Section 16 4 October 2026

Finite complete theorem

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

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16 Finite complete theorem

Let $X=x-R$ . Define $t_a=1.22$ , $t_b=1.28$ , $T_H=3$ and

I_0=[-8,8],\ I_1=[16,32],\qquad \mathcal R_0=[-10,10],\ \mathcal R_1=[14,34].

The record symbol is $0$ or $1$ in the corresponding outer region and $\perp$ elsewhere. The relative rotor classifier is halfway between the nominal sector centers. No observation is performed at $t_a$ ; it identifies the earlier actual outcome which the pointer records.

Theorem 16.1 (Complete direct ordinary record)

Assume the stated conservative current flow on the ordinary model, the exact conserved internal sectors, the specified initial product waves and actual-law family, and the prescribed-profile realization of the periodic-model section. Uniformly over every normalized qubit/reference input, every admitted actual source and clock conditional law, and every $t\in[1.28,3]$ ,

\boxed{ \TV\!\left(\mathcal L(\mathfrak r(Z_t)), \operatorname{Bernoulli}_{\{0,1,\perp\}}(p_{\rm tar})\right) \le\varepsilon_{\mathrm{per}}=.00443484008607784720002304<.01.}

Moreover,

\boxed{ P\{Z_t\in\mathcal R_{\ell(X_{1.22})}\ \forall t\in[1.28,3]\} \ge1-.00232725479707784720002304.}

Proof

Exact relative-coordinate reduction and spectator independence leave the active initial law $f_0(X,Z,S)$ . Its density is bounded by16 times the initial active wave reference. The isolated rotor's periodic-flow event is calibrated by $1/(2N)+D^2$ . The current-sensitive quiet-prefix calculation changes its actual label by at most $8Nr+16I_{\rm pre}/r$ . The source-free sector factorization is used explicitly; it is not inferred from a small source error.

The soft-classifier proof copies the actual coupled earlier label with error $C_{16}$ , and the pointer-band current gives the whole holding error $H_{16}$ . Add the same-support actual-law neighbourhood and target-projector error once. The proof compares the ordinary record to its own actual earlier label and then to the Born target. It does not need a small difference from squared-controller's individual trajectories.

For the simultaneous history claim, only copying, holding and the actual-law neighbourhood are required. All bounds concern one Hamiltonian and one parameter family. Source and preparation-clock conditionals integrate exactly because the event is independent of their initial values; they are not averaged under an assumed physical Born law.

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Contribution	Certified bound
Cell averaging	$1/512=.001953125$
Wrong-side rotor probability	$D^2=.000052316289$
Clock-prefix label change	$.000002144$
Actual historical copy $C_{16}$	$.00222725479707774720002304$
Whole-interval holding $H_{16}$	$10^{-16}$
Actual-law neighbourhood	$.0001$
Target-projector calibration	$.0001$
Complete sum	$.00443484008607784720002304$

Here $D=.007233$ , $Q=.003983$ , $e_A=2\times10^{-13}$ , $r=10^{-9}$ , and $I_{\rm pre}\le6\times10^{-18}$ . The exact record fraction is

\frac{1732359408624159062509}{390625000000000000000000}.

The unused direct probability margin is $.00556515991392215279997696$ . It could be spent on a separately proved microscopic field-realization history error; it is not such an error certificate.

Corollary 16.2 (Joint periodic-record law)

With the same assumptions and parameters, the law of the actual earlier label and its entire symbolic holding path has distance at most $0.00443484008607784720002304$ from the ideal constant path with the specified calibrated Bernoulli label.

Proof

For a base law, calibration and prefix transfer cost $1/512+D^2+0.000002144$ . Copying and holding cost $C_{16}+10^{-16}$ . Apply Lemma 2.1 with the isolated label and these actual-history events. The actual-law neighbourhood costs $10^{-4}$ once on the full path law because the same flow is used. The target-projector allowance costs another $10^{-4}$ . Their sum is the displayed bound. This strengthens the presentation to a joint observable using the already proved common-event argument, without inferring path stability from norm error alone.

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16.1 A genuinely nonequilibrium admissible stock

On the readiness box take $f_0=(1+0.8X)\one_{[-1,1]^2\times[0,1/4]}$ . Its mass is one, maximum is $1.8$ , directional variations are $(1.8,1,8)$ and rotor half-probability is $0.7$ . Its clock is confined to one half of a symmetric wave packet, so the initial full-law TV distance from the wave measure is at least $1/2$ . An admitted source conditional is $(1+\tfrac12\operatorname{sgn}u_1)|\zeta|^2$ . These are examples, not new assumptions imposed on the whole class. Invertibility of the full flow preserves fine-grained TV. Calibration of a coarse record therefore does not prove that the full ensemble equilibrates or that previously retained information is destroyed.