What changed in the dependency route

J. A. Geiger’s Extending Precision of the Physical Constants uses a count-and-charge construction to form fundamental gamma and calculate electric, magnetic and Coulomb quantities. The historical auxiliary electromagnetic gamma route is retained in the research record as a different construction. Its measured electromagnetic input must not be silently carried into the new route or described as the same parameter. [1] [2]

The methodological statement is precise: measured vacuum permittivity, vacuum permeability and Coulomb coupling are not direct numerical inputs to the current fundamental-gamma calculation. The retained measured source is the 2022 CODATA inverse fine-structure recommendation, with exact SI speed of light and Planck constant supplying reference values. The adopted momentum and charge realizations add substantive normalization premises. [1] [4]

The Internal Frame is the discrete configuration domain. Physical dimensions belong to realized System Frame quantities through the Frames mapping. A normalized gamma parameter and a mapped charge-squared realization must therefore retain distinct labels and units, even when their numerical coefficients coincide in the chosen SI convention. [1]

Definitions and charge-coordinate normalization

Let ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ be the invariant dimensionless coefficient. The fundamental length ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ has units m, the fundamental mass ${\mathrm{𝑚}}_{\mathrm{𝑓}}$ units kg, and light speed $\mathrm{𝑐}$ units m s⁻¹. Let ${\mathrm{𝑒}}_{\mathrm{𝑓}}$ denote the fundamental charge representation in the SI realization used by this manuscript, with units C. Equation (1a) states its numerical-coordinate definition; dividing by each displayed unit removes dimensions. [1]

Equation (1a) is the source’s Equation (30). It does not equate a dimensionful length times speed squared directly with charge. It uses fixed SI unit references and assigns the resulting charge coordinate under the stipulated realization. That prescription is not established by dimensional analysis alone. [1]

The signed coefficient-count remainder ${\mathrm{𝑄}}_{{\mathrm{𝑛}}_{\mathrm{𝜃}}}$ is forty-two times the coefficient minus its nearest whole unit, with nearest integer 137 on the selected branch. The charge-valued residual ${\mathrm{𝑛}}_{\mathrm{𝑒}}$ is formed from eighty-four times the numerical coordinate of the fundamental charge minus the nearest whole unit of forty-two times that coordinate, then multiplied by 1 C. It has units C here; it is not a primitive integer availability count. The Planck-like charge ${\mathrm{𝑒}}_{\mathrm{𝑝}}$, also in C in this SI formula, is Equation (2b). [1]

The coefficients and cubic form in Equation (2b) belong to the adopted charge construction; they are not obtained merely by choosing units. At the tiny charge coordinate in Equation (1a), the nearest-whole-unit term in the charge residual is zero. This nearest operation and the nearest integer 137 in the coefficient remainder act on different arguments and must not be interchanged. [1]

Normalized gamma and mapped gamma have different units

Let ${{\mathrm{𝛼}}_{\mathrm{𝑝}}}^{\mathrm{–1}}$ denote the dimensionless Planck-like inverse fine-structure form. The positive electromagnetic count parameter is ${\mathrm{𝑛}}_{\mathrm{𝐿}}$ and its Diophantine realization residual is ${\mathrm{𝑄}}_{\mathrm{𝐿}}$, given by the square root of one plus the squared count parameter, minus that parameter. This route uses the specified fine-structure count, with count parameter times the invariant coefficient equal to 276. Define normalized fundamental gamma ${\mathrm{\gamma }}_{\mathrm{𝐹}}$ by Equation (3c). [1] [3]

Every factor on the right of Equation (3c) is dimensionless. Its mapped electromagnetic realization ${\mathrm{𝛾}}_{\mathrm{𝐹}\mathrm{,}\mathrm{sys}}$ is introduced separately by Equation (4d). The upright sys qualifier identifies this physical representation, not another fitted coefficient. [1]

Table 1 is specific to the current normalized SI formulation. Older papers sometimes attach a fundamental-gamma name directly to a charge-squared realization. Their equations must be read in their own representation and dimensions rather than combined solely because the symbols look alike. Primitive configuration-domain quantities do not acquire SI charge units through that name. [1]

The forward calculation and its charge residual

The realized classical charge ${\mathrm{𝑒}}_{\mathrm{𝑐}}$ follows by applying the finite-count normalization to the Planck-like charge, as in Equation (5e). It is the forward result at the specified fine-structure count, not the exact SI elementary charge inserted as an input. [1]

The 2022 inverse fine-structure source 137.035999177(21) fixes the coefficient as 3.26239030504(25) and the fine-structure count as 84.6005456716(65). The adopted SI momentum realization uses that coefficient as its numerical coordinate in kg m s⁻¹. This is a normalization premise, not a conclusion of dimensional analysis. The exact SI speed of light and Planck constant complete the upstream reference chain. [1] [3] [4]

Table 2 reproduces the current manuscript’s forward column. For this Brief, the full chain was evaluated with 90-digit working arithmetic, retaining the unrounded coefficient and count residual. Varying the single inverse fine-structure source through its standard uncertainty reproduces the displayed source sensitivities. Additional working digits are numerical precision, not new measurement information.

The forward charge is lower than the exact SI reference by about 1.39 × 10⁻²⁸ C, about 0.87 parts per billion relative to the exact value. Dividing that residual by the propagated source uncertainty gives about 4.6. The manuscript explicitly reports this residual. The ratio omits unquantified model uncertainty and is not a complete statistical significance or an accuracy ranking. [1] [4]

The electromagnetic family uses mapped charge-squared gamma

Let ${\mathrm{\varepsilon }}_{\mathrm{0}}$ denote the electric constant, or vacuum permittivity, in F m⁻¹, and $\mathrm{\pi }$ the dimensionless circular constant. Equation (6f) uses the mapped parameter with units C², together with the realized fundamental references. Substituting dimensionless gamma into that formula without its charge-unit factor would fail dimensional consistency. [1]

The denominator in Equation (6f) has units kg m³ s⁻², so charge squared divided by it has units C² s² kg⁻¹ m⁻³, equivalent to F m⁻¹. This check identifies which gamma realization is required. It does not establish the underlying charge construction physically.

Vacuum permeability then follows as the reciprocal of permittivity times light speed squared, and Coulomb’s constant as the reciprocal of four pi times permittivity. The current source uses the same fine-structure count and mapped gamma for the full family. They are linked outputs, not three independent tests. [1]

Table 3 gives conditional MQ outputs, not replacement recommended constants. Their standard uncertainties propagate the same fine-structure input with count and realization premises fixed. Checking the reciprocal electromagnetic identities tests internal consistency; the same derived outputs cannot be treated as independent confirmations of the gamma construction. [1]

Exact-charge inversion is a separate route

The manuscript also solves for an interaction count using the exact SI charge as an input. This gives a charge-inversion count of 84.60159(23), distinct from the fine-structure count 84.6005456716(65). The corresponding mapped gamma is 2.1984688037(12) × 10⁻³⁷ C². The inverse route returns the exact input charge by construction. Its count is not the one used for Table 3. [1]

Figure 2 prevents a comparison from silently changing its input. Using exact charge to select a new count is a legitimate inverse calibration, but its agreement with that same charge is not independent validation. The current source keeps both routes visible and labels their roles. [1]

Calibration, covariance and the remaining test

Removing direct electric-constant feedback improves the transparency of the calculation. It does not remove the fine-structure source, exact SI reference definitions, momentum normalization, selected count branch or cubic charge premises. A complete test must evaluate those dependencies and retain covariance between quantities with shared inputs. [1] [5]

The manuscript’s separate substitution of the Morel rubidium-recoil input is a source-sensitivity calculation. That experimental result contributes to the CODATA adjustment, so the two input paths are not statistically independent. Their substitution alone does not rank framework accuracy or turn all dependent outputs into independent observations. [1]

An independent physical claim would need a predeclared observable and calibration route, comparison data whose shared dependencies are accounted for, and a treatment of model discrepancy. The present reproduction checks formulas, units and propagated source sensitivity. It does not supply an independent laboratory determination of all electromagnetic constants.

Chronology and reporting context

October 13, 2026 is the archive’s recorded source-version date, not an established first discovery or original announcement date. This Brief was prepared October 27, 2026 using the current online manuscript and its qualifications. Its current normalized and mapped gamma definitions must not be retroactively substituted into historical gamma formulas without checking their units and assumptions. [1]

Journal publication of this current manuscript version is not established. For an interview, request the full source-input ledger, numerical-coordinate conventions, charge construction, exact count used for each result and common-input uncertainty propagation. Ask what observation tests the model beyond closure identities and how the forward charge residual is addressed without replacing its prediction by exact-input recovery.