Signal True Always True — Mathematical Kernel

This document presents the **Mathematical Kernel** of the *Signal True Always True* framework (GUT).The purpose of this kernel is to extract and formalize the **minimal, rigorous, and falsifiable core** of the theory, independently of its full ontological and physical extensions. It is designed as a **compression** of the complete GUT framework, preserving its essential axioms, operators, and conservation principles while remaining fully self-contained and mathematically well-defined.The kernel introduces:- A coherence-first relational substrate,- Conditional fractalization governed by explicit thresholds,- The FRAC expansion operator and associated projection structures,- A conserved information functional linking coherence and effective entropy,- A reproducible empirical strategy across cosmology, AI learning dynamics, and financial stress proxies.Empirical sections are intentionally limited to **comparative and falsifiable tests**, emphasizing structural competitiveness rather than parameter optimization. The kernel demonstrates that the coherence-driven FRAC projection is empirically comparable to standard models (e.g., ΛCDM) at first order, while remaining distinct and testable.This work does **not** attempt to reproduce the full ontological scope of the GUT (such as emergent spacetime structures or quantum–classical bridging). Those elements are deliberately omitted to ensure clarity, mathematical rigor, and reproducibility. They are treated in the full GUT framework, to which this kernel serves as a formal and empirical anchor.The Mathematical Kernel is intended to function as:- a standalone, arXiv-compatible reference,- a rigorous entry point for peer evaluation,- a stable citation target for future extensions of the GUT framework.The complete GUT theory, including ontological structure and extended invariants, is documented separately and should be cited alongside this kernel when discussing the full theoretical scope.