The Holographic Standard Model: Geometric Particle Manifestation and Force Unification in the Complex-τ Plane

This paper extends the holographic geometric framework of Papers I–VI into the complex-τ plane, deriving a unified geometric description of the Standard Model. The real axis τ_R governs classical dissipative evolution; the imaginary axis τ_I governs quantum unitary phase rotation. The multiplicative conservation law C(τ)C(−τ) = C₀² acts as the single governing constraint on all physical states. Principal results: (1) A geometric manifestation window 1 < C(r)/C₀ ≤ 4 is derived without empirical nuclear input, bounding the stable material world from U-238 to S-32, with Fe-56 at the distinguished value C/C₀ = e. (2) A complete seven-zone classification of particle states is established, with zone boundaries at C/C₀ ∈ {1, 4, 7.28}; the upper boundary coincides exactly with the Schwinger critical field and the entanglement scale R* of Paper VI. (3) The four fundamental interactions are identified as geometric objects: gravity as the global conservation law, the strong force as real-axis topological nodes, electromagnetism as imaginary-axis unitary rotation, and the weak force as transient saddle-point bridges. (4) Parity violation is derived as a geometric theorem via the identification R̂(τ_I) = exp(2iτ_I γ⁵), with the weak interaction coupling exclusively to left-handed spinors as a consequence of the thermodynamic uniqueness of τ = 1 established in Paper I. (5) The Dirac mass term is shown to be the four-dimensional expression of C(τ)C(−τ) = C₀², providing a geometric reinterpretation of the Higgs mechanism. All results are zero-parameter.