One coefficient and three quantity roles

Let ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ denote the invariant dimensionless MQ coefficient and ${\mathrm{𝛩}}_{\mathrm{𝑠}\mathrm{𝑖}}$ its plane-angle realization. In the adopted representation, the number of radians in the angle is the value of the coefficient. Equation (1a) writes the source definition in that representation. A radian is the coherent SI unit for plane angle and identifies the quantity kind. [1] [2]

Let ${\mathrm{𝑝}}_{\mathrm{𝑠}\mathrm{𝑖}}$ denote the linear-momentum realization in kilograms metres per second. The source assigns it the same numerical coefficient in the adopted SI representation. Let ${\mathrm{𝐶}}_{\mathrm{𝑝}\mathrm{𝜃}}$ denote the conversion factor in kilograms metres per second per radian. Equation (2b) makes the dimensional link explicit. The coefficient, angle and momentum play different roles in the Frames mapping. [1] [2]

Table 1 separates the two multiplications by radius. For arc length, use the numerical angle in radians and multiply by the radius in metres. For angular momentum, multiply the radius by the tangential component of linear momentum. The latter product has units of kilogram metre squared per second, equivalently joule second. [1] [2]

The half-action identity

Let ${\mathrm{𝑙}}_{\mathrm{𝑓}}$, ${\mathrm{𝑚}}_{\mathrm{𝑓}}$ and ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ denote the fundamental length, mass and time in metres, kilograms and seconds. Let $\mathrm{ħ}$ denote the source’s reduced action in the upper-count regime, in joule seconds. Equation (3c) gives the source normalization used in Section 3.3, with the momentum equal to one half of the fundamental mass times the fundamental length divided by the fundamental time. [1] [2]

Multiplying Equation (3c) by the fundamental length cancels the length in the action quotient and yields Equation (4d). The factor of one half is already present in the momentum normalization. The circle provides the selected radius and a geometric setting for interpreting the product. [1]

Figure 1 shows the angle and arc, and adds a tangent to make the angular-momentum interpretation explicit. Angular momentum about the center is the cross product of the position vector and linear momentum. For a momentum perpendicular to the radius, its magnitude is exactly the product in Equation (4d). A radial momentum contributes zero about that center. A general direction requires the component perpendicular to the radius. The geometric arc alone supplies neither a trajectory nor a momentum direction. [1] [2]

Which reduced action appears in the calculation

The source marks the reduced action in Equation (3c) as an upper-count quantity. Its separate action discussion distinguishes the upper-count calculation from the conventional reduced Planck constant obtained by dividing the exact SI Planck constant by twice pi. Keep the regime and calibration attached to the action when taking half of it. The identity is exact within the stated normalization, regardless of how a numerical value for that action is obtained. [1] [4]

Section 3.3 also gives a gravitational form for the momentum realization. Equating it to the action form recovers the conventional algebraic shape of the Planck length relation under the source assumptions. Since the two forms enter the same normalization, the cancellation in Equation (4d) cannot count as an independent measurement of either the momentum or the action. [1] [2]

What the result would take to test

The radius–momentum product applies to a specified geometric assignment. Intrinsic quantum spin is a property described by a spin operator and its states. A shared unit or a factor of one half leaves those physical identifications open. To predict a measurement, specify the physical state, the relevant operator or orbital trajectory, the realized radius and how a detector would resolve the predicted result. A comparison would also need the dependencies of the calibration and an uncertainty model. [1] [2] [5]

September 8, 2026 is the archive’s completed-source date. This Brief uses the current public Section 3.3 notation and distinguishes the exact product derived from Eq. (32) from the arc-length illustration in Eq. (33) and Figure 2. The drawings are geometric illustrations. Journal publication of the cited manuscript versions is unestablished. [1] [2]