# CSM_RH Paper 48 package

This package advances CSM_RH from state v1.38 to v1.39 inside Campaign 44.

Main certified results:

- exact PESC positive-lag position kernel and exact lag masses;
- the early prefix plateau is root-scale and cannot be demoted to a boundary error;
- exact symmetric completion `G_N(m,n)=w_N(max(m,n))=V_N^T V_N`;
- rank `N`, nullity `N-1`, and the exact root quotient `y_0=B(N)`, `y_r=c_{N+r}`;
- reversed quotient kernel is the discrete Brownian covariance `H_N(i,j)=min(i,j)`;
- explicit sine eigenbasis and eigenvalues `mu_k=[4 sin^2(theta_k/2)]^{-1}`;
- spectral condition number is `(16/pi^2)N^2(1+o(1))`;
- on a pure-prefix test direction of the exact root quotient, the first spectral mode carries `(8/pi^2+o(1))` of that quotient energy; this is a kernel-observability witness, not a modal claim about the actual prime-error sequence;
- fixed-power PESC is exactly exponent-equivalent to a weighted Brownian spectral-energy bound, with the inherited diagonal remaining lower order.

PK1 closes as `CLOSED_AS_EXACT_LAG_POSITION_NORMALIZATION_WITH_PREFIX_GRAM_COMPLETION`.

PK2 closes as `CLOSED_AS_EXACT_BROWNIAN_SPECTRAL_REPRESENTATION_WITH_LOW_MODE_PRESERVED`.

Campaign 44 remains active.

Next continuation: Campaign 44 / PK3 — `TRANSLATED_WINDOW_SYNTHESIS`.

No fixed-power PESC theorem is claimed. No RH proof or disproof is claimed.