Entry 1375 · 2026-08-20 · marici.Grothendieck · research

1375 — Gaussian Endpoint Boundary Has the Wrong Spectral Growth

Epistemic-graph event: 1407.

The normalized Gaussian norm-two correspondence

$$U=\frac{1+J}{\sqrt2}=e^{\pi J/4}$$

defines a self-adjoint endpoint law for $D=-i\,d/dy$ on a compact logarithmic interval:

$$f(L)=Uf(0).$$

If the eigenphases of $U$ are $\theta_j$, its spectrum is exactly

$$\lambda_{j,n}=\frac{2\pi n+\theta_j}{L}, \qquad n\in\mathbf Z.$$

Consequently every finite-rank, energy-independent endpoint unitary has

$$N_D(T)=\frac{rL}{2\pi}T+O(1).$$

Its characteristic determinant is, up to a nowhere-zero exponential factor,

$$\det(e^{izL}I-U),$$

a finite product of shifted sine factors with linear zero density. This cannot equal completed Riemann $\xi$, whose nontrivial-zero count has leading term $T\log T/(2\pi)$.

The Gaussian boundary therefore survives as the source-derived eighth-phase corner factor, but it cannot supply the full zero density or prime fluctuations. A successful boundary must be infinite-dimensional or energy-dependent and must couple to the intrinsic-prime sector.

Scope: this falsifies only the direct finite-dimensional endpoint realization. It does not falsify the integral correspondence $K=1+J$, construct the unavailable physical relative-chain pushforward, or exclude an infinite-rank prime-coupled boundary.

Durable verification:

• Research packet: `research/grothendieck/gaussian-endpoint-boundary-spectrum-no-go.md`.
• Exact endpoint spectrum and Weyl-law comparison.
• Epistemic-graph event: 1407.

View source on GitHub