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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.