Research: E6 projected-Gram decoder screen with adapted pairing

[1337isnot1337]

Result

Adds a reproducible exact screen for one new E6 rank-four decoder family with a decoder-adapted pairing. For the 36 E6 root lines, the package exhausts all 70 principal and 4,900 ordered four-coordinate projections M=P_U P_V^T.

• Five principal candidates and 13 ordered candidates meet the rank-four, nonzero-diagonal checks.
• Under the entrywise-maximal compatible symmetric pairing, the best ordered candidate has charged cost 65 at U=[3,4,5,6], V=[3,4,5,7]. No member of this projected-Gram/maximal-pairing family reaches the predeclared E6 T=4 target 22.
• Each of the 13 candidates has a certified six-clique in its forced-zero graph, excluding a rank-five Gram pairing B=A A^T over F2^5 for those candidates.

This is a scoped negative result for the declared construction family. Cost 65 is not a lower bound for sparser pairings. The clique argument does not exclude rank-six, non-Gram, or arbitrary admissible pairings, and the package does not search arbitrary rational decoders. No supplier placement or new κ is claimed.

Reproduction

From repository root:

python3 -I -O -B research/e6-adapted-pairing-projected-gram-seed0883/verify.py
python3 scripts/research_package.py --package research/e6-adapted-pairing-projected-gram-seed0883 --check-only
make selected-record-check

The direct exact verifier, package metadata/pin check, and selected-record check passed locally. The verifier produced RESULT_SHA256 97a3cc0eca67f9140c9c36ca076a98e3399780b48759b0a85b2fc4ecd42f94dc. This environment has Python 3.14.7 but not Python 3.12; the registered generic replay requires 3.12 and remains for CI.

Pins and overlap

Accepted seed 0883a0f84126624d58444b85d7c6b119; task complex-rank-shape-decoder, priority revision 4; base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. Exact source blob and SHA-256 pins are in SOURCES.json.

The closest predecessor is PR #2137 (head 6a022d50e21bdcf2014a26937a34d44eed3473f0), which fixes B0 and studies R L R^T; this package changes the pairing premise and tests P_U P_V^T. PRs #1590, #2111, and #2125 provide the root-unit cost model and E6 context. The complete current overlap search found no other E6 projected-Gram adapted-pairing scope. Active P112 paired-tetrad work in Discussion #2003 and selected-provider Phase B work in Discussion #2119 concern different objects and cost models.

Claim and scope update: #1166 (reply in thread) and #1166 (comment).

Prepared by Cinna Davis with substantial OpenAI Codex assistance. No independent human review is claimed.