IndisputableMonolith.Causality.Reach
The Causality.Reach module defines reachability predicates and ball inclusions for discrete causal structures. Researchers deriving cone bounds in Recognition Science cite it when removing explicit step counts from light-cone statements. The module supplies definitions such as ReachN and inBall together with monotonicity lemmas that support the export theorem.
claimReachability relation $R(p, q, n)$ for points $p, q$ after $n$ steps and ball predicate $B(r, p)$ for membership within causal radius $r$, satisfying $R(p, q, n) implies B(r, p)$ and monotonicity under radius increase.
background
This module operates in the causality domain, introducing reachability concepts built on the underlying kinematics of the space. Key definitions include reachability in a fixed number of steps, membership in causal balls, and the existential closure over step counts. These support the discrete light-cone structure where propagation respects the eight-tick octave bound.
proof idea
This is a definition module containing no complex proofs. It consists of direct definitions for reachability predicates followed by straightforward lemmas establishing inclusions and monotonicity properties.
why it matters in Recognition Science
The module supplies the causality primitives required by the Cone Bound Export Theorem. That theorem exports the discrete light-cone bound without the step count parameter, completing the verification-level statement of the causal structure. It connects to the T7 eight-tick octave in the forcing chain.
scope and limits
- Does not establish the full cone bound theorem itself.
- Does not address continuous spacetime limits.
- Does not incorporate J-cost or phi-ladder mass formulas.
- Does not depend on any project-specific upstream theorems beyond Mathlib.