[petalas]

Summary

Scoped written proof with exact finite corroboration. For PR #5436's growing-support tensor annihilator, this determines every coefficient denominator and proves sharp quadratic maximum bit height and cubic storage for its primitive Gaussian-integer coefficient list.

alpha=(3+4i)/5,  n=f+1,  S=f(f+1)/2
p_f(z)=product_(k=0)^f(z-alpha^k)=sum_j c_j z^j
v_(2+i)(c_j)=binom(n-j,2)
v_(2-i)(c_j)=binom(j,2)-S
least positive integer denominator of c_j = 5^(S-binom(j,2))

The primitive normalization is (2-i)^S p_f. Its maximum coefficient needs Theta(f^2) bits and its explicit dense coefficient list needs Theta(f^3) bits. Every integral annihilator of degree at most f+1 is an integral multiple of that normalization. The first verified arithmetic result ended the investigation.

Credit zxv's #5436 for the fixed-generator escape, existing upper bound and coefficient-pricing question; Anthony Chefles for classical multi-copy independence; classical Gaussian valuation methods; and the source's retained native/control contributors. The new scoped theorem and checker were produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before: #5436 gives an O(f^2) per-coefficient upper bound and leaves coefficient pricing open. After: exact valuation and denominator profiles establish necessity, a primitive normalization and the dense-storage lower bound.

Local CPython 3.12.15 standard-library verification passes 592 coefficient checks across f=1,...,32, 1,020 independent subset terms and 42 rejection controls. Both registered warning-strict normal and optimized commands pass. Committed-byte/source-history preflight, official export integrity/input coverage, and both read-only Git-free exported replays pass. Each timed standalone command took less than one second locally. Receipts are generated on stdout and bind the five package files and both source documents.

Exact reproduction from the repository root:

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

Any CPython 3.12 interpreter with the standard library can replace the absolute interpreter path. The direct checker prints PASS_TENSOR_ANNIHILATOR_HEIGHT; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations

The theorem fixes alpha and consecutive degree f+1. Different supports, higher-degree identities, approximation, compressed/factored generation and real productive consumers remain open. The storage lower bound concerns an explicit dense binary coordinate list, not every representation or native program. The predecessor's tensor/source verifiers were not replayed. No physical coefficient implementation, complete recursive invoice, supplier admission, selected cold replay, compiler/tape/ownership/recovery proof, Lean or Docker check was run. No new kappa or selected-frontier change.

Open next questions

• Specify a productive consumer with factored or streamed coefficients, then price arithmetic, passes and restoration. No such consumer is supplied here.
• Fix another support of f+2 integer powers and derive exact Vandermonde-minor heights. This proof's consecutive-degree uniqueness does not optimize that support.

Related open work

No open PR in the refreshed account list studies this consecutive-power Gaussian annihilator's coefficient denominators or primitive height. The refresh at 2026-10-27 14:55:09 UTC covered 13 other open account PRs created since the advance CLAIM: #5930, #5931, #5932, #5933, #5935, #5936, #5937, #5938, #5939, #5940, #5941, #5942, #5943. Merged #5436 is the direct predecessor; merged #5441 and #5446 treat different transfer-table and paid-history contracts.

• Sharpen averaged cyclotomic gap from 8ell^2 to 3ell^2 (no new kappa) #5930 studies averaged cyclotomic coefficient heights and circle-gap bounds. This concerns a different polynomial family and computes no shared coefficient value.
• Prove exact Cayley path denominators (no new κ) #5941 determines rational signed-path Cayley matrix denominators. This concerns a different matrix family and computes no shared denominator value.
• Prove a smaller equal-energy packing radix: 726 versus 1024 #5943 changes a sufficient integer-packing radix for equal-energy Gaussian rows. It supplies no tensor-annihilator coefficient-height claim.

No extra mathematical computations or predecessor replays were run to compare these contributions. No same-instance numerical disagreement was found.

CI notes

Both hosted checks passed at exact head ac7d0e5a7e6918336eaf552644d14d09840d33dc in run 40523916711: plan in 4m55s and replay (research/tensor-annihilator-height) in 2m10s. The registered package replay executes the complete warning-strict normal and optimized checks. Local mathematical replay, metadata/source pins, committed bytes/source history, official export integrity and both read-only Git-free exported replays also pass. Hosted execution certifies the bounded registered checks; independent human and formal proof-kernel review of the all-width written proof remain unclaimed.

Merge danger

Two-way door. Blast radius: one isolated research package with an opt-in replay manifest. No shared code or selected evidence changes.