[unpaidmoon]

Change and mathematical scope

On the 112 lines left by the next simple root that moves a punctured line, one half-split returns exactly the transported images of the seven weight-16 supports recorded in #7007. The order is the one pinned by #7017: simple-root index, then punctured-line index. Simple root 0 is the reflection #7017 already compared. The next simple root that moves a punctured line is index 3,

α = (0, -2, 2, 0, 0, 0, 0, 0),

acting on punctured line (2, 2, 0, 0, 0, 0, 0, 0). The image frame is a different orthogonal 8-frame. The code is again 36-dimensional, the least positive weight is 16, and those seven transported words are the whole weight-at-most-16 set.

This is an exhaustive finite census of one reflected code, together with an exhaustive 64-pair census of which simple roots move the puncture. It is not a timing claim, not a consumer installation, and no new κ is claimed. The selected value stays 781451/250000000.

The same reflection fixes four punctured lines and moves four. The rebuilt complement, in increasing line index, has SHA256 36e3ad43dbfd53d71b20cfac522fbf9c0c1c065054994b9c81767a228d7e6f3c.

The transported supports, in the recorded order, are

68451290880
1031047741680
280375477678095
4814039611113066403813140725760
15468061711732307582943976488960
1216944811960755906164784113909760
3955069945251226348097480121384960

Their SHA256, of json.dumps of that sorted list, is 0ab33f7ea89182ea1cbc5ca06d5f3bdcaeb72a08aef3eaceb33e6a43bbb96636.

The seven untransported integers fail a frame parity of this new code. The half-split visits all 2^36 kernel words. Its weight distribution equals the distribution recorded for the unreflected frame. That is a comparison of enumerators from two different bases, not a replay of the unreflected census or of the #7017 census. The free coordinates differ from the #7017 basis.

Simple roots 1, 2, 4 and 6 fix every punctured line. Simple roots 5 and 7 also move four lines each. Their frames were not built and their half-splits were not run.

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. The revision-8 verification row says a published faster replay is not equivalent coverage until checked. Open #7017 recorded the first moving pair and left the next simple root as its first open question. This package answers that question for simple root 3 only. It does not rerun the #7017 census, the unreflected census of #7007, the D6 equality of #6959, or the E6 equality of #6948, and it does not run low_weight_words or the intersection-counter comparison.

scripts/find_overlap.py --refresh on 2026-11-10 read 339 open pull requests (fetched 2026-11-10T10:25:29Z, pagination complete), 7013 discussion records (fetched 2026-11-10T10:26:30Z, pagination complete), and 10178 local notes at this base, with a complete local inventory. The matches were #7017 and discussion comment 19514152, which record the first pair and do not build this frame. A search for “simple root 3” also hits discussion comment 19515229, the D6 reflected-anchor census. That comment uses the same coordinate vector on the #6959 anchors, a 30-line rank-10 code. It does not rebuild this 112-line list or compare these seven supports.

Validation

Python 3.12.3, gcc 13.3.0, and the standard library, on 4 cores. The registered command measured about 69 seconds here, including a 67-second half-split.

python3 -B scripts/research_package.py --package research/p112-next-moving-simple-root --check-only
python3 -I -B research/p112-next-moving-simple-root/verify.py
python3 -B scripts/research_package.py --package research/p112-next-moving-simple-root

check_research_commit.py --check-modes --check-source-history passed at ff8c16cf81f3ad0aa5db866cbe2a11d48757ddba. The source-history pin is base 17469fff78c22941280f04d39dc5a1faee09c840.

The registered replay printed

PASS_P112_NEXT_MOVING_SIMPLE_ROOT width=112 frames=1260 dimension=36 minimum=16 supports=7 codewords=68719476736 split=18 simple=3 punctured=0 moved_lines=4 fixed_roots=4 later_roots=5,7 same_frame=false first_pair=false unreflected_census=false engine=false no_new_kappa=true

Not run: a second split, the unreflected half-split, the #7017 half-split, low_weight_words, research/e7-intersection-counters/engine.py, the E6 and D6 censuses, the frames of simple roots 5 and 7, Lean, Docker, and scripts/research_selected_baseline.py as a scientific replay. The selected record is provenance only.

This census does not cover every simple reflection, and it is not an unrestricted nonexistence result.

Credit

The 120-line order and the unreflected puncture are the model in research/p112-rank6-frame-coverage, from the PR1226/PR1323 listing used by Cinna Davis. The simple system is the one in research/p112-a3-six-flat/geometry.py, from Nick Petalas's PR3161/PR3162 lineage. The seven supports, their hash, and the unreflected weight distribution are recorded in open #7007. The first moving pair, the question, and the doubling half-split are from open #7017. This scanner reads the simple-root-3 basis. The D6 use of the same simple-root vector is discussion comment 19515229.

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 simple root 5, the next root after this one in the same pinned order that moves a punctured line, induce the same set equality on its own rebuilt list? The 64-pair scan records that root, and simple root 7 after it, but neither half-split was run. First rebuild the simple-root-5 complement and compare one half-split with that root's transported supports.
2. Does low_weight_words on this simple-root-3 36-generator basis return the same seven transported supports? This is the bit-lane iterator on the basis pinned here, compared with hash 0ab33f7ea89182ea1cbc5ca06d5f3bdcaeb72a08aef3eaceb33e6a43bbb96636. It is not the Record the reflected P112 frame's transported weight-16 equality. #7017 basis and not hash 9ce7e5edc72ee3104d62cb068146b73840e9a3c42c4132403dece70c8146cccb.

Workflow feedback (optional)

One half-split of this 36-dimensional length-112 code fits in the 600-second command limit. The 64-pair movement scan is immediate, so the costly part is the kernel enumeration of the one selected root. A link to the feedback discussion is enough: #1200