[unpaidmoon]

Change and mathematical scope

Hosts frame 1 of the open #7014 104-by-13 witness as a product of root-line labels, charges every new monomial of that word, and intersects those monomials with the 40 recorded for frame 0 by #7031. No new κ. The selected record stays 781451/250000000.

Frame 1 is the odd-graph clique 13, 26, 131, 148, 262, 273, 392, 415. The only letters identically 1 on these eight lines are the eight frame letters. Their linear parts are the eight vectors of one affine hyperplane in a 4-dimensional space. A subset hosts the row exactly when those vectors span the space, so there are 56 hosting words of length 4 and every subset of length 5, 6, 7, or 8, and none shorter. All 149 hosting products equal

(1 + x1 + x3)(1 + x0 + x1 + x4)(1 + x1 + x2 + x5 + x6)(1 + x0 + x1 + x5 + x7).

The lexicographically first word uses ports 13, 26, 131, 262. The expansion has 120 square-free monomials. Eight degree-1 monomials and the constant are the degree-at-most-1 part. The other 111 have degrees 2, 3, and 4 (counts 27, 47, 37).

Of those 111, 31 lie in the 40 monomials recorded for frame 0. The additional bill, once those 40 are paid, is the other 80. Nine recorded monomials are absent from frame 1. The joint degree-at-least-2 support of the two words has 120 monomials. Fifty-one of the 111 contain x3; the recorded polynomial does not use x3, so all 51 sit in the additional 80. The pairing rank of the 8-by-8 all-ones support is 1 and is not added.

The hosting classification and the length count are written proofs. The chart identity, the admissible-letter test, the 255 subset products, and the boolean expansion (cross-checked by the Möbius transform of the 16-point flat) are exhaustive finite censuses. 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. Frames 2 through 12 are not expanded. The frame-0 census is not rerun; the recorded 40 are the published closed form, locked by ANF hash 26803d29aa3141a357da4aa551195fe6378d3f2e1cba0c4291a4dedbce6ded2a.

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 open question 1 of #7031, for frame 1 only. The intersection is a prerequisite for pricing two rows together. It is not a signed histogram and not a supplier admission.

Nearest overlap is #7031 (head 531ecca652dde63c6010cd6097cb55a994367ce7) and its finding at discussion comment 19514167, which ask this question and do not answer it. #7014 (head 949c0ada285aae619468ae49317c8d9f4df130f7) factors the window and does not host a row. A refreshed overlap search on 2026-11-10 (339 open PRs, 7013 discussion records, complete local note inventory at the base commit) returned only those items for this question. data/frames.json is the #7014 witness as vendored by #7031, hashed c75915c18fc06ac19cc90fb9911209517c071862fdf401b998d5075efc0cda0d. Ports and the E8 line map are the #6933 inputs pinned in SOURCES.json.

Validation

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

python3 -B research/p104-frame1-monomial-intersection/verify.py
python3 -I -O -B research/p104-frame1-monomial-intersection/verify.py
python3 -B scripts/research_package.py --package research/p104-frame1-monomial-intersection
python3 -B scripts/check_research_commit.py --package research/p104-frame1-monomial-intersection --check-source-history

Observed: PASS_P104_FRAME1_INTERSECTION word=13,26,131,262 monomials=120 new=111 intersection=31 additional=80 joint=120 absent_from_frame1=9 and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. Source-history check passes at e7271f9d709149590071f11a29dfe702d427b59a. Failure controls delete one canonical letter, count the 14 length-4 misses, reject a rank-3 prefix, delete the constant monomial, and multiply by a non-frame letter.

Not run: the frame-0 product census, a sum-of-letters search, frames 2 through 12, the selected-supplier replay, make frontier6030-verify, and Lean.

Credit

Frame partition: unpaidmoon, PR #7014. Frame-0 hosting and the recorded forty: PR #7031. 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 2 the same way and intersect its degree-at-least-2 monomials with the joint 120-monomial support of frames 0 and 1. The two-row union is not yet a thirteen-row bill.
2. Can one circuit charge each of the 31 shared monomials once and still host both rows? The intersection compares two separate expansions and does not exhibit a joint word.

Workflow feedback (optional)

#1200