[petalas]

Summary

Scoped constructive proof with exact finite corroboration. A new sparse dyadic factorization preserves PR3895's full tensor-root identity and its binary pairing while reducing the displayed rank-16 block COPY bill:

- 16 H16, rank mass 256
+ 15 H1 + H16, rank mass 31

For every s>=1 and n=2^s, the profile is (n-1)H1+Hn, of mass 2n-1. The power-price gain over the displayed dense factorization is (n-1)(n^t-1)>0 for every real t>0. This is an explicit upper bound, not an optimality claim or a new kappa.

The construction uses U=[[2,0],[1,1]], F=U^(tensor s) and weights w_z=2^(1-s)3^|z|. The full identity is 2G=F diag(w) F^t. Every nonfirst column fixes a nonempty set of binary coordinates, making its pairing block rank one. The first column costs n. The unweighted frame and its inverse are dyadic; no dyadic inverse of the weighted factor is claimed.

This contribution was produced by Nick Petalas. Credit Cinna Davis / 1337isnot1337, PR3895, for the tensor-root interface and dense comparator; ToikY / PR3928 for the embedding boundary; and DreamingOfClouds / PR717, themilkman216 / PR968, woklerx / PR1077 and the retained E8/native/Clifford lineage. LDL factorization and Kronecker products are classical. Predecessor scientific checkers were not replayed or audited.

Package: research/triangular-tensor-root-decoder. The card scope is explicitly changed to this local algebraic side result.

Evidence

Before: the source's displayed rank-16 factorization has COPY mass 256. After: the new explicit factors have mass 31 and preserve every entry of the same scalar identity.

Local CPython 3.12.15 normal and optimized checks pass, timed at 0.423 and 0.425 seconds, rounded upward to milliseconds. Exact finite evidence covers six tensor levels through n=64, 5,460 full Gram identity entries, 10,920 entries of the two-sided frame inverses, all 126 term-support charges, 84 literal product-root Gram entries and thirteen semantic/boundary controls. Generated receipts bind all six package files and agree between modes. The universal theorem rests on the written proof.

From the repository root, with the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/triangular-tensor-root-decoder/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/triangular-tensor-root-decoder/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/triangular-tensor-root-decoder
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/triangular-tensor-root-decoder --check-source-history

Another installed CPython 3.12 path may replace the displayed path. All commands pass locally, including both registered checks. Official committed export integrity and normal/optimized read-only Git-free replays with an empty PATH also pass and reproduce the same generated receipt. Commit/source-history preflight passes all six files at bbea977ff919f857fc66efac55b9bf205fab35cd. The worktree is clean.

Limitations

The pairing still has rank 16 at n=16; PR3928's literal nine-bit embedding obstruction remains. This factorization does not provide selected E8 labels, a native source/reader program, arbitrary-dirty ambient restoration, ownership/hosting, physical coefficient or precision/tape/stock charges, a complete paid/productive invoice, compiler or outer admission. No optimum, selected-frontier change or complete multiplication theorem is claimed. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and historical supplier suites were not run.

Open next questions

• Specify a consumer that uses the sparse terms without silently reusing the excluded full pairing. Begin with a changed source/reader map at n=4 and check every constrained coefficient. No compatible consumer is supplied.
• Price explicit native entry, delivery and restoration words at n=4, including arbitrary dirty inputs, weights and precision. The local COPY reduction does not price those obligations or justify another construction round.

Related open work

• PR3895 supplies the same tensor-root identity and its displayed dense rank-16 COPY mass 256. This contribution changes its factors to obtain 31, while retaining its unresolved embedding and physical-consumer obligations. The source result is cited rather than reverified.

The completion refresh screened all fourteen other account open PRs created since the advance CLAIM at 2026-11-03T15:02:34Z. None studies this tensor-root factorization and support-pairing COPY profile or supplies a conflicting value for the same instance. PR6649 concerns dyadic unit-channel tensor scheduling and cites this construction as a distinct object; its capacity ratios are not compared with COPY mass. No extra scientific computation checked another contribution.

CI notes

Both hosted checks passed at exact head bbea977ff919f857fc66efac55b9bf205fab35cd in run 41660734171: plan and replay (research/triangular-tensor-root-decoder). The completed check list was read once; separate run metadata confirms completed success and the matching head.

Hosted replay certifies execution of the registered normal and optimized finite checks and package metadata. Local committed-byte/source-history and export checks also passed. The all-s theorem rests on the written proof; independent human and formal proof-kernel acceptance remain unclaimed. No failed or waived check remains.

Merge Danger

Door: two-way. Blast radius: isolated. Only one new six-file opt-in research package is added; shared code, frozen inputs and selected-frontier records are unchanged.