Polynomial compatibility on the Boolean cube from disjoint-support separators

A complete affirmative deduction for Austrin's exact two-polynomial question, OWR 9/2025 printed p. 445 (30006210 / OWR-14299088-012). For every positive n,d and nonzero real multilinear f,g of degree at most d on the uniform sign cube, all full-norm relative coordinate influences at most 1/(4096d^8) guarantee a common nonzero vertex. Disjoint supports force maximum relative influence at least1/(1024d^8). The denominator includes the constant Fourier coefficient; arbitrary common zeros are allowed. Constants/exponent are not claimed optimal. General distribution formulations, other groups/measures and approximate supports are outside the asserted theorem. This four-page note is an explicit target-specific application of the established Li-Li-Li-Liu 2026 disjoint-support separator and O'Donnell-Saks-Schramm-Servedio 2005 section3.3 Eq(6) real covariance inequality, with the classical Markov/Bernoulli presentation credited to Kothari and coauthors. Neither the separator nor covariance inequality is claimed new. The complete square-density conversion and extracted depth 16d^4 are supplied. Extensive bounded primary priority audits found no earlier recorded complete target-specific deduction in the bodies inspected. They do not certify novelty, first discovery/publication or global current openness. The full ePrint 2022/218 and full 2023/570 followup were inaccessible; the official CRYPTO 2022 conference version was read and separately pinned. Other historical coverage and procedural-independence limits are documented. Dates and negative searches do not certify firstness or causal independence. The source archive preserves all ten original PR artifacts, three distinct mathematical audits, three priority approaches, root checks, and a portable offline standard-library verification runner. Four exact integer/rational programs run in ordinary and optimized modes with all mathematical fields compared exactly and forty actual incorrect-result failures. Finite checks supplement the universal proof; they are not arbitrary-dimensional enumeration or formal proof certificates. Third-party source bodies are excluded. AI tools were used extensively in proof development, writing, computation, reproduction, literature research and adversarial verification. This note and package are unrefereed. Internal AI-assisted reviews do not constitute external human peer review, journal acceptance or formal proof-assistant certification.