Quadratic Persistence and Emergent Geometry: A Coherence–Selection Account of Inner-Product Structure (IM-C004, v0.3)

This document is part of the Informational Mechanics (IM) framework and is published in the IM-C (Foundations / Axioms / Ontology) series within the IM v0.3 corpus. The paper develops two structural constructs within the admissibility framework of Informational Mechanics: a derived regime principle (Quadratic Persistence Principle, QPP-L; “L” denotes linear coherence regimes) and a non-axiomatic organizing hypothesis (Emergent Geometry Hypothesis, EGH). Neither construct introduces new axioms. QPP-L formalizes conditions under which quadratic persistence arises as a structural consequence of admissibility constraints within linear coherence regimes admitting orthogonal decomposition and representation-invariant comparability. EGH proposes that inner-product geometry can be understood as a selection-stable representation structure in regimes where quadratic persistence supports stable comparability under noise-limited coherence selection. The analysis does not introduce new physical dynamics, empirical fits, or ontological postulates. Established physical theories (e.g., General Relativity, Quantum Mechanics, Quantum Field Theory, and ΛCDM) are treated as effective descriptions within their validated regimes. The contribution of this work is foundational and classificatory: it clarifies the structural conditions under which geometric language becomes scientifically meaningful, and it defines explicit stress-test pathways under which QPP-L or EGH would require refinement in the presence of demonstrated regime counterexamples.