The complete interaction and its record

Let $\mathrm{𝐸}$ denote the set of complete admissible interaction configurations for a fixed preparation and apparatus context. In MQ the Internal Frame carries conserved count structure and the System Frame carries the realized descriptions. The Frames mapping supplies the admissibility and representation structure. Let ${\mathrm{𝑀}}_{\mathrm{𝑐}}$ denote the completed contextual realization, assumed single-valued for each interaction in the set. [1]

Let ${\mathrm{𝐴}}_{\mathrm{𝑘}}$ denote the measurable region for record index k in the realized outcome space. The regions are pairwise disjoint and exhaustive. A record label is assigned by membership of the realized configuration in a region. Exhaustiveness gives a label for every completed configuration and disjointness gives at most one. Section 3.2.1 uses both conditions to obtain exactly one label. [1]

This conditional uniqueness identifies the role of the apparatus and its record map. The partition must distinguish the physical records relevant to the question and the realized configuration must be well defined for the complete interaction. The result leaves the detailed detector interaction to the physical realization model. Weights across alternatives alone cannot choose the map or supply the dynamics that produce a stable record. [1] [2]

How the count measure becomes a distribution of records

Let ${\mathrm{𝜇}}_{\mathrm{𝐸}}$ denote the count measure on the complete interaction set and ${\mathrm{𝑃}}_{\mathrm{𝑘}}\left(\mathrm{𝑡}\right)$ the probability of record index k at the fixed preparation and context, with any time argument suppressed. The total measure must be finite and positive. The numerator in Equation (1a) is the measure of the preimage of the record region. That preimage is the set of all completed interactions mapped into the region. The inverse-image notation allows multiple interactions to give the same record. [1]

Equation (1a) is the count-measure expression in the source’s Eq. (32). The pushforward gives each record the combined normalized weight of all interactions carrying that label. The source equates these record probabilities with normalized realized availability when that availability is induced by the same count measure and realization. Under repeated sampling with the appropriate identically prepared conditions, relative frequencies converge to the induced probabilities. An observed finite sample can fluctuate around them. [1]

A finite example of restriction and renormalization

For Table 1, assume twelve equally weighted complete configurations in one context and two exclusive records. Eight configurations lead to record A and four to record B. Also assume a compatible binary question about a retained feature, with two yes configurations in class A and three in class B. The table is a worked example of the measure calculation. A physical detector design would have to establish such a partition and the retained feature independently.

Before learning the record, the feature has probability five twelfths. After recording A, retain its eight configurations and normalize within that class. The joint prior weight of A and yes is two twelfths and the prior weight of A is eight twelfths. Dividing gives the updated probability in Equation (2b). After B, the corresponding probability is three quarters.

Before record · A 2/3 · B 1/3

After record A · A 1

Figure 1 shows the normalized class weights before and after the ideal update. Conditioning requires a class of positive prior weight. A zero-weight record would leave the division undefined and would call for checking the preparation, measurement or model. In a finite integer count example the weights are rational. Extending this to arbitrary ideal continuum probabilities requires the source’s additional completion assumptions. [1]

Physical records, repeatability and scope

The source’s Eqs. (33)–(34) restrict the predictive density to the region associated with the realized record and normalize there. For an ideal repeatable measurement, retained support lies entirely in the recorded class. A second measurement yields that class with certainty under the ideal assumption that the relevant support and record partition are preserved. A changed apparatus, intervening evolution or a disturbing instrument requires its own physical update model. [1] [2]

The explanatory sequence separates what happens in one run from what is predicted across runs. The complete contextual realization supplies the first. Its pushforward measure supplies the second. The restriction uses the known record to update later predictions. The source assigns definite records within these premises. Agreement with ordinary conditional probabilities gives a consistency check, while empirical discrimination among interpretations would require an independently testable difference. [1] [3]

September 5, 2026 is the recorded archive snapshot. The current manuscript provides the detailed construction cited here and may incorporate later qualifications. This Brief was prepared November 3, 2026. The worked counts are illustrative and the cited result is theoretical. [1]