Trace-Class Følner Selection and Observability Barriers in Localized Weil–Parent Systems

## Overview This paper develops three operator-theoretic mechanisms for soft-to-hard arithmetic selection and applies them to a localized Weil--parent transfer problem. The first mechanism is a mass-retaining trace-class Følner selection principle: a positive trace-class operator on $\ell^2(\mathbb Z^r)$ with small normalized commutators against fixed prime shifts admits a threshold set that simultaneously retains a prescribed fraction of the soft mass and has small normalized arithmetic boundary. The second mechanism is a position--modulation coercivity theorem. On a fixed compact physical interval, commutators with two or more distinct prime modulations are quantitatively equivalent, in Hilbert--Schmidt norm, to the position commutator. This converts prime-label covariance into physical localization. The third mechanism is an explicit finite-rank phase-space compression theorem for weighted-exponential source families. If $q$ atoms form $M=o(q)$ arithmetic components, a cellwise Taylor construction produces a projection $P$ satisfying $$\operatorname{rank} P = O(q),\qquad\operatorname{Tr}(P S_{\rm phys}) \ge (1-\varepsilon^2)q,$$ and $$\|[X,P]\|_{\rm HS}^2=O\!\left(\frac{M^2}{q}\right)=o(q).$$ These three results are independent of zeta-zero input. ## Localized-Weil application The operator-theoretic mechanisms are then inserted into a localized Weil--parent framework motivated by first-failure and genuine-zero completion problems. Exact prime-step geometry shows that an arithmetic Følner cloud cannot remain inside a fixed translation window. Under the explicit first-failure scale assumptions used in the paper, the component geometry yields an anchored-rich versus one-sided-unanchored routing alternative. On the absolute-gauge side, the paper proves that an exact genuine dyadic completion has a side-corrected boundary deck with $$\operatorname{Tr} B_{\partial}=q-O(\log T),$$ so the conserved boundary mass is asymptotically macroscopic. The obstruction is instead one of observability. Uniformly localized $R$-channel observations see at most $$O(R\log T),$$ while even an optimistic span of the first $R$ unshifted Carleman modes sees only $$O\!\bigl(\log T(1+\log R)\bigr).$$ Consequently, in the canonical lower-scale regime $q\gtrsim(\log T)^2$, recovering a fixed positive fraction of the boundary deck through the unshifted Carleman tower requires polynomial-in-$T$ observation rank. ## Structural conclusion The paper therefore separates two issues that can otherwise be conflated: - arithmetic small-boundary production can be reduced to soft covariance and component geometry;- the remaining obstruction lies in absolute-gauge parent observability and component routing. The resulting theorem is a structural obstruction statement for the present fixed-response / hereditary architecture. It is not a proof of the Riemann hypothesis. Several interfaces remain explicitly open, including genuine parent realization from a hypothetical first failure, exclusion of the one-sided unanchored branch, identification of retained selector-source overlap with contradiction-carrying signed energy, and heterogeneous-depth extensions. ## Mathematical context The localized-Weil background is placed in the line of Weil's explicit-formula positivity, the localized and variational formulations of Yoshida and Bombieri, the operator- and trace-formula framework of Connes--Consani, and Suzuki's closed localized quadratic-form realization. The paper also relates its operator core to operator Følner theory, trace ideals, discrete coarea and isoperimetry, and classical time-frequency localization. A recent certified prime-7 localized-Weil calculation is cited only as a concrete companion application of finite-rank and Feshbach reduction in the same underlying operator setting; it is not a proof dependency of the present work. ## Version 3.0 Version 3.0 is a major public revision of the earlier release. The theorem core is preserved, while the paper has been substantially strengthened in presentation and provenance. In particular: - the title and Introduction have been reorganized around the standalone operator-theoretic results;- the active-prime hypothesis needed for the one-sided routing theorem is now stated explicitly;- the trace-class pullback and soft-to-hard extraction hypotheses are presented more transparently;- the localized-Weil historical and operator-theoretic provenance has been expanded;- the related-work discussion now connects the paper more directly to operator Følner theory, trace ideals, phase-space localization, and the Weil--Yoshida--Bombieri--Connes--Consani--Suzuki lineage;- the scope of the structural obstruction theorem and the remaining open interfaces are stated explicitly. No Riemann-hypothesis conclusion is claimed.