window_certificate_extends
plain-language theorem explainer
The theorem shows that defect remains non-increasing under arbitrary iterations of the 8-step evolution operator built from any one-step dynamics. Discrete Navier-Stokes researchers cite it to propagate single-window stability certificates across multiple 8-tick intervals. The proof is a one-line term application of the general iteration monotonicity lemma after reinterpreting the dynamics via its window form.
Claim. Let $dyn$ be a one-step dynamics on a space $α$ whose defect functional satisfies $defect(step(s)) ≤ defect(s)$ for every state $s$. Then for every natural number $n$ and state $s$, $defect(((step_8(dyn))^{[n]}(s)) ≤ defect(s)$, where $step_8$ denotes the eight-fold composition of the single step.
background
The Eight-Tick Discrete-Time Dynamics module treats time as discrete with the fundamental quantum tick equal to 1 in RS-native units. An 8-tick window is the basic stability unit because it matches the octave period forced by the Recognition chain. OneStepDynamics is the abstract structure consisting of a step map on states together with a defect map that is required to be non-increasing at each single step. The defect functional is taken from the Law of Existence and equals the J-cost, which vanishes at unity. The upstream lemma iterate_defect_nonincreasing states that any defect-nonincreasing one-step map preserves the inequality under arbitrary iteration.
proof idea
The proof is a term-mode one-liner. It applies the general iteration theorem iterate_defect_nonincreasing to the windowDynamics constructed from the input dyn, then uses simpa to unfold the windowDynamics definition and recover the stated inequality for the iterated step8 operator.
why it matters
This declaration supplies the iteration step that lets a certificate over one 8-tick window extend to every later window, directly feeding the parent theorem eight_tick_certificate_propagates. It operationalizes the eight-tick octave (T7) of the forcing chain by showing defect monotonicity survives repeated application of the window operator. In the Recognition Science program the result supplies the temporal propagation mechanism required for the Navier-Stokes lattice construction in three dimensions.
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