Rule out rank-four E6 B0 coordinate-form decoders

[1337isnot1337]

Result

For the 36 E6 root lines with the fixed B0 mask, consider every rational matrix L in the root-coordinate family M = R L R^T, including nonsymmetric L. Here the rows of R are exact Q^6 coordinates in a pinned E6 basis. The 540 ordered distinct orthogonal-pair zero constraints on the 36 entries of L have exact rational rank 35 and one-dimensional kernel QG, where G is the basis Gram matrix. Since rank_Q(G)=6 and the diagonal is nonzero, no exact decoder in this family has four or fewer totals.

The result is scoped to factors p_j=R u_j and q_j=R v_j. It does not rule out arbitrary rational E6 decoders, other admissible pairings, or any stronger multiplication bound.

Cost screen

The E6 per-total floor is w=5, so a four-total candidate has a floor-only bill of 20 if all other costs are granted free. The pinned E6 T=4 windows require 21, 22, or 23. This favorable screen says the family was worth testing; it is not a realized decoder cost. No rank-at-most-four form survives, and no κ claim is made.

Evidence and reproduction

The package generates the E8/E6 line sets with exact integer vectors, derives coordinates over Q, row-reduces the full constraint system, and records root/basis/coordinate/constraint hashes. Positive control L=G passes all orthogonal zeros and has diagonal eight. Negative controls show that L=0 fails exactness and G+E_00 violates an orthogonal-pair zero (260 violations).

Local Python 3.14.7 runs passed:

• python3 -I -O -B research/e6-coordinate-bilinear-form-seed8e9cf2/verify.py
• python3 -I -O -B research/e6-coordinate-bilinear-form-seed8e9cf2/test_verify.py (3 groups)
• python3 scripts/research_package.py --package research/e6-coordinate-bilinear-form-seed8e9cf2 --check-only
• make selected-record-check
• git show --check HEAD

The generic registered replay requires Python 3.12, which is not installed in this environment; CI is the remaining replay check. The full selected-certificate replay was not run.

Pins and overlap

Base main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; contribution head: 6a022d50e21bdcf2014a26937a34d44eed3473f0. The package pins the selected record, task catalogue, selected review, source manifest, certificate, E6 window source, and ADE predecessor source. Selected-result reproduction remains make frontier1094-verify.

Claim: #1166 (reply in thread). The complete overlap cache refreshed at 20:33 UTC had 393 open PRs and 802 Discussion records; exact-term search had no external matches. Adjacent work: #2111 concerns T=3 and its lift question; #2125 reports standard-Gram costs and expressly leaves E6 windows open. This contribution classifies a narrower rational coordinate-form family.

Predecessor credit: DreamingOfClouds' #1590/#2111/#2125 root-unit work and the inherited #446/#595 cost model. Prepared for Adam Gall with substantial OpenAI Codex assistance. No independent human review is claimed.