PossibilityBall
The PossibilityBall definition equips modal geometry with closed balls of radius r around a configuration c under the modal distance. Modal theorists studying reachability and actualization paths cite it when defining neighborhoods in configuration space. It is realized directly as the set of all y satisfying modal_distance(c, y) ≤ r.
claimThe closed ball of radius $r$ centered at configuration $c$ is the set $B_r(c) = { y : d(c,y) ≤ r }$, where $d$ is the modal distance on the space of configurations.
background
The ModalGeometry module works on the space of configurations (Config) imported from the ILG gravity model. Modal distance is the metric on this space induced by the possibility structure; it inherits non-negativity, symmetry, and zero self-distance from the Identity axiom in LogicAsFunctionalEquation and the zero-cost identity event in ObserverForcing. The local setting is the modal layer that equips possibility spaces with geometry to track actualization.
proof idea
This is a direct set-comprehension definition. The ball is written as the collection of all configurations y such that modal_distance c y ≤ r. No lemmas or tactics are invoked; the body is the primitive notation for the closed ball in the modal metric.
why it matters in Recognition Science
PossibilityBall is the immediate prerequisite for the lemma identity_in_ball, which places the identity configuration inside every sufficiently large ball. It supplies the basic neighborhood structure for modal geometry and thereby supports connectedness and curvature statements in the same module. The construction sits inside the modal extension of the forcing chain (T0–T8) and the Recognition Composition Law.
scope and limits
- Does not supply an explicit formula or bound for modal_distance.
- Does not prove the triangle inequality or other metric axioms.
- Does not restrict the admissible values of r or the type of configurations.
- Does not link the ball to the phi-ladder or mass formulas.
formal statement (Lean)
78def PossibilityBall (c : Config) (r : ℝ) : Set Config :=
proof body
Definition body.
79 {y : Config | modal_distance c y ≤ r}
80
81/-- Identity is in every sufficiently large possibility ball.
82
83 **Note**: The radius r must be large enough to contain the path from c to identity.
84 For r > modal_distance(c, identity), the identity is guaranteed to be in the ball. -/