Rare-Edge Survival Recurrence
Abstract
Avoiding paths for the single-peak parity kernel satisfy an exact cubic survival recurrence.
Definition 1.1 (Avoiding-path survival mass).
Formalization. D5/S3/Estimation/TimeArrow/SinglePeakRareEdgeSurvival.survival (✓ std3).
Source. Repository-derived.
Commentary.
The survival mass at time T is the total uniform-start weight of explicit paths x_0, …, x_T that never traverse an edge from the opposite region Z to the peak region H. Each path weight is the product of the single-peak kernel along its T transitions.
Theorem 1.2 (Exact survival recurrence).
Proof. Machine-checked in Lean as D5/S3/Estimation/TimeArrow/SinglePeakRareEdgeSurvival.survival_recurrence (✓ std3). ∎
Source. Repository-derived.
Commentary.
Assume the sign function takes only the values 1 and -1, the peak has sign 1, the state space has 2M elements with M positive signs, M is at least 2, and q = r/(M-1). Put N = |X|, p = 1/(2N), and epsilon = 1-r, written as varepsilon in the display.
The mass starts at one. One forbidden edge is possible after one step and two placements are possible after two steps, giving s_1 = 1-p and s_2 = 1-2p. For every later horizon, the killed three-region transition operator obeys its cubic identity, yielding the displayed recurrence.
The bridge from state paths to the three-region recursion appends one last state to every path, separates the final transition, and sums its kernel weight over the peak, bulk, and opposite regions.