The exact dependency comes first

Let ${\mathrm{𝜇}}_{\mathrm{0}}$ be vacuum magnetic permeability in H m⁻¹, ${\mathrm{\varepsilon }}_{\mathrm{0}}$ vacuum electric permittivity in F m⁻¹ and $\mathrm{𝑐}$ vacuum light speed in m s⁻¹. In the present SI, light speed is exactly 299792458 m s⁻¹. Equation (1a) is the conventional identity retained by the MQ calculation. [1] [4] [5]

Thus a specified electric result fixes the magnetic result at exact light speed. Physically distinct electromagnetic roles do not imply independent numerical information. The SI revision effective May 20, 2019 fixed elementary charge and Planck’s constant; permeability ceased to have its former exact assigned value. Its recommended value now depends on measured fine-structure information. “Exact identity” and “exact numerical permeability” are different claims. [4] [5]

What the current forward MQ route supplies

The current constants manuscript constructs the electric output through a forward count-and-charge realization and fundamental gamma. This differs from a historical auxiliary electromagnetic parameter resolved using measured constants. The forward route removes those electric and magnetic values as direct numerical inputs, but retains the inverse fine-structure calibration, exact SI references and the count, momentum and charge-realization conventions. [1] [2] [3]

The fundamental physical references are ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ in metres, ${\mathrm{𝑚}}_{\mathrm{𝑓}}$ in kilograms and ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ in seconds. The momentum realization ${\mathrm{𝑝}}_{\mathrm{𝑠}\mathrm{𝑖}}$ has units kg m s⁻¹. Normalized fundamental gamma ${\mathrm{\gamma }}_{\mathrm{𝐹}}$ is dimensionless; its mapped physical realization ${\mathrm{𝛾}}_{\mathrm{𝐹}\mathrm{,}\mathrm{sys}}$ has units C². The latter is used in the dimensional expressions below and is not a separately fitted parameter. Let $\mathrm{\pi }$ denote the upright dimensionless circle constant. [1] [2]

Substituting Equation (2b) into Equation (1a) cancels the light-speed square and yields the first expression in Equation (3c). The source’s dimensional fundamental relation equates fundamental length times fundamental mass to twice momentum times fundamental time, giving the second expression. A dimensionless coefficient cannot replace the dimensional momentum merely because an adopted SI numerical coordinate happens to coincide. [1] [2]

Both numerator forms in Equation (3c) have units kg m; dividing by C² gives H m⁻¹. The Internal Frame is the dimensionless count-configuration domain. Physical measures and charge-squared gamma belong to the System Frame realization through the Frames mapping. Giving the configuration gamma physical charge units would conflate these roles. [1] [2]

Reproduce the current numerical family

Table 1 identifies the retained calibration. A 90-digit fixed-branch evaluation reproduced the forward count, charge and gamma route and reran it at the common inverse fine-structure input shifted by its standard uncertainty. The quoted conditional sensitivities retain that one shared uncertain input. They do not include uncertainty in the physical realization convention or model discrepancy. [2] [4]

Table 2’s MQ permeability exceeds the printed CODATA reference by about 2.1752 × 10⁻¹⁵ H m⁻¹. That difference is a numerical residual, not a complete statistical significance. The two descriptions share metrological information and need a cross-covariance treatment; the MQ model discrepancy is unquantified. The forward route is also distinct from the source’s separate exact-charge inversion, which recovers a defining input by construction. [1] [2] [4]

The public magnetic page contains mixed numerical versions. Its current figure gives the reproduced value in Table 2, while older prose retains 1.2566370635(18) × 10⁻⁶ H m⁻¹ and a 1.93σ comparison. This Brief does not merge those uncertainties or reuse that quoted significance. The updated figure and current forward calculation control the displayed numerical ledger. [1] [2]

Why the product has no propagated input uncertainty

Multiplying Equation (1a) by its denominator gives Equation (4d). This identity is satisfied for every positive electric input at fixed light speed. Recovering it therefore checks algebraic closure, not the physical accuracy of the electric prediction. [1] [3]

The common-input relative standard uncertainty is about 2.2152 × 10⁻¹⁰ in each of the MQ electric and magnetic outputs. If those propagated errors were falsely treated as independent, the relative uncertainty assigned to Equation (4d)’s product would be about 3.1328 × 10⁻¹⁰. With the dependency retained, the product is exactly constant and its input-propagated uncertainty is zero. This cancellation does not assign zero uncertainty to the model’s physical premises. [2] [6]

Reciprocal sensitivity and covariance

At fixed light speed, differentiating Equation (1a) gives Equation (5e). The derivative is exact. Its use in ordinary first-order uncertainty propagation is a linearized calculation; that propagation order should not be confused with an exact finite-variation formula. [6]

The electric and magnetic perturbations have opposite signs. Their first-order relative standard uncertainties have the same magnitude, and their first-order covariance is negative: multiply the electric variance by the negative magnetic-to-electric ratio. Under this single-input linearization, their correlation is minus one. This is a statement about the propagated model, not a measured statistical correlation of independent apparatus results. [6]

Figure 2 shows why the exact finite response is not a symmetric percentage rule, even though its tangent reverses equal relative changes at first order. The propagated uncertainty here is many orders of magnitude smaller than the selected 1% illustration. If several genuinely uncertain inputs were used, their full covariance would have to be propagated before assessing the outputs. [6]

The Coulomb row carries the same electric information

Let ${\mathrm{𝑘}}_{\mathrm{𝑒}}$ be Coulomb coupling in N m² C⁻². Its conventional reciprocal relation to permittivity, combined with Equation (1a), gives Equation (6f). Magnetic permeability and Coulomb coupling are exact positive multiples of one another when light speed is fixed. [1] [2] [3]

Table 3 explains why physically different constants can supply only one inherited direction of numerical variation in this calculation. The first-order output covariance is rank one when a single uncertain input drives the nonconstant outputs. Replicating the same information in several parameterizations does not multiply the amount of independent evidence. [2] [6]

What an independent test would require

A prospective test needs a defined observable, an independently justified source and uncertainty chain, and an apparatus model that connects the proposed realization to the recorded signal. A conventional magnetic-field illustration or a numerical permeability row does not supply that measurement. Reusing the same electromagnetic adjustment or recovering an exact SI input must be disclosed. [1] [7]

A useful analysis should freeze the count branch and realization convention, state all direct inputs and report their covariance. It should distinguish source-input uncertainty, numerical precision, metrological cross-dependence and physical model discrepancy. An agreement claim remains conditional until those distinct contributions are addressed. [2] [4] [6]

Chronology and source requests

October 13, 2026 is the recorded source-version date used in this retrospective archive, not an established original discovery, announcement or journal publication date. This Brief was prepared November 2, 2026 using the current public figure and forward constants calculation. Journal publication of these current MQ manuscript versions is not established. [1] [2]

For an interview, request the versioned numerical ledger and joint covariance before comparing uncertainties or counting confirmations. Ask which magnetic observable is predicted independently of the electric calibration. The result delivered here is a reproducible linked calculation with retained dependencies, not a new measurement of vacuum permeability.