# Section 1: The problem and the contribution

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## 1 The problem and the contribution

 An account of a measurement should explain not only the frequency of its final display but also what that display records. A pointer may have a suitable one-time distribution while subsequently changing sides, or while failing to copy the system's earlier position. We therefore specify an earlier configuration event and a whole holding interval. The statistical conclusion concerns that joint observable.

The two constructions below separate three mathematical requirements: calibration of the earlier event, physical copying under the same dynamics, and retention at every time of a finite interval. Their error estimates are uniform over stated input and initial-law classes. They are not claims of universal relaxation from an arbitrary configuration law.

Quantum equilibrium, subsystem statistics and Bohmian measurement theory have a substantial established literature [[2](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-DGZ), [3](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-Operators)]. The method of arbitrary functions has also been applied to quantum probabilities: Bonds and collaborators study a random perturbation of a double-well model in long-time and small-$\hbar$ limits [[1](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-Bonds)]. Our finite apparatus statements instead concern restricted configuration ensembles and explicit recorded histories. Inverse engineering of harmonic transport is established [[12](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-Tor)]; it is used here with an autonomous quantum clock and a separate historical-current estimate. Coarse-grained relaxation studies [[13](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-ValentiniWestman)] address a different statistical question. We make no claim that any of these general ideas originates in this work.

The contribution being consolidated is the particular assumption-to-record chain, including its finite parameters and its current-sensitive error accounting. It consolidates the author's earlier research; the numerical constructions are not presented as new discoveries in this revision. Its relationship to the author's equilibrium and hybrid pilot-medium papers is comparative: those papers use different initial laws or dynamics and do not supply missing premises here [[10](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-RodgersMonograph), [8](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-RodgersMassive), [7](/quantum-measurement/research/nonequilibrium-records/bibliography#bib-RodgersPilot)].



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### 1.1 What is and is not a physical claim

 Throughout, the wave obeys a specified Schrödinger equation and actual configurations obey its guidance law. These are the constitutive assumptions. The potentials and coherent entrance stock are supplied mathematically. “Autonomous” means the complete displayed Hamiltonian has no externally switched laboratory-time parameter. It does not mean that its coefficients, supports and stock have been manufactured with established interactions. The intrinsic-circle model has an explicitly finite recoil coordinate. The radial model is an exact two-body $s$-wave reduction, but its clock and pointer remain effective scalar coordinates. Adding a field or a material support changes the model and requires a new same-model history proof.