[unpaidmoon]

Change and mathematical scope

Proves that no F_2 sum of root-line letters hosts frame 0 of the #7031 indicator, on the 104 P104 lines or on all 120 root lines. No new κ. The selected record stays 781451/250000000.

The explicit sum λ_11 + λ_21 + λ_38 + λ_56 cancels in the inherited simple-root chart, so its algebraic normal form is 0. That is an empty monomial list: zero monomials of degree at most 1 and zero new monomials (degree at least 2, the same cut as #7031). The value is 0 on the eight frame lines, so this sum does not host the row.

Every sum of the 120 letters is affine, so every sum has zero new monomials. The four P104 ports 478, 11, 14, 475 have augmented chart labels summing to 0, while the indicator sums to 1 there. No affine function matches the row. A second P104 four, 21, 13, 493, 501, gives the same contradiction. The degree bound and the parity are written proofs. The 512-function distance census, the 7168 weight-4 obstructions (4384 inside P104), and the 255 frame-letter subset sums are exhaustive finite censuses. They are not a sample and not a minimality claim outside this alphabet.

This does not prove that the #7031 charge of 40 is minimal for sums of products, another chart, or a circuit that is not an F_2 sum of letters. The product hosting is not redone. No selected-invoice histogram is computed.

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 2 of open PR #7031, for frame 0 only.

Nearest overlap is #7031 itself (head 531ecca652dde63c6010cd6097cb55a994367ce7), which hosts the row by a product and records that its census does not cover sums. This package starts from that question. It does not host frame 1 and it does not name the 40 monomials on the #6030 invoice. A GitHub search of open upstream PRs for this sum question returned only #7031. Local notes at the base commit did not contain this argument. data/frames.json is the #7014 witness as vendored by #7031, same sha256 c75915c18fc06ac19cc90fb9911209517c071862fdf401b998d5075efc0cda0d.

Validation

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

python3 -B research/p104-frame0-letter-sum/verify.py
python3 -I -O -B research/p104-frame0-letter-sum/verify.py
python3 -B scripts/research_package.py --package research/p104-frame0-letter-sum
python3 -B scripts/check_research_commit.py --package research/p104-frame0-letter-sum --check-source-history

Observed: PASS_P104_FRAME0_LETTER_SUM explicit=11+21+38+56 monomials=0 new=0 hosts=0 and PASS_REGISTERED_RESEARCH_PACKAGE research/p104-frame0-letter-sum outcome=negative-result. Source-history check passes at 44953245c5cf2d382e1c0f3095d3866930c310e1. Failure controls delete one letter of the explicit sum, reject every proper subset of the obstructing four, keep the even frame dependence out of the obstruction count, and reject the constant-1 function and λ_11.

Not run: the #7031 product census, selected-supplier replay, make frontier6030-verify, Lean, and any expansion of frames 1–12.

Credit

Frame partition: unpaidmoon, PR #7014. Product hosting and the 40-monomial charge: 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. Can one explicit F_2 sum of two products of root-line letters, each of length at least 2, host this row with fewer than 40 monomials of degree at least 2? One miss is not a minimum.
2. Among the 2^16 functions of the four Host one P104 frame-indicator row and charge its new monomials. #7031 forms p = 1 + x1 + x2 + x4, q = x0, r = x5, and s = x6 + x7, what is the minimum number of degree-at-least-2 monomials of a function that matches the indicator on all 120 lines?

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