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 $[\theta, 1]$, $0 < \theta < 1$, and allow only positive dilation together with the two characters of reflection. For the associated prime-power statistic $\mathcal{A}_{\Xi, h}(a)$, the paper proves the exact inverse growth law $$\frac{1}{2}\limsup_{a \to \infty} \frac{1}{a} \log\left(2 + \left\vert{}\mathcal{A}_{\Xi, h}(a)\right\vert{}\right) = \delta_*(\Xi),$$ where $\delta_*(\Xi)$ 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 $\mathcal{R}_{\zeta, h}(a)$ satisfies $$\frac{1}{2}\limsup_{a \to \infty} \frac{1}{a} \log\left(2 + \left\vert{}\mathcal{R}_{\zeta, h}(a)\right\vert{}\right) = \delta_*(\zeta),$$ 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 $1 \le p < \infty$, its weighted $L^p$-abscissa is $$\sigma_p = 2\delta_*(\zeta),$$ and its cumulative moments satisfy $$\limsup_{A \to \infty} \frac{1}{A} \log \int_0^A \left\vert{}\mathcal{R}_{\zeta, h}(a)\right\vert{}^p \, da = p \cdot \delta_*(\zeta).$$ After the multiplicative change of variables $X = e^{2a}$, 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, $$\mathcal{R}_{\zeta, h_{\mathrm{hb}}}(a) = -2\left\{ E_1(e^a) - 2E_1(e^{3a/2}) + E_1(e^{2a}) \right\}.$$ 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