Dilation-Orbit Detection of the Generalized Riemann Hypothesis: Exact Arithmetic Growth, Moment Spectra, Polar Renormalization, and Filter Rigidity

We study dilation-orbit restrictions of the Weil quadratic form for reflection-Hermitian L-data with real logarithmic-derivative coefficients. Fix a nonzero real finite-step master profile h with exact support hull [θ, 1], 0 < θ < 1, and allow only positive dilation together with the two characters of reflection. For the associated prime-power statistic AΞ, h(a), the paper proves the exact inverse growth law 1/2a → ∞ 1/a log(2 + AΞ, h(a)) = δ_*(Ξ), where δ_*(Ξ) is the maximal transverse displacement of the designated nontrivial zero set from the critical line. Consequently, GRH is equivalent to subexponential growth of a single fixed-profile dilation observable. The same orbit yields eventual Weil-positivity criteria and exact exponential laws for the corresponding positivity defects. For the Riemann zeta function, the pole at $s = 1$ is removed by an explicit one-sided polar renormalization. The resulting observable Rζ, h(a) satisfies 1/2a → ∞ 1/a log(2 + Rζ, h(a)) = δ_*(ζ), giving an explicit fixed-profile criterion equivalent to the Riemann hypothesis. Version v2.0 further develops an exact moment theory for this renormalized detector. For every 1 ≤ p < ∞, its weighted Lᵖ-abscissa is σₚ = 2δ_*(ζ), and its cumulative moments satisfy A → ∞ 1/A log ∫₀A Rζ, h(a)ᵖ \, da = p · δ_*(ζ). After the multiplicative change of variables X = e²ᵃ, this yields an exact logarithmic-energy law and quantitative near-RH implications from mean bounds for the detector. For the canonical half-band profile, scale differentiation also gives an exact three-scale identity in terms of a weighted prime-number-theorem remainder, R_ζ, hhb(a) = -2\ E₁(eᵃ) - 2E₁(e3a/2) + E₁(e²ᵃ) \. Finally, the paper proves rigidity results showing that finite fixed dilation mixtures cannot cancel the largest dilation exponent, and that positive quadratic post-processing with only subexponential conditioning preserves the same defect exponent. These results delineate the limits of fixed-filter and coercive positive-energy improvements within this dilation-orbit framework. The paper also establishes reciprocal Stieltjes positivity and a universal cubic reciprocal-tail obstruction for real compactly supported $BV$ profiles, together with an extremality result for interval profiles among finite-step filters with fixed support endpoints. Version: v2.0 — Zenodo public research release. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Generalized Riemann hypothesis; Riemann hypothesis; Weil quadratic form; explicit formula; automorphic L-functions; dilation orbit; polar renormalization; exact zero-width recovery; L^p moment spectrum; logarithmic energy; prime number theorem error; weighted prime sums; zero-free regions; Laplace transform; pole propagation; support-filter rigidity; finite dilation filters; positive quadratic energy; Stieltjes positivity; reflection-Hermitian L-data