Localized Weil Positivity, Outer-Shell Hankel Geometry, and a Box-Parity GRH Criterion for a Degree-64 Triple Symmetric-Cube L-Function

Description This paper develops a support-local analysis of the Weil quadratic form for a fixed degree-64 triple symmetric-cube $L$-function $L_{333}(s)$, with the aim of identifying how far compact-support positivity and finite-interval Hankel methods can reduce the generalized Riemann hypothesis (GRH) problem. The main results include: support monotonicity and a summable dyadic regularization scheme for the localized Weil form; a subcritical boundary-packet positivity theorem with transition scale $$w_c(Y) \asymp \frac{\log Y}{\sqrt{Y}};$$ precise obstructions to fixed-profile reduction, canonical single-scale localization, and unrestricted Bloch completion; a Paley–Wiener / finite-interval Hankel quotient formulation together with a quotient-invariant outer-shell tent functional; a differential three-scale transform and a growth-to-zero-strip contraction principle; an explicit outer-band box-parity criterion $$\mathrm{GRH}(L_{333}) \Longleftrightarrow P_+(a) \ge 0, \quad P_-(a) \ge 0 \qquad (a > 0),$$ together with the equivalent scalar determinant criterion $$\Delta(a) = P_+(a)P_-(a) \ge 0 \qquad (a > 0);$$ recovery of the signed good-prime coefficients from derivative cusps of the parity observables; a three-scale prime-power cusp measure whose Laplace transform is an explicit linear combination of scaled values of $\log L_{333}$; and the dyadic shell criterion $$\mathrm{GRH}(L_{333}) \Longleftrightarrow \log \left( 2 + \left\vert{} \sum_{e^a \le n < e^{2a}} \frac{\Lambda_{333}(n)}{\sqrt{n}} \right\vert{} \right) = o(a).$$ The analysis shows that localized Weil geometry can compress the full positivity problem to explicit one-parameter scalar observables, but does not eliminate the signed arithmetic content. In particular, the prime coefficients reappear as singular cusp data on the support parameter, and the remaining arithmetic obstruction is equivalent in strength to GRH. No proof of GRH is claimed. The purpose of the paper is to identify a sharp structural and criterion-theoretic boundary for the localized-Weil/Hankel approach and to isolate the arithmetic information that any further proof strategy would still have to control. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords L-functions; GRH; Weil quadratic form; Hankel operators; Paley–Wiener space; triple symmetric-cube; localized positivity; prime-power cusp measure; parity observables