The publication and the narrower question here

Jody A. Geiger’s Resolving the Hubble tension 68.261 and 73.510 km s⁻¹ Mpc⁻¹ appeared in AIP Advances 15, 075134, DOI 10.1063/5.0269029, on July 22, 2025. The publisher records receipt on March 7 and acceptance on June 12, 2025. This Brief concerns the common baseline, rather than treating publication itself as evidence that the observational tension has been resolved. [1]

The paper starts from an age-based baseline and associates two mapping normalizations with CMB and Cepheid-based inferences. The baseline and both branches are model calculations. Their physical association with observational procedures is the author’s interpretation and is not a classification supplied by the observational teams. [1] [7] [8]

Exact rate relations in current notation

Let ${\mathrm{𝐴}}_{\mathrm{𝑈}}$ denote the realized universe age in seconds, ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ the dimensionless MQ invariant coefficient and $\mathrm{\pi }$ the dimensionless circle constant. Let ${\mathrm{𝐻}}_{\mathrm{sys}}$ and ${\mathrm{𝐻}}_{\mathrm{int}}$ denote the two physical realized comparison rates labeled for the System Frame and Internal Frame, respectively. Equations (1a) and (2b) are their inverse-second forms. Astronomical rate units are applied afterward. [1] [2]

The Internal Frame is the discrete count-configuration domain; the System Frame is the observable relational realization induced by the Frames mapping. Both quantities in these equations are physical realized comparison rates. Their labels do not assign an SI frequency to a primitive pre-realization domain. The historical paper’s overloaded dimensional notation is clarified here without identifying a dimensionless coefficient with its momentum or angular realization. [2]

The formulas express the rate normalization discussed in the published Section III H using the later dimensionally explicit convention. They do not transfer every later calibration or ontology statement into the 2025 paper. The lettered equation labels are local to this Brief. [1] [2]

Why the baseline is the midpoint

Adding the two rates cancels the circle-constant contributions and leaves twice the inverse age. Equation (3c) gives the exact midpoint identity. It follows from the definitions; it is not an independently fitted or measured third rate.

Subtracting and dividing the definitions give Equations (4d) and (5e). The separation carries inverse-time units; the ratio is dimensionless. A dimensionless circle-coefficient offset must not be reported as a physical Hubble-rate difference without its common normalization.

The ratio cancels age algebraically, but the two endpoint values used to form it are still the same model outputs or observations. A ratio, a difference and a midpoint cannot be counted as three new datasets. Holding the coefficient fixed, an age change rescales both rates and their separation together while leaving the ratio unchanged.

A reproducible historical rounded-input example

Use 13.794 billion Julian years for the rounded historical age and 3.26239030392 for the historical numerical coefficient in the dimensionless rate ratios. One Julian year is exactly 31,557,600 seconds. Use the paper’s rounded megaparsec conversion, 3.08567758 × 10¹⁹ kilometres. These are stated illustrative inputs, not a reconstruction of every hidden spreadsheet digit or an assertion that all displayed coefficient digits were independently measured. [1] [9]

The age becomes 435,305,534,400,000,000 seconds. Divide one by that duration to obtain the inverse-second baseline, then multiply by the kilometre length of a megaparsec to express it in km s⁻¹ Mpc⁻¹. Multiply that baseline by the two coefficients in (1a) and (2b) to obtain the branches. Table 1 records the resulting rounded calculations.

The calculation rounds to all three historical displayed rates. Its midpoint is exact before decimal rounding. Averaging the two published three-decimal endpoints instead gives 70.8855, differing from the printed 70.885 baseline by 0.0005. That final-digit effect is expected when separately rounded numbers are recombined; it is not evidence for an additional offset. Figure 1 uses the worked central values before displayed rounding.

How changing the age moves both branches

Table 2 changes the adopted age by one percent in each direction while fixing the coefficient and unit conversion. This is a sensitivity exercise, not an uncertainty interval, an observational range or a fit. Both rates and the baseline change by the reciprocal age factor. Figure 2 shows the same conditional dependence.

If age alone were uncertain with the coefficient held fixed, its contribution would correlate the two rate errors positively. At fixed age, changing the coefficient instead moves the branches oppositely while preserving their midpoint. The complete covariance must propagate the actual shared input chain; it cannot be inferred by treating two quoted marginal errors as independent.

Historical observational comparisons are separate from the baseline identity

Table 3 reproduces the selected observational entries exactly as displayed in the 2025 paper’s Table I. The CMB entry is tied to the SPT-3G 2018 TT/TE/EE analysis cited there. The asymmetric ladder entry is the selected Pantheon+/SH0ES combination, not a universal summary of every distance-ladder analysis. Their external papers contain model- and combination-specific inferences. [1] [7] [8]

The ladder entry is asymmetric, and no symmetric Gaussian error has been invented for it here. The central residual of 0.110 is slightly larger than a literal 0.1 threshold, although small compared with the quoted observational uncertainty. This Brief uses the printed table values rather than repeating the abstract’s blanket less-than-0.1 wording as exact arithmetic. No joint significance or theory probability is calculated from these rows.

CMB analyses infer a present-day rate through a cosmological model fitted to radiation observables; they do not directly measure today’s expansion at the time of last scattering. Cepheid-calibrated supernova analyses use a distance and luminosity calibration chain. Agreement with selected compressed parameter estimates does not itself show that those procedures realize the proposed MQ frame assignments. [7] [8]

Model normalization is not a general consequence of flatness

An inverse-age baseline is part of the MQ expansion prescription. In general cosmology, the age is an integral over the expansion history. Spatial flatness alone does not require the current expansion rate times the universe age to equal one. A complete physical comparison must therefore specify the expansion history, distances and relevant dynamics rather than treating the baseline identity as a universal geometric law. [6]

The current revised MQ source reconstructs its age from measured CMB temperature and other adopted inputs with a fixed temporal-normalization premise. That age is derived rather than a new independent age measurement. The premise and unquantified model discrepancy are not included in the small input-propagated rate uncertainties. These later qualifications limit the interpretation of a precise numerical baseline without silently revising the historical calibration. [2] [4]

Versions, evidence and reporting scope

The historical title values remain 68.261 and 73.510. The linked current preprint uses the revised pair 68.259(19) and 73.508(20), with conditional input uncertainties. Those later values and later observational comparisons belong to their own versions. This historical reconstruction does not attach new error bars to the old rounded rates or present the selected 2025 comparison as a current census. [2] [5]

An older concordance score must not be reinterpreted as a discovery significance or multiplied across the two dependent branches. A stronger test requires the covariance, source versions, age-normalization assumptions and an observable-level treatment of CMB and distance-ladder data. The common baseline is useful for locating those dependencies; it is not an additional empirical confirmation. [2] [4]

For a journalist’s reproducibility request, identify the published equations, age inputs, time normalization, unit conversion and observational model in each comparison. Ask which quantities were frozen or inferred and how their covariance was propagated. The calculations in this Brief reproduce the historical displayed rates under stated rounded inputs; they do not fit the original cosmological likelihoods.

July 22, 2025 is the journal publication date. This Brief was prepared October 26, 2026. Later terminology clarifies the account without assigning a new discovery or independent validation to the publication milestone.