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Geometric Energy Spectrum and Redshift from Temporal Curvature: A C3–Phase Formalism Approach Bora Akta¸s & ChatGPT (co-author) October 2025 Abstract This work develops a geometric model in which temporal curvature directly determines the quantum energy spectrum. Within the C3–phase formalism, time and energy are treated as conjugate geometric operators acting on curved phase channels. The resulting framework replaces the classical concept of random uncertainty with a curvature-based necessity, where redshift and spectral deformation emerge as direct consequences of time curvature differences. The model demonstrates that a curved temporal metric reproduces observable redshift relations without invoking spatial expansion, providing a unified geometric interpretation bridging quantum and cosmological scales. 1. Introduction Standard quantum mechanics treats time as an external parameter and energy as its conjugate observable. However, in curved phase geometries, time and energy must be defined on equal footing: both are directions within the same geometric manifold. In the C3phase algebra (j3=−1), the triadic structure allows the temporal axis to possess curvature, encoded in its operator spectrum. The goal of this paper is to link that curvature directly to measurable quantities such as energy levels and redshift. This construction arises naturally from earlier results in which the time operator ˆ Twas defined as C3–Hermitian, cyclic, and geometrically closed ( ˆ T3=−τ3I). Here we extend that approach by coupling ˆ Tand ˆ Hthrough a common curvature tensor. 2. Mathematical Framework In the C3–phase manifold, the time and energy operators belong to the same geometric basis: (ˆ T, ˆ H)∈ PC3={jpj2q|p, q ∈ {0,1,2} }. The generalized commutation relation reads [ˆ T, ˆ H]3=iℏI+εˆ Cj−j2,(1) where ˆ Cj−j2is the phase curvature tensor, quantifying the local twist between time and energy directions. The corresponding phase-space metric is ds2= (dT)2+ (dE)2+α dT dE, α =j−j2,(2) introducing a cross term dT dE that represents the intrinsic time–energy coupling. In the flat limit (α= 0), this reduces to the conventional Schr¨odinger metric, while nonzero αencodes curvatureinduced mixing between the two axes. 1 The C3–Schr¨odinger equation with temporal curvature then reads (jℏ∂t+j2γ∇2+µ+ Φt)ψ= 0,(3) where Φtrepresents the temporal potential associated with local curvature Rt. 3. Geometric Energy Spectrum The time operator’s eigenvalues, λn=ℏ µeiπ(2n+1)/3, n = 0,1,2, define three distinct temporal curvature modes corresponding to the C3phase channels. We now postulate that the local energy spectrum couples to these curvature modes: En=E0(1+ε R(n) t),(4) where R(n) tdenotes the curvature eigenvalue associated with each temporal channel. The expectation value of the energy becomes ⟨E⟩=E01+εℜ⟨κj−j2⟩,(5) with κj−j2representing the projected curvature operator along the visible phase axis. Thus, the energy spectrum acquires a geometric correction proportional to temporal curvature. 4. Redshift as Temporal Curvature In this framework, redshift is not due to spatial recession but a relative difference in temporal curvature. Let R(s) tand R(o) tdenote the source and observer time curvatures, respectively. Then the redshift relation becomes 1+z=e∆Rt=e(R(s) t −R(o) t).(6) The observed frequency shift is therefore a direct measure of temporal curvature contrast. This leads to a reinterpretation of cosmological redshift: galaxies appear receding because their local time potentials differ, not because space itself expands. Equation (6) predicts that small curvature differences produce a linear redshift regime, while large curvature gradients yield exponential saturation, consistent with high-zobservations. 5. Discussion The curvature-dependent uncertainty relation ∆T∆H=ℏ 2eκtRt+κxRx implies that time curvature acts as a scaling factor for quantum uncertainty. When Rt→0, uncertainty approaches its classical limit ℏ/2, and both space and time flatten — the system classicalizes. Conversely, when Rtgrows, phase trajectories contract, producing energy quantization and redshift simultaneously. This mechanism provides a geometric link between quantum discreteness and cosmological expansion. The tensor ˆ Cj−j2governs this coupling: ˆ Cj−j2=j ∂T−j2∂H, binding visible and hidden channels of time and energy evolution. 2 6. Conclusion We have shown that: 1. Temporal curvature directly determines the quantum energy spectrum. 2. Redshift arises naturally as an exponential of curvature difference. 3. The C3phase metric provides a unified measure for time and energy, replacing randomness with geometry. 4. In the limit of vanishing curvature, classical dynamics and flat spacetime are recovered. This approach thus establishes a geometric bridge between quantum mechanics and cosmology, where energy quantization and cosmological redshift become manifestations of the same temporal curvature field. Appendix A: Temporal Curvature Operator For small curvature, Φt=ℏ2 2µRt+O(R2 t), and the associated Hamiltonian correction is δˆ H=εℏ2 2µRt(j−j2). This modifies energy eigenvalues by ∆En=ε E0R(n) t. Appendix B: C3Transform and Phase Projection The projection operators on visible and hidden channels are Πvis =1 3(1+j−j2),Πhid =1 3(1+j+j2). Expectation values of observables decompose as ⟨ˆ O⟩=⟨ˆ O⟩vis +ε⟨ˆ O⟩hid. Appendix C: Observational Implication Using (6), a curvature gradient of ∆Rt∼10−3produces a redshift of order z∼10−3, matching typical nearby galaxies. Hence, cosmological redshift can be reinterpreted as a differential temporal curvature effect, consistent with the geometric phase of the C3manifold. 3