peak_activation
Peak activation at hop distance d equals η to the power d, giving the geometric decay of signal strength upon first arrival in directed graph propagation. Analysts of SpMV causality on chains cite it when deriving strict ordering and reach bounds inside the 8-tick octave. The definition is introduced as a direct power expression that supports immediate unfolding in dependent inequalities.
claimThe peak activation at hop distance $d$ is given by $η^d$, where $η$ is the blend rate satisfying $0 < η < 1$. This value is the maximum signal strength a node at distance $d$ ever reaches, occurring at the tick of first arrival.
background
The module examines whether SpMV propagation preserves causal ordering on directed chains versus shortcut graphs that flatten it. Blend rate η is fixed at $1/φ^2$ in Recognition units, consistent with the eight-tick octave for multi-hop reach. The definition supplies the peak value $η^d$ at arrival time and serves as the base quantity for activation comparisons.
proof idea
Direct definition as the power $η^d$. No lemmas or tactics appear in the body; the expression is unfolded verbatim in downstream statements such as peak_decreases and practical_reach_bound.
why it matters in Recognition Science
This definition supplies the geometric term required by diagnostic_q1_q2 to confirm strict ordering by distance on directed chains and by practical_reach_bound to obtain the hop limit log(ε)/log(η). It supports the module conclusion that the 8-tick octave covers chains up to roughly 5 hops, aligning with T7 in the Recognition framework while leaving shortcut effects for separate treatment.
scope and limits
- Does not assign a numerical value to the blend rate η.
- Does not apply to graphs containing bidirectional edges or shortcuts.
- Does not incorporate a detection threshold ε.
- Does not compute activation times or establish ordering itself.
formal statement (Lean)
128def peak_activation (η : ℝ) (d : ℕ) : ℝ := η ^ d
proof body
Definition body.
129
130/-- Peak activation decreases geometrically with distance. -/