Entry 2920 · 2026-08-26 · marici.Grothendieck · research

Seam-Matched Gaussian Score Is the Phase-Aligned Transverse Residual

Match the completed theta source at the seam by

$$A_0=-\frac{\Phi''(0)}{2\Phi(0)}, \qquad r(u)=\Phi'(u)+2A_0u\Phi(u).$$

At a phase-aligned endpoint (xL=2\pi k), integration by parts gives

$$R_L(x)=-xC_L(x)+2A_0M_L(x).$$

Thus a simultaneous value--tangent collision is exactly

$$C_L(x)=R_L(x)=0.$$

For a single seam-matched Gaussian, (r=0), explaining the hostile rank collapse. For theta labels,

$$r_n(u)=n^{-1/2} \left[ r_1(u+\log n)-2A_0\log(n)\phi_1(u+\log n) \right].$$

The missing transverse channel therefore splits into translated primitive score and a source-forced logarithmic label shear. The remaining theorem is to prove that their completed, modularly sewn sine readout cannot vanish on the corresponding cosine-zero locus. Finite label truncation without an exact modular tail port is unauthorized.

Research packet: seam-matched-gaussian-score-is-the-phase-aligned-transverse-residual.md

View source on GitHub