Mesoscopic Zero Clouds in Bandwidth-One Weil Certificates: Sparse Prime-Log Ancestry, Framed Zero Modes, and a Two-Price Fragmentation Barrier

## Overview

This Version 2.0 develops a new theorem architecture for the arithmetic-selection and observability bottleneck that arises in localized Weil-to-parent source systems. The main question is no longer only whether a large genuine zero cloud must pay one of two fragmentation prices, as in Version 1.0, but how a parent-side soft operator can produce a large arithmetic Følner set and why the presently available first-failure boundary observables still fail to route enough absolute-gauge information back to that set.

The paper therefore has two complementary parts. The first is constructive: it turns soft prime covariance and position localization into hard arithmetic selection. The second is obstructive: it proves that the full boundary deck may carry macroscopic trace while fixed-gauge and unshifted-Carleman observation channels recover only a vanishing fraction of it at the natural lower scale.

## Main results

### 1. Arithmetic Følner escape and soft-to-hard selection

For an exact prime-step ancestry cloud, directional arithmetic boundary is the number of maximal prime runs. As a consequence, a Følner cloud with

$$
b(S)=o(|S|)
$$

cannot remain macroscopically concentrated in any fixed translation window: every bounded window contains only $o(|S|)$ selected atoms.

The paper then proves a mass-retaining trace-class coarea theorem. If $B\ge 0$ is a positive trace-class soft selector with small normalized commutators against the active prime shifts $U_p$, then one can choose a single threshold set $S$ that simultaneously retains a prescribed fraction of the $B$-mass, has cardinality comparable from below to the effective rank of $B$, and satisfies a small normalized arithmetic boundary estimate. Thus hard Følner selection can be produced from operator covariance rather than imposed as an external combinatorial hypothesis.

### 2. Physical position localization as the prime-covariance criterion

For at least two active prime modulations $M_p=e^{-ih_pX}$ on a compact physical interval, the paper proves the two-sided coercive comparison

$$
c_*\|[X,A]\|_{HS}^2
\le
\sum_p \|[A,M_p]\|_{HS}^2
\le
C_*\|[X,A]\|_{HS}^2.
$$

Hence, for finite-rank selectors, small prime leakage is equivalent to small position leakage. This converts the arithmetic selection problem into a physical localization problem.

The covariant parent-filter pullback then transfers physical modulation leakage to the coefficient lattice while preserving the relevant parent load. This provides the operator bridge needed to apply the soft coarea theorem to genuine source systems.

### 3. Fragmentation-scale phase-space compression and anchored routing

For weighted-exponential source atoms divided into $M$ arithmetic components and total atom count $q$, a self-contained cellwise Taylor construction produces a source-macroscopic position-localized projection $P_{\delta,\varepsilon}$ satisfying

$$
\operatorname{rank} P_{\delta,\varepsilon}
\le
C_\varepsilon\left(q+\frac{M}{\delta}\right),
$$

$$
\operatorname{Tr}(P_{\delta,\varepsilon}S_{\rm phys})
\ge
(1-\varepsilon^2)q,
$$

and

$$
\|[X,P_{\delta,\varepsilon}]\|_{HS}^2
\le
\delta^2\operatorname{rank} P_{\delta,\varepsilon}.
$$

Choosing $\delta=M/q$ shows that $M=o(q)$ is enough to generate the position-localized selector required by the arithmetic-selection mechanism.

At the canonical lower-scale input

$$
q\ge c_0(\log T)^2,
$$

while the directly anchored atom count is only $O(\log T)$. Therefore, if a positive fraction of the cloud lies in anchored components, those components automatically satisfy the sublinear-fragmentation condition and activate the selector theorem. If not, a macroscopic family of completely unanchored components survives on one side. This isolates the precise remaining routing branch.

### 4. Macroscopic boundary mass versus observation rank

On the first-failure side, the genuine dyadic completion carries a macroscopic boundary deck:

$$
q-O(\log T)
\le
\operatorname{Tr} B_\partial
\le
q.
$$

Thus the obstruction is not a lack of boundary mass. It is a lack of observable rank.

The established fixed-gauge causal response samples only $O(\log T)$ total boundary energy on a genuine packet. More generally, any $R$ uniformly localized observation channels see at most $O(R\log T)$ deck trace.

The paper then studies an optimistic unshifted Carleman tower. After exact Jacobi orthogonalization, the first $R$ modes satisfy the visibility bound

$$
\mathcal O_R(S)
\ll
\log T\,(1+\log R).
$$

Consequently, observing a fixed positive fraction of a $q$-scale boundary deck requires

$$
R
\ge
\exp\!\left(c\frac{q}{\log T}-C\right).
$$

Since the paper only needs the lower-scale estimate $q\ge c_0(\log T)^2$, this already forces

$$
R\ge T^{c'}
$$

for some $c'>0$. Thus polylogarithmic Carleman truncations are asymptotically insufficient for macroscopic deck observation.

## Structural conclusion

The paper separates two issues that were intertwined in the earlier zero-cloud formulation.

* Arithmetic small-boundary extraction is no longer a purely external hypothesis: within the present common-depth framework it can be reduced to soft prime covariance, physical position localization, and sublinear component geometry.
* The surviving obstruction is parent routing with absolute gauge. The currently analyzed fixed-response, hereditary, and unshifted-Carleman mechanisms either carry macroscopic but translation-insensitive information, or retain absolute-gauge sensitivity at insufficient observation rank.

Accordingly, a proof-facing continuation would require a genuinely new non-hereditary mechanism that simultaneously carries macroscopic load, absolute translation gauge, same-atom provenance, and component-routing information.

## Relation to Version 1.0

Version 1.0 established a two-price fragmentation barrier for mesoscopic zero clouds and isolated four open upstream/downstream gates. Version 2.0 does not merely append estimates to that framework. It reorganizes the problem around a soft-to-hard arithmetic selection theorem, a physical position/modulation bridge, a self-contained phase-space compression theorem, and a quantitative observability-rank barrier. The earlier two-price theorem remains part of the background research line, but the central message of Version 2.0 is the separation between **arithmetic selection**, which can be made constructive under covariance/localization hypotheses, and **absolute-gauge parent observability**, which remains the principal structural obstruction in the present architecture.

## Scope

The results are conditional on the stated localized Weil/parent hypotheses and on the imported genuine-ancestry/first-failure inputs where explicitly indicated. The paper does **not** derive a full genuine parent realization from a hypothetical first failure, does **not** close the one-sided unanchored branch, and does **not** prove the Riemann Hypothesis. The structural barrier applies to the fixed-response/common-depth mechanisms analyzed in the paper and is not claimed as a universal impossibility theorem for all future approaches.