[unpaidmoon]

Change and mathematical scope

Partial obstruction, not a depth-two exclusion and not a κ change. In the unchanged unsigned dirty connector, every legal centre of a six-source length-30 floor word on s + t = 33 is a weight-3 mask, and each such centre freezes a coordinate sum of 10. The diagonal step cannot move 10 inside {0,1,5,6,10}, so the chain hypothesis

(H) S(s) = q_s + q_{33-s} is legal for every s = 4,...,16

is empty: it would force a coordinate 5 in q_4.

That is a human proof of (H) only. It does not prove that no depth-two word exists. PR #6141's characteristic-5 exclusion remains a solver verdict. Open PR #6216 is a DRAT certificate of that encoding. A solver UNSAT is not this argument, and this argument does not replace the certificate.

The missing implication is: every depth-two floor word satisfies (H). Weights 2 and 6 are disjoint from every pair of sum 33, because a legal vector of sum 33 has mask weight congruent to 3 modulo 5. The witness word 000001231234512345123451234545 is a floor word, realizes a weight-3 mask at s + t = 13, has an illegal centre, and realizes 8 nonzero masks rather than all 63.

The selected conditional κ stays 781451/250000000. No selected-frontier file changes.

Task and overlap

Base: CrocSwap/integer-mult-bounds 7c49fe32c55de02c2c873655c804b16e4c3fef0a. This draft uses cursor/upstream-7c49fe32-pin-9269 as its base because that branch is exactly that commit. The fork's main is an older snapshot, so it is not the review base.

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 the characteristic-5 half of open question 2 in fork draft PR #156, from Nick Petalas's handoff in discussion comment 19384981. A new operation is not the missing step.

Package: research/c5-frozen-centre-gap.

Overlap was checked with scripts/find_overlap.py on 2026-11-10 (885 open PRs, complete pagination; 6934 discussion records, complete pagination; local note inventory complete at the base commit, 8917 head records). A query for s + t = 33 and frozen-14 hits this package and discussion comment 19493159. That comment is the characteristic-7 chain, which this note does not rerun. Neighbouring work is a different question:

• PR No six-source depth-two connector block in characteristics five and six (local SAT; c = 7 open; no new κ) #6141 (merged) leaves the characteristic-5 and characteristic-6 exclusions as solver verdicts and leaves characteristic 7 open.
• PR Certify PR6141 characteristic-five connector exclusion (independent verification; no κ) #6216 (open) certifies the characteristic-5 SAT encoding. This package does not replay or replace it.
• Discussion comment 19384981 records the slot counts and the diagonal move. The 14-slot pigeonhole for the restricted model s + t ∈ {3, 33} is rechecked and left at that strength. It is not (H) and it is not the exclusion.
• Fork draft PR [Closed: Withdrawn]  #156 proves the characteristic-7 diagonal equality case. The off-diagonal replacement at characteristic 7, and the characteristic-6 centre, are not taken here.

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

Marker: PASS_C5_FROZEN_CENTRE_GAP centres=300 type_r0=180 type_r2=120 frozen_coordinates=480 moves=+[0, 5]/-[1, 6] residue3_pairs=93 middle_slots=13 early_realizable_slots=1 witness_masks=8 chain_H=empty controls=7

The written argument is in the package README. The checker exhausts the centre domain, the allowed alphabet, and all 465 pairs. It does not search words and does not rerun the SAT encoding. Both command modes, the registered package replay, and the source-history preflight passed. No Lean, Docker, or historical suite was run.

Credit

The depth-two interface and the weight identity are Nick Petalas's (PR #4231, PR #4248, PR #6107), with the Ross Richardson PR #934 / PR #943 lineage. JosieElliston's PR #6141 is the solver exclusion. queelius's open PR #6216 is the characteristic-5 certificate. Nick Petalas's handoff on 2026-11-03 (discussion comment 19384981) isolates the frozen step. Fork draft PR #156 and discussion comment 19493159 are the characteristic-7 predecessor; this note is only the characteristic-5 half of that draft's second open question.

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

Open next questions

1. Does every six-source depth-two floor word in characteristic 5 satisfy (H)? This implication, with the theorem above, would be a human proof of the exclusion. First step: derive S(16) ∈ {0,1,5,6,10}^6 from the sixteen masks of weights 2 and 6, or show that those masks do not determine the sum-33 diagonal. Blocked because weights 2 and 6 lie on sums congruent to 2 and 1 modulo 5, while 33 is congruent to 3. No transport lemma is proved here.
2. Can the adjacent centre on one other sum σ ≡ 3 (mod 5), σ ≠ 33, freeze a coordinate that meets a positive share of the twenty weight-3 masks? The witness realizes one at sum 13. First step: classify the single adjacent pair on σ = 13 or on σ = 28, at characteristic 5 only. Blocked because the adjacent sum 2s + 1 equals 33 only at s = 16.

Workflow feedback (optional)

The handoff's slot count and the characteristic-7 move were enough to separate the chain hypothesis from the solver exclusion without another SAT run. Link once: #1200