[Dugongue]

Construct the actual periodic Gaussian inverse by splitting its rounded grid into complete Beatty runs. Each run is an explicit diagonal similarity of a Gaussian Toeplitz matrix with a two-convolution inverse. Truncating only cross-run interactions leaves a boundary system; a cyclic block solve and two run-inverse applications recover the result without a dense inverse-update product.

In the stated regime, the inverse-adapter upper bound improves from O(V p^(3/8)) to O(V p^(1/4) log p), after charged input-independent exp(o(p)) preparation, with O(p)-bit words. The proof includes the periodic seam, truncation, conditioning, intermediate precision, convolution arithmetic and tensor error accumulation. The native corollary uses an explicitly extended Word RAM with integer division and fixed-point rescaling. This is an inverse-component result; no higher complete multiplication kappa is claimed.

Adds five mathematical Markdown documents only. The accompanying notes prove a common-gauge convolution algebra with an explicit carry tensor, normalized geometric-annihilator bounds, and complete cost/precision/recurrence assembly statements. These are informal proofs with manual mathematical cross-review; no numerical experiments, Lean verification or independent human acceptance are claimed.

The notes reuse the shared DAG and linked Lean sources on total positivity, symplectic mode support and balanced coevaluation, exact tensor execution, polynomial Toeplitz products, FastPoly, tcslib, Mpfx and CatCrypt. The inherited Gaussian formulas, banded inverse and bilateral LDU retain their Harvey–van der Hoeven/source08 and contributor credits. The coded-interface lineage credits THU New Media Research Center, Ali Romman, Dugongue, Dillon Amburgey, Cinna Davis, Ryanson Jonathan, Cao Yifan, Jacob Alan Sussman, Chris R/Clear1nsight and zxv. Full source-relative citations and scope distinctions are in the notes. Substantial OpenAI Codex assistance from four workers; classical algebraic and numerical identities are not claimed as inventions.

Mathematical review can start at analysis.md A11 and complexity.md Theorem 11. Only research/beatty-run-gaussian-inverse is added; the selected frontier is unchanged.