Formal Proof Chain of the σ-Calculus

We provide a full formal derivation of the σ-Calculus: starting from the directed-tension postulate we prove the emergence of Einstein–Hilbert gravity, global unitarity, singularity theorems, a stable Friedmann cosmology and, via a Lyapunov C-function, the universal RG attraction to the Planck fixed point (G,ℏ,c), including mesh-independent convergence and gauge-field extensions. Updates Appendix B. Adds the Lyapunov method, which is useful for obtaining stability certificates (aerospace standards and ISO 26262 spectral-proof compliance). - Expanded. Added: A scale-dependent renormalization group (RG) flow package (sigma-block_cosmology_papersv3.zip) with an analytically derived constant C in the σ–block framework. (Source link https://github.com/Sigma-Calculus/block-scaling-rg-coefficient/tree/main) Added: README.txt describing latest changes. Notably, the Lyapunov C-function used in the RG analysis is defined independently of any energy or action functional, relying solely on the combinatorial and spectral properties of the discrete curvature field. 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.