Theory of Epistemic Abductive Geometry(TEAG): A Unified Theory of Admissibility-Driven Inference Across Dynamical Systems, Measure Theory, and Language

Abstract

We introduce the Theory of Epistemic Abductive Geometry (TEAG), a framework for non-Bayesian inference grounded in admissible-support contraction under possibility theory. The central object is the TEAG quintuple \( \mathcal{E} = (H, \pi, \{H_\alpha\}_{\alpha\in(0,1]}, C, A) \), where evidence acts by contracting the geometry of admissible hypotheses rather than redistributing probabilistic belief mass. The falsification boundary is a tropical variety — exactly. Under the log-admissibility transformation \( \Phi(h) = -\log\pi(h) \), the canonical TEAG conjunctive update becomes tropical addition in the max-plus semiring: \( \Phi^+(h) = \Phi^-(h) \oplus \psi(h) = \max\!\bigl(\Phi^-(h),\,\psi(h)\bigr), \) where \( \psi(h) = -\log\kappa(y\mid h) \) is the surprisal of hypothesis h under observation y. The falsification boundary is the tropical variety of this polynomial: \( \mathcal{F} = \bigl\{h \in H : \Phi^-(h) = \psi(h)\bigr\}. \) This is the exact locus dividing surviving from falsified hypotheses: h is falsified if and only if \( \psi(h) > \Phi^-(h) \); it survives if and only if \( \Phi^-(h) \geq \psi(h) \). Within the class of possibility-theoretic recursive inference systems, this is, to the best of our knowledge, the first exact algebraic expression of Popper's falsification criterion: the boundary is the zero set of a tropical polynomial, determined entirely by the geometry of the prior impossibility and current surprisal fields. Main results. 1. Epistemic Contraction Theorem. Contraction is tropical addition: \( \Phi^+ = \Phi^- \oplus \psi \). Posterior α-cuts satisfy \( H_\alpha^+ = H_\alpha^- \cap E_\alpha(y) \): geometric intersection, not belief redistribution. The falsification boundary is the tropical variety \( \mathcal{F} \). 2. Possibilistic Cramér–Rao Bound (PCRB} For any filter in the class \( \mathcal{F} \) of epistemically admissible, contraction-based recursive estimators satisfying Axioms 2.1–2.5: \( \mathcal{E}_{\pi,k|k} \geq \mathcal{E}_{\pi,k|k-1} + \tfrac{n}{2}\log(1-I_k) \), where \( I_k \) is the Choquet integral of per-hypothesis surprisal against the prior possibility capacity. Within this class, the ESPF [28] is the unique filter achieving this bound with equality, and is therefore the unique minimax-entropy-optimal set-based recursive estimator under bounded epistemic uncertainty. 3. Tropical Hamilton–Jacobi structure (summary). The TEAG update is structurally consistent with a tropical Lagrangian \( L = T - V \), Legendre transform to a tropical Hamiltonian equal to the surprisal field, and a Hamilton–Jacobi equation whose solution is the tropical addition rule. The Euler–Lagrange equations on the epistemic manifold yield geodesic motion with explicit Levi–Civita connection and Christoffel symbols. This structure is interpretive and consistent with the axioms; full derivations are in the companion paper [31]. Taken together, this structure admits a precise interpretation: the TEAG update rule is a max-plus dynamical system whose governing equations have the same algebraic form as the Hamilton–Jacobi equations of classical mechanics, instantiated on hypothesis space rather than physical space. 4. Gaussian collapse. Probability theory is the collapse limit of TEAG as epistemic width \( W \to 0 \): Choquet converges to Lebesgue, the ESPF recovers the Kalman filter, and \( \mathcal{E}_\pi \to \tfrac{1}{2}\log\det\Sigma + \mathrm{const}(n) \). Probability is earned by evidence, not assumed. Epistemic neutrality and knowledge-system synthesis. Because TEAG's axioms require only a hypothesis space, a possibility field, and a contraction operator — not a probability measure, a likelihood function, or a frequentist grounding — heterogeneous knowledge systems can each instantiate the TEAG quintuple independently. Their joint admissible support intersection is the locus of coherence: the set of hypotheses neither system has falsified. No transformation of one system into the other's representational primitives is required. The composition theory (Section 6) formalizes the coupling architecture. Four instantiations provide the unifying structure: the ESPF [28] for recursive state estimation; the Geometry of Knowing [29] for measure-theoretic collapse; the minimax-entropy optimality proof [30]; and the Possibilistic Language Model (PLM, forthcoming [32]).