Theoretical analysis demonstrates a dual mathematical and physical-optical architecture for the Riemann zeta function, establishing bridges between number theory and nonlinear optics.
We integrate two frameworks for the Riemann zeta function into a single coherent theory. Part A (The Point of Equilibrium) develops the pure-mathematical architecture in the recentred coordinate w = s − ½: the well at the PoE (vacancy value ζ(½) = −1.4603, tilt ζ′(½) = −3.9226, saddle ζ″(½) = −16.008), the fold (the ξ-completion as the crease where + and − balance, with height reversal |M(α + iγ)| ≈ (2π/γ)^α), five independent rigidities of σ = ½ (algebraic, analytic, group-theoretic, metric, variational), and the Parity Split Law ((ζ′/ζ)^(2n)(½) = 2^(2n)·(2n)!·[λ(2n+1) + β(2n+1)], verified to 10⁻³⁸). Part B (The 0–1 Light) translates this architecture into a physical-optical correspondence: primes map to emission lines at frequencies f_p = U ln p, the well becomes the potential landscape of a nonlinear fibre bench, and the sieve emerges dynamically from the concavity of the logarithm — composites carry a phase mismatch Δk = −2 ln p ln q that suppresses their amplification while primes (Δk = 0) are attractors. Nine bridge structures connecting the two frameworks are identified, of which the strongest is the well-asymmetry bridge: the Light paper's measured ratio |ζ(0.7+iγ₁)|/|ζ(0.3+iγ₁)| = 0.850 is shown to equal the Doorway Theorem's |M(0.2+iγ₁)| to eight significant figures. All claims classified proved/measured/conjectured/interpretive. No proof of RH is claimed.
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Timothy Desmond (2026) studied this question.
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