Low-Entropy Evolutionary Fingerprint Axiom: Invariants and Constants Layer

We introduce the Low-Entropy Evolutionary Fingerprint Axiom, which states that for any physically realizable system capable of sustained evolution under perturbations, there exists a non-empty set of steady-state evolutionary invariants (“fingerprint constants”) that is independent of implementation details and robust to scale choices. These invariants are not assumed as priors nor injected as design goals; they are the compressed residues left by feasible evolutionary trajectories under hard constraints and settlement. “Steady-state evolutionary invariants” denotes an audit-level residue of the trial–constraint–settlement chain, not a mechanistic account. We provide a bridge from axiom-level invariants to an Engineering Audit Layer centered on the axis Steady State — Structure — Constants — Geometricization. The core claim is structural: when constraints are reusable across scales and the system must preserve compositional closure under perturbations, the minimal-entropy representation that remains stable and auditable tends to become geometric/spectral (embeddings, metrics, eigenmodes). This module defines a constraint-first, implementation-agnostic specification: input fields (objectives, constraints, perturbations, resources, scale ladder, reality-state environment), output artifacts (geometricized representations, gates, failure criteria, recovery actions), and a minimal audit panel (compression/effective dimension, spectral concentration, cross-scale consistency, perturbation robustness, and settlement feasibility). The result is a survival-grade engineering layer that explains where geometric memory comes from in both real systems and deep models without forcing a single technical instantiation.