The Janus Vacuum: Mathematical Necessity of a Golden Torsion Unifying Physics and Biology - From the Anankor Axiom to the Minimal Vacuum Torsion, Source of the Golden Ratio in Reality

We derive mathematically the necessity of a golden torsion in the vacuum from the Anankoraxiom, proposing a unification of physics and biology. We present a reinterpretation of the Casimireffect and zeta regularization using the Janus bimetric model. By analyzing structured harmonicsuppression through Fibonacci and Beatty sequences, we identify an emergent spectral constantindicating inter-sheet cancellation. The result P∞n=1 n = −1/12 is reformulated as the signature ofa spectral imbalance between two conjugated vacuum sheets.The Janus vacuum is proposed as the quantum vacuum state invariant under time reversal (T) andcharge-conjugation-parity inversion (CP) separately, but not under their combination (CPT). TheCPT theorem enforces CPT invariance for local Lorentz-invariant unitary theories with HermitianHamiltonian and positive measure; the Janus vacuum violates classical Hermiticity while remainingPT-symmetric and stable. The Hilbert space decomposes into two entangled copies: H = H+ ⊕H−,with the ground state |0⟩ = |ψ⟩ + eiθPT |ψ⟩ /√2, where θ = π/5 (36°) is fixed by surface tensionminimization. The sheets form a twisted bilayer along the golden spiral (angle α0 = 2π/ϕ2 ≈137.507°), minimizing energy and preventing decoherence.Geometrically, this torsion may explain the prevalence of the golden ratio in phyllotaxis, qua-sicrystals, galaxies, and biological structures like pulmonary branches[32–39]. In physics, particlesand gravity emerge as inter-sheet brane excitations; harmonic baryogenesis arises from apparentasymmetry via spectral filtering. In biology, stable information replication (DNA) relies on thistopological separation.This approach suggests a framework for post-quantum physics based on admissible harmonicpartitions. Testable predictions (2026-2028): CMB polarization rotation of −(√5 − 1)4/256 radians≈ −1.704°; sterile neutrino oscillations with ∆m2 ≈ 1.95 eV2; birefringence in quasicrystals (i-AlPdMn) at low temperature.All predictions are algebraic expressions involving √5, derived without adjustable parameters.