[unpaidmoon]

Change and mathematical scope

No new κ. The selected conditional exponent stays 781451/250000000. This package does not edit certificates/selected-result.json.

On the orthogonal E8 pair obtained by reflecting the pinned D6 anchors of #6959 by one simple root, every coordinate permutation that carries the image pair back to the pinned pair transports the partner supports onto the recorded set. The support-set SHA256 is 23a7b374a6bc62bc2aca47843d52a01fa8e791ed3a98085133829d816c5094e8.

The simple root is index 3 in the system pinned by research/p112-a3-six-flat/geometry.py,

α = (0, -2, 2, 0, 0, 0, 0, 0).

It sends (2, 2, 0, 0, 0, 0, 0, 0) to (2, 0, 2, 0, 0, 0, 0, 0) and (2, -2, 0, 0, 0, 0, 0, 0) to (2, 0, -2, 0, 0, 0, 0, 0). Keeping the second anchor fixed destroys orthogonality. The reflected 30-line list has SHA256 bcc5dc15f642aca7bdcc1249d2588869b6510aa8e08cde8688fe340c665435b5. Its 15 matching frames have F2 incidence rank 10. All 466 vectors of weight at most 2 meet the frame parities exactly on zero and the 15 partner masks.

itertools.permutations visits each of the 8! = 40320 coordinate permutations once, for each of the eight simple roots. Root 3 is the first that moves the pair and admits a coordinate permutation. The carrying permutations are exactly the 2! · 6! = 1440 permutations σ with {σ(0), σ(1)} = {0, 2}, acting by (P_σ v)_i = v_{σ(i)}. Each one transports the partner masks onto

3, 24, 36, 768, 1152, 2112, 98304, 147456, 270336, 528384,
50331648, 75497472, 138412032, 270532608, 537919488

Simple root 0 moves the second anchor to (1, -1, 1, 1, 1, 1, 1, -1). That vector and either pinned anchor have different absolute-coordinate multisets, so no signed coordinate permutation carries the image onto the pinned pair. The unsigned census finds 0 of 40320 permutations. This is scoped to monomial signed permutations. It is not a statement about the rest of the Weyl group, and the spinor complement's frame code is not built.

This is an exhaustive finite census of those two finite domains. It is not a census of all 3780 orthogonal anchor pairs, and it is not a run of the bit-lane iterator.

Task and overlap

Base 17469fff78c22941280f04d39dc5a1faee09c840. Task card replay-runtime, scope refined. The card asks for a measured speedup that keeps coverage. The revision-8 verification row says faster output is not the same coverage until checked. Merged PR #6959 checked one pinned D6 pair and left this reflection as open question 1.

Live inputs are research/d6-punctured-frame-code/ROOTS.json and research/p112-a3-six-flat/geometry.py. The selected record is provenance only.

Ordered overlap check on 2026-11-10 with scripts/find_overlap.py --refresh --include-closed. Coverage: 339 open pull requests, including drafts, fetched 2026-11-10T09:50:29Z, pagination complete; 6175 closed pull requests, fetched 2026-11-10T09:52:16Z; 7007 discussion records, fetched 2026-11-10T09:51:28Z, pagination complete; 10180 local notes at that head, inventory complete. The matches were the pinned D6 package, merged PR #6959, and discussion comment 19507177. Those state the question and do not reflect an anchor. Open #7007 is the P112 weight-16 equality. Merged #6948 is the E6 equality. Neither root list is this one. Open #7017 is the reflected P112 frame, a different root list.

Validation

Python 3.12, standard library, from the repository root. Both registered commands passed in this session.

python3 -B scripts/research_package.py --package research/d6-reflected-anchor-pair --check-only
python3 -B scripts/research_package.py --package research/d6-reflected-anchor-pair

The registered commands are python3 -I -B and python3 -I -O -B on research/d6-reflected-anchor-pair/verify.py. The headline is

PASS_D6_REFLECTED_ANCHOR_PAIR simple=3 permutations=1440 width=30 frames=15 dimension=20 supports=15 vectors=466 hash_match=true spinor_coordinate_permutations=0 engine=false no_new_kappa=true

Failure controls: dropping one support changes the hash; clearing the top bit of a partner mask produces a weight-1 word outside the screened set; the mask of lines 0 and 2 is outside the set; the identity permutation does not carry the image pair; an unreflected partner is not orthogonal to the reflected anchor.

Not run: low_weight_words, the 2^20 Gray walk, research/e7-intersection-counters/engine.py, the E6 and P112 censuses, the spinor complement's frames, the coordinate transport of the non-simple root (0, 2, 0, -2, 0, 0, 0, 0), Lean, Docker, and scripts/research_selected_baseline.py as a scientific replay.

Credit

The pinned D6 lines, the partner-support hash, and the open question are PR #6959, research/d6-punctured-frame-code, discussion comment 19507177. The simple system is research/p112-a3-six-flat/geometry.py, from Nick Petalas's PR3161/PR3162 lineage. Ali Romman's bit-lane iterator from PR #5729 is not rerun.

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

The structured questions are in research/d6-reflected-anchor-pair/README.md.

1. Does the image pair of simple root 0 have the same weight-at-most-2 equality on its own 30-line complement? No coordinate permutation carries that pair to the pinned anchors. First screen all 466 vectors of weight at most 2 on the lines orthogonal to (2, 2, 0, 0, 0, 0, 0, 0) and (1, -1, 1, 1, 1, 1, 1, -1).
2. Does the coordinate root (0, 2, 0, -2, 0, 0, 0, 0), which lies outside the simple system, induce the same transported equality? The package checks that this root is an E8 line, is not simple, and moves the pair onto a different orthogonal pair. It does not transport that image.

Workflow feedback (optional)

The predecessor named the support hash and the reflection step in one open question, and the pinned simple system made the choice of root a finite census rather than a search. A link to the feedback discussion is enough: #1200