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 Documets – Supplementary proofs are published as standalone documents and not stacked as versioned uploads. Documet date reset to initial publication date (as requested). A minor update has been made to the appendix. To Fix: Axiom NumberingLaTEX: ω_ij should be read as: σ_ij (replace any occurence of ω by σ) Note: 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.