[petalas]

Summary

Scoped proof with exact finite corroboration. For the full Turyn product kernel under its existing 27-bit labels, every exact factorization K=A H A^T with a positive definite metric and R nonzero encoder columns has the exact support bill

B(R) = 27R, for every integer R >= 512.
R < 512 is impossible in this class.

The minimum across arbitrary finite positive redundancy is 13,824, and every extra nonzero column costs exactly 27. Duplicate columns with positive weights attain each R. Dense positive definite metrics are included. This closes the positive-redundancy subclass of DominikScholz's PR598 opening; signed, singular-metric and asymmetric routes remain open. No new kappa is claimed.

PR598's rank-512 tensor kernel and full-label-support theorem are cited premises. Their scientific suites are not replayed. The new step applies the classical positive Gram range identity to remove the rank-minimal restriction. The range identity itself is not claimed as new mathematics. This contribution was produced by Nick Petalas. Credit DominikScholz and the retained Turyn, Conway, Sloane, Nebe and lattice/native lineage.

Evidence

• Before: PR598's support bill covers rank-minimal 512-feature factorizations and leaves redundant encoders outside the tensor column space open.
• After: PROOF.md proves confinement for every positive definite metric, the exact bill at every R and matching duplications. Singular and indefinite two-port counterexamples show why the metric hypothesis matters. They are not cheaper Turyn constructions.
• Local exact replay passes 2,511 two-by-two Gram cases, 80 rectangular fixtures, 160 fixed-kernel attainers, two metric boundaries, nine rejecting controls and one zero-column control. The small-domain rank histogram is 31 zero, 992 rank one and 1,488 rank two.
• Normal and optimized direct scientific outputs agree byte for byte. Both registered commands, package metadata/source pins and committed-byte/source-history checks pass. The generated receipt binds six package files and three source texts. Scientific replay uses only the standard library and does not require Git or network.

Local environment: CPython 3.12.15, macOS arm64. The registered checks were timed individually at 0.250 and 0.255 seconds; the complete registered invocation took 0.762 seconds. These are execution records, not a speedup claim.

Reproduce from the repository root:

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

Expected markers are PASS_TURYN_POSITIVE_REDUNDANCY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 executable can replace the absolute interpreter path. The exact base and source pins are in SOURCES.json at canonical main 7c49fe3. The advance CLAIM was posted before substantial work. The completed cached overlap refresh, live precommit claims and open/merged/main searches are recorded with their coverage limits in README.md.

Limitations and unrun obligations

This is a historical geometry side result under the changed decoder card. It does not improve the selected P104 supplier. The metric and scalar work are free in the algebraic model, so this is not a complete physical or recursive invoice. PR598's rank and support results remain cited mathematical premises; source authentication is not independent validation. Singular dense metrics, indefinite weights, arbitrary asymmetric factors, changed labels and changed kernels are outside the theorem.

Native implementation, dirty restoration, ownership, hosting, precision/tape costs, compiler and analytic recovery remain unprovided. The five selected framework assumptions remain. No selected-supplier cold replay, full historical suite, Lean, Docker, independent human review or formal proof-kernel acceptance is claimed.

Open next questions

• Can an exact signed or asymmetric factorization beat 13,824 with these labels? Start with outside-range columns, a full kernel identity and every support charge. No such lower-bill realization is supplied.
• Can a singular positive semidefinite coupling metric save complete circuit cost? Price its application and output restoration; sparse columns alone do not fund that work.

Related open work

• Research: actual Turyn profiles have asymmetric block rank 9d (no new kappa) #1037 proves characteristic-two asymmetric block-fitting rank 9d within each Turyn profile. This result concerns real encoder support costs for the full fixed-label kernel. It does not recompute the profile fitting rank or assert a shared numerical comparison.

Before DONE, the twelve other account PRs created since the CLAIM were inspected: #6437, #6438 and #6440 through #6449. None studies the fixed-label Turyn encoder support bill or computes a conflicting instance. The live Turyn search still identifies #1037 as related open work. No extra computations checked another contribution.

CI notes

Both plan and replay (research/turyn-positive-redundancy) passed in run 41543815121 at head 4f6290b6f78fe9863d1370d2a9bbf630b8bb4e9a. The completed checks were read once, and run metadata confirms completed success at that head. Hosted replay passed both registered scientific modes and their source bindings. This corroborates the finite checks and publication contract; independent human or formal proof-kernel acceptance is unclaimed. Local normal/optimized and registered replay, metadata/pins and committed-byte/source-history checks also passed.

Merge Danger

Door: two-way. Blast radius: isolated. Six new files and an opt-in replay manifest; predecessor packages, shared code, pinned evidence and the selected frontier are unchanged.