Scaling of Permanent Blocking in Nonlinear Cascades with Selective Reset

A periodically driven cascade can stop accepting requests while the forcing continues. Westudy a feedforward chain of decaying stages with power-law bounded transfers. Requestsarrive with period T and are accepted while the readout stays below a threshold. Acceptancekeeps a fraction η of the first k stages and admits an input of amplitude A during a windowof width ε that ends δ before the next request. Refusal suspends the reset and admits theinput for the whole period. Starting from rest, requests are eventually refused forever exactlywhen A exceeds a critical amplitude A∗. Every request is accepted at A = A∗. We show thatA∗ is of exact order 1/(εW ), with constants independent of ε, δ and η. The weight W is themaximum of k + 1 contributions, one from each partially reset stage and one from the firstfully retained stage. The channel exponents are sums of reciprocals of cumulative productsof the transfer powers. The blocking exponent can depend on their order and can be set byan intermediate residual. For fixed parameters, the monomial envelopes attain the extremethresholds, with exact ratio QN −1j=0 (hj /ℓj )1/Pj+1 , where Pj+1 is the product of the first j + 1transfer powers. The flow results and the principal threshold and scaling results are verifiedin Lean 4 with Mathlib