Para-Dimensional Chronodynamics: A Formal Geometric Framework for Quantum Phase, Gauge Fields, and Singularity Regularization

We present Para-Dimensional Chronodynamics (PDC), a speculative geometric framework in which the real temporal coordinate t is extended to a complex manifold C^1, τ = t + iχ. Under analyticity assumptions, the Cauchy-Riemann condition ∂Ψ/∂t = i∂Ψ/∂χ is reinterpreted such that the Hamiltonian appears as translation generator along hyper-temporal axis χ. Compactification χ ~ χ + 2πR_χ yields discrete Fourier modes with quantized momentum nħ/R_χ, postulated to map to electric charge. 5D Kaluza-Klein reduction of metric ĝ_AB yields Maxwell action and gauge transformation A'_μ = A_μ + ∂_μΛ, with covariant derivative D_μ = ∂_μ - iqA_μ emerging from factorization. A hyper-temporal Higgs-type stabilizer field Φ with V(Φ) = -μ^2|Φ|^2 + λ|Φ|^4 is introduced to remove tachyonic modes, yielding M_n^2 > 0. A phenomenological Renormalization Group flow α(E) = [1+(E/E_P)^2]^-γ with γ = α_EM(E_P/E_ew)^2 ≈ 1.794×10^31 is proposed as geometric filter. At low energies, this suppresses vacuum energy density by ~10^-120, offering a mechanism for Cosmological Constant Problem. Modified Einstein equations include scale-dependent Λ(E). Black-hole singularity regularization is proposed via f(r,E) = 1 - 2GM/(c^2 r)[1+(E/E_P)^2]^-γ, leading to effective Kretschmann scalar K_eff → 0 at high curvature. Photon sphere correction r_ph = 3GM/c^2[1+(E/E_P)^2]^-γ is Planck-suppressed and not testable with current EHT. Section XII discusses explicit limitations: status of complex time, circularity in Schrödinger reinterpretation, fractional metric status, phenomenological nature of RG flow and vacuum suppression, and need for covariant derivation. This work was conducted as self-directed independent research by a high school student without institutional affiliation. All derivations developed through independent study. Database includes: main paper, supplementary derivations, numerical verification code.