Phase–Channel Decomposition ( 𝐶𝑛 Algebras) as a Bridge between Number Theory, Quantum Mechanics, and Relativity

This collection of works develops a unified phase–channel framework that connects number theory, modular geometry, and fundamental physics. The central idea is the decomposition of wave dynamics into visible and hidden phase channels, inspired by the dual periods of elliptic curves. Within this approach: Number theory and modular forms: Phase–channel decomposition (CnC_{n}Cn algebras) reorganizes power series by exponents modulo nnn, producing hypergeometric kernels whose Picard–Fuchs equations link directly to modular families with complex multiplication (CM). Evaluations at CM points recover Ramanujan- and Chudnovsky-type series for 1/π1/\pi1/π, and extend naturally to other constants such as ζ(3)\zeta(3)ζ(3) and the Catalan constant. The framework clarifies why discriminants such as D=163D = 163D=163 and D=331D = 331D=331 yield extraordinary convergence through phase locking and CnC_{n}Cn-symmetry. Elliptic periods and geometry: The visible period K(k)K(k)K(k) and invisible period iK′(k)iK'(k)iK′(k) are interpreted as phase channels forming a pseudo-Hermitian structure. Diagnostics such as the contrast law κ=tanh⁡ρ\kappa = \tanh \rhoκ=tanhρ and the pseudo-metric N=∣ω1∣2−∣ω2∣2N = |\omega_{1}|^{2} - |\omega_{2}|^{2}N=∣ω1∣2−∣ω2∣2 connect data directly to hyperbolic geometry. Carrier decompositions (C3/C4/C5C_{3}/C_{4}/C_{5}C3/C4/C5) reveal multi-phase dynamics beneath apparently single-frequency observables and provide modular parametrizations with clear geometric meaning. Quantum mechanics and uncertainty: Extending Schrödinger dynamics to C3C_{3}C3 introduces hidden channels that conserve the full norm but inflate variances when projected onto the visible subspace. This leads to a variance decomposition theorem in which Robertson-type uncertainty bounds emerge as projection artefacts, reframing uncertainty as epistemic rather than ontological. Collapse is reinterpreted as information stored in hidden channels, with proposed experimental tests in Ramsey interferometry, multi-slit setups, and quantum tomography. Relativity and unification: The modular parameter τ=ω2/ω1\tau = \omega_{2}/\omega_{1}τ=ω2/ω1 is promoted to a spacetime field, yielding a nonlinear sigma model into the hyperbolic plane with positive-definite stress and natural coupling to Einstein gravity. The Unified Phase–Channel Equation (UPCE) recovers pendulum dynamics, Klein–Gordon quantization, and relativistic redshift as different limits of the same underlying structure. Curvature is expressed as R∝−□ln⁡ℑτR \propto -\Box \ln \Im \tauR∝−□lnℑτ, providing a geometric bridge between microscopic uncertainty and macroscopic gravity. Experimental outlook: The framework suggests signatures in atomic clocks, interferometers, Ramsey fringes, triple-slit experiments, and gravitational-wave phase shifts, offering concrete ways to test the hidden-channel hypothesis. Taken together, these contributions establish phase–channel decomposition as a unifying analytic language connecting elliptic curves, modular invariants, Ramanujan-type identities, quantum measurement, and relativistic geometry. Keywords: phase–channel decomposition, CnC_{n}Cn algebras, elliptic periods, modular forms, Ramanujan–Chudnovsky series, hypergeometric functions, complex multiplication, hidden channels, uncertainty principle, sigma models, curvature, quantum foundations, interferometry, gravitational waves