Algebraic Congruence Laws and Multi-Base Expansions of the Positive Overflow Spectrum

This paper explores the deep mathematical structure and physical mapping of the positive overflow phenomenon (algebraic solution c₂ > 1), based on the reflecting fixed point conditions in the Theory of Algebraic Spectra of Equilateral Polyhedra. Unlike the "single-point negative overflow" generated by the self-generating closed loop, positive overflow constitutes a continuous, tunable Lorentz factor Γ spectrum. Through complex analytic continuation (real part phase shift α = 0), the positive overflow solution can precisely correspond to the complete velocity range from weak to extreme relativity. Based on the 18,666 legal angle data less than 90° obtained from three iterations, this paper proposes the "Decimal-Golden Ratio Congruence Law": Γ = 10k + N + n/φᵐ This paper provides strict canonical decomposition rules and the dominant statistical analysis of m = 3 (three-dimensional resonance). The study further introduces half-integer powers, discovers the velocity ratio operator V = nφ⁻ᵐ, and reveals the "triple alignment" of the Golden Rhombus (63.43495°) in geometric self-fitting, algebraic fixed points, and the Lorentz intersection of positive and negative overflows. For high-order mantissas, this paper proves the limitation of single-base representations and establishes a multi-base composite expansion theory and complexity hierarchy based on Fibonacci bases: φ⁻ᵐ = Fₘφ - Fₘ₊₁. Combining these findings, the paper further reveals the "Threshold Shift Law of Algebraic Bases": any algebraic base V < 1 can excite discrete physical energy level ladders through integer shifts N, and undergoes "topological short-circuiting" at specific nodes (e.g., 4 + 4/φ³ = 8/φ).