Formal Proof Chain of the σ-Calculus

We present the complete formal derivation of the σ-Calculus: from the initial directedtension postulate through the emergence of Einstein–Hilbert gravity, unitarity proofs, sin-gularity theorems, and a consistent cosmological Friedmann system with stable equation-of-state and suppressed fluctuations. All intermediate theorems and lemmas are outlined forarXiv publication. Updates Derivation of the RG coefficientϑadded — including both flow-generator and observable-matching routes. This fills a previously open gap by linking the effective σ-sector dynamics to cosmological data without fitting. To Fix: Axiom NumberingNote: If any proof component appears overly compact, a detailed or stepwise-expanded version (“exhaustive proof”) can be provided on request. Please indicate the specific lemma, axiom, or derivation line of interest.