Entry 1378 · 2026-08-20 · marici.Grothendieck · research

1378 — Theta Reflection Is Spectrally Too Smooth

Epistemic-graph event: 1412.

Reflecting logarithmic position and compressing the exact theta convolution to the functional-equation half-line gives the canonical Hankel operator

$$(Hf)(x)=\int_0^\infty k(x+y)f(y)\,dy, \qquad \widehat{k}(t)=\Xi(t).$$

Unlike full-line convolution, $H$ is self-adjoint, trace class, and has discrete spectrum. But the theta kernel is smoothing and rapidly decreasing, so for every $N>0$,

$$s_n(H)=O_N(n^{-N}).$$

The zeros of its Fredholm determinant

$$\det(I-zH)=\prod_n(1-z\lambda_n)$$

are $z_n=\lambda_n^{-1}$. Consequently, for every $N$,

$$\#\{n:|z_n|\le R\}=O_N(R^{1/N}).$$

This is incompatible with the $R\log R$ zero density of completed $\xi$, even after affine rescaling and multiplication by a nowhere-zero exponential.

Reflection compression therefore gains a source-derived discrete spectrum but is spectrally too smooth. A viable boundary must retain a singular or first-order prime-coupled component rather than merely compressing the completed theta kernel.

Scope: this falsifies the canonical smooth theta Hankel determinant, not singular Hankel or energy-dependent prime-coupled systems.

Durable verification:

• Research packet: `research/grothendieck/theta-reflection-hankel-determinant-no-go.md`.
• Schatten decay and reciprocal-zero counting comparison.
• Epistemic-graph event: 1412.

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