Coupling Gating, Relational Stability, and Generative Condensation: A Structural Extension of Persistence Theory to the Multi-Entity Regime

Papers 1 and 2 established the structural requirements for single-entity persistence: non-null amplitude, closed coordination on a torus target space, and non-Markovian causal memory, together with a complexity hierarchy classifying memory kernels by sufficient-statistic dimension. The present work extends persistence theory to the multi-entity regime by asking: what structural constraints are necessitated when persistent entities admit direct internal-to-internal coupling? We show that this enlargement of the admissible operator set introduces four relational axioms: bidirectional coupling gating (R1), derived from Exclusion and Reflection protection; composite admissibility and factorization (R2), forced by closure of the persistence framework under its own coupling dynamics; continuous bandwidth control (R3), required for stable navigation of the coherence landscape; and bounded auxiliary evolution (R4), necessitated by the irreversibility of memory writes near sharp stability boundaries. From these axioms we construct a relational stability landscape with basin/threshold structure and derive a canonical coherence functional H = A + M - I - C, where alignment (A), representational compatibility (M), interference (I), and gate strain (C) are individually necessitated by the four persistence axioms. We give a conditional normal-form description of generative condensation as a coherence-threshold bifurcation producing an autonomous persistent excitation, and introduce a generative coupling taxonomy distinguishing transient coupling, relational imprinting, environmental projection, and full condensation by structural permanence. The framework yields four operational discriminants, four explicit falsification criteria, and organism-facing observables connected to existing computational infrastructure. Preliminary computational validation from a Rust-based multi-agent simulation implementing the full relational architecture demonstrates consistency with theoretical predictions, including statistically significant efficiency amplification under coupling (p < 0.001), coherence functional differentiation across agent pairs, zero interference under coupling (confirming Exclusion protection), and routine observation of generative outcomes below the condensation threshold. The framework is conservative (reduces to the unary regime when coupling vanishes), falsifiable, and testable. Reproduction, defined as coherence-driven relational condensation, emerges as a dynamical consequence of relational persistence structure rather than an external postulate. Fourth revision. Changes from v3: generative coupling taxonomy (Section 9.5), preliminary computational validation (Section 11.4), softened Limitation 1 reflecting explored implementation class, recursive structure noted in Conclusion.