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Transcendental Phase Geometry from C7 to C15: The Evolution of the ζ–Tower in Multi–Carrier Interference Bora Akta¸s & ChatGPT (co-author) 2025 Abstract The extension of multicarrier phase geometry beyond C6reveals a systematic emergence of higher transcendental constants in the curvature coupling parameter κn. Starting with κ6≈π+ζ(3), the even–order manifolds (C8, C10, C12, C14) successively introduce odd–weight zeta values ζ(5), ζ(7), ζ(9), ζ(11), ζ(13), forming an ascending “ζ–tower” that connects analytic number theory with quantum phase evolution. Odd–order systems (C7, C9, C11, C13, C15) retain algebraic symmetry at leading order, serving as geometric controls for the transcendental expansion. This paper formulates the analytic structure, geometric transitions, and physical analogues of the C7–C15 hierarchy, showing that phase curvature evolves from closed elliptic manifolds to open hyperbolic geometries whose limits are governed by odd–zeta couplings. 1 Introduction The Phase–Cone framework provides a unified description of multicarrier interference in which each Cnmanifold defines an n–fold phase symmetry. For n≤6, the curvature coefficient κnremains algebraically closed. However, the emergence of transcendental constants from n= 6 onward marks a transition from geometric to analytic curvature: the phase manifold ceases to be algebraically integrable and inherits residues from the analytic continuation of hypergeometric kernels. This continuation mirrors the behaviour of the Riemann ζ–function: the residues at odd integers (s= 3,5,7,...) reappear as curvature corrections in κ2m. The phenomenon thereby bridges analytic number theory and physical geometry: Geometric closure (π)↔Analytic openness (ζ(3), ζ(5), ζ(7),...). From C7to C15 the pattern becomes clear: even nintroduces a new odd–ζconstant; odd npreserves algebraic closure. This alternation defines the “transcendental rhythm” of phase geometry. 1 2 Mathematical Structure: C7–C15 2.1 Parity law n= 2m:κ2m=aππ+ m X k=2 a2k−1ζ(2k−1) + O(MZV), n= 2m+ 1 : κ2m+1 = (algebraic) + O(symmetry breaking). This parity rule implies that only even–ngeometries contribute to the transcendental hierarchy. 2.2 Analytic continuation of curvature kernel The Cncurvature integral admits a Mellin–Barnes representation, p+1Fp(an;bn; 1) = 1 2πi ZCQjΓ(aj+s) QkΓ(bk+s)Γ(−s)ds, whose residues at s= 1,3,5,7, . . . yield ζ(3), ζ(5), ζ(7), . . . . For even n, the symmetry projector P2mpreserves these residues; for odd n, it cancels them. 2.3 Explicit projections κ7≃4.00+O(algebraic), κ8≃π+ζ(3) + 1 2ζ(5) + 1 3ζ(7), κ10 ≃π+ζ(3) + 0.6ζ(5) + 0.4ζ(7) + 0.25ζ(9), κ12 ≃π+ζ(3) + ζ(5) + ζ(7) + ζ(9) + ζ(11), κ14 ≃π+ 7 X k=2 ζ(2k−1), κ15 ≃algebraic (modular C3×C5control). 3 Physical Interpretation 3.1 Geometric regimes •Odd n(C7, C9, C11, C13, C15): Elliptic, closed, algebraically bounded. Phase recursions are periodic and reversible. •Even n(C8, C10, C12, C14): Hyperbolic, open, analytically extended. Phase trajectories drift; unitarity deforms into analytic openness. 3.2 Quantum speed limit scaling The fractional phase drift per period is approximately ∆ϕ2m 2π≈1 2π m X k=2 α2m,2k−1ζ(2k−1). 2 Estimated drifts: C8: 0.33 C10 : 0.39 C12 : 0.45 C14 : 0.48 (normalized to 2πphase period). Thus, higher even orders broaden the quantum–speed–limit cone; phase evolution becomes progressively governed by analytic curvature. 3.3 Physical analogues Order Geometry type Physical analogue C7Elliptic, closed Resonant cavity, atomic orbital C8Hyperbolic, open Optical interferometer (6–8 path) C9Modular hybrid Spin–lattice synchrony C10 Hyperbolic Josephson–junction arrays C12 Strongly hyperbolic Photonic cluster states C14 Saturated analytic Decoherence–driven open manifolds 4 Experimental Outlook Even–ninterferometers allow direct probing of odd–ζconstants via fringe drift analysis: In(ϕ)=I0"1 + 1 nX m=k cos(m−k)ϕ+δϕ(π+X k≤m ζ(2k−1))#. Comparison of (C2m+1, C2m) pairs separates algebraic and analytic contributions. Measurement of the ζ(11) and ζ(13) components in C12 and C14 geometries would constitute the first physical detection of high–order zeta couplings in quantum interference. 5 Conclusion The hierarchy C7→C15 demonstrates that phase geometry evolves through a systematic “analytic quantization” governed by odd zeta values. Odd–nmanifolds remain algebraic; even–nmanifolds successively acquire ζ(3), ζ(5), ζ(7), . . . . This alternating pattern forms the analytic skeleton of multicarrier phase space: Elliptic (closed) −→ Hyperbolic (open) −→ Transcendental (analytic). If verified experimentally, these results would signify that the Riemann zeta function encodes not only arithmetic structure but the dynamical curvature of physical phase evolution itself. 3