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Unrestricted Half-Power Obstruction and Polynomial Certificates for the Vámos Quartic

Scott Schubr

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Abstract

Complete monotonicity of an inverse polynomial power is equivalent to a positive Laplace representation. We prove failure at the critical half power for the specialized four-variable Vámos quartic and every interior stable deformation hλ = 4e₁e₃ + λa²b², 0 < λ < 64/27. A negative correction to the inverse transform is isolated outside the support of its positive leading term; uniform estimates on complex inversion planes justify its return to the full four-variable inverse. For the original quartic the fixed-test remainder is O(√q(1 + |log q|)) as the inverse-section coordinate q tends to zero. The obstruction persists for a strict exponent interval above 1/2, uniformly on compact interior parameter sets. Polynomial squares also detect the finite correction measure at every interior parameter, with locally bounded detection degree. For the original quartic this degree is exactly 54: every nonzero polynomial of degree at most 53 has positive square integral, while an explicit integer polynomial of degree 54 has negative square integral. The associated 55-dimensional moment matrix has inertia (54,1,0), and the exact degree 54 persists on an open parameter interval about the original quartic. Rational moments, fraction-free elimination and independent formal-series reconstruction give portable certificates. At the sharp stable parameter 64/27, an exact formula describes the nonnegative first correction outside the leading support.

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Edition files

Available editions

  1. Edition 5 · 2026-10-31 · public
  2. Edition 4 · 2026-10-29 · public
  3. Edition 3 · 2026-10-27 · public
  4. Edition 2 · 2026-10-25 · public
  5. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Analysis and Spectral Theory
Keywords
Vámos quartic, complete monotonicity, inverse Laplace transform, signed measures, exact moment certificates
Manuscript date
2026-10-25
Reader reuse
none

Scope

Unrestricted half-power failure for 0 < λ < 64/27; compact-parameter exponent gap; quantitative fixed-test inversion remainder O(√q(1 + |log q|)) at λ = 1; finite polynomial-square detection throughout the interior with locally bounded degree; exact first degree 54 at λ = 1 and nearby parameters; endpoint first-correction density on y > 1.

Disclosures

OpenAI ChatGPT/Codex assisted with literature discovery, mathematical exploration and proof development, manuscript drafting, verification-code development and internal critical checking. The author assumes responsibility for all content. Scott Schubr owns Luminary Oddities, which operates the Research Archive hosting this preprint. This work received no specific grant from any funding agency, commercial or not-for-profit sector.

Reproduction

The 18-page manuscript and complete editable source are accompanied by three exact verifiers, the integer polynomial certificate and build instructions. The standard-library analytic verifier checks 30 algebraic identities and bounds. Two finite implementations reconstruct the 55 determinant signs and degree-54 negative square; the alternative uses python-flint 0.9.0. README.md documents every command.

References and related resources

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