We examine a Measurement Quantization (MQ) representation of quantum position and momentum information. Discrete configurations belong to the Internal Frame, while observable quantities are realized in the System Frame through the Frames mapping. We connect distinct angular and momentum realizations, compare calculated angles with a selected theoretical X-ray model branch, and recover reference-scale consistency relations. For the uncertainty assessment, all interaction-dependent quantities use the electromagnetic demarcation, leaving a finite count factor in the joint spread constraint. These results do not supply a measurement dynamics or establish fundamental nondeterminism.

Keywords: Determinism, Quantum Behavior, Discrete, Measure, Quantum Information, Uncertainty Principle

Using MQ [1–7], we examine lower and upper count bounds and their role in available measurement information. The fundamental reference length l_f, mass m_f, and time t_f have units m, kg, and s. The Internal Frame is the discrete configuration domain whose relational structure is encoded by integer counts relative to these references. The System Frame is the non-discrete encoding in which observable quantities are continuous relational separations. The Frames mapping is the kinematic transformation between these domains; it preserves realization consistency and imposes closure conditions on relational configurations. These domains are not additional inertial or gravitational observer frames (for the historical framework discussion, see [7], Appx. K).

The fundamental references and count relations are discussed in [7], Appx. I and J. Here deterministic behavior means that a specified state and evolution law determine a unique subsequent state. The position–momentum uncertainty relation [8,9] constrains joint state spreads. We examine its MQ count representation; this kinematic constraint does not establish an evolution law or the cause of quantum uncertainty.

State spreads describe the distribution of measurement outcomes for a specified quantum state. They are not, by definition, an instrument error or ignorance of a definite classical trajectory [9]. Questions about when a measurement record becomes available require a state, evolution and measurement protocol beyond the count inequality.

The comparison also bears on the relationship between mathematical description and physical observation. We examine how reference measures and count relations encode quantities with distinct physical dimensions. The proposed correspondence is assessed through explicit realization maps and their assumptions.

We first compare the invariant numerical coefficient appearing in angular and momentum realizations at the stated bound. Their common coefficient does not identify their physical dimensions. The explicit conversion between the realizations permits comparison with a quantum-optical model while preserving the distinction between angle and momentum.

Within this account, fundamental references have no intrinsic curvature. Curvature-like effects belong to the realized System Frame description, rather than to continuous geometry within the Internal Frame. This distinction is used when interpreting the finite-count gravitational relation below ([7], Appx. C and K).

This paper combines classical expressions with MQ count relations for realized gravitational curvature, the speed of light, escape velocity, and the uncertainty bound. The reductions test the consistency of the stated reference scales and normalization; their physical interpretation depends on the Frames mapping and the applicable count domain.

The introductory Measurement Quantization paper [7] presents the underlying derivations and comparisons with measurement data, together with the appendices referenced here. We distinguish those comparisons from the input-dependent calculations in this paper. Current frame and realization meanings are those stated above. The cited publication remains the historical source of the appendix discussions; later notation clarifications do not change its published text. A preprint is available in Online Resources.

Earlier versions used the 2018 CODATA recommended values [10]. CODATA is the Committee on Data of the International Science Council; the numerical calculations revised here use its 2022 adjustment [11].

Two spreadsheets are also listed in Online Resources. Supplement 1 contains calculations for the introductory paper and its appendices; Supplement 2 collects calculations for this paper and other MQ applications. The values revised here use the fixed 2022 inputs identified in [11], rather than inheriting an earlier spreadsheet input edition. The common input chain must be taken into account when judging whether a numerical comparison is independent.

The results distinguish the MQ realization definitions, algebraic consistency reductions, and the state-spread constraint. We assess each at its stated normalization and separate calculated values from independent source results.

At the stated bound, the MQ momentum realization and angular realization inherit the same invariant numerical coefficient. Momentum and angle remain dimensionally distinct, as made explicit in (6). We use this relation to organize the comparison of MQ calculations and modeled photon angles; the comparison does not by itself prove that physical observation and mathematical description are identical.

The dimensionless realized measure counts n_L, n_M, and n_T normalize length, mass, and duration by their respective fundamental references. These realized ratios need not all be integer Internal Frame configurations. When different properties of one phenomenon require separate length counts, we retain distinct notation. This distinction is essential for the position and momentum factors in the uncertainty discussion.

We begin with the invariant dimensionless coefficient θ_si and its angular realization Θ_si, in rad ([7], Appx. R). The momentum realization p_si has units kg m s⁻¹, and defines the dimensional conversion factor, in kg m s⁻¹ rad⁻¹. The speed of light c is exactly 299792458 m s⁻¹ in the International System of Units (SI) [11]. Upright G, in m³ kg⁻¹ s⁻², and ħ, in J s, denote formal MQ upper-count quantities. They provide a limiting normalization for the reference-scale reductions in (2)–(5); upright ħ is not a physically realized macroscopic Planck constant. The Informativity differential denotes the non-Lorentz contraction of realized spatial separations under the Frames mapping ([7], Appx. D). The electromagnetic realization is introduced before (7) and used for the optical comparison and uncertainty bound. Equation (1) uses the 2022 inverse fine-structure input [11]. Parenthetical digits give standard uncertainty in the last displayed digits.

We use the MQ finite-count gravitational relation ([7], Appx. C) to obtain the momentum realization. In (2)–(5), p is the alias for p_si; r is the realized separation in m. The positive, dimensionless fractional count residual has the exact form . Italic G denotes the finite-count coupling at r, with the same units as its upper-count limit. Writing r as the length count times the fundamental length exposes the common count factor. In the limit that product tends to one half, giving the final expression in (3) with upright G. This is a limit of the count geometry, not an identification of finite and upper-count couplings at every radius.

Using ([7], Appx. F), l_fm_f=2p_sit_f ([7], Appx. V), and ħ=2p_sil_f, the following reductions give the momentum realization.

The angular and momentum realizations share an invariant numerical coefficient but retain distinct dimensions. Their relation uses the conversion factor defined above.

For the electromagnetic assessment, MQ uses the interaction count , calculated from the 2022 inverse fine-structure input [11]. This is a normalization at the electromagnetic demarcation, not the macroscopic optical path length. Italic ℏ denotes the electromagnetic reduced Planck constant, with ; the corresponding gravitational realization is G=2Q_Ln_LG. Both quantities therefore use the same interaction count. The two momentum expressions give (7), and their common count factor cancels to yield the length form (8), associated with the conventional Planck-length expression [12].

Equation (8) recovers the fundamental-length form using the electromagnetic values of both G and ℏ. Their product equals the product of the formal upper-count quantities because their reciprocal count factors cancel. The dependent substitutions establish consistency of the adopted MQ reference scale rather than an independent measurement of the constants.

For the angular comparison, we use the theoretical model of Shwartz and Harris, Polarization Entangled Photons at X-Ray Energies [13]. Their reported angles are numerical solutions of phase-matching and nonlinear-current-density equations. They are not measurements of the equal-angle Bell-state configuration. Related work on X-ray down-conversion and wave mixing is listed in [14,15].

We select the positive equal-angle branch in Table I of [13] for the symmetric Bell state appearing in (9). The model uses diamond, its (111) reciprocal-lattice vector, a 25 keV pump, and degenerate signal and idler energies of 12.5 keV. The pump polarization is perpendicular to the scattering plane, and the selected pump angle is 3.02079 rad. The function f denotes the common signal and idler angle returned by the model on this specified branch. H and V denote polarization in and perpendicular to the scattering plane, respectively; the subscripts s and i distinguish signal and idler. The angles θ_p, θ_s, and θ_i are measured relative to the atomic planes normal to the reciprocal-lattice vector. The Bell-state notation follows [13,16]; Bell-theorem context is discussed in [17].

Tables 1 and 2 evaluate the MQ angular expressions using the 2022 calculation of the invariant coefficient, with π the exact mathematical circle constant. Table 3 reproduces the two equal-angle branches reported in Table I of [13]. On the positive branch, the MQ angle rounds to the reported model value 3.26239 rad. This comparison is between a calculated MQ realization and a theoretical optical-model result; it does not supply an experimental uncertainty or establish the additional MQ digits independently.

The − and + branch labels refer to the sign of the common signal and idler angle. The comparison values are theoretical results in Table I of [13].

Both tables use the same MQ coefficient and therefore share its uncertainty; they are equivalent calculations, not independent predictions.

The angular comparison with the selected Shwartz–Harris branch is described in (9) and Table 3 [13]. The paper does not report a ±2 μrad experimental uncertainty for that model angle.

The displayed uncertainties propagate the 2022 inverse fine-structure input [11]; their shared input does not provide independent corroboration.

We next examine several algebraic reductions involving the fundamental references. Substitution of dependent expressions can produce the identity 1=1. Such a reduction checks internal consistency under its assumptions; it does not establish that all laws of nature are geometric identities or supply an independent physical test.

Some relations connect two reference measures. The speed of light relates length and time through ; the ratio of m_f to t_f has different dimensions. We keep these dimensional roles distinct when making substitutions.

Other relations involve all three reference measures and reduce to the fundamental relation connecting the fundamental length and mass through the momentum realization and fundamental time ([7], Appx. V).

A third class of relations cancels the reference measures and leaves count constraints ([7], Appx. A and H). Heisenberg’s uncertainty principle provides one example ([7], Appx. B). We first write the fundamental measures in terms of the upper-count gravitational coupling, momentum realization, and speed of light ([7], Appx. W).

Using the upper-count coupling G ([7], Appx. F) and , we can combine (10–12) to recover the fundamental relation (13).

Substituting the already established momentum realization into (13) cancels the same fundamental references on both sides, giving (14) and (15).

Fundamental Gamma γ_F is a dimensionless configuration scalar based on the elementary-charge realization [18]. It describes the Frames mapping from the Internal Frame configuration domain to observable System Frame quantities. Here e_f denotes the unit-free fundamental charge configuration quantity before its SI realization. Mixed left-floor/right-ceiling brackets denote nearest-whole-unit realization of the enclosed configuration-domain number; they do not denote ordinary floor or ceiling alone. These operations act before a charge is expressed in coulombs. We write δ_θ:=42θ_si-⌊42θ_si⌉ for the signed dimensionless Diophantine realization residual in the charge construction, distinguishing it from the whole-unit count 42. The charge-based mapping scalar is

Equation (16) expands the charge residual in the first-principles mapping expression. The coefficients 3, 7 and 14 and the cubic order implement the charge construction in [18], Section 3.5. Physical dimensions enter through the adopted observable realization developed below.

Within this implementation, a realization class groups counts with repeating nearest-whole-unit behavior; a primitive generator is the first retained count of such a class. The construction selects the nontrivial proper divisors of 42. The count 3 is a primitive generator; 6 repeats its class without adding independent information. The count 7 generates a second stable class, 14 supplies its first independent repeated state, and 21 is its terminal member before 42. The non-repeating count 2 is excluded. The resulting basis assigns 3, 7 and 14 to the linear, quadratic and cubic contributions. We adopt this selection rule and cubic order as implementation premises. The source rationale explains their roles, while a general admissibility or uniqueness theorem remains a subject for further research [18]. At zero signed residual only the polynomial correction vanishes; the displayed divisor remains 84 plus 42/277.

The gravitational construction is independently expressed through the count geometry and the loss associated with the fractional residual at successive fundamental-time increments ([7], Appx. C). Both constructions use the same fundamental references. When those references cancel in a motion or inertial-mass relation, the resulting constraint concerns count geometry within MQ ([7], Appx. AU–AW). This removal of reference scales must be distinguished from an independent empirical test of the proposed physical realization.

For comparison with electromagnetic constants, the present SI fixes elementary charge e, in C, and Planck constant h, in J s, exactly. The dimensionless fine-structure constant α remains experimentally determined. Vacuum magnetic permeability μ₀, in N A⁻², is derived from that constant together with the exact SI quantities, and vacuum permittivity and Coulomb’s constant follow from the same dependent family [11]. We adopt the normalization developed in the revised Unification derivation and supported by the Bohr-radius and charge constructions [7,18,19]. A mapped MQ charge can therefore be compared with the defined charge scale or used to calculate permeability. Such a comparison retains the covariance of shared adjustment inputs. A historical charge measurement requires its original unit realization and input edition.

Here a₀ denotes the Bohr radius in m, m_e the electron mass in kg, and e_c the mapped classical charge predicted by the elementary-charge construction, in C. We distinguish γ_F,sys, the mapped charge-squared realization in C², from the dimensionless configuration scalar in (16). The conventional Bohr-radius identity and the MQ count form give ([7], Appx. BL; [11,19])

Equating the two Bohr-radius representations and using the fundamental speed relation gives the electromagnetic action normalization already used in (7)–(8) and in the uncertainty bound

The Planck-like charge is the charge realization before normalization to the classical value. The charge construction applies twice the residual-count product to perform that normalization. Combining this factor with the inverse fine-structure relation and (16) fixes the adopted mapped gamma realization

Vacuum permittivity ε₀, in F m⁻¹, is evaluated with the predicted charge in the conventional electromagnetic identity [11]. Substitution of the action normalization and the fundamental momentum relation gives

Multiplying the permittivity relation by the upper-count gravitational normalization in (10) cancels the fundamental length and speed scales. The following identity uses upright G, rather than the finite-count electromagnetic gravitational coupling

Equations (17)–(21) specify the adopted normalization and the resulting coupling identity. They do not establish independent field dynamics or validate the realization-class selection. The electromagnetic action and gravitational coupling used together in (7–8) and (26–28) remain evaluated at the same electromagnetic interaction count.

The supplied concordance Table VII of [19] reports agreement sigma as a normal-equivalent transformation of its reported agreement tail probability, distinct from a discrepancy in standard uncertainties. Its charge row reports an agreement sigma of 2.30; this cannot be interpreted as a charge discrepancy of 2.30 standard uncertainties or as discovery significance. In the fixed 2022 chain, the mapped charge is 1.6021766326122 × 10⁻¹⁹ C with input-only standard uncertainty 3.0022 × 10⁻²⁹ C. The difference from the exact SI charge is 4.62 conditional standard uncertainties. The permeability comparison has the same conditional discrepancy when the shared inverse fine-structure input is propagated through the difference. These results condition on the charge mapping and fixed coefficients adopted as implementation premises. The broader uncertainties in the source-reported concordance benchmark are not used in this fixed 2022 calculation, and no uncertainty is assigned to the exact defining SI charge. Reproduction of that benchmark’s uncertainty model and independent testing of the adopted charge mapping remain subjects for further research. Shared electromagnetic identities and reused adjustment inputs cannot be aggregated as independent evidence for MQ.

For the escape-velocity example, (22) uses the classical upper-count coupling, with M the gravitating mass in kg and r the radius in m. Substituting mass and radius as their fundamental measures times counts gives the classical count form. At ([7], Appx. I), setting the escape speed equal to c gives (23–25). The symbol v with subscript e denotes the escape speed in m s⁻¹.

For the quantum uncertainty example, σ_x and σ_p denote the state standard deviations of a position component and its conjugate momentum component, in m and kg m s⁻¹ [8,9]. The position-related count n_Lo normalizes the observed separation spread, while n_Lm normalizes the displacement spread entering the velocity description. We use σ_x=n_Lol_f and for a fixed-mass, nonrelativistic translational model and a positive reference duration. This parametrization of the momentum spread does not apply to photons by replacing momentum with rest mass times velocity. The electromagnetic value of ℏ is the one introduced before (7), and the gravitational realization used to resolve it has the same interaction count. Substitution into the saturated bound gives (26). The interaction count in Q_Ln_L is distinct from the two spread counts. Dividing by the positive duration count gives the product condition (27), rather than the upper-count value one half. This does not assign the counts separately or require a light-speed velocity spread.

The reference-scale and escape-velocity reductions are algebraic consistency checks. The uncertainty relation instead constrains two state spreads and retains its electromagnetic normalization. Reducing a relation to counts preserves the physical meaning of those spreads; it does not turn a quantum-state constraint into an outcome-selection law.

The quantum discussion below concerns the information encoded by the state-spread constraint. A statistical description of outcomes is compatible with that constraint, but no measurement dynamics follows from this algebraic translation.

Using the same spread definitions and electromagnetic interaction count, the position–momentum inequality gives (28) [8,9]. Equality is the saturated case in (26). The factor multiplying the three spread-related counts is calculated as 1.999930145668961(11) from the 2022 inputs [11]; replacing it by 2 would silently restore the upper-count normalization. The two length factors remain separate spread scales.

The counts in (28) have distinct roles. The position-related observed-separation count n_Lo represents the first factor, while n_Lm enters the displacement-to-duration ratio in the momentum factor. The mass and duration counts retain their meanings given above. The former gravitational-versus-flat-curvature distinction does not determine these two length counts, and their magnitudes need not be equal.

As a numerical example, with m_n denoting the 2022 neutron mass, division by the recalculated fundamental mass gives , a dimensionless realized mass count [11]. Its standard uncertainty includes the National Institute of Standards and Technology (NIST) correlation between neutron mass and inverse fine-structure constant [20]. This realized ratio does not contradict the integer configurations defining the Internal Frame. It supplies only the mass factor; the state-dependent position and momentum spreads must still be specified before (28) can be evaluated.

Within the stated nonrelativistic model, the displacement-to-duration factor parametrizes the momentum spread. Equation (28) constrains it together with the distinct position-related count and mass count. It does not determine these factors separately.

The uncertainty inequality specifies a joint constraint on the two realized spreads. It does not by itself specify when information becomes available or select an outcome. Changes in the attainable spreads must be assessed with the state and measurement protocol specified.

The count configuration and its observable realization must also be distinguished when discussing determinism. The position–momentum inequality does not prove that the underlying Internal Frame evolution is nondeterministic, nor does it prove that uncertainty arises from a lack of classical information. It constrains the joint state spreads of the realized description under the assumptions stated before (26) [9].

To determine individual measurement outcomes, a model would additionally need a specified state evolution and measurement rule. Neither is obtained from the count inequality. This separates the present kinematic result from a solution of the quantum measurement problem.

The present equations do not establish whether a system can influence future outcomes by favoring an optimization criterion. Such a proposal would require a specified dynamical mechanism and supporting evidence beyond the algebraic reductions considered here.

The MQ account developed here connects distinct angular and momentum realizations through a dimensional conversion, recovers reference-scale consistency identities, and expresses the position–momentum bound using separate length counts. The optical comparison is with a specified theoretical branch. The uncertainty assessment uses electromagnetic ℏ and its matching gravitational realization; its finite interaction-count factor remains in the count inequality.

A statistical description remains appropriate when a quantum state does not specify a unique measured outcome. The count inequality organizes its position and momentum spreads without supplying a measurement dynamics or proving that uncertainty is caused by missing classical information. Numerical consistency with calibration inputs and a rounded theoretical model value does not establish an independent empirical prediction.

The electromagnetic implementation adopts the realization-class selection and the charge normalization as premises pending further research [18,19]. Its electromagnetic comparisons are conditional consistency tests. Source-reported concordance benchmarks are not treated as independent validation of those premises, and this paper makes no aggregate discovery-significance claim from their agreement statistics.

The symbol entries below collect the notation used in this paper. Their meanings follow the definitions introduced at first use; they do not introduce additional physical identifications.

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

The numerical calculations use the 2022 CODATA adjustment [11]. The MQ source relations and supplementary resources are identified in Methods.