ONE AXIOM : Decision as Domain Change

ONE AXIOM : Decision as Domain Change — Freedom, Coherence Gradient, and the Limits of Choice Classical decision theory (von Neumann–Morgenstern 1947, Savage 1954) models decision as the selection of a utility-maximising element from a fixed outcome space $\Omega$, with the Axiom of Choice guaranteeing existence of a choice function $f: \Omega \to X$. This document proves that this framework embeds a structural category error. The formal object. A decision is defined as a morphism $\Lambda: D \to D'$ between formal domains equipped with an Ontological Coherence Field (OCF), subject to four jointly necessary conditions (Theorem DC): (DC1) $F(D) > 0$ (freedom functional, Document 000). (DC2) $\nabla\heartsuit \neq 0$ (coherence gradient, Documents 0A/0B). (DC3) $Det(D) = 0$ (no blocking constraint). (DC4) PSP-compatibility (the morphism is evaluable by the Possibility Selection Principle, Document 0B §7.1). The structural cost of any decision is $K_{\mathrm{coh}}(D'|D) \geq \delta_\alpha = 1/192$ — the minimum quantisation unit (Proposition FM: lower bound of KC is tight), defined as the global primitive step unit (Document 0M Capacity Invariant, Document 0A uniqueness of $G = S_4 \times \mathbb{Z}_2^3$). Decision competence as the capacity to shape the future. The joint DC conditions formalise what it means for an agent to be structurally capable of changing its own domain — that is, of making the future differ from the present in a coherent, non-arbitrary way. This is not preference satisfaction within a given frame; it is the structural prerequisite for the frame itself to change. Decision competence is therefore not a property of the outcome space but of the domain: it requires available freedom ($F > 0$), a direction of coherence increase ($\nabla\heartsuit \neq 0$), and the absence of deterministic lock-in ($Det = 0$). When all three hold, Theorem ALT guarantees that a cost-incurring morphism to a genuinely distinct domain $D'$ exists — one that expands or preserves global freedom $F(D') \geq F(D)$ despite the local cost $\Delta\heartsuit_{\mathrm{loc}} < 0$. Decision competence is therefore the structured capacity to bear irreversible cost in the presence of outcome uncertainty, in exchange for a domain in which the coherence gradient is positive again. Main results (34 proven, 1 structural): Theorem NDT (Non-Decision Theorem): $\exists f \text{ via AC} \nRightarrow \exists \Lambda$ compatible with D1 and PSP. Proven by formal vNM countermodel (utility saturation: $f^*$ maps $\Omega \to \{x_1\} \subseteq \Omega$, no new OCF($D'$) generated) and Banach–Tarski structural illustration (non-measurable pieces $\implies \sigma|_{P_i}$ undefined $\implies$ NRP $\implies$ D1 fails). The gap is categorical, not epistemological. Lemma NRP (Non-Recoverable Projection): Reversal morphisms drop the $\sigma$-coordinate required for PSP-evaluability; they are forgetful projections, not D1-decisions. Novel DT extension: state projection $c: S \to S'$ non-injective on decision-relevant features produces decision invalidity (EU operator undefined), not merely accuracy degradation — the first formal characterisation of state-representation constitutivity in MDP/DT theory. Theorem ALT (Andréia Lift): When DC1–DC3 hold, $\exists \Lambda: D \to D'$ with $\Delta\heartsuit_{\mathrm{loc}} < 0$ and $F(D') \geq F(D)$. Coherence Complexity $K_{\mathrm{coh}}(D'|D) := T_{\mathrm{min}}(D \to D') \cdot \delta_\alpha$ is the universal cost measure; it satisfies reflexivity, positivity, triangle inequality, and integer quantisation in units of $\delta_\alpha = 1/192$. Theorem CLF (5-Category Classification): Exhaustive partition of agent states by decision accessibility. Categories 1–4 are formally blocked from T6-transformations (Document 7B) by NDT or $Det = 1$; Category 5 satisfies all DC conditions. Corollary EXP-1 [DT] (novel): A complete Q-map $r: S \times A \to \mathbb{R}$ does not imply T6-access. This is a formal impossibility: reward-maximisation (Q-learning, UCB, posterior sampling) and domain-change are categorically distinct objectives. First formal result of this type in RL/DT theory. Falsifiable: Prediction [3M-P7]. Theorem GSC (Great Short-Circuit): Gödel incompleteness, Tarski undefinability, the Turing halting problem, and NDT are four instances of one structural isomorphism (domain-closure pattern). Axiom M escapes by construction: $M = \Phi(S,M)$ operates at the meta-level of all domains. Theorem CPS-D: A Complex Predictive System (CPS) is a proper D1-domain; inter-CPS contact requires a meta-language that is itself a D1-domain. Predictions: Nine predictions registered a priori with explicit kill conditions (P1–P9), covering: Andréia Signature (P1), Optionality Decoupling (P2), Sunk Cost Immunity in Category-5 agents (P3), CPS Contact Impossibility (P4), $K_{\mathrm{coh}}$ quantisation (P5), T6 $\sigma$-access requirement in AI benchmarks (P6), RL scaling gap for T6 (P7), GSC Closure (P8), Legacy Coherence Trap (P9). Residual novelty: Nine objects with no counterpart in 18 surveyed frameworks (L.A. Paul, Sen, Kreps–Porteus, Friston, McClennen, Frankfurt, AGM, Gigerenzer, Elster, Rawls, Von Weizsäcker, Gilboa–Schmeidler, Lewis/Gibbard–Harper, Kahneman–Tversky, and the classical vNM/Savage/MDP/CTD core): the decision morphism $\Lambda$, the joint DC conditions, $K_{\mathrm{coh}}$, the partial functor $F_{\mathrm{GSC}}$, the $\sigma$-coordinate (Lem. NRP), the five-category classification, Theorem NDT as a formal impossibility, state-representation constitutivity (Lem. NRP [DT]-track), and T6-access impossibility under pure EU/RL maximisation (Cor. EXP-1 [DT]-track).