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 Appendix B. The Block–Laplacian Theoremin the σ–Calculus (new section added to the Appendix VI) To fix: Numbering (LaTeX)To fix: Remove 14.1 and keep 14.3 (two versions of the same topic).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.