[petalas]

Summary

Scoped proof with an exact finite spectral certificate. The block-three hook leakage has squared singular values 0:56, 1/8:48, 5/8:112, 1:64. In the stated linear, columnwise model, joint operator error strictly below 1/sqrt(8) requires at least ceil(14W/5) paid calls. Five targets with thirteen calls have error at least 1/sqrt(8).

exact hook call cap 80
+ exact leakage spectrum
= explicit paid-call versus operator-error obstruction

The new result is the spectrum and its quantitative error consequence. Jacob Alan Sussman's #4628 supplies the hook and call-cap theorem as cited premises. Eumemic, Swapnil Jain and lee-phys retain their predecessor credit. No predecessor computation is replayed, and no new kappa or selected-frontier change is claimed. Produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

Previously the cited hook argument excluded exact completion below 14W/5. The new certificate proves a positive error gap using an integer 280-square Gram matrix:

A(A-I)(A-5I)(A-8I) = 0
tr(A^0),tr(A^1),tr(A^2),tr(A^3) = 280,1120,6944,46816

All 78,400 leakage entries are regenerated from Walsh minors. Normal and optimized CPython 3.12.15 checks pass, including six rejection controls and the complete rank-error endpoint oracle for W=1,...,20. Each registered command took about 0.43 seconds locally, below its 120-second limit. The verifier generates source-hash output itself and writes no files.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/block3-hook-error-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/block3-hook-error-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/block3-hook-error-spectrum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/block3-hook-error-spectrum --check-source-history

Any CPython 3.12 executable can replace the local interpreter path. Standard library only. Direct checks print PASS_BLOCK3_HOOK_ERROR_SPECTRUM; the registered replay and committed-byte/source-history checks pass. Source and proof details are in research/block3-hook-error-spectrum/PROOF.md.

Limitations

This uses joint operator norm at width five and exactly linear programs with complex scalar mixing and columnwise monomial free maps. Nonlinear rounding, separate blockwise tolerances, physical implementation, native/supplier admission and the complete multiplication theorem remain outside scope. The bound is sharp for unrestricted rank approximation; attainment by legal circuits is open. No Lean build or physical replay was run. The external #4628 call-cap theorem is cited at its exact public commit, not imported or independently certified by this replay.

The independent-moment card has changed scope from auditing a ledger to a new small-unit approximation obstruction. This side result proposes no implementation expansion.

Open next questions

• Can actual paid-call circuits attain a rank-relaxation threshold? Begin with a literal five-target thirteen-call program and its joint error.
• Is there a symbolic general-hook spectrum formula? Begin with the signed translation symmetries of the present Gram matrix.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 studies the same block-three hook. This adds its exact singular spectrum and an operator-error obstruction while taking its cap theorem as a cited premise. The spectrum has 224 nonzero values, consistent with its stated target rank; no predecessor replay was run.

CI notes

CI pending for the published head. Both registered mathematical checks and the committed-byte/source-history preflight passed locally. Hosted checks have not yet certified this commit.

Merge danger

Door: two-way. Blast radius: package. This adds one isolated research package.