Section 11 4 October 2026

Reduced active law and periodic averaging

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131081

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11 Reduced active law and periodic averaging

Use the finite Hamiltonian and initial data of the periodic-model section. The recoil and auxiliary calculation above proves the exact relative dynamics and spectator reduction. The active initial reference density is

q_0(X,Z,S)=\tfrac12|\gamma(Z-z_0)|^2|\varphi(S)|^2.

On the readiness box, $q_0>1/8$ . Indeed $\exp((1.001)^2)<11/4$ , $\sqrt\pi<9/5$ , $Z_c<4/5$ , and $2(11/4)(9/5)(4/5)<8$ . Thus every active actual law is dominated by $16q_0$ . This reference is only a proof measure.

Lemma 11.1 (Periodic event averaging)

Let $A$ be any measurable $h$ -periodic subset of $\T_2$ , with volume fraction $\theta$ . For a normalized actual marginal $f_X$ of zero-extension variation at most $V_X$ ,

|P_{f_X}(A)-\theta|\le\frac{hV_X}{8}=\frac{V_X}{4N}.

Proof

The mean-zero function $\one_A-\theta$ has an $h$ -periodic primitive of oscillation at most $h\theta(1-\theta)$ . Subtract its midrange value; its sup norm is at most $h\theta(1-\theta)/2\le h/8$ . Integration by parts against the circular BV derivative proves the estimate. Its variation is bounded by the given zero-extension variation, including boundary jumps. No smoothness of the event boundary and no equilibrium actual law is needed.

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