[unpaidmoon]

Change and mathematical scope

Hosts frame 0 of the open #7014 104-by-13 witness as a product of root-line labels and charges the new monomials of that word. No new κ. The selected record stays 781451/250000000.

Frame 0 is the odd-graph clique 11, 21, 38, 56, 140, 146, 161, 191. A product in F_2 is 1 only when every factor is 1, so every letter of a hosting word is identically 1 on these eight lines. The only such letters, among all 120 root lines, are the eight frame letters. An exhaustive census of the 255 nonempty products finds no hosting word shorter than length 4, and 56 hosting words of length 4. Every one of those 56 words expands, in the inherited simple-root chart, to the same 48 square-free monomials

(1 + x0)(1 + x1 + x2 + x4)(1 + x5)(1 + x6 + x7).

The lexicographically first word uses ports 11, 21, 38, 140. Eight monomials have degree at most 1. The other 40 have degrees 2, 3, and 4 (counts 17, 17, 6). Those 40 are the new charges for this row. The pairing rank of the 8-by-8 all-ones support is 1 and is not added; it is a different unit.

The factor restriction and the reduction to the four-factor form are written proofs. The census, the 120-by-120 chart identity, and the agreement of the 56 expansions are exhaustive finite checks. This is not a heuristic. It is not a lower bound for sums of letters, another chart, or a non-product circuit. It is not a selected-invoice histogram, so it does not beat the selected profile.

The other twelve frames are not expanded. A monomial shared with one of them would make a joint bill smaller than the sum. That comparison is not made.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840. Task card complex-rank-shape-decoder, priority revision 8. Scope is changed: the card asks whether a decoder or coframe change improves the full signed moment; this package answers the prior question posed by open PR #7014, for frame 0 only. The 40 monomials are a prerequisite for pricing the rows together. They are not a signed histogram and not a supplier admission.

Nearest overlap is #7014 itself (head 949c0ada285aae619468ae49317c8d9f4df130f7) and its handoff at discussion comment 19507691. That pull request factors the window; it does not host a row or charge monomials. data/frames.json is that witness, vendored and hashed (c75915c18fc06ac19cc90fb9911209517c071862fdf401b998d5075efc0cda0d). This package does not reprove the rank-13 factorisation. Ports and the E8 line map are the #6933 inputs pinned in SOURCES.json. A refreshed overlap search on 2026-11-10 (317 open PRs, 6973 discussion records, complete local note inventory at the base commit) returned only #7014 and that handoff comment for these terms.

Validation

Python 3.12, standard library only. From the repository root:

python3 -B research/p104-one-frame-indicator-word/verify.py
python3 -I -O -B research/p104-one-frame-indicator-word/verify.py
python3 -B scripts/research_package.py --package research/p104-one-frame-indicator-word
python3 -B scripts/check_research_commit.py --package research/p104-one-frame-indicator-word --check-source-history

Observed: PASS_P104_ONE_FRAME_INDICATOR word=11,21,38,140 monomials=48 new=40 and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. Source-history check passes at 531ecca652dde63c6010cd6097cb55a994367ce7. Failure controls delete one canonical letter, count the 14 length-4 misses, delete the constant monomial, and reject a non-frame letter.

Not run: selected-supplier replay, make frontier6030-verify, Lean, and any expansion of frames 1–12.

Credit

Frame partition: unpaidmoon, PR #7014. Ports and the odd-graph window: PR #6933. Label pairing: the inherited simple-root chart. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

1. Host frame 1 the same way and intersect its degree-at-least-2 monomials with these 40. A shared monomial is the first place two rows can interact.
2. Decide whether a sum of letters can host this row with fewer than 40 new monomials. The product census does not cover sums.
3. Name these 40 monomials as primitives on the selected Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030 invoice before asking whether thirteen rows beat the profile. No κ change is available until that naming exists.

Workflow feedback (optional)

#1200