What the reference set contributes

The research milestone is a reproducible set of dimensional MQ references with a coherent input ledger. Different manuscripts can then evaluate their dependent quantities without mixing input editions or treating repeated calibration information as independent evidence. The same central values can support a correct or an incorrect uncertainty budget depending on whether their shared dependence is retained. [1] [2]

MQ begins from discrete measure and derives count relationships with a proposed physical realization. The Internal Frame describes count configurations; the System Frame expresses realized measures through the Frames mapping. Deriving the form of a relationship, selecting a coefficient corresponding to our universe and assigning dimensional reference units have distinct roles. The numerical profile here fixes the latter choices while retaining the source’s count premises. [1] [2]

Quantities and the defining reference relations

Let ${\mathrm{𝑙}}_{\mathrm{𝑓}}$, ${\mathrm{𝑚}}_{\mathrm{𝑓}}$ and ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ denote fundamental length, mass and time, in m, kg and s. Let $\mathrm{𝑐}$ denote vacuum light speed, exactly 299792458 m s⁻¹ in the SI. Let ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ be the dimensionless MQ coefficient, and ${\mathrm{𝑝}}_{\mathrm{𝑠}\mathrm{𝑖}}$ its momentum realization in kg m s⁻¹. The source’s adopted SI normalization assigns the latter the same numerical coefficient, 3.26239030504(25), without identifying the two dimensions. Equation (1a) uses the dimensional momentum quantity. [1] [2] [3]

Let ${\mathrm{𝑚}}_{\mathrm{𝑒}}$ be electron mass in kg, ${\mathrm{𝑎}}_{\mathrm{0}}$ the Bohr radius in m and ${\mathrm{𝛼}}_{\mathrm{𝑝}}$ the dimensionless Planck-like fine-structure coupling. Equation (2b) gives the source’s orbital length relation. Its numerator has units kg m and division by fundamental mass gives metres. The reference set does not assign momentum units to the dimensionless coefficient to repair this equation. [1] [2]

The time reference follows from length and the exact speed input in Equation (3c). Thus length and time share their entire relative calibration dependence in this profile. The local equation labels in this Brief are distinct from the source’s labels; the reference calculation appears in Eqs. (26)–(30) of the fundamental-measures manuscript. [1]

Figure 1 follows the common input into the coefficient, its specified dimensional realization and the three references. A branching calculation diagram does not mean each branch carries a separate experiment. Exact SI anchors and fixed prescriptions remain part of the calculation even when they add no propagated input uncertainty. [1] [2]

Reduce the orbital product before propagating uncertainty

Let ${{\mathrm{𝛼}}_{\mathrm{𝑐}}}^{\mathrm{–1}}$ denote the dimensionless classical inverse fine-structure input, 137.035999177(21). Let $\mathrm{ħ}$ denote the conventional reduced Planck constant used for calibration, in J s. Its SI value is exact through the defining Planck constant divided by twice the circle constant. This conventional input must be distinguished from the different MQ upper-count action output. Equation (4d) is the conventional orbital identity used in the current source. [1] [2] [3] [5]

Substituting Equation (4d) into Equation (2b) removes electron mass and Bohr radius as separate adjusted inputs to this numerical profile. Multiplying their individually rounded recommendations and adding their uncertainties as if independent would discard their adjustment dependence. The symbolic appearance of two quantities does not itself identify two independent sources of uncertainty. [1] [2]

The calibrated Planck-like coupling and momentum are also functions of the same inverse fine-structure input. The source uses the fixed nearest-whole-unit branch 137 in its coefficient inversion. That branch stays unchanged across the stated input interval. Propagation must rerun the complete dependent sequence; holding a downstream quantity fixed while varying its parent would define a different calculation. [1] [2]

Table 1 distinguishes the input’s role from the law’s form. A measured selector is not automatically the axiom from which the relationship was derived. Its uncertainty still enters numerical outputs, and the adopted normalization is not an independent momentum measurement. That separation is central to interpreting the claimed precision. [1] [2]

Reproduced references and their signed response

The calculation used 90-digit arithmetic and repeated the fixed-branch sequence at the input shifted up and down by its stated standard uncertainty of 0.000000021. Half the difference between those outputs supplies the local propagated input uncertainty. Displayed values are rounded only after calculation. Parentheses in Table 2 indicate standard uncertainty in the last displayed digits. [1] [2]

Table 2’s responses are local, rounded sensitivities for the one-input profile. The signs matter. Length and time decrease together while mass increases. In the first-order covariance description this produces a rank-one uncertainty matrix, positive correlation for length with time and negative correlations for mass with either. Those first-order correlations describe calibration propagation, not a separately measured joint distribution of microscopic lengths and masses. [1] [2]

The absolute standard uncertainties are about 1.23868 × 10⁻⁴⁵ m, 1.66794 × 10⁻¹⁸ kg and 4.13180 × 10⁻⁵⁴ s, respectively. Printing those small errors does not establish experimental access at those scales. The intervals omit uncertainty in the count premises and physical realization prescriptions. [1] [2]

Exact identities explain cancellation and residual dependence

The fundamental relation in Equation (5e) follows directly by multiplying Equation (1a) by length and using Equation (3c). It is an algebraic relationship within the same calibrated reference set, not a fourth independent measurement. [1] [8]

Combining Equations (2b) and (4d) gives Equation (6f). The length–mass product therefore retains the source dependence of the two coupling quantities. Their nearly compensating changes leave a small nonzero uncertainty. The fact that the reduced Planck input and light speed are exact does not make that product exact. [1] [2]

Worked products and ratios

Table 3 and Figure 2 recompute three combinations using the same shifted input. The contrasting column deliberately discards covariance and combines the individual relative uncertainties in quadrature. It is shown as an error demonstration, not as an alternative valid budget for this source profile. [1] [2]

Length divided by time returns 299792458 m s⁻¹ exactly under Equation (3c). Assigning independent errors to numerator and denominator creates a spurious uncertainty on this input recovery. This cancellation is a property of the defining calculation. A laboratory speed determination has its own realization effects and uncertainty. [1] [3] [5]

The length–mass product is 3.51755007940 × 10⁻⁴³ kg m at the central profile. Its relative input uncertainty is much smaller than the individual reference uncertainties because their dominant variations oppose one another. It is not zero, as Equation (6f) shows. By contrast, in mass divided by length those opposite shifts add, so ignoring their dependence understates the uncertainty. Shared input does not always mean a smaller final error. [1] [2]

For a general measurement model, the GUM’s propagation framework retains covariance between correlated inputs. Here the complete one-input recalculation already carries that dependence. Additional uncertain inputs would require their joint information and sensitivities; the rank-one description would no longer be sufficient merely because it worked for the present profile. [4]

What the reported uncertainty includes

The reported uncertainties include sensitivity to the specified fine-structure calibration with exact SI anchors and source prescriptions fixed. They do not include a quantified discrepancy between the physical world and the proposed realization map. That missing contribution has not been measured to be zero. Narrow conditional intervals therefore cannot be read as a complete statement of physical accuracy. [1] [2]

Other calculations can add independent observational inputs, such as a measured temperature or an apparatus quantity. Those inputs need their own uncertainty and any cross-dependence. The current three-reference budget must not be carried into an extended model as if it already included all such sources. Conversely, reintroducing electron mass and Bohr radius as independent uncertainties after the orbital reduction would count the adjustment information incorrectly. [1] [2]

The constants manuscript also examines a separate rubidium-recoil fine-structure input as a source-sensitivity exercise. It reruns the chain with that source and its uncertainty. That comparison is not a new authoritative calibration here, and the two columns are not assumed statistically independent. Changing input editions or experiments requires updating the entire dependent set, not selecting favorable rows from different versions. [2]

What consistency establishes and what requires a test

Recovering an input or satisfying the fundamental relation checks the implementation of the stated dependencies. Reproducing the same reference set in the Planck-root construction provides another algebraic representation under compatible upper-count quantities. Neither operation adds an independent measurement or proves the physical discreteness of measure. Conventional Planck units and their metrological uncertainty budget retain their separate definitions. [1] [8]

An independent test must specify a physical observable, its realized apparatus or astronomical model, the frozen calibration and premises, and data that add information beyond the input already used. Comparisons across derived constants need the complete covariance of predictions and references, including shared information. A collection of related residuals cannot be multiplied into a global significance as if each were a separate experiment. [1] [2] [10]

The bounded-domain discussion linked from the original release concerns the admitted count domain. It is useful theoretical context, but it supplies neither a laboratory determination of these references nor an uncertainty budget for practical measurement at their scales. [9]

Chronology and reporting guidance

October 13, 2026 is the recorded source-version date for this retrospective archive. It is not established here as the first discovery, original announcement or journal publication date. This Brief was prepared November 2, 2026 against the current public technical sources. Journal publication of those current versions is not established. [1] [2] [8] [9]

For interviews, request the unrounded reference set, source edition, branch prescription and coherent input ledger. Ask which quantities select the numerical realization and which relations derive its form. Preserve the distinction between source-input uncertainty, practical realization uncertainty and model discrepancy. A complete story should report both the reproducibility achievement and the independent physical tests still needed.