PRH | Aux | 4.2 • Fermat via Geometry, Blur, and Fourier

We develop a Fourier-analytic “blur’’ framework for the geometric avatar of a putative Fermat identity. The target is the cube ring \(\mathcal R_{a,c}=Q_c\setminus Q_a\) with \(Q_L=[-L/2,L/2]^n\) and integers \(n\ge3,\ 0<a<c\). We ask whether \(\mathcal R_{a,c}\) can be exactly realized by the superposition \(f_{s,\mu}=\chi_{B_s}*\mu\) of \(b^n\) congruent axis–aligned cubes \(B_s=[-s/2,s/2]^n\) (“builders’’). Two placement–independent spectral obstructions survive Gaussian blur (i) the zero–hyperplane wall (\(s\mid a\) and \(s\mid c\) are necessary), and (ii) the DC wall (\(s^n b^n=c^n-a^n\) is necessary). Under these arithmetic walls we study the spectral ratio \(g(\xi)=\widehat{\chi_{\mathcal R_{a,c}}}(\xi)/\widehat{\chi_{B_s}}(\xi)\). Exact equality would force \(g\) to be the Fourier transform of a finite positive measure, hence positive–definite (PD). Along a coordinate axis we compute the \(3\times3\) Toeplitz Gram determinant with sharp Taylor remainders and prove PD–failure on a large, explicit parameter region; what remains is a compact small–\(x\) strip.

The new contribution of this note is an analytics–to–certificate reduction tailored for implementation (e.g., in C#): we rigorously shrink the search to a finite set of steps \(\tau\) and to finitely many parameter boxes, within which an interval arithmetic check of \(\Delta_g(t)<0\) (with explicit Taylor remainders) certifies PD–failure. This leaves a finite, fully explicit computer–verifiable certificate as the only remaining step. We also prove a large--\(n\) simplification that makes this finite certificate particularly shallow for all \(n>50\)
(fewer \(\tau\) values and coarser boxes suffice), though still finite rather than purely analytic.