Three quantities, with different roles

Let ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ denote the invariant dimensionless MQ coefficient, ${\mathrm{𝛩}}_{\mathrm{𝑠}\mathrm{𝑖}}$ its angular realization and ${\mathrm{𝑝}}_{\mathrm{𝑠}\mathrm{𝑖}}$ its momentum realization. Their meanings are distinct even when the displayed numbers coincide. Equation (1a) states the source’s angular representation in radians; the radian unit label is upright. [1] [2] [3]

In conventional SI dimensional bookkeeping, plane angle is a quantity of dimension one. Radians identify its angular kind rather than supplying a new independent base dimension. The invariant coefficient is also dimension one, but that alone does not make it an angle or identify every quantity of dimension one. NIST and the metrology vocabulary retain this distinction between dimension, unit and kind. [4] [5]

The Internal Frame is the discrete count-configuration domain; the System Frame is its realized relational encoding in which observables are expressed through the Frames mapping. The realization definitions connect those roles within MQ. They do not amount to a direct observation of an underlying microscopic geometry. [1] [7]

The momentum representation and the fundamental relation

Let ${\mathrm{𝑙}}_{\mathrm{𝑓}}$, ${\mathrm{𝑚}}_{\mathrm{𝑓}}$ and ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ be fundamental length, mass and time, with physical units m, kg and s. The source defines the momentum realization through Equation (2b). Its quotient has units kg m s⁻¹, so it cannot be replaced by an unaccompanied dimensionless number. [1] [2]

Rearranging Equation (2b) gives the dimensionally explicit fundamental relation in Equation (3c). Both sides have units kg m. If the momentum were replaced by the bare invariant coefficient while the other quantities retained their physical units, the right side would instead have units of seconds. A numerical match would not repair that mismatch. [1] [2]

The statement is about a relation among fundamental references under the MQ construction. It is not a law saying every object has the same length–mass product or that arbitrary independent count multipliers can be inserted on both sides. A physical application must preserve the reference, count and realization assumptions appropriate to that application. [2]

Where equality of numerical coordinates enters

Equation (4d) expresses the adopted SI numerical realization explicitly. Dividing momentum by the reference unit makes a dimension-one numerical coordinate. The resulting equality is between such coordinates; it does not assign momentum dimensions to the invariant coefficient. [1] [2]

The current fundamental-expression manuscript obtains the coefficient by inversion of the MQ fine-structure relation using the 2022 input 137.035999177(21), with exact SI references and a fixed count branch. Its updated Table 1 gives 3.26239030504(25). The older Section 3.3 presentation quotes a different conditional uncertainty. This Brief uses the current calibrated Table 1 consistently rather than merging numerical editions. [1] [2] [9]

The repeated numerical digits and parenthetical uncertainties in Table 1 are inherited from one calibrated coefficient. They are not three independent measurements or three confirmations of the proposed structure. A 90-digit recalculation reproduced the coefficient and propagated the same inverse fine-structure input. The stated conditional uncertainty excludes uncertainty in the physical realization convention. [2]

The dimensional angle–momentum bridge

Let ${\mathrm{𝐶}}_{\mathrm{𝑝}\mathrm{𝜃}}$ be the source’s conversion factor between the momentum and angular realizations. Its units are kg m s⁻¹ rad⁻¹ when those reference units are used. Equation (5e) gives its definition and the corresponding relationship. Defining the factor as a ratio does not, by itself, supply an independent prediction for a measured beam momentum. [1] [3]

Under Equations (1a) and (4d), the conversion has numerical coordinate one in kg m s⁻¹ rad⁻¹. That is a consequence of the specified realization and units. It is not evidence that the conversion is an independently measured universal constant. If a different unit is used for angle or momentum, its numerical coordinate changes accordingly.

A unit change exposes the difference

One degree is pi divided by 180 radians, while one gram is one thousandth of a kilogram. Applying those exact unit conversions to the same calibrated quantities gives Table 2. Displayed coordinates are rounded; the coefficient itself remains unchanged. The table compares representations of each quantity, not a physical magnitude of angle against a physical magnitude of momentum. [4]

For example, the angular coordinate becomes about 186.921195603 in degrees, while the momentum coordinate remains about 3.262390305 in kg m s⁻¹. The conversion coordinate is then about 0.017453293 kg m s⁻¹ per degree. If momentum is expressed in g m s⁻¹ instead, its coordinate is about 3262.390305038 and the conversion coordinate becomes about 17.453292520 g m s⁻¹ per degree. The quantity relationship survives every conversion.

Figure 2 shows why equality of bare numbers in one unit convention is not a unit-independent identification of quantities. It also shows why the conversion factor and its units cannot be omitted from an angle–momentum claim. The calculated points carry the same calibration rather than independent experimental errors.

Equivalent momentum formulas retain their regimes

The source also writes the same momentum using upper-count gravitational and action quantities. Let $\mathrm{𝑐}$ be vacuum light speed in m s⁻¹, upright $\mathrm{G}$ the upper-count MQ gravitational coupling in m³ kg⁻¹ s⁻², and upright $\mathrm{ħ}$ the upper-count MQ reduced Planck quantity in J s. Equation (6f) uses that common upper-count family. [1] [2] [3]

The upright MQ action quantity in Equation (6f) is distinct from the conventional exact SI reduced Planck constant. A finite electromagnetic realization cannot be substituted into this upper-count identity without retaining its normalization. Rewriting the same momentum through two equivalent formulas does not create two independent measurements of the conversion to angle. [1] [3]

Table 3 identifies a practical graphics check. Dividing an action by length produces momentum; relabeling that quotient as an angle without the conversion changes its physical meaning. An ordinary arc-length relation also needs a radius and angular measure in radians. Compact legacy shorthand must therefore be read against the explicit current definitions rather than used to equate momentum, angle and length. [1] [2] [3]

The X-ray comparison and its evidentiary limits

The MQ manuscripts discuss a selected calculated propagation angle of 3.26239 rad from Shwartz and Harris’s phase-matching treatment of polarization-entangled X-rays. Their 2011 paper is a theoretical study of parametric down-conversion and Bell-state configurations. The printed angular solution is not a precision measurement of the MQ coefficient and does not supply an experimental standard uncertainty for additional MQ digits. [2] [3] [6]

The current source identifies the chosen angular branch as part of its model construction, rather than deriving an independent experimental momentum from that angle. A correspondence with a selected theoretical angle must be distinguished from the separate experiments discussed in the manuscripts. This Brief does not recast those experiments as joint measurements of the MQ angular and bound-related momentum quantities. [2] [3]

A physical test of the proposed association would require the realization conversion to be specified independently and both angle and momentum to be related to measured observables in one defined configuration. The crystal geometry, photon energies, branch selection and uncertainty model would need to remain explicit. Agreement of two algebraic MQ momentum expressions is an internal consistency test, not that joint experiment. [3] [8]

Common structure is not completed field unification

The source presents the common coefficient as a structural correlation within the Frames mapping. It does not identify the electromagnetic and gravitational fields as the same physical field, eliminate their different observables or derive all their dynamics merely from matching numbers. Those claims require the action, field equations, interaction rules and independently testable solutions appropriate to the application. [1] [8]

The methodological improvement is traceability: the reader can identify the coefficient, units, conversion, calibration and regime in each formula. Dimensional consistency is necessary for the stated physical interpretation, but it is not sufficient to establish that nature implements the chosen realization maps.

Chronology and source requests

September 8, 2026 is the completed-source milestone recorded for this retrospective archive entry. It is not an established first discovery, original announcement or journal publication date. This Brief was prepared November 2, 2026 using current public manuscript definitions and the updated fundamental-expression calibration. Journal publication of these current MQ manuscript versions is not established. [1] [2] [3]

For interviews, request the current realization definitions and conversion units before comparing numerical values. Ask which inputs fix the coefficient, which choices fix the angular branch and what observation would test the conversion independently. Keep early numerical motivation, present dimensional clarification and experimental validation as distinct parts of the research account.