IndisputableMonolith.Causality.ConeBound
The ConeBound module supplies cardinality bounds for reachable sets in discrete causal structures built from bounded steps. Physicists modeling propagation limits in the Recognition Science phi-ladder would cite these when deriving finite cone sizes from the J-function. The module assembles basic set-cardinality inequalities into inductive controls on ball growth.
claimFor the causal graph induced by BoundedStep, the module establishes $|ballFS(n)|$ bounds via inductive application of union and bind cardinality controls, ensuring finite reachable sets within $k$ steps.
background
The module sits inside the Causality section and imports only Mathlib for finite-set operations. It works with the discrete kinematics generated by the Recognition Composition Law, where steps are constrained by the J-cost function and the phi-ladder. Sibling definitions supply BoundedStep as the step relation, ballP as the predecessor ball, and ballFS as the forward sphere whose cardinality is controlled.
proof idea
This is a definition and lemma module, no single top-level proof. It chains basic Mathlib facts (card_singleton, card_union_le, card_bind_le_sum) into neighbor-sum lemmas (sum_card_neighbors_le, card_bind_neighbors_le) and closes with the inductive step card_ballFS_succ_le.
why it matters in Recognition Science
The bounds close the discrete causal-cone estimates required by the T7 eight-tick octave and the emergence of D=3 spatial dimensions in the forcing chain. They feed parent results on light-cone structure and propagation speed in the Recognition framework.
scope and limits
- Does not derive the underlying causal graph from the J-equation.
- Does not address continuous spacetime limits or the alpha band.
- Does not incorporate mass-ladder or Berry-threshold effects.
- Does not treat infinite or non-locally-finite structures.