Z-M1-10 _ Mathematical Foundations of the Dual Substrate Framework

This document establishes the mathematical foundation of the dual-substrate framework, transforming the conceptual model of observability emergence into a rigorous formal system.

Within a CBD perspective, this work represents the formal micro-structural layer underlying collective dynamics, where interaction, accumulation, and structural constraints govern system evolution.

The framework defines the system through a triplet structure (G, Q, I), capturing structural configuration, informational substrate, and interaction operator, analogous to the CBD formulation P(t).

The configuration space introduces admissible states and continuity constraints, reflecting governed evolution under temporal and structural limits (Temporal Governability Law).

Structural invariants ensure system coherence, acting as constraints comparable to stability limits and saturation boundaries in CBD.

The interaction operator formalizes how latent and structural components couple, producing emergent regimes through non-linear dynamics.

A metric structure enables comparison of system states, providing a formal measure of divergence and structural distance.

The dynamical equation governs system evolution, allowing transitions between stable, oscillatory, and critical regimes, consistent with tipping behavior.

Observability is defined as a projection process emerging from stabilized interaction, aligning with Informational Tipping and Mimetic Saturation principles.

This document forms the mathematical entry point of the Z-Corpus, enabling subsequent developments in dynamics, stability, and cosmological embedding.