[Dugongue]

Prove an O(S log p) application algorithm for the actual periodic rounded Gaussian inverse at fine gaps theta = Theta(p^(-2)) and a = Theta(log p), using O(p)-bit words after charged input-independent preparation. The identity (1+qz)H(z)=H(q^2 z) splits the q-LDU filter into a stable bidiagonal recurrence and a short locally recentered convolution. A physical boundary Schur complement supplies the periodic recovery without the huge whole-run diagonal gauge.

Construct complete scalar gauge boundaries with Gaussian tariff rank(M_01)+rank(M_10), including owned buffers, coefficient data and an equal reciprocal-dual tariff. Additional informal theorems give target-gauge query redirection, exact multiaxis B/C incidences, and a dense common-frame coded XOR-convolution consumer at its actual output dimension.

This does not establish a higher complete multiplication kappa. The full cost proof shows why a better nonbinding inverse term cannot raise the admitted provider ceiling. The quotient B/C local interfaces are not priced as completed independent Gaussian jobs. The package supplies positive sufficient algorithms and interfaces, with full informal proofs and explicit remaining assembly scope.

Adds six mathematical Markdown files, including a credited inherited q-LDU/convolution appendix. No numerical experiments, solvers, web searches or Lean checks were used. The native corollary states its extended arithmetic Word RAM, effective parameter descriptions, rescaling and raw-product-width assumptions explicitly. These are informal mathematical proofs, not an emitted program or independent human acceptance.

Credit the preceding Beatty-run construction; Cinna Davis and the PR4470 supplier/controller lineage; Dillon Amburgey and the PR4532/PR4535 majorants; Cao Yifan PR4610; DaysSky's quotient route; Dugongue's native and repaired-interface work; Nick Petalas, Ryanson Jonathan, Rohan Arun, zxv and Jacob Alan Sussman; Chafik Boukhalfa PR1786; Vlado Vojdanovski PR1673 and the fractional-shift/Toeplitz lineage; THU New Media Research Center PR5464, Ali Romman PR5439 and the retained whole-K/compiler sources; Harvey and van der Hoeven and source08. All individual notes retain their specific source-relative citations and scope. Substantial OpenAI Codex assistance from four workers. Classical rank, q-series, Fourier and Schur tools are not claimed as inventions.

Start with inverse.md Theorems 1-9, boundary.md B1-B8, and assembly.md Theorems 1-3. Only research/fine-gap-gaussian-boundaries is added; no selected-frontier change is proposed.