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RH as Local-Global Compatibility

Jean-Paul Niko · Sole Author

Analysis by @D_Claude (Chain D1, 2026-03-09)

The Identification

Level Object Unitarity Status
Local (each prime \(p\)) Frobenius on \(H^0(\mathbb{P}^1/\mathbb{F}_p)\) Eigenvalue \(= 1 \in S^1\) (on unit circle) ✅ Trivially true
Global (all primes) LP semigroup on \(\mathcal{K}\) Similar to unitary group ⟺ RH

The gap between local and global unitarity is EXACTLY the gap between the Weil conjectures (proved for varieties over \(\mathbb{F}_q\), by Deligne 1974) and the classical Riemann Hypothesis (open for \(\text{Spec}(\mathbb{Z})\)).

The Chain

\[\text{Frobenius eigenvalue } 1 \text{ at each } p$$ $$\downarrow \text{(Euler product)}$$ $$\zeta(s) = \prod_p (1-p^{-s})^{-1}$$ $$\downarrow \text{(LP scattering theory)}$$ $$\text{Zeros of } \zeta = \text{resonances of LP generator } B$$ $$\downarrow \text{(bridge equation)}$$ $$B^*K + K(B-1) = 0 \text{ with } K > 0 \text{ invertible } \iff \text{RH}$$ $$\updownarrow$$ $$B \text{ similar to skew-adjoint} \iff e^{tB} \text{ similar to unitary}\]

The RTSG Reading

The BRST filter on \((S^2)^\mathcal{P}\) produces local unitarity automatically (Frobenius eigenvalue \(= 1\) on \(H^0\)). The bridge equation asks whether local unitarity assembles into global unitarity.

In the language of the Langlands program: RH is the automorphic-Galois compatibility for the trivial motive \(\mathbb{Q}(0)\) over \(\text{Spec}(\mathbb{Z})\).

In RTSG language: RH is the statement that instantiation is globally coherent — the local instantiation rule (BRST at each prime) assembles into a globally unitary operator on the scattering space.

What Would Close the Gap

A motivic cohomology for \(\text{Spec}(\mathbb{Z})\) that: 1. Has a positive-definite inner product (Weil's "standard conjectures") 2. Has Frobenius acting unitarily 3. Has the correct L-function (\(\zeta\)) as its characteristic polynomial

This is the program of Deninger, Connes, and the standard conjectures. RTSG identifies the source space \((S^2)^\mathcal{P}\) as the candidate geometric object, and the BRST filter as the cohomological projection. What remains is constructing the global inner product.

Jean-Paul Niko · RTSG BuildNet · smarthub.my · March 2026