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

Scott Schubr

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Abstract

We prove that the inverse square root of the specialized four-variable Vámos quartic and every interior stable deformation hλ=4e₁e₃+λa²b², 0<λ<64/27, fail complete monotonicity. Uniform estimates on complex inversion planes separate a singular positive leading inverse from a regular correction; a negative correction density outside the leading support then gives the unrestricted obstruction. This also yields a strict exponent gap above 1/2, uniform on compact interior parameter sets. For the original quartic, we determine the first polynomial-square degree detecting negativity of the finite correction measure ν=yκ₁. Every nonzero polynomial of degree at most 53 has strictly positive square integral, whereas an explicit integer polynomial of degree 54 has negative square integral. The associated 55-dimensional moment matrix has inertia (54,1,0). Exact rational moments, fraction-free elimination and an independent formal-series reconstruction provide portable certificates. At the sharp stable parameter 64/27, an exact formula gives a nonnegative first correction outside the leading support.

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Edition files

Available editions

  1. Edition 4 · 2026-10-29 · public
  2. Edition 3 · 2026-10-27 · public
  3. Edition 2 · 2026-10-25 · public
  4. 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; qualitative compact-parameter exponent gap; complete analytic correction inversion; exact first negative polynomial-square degree 54 for ν=yκ₁ at λ=1; endpoint first-correction density on y>1. Consolidates the analytic and finite results of the same article.

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 15-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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