Applications of the σ-Calculus

We present a application of the σ-Calculus, a discrete-to-continuumframework encoding causal and metric structure in a single tension variable σij . From threecore axioms—primitive directed tension, Planck-scale normalization, and bounded trianglestability—we derive continuum geometry via a Regge-deficit construction reproducing theEinstein–Hilbert action. We prove unitarity of the Wick-rotated path integral, establishsingularity theorems at the σ-horizon, and formulate a self-consistent cosmology with anRG-driven exponent, stable equation-of-state, suppressed σ-fluctuations, and full Friedmannconsistency. Matter coupling via variation of the emergent metric yields the standard energy-momentum tensor, and discrete Euler–Lagrange variation recovers Newtonian, linearizedEinstein, and Schr¨odinger dynamics. Detailed proofs are provided in the appendix, ensuringa self-contained presentation suitable for arXiv submission.