The Epistemic Support-Point Filter as a Tropical Hamilton–Jacobi System: Wavefront Propagation and Possibilistic Inference

We prove that the predict–update recursion of the Epistemic Support-Point Filter satisfies a tropical Hamilton–Jacobi equation in the max-plus semiring, and derive the complete dynamical geometry that follows from this identity.Two scalar fields live on hypothesis space. The impossibility field encodes accumulated epistemic history: it is zero where a hypothesis enjoys full prior support and grows without bound as evidence withdraws that support. The surprisal field encodes the tension between each hypothesis and the current observation in whitened innovation space. The conjunctive update produces the posterior impossibility field as the pointwise max-plus upper envelope of these two fields. This equality follows from a one-line algebraic identity relating the minimum of two possibilities to the maximum of their negative logarithms. It is not a modeling choice and not an analogy — it is an algebraic fact about the log-admissibility transformation.The falsification boundary — the tropical variety of the resulting two-term polynomial, where both fields achieve the maximum simultaneously — is the exact locus dividing surviving from falsified hypotheses. 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 determined entirely by the geometry of the prior impossibility and current surprisal fields. A scalar example derives every quantity in closed form, making the front geometry fully visible without probabilistic machinery.The tropical Hamilton–Jacobi structure is not introduced as a framework imposed on the filter. It is forced by the TEAG axioms: Popperian contraction forces the max-plus operation in log-admissibility coordinates, and the evidence-referencing condition forces momentum independence of the Hamiltonian. Given these axioms, no alternative update structure exists. The ESPF predict–update recursion is the Lax–Oleinik operator of max-plus optimal control — the unique one-step solution operator of the tropical Hamilton–Jacobi equation for this class of Hamiltonians.The Possibilistic Cramér–Rao Bound emerges as a minimum action principle: no measurement can compress the surviving support below the PCRB floor per update. The zero-temperature limit of the classical Hamilton–Jacobi equation — passing from probabilistic log-sum-exp aggregation to possibilistic max aggregation — recovers this framework exactly, making precise the passage from Bayesian to possibilistic inference as a thermodynamic degeneration. The whitened minimax medioid is proved to be the geodesic attractor of the surviving well: the unique support point nearest the center of the PCRB-defined epistemic geoid under the whitened innovation metric. The term wavefront denotes level-set evolution under max-plus dynamics; no physical medium is assumed. The gravitational language used throughout reflects equivalence of governing equations, not shared physical ontology.These results constitute the dynamical foundation of the Theory of Epistemic Abductive Geometry [4].