Reconstructing Claude's $1.04/1.26/1.70$ Support Ladder

Claude's Remark 1.1 only says “roughly” in giving $70\%\to\sigma\approx1.04$, $80\%\to\sigma\approx1.26$, $90\%\to\sigma\approx1.70$; this document works out the mechanism that generates these three numbers — it generalizes the support of the Montgomery–Taylor/CCLM one-delta extremal problem from $\sigma=1$ to arbitrary $\sigma$, and uses a Lagrange-multiplier / reproducing-kernel argument to obtain the closed form $q(\sigma)=1-1/\langle\mathbf1,A_\sigma^{-1}\mathbf1\rangle$, where for $\sigma\le1$ the kernel is untruncated and degenerates to the cosine closed-form solution of Claude §7.1; for $\sigma>1$ the kernel becomes a truncated triangular-kernel Fredholm equation, and the $\sigma\le1$ formula cannot simply be extrapolated (exactly the pitfall hit earlier when computing 90%). Using a midpoint-Nyström numerical reconstruction: $\sigma_{70}\approx1.04263$, $\sigma_{80}\approx1.25785$, $\sigma_{90}\approx1.70146$ — matching the paper's rough values almost exactly. The document also draws a clear boundary: the extended values $\sigma_{95}\approx2.26079$ and $\sigma_{99}\approx4.1872$ are a numerical reconstruction this research carried out under the same structure, not a new theorem stated in Claude's paper. Finally, it reformulates the entire ladder as a “proportion ↔ information-bandwidth” map, and splits out two entirely distinct proof obligations: the extremal requirement ($\sigma_q$ itself, already numerically reconstructible) and arithmetic realizability (whether support $\sigma$ can actually be legitimately supplied — this is the real number-theoretic wall).