What has changed since the published proposal

The 2025 AIP Advances paper proposed two realized rate assignments for a common expansion geometry. Its historical values were 68.261 and 73.510 kilometres per second per megaparsec. The current revised source gives 68.259(19) and 73.508(20), with conditional input uncertainties and stronger evidence qualifications. Neither set is itself an independent astronomical measurement. [1] [4]

All rate values in Table 1 use kilometres per second per megaparsec. The current uncertainties propagate declared inputs while holding the temporal-normalization premise fixed. Historical rounded values are not assigned new error bars. Parentheses in the current source refer to one standard input-propagated uncertainty in the final printed digits. [1]

The two realized rate assignments

Let ${\mathrm{𝐻}}_{\mathrm{sys}}$ and ${\mathrm{𝐻}}_{\mathrm{int}}$ denote the System Frame and Internal Frame-resolved physical comparison rates. Let ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ be the dimensionless invariant MQ coefficient, $\mathrm{\pi }$ the dimensionless circle constant and ${\mathrm{𝐴}}_{\mathrm{𝑈}}$ the realized System Frame universe age in seconds. Equations (1a) and (2b) give inverse-second rates before conversion to astronomical units. These labels are local to the Brief. [1]

The Internal Frame is the discrete count-configuration domain; the System Frame is its observable relational realization. The Frames mapping connects them. Both displayed rates are physical realized comparison quantities, with inverse-time units. Their labels do not assign an SI frequency to the primitive pre-realization domain, and they do not mean that an observational dataset has independently detected either frame. [1]

For a central arithmetic example, take the coefficient 3.26239030504 and the reconstructed age 13.794339221862558 billion Julian years. One Julian year is exactly 31,557,600 seconds. The resulting inverse-second rates are about 2.212122057 × 10⁻¹⁸ and 2.382239051 × 10⁻¹⁸. Conversion using one megaparsec of about 3.08567758149 × 10¹⁹ kilometres gives rates rounding to 68.259 and 73.508. This reproduces central arithmetic under the stated inputs; the extra digits do not create new measurement precision. [1]

The age reconstruction uses measured CMB temperature and the fine-structure calibration, together with a fixed realization normalization. The age is derived, not an additional independent datum. Its absolute temporal normalization remains unverified, and its uncertainty and model discrepancy are excluded from the small quoted input errors. Conditional precision must not be presented as a complete theory uncertainty. [1]

Midpoint, separation and ratio share the same inputs

Equations (3c)–(5e) follow algebraically from the two rate assignments. They show that the midpoint, separation and ratio are related outputs, rather than additional independent predictions to multiply into a global evidence claim.

The midpoint is an MQ normalization, not a universal identity between the Hubble constant and cosmic age in arbitrary cosmologies. The physical separation has rate units; it is distinct from the dimensionless circle-coefficient offset used in the source’s mapping construction. Common age changes shift both model rates together. Uncertainty in their difference therefore cannot be obtained by simply adding their individual variances as if independent. [1]

Which external results are being compared

The selected observational values are fixed to explicit source versions. Camphuis and collaborators’ version 2, dated April 1, 2026, gives SPT-3G D1 66.66 ± 0.60 and the combined CMB-SPA result 67.19 ± 0.38. CMB-SPA combines SPT, ACT and Planck information. These are base-ΛCDM inferences from CMB primary spectra and lensing, not direct measurements of today’s rate at last scattering. The combined sample contains SPT information and is not independent of the SPT-only row. [2]

Version control matters. The first arXiv version used 67.24 ± 0.35 for the combined result; the version cited here uses 67.19 ± 0.38. A historical or updated comparison must identify the exact version and combination rather than transfer a value across editions without notice. [2]

Breuval and collaborators’ 2024 four-anchor SH0ES result is 73.17 ± 0.86. The Small Magellanic Cloud was added to the geometric foundation of the Cepheid and Type Ia-supernova ladder. This combined result is distinct from their SMC-only value of 74.1 ± 2.1. The present comparison uses the combined four-anchor inference. [3]

Figure 1 shows the three external estimates and the two MQ assignments on the same rate axis. The very short MQ error bars represent conditional propagated inputs, not a complete uncertainty budget for the proposed mechanism.

Reproduce the individual displayed-value diagnostics

Table 2 uses the corresponding MQ central value minus the observed value. For this diagnostic only, the denominator is the square root of the sum of their two quoted variances. The calculation treats the displayed errors as symmetric Gaussian standard uncertainties, assumes zero model–observation cross-covariance and excludes unquantified theory discrepancy. These assumptions define a transparent arithmetic check, not a complete likelihood.

For the SPT-only row, the residual is 1.599 and the combined scale is 0.600301, giving 2.663665, or 2.66 at two decimals. The archive source review stated about 2.67. Ignoring the tiny MQ uncertainty gives 2.665, explaining the rounding-level difference. The combined CMB and ladder rows give 2.81 and 0.39 at two decimals. This Brief reports the explicit denominator rather than implying a new data reduction.

A standardized residual is neither a probability that MQ is true nor a discovery significance for the proposed mapping. The CMB-derived estimates assume a particular cosmological model. MQ must specify its own acoustic, distance and growth predictions before it can replace that inference model. Comparing a model rate to another model’s compressed parameter estimate is a useful diagnostic, with that limitation retained. [1] [2]

Why the separation diagnostic is not an extra test

Table 3 uses the rounded model rates and the four-anchor-minus-CMB-SPA observed pair. For the observational difference alone, independent quoted errors give a scale of 0.940213. The absolute separation residual of 0.731 is 0.777484 times that scale. This reproduces the archive’s approximately 0.78 diagnostic without inventing an uncertainty for the correlated MQ split.

The complete variance of a difference includes the two individual variances and subtracts twice their covariance. The MQ covariance must be propagated from its common inputs; rounded marginal errors do not determine it. A joint comparison also requires any observational covariance and model–observation cross-dependence. The value 0.78 above deliberately uses the external scale only and is not presented as such a completed joint result.

Figure 2 explains the logical point. A difference removes a common shift in both endpoints. Two pairs can have similar separations while their absolute normalizations differ. Here the displayed MQ midpoint is 70.8835 and the external-pair midpoint is 70.1800. Neither midpoint is a third independent Hubble observation.

What a stronger assessment would require

A complete assessment must propagate the shared MQ calibration and temporal-normalization assumptions, specify an observable-level expansion and perturbation model, and confront the original CMB and distance information with an appropriate joint likelihood. Dataset overlap, astrophysical systematics and cross-calibration must remain visible. The branch assignments are hypotheses to test, not labels supplied by the observational teams. [1] [5] [7]

The update does not establish that any one discrepancy isolates a failure mechanism. Nor does it justify selecting only the favorable ladder match. The larger lower-branch differences are relevant observational pressure on the stated numerical proposal. Their physical interpretation depends on the shared normalization and the inference model as well as on the data.

Earlier descriptions of a 3.35-sigma concordance should not be repeated as a discovery significance or a probability for MQ. Re-expressing a goodness-of-agreement statistic does not turn it into evidence against a specified null model. Laboratory, galaxy and Hubble comparisons with shared inputs and different likelihoods cannot be multiplied into a global theory probability. [1] [5]

Dates, versions and reporting requests

The original paper was published July 22, 2025. September 27, 2026 is the archive’s source-review snapshot, not an established discovery or journal date for this update. This Brief was prepared October 26, 2026. The SPT comparison is pinned to version 2 and the SH0ES result to the four-anchor analysis; no claim is made that this short selection is a census of every current expansion estimator. [2] [3] [4]

For an interview, request the exact rate calculation, age inputs, full covariance and observational source versions. Ask which quantities were fitted or frozen and whether the acoustic and distance observables were modeled jointly. The reproducible arithmetic in this Brief is a documented comparison of displayed numbers, not a new cosmological reanalysis.