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

This paper develops a quantitative rigidity theory for finite weighted-exponential source clouds modeled by genuine nontrivial zeros of the Riemann zeta function within a fixed-rank prime-log module.

The main result is a two-price fragmentation barrier. Under exact framed prime-log ancestry and a vanishing normalized prime boundary, a macroscopic fragmented zero cloud with \(q\) atoms and \(M\) arithmetic components must pay at least one of two analytic costs:

$$
\mathfrak E_S^2
\ge
\left(
\frac{p_{\min}}{4\ell^2}+o(1)
\right)\frac{q}{M},
$$

or

$$
\mathfrak B_{\mathrm{comp}}^2
\ge
\left(
\frac{p_{\min}^2}{256}+o(1)
\right)\frac{q^2}{M}.
$$

Here \(\mathfrak E_S\) is a Stein-stripped framed differential leakage and \(\mathfrak B_{\mathrm{comp}}\) is a within-component Gram defect.

The proof proceeds through a sequence of structural rigidity mechanisms. An endpoint-tapered atomic Hilbert–Schmidt large-sieve estimate forces mesoscopic delocalization of any macroscopic genuine-zero source. Exact prime-log ancestry then yields a zero-capacity bound that excludes dense full coefficient boxes and forces sparse arithmetic occupancy. Under a small prime-boundary hypothesis, the selected coefficient cloud must fragment into logarithmically many connected components. The associated graph Laplacian produces exact component zero modes. A retained carrier frame restores the absolute ordinate coordinate lost at the pair-Gram level, converts prime-log shifts into an exact commutator relation, and transports the resulting coefficient-space leakage to a first-order weighted-exponential differential operator. A weighted point-trace argument then yields the final differential-energy versus Gram-defect alternative.

The novelty lies not in the classical ingredients separately, but in their quantitative coupling: zero counting and prime-log sparsity are converted into arithmetic fragmentation, framed zero modes, and ultimately a source-side analytic obstruction.

The paper does not prove that the assumed genuine-zero realization or the small prime-boundary condition follows from a hypothetical first failure of Weil positivity, and it does not claim a proof of the Riemann hypothesis. Its results are conditional rigidity theorems for the explicitly defined realization class.

Version v1.0.