in A = {0, 1, 5, 6, 10} is 10, so each of those coordinates is frozen. It is a mask bit 0 and it never sits on the centre letter. All 120 vectors extend to floor words. The one-step neighborhood has 540 legal images (360 stationary, 180 moving). Every legal image keeps the coordinates equal to 10.

If every slot S(s) for s = 14, ..., 21 is legal, those same 10s survive to q_14. A prefix of length 14 can contain a 5, so the numerical stop used for sum 33 does not apply. Emptiness of that eight-slot chain is open. The sums σ ∈ {3, 23, 53} are not classified.

This is not a depth-two exclusion and not a change to κ.

Task and overlap

Base: CrocSwap/integer-mult-bounds e54523517d0f03e58352e647c403a5af3e525a57, fetched at session start and again before submission. Both reads were that commit. The fork's main is not that commit, so this draft uses cursor/upstream-e545235-base-5a3f 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 1 of PR CrocSwap#7044 at the single named sum σ = 43. A new operation is not the missing step.

Package: research/c5-sum43-adjacent-ten.

Overlap: local scripts/find_overlap.py --offline at the base commit searched the complete note inventory (10178 head records) and did not find a classification of (21, 22). The remote snapshot for that command was absent. A full title list of the 500 open upstream PRs, paginated through an empty last page, contains this question only as open PR CrocSwap#7044, which asks it and does not answer σ = 43. The other c = 5 centre titles on that list are PR CrocSwap#7038 and PR CrocSwap#7058. PR CrocSwap#7058 leaves the adjacent-centre question untouched. An earlier body search for adjacent centre returned those three PRs; the search API then rate-limited, so this is not a second full-text refresh. The last 15 comments on discussion CrocSwap#1166 at session time do not mention this sum.

Neighbouring work left in place:

• PR Prove the c=5 adjacent centre on sum 13 freezes no coordinate. CrocSwap/integer-mult-bounds#7044 proves the sum-13 adjacent centre freezes no coordinate and leaves σ ∈ {3, 23, 43, 53} open. This package takes 43 only. Questions 2 and 3 of that PR are not taken.
• PR Record the c=5 centre freeze and the gap to the depth-two exclusion. CrocSwap/integer-mult-bounds#7038 proves the sum-33 centre freeze. It is not rerun.
• PR No six-source depth-two connector block in characteristics five and six (local SAT; c = 7 open; no new κ) CrocSwap/integer-mult-bounds#6141 remains the solver exclusion. Merged PR Certify PR6141 characteristic-five connector exclusion (independent verification; no κ) CrocSwap/integer-mult-bounds#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 2026-11-11:

python3.12 -I -W error -B research/c5-sum43-adjacent-ten/verify.py
python3.12 -I -O -W error -B research/c5-sum43-adjacent-ten/verify.py
python3.12 -B scripts/research_package.py --package research/c5-sum43-adjacent-ten --check-only
python3.12 -B scripts/research_package.py --package research/c5-sum43-adjacent-ten
python3.12 -B scripts/check_research_commit.py --package research/c5-sum43-adjacent-ten --check-source-history

Marker:

PASS_C5_SUM43_ADJACENT_TEN sigma=43 centre=(21,22) centres=120 shape0=60 shape2=60 frozen10=300 min10=2 max10=3 weight3=120 steps=540 stationary=360 moving=180 slots=8 earliest_prefix=14 capacity_has_5=yes controls=6

The six controls are the near-miss centre (5, 5, 5, 5, 1, 0), the eight sum-43 slots and the unique adjacent pair (21, 22), a length-14 vector that contains a 5, the shape-0 witness word, the shape-2 witness word, and a count-feasible subtraction of a 10 that leaves A. Registered package replay and the source-history preflight passed. No Lean, Docker, or historical suite was run.

Evidence class: (1) a proof by the count equations at prefix total 21 and the one-step case split; (2) an exhaustive finite census of {0, ..., 5}^6 and of the realizable one-step neighborhood; (3) two floor-word witnesses. The census is not a heuristic search. A witness is not an unrestricted exclusion.

Credit

The depth-two interface is Nick Petalas's (PR CrocSwap#4231, PR CrocSwap#4248), with the Ross Richardson PR CrocSwap#934 / PR CrocSwap#943 lineage. PR CrocSwap#6107 supplies the weight identity. JosieElliston's PR CrocSwap#6141 is the solver exclusion. queelius's merged PR CrocSwap#6216 is the characteristic-5 DRAT certificate. Open PR CrocSwap#7038, head 3240f7d58d608a97449d4c07f04e8995b9761c15, is the sum-33 centre freeze. Open PR CrocSwap#7044, head 53b56df645f591ea26da3c6f5b478dd88c50e1f8, is the sum-13 classification and the question answered here for σ = 43 only.

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

Open next questions

The structured questions are in research/c5-sum43-adjacent-ten/README.md.

1. Does any floor word make S(s) legal for every s = 14, ..., 21 on s + t = 43? The centre lemma says any such word carries at least two fives in q_14. First step: classify S(19) from a shape-2 centre. Blocked because those prefixes differ by more than one letter.
2. Does the adjacent centre on σ = 23 contain a coordinate equal to 10? First step: the count equation at s = 11. Blocked because the prefix total is not 21.
3. Does the adjacent centre on σ = 3 contain a coordinate equal to 10? First step: the count equation at s = 1. Blocked because that prefix length is a different equation.
4. Does the adjacent centre on σ = 53 contain a coordinate equal to 10? First step: the count equation at s = 26. Blocked because that prefix length is a different equation.

Workflow feedback (optional)

Naming σ = 43 before the case split kept the other three prefix totals out of the census. Link once: CrocSwap#1200