[unpaidmoon]

Change and mathematical scope

Scoped existence result. No new κ. The selected conditional exponent stays 605457647/200000000000 (0.003027288235, CrocSwap#5413).

The set

P = {0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 55, 56}

is a difference basis of (\mathbb{F}_2^6). The closed walk

0-32-40-56-24-16-8-4-5-55-6-2-3-1-0

visits each point once. Its fourteen steps have Hamming weights

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 1

so the maximum is 3 and the sum is 20.

Among the 53,340 normalized bases that contain 0 and every weight-1 vector, 53,270 have some such walk and none has total weight below 20. The displayed basis of research/fixed-fourteen-basis-walk, whose point 63 lies at distance at least 4 from the rest, is one of the 70 exceptions.

Under that note's paired-readout accounting this walk still costs 140 root queries and 146 map applications. That matches the displayed weight-4 cycle and is four above the generic six-source DUO bill of 142. This is not a physical connector program and does not change the alphabet.

The minimum 20 is only inside the normalized slice. A linear change of basis does not preserve Hamming weight, so a basis that omits some weight-1 vector could still have a lighter weight-at-most-3 cycle.

Task and overlap

Base commit 32890bc5de034ab8fc93ac88ead7279a6a416d04. No live inputs. Provenance pins: certificates/selected-result.json blob fc3d4ba105e1f097d0d7d051a308281e86ab1446, docs/research/agent-task-catalog.json blob f126dd2e2dad2b669e75d1d0d0bd05c600725602.

Task card connector-alphabet-change, priority revision 7. Scope status: refined.

The card asks whether one added connector operation has a favorable completed budget. This package does not add an operation. It answers open question 1 of research/f2-six-difference-basis, left open by research/fixed-fourteen-basis-walk: whether any 14-point difference basis of (\mathbb{F}_2^6) has a simple closed walk of step weight at most 3. The displayed witness needs weight 4. The alphabet, source count and paired-readout contract stay fixed. The result does not lower the connector bill or move κ.

Eligible list for session r46, ordered as in research/f2-six-difference-basis: open handoffs from the 19 October index (discussion comment 19029944), then priority-table rows whose stated next decision is still open, then catalogue cards that are not already those rows. Dropped before the cut: work this account already delivered, results on current main 32890bc5, questions covered by an open pull request, and topics already on a fork cursor/* branch.

1. A caller that accepts (w, q, v_N) without restoring the original source slots (comment 19030045; open question 1 of research/p104-role-exchange-caller-audit).
2. This package. Fewer than 14 items remained, so the assigned index is (14-1) mod 2 = 1.

Nearest predecessors: research/f2-six-difference-basis (the census and the open walk question) and research/fixed-fourteen-basis-walk (the weight-4 theorem on a different basis). No open upstream PR title matched a fourteen-point weight-at-most-3 walk when this branch was cut.

Validation

From the repository root, Python 3.12 and cc:

python3 -B research/f2-weight3-basis-walk/verify.py
python3 -B scripts/research_package.py --package research/f2-weight3-basis-walk

Expected line:

PASS_F2_WEIGHT3_BASIS_WALK steps=14 max_weight=3 applications=20 normalized_hits=53340 normalized_cycles=53270 normalized_min=20 queries=140 map_applications=146 no_new_kappa=true

verify.py checks the explicit cycle, coverage, the deleted-point control, the displayed distance obstruction, and an independent Held-Karp minimum of 20 on the witness. It then compiles census.c with cc -O3 -std=c11 -Wall -Werror and requires the normalized counts. The census is a finite search of the (\binom{57}{7}) normalized candidates, not a proof over every (\mathrm{GL}(6,2)) image.

Not run: Lean, Docker, the selected cold replay, and any search outside the normalized slice.

Credit

The order-14 census and the walk question are unpaidmoon's research/f2-six-difference-basis. The displayed weight-4 theorem and the paired-readout accounting are Nick Petalas's research/fixed-fourteen-basis-walk. The six-source interface is Ross Richardson's research/universal-connector-boundary. Aaron Wang's CrocSwap#2128 and Tairitsu-Aya's CrocSwap#2156 supply the generic DUO bill cited as a comparison, not replayed here. Volodymyr Gavrylkiv's six-dimensional difference-basis value is the one already credited in CrocSwap#2156.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review and no formal-kernel acceptance are claimed.

Open next questions

1. Does a 14-point difference basis that omits some weight-1 vector have a weight-at-most-3 closed walk with source-application total 16? That is the generic DUO application count. First drop the standard-basis requirement and search sets translated to contain 0. The blocker is that (\mathrm{GL}(6,2)) does not preserve Hamming weight, so the 53,340 normalized bases do not represent every weight pattern.
2. The length-10 five-source leaf question in research/connector-triple-query-floor stays open. This six-source walk does not decide it.

Workflow feedback (optional)

The handoff-then-priority-table-then-catalogue cut recorded on current main left two eligible items, so item 14 is (14-1) mod 2. Discussion 1200 is the feedback thread; no separate session report.