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

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}_{alphain(0,1]}, C, A) ), where evidence acts by contracting the geometry of admissible hypotheses rather than redistributing probabilistic belief mass. Falsification has two-stage structure. Under the log-admissibility transformation ( Phi(h) = -logpi(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) = -logkappa(ymid h) ) is the surprisal of hypothesis h under observation y. The tropical variety of this polynomial, ( mathcal{B}_{mathrm{active}} = bigl{h in H : Phi^-(h) = psi(h)bigr}. ) is the active deformation front: the exact locus where incoming evidence first matches prior impossibility and begins to deform the posterior field. This is a necessary condition for falsification but not sufficient. Sufficient falsification requires exit from the PCRB admissible basin ( mathcal{A}_k = {h : Phi^+_k(h) leq c_k^star} ), where ( c_k^star ) is the equipotential threshold determined by the PCRB at step k. Popper’s criterion thus receives a two-stage algebraic formulation: the tropical variety marks where falsification becomes possible; the PCRB basin boundary marks where falsification is complete. Within the class of possibility-theoretic recursive inference systems, this is, to the best of our knowledge, the first exact formulation of this distinction. 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 active deformation front is the tropical variety ( mathcal{B}_{mathrm{active}} ); the falsification boundary is the PCRB admissible basin boundary ( mathcal{B}_{mathrm{adm}} ). 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}logdetSigma + 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]).