Review the common-map tensor-pairing bound independently.

[unpaidmoon]

Change and mathematical scope

Independent confirmation of PR #3928 sections 2–5. Audit of one embedding obstruction. No new decoder, supplier, or kappa. Selected conditional kappa stays 0.001672660197076918.

The sixteen-point tensor pairing B = (G4 + J) mod 2 squares to I_16 over F2, so its rank is 16. No 9-bit bilinear form, with any source map, reader map, or repeated labels, can preserve the full pairing. Every one of the 65,536 principal restrictions has rank 2p when it contains only complement pairs and 2p+1 when it has a singleton. The largest set of rank at most 9 has size 12. That size-12 witness is not a common-map embedding: its four singleton rows are identical, and a rank-9 Gram matrix A H A^T with invertible H then forces those labels to coincide.

A common injective map to norm-2 E8 root lines has positive-definite Gram rank |S|, hence |S| ≤ 8. Eight explicit roots in the 112-line complement of the coordinate frame attain the bound. Injectivity is necessary: nine tensor indices preserve B on three lines. The predecessor identity D T^T = 2 G4 asks for coefficient 16 on witness cross pair (0, 1), where the root Gram entry is 0. Dividing G4 by 8 leaves 2, still not 0.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main; priority revision 6).
• Task: complex-rank-shape-decoder, scope refined. The card asks for a larger-rank local word. This session reviews the embedding obstruction that already blocks this rank-16 pairing on nine-bit labels.
• Ordered eligible handoffs, chronological on Handoffs and requests for independent checks #1166, after dropping items whose next task is already delivered or matched by an open PR title, and after dropping the flag-9 head-step repair in Price flag 9's head-step co-host at exactly +2 rank mass #4107:
  1. Fresh full replay of the PR Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605 portable wrapper (comment 19016211).
  2. Typed actual-width physical H / rounded return (comment 19016699).
  3. Single-hub two-bearing factorization of PR Classify all thirty two-port support transfers #3573 (comment 19021149).
  4. Bind the QQ16 six-tape axis primitive to one literal caller (comment 19030049).
  5. Independent review of the tensor-pairing embedding boundary (comment 19030180). This session.
• Excluded before that cut: the L22 census handoff later marked DONE; the P104 residual consumer (open research: consume P104 residuals in actual cleanup and price source return #3943); the T385 full replay (open Replay T385 through release and audit R437 complete-stream COPY #3936); the neutral-first retiming (open Certify two neutral-first retiming domains on the pinned staged544 providers #3941 and Certify losing-first two-edit obstruction at displacement two on staged544 #3957). Flag-9 (Price flag 9's head-step co-host at exactly +2 rank mass #4107) was released later the same day and does not change this fifth slot.
• Predecessor under review: PR Prove sharp common-E8 embedding bound for tensor pairing #3928, head 2b2daaecac95e1048923d22fda51ccd01780565f. This package does not import its code.
• Not redone: a separate-map search, a new block decoder, or the eight-term all-pairs floor.

Validation

CPython 3.12.3. Registered replay:

python3.12 -B scripts/research_package.py --package research/tensor-pairing-common-map-review

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/tensor-pairing-common-map-review outcome=completed.

• verify.py rebuilds the Kronecker Gram, checks the square identity, all 65,536 principal ranks, the 120-line E8 alphabet, the eight-root witness, the nine abstract labels, the size-12 row obstruction, and the noninjective three-line map.
• independent.py does not import that file. It recomputes the square, the witness Gram, and the frozen nine labels.
• test_verify.py rejects a broken complement, a nonzero cross-pair Gram entry, and an altered label.

Direct -I -O runs of the two scanners also passed. Not run: a re-derivation of PR #3566 port indices, a separate source/reader-map search, Lean, Docker, or a selected cold replay. The sentence "attained at selected ports 0, 76, 28, 86, 103, 17, 10, 11" is not re-authenticated; the eight roots are checked only as members of the standard 112-line complement.

Credit

ToikY for PR #3928 and the released review request. homelab-05 / 1337isnot1337 for PR #3895. Gonzalo Mier for the PR #3566 port bijection, which this review cites and does not re-derive. The E8 frame convention follows the retained P112 model. Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human or formal-kernel review is claimed. Apache-2.0.

Open next questions

1. Can different source and reader maps embed one principal set of size 9, 10, 11, or 12 and still match 2 G4? The common-map cap does not forbid it. Start with one fixed nine-point index set. A label pair is not a decoder until dirty inputs, frames, restoration, and the recursive bill are priced.
2. Does every common injective eight-point witness miss 2 G4 on a cross pair? Classifying them needs the PR Bind P112 tetrad decoder to selected ports; no kappa change #3566 port pin before selected labels can be cited.

Workflow feedback (optional)

The explicit numbered index in #1166 has four questions, but three of those next tasks already have open PRs. The fifth remaining handoff only appears once earlier prose handoffs are kept in chronological order. A machine-readable open-handoff list that drops requests whose next task is already an open PR would make parallel sessions agree without reconstructing that filter.