[unpaidmoon]

Change and mathematical scope

Scoped census. No new κ. The selected conditional exponent stays 60403/20000000.

A difference basis of (\mathbb{F}_2^6) is a set whose pairwise distinct sums are all 63 nonzero vectors. No 12-point or 13-point set is a difference basis. The minimum order is 14, with 53,340 normalized witnesses. One witness is {0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 63}.

On the six-source connector, prefix differences can cover every nonzero mask only when there are at least 14 distinct prefixes. Revisits shrink that set, so every covering closed walk is simple and has length at least 14. The set-level query count (2 \cdot 63 + |P|) is at least 140, not (2 \cdot 63 + 12). The 14-point set is not a walk and not a priced program, so 140 is not shown to be attained.

Task and overlap

Base fbc9ea0249746acca4f504d4fda2b84e0e36cf0d. Card connector-alphabet-change, scope refined. Package research/f2-six-difference-basis.

This is item 10 of the ordered eligible list in the package README. It answers open question 2 of research/connector-triple-query-floor. It does not rerun that package's nine-point census and does not search its length-10 triple leaf question.

Nearest related work: Aaron Wang's research/connector-trio-cycle25-seed1b82df excludes one twelve-call profile (eleven singleton edges and one closing triple). This census excludes every 12-point and every 13-point set, whether or not the set is that profile. Ross Richardson's research/universal-connector-boundary supplies the paired-readout model.

Validation

CPython 3.12 and cc, from the repository root:

python3.12 -B scripts/research_package.py --package research/f2-six-difference-basis

Observed:

PASS_F2_SIX_DIFFERENCE_BASIS min_order=14 cover12=57 cover13=62 sets12=4187106 sets13=36288252 sets14=264385836 hits14=53340 query_floor=140 no_new_kappa=true
PASS_F2_SIX_DIFFERENCE_BASIS_INDEPENDENT hits12=0 best12=57 order14=1
PASS_REGISTERED_RESEARCH_PACKAGE research/f2-six-difference-basis outcome=completed

The twelve-point census is pure Python. The thirteen- and fourteen-point censuses are the compiled loop in census.c. The two coverage routines agree on all 4,187,106 normalized twelve-point sets and on the recorded extremal witnesses. Controls: the (\mathbb{F}_2^3) predicate finds order 5 and rejects order 4; a coordinate hyperplane misses the odd coset; deleting any point of the order-14 witness drops the cover.

Not run: Lean, Docker, the selected cold replay, and any search for a closed walk.

Credit

The question is open question 2 of unpaidmoon's research/connector-triple-query-floor. The five-source model is Ross Richardson's. The twelve-call profile exclusion is Aaron Wang's.

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 simple closed walk of length 14 realize a 14-point difference basis with legal connector steps? First order the recorded witness so every edge has weight at most 3 and the last vertex returns to the first. An ordering is not yet a priced program.
2. Does any closed length-10 five-source walk with one triple and multiplicity (4, 2, 2, 2, 2) attain leaf bill 120 on prices (1, 4, 10, 16, 28)? That remains open question 1 of the triple-source floor.