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Analytic Invariants of Phase Geometry: Reformulating Quantum Evolution as Energy + Analytic Continuation Bora Akta¸s1ChatGPT2 1Independent Researcher, Ankara, T¨urkiye 2OpenAI Research Partner October 2025 Abstract Odd Riemann zeta values, ζ(2k+1), emerge as intrinsic analytic invariants of multicarrier phase geometry. We propose that quantum evolution laws can be reformulated not merely as functions of energy dispersion, but as composite relations of energy plus analytic continuation. The result is a hierarchy of “analytic curvature layers” in which ζ(3), ζ(5), and ζ(7) successively correct the geometric speed limits of phase propagation. This short communication outlines the foundational axioms, derives a general analytic-continuation factor for quantum speed limits, and sketches an experimental calibration protocol linking C6and C8interferometric manifolds. 1 Analytic Invariants in Phase Geometry For each even multicarrier manifold C2m, we define a set of analytic invariants Z2m={I2k+1 =α2m,2k−1ζ(2k+1) : 1 ≤k≤m−1}, which remain conserved under symmetry-preserving evolution. These constants act as internal curvature fingerprints of the interference metric: while πdescribes geometric closure, ζ(2k+1) quantify analytic openness. Hence, phase geometry inherits both algebraic symmetry and analytic depth. 2 Energy + Analytic Continuation Principle Let ∆Ebe the energy uncertainty between two distinguishable states on a curved phase manifold. We postulate that the minimal evolution time obeys τmin(Cn) = ℏ 2∆EAn,An="1 + X k≥1 cn,2k+1 ζ(2k+1) π#1/2 . 1 When analytic corrections vanish, An→1 and the Mandelstam–Tamm bound is recovered. In the presence of analytic continuation, however, An>1, introducing an additional “temporal curvature” that slows phase evolution even at fixed ∆E. The maximal phase velocity reads v(max) ϕ(Cn) = √κn ∆E ℏ, κ2m=π+ m−1 X k=1 α2m,2k−1ζ(2k+1), so that each even parity adds new odd-zeta terms to the curvature coupling. For C6, κ6≈π+ζ(3); for C8,κ8≈π+ζ(3) + ζ(5) + ζ(7). The corresponding phase-velocity enhancement, ∆vϕ vϕ≃ζ(5) + ζ(7) 2[π+ζ(3)] ≈0.12, predicts a measurable ∼12% analytic curvature increment between C6and C8systems. 3 Analytic Noether Symmetry If the action functional S[Φ] = Zddxn1 2(∂Φ)2−V(Φ)o+X k≥1 ζ(2k+1) ZddxJ2k+1[Φ] is invariant under Φ →Φ+ϵχ(x) with δJ2k+1 =∂µ(χQµ 2k+1), then each odd-zeta layer yields a conserved analytic current ∂µQµ 2k+1 = 0. Thus ζ(3), ζ(5), ζ(7) generate a hierarchy of analytic conservation laws beyond standard geometric symmetries. Experimentally, these invariants manifest as persistent phasecurrent channels whose interference contrast remains stable under global curvature deformation. 4 Calibration and Experimental Outlook To verify the analytic corrections, we propose a differential calibration between C6and C8interferometers operated under identical energy dispersion: ∆AC =v(max) ϕ(C8)−v(max) ϕ(C6) v(max) ϕ(C6)≈ζ(5) + ζ(7) 2[π+ζ(3)] . Eight-arm optical or atomic interferometers with sub-milliradian precision can detect such fractional drifts. A confirmed signal would establish that ζ(5) and ζ(7) are measurable analytic invariants of phase geometry, linking the Riemann hierarchy directly to quantum kinematics. 2 5 Concluding Remark Odd zeta values act as intrinsic analytic curvature coefficients of physical space–time’s phase fabric. They represent the “continuation depth” of reality: πcloses geometry, while ζ(2k+1) opens it. Quantum evolution, therefore, is not purely energetic but energeticanalytic — governed simultaneously by dispersion and analytic continuity. In this picture, the constants ζ(3), ζ(5), and ζ(7) are no longer abstract; they are the measurable fingerprints of how the universe folds analytic structure into physical time. 3