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

Any inference system satisfying the TEAG axioms must obey a tropical Hamilton–Jacobi equation in the max-plus semiring, and the Epistemic Support-Point Filter (ESPF) is that solution. This paper proves the necessity and derives the complete dynamical and causal geometry that follows. The necessity has three steps, each forced. Popperian contraction requires that evidence can only increase impossibility, never decrease it: under the log-admissibility transformation this forces the max-plus operation. The evidence-referencing axiom requires that survivor selection depends only on innovation geometry, not on prior structure: this forces the Hamiltonian to be momentum-independent. A momentum-independent tropical Hamiltonian forces the Lax–Oleinik reduction to a pointwise max. The result is unique: no alternative update structure consistent with these axioms exists. The ESPF is not one admissible filter among many — it is the unique admissible inference dynamics under TEAG. Two scalar fields live on hypothesis space. The impossibility field \( Phi_\varnothing = -\log\pi \) encodes accumulated epistemic history: zero where a hypothesis enjoys full prior support, growing without bound as evidence withdraws that support. The surprisal field \( \Phi_S(h) = \tfrac{1}{2}\|L_e^{-1}(y - g(h))\|^2 \) encodes the tension between each hypothesis and the current observation in MVEE-whitened measurement space. The conjunctive (Popperian) update produces the posterior impossibility field as their pointwise max-plus upper envelope: \( \widetilde{\Phi}_\varnothing = \Phi_\varnothing \oplus \Phi_S = \max(\Phi_\varnothing, \Phi_S). \) This equality follows from \( -\log\min(a,b) = \max(-\log a, -\log b): \) it is an algebraic identity, not a modeling choice or an analogy. The active deformation front — the tropical variety of this two-term polynomial, where both fields achieve the maximum simultaneously — is the exact locus where evidence begins to deform the posterior impossibility field. It is a necessary condition for falsification. Sufficient falsification requires exit from the PRCB-admissible basin, whose threshold is determined by the PRCB at each step. A scalar example derives every quantity in closed form, making the front geometry and basin structure visible without probabilistic machinery. The ESPF predict–update recursion is the Lax–Oleinik operator of max-plus optimal control: the one-step solution operator of the tropical Hamilton–Jacobi equation, with the surprisal field as Hamiltonian and the Possibilistic Cramer–Rao Bound (PCRB) as the minimum action per update cycle. This structure is not chosen: Proposition 5.2 proves it is forced by the TEAG axioms — specifically, Popperian contraction forces the max-plus operation, and the evidence-referencing condition forces momentum independence of the Hamiltonian. No alternative update structure consistent with these axioms exists. Falsification is wavefront propagation: the surprisal field radiates outward from each observation, and a hypothesis enters the active deformation front at the moment the surprisal wavefront overtakes the prior impossibility field. The active deformation front is the epistemic Lagrange point — the locus of exact balance between prior epistemic history and current evidence tension — and is a necessary condition for falsification. Sufficient falsification occurs when the wavefront has pushed a hypothesis outside the PRCB-admissible basin: the isotropic equipotential region whose threshold is governed by the PRCB at each step. The term \emph{wavefront} denotes level-set evolution under max-plus dynamics; no physical medium is assumed. The gravitational language is structural: it reflects equivalence of governing equations, not shared physical ontology. The PCRB emerges as a minimum action principle: no measurement can compress the surviving well below the PCRB floor per update. The zero-temperature limit of the classical Hamilton–Jacobi equation — passing from log-sum-exp (probabilistic) to max (possibilistic) 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 in the MVEE-whitened metric. The correct geometric primitive of epistemic phase space is identified as a contact manifold rather than a symplectic one: the irreversibility of Popperian falsification forces a contact structure, the PCRB is a contact energy floor, and the ESPF implements a contact Hamiltonian system with discrete projection onto the admissible basin. These results constitute the dynamical foundation of the Theory of Epistemic Abductive Geometry (TEAG) (Jah, 2026b).