Paper 2026/1586

Perturbation of Hankel moment singular values and supersingular endomorphism rings via CVP: a $p$-adic super-resolution law and a fully computed pipeline

Abstract

We give two rigorous results in post-quantum algebra with a $p$-adic strengthening and a complete reproducible pipeline. Part I proves an explicit sufficient noise bound under which the Hankel matrix of the power moments of supersingular $j$-invariants deterministically recovers the nodes, with the propagation constant written through the Vandermonde condition number (Weyl, Bauer-Fike); non-archimedeanly, the Teichmüller lift makes the Vandermonde matrix unimodular ($\mathrm{cond}_p = 1$ for every $L$) and an exact super-resolution law gives the $p$-adic precision loss as $2\sum_{i<j} v_p(x_i - x_j) + \sum_i v_p(c_i)$, a sharp analogue of Moitra's bound. Part II reduces a $\mathbb{Z}$-basis of $\mathrm{End}(E)$ to a rank-4 CVP and recovers the exact Gram matrix of the norm form in $\mathrm{poly}(\log p)$ time via Weil-pairing discrete logs (Shor; in the smooth regime actually run, the logs are classical Pohlig-Hellman), after which fixed-dimension LLL yields a canonical basis and a $\lambda_1$-criterion reads the node type. Both parts are joined by an end-to-end theorem and fully computed on real numbers: node recovery, real Vélu chains with measured degree, deterministic KLPT construction with the ideal-to-isogeny and smoothing steps, reading the torsion action in $\mathbb{F}_{p^4}$ and via Weil pairing + Pohlig-Hellman without an $O(N)$ table, true LLL + Fincke-Pohst, and the classification of all three nodes of $B_{23,\infty}$. The last KLPT heuristic (polynomial running time) is replaced by an explicit hypothesis PRH and a conditional theorem; PRH is shown to be exactly a Titchmarsh-type shifted-prime divisor sum with positive singular series, provable under GRH for fixed $p$, with uniformity in $p$ an identified open problem. Companion code (23 modules) verifies every numerical claim.