[petalas]

Summary

Prove exact coefficient valuations for every minimum-term integer-power support of the fixed tensor generator D=diag(1,(3+4i)/5). Claim type: scoped written proof with exact finite corroboration, produced by Nick Petalas. This answers the arithmetic-order part of Open next question 2 in merged #5934.

For n=f+1, exponents m_0<...<m_n and positive integer gaps delta_s, the primitive Gaussian-integer relation has

v_(2+i)(c_j) = sum_(s=j)^(n-1) (n-1-s) delta_s
v_(2-i)(c_j) = sum_(s=0)^(j-1) s delta_s

total valuation mass
  = sum_s [n-1+2s(n-1-s)] delta_s
  >= 2 binom(n+1,3), with equality exactly for consecutive supports.

Thus arbitrary nonconsecutive supports, including Laurent supports and arbitrarily high degree, cannot reduce the quadratic maximum coefficient bit height or cubic explicit coefficient-list storage order at the same f+2 term count. The exact actual-modulus optimum remains open. No new kappa, physical supplier or selected-frontier change is claimed.

The new proof, verifier and reproduction guide are isolated under research/sparse-tensor-annihilator-heights/.

Credit Nick Petalas #5934 for the valuation method, consecutive-support theorem and explicit question; zxv #5436 for the fixed-generator tensor escape and coefficient-pricing problem; Anthony Chefles for classical multi-copy independence; and the retained native/control lineage. Vandermonde and Gaussian-integer methods are classical. Predecessor results are cited premises; no predecessor scientific verifier was replayed. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before, Prove exact tensor annihilator coefficient heights (no new κ) #5934 determines only consecutive-power coefficient heights. After, the written theorem determines both split-prime valuations for every integer exponent support and proves the sharp weighted-gap mass minimum.
• The new census checks 246 supports, including 240 nonconsecutive ones, by signed Vandermonde cofactors and a separate exact rational kernel solver. It checks 1,190 coefficient occurrences, 944 root residuals and 1,776 literal lifted diagonal entries. The sample and rejection controls are additional to those census counts.
• Thirteen controls reject invalid supports, altered or missing coefficients, nonprimitive scaling, a zero coefficient and conjugated coefficients. All checks remain enabled under optimized Python.
• Both registered warning-strict commands pass on CPython 3.12.15 with identical generated six-file hash-bound receipts. They took 0.256 and 0.258 seconds locally, below their 60-second limits. These are check durations, not a performance comparison.
• Metadata/source pins, in-memory syntax compilation, whitespace, exact committed bytes/modes/source-history ancestry, official seven-member export integrity and input coverage pass. Both read-only Git-free exported replays with an empty executable PATH agree with the local receipt.

Exact commands from the repository root:

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

Any CPython 3.12 interpreter can replace the local path. The direct marker is PASS_SPARSE_TENSOR_ANNIHILATOR_HEIGHTS; the registered runner returns PASS_REGISTERED_RESEARCH_PACKAGE. Direct replay writes nothing and requires no Git, sibling package or dependency beyond the standard library.

Advance CLAIM. Canonical main was refreshed before submission and remains e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8. The selected #7000 binding control passed; no selected scientific replay was run. The card's scope change to this arithmetic side result and exact provenance are declared in TASK.json and SOURCES.json.

Limitations

The theorem fixes the generator, exact equality, Gaussian-rational coefficient field and minimum term count. It optimizes split-prime valuation mass, not the full complex-modulus objective. More terms, approximate cancellation, other generators and factored or streaming descriptions are outside the claim. It supplies no productive physical consumer, coefficient-generation price, precision, field-pass, ownership/restoration, tape, compiler or outer invoice. The five selected framework assumptions remain. Selected cold replay, predecessor scientific suites, Lean and Docker are unrun and unclaimed. No registered mathematical command was deferred to CI.

Open next questions

• Do consecutive supports minimize the actual maximum primitive coefficient modulus? Start at f=2 by controlling the extra Gaussian prime factors of the four cofactors. The valuation optimum alone does not decide this.
• Can a productive consumer use a factored or streamed relation with a favorable complete physical bill? Specify its endpoint and price arithmetic, passes, precision and restoration; an identity producing zero is not that consumer.

Related open work

The refreshed open-PR body corpus contains no other result on this fixed generator's sparse-support coefficient valuations. Related broader tensor-interface work is distinct:

• #7249 bounds approximate Gaussian tensor-child fusion through rectangular adapters. Its target and norm-error quantities differ; no coefficient-height value is shared.
• #7172 studies doubled centre-Gram stabilization and its annihilator. It uses a different operator and computes no shared valuation mass.

The completion refresh read the eight other open account PRs created since the CLAIM: #7247, #7248, #7249, #7251, #7252, #7253, #7254 and #7255. None studies this fixed generator's coefficient-height or split-prime valuation quantity. The broader tensor relationships are recorded above and below; no additional scientific computations checked another result.

• #7247 counts contiguous blocks in a rounded tensor image. Its layout quantity differs from coefficient heights; no value is shared.
• #7252 determines finite-field multiplication-tensor stabilizer determinant images and orbit counts. Its tensor and arithmetic quantity differ; no valuation value is shared.

Sparse local-note and unpublished-work coverage remains incomplete; the package records the search limits.

CI notes

CI did not certify this package. At unchanged head 025e0a934635dbcd0e1321dba36decdd96235c9a, run 43023829856, plan job 130533389872, failed in Onboarding helper tests. I read the completed checks once and inspected the job log directly. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The assertion at tests/test_research_current_status.py:58 expects the selected reproduction command in the current section of docs/ci-verification.md. The logged section still describes #6030 and make frontier6030-verify. Direct inspection of canonical base e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json instead names make frontier7000-verify. The test, guide and selected record are byte-identical between that base and this PR. This is a repository-level navigation mismatch.

The dependent replay job 130535289845 was skipped, so there is no hosted scientific replay evidence for this package. The local normal/optimized registered checks, source preflight and portable export results above remain the available execution evidence.

I directly verified the same sole failing test, missing-command assertion and 760-test/one-failure/24-skipped totals in all three comparison logs. Each failed step is Onboarding helper tests:

• Run 42905094314, plan job 130140894651.
• Run 42903313790, plan job 130134720799.
• #7247 run 43023514635, plan job 130532328734.

No fix, retrigger commit, push or workflow rerun was made. The repository-level repair remains with maintainers. Independent human and formal proof-kernel review are unclaimed.

Merge Danger

Door: two-way. Blast radius: one isolated six-file research package. No shared source, predecessor evidence or selected-frontier file changes.