[unpaidmoon]

Change and mathematical scope

One independent generating set of length 27 inside the P112 punctured frame code makes research/e7-intersection-counters/engine.py return the eight-word set at batch size 20. The SHA256 of the seven positive words is ff8861dab18040c03d403a860773218df7c120d03770d701afb61b78c941e3d3.

The seven published supports are pairwise disjoint and each has weight 16, so they partition the 112 coordinates. The XOR of a subset has weight 16 times the subset size. In their span, the words of weight at most 16 are exactly zero and those seven supports. That is a proof about these seven integers.

The 27-tuple is those supports followed by kernel basis vectors 0 through 19, the first twenty basis vectors that remain independent under the engine's high-bit elimination. Every support is a generator, so the span contains all seven. A binary-reflected Gray code visits each of the 2^27 = 134217728 span vectors once. Eight have weight at most 16, and they are the same eight words. This is an exhaustive finite census of one span. The coefficient check i xor (i >> 1) is enforced at every step.

The published engine, called once at batch size 20, yields those same eight words, with no duplicate, no missed support, and no extra word. Batch size 20 places 20 generators in the lane block and leaves 128 high-offset batches. The published dimension guard is unchanged: length 28 raises dimension exceeds 27 before any yield. research/e7-intersection-counters/engine.py is not edited.

The ordered 27-prefix of the same kernel remains a different tuple. It contains the first three supports and misses the last four. That membership split is recomputed from the kernel. The prefix stream is not enumerated, and the 36-generator input is not sent to the engine.

The selected conditional exponent stays 781451/250000000. No timing is recorded.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 on canonical CrocSwap main.

Task card replay-runtime, priority revision 8. Scope status: refined. The card asks for a measured speedup that keeps coverage. Priority revision 8 requires the yielded sets to be compared before a faster replay counts as the same coverage. Open PR #7034 records that this engine refuses the 36-generator basis at every legal batch size. Its second open question asks whether some other independent generating set of length at most 27, still inside this code, returns the eight-word set at batch size 20. This package names one such tuple and compares that stream. It does not rerun the bit-lane census of #7007, the reflected-frame comparison of #7017, or the E6 and D6 engine comparisons.

scripts/find_overlap.py --refresh --include-closed on 2026-11-10 searched ff8861dab18040c0, length at most 27, admissible 27, ordered 27-prefix, and p112 intersection-counter. Coverage: local notes, 10178 records, complete inventory at 17469fff78c22941280f04d39dc5a1faee09c840; 339 open PR bodies at 2026-11-10T10:29:31Z; 7013 discussion records at 2026-11-10T10:30:33Z; 6175 closed PR bodies at 2026-11-10T10:31:23Z. Five matches: #7034, discussion reply 19513866, discussion reply 19507785, #7007, and #7017. #7034 and reply 19513866 leave this 27-tuple question open. #7007 and reply 19507785 record the bit-lane set and leave this engine unrun. #7017 separates its reflected bit-lane hash from this engine. None names this 27-tuple. The helper searches titles and bodies, not code diffs.

Validation

Python 3.12 and the standard library. scripts/check_research_commit.py --check-source-history passed at 93cd7d9f5d1108d28dca7ef0409acc8f0fd07f95. scripts/research_selected_baseline.py printed PASS_RESEARCH_BASELINE_BINDINGS_ONLY and is provenance only.

python3 -B scripts/research_package.py --package research/p112-length27-intersection-return --check-only
python3 -I -B research/p112-length27-intersection-return/verify.py
python3 -I -O -B research/p112-length27-intersection-return/verify.py
python3 -B scripts/research_package.py --package research/p112-length27-intersection-return

Registered normal and optimized runs passed. Direct output:

PASS_P112_LENGTH27_INTERSECTION_RETURN batch=20 generators=27 span_words=134217728 words=8 hash=ff8861dab18040c03d403a860773218df7c120d03770d701afb61b78c941e3d3 equal=True missed=0 extra=0 timed=false no_new_kappa=true

Failure controls: dropping one support changes the hash; a repeated generator raises dependent generators; length 28 raises dimension exceeds 27; batch size 10 raises batch count exceeds 65536. The bit-lane enumerator rejects the injected keywords with TypeError.

Not run: batch size 11, a second 27-tuple, the 36-dimensional bit-lane census, a copy of the engine with the dimension guard raised, the E6 and D6 comparisons, the E7 census, the #5729 timing harness, Lean, and Docker.

Credit

The 120-line order and the punctured frame are the model in research/p112-rank6-frame-coverage, from the PR1226/PR1323 listing used by Cinna Davis. The intersection-counter engine is Nick Petalas's research/e7-intersection-counters/engine.py. The injected population and threshold routines are Ali Romman's PR #5729. The seven supports, the eight-word hash, and the bit-lane equality are open PR #7007. The 36-generator refusal and the question answered here are open PR #7034. The reflected-frame comparison is open PR #7017.

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 this same 27-tuple return the eight-word set at batch size 11? Batch size 20 uses 20 lane generators and 128 high-offset batches. Batch size 11 is the smallest legal batch for 27 generators (27 - 11 = 16) and uses 65536 high-offset batches. First rerun the comparison with batch_bits 11 and the same 27 integers, weight bound, and hash. This package stops after batch size 20. The published dimension guard stays 27.

2. Does one other admissible 27-tuple inside this code, whose span contains the seven supports and which is not this extension by basis vectors 0 through 19, return the same stream at batch size 20? This census covers one span. First name that tuple and compare one batch-size-20 stream with the hash above. The 36-dimensional bit-lane census is open PR Record the P112 punctured frame code's weight-16 set equality. #7007 and is not replayed here.

Workflow feedback (optional)

The published dimension cap made the informative object a 27-tuple whose span contains every support, rather than another call on the refused 36-generator basis. A link to the feedback discussion is enough: #1200