From Finite Certificates to Uniform Coercivity in Localized Weil Eight-Mode Comparisons

## Version v1.0 We prove a computer-assisted uniform coercivity theorem for the parity-complete eight-mode comparison arising from a localized Weil quadratic form throughout the arithmetic cell $$5.01 \le X < 7.$$ The main result is $$D_{X,8} \succeq 0.20 I\qquad\text{for every real } X \in [5.01,7).$$ Here the interval is an arithmetic cell in the sense that the active prime-power ledger remains fixed throughout the open right-hand endpoint; in this case the active prime powers are $$\mathcal Q(X)=\{2,3,4,5\}.$$ The central problem is to pass from finitely many rigorous local matrix certificates to a theorem valid on the entire parameter interval. A finite collection of verified anchor values alone does not provide such a conclusion. The proof uses a fixed-target Loewner propagation principle for parameter-dependent Hermitian matrices. Instead of differentiating a smallest generalized eigenvalue, the argument propagates a matrix gap of the form $$\tau M(X)-S(X)$$ in a moving positive metric. Two-sided derivative inequalities control the variation of both the metric and the target gap. The finite computation is organized adaptively. The interval is first divided into 9,046 exact rational root cells. Two independent derivative families are then certified at their minimal dyadic depths. For the rank-two derivative family, $$\max_j d_j^{R}=3,$$ with depth distribution $$(1902,2786,3483,875).$$ For the finite-middle derivative family, $$\max_j d_j^{F}=2,$$ with depth distribution $$(6905,1719,422).$$ The certified rootwise relation $$d_j^{F}\le d_j^{R}$$ holds for every root. Consequently the two derivative systems share the same adaptive refinement as the rank-two system, and the common derivative cover contains exactly $$28{,}406$$ rational dyadic leaves. This cover is then aligned with a mesh of 2,001 anchor points. Exactly 1,982 derivative leaves cross an anchor midpoint. Splitting only these leaves produces an anchor-compatible master cover with $$30{,}388$$ leaves. The anchor prices and derivative inequalities propagate to every point of each master leaf. The resulting whole-cell relative prices satisfy $$\rho_{\mathrm{fin}}(X)<0.53,\qquad\rho_{\mathrm{exp}}(X)<0.22.$$ The rank-two channel also carries a rigorously bounded residual contribution. After this residual is included, the corrected tail price satisfies $$\rho_{\mathrm{tail}}(X)<\frac{113356755619}{500000000000}=0.226713511238.$$ The remote complement has the certified uniform lower bound $$R(X)\succeq 0.779I.$$ Hence the final Schur reserve is bounded below by the exact rational quantity $$\frac{779}{1000}-\frac{53}{100}-\frac{113356755619}{500000000000}=\frac{11143244381}{500000000000}>0.$$ This yields $$D_{X,8}\succeq0.20I$$ throughout the full cell $5.01\le X<7$. The analytic contribution is the passage from finite local certificates to a continuum coercivity theorem. The principal ingredients are: - fixed-target Loewner propagation in a moving positive metric;- separate relative pricing of independent coupling channels;- adaptive minimal-depth certification on exact rational cells;- hereditary common-cover inheritance;- exact alignment with the anchor geometry;- a final rational Schur-complement closure. The theorem-critical finite computations are performed using rigorous outward-rounded interval arithmetic and exact rational checks. The numerical certificates establish local directed inequalities; the implication from those finitely many inequalities to the whole-interval theorem is analytic. Some constants used at the frozen block-reduction stage, including the remote-complement lower bound and the residual-channel bound, are inherited inputs from the sealed reduction underlying the comparison operator. The whole-cell metric, anchor, derivative, adaptive-cover, and propagation certificates are verified over the complete interval. No endpoint value is used as a substitute for a whole-cell inequality. The theorem concerns the parity-complete eight-mode comparison operator on the arithmetic cell $5.01\le X<7$. It is not a proof of the Riemann Hypothesis and does not assert positivity of the full localized Weil form for arbitrary support ranges. ## Verification Supplement The complete verification supplement is archived separately on Zenodo: **DOI:** 10.5281/zenodo.22127800 The verification archive contains the directed interval certificates, exact root-depth maps, common- and master-cover data, corrected tail and Schur certificates, dependency checks, and final validation material required to reproduce the computer-assisted part of the proof. The verification archive has SHA-256 `bd5d9e23e7dede9a1076d51726aba2b82e55d9921e845ac76b0a64d8c6abba3e` and its final whole-cell seal has SHA-256 `4b9c461c765725291421d3ddada382761c5a4f588bb95c7206ffd5a75dbd97ee`. **Version:** v1.0 **Author:** Lee Byoungwoo