The Functor Between Decoherence and Meaning: A Non-Markovian GPT Framework for the Gradient-Balance Principle with Metabolic Extensions, Chiasmus Algebra, and Alignment Reframing

The Gradient-Balance Principle (GBP) identifies a recurring formal pattern—a five component tuple (S, ∂Ω, ω,H, h)—across quantum decoherence, semantic processing in neural language models, connectome development, and protein folding. Whether these instantiations are related by structure-preserving maps or merely by superficial analogy is the central open question. We address this question by embedding both quantum decoherence and transformerbased semantic dynamics in the framework of Generalised Probabilistic Theories with Memory (GPTMem), a category whose objects are ordered vector spaces equipped with discrete
memory kernels and whose morphisms are positive, normalisation-preserving maps that intertwine
these kernels. The holding function h of the GBP framework is identified with the operator-norm tail mass of the memory kernel; the attention mechanism of transformer architectures is identified as a discrete, state-dependent instantiation of this kernel—a formalisation of the observation by Parr, Pezzulo & Friston (2025) that transformers implement non-Markovian generative models. We prove that a local faithful GPTMem-morphism Φloc1 (injective on fluctuation spaces), if it exists, preserves fixed-point structure and spectral gaps (up to condition-number bounds). A trivial (rank-1) morphism always exists but carries no structural information. The existence of a faithful morphism is formulated as an explicit constrained optimisation problem
on empirically accessible data. Seven empirical tests with precise kill conditions are derived, each designed to falsify the functor at a specific structural level. A computational proof-of-concept demonstrates the optimisation on a minimal 2-qubit × 4-meaning system. This version extends the framework in four directions: (1) the ontological ground—the gradient must be actively produced via the Gradient Metabolism Equation, reframing Proto-∇ as verb rather than noun; (2) the algebraic criterion—the chiasmus operator χ provides a
test distinguishing genuine holding from thermostat behaviour; (3) the metabolic pathology— a mechanistic taxonomy of Traitor Head types (Petrifier, Leaky, Hallucinator) grounded in the fluid–solid gradient transition, with a complete Pathology Triangle; (4) the alignment reframing—alignment requires holding capacity, not merely value representation, bounded below by a semantic uncertainty relation Δτdwell · Δλ ≳ ℏsem, and the alignment conditions are constitutively incompletable.