modal_distance
plain-language theorem explainer
The modal distance definition equips pairs of configurations with a real-valued measure given by the J-transition cost on their value and position fields. Researchers constructing the geometry of possibility spaces cite it when defining the metric on modal manifolds. The definition is a direct one-line delegation to the imported J_transition function.
Claim. The modal distance between configurations $c_1$ and $c_2$ is $d(c_1,c_2)=J_*(c_1.v,c_2.v,c_1.p,c_2.p)$, where $J_*$ denotes the J-cost evaluated at the transition state between the two configurations.
background
Configurations are records carrying a real value and a position index; they label points in the possibility space. The J-transition function computes the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently cosh(log x)-1) between a pair of such points, drawing on the Recognition Composition Law. The ModalGeometry module imports this function together with the Config structure from the Possibility module to equip the space with a metric.
proof idea
One-line definition that applies J_transition directly to the value and position fields of the two input configurations.
why it matters
The definition supplies the metric used by ModalManifold, PossibilityBall, and the lemmas modal_distance_nonneg, modal_distance_self, modal_distance_symm, identity_in_ball, and constructive_at_zero. It realizes the possibility metric required by the modal geometry construction, linking to J-uniqueness (T5) and the forcing chain. Downstream results rely on it to establish non-negativity, symmetry, and ball membership.
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