The physical question and the proposed contribution

J. A. Geiger’s Measurement Quantization: A Count-Based Resolution of the Quantum Measurement Problem develops a shared-preparation account of entangled records. Its contribution is an explanatory chain from conserved composite count availability to contextual realized probabilities and an ideal analyzer response. Section 3.7 and Appendix G present the Bell construction; Appendix H treats measurement conditioning. [1] [2]

A conventional Bell experiment compares paired binary records obtained at separated stations. Each station has two possible analyzer settings. The CHSH statistic combines the four setting-pair correlations. Its violation excludes the relevant Bell-local factorizable model class under the corresponding measurement-independence and experimental assumptions. It does not, by itself, identify a unique microscopic interpretation. [3] [4]

A joint preparation before spatial encoding

In MQ, the Internal Frame is the discrete count-configuration domain. Its relational cells are not little positions in an already existing space. The System Frame is the realized relational encoding in which spatial separation, angle and detector records are expressed. The Frames mapping realizes the count structure as that observable description. [1] [2]

The paired preparation carries joint availability. It need not decompose into independent subsystem measures even after the measured subsystems are spatially separated. A many-to-one realization map can discard distinctions between underlying configurations, but that fact alone cannot determine the Bell correlations. The composite availability, contextual record partitions, normalization and angular response are also required. [1] [2]

The source retains preparation-setting independence. A setting-dependent partition used to assign records is distinct from changing the preparation distribution in response to a later analyzer choice. This distinction matters because contextuality by itself does not specify which premise of a Bell model has changed. [1]

Joint probabilities and the correlation

Write ${\mathrm{𝑃}}_{\mathrm{MQ}}\left(\mathrm{𝛼}\mathrm{,}\mathrm{𝛽}\mathrm{\mid }\mathrm{𝑎}\mathrm{,}\mathrm{𝑏}\right)$ for the dimensionless joint probability of the two binary outcomes, labeled by the Greek letters alpha and beta, each taking +1 or −1. Its final two arguments label the left and right analyzer settings. These outcome labels are local to the Bell calculation and do not denote a fine-structure coupling. The four possible outcome pairs are ++, +−, −+ and −−. Their normalized probabilities obey Equation (1a). [1]

Equation (1a) follows from the source’s normalized contextual joint availability. Normalization is required for a probability distribution; the physical identification of that distribution with record frequencies relies on the completed realization and record partition developed earlier in the manuscript. The labels here are Brief labels, not the manuscript’s equation numbers. [1] [5]

The dimensionless correlation $\mathrm{𝐸}\mathrm{(}\mathrm{𝑎}\mathrm{,}\mathrm{𝑏}\mathrm{)}$ is the mean product of the two binary records. Equal outcomes contribute +1 and opposite outcomes contribute −1. Grouping the source’s weighted sum in Appendix G.7 gives Equation (2b). [1]

A value of +1 means perfect agreement, zero means equal total probability for agreement and disagreement, and −1 means perfect disagreement. A non-product joint distribution is not by itself evidence of entanglement. A classical mixture of shared source states can also correlate records. Bell discrimination concerns whether one measurement-independent Bell-local factorized model can reproduce the complete setting family. [1] [3]

What fixes the analyzer-angle dependence

The worked benchmark selects an ideal lossless two-channel linear-polarization analyzer and a symmetric preparation. At equal analyzer axes, ++ and −− each carry one half of the total availability, while the opposite-outcome channels carry zero. These source weights specify the preparation class. They are not a claim that every entangled state has this particular correlation sign. [1] [2]

Appendix G.11 represents the two normalized channel coefficients as a real amplitude pair after removal of a common phase. Analyzer rotation acts linearly and preserves their squared norm. The map is continuous, is the identity at zero angle and composes under successive angle additions. Orientation-preserving two-dimensional rotations then give cosine and sine coefficients. [1]

The calibration between the representation’s rotation parameter and the physical analyzer angle is an essential premise. Norm preservation and continuity alone do not set that angular calibration. Squaring the resulting coefficients represents the probability identification already developed in the count-to-record construction. The source therefore does not insert a separate Malus probability law at this step, but it does retain the derived amplitude structure and the stated ideal analyzer specialization. [1]

For an analyzer-axis difference, each equal-outcome joint channel receives one half times its cosine squared. Each opposite-outcome channel receives one half times its sine squared. Summing the equal channels and subtracting the opposite channels gives the cosine of twice the difference. These are the source’s ideal linear-polarization formulas, not a new fit to detector data. [1] [2]

Figure 1 makes the two different statistical questions visible. The relationship between paired outcomes changes with the angle difference, although a detector’s own distribution remains balanced. At 22.5°, the equal-outcome total is 0.853553 and the opposite-outcome total is 0.146447, rounded to six decimals. Each individual joint channel is half its corresponding total. [1] [2]

An exact four-setting calculation

The dimensionless statistic ${\mathrm{𝑆}}_{\mathrm{CHSH}}$ adds the correlations for left-first/right-first, left-first/right-second and left-second/right-first, then subtracts left-second/right-second. Choose left settings 0° and 45°, and right settings 22.5° and −22.5°. Table 1 gives the exact probabilities from the source’s angular law. Degrees are used for readability; the equivalent source angles are zero, one quarter pi, one eighth pi and minus one eighth pi radians. [1]

Table 1 specifies ideal realized probabilities, not event counts. No finite sample is asserted to realize its irrational entries exactly. Every row sums to one. Either local outcome has probability one half because its two compatible joint channels sum to one half. The first three correlations are positive square root of two divided by two; the fourth is its negative. Subtracting that fourth correlation adds the final positive contribution. Equation (3c) is therefore the ideal optimal-setting value, not a bound attained at arbitrary angles. [1]

The rounded value is 2.828427. The measurement-independent Bell-local factorized class has a maximum magnitude of two, and the quantum CHSH maximum is two times the square root of two. The local condition constrains the full setting family; it is stronger than observing that local singles remain balanced. [1] [3]

No-signaling, conditioning and Bell factorization

Operational no-signaling requires the local marginal to be independent of the distant setting. In Table 1, adding ++ to +− always gives one half for the left + outcome, and adding ++ to −+ does the same for the right + outcome. The other two local marginals are also one half. This verifies no-signaling for the ideal family without assuming that its joint distribution factorizes. [1] [2]

At a 22.5° difference, conditioning on the right + record gives a left + probability of 0.853553, rounded, rather than the unconditioned one half. That conditional probability divides the ++ joint weight by the right + marginal. A reader can use it only after learning which remote record occurred. Sorting coincident records later is not a change in the detector’s unconditioned distribution and is not a controllable superluminal channel. [1] [2]

Figure 2 illustrates the source’s restriction-and-renormalization account of subsequent predictions. It does not specify a unique physical detector instrument or prove a primitive pointer-writing trajectory for arbitrary input preparations. An outcome probability law and a complete interaction producing each individual stable record are distinct technical requirements. [1]

Bell factorization conditions on an appropriate complete source variable and requires a product of local response probabilities, together with the preparation-setting independence used in the CHSH argument. Merely showing that the observed joint distribution is not the product of its observed marginals would not exclude such models, since averaging shared local causes can generate classical correlation. The four-setting CHSH violation supplies the relevant stronger test. [1] [3]

The MQ source does not provide a Bell-local hidden-variable completion of that kind. Its count adjacency locality is a statement about redistribution in its relational domain, not Bell local causality for spacetime-separated records. Operational no-signaling should also not be promoted to a claim about parameter independence conditioned on an additional primitive variable that the model has not specified. [1]

Nor does no-signaling alone force the quantum maximum. Abstract no-signaling probability families can reach a CHSH magnitude of four. The two-channel coherent amplitude structure and analyzer response do substantive work in obtaining the quantum-compatible value. Balanced marginals alone are an insufficient derivation of the result. [1]

Independent evidence and the remaining discrimination

Hensen and colleagues’ 2015 Nature experiment used electron spins separated by 1.3 km. Across 245 event-ready trials it reported a CHSH value of 2.42 ± 0.20. Fast setting selection, spatial separation and efficient readout addressed the locality and detection loopholes. This is independent experimental evidence against the tested Bell-local class under the experiment’s stated assumptions. [4]

That spin experiment is not an implementation of the MQ ideal photon-polarization example. Its finite observed statistic must not be equated with the lossless theoretical maximum or treated as a new MQ-specific confirmation. Recovering the same quantum-compatible correlations does not distinguish MQ from standard quantum theory. [1] [4]

Real polarization apparatus also requires measured source visibility, optical loss, detector efficiency, background counts and an explicit trial-selection rule. A fair comparison must model these effects without silently conditioning away unrecorded events. Appendix G.12 explicitly separates its ideal dichotomic benchmark from Bell inequalities and detector models that handle finite detection efficiency. [1]

A discriminating MQ test would require a further observable prediction whose distribution differs quantitatively from standard quantum theory, with specified apparatus response, calibration, uncertainty and selection rules. No such distinction is supplied merely by the ideal CHSH arithmetic reproduced here. The present calculation checks internal probability consistency and the recovered benchmark, not the count ontology experimentally. [1] [6]

Chronology and interview context

September 5, 2026 is the archive’s recorded source or audit snapshot. It is not established here as the first discovery date or original announcement date. This Brief was prepared October 28, 2026 using the current public manuscript. Current qualifications and analyzer-rotation wording should not be silently attributed to the earlier snapshot. Journal publication of this manuscript version is not established. [1]

For reporting, request the precise preparation class, completed record mapping, preparation-setting independence argument and physical angle calibration. Ask how the apparatus-specific response and individual detector records are obtained, which locality statement is retained and what additional prediction would separate the proposal from quantum correspondence. These questions locate the scientific contribution without turning an interpretation into a new experiment.