From Markov Chains to Minkowski Space: Lorentz Invariance, Quantum Measurement, and Gravitational Analogs

We show that Möbius coupling on the eigenvalue spectra of doubly stochastic Markov matrices, f (λ, v) = (λ+v) / (1+λv) with period function g (λ) = (1−λ²) ^ (−1/2), reproduces the complete mathematical structure of special relativity, provides a new derivation of the Born rule, and yields 2D gravitational analogs. While the Möbius–Lorentz connection is classical (Klein, 1893; Penrose (ii) 4×4 doubly stochastic matrices have all-real eigenvalue fraction exactly 1/3 (the Foss trichotomy — unique to n=4, giving 3+1 dimensions) ; (iii) the only two group-forming couplings correspond bijectively to the two Poincaré Casimir invariants (mass and spin) ; (iv) the Born rule P = |ψ|² follows from the Perron-Frobenius theorem via Wick rotation, with Osterwalder-Schrader reflection positivity ensuring rigor; (v) SL (2, C) spinor structure with spin-statistics and CPT emerges from the double cover; and (vi) position-dependent coupling on agent lattices produces Ricci curvature scaling as r^ (−1. 91) (within 5% of the Schwarzschild r^ (−2), in a 2D proof of concept). Extensions yield: a Hawking temperature analog (divergent at the spectral gap horizon), Bekenstein-Hawking area-entropy scaling with logarithmic corrections, Bell inequality violations up to the Tsirelson bound (2√2) via unital CPTP maps, quadratic Lorentz-violation corrections consistent with Fermi-LAT bounds, a cosmological constant Λ ∝ 1/N³ requiring N ~ 10⁴¹ agents (resolving the hierarchy problem without fine-tuning), and S₄ gauge group structure matching one Standard Model generation. We present 28 theorems, 5 results, 3 conjectures, and over 50 numerically verified phenomena from a single algebraic identity. The mathematical results are independent of any interpretive framework. PACS numbers: 03. 30. +p, 02. 50. Ga, 03. 65. Ta, 04. 20. Cv