Abstract

We investigate a phenomenological rotating black-hole-universe (BHU) framework that links cosmic thermal evolution with distance observations and independent spectroscopic tests. Its central proposal is a change of cosmic geometry rather than the addition of a rotation parameter to the usual irrotational Friedmann-Lemaître-Robertson-Walker (FLRW) background. A physical description of cosmic rotation requires a specified observer congruence, nonzero vorticity, and a consistent spacetime geometry. By interpreting Hubble’s law within its rotating cosmic geometry, the BHU framework identifies the cosmic angular rate with the Hubble rate, $\omega =H$ , and connects the Planck and Hubble scales through a thermal relation with constant $H/T^2$ . Together with the adopted temperature-scale-factor identification, this relation motivates a continuously decelerating expansion history. The observational programme combines supernova distance comparisons with independent tests of thermal evolution, galactic mass-velocity relations, and atomic spectra. For Pantheon, the reported weighted RMS residuals are 0.1348 mag for the auxiliary evolving-partition prescription and 0.1336 mag for the selected flat-ΛCDM reference. For 22 Union3 bins, the bounded law and the reference model give ${\chi }^2=24.902$ and 26.411, respectively. These retained comparisons motivate further tests but do not by themselves establish the proposed rotating mechanism. We also propose reverse distance-redshift tests in which independently calibrated Type Ia supernova luminosity distances are used to infer redshifts and, under the adopted identification, their associated scale factors. The bounded-sinh law uses $z_c=z/\left(1+z\right)$ and the fixed geometric coefficient 2/3; no separate 1/3 term enters the formula. A fixed-parameter ΛCDM distance law can likewise be inverted to predict $z$ and $1+z$ for comparison with spectroscopy. Such inversions are not independent evidence when the fitting data are reused. A genuinely independent test requires an absolute-magnitude calibration from a separate nearby sample, an independently fixed $H_0$ , and explicit treatment of uncertainties, local motions, and redshift-dependent photometric corrections. Although $1+z=1/\left(1-z_c\right)$ is an exact identity, the central physical question and point of debate examined here is which quantity—the bounded expression $z_c=z/\left(1+z\right)$ or the conventional unbounded expression $z$ —should be adopted as the physically fundamental variable for defining and interpreting cosmic redshift. Repeated spectroscopy of distance-ordered source pairs provides a complementary test of explicitly specified redshift-drift predictions. A 50-year sensitivity assessment distinguishes instrumental resolving power from line-shift precision and quantifies the demanding stability required for such measurements. The proposed quadratic thermal and atomic scalings are presented with their separate assumptions and observational consequences. The result is a testable research programme rather than a completed cosmological theory: a covariant rotating description and joint agreement with independent observations remain essential.

Keywords

Cosmic Acceleration, Pantheon, Union3, Redshift Drift, Bounded Redshift, Black-Hole Universe, Galactic Rotation, Super-Gravity, Millimetre Spectroscopy

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Overview and Notation

This paper examines whether a bounded, rotating cosmic construction can connect thermal evolution, supernova distances, galactic dynamics, and spectroscopic tests. It presents a phenomenological programme whose proposed relations can be compared with observations, while a unified spacetime derivation of those relations remains a development goal. Part II introduces the physical questions and observational tests; Part III supplies the detailed construction. The following guide introduces the notation used in both parts.

BHU and the reference cosmology.

BHU means black-hole universe. FLRW means Friedmann-Lemaître-Robertson-Walker, the homogeneous and isotropic cosmological geometry. ΛCDM denotes the reference model with a cosmological constant Λ and cold dark matter.

Expansion, rotation, and the cosmic boundary.

$H$ is the Hubble rate and $\omega$ the proposed angular rate. $R_H=c/H$ and $M_H=c^3/\left(2GH\right)$ are the Hubble radius and the mass assigned to its critical-density sphere; $c$ is the speed of light and $G$ the gravitational constant. Their unit compactness $2G{M}_H/\left(c^2{R}_H\right)=1$ is an algebraic identity, not by itself a proof of a black-hole spacetime. Vorticity describes local rotation of an observer congruence; shear describes its shape-changing distortion.

Thermal evolution and Planck quantities.

$T$ is cosmic temperature, $k_B$ is Boltzmann’s constant, $a\left(t\right)$ is the scale factor, and $q=-a\ddot{a}/{\dot{a}}^2$ is the deceleration parameter under the identification $H=\dot{a}/a$ . A dot denotes differentiation with respect to the specified time. $M_{\text{P}l}$ is the Planck mass; this paper uses its associated radius $R_{\text{Pl}}=2G{M}_{\text{Pl}}/c^2$ . A subscript 0 denotes the present epoch. The relation $H\propto T^2$ implies $q=1$ only with the additional temperature-scale-factor and time identifications.

Redshift and distance.

$z$ is spectral redshift, and $z_c=z/\left(1+z\right)$ is its bounded reparameterization. $D_C$ denotes the adopted comoving-distance quantity, $D_L$ is the luminosity distance, and $\mu =5{\log }_{10}\left[D_L/\text{Mpc}\right]+25$ is the distance modulus. A megaparsec (Mpc) is a distance unit; mag denotes the logarithmic magnitude unit. The velocity-shaped bounded distance proxy must be distinguished from an independently measured galaxy speed.

Mass fractions and observational comparisons.

$f_{\text{Λ}}$ and $f_m$ denote the proposed complementary mechanical fractions, whereas ${\Omega }_i$ denotes a density parameter or an explicitly specified auxiliary coefficient. The same symbol does not make the constructions physically identical. CMB is the cosmic microwave background and BAO the baryon acoustic oscillation scale. Pantheon and Union3 are Type Ia supernova compilations. Weighted RMS describes residual scatter; ${\chi }^2$ measures residuals with the adopted uncertainties or covariance. Neither statistic alone determines a physical mechanism.

Source pairs and galactic tests.

$G_1$ and $G_2$ label two astronomical sources (not gravitational constants); $\text{Δ}{z}_i$ is the change in a source’s measured redshift between observing epochs. Its interpretation requires a spectroscopic propagation model. $M_b$ is baryonic mass, $M_d$ the proposed effective non-particle mass supplement, and $M_T$ a threshold mass. “Super-Gravity” names the particular galactic scaling proposed here, not an established new force.

1. Introduction

The discovery papers of 1998-1999 established that distant Type Ia supernovae are dimmer than expected in a simple matter-dominated cosmology ([1] [2]). Within a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) analysis, the observed distance-redshift relation is well described by a late-time component with negative effective pressure. The resulting ΛCDM framework also provides a successful joint description of the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and large-scale structure ([3]-[5]). Nevertheless, the physical nature of the dark sector remains unresolved, and tensions among determinations of the present Hubble parameter motivate continued examination of alternative descriptions ([6] [7]).

Recent peer-reviewed reanalyses have challenged the uniqueness of the acceleration inference. Nielsen et al. found only marginal statistical evidence for acceleration in Type Ia supernova data [8]. Son et al. concluded that correcting for supernova progenitor-age bias aligns the data with a non-accelerating universe [9], while Sah et al. reported that the corrected Pantheon+ compilation indicates deceleration [10]. Wiseman et al., while defending the standard measurement methodology, acknowledged that the physical cause of the acceleration and the nature of dark energy remain unknown [11]. Andrei, Ijjas, and Steinhardt demonstrated, within a physically rigorous quintessence framework reviewed by Nobel Laureate Saul Perlmutter, that the same supernova data are also consistent at the $2\sigma$ level with a universe whose expansion is approaching its end [12]. Together, these results motivate the present examination.

This article separates two questions that are often combined. The first is observational: what distance-luminosity curve is encoded by the supernova data? The second is interpretive: does that curve uniquely require accelerated expansion driven by a cosmological constant? Pantheon and Union3 provide an unusually clean way to examine this distinction. If a compact alternative distance law produces a comparable residual structure, the data remain valuable, but their physical interpretation is not unique.

A second distinction concerns historical and live observations. A supernova Hubble diagram combines photons emitted at many past epochs. It is not a repeated measurement of the same galaxy’s redshift increasing from one observing date to another. We therefore formulate a simple operational test involving two galaxies, $G_1$ and $G_2$ , at different distances. Hubble’s law fixes the ordering of their recession speeds; the addition of global acceleration fixes the ordering of their recession accelerations. Repeated telescope spectra can then search directly for the corresponding distance-ordered temporal change.

The bounded-redshift black-hole-universe (BHU) program developed in earlier work ([13]-[17]) provides the alternative framework studied here. Its Hubble-scale and thermodynamic ideas are retained, but the present paper is deliberately organized around the two empirical questions above. Our objectives are:

1) State the minimal BHU assumptions without conflating identities and physical postulates;

2) Present the Pantheon and Union3 distance fits in a common notation;

3) Derive the $G_1$ - $G_2$ live-observation logic directly from distance, speed, and acceleration;

4) Compare that expectation with published repeat-telescope measurements;

5) Formulate the galactic Super-Gravity scaling and its merger test;

6) Formulate an independent cosmic-dawn test using hydrogen spectra and geometric-mean black-hole thermodynamics;

7) Identify the evidence already supporting BHU, its distinctive predictions, and the observations that can provide decisive confirmation.

The paper is organized into three parts. Part I contains the Introduction. Part II presents the general considerations, the live $G_1$ - $G_2$ test (Section 3), the additional observational tests (Section 4), and the development programme and reported achievements. Part III develops the foundations (Section 9), earlier BHU literature (Section 10), thermal coupling (Section 11), mass partitions (Section 12), bounded kinematics (Section 13), supernova comparisons (Section 14), and geometric closure (Section 16).

2. General Considerations on Cosmic Acceleration

2.1. Published Challenges to Permanent Acceleration

The inference of acceleration from Type Ia supernovae has also been repeatedly re-examined. Nielsen, Guffanti, and Sarkar found only marginal evidence for acceleration in an earlier likelihood treatment of a large supernova sample [8]. This did not invalidate the supernova measurements; it demonstrated that the inferred significance depends on the statistical population model and treatment of intrinsic dispersion.

A different challenge concerns the future rather than the present sign of acceleration. Andrei, Ijjas, and Steinhardt constructed a rapidly descending dark-energy model that remains consistent with current observations while allowing the present accelerated phase to end and expansion eventually to reverse into slow contraction [12]. Its relevance here is logical: historical distance data do not uniquely establish indefinite future acceleration. Even within relativistic dark-energy cosmology, models with very different future evolution can remain observationally degenerate over the measured interval.

Luu, Qiu, and Tye have developed a second quantitative example using an ultralight-axion dark-energy model fitted to recent DES and DESI indications of evolving dark energy [18]. Within the allowed but substantially degenerate parameter region, their analysis finds a high probability for a negative cosmological constant. Their benchmark solution has a total cosmic lifespan of approximately 33.3 billion years: expansion eventually reaches a maximum, reverses, and ends in a Big Crunch. Within that fitted model, future cosmic deceleration, turnaround, contraction, and collapse follow unambiguously; an eternally accelerating universe is excluded. The remaining uncertainty is empirical rather than logical: whether the evolving-dark-energy indication is confirmed and whether nature realizes the proposed axion model. Thus the result provides strong contemporary support for a finite expansion history and an eventual decelerating phase, while the separate question of whether the present-day deceleration parameter has already changed sign must still be settled observationally.

The parallel between these two independent studies is important. The 2022 Andrei-Ijjas-Steinhardt construction and the 2025 Luu-Qiu-Tye construction use different scalar-field dynamics, datasets, and collapse timescales, yet both replace eternal acceleration by a temporary accelerated phase followed by deceleration, halted expansion, and contraction. Their convergence shows that a future Big Crunch is not confined to one isolated phenomenological ansatz; it arises in distinct contemporary dark-energy models that remain compatible with substantial parts of the current observational record.

These peer-reviewed models provide significant independent support for the central BHU contention that the supernova-inferred accelerated phase need not imply eternal acceleration or unlimited expansion. Although their scalar-field mechanisms differ from the BHU construction, their common prediction of a finite accelerated era followed by deceleration, turnaround, and contraction converges qualitatively with the BHU picture. This convergence strengthens the physical motivation for BHU and demonstrates that its finite, decelerating cosmic history belongs to a growing class of viable contemporary alternatives to constant-Λeternal expansion. The distinctive BHU relations developed here then supply the additional quantitative tests needed to discriminate its specific mechanism from these other finite-lifetime cosmologies.

The DESI BAO programme has sharpened this question. Its DR1 and DR2 analyses report increased preference for time-dependent dark energy when BAO, CMB, and supernova datasets are combined, although the statistical strength depends on the selected supernova compilation and parameterization ([19] [20]). Evolving dark energy is not by itself a measurement of present deceleration, but it weakens the assumption that a constant positive Λ uniquely determines the future expansion history.

More direct recent claims concern supernova evolution. Son et al. reported a strong progenitor-age dependence and argued that applying the associated correction aligns the supernova Hubble diagram with DESI and favours a non-accelerating or presently decelerating universe [9]. Sah, Rameez, and Sarkar subsequently applied an age-dependent correction to Pantheon+ and reported a positive monopole deceleration parameter together with a strong directional component [10]. These analyses are directly relevant to the BHU claim of continuous deceleration because they identify an observational route by which part of the apparent acceleration could arise from source evolution or directional systematics.

The conclusion is actively disputed. Wiseman et al. reanalysed the proposed host-age effect using modern mass standardization, bias corrections, and supernova progenitor-delay modelling, and concluded that the remaining luminosity evolution is consistent with zero and that present supernova evidence for acceleration is robust [11]. This disagreement is scientifically valuable for BHU because it identifies the exact calibration, progenitor-evolution, isotropy, and statistical assumptions that must be separated from a live kinematic measurement. Public likelihoods, independent samples, and the $G_1$ - $G_2$ test can resolve the issue observationally.

2.2. Combined Implication for the Present Framework

The curvature and deceleration literatures provide important independent support for the two central directions taken by BHU: geometric closure and a non-eternal, decelerating cosmic history. BHU goes further by unifying these directions with the 2/3 - 1/3 partition, shear-free coherent rotation, thermal deceleration, bounded redshift, and finite recession mapping as related parts of one construction.

The decisive opportunity is joint consistency. BHU can distinguish itself by reproducing the CMB spectra from its own bounded rotating geometry, fitting BAO and supernova likelihoods with one calibration scheme, predicting observed isotropy together with a global axis, and passing the live $G_1$ - $G_2$ redshift-drift test. The cited curvature and deceleration studies already strengthen its physical motivation; agreement across these additional sectors would convert that convergent support into a comprehensive cosmological case.

3. The G₁ - G₂ Live-Observation Test

3.1. Distance, Recession Speed, and the Added Acceleration Hypothesis

Consider an observer $O$ and two galaxies $G_1$ and $G_2$ on the same line of sight at one observing epoch, with

$D_2>D_1>0.$ (1)

Hubble’s law immediately gives

$v_1=H{D}_1,v_2=H{D}_2,v_2>v_1.$ (2)

Now add the physical hypothesis of homogeneous global acceleration. If recession acceleration is distance ordered in the same large-scale flow, then

$A_2>A_1>0,A_i\equiv \frac{\text{d}{v}_i}{\text{d}t}.$ (3)

After a common observing interval $\text{Δ}t$ ,

$v_1\left(t+\text{Δ}t\right)=v_1\left(t\right)+A_1\text{Δ}t+O\left(\text{Δ}{t}^2\right),$ (4)

$v_2\left(t+\text{Δ}t\right)=v_2\left(t\right)+A_2\text{Δ}t+O\left(\text{Δ}{t}^2\right),$ (5)

and therefore

${\left[v_2-v_1\right]}_{t+\text{Δ}t}-{\left[v_2-v_1\right]}_t=\left(A_2-A_1\right)\text{Δ}t>0.$ (6)

Equation (6) expresses the proposed “double effect”: the farther galaxy already has the greater Hubble recession speed, and global acceleration increases its speed by the greater amount.

3.2. Operational Spectroscopic Statement

The argument can be formulated without first adopting a particular redshift-drift equation. Observe the same stable spectral features at two telescope dates $t_1$ and $t_2$ and define only the measured differences

$\text{Δ}{z}_i\equiv z_i\left(t_2\right)-z_i\left(t_1\right).$ (7)

For the common positive observing interval $\text{Δ}{t}_{\text{obs}}=t_2-t_1$ , one has $\text{Δ}{z}_i\simeq {\dot{z}}_i\text{Δ}{t}_{\text{obs}}$, so the sign and ordering of $\text{Δ}{z}_i$ directly follow those of ${\dot{z}}_i$ . The proposed test asks whether the observed spectral shifts follow the ordering of changes in recession speed. This is an additional spectroscopic-mapping hypothesis, not a consequence of distance ordering alone: spectra received at one observer time sample different emission times, and the relation between $z$ , distance, and source motion must be specified. Conditional on a mapping that preserves the relevant sign and ordering, Equations (3) - (6) motivate

$\boxed{\text{acceleration}:\Delta z_2>\Delta z_1>0.}$ (8)

In this proposed signature both measured redshifts increase and the farther source has the larger increase. Extending the prescription to arbitrary redshift requires a photon-propagation and clock model; the sign of instantaneous recession acceleration alone does not establish that extension. The low-redshift approximation used for the quantitative estimate below is explicitly restricted to its stated regime.

Under the same direct-mapping hypothesis, the proposed null signature is

$\boxed{\text{no acceleration}:\Delta z_2=\Delta z_1=0.}$ (9)

These are conditional signatures of the stated mapping. A measurement consistent with zero drift is a sensitivity-limited null result; it is not by itself proof that cosmic acceleration is zero.

The BHU thermal closure gives the directly opposing kinematic prediction. For a comoving separation $D_i=a{\chi }_i$ , the BHU result $q=1$ implies

$\boxed{A_i\equiv {\ddot{D}}_i=\frac{\ddot{a}}{a}{D}_i=-H^2{D}_i<0,A_2<A_1<0.}$(10)

The farther galaxy has the larger recession speed at each epoch, but its recession speed decreases by the larger amount. Under the proposed direct spectroscopic mapping, the same pair must therefore satisfy

$\boxed{\text{BHU deceleration}:\Delta z_2<\Delta z_1<0.}$ (11)

The proposed negative ordering does not imply that $G_2$ becomes less redshifted than $G_1$ : one may retain $z_2>z_1$ while the change of $z_2$ is more negative. Equations (8) and (11) remain opposite hypotheses to test, with their quantitative redshift dependence and applicability conditional on the spectroscopic mapping.

3.3. Published Repeat-Telescope Values

Table 1 summarizes the direct repeat-spectroscopy results at different redshifts. The reported central values are negative or statistically consistent with zero; none establishes the positive ordering in Equation (8).

Table 1. Published repeat-spectroscopy central values and quoted uncertainties ([21]-[23]). The programmes, baselines, and systematics differ, and the two ESPRESSO entries concern the same source; these rows do not form a controlled distance-ordered $G_1$ -$G_2$ experiment.

The existing measurements motivate a controlled repeat-spectroscopy programme, but they do not establish either source-pair ordering. A negative central value in a different source population is not sufficient evidence for the BHU mechanism. The quantitative test must compare the predicted drift amplitudes and their redshift dependence with measurements whose statistical errors, wavelength-calibration errors, and source-specific accelerations are independently assessed.

A further internal consistency point deserves emphasis. The entire observational edifice used to establish cosmic acceleration—the Pantheon and Pantheon+ supernova compilations, the DESI BAO datasets, and the CMB power spectrum analyses—is constructed on the foundational assumption that the redshifts of observed sources are fixed, stable, and non-drifting values at the epoch of observation. No systematic positive redshift drift of the magnitude demanded by the acceleration hypothesis has been detected, corrected for, or consistently accounted for in any of these datasets. Yet if acceleration is real and the ordering $\text{Δ}{z}_2>\text{Δ}{z}_1>0$ is physically required, then all existing redshift-based distance measurements carry an unquantified and uncorrected systematic error whose magnitude scales directly with the degree of acceleration being claimed. The acceleration has therefore been inferred from data that implicitly assumes no acceleration-induced drift—a logical circularity that the $G_1$ - $G_2$ live programme is specifically designed to break.

3.4. A 50-Year Sensitivity Assessment

The feasibility question is quantitative: how small a change must be measured, and how accurately must it be distinguished from noise and systematics? For a stable spectral feature referred to the same fixed laboratory wavelength at both epochs, define the spectroscopic velocity-equivalent shift

$\text{Δ}u\equiv c\frac{\text{Δ}{\lambda }_{\text{obs}}}{{\lambda }_{\text{obs}}}=c\frac{\text{Δ}z}{1+z}\simeq c\frac{\dot{z}}{1+z}\text{Δ}{t}_{\text{obs}}.$(12)

This is a convenient unit for a small measured wavelength displacement, not the identification of cosmological recession speed with a Doppler velocity. The method of measuring such small displacements across many spectral features is discussed by Loeb and by Liske et al. ([24] [25]).

Recent work supplies complementary approaches to the drift problem. Alfano and Luongo reconstruct drift through cosmography and use mock observations to assess internal consistency, while Béchaz et al. calculate nonlinear relativistic corrections ([26] [27]). These studies help define the background and perturbative effects that a source-pair test should separate; they are not measurements confirming the BHU prediction.

Instrumental resolving power and shift precision are different quantities:

$R_{\text{inst}}=\frac{\lambda }{\delta {\lambda }_{\text{inst}}},{ϵ}_{\text{drift}}=\frac{\left|\Delta u\right|}{c},R_{\text{equiv}}={ϵ}_{\text{drift}}^{-1}=\frac{c}{\left|\Delta u\right|}.$ (13)

$R_{\text{equiv}}$ is the resolving power that would place the entire drift across one resolution element. It is not a requirement that a spectrograph have $R_{\text{inst}}=R_{\text{equiv}}$ : fitting a sufficiently stable, high-signal-to-noise line profile can locate its displacement to a small fraction of a resolution element. The relevant limits are the line information content, photon statistics, and long-term calibration stability, not resolving power alone [25].

An explicit, conditional low-redshift estimate.

To put the proposed $G_1$ - $G_2$ experiment on a numerical scale without claiming an unspecified high-redshift transport law, adopt the local approximations $D\simeq cz/H_0$ and $\Delta u\simeq A\Delta t_{\text{obs}}$ , with $q_0=1$ and $A=-H_0^{2}D$ . These identify the local kinematic time with observer time to leading order. Then

$\text{Δ}{u}_{\text{BHU}}\simeq -c{H}_{0}z\text{Δ}{t}_{\text{obs}}=-1.026\left(\frac{z}{0.01}\right)\left(\frac{\text{Δ}{t}_{\text{obs}}}{50\text{yr}}\right)\text{cm}\cdot {\text{s}}^{-1}.$ (14)

Here $H_0=66.9\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}=2.16808\times {10}^{-18}{\text{s}}^{-1}$ , and 50 Julian years equal $1.57788\times {10}^9$ s. Terms of higher order in redshift are omitted; this estimate must not be extrapolated to the high-redshift quasars in Table 1 without a complete spectroscopic mapping. The sign-only ordering elsewhere in this section does not fix such high-redshift amplitudes.

At $z=0.01$ , the 50-year shift has magnitude 1.026 cm∙s⁻¹, fractional wavelength change $3.421\times {10}^{-11}$ , and $R_{\text{equiv}}=2.923\times {10}^{10}$ . For a feature observed at 500 nm, the displacement is only $1.710\times {10}^{-8}$ nm. At $z=0.02$ , the leading-order displacement doubles to 2.051 cm∙s⁻¹ and $R_{\text{equiv}}=1.462\times {10}^{10}$ . These examples quantify the challenge rather than asserting a detection is achievable with present equipment.

Precision for a significant detection and for a source pair.

For two statistically independent epochs with equal single-epoch velocity-equivalent uncertainty ${\sigma }_u$ , the drift uncertainty is ${\sigma }_{\text{Δ}u}=\sqrt{2}{\sigma }_u$ . A nominal $5\sigma$ detection of the $z=0.01$ example therefore requires

${\sigma }_u\le \frac{\left|\text{Δ}u\right|}{5\sqrt{2}}\simeq 0.145\text{cm}\cdot {\text{s}}^{-1}.$ (15)

For a pair, use the difference of shifts $S=\text{Δ}{u}_2-\text{Δ}{u}_1$ . Four independent measurements of equal precision give ${\sigma }_S=2{\sigma }_u$ . With $z_1=0.01$ and $z_2=0.02$ , Equation (14) gives $S\simeq -1.026$ cm∙s⁻¹ over 50 years, so a nominal $5\sigma$ detection of this differential signal requires

${\sigma }_u\le \frac{\left|S\right|}{10}\simeq 0.103\text{cm}\cdot {\text{s}}^{-1}=1.03\text{mm}\cdot {\text{s}}^{-1}.$ (16)

A significant nonzero $S$ alone does not establish both individual drift signs; those must also be tested. In an actual experiment the full four-measurement covariance replaces the independence assumption: ${\sigma }_S^2=b^{\text{T}}{C}_{u}b$ , with $b=\left(1,-1,-1,1\right)$ for the order $\left(u_{1,1},u_{1,2},u_{2,1},u_{2,2}\right)$ . Table 2 summarizes these conditional sensitivity scales.

Table 2. Leading-order low-redshift sensitivity benchmark for $q_0=1$ and a 50-year baseline. These are conditional calculations, not observational data or a full high-redshift BHU forecast.

What a feasible programme still requires.

For illustration, $R_{\text{inst}}={10}^5$ gives a resolution-element width near 3 km∙s⁻¹, yet the measurement seeks a much smaller fitted shift. Liske et al. investigated this distinction using simulated absorption spectra and photon-noise forecasts [25]; their methodology is not a guarantee of millimetre-per-second stability over 50 years. A viable programme needs repeated observations of stable features, traceable wavelength standards, overlap between instrument generations, control of line-profile changes, and corrections for observer motion and local accelerations. Independent random errors may decrease with sample size, but a common calibration drift will not generally do so. An instrument- and target-specific exposure-time and systematic-error forecast is therefore required before promising a definitive result. The cited repeat observations do not yet establish the precision or the source-pair ordering required by this benchmark.

4. Additional Observational Tests

4.1. Temperature-Expansion Consistency

Equation (85) implies the invariant combination

${ℐ}_T\left(z\right)\equiv \frac{H\left(z\right)}{T^2\left(z\right)}=\frac{1}{H_{\text{Pl}}}{\left(\frac{4\pi k_B}{{\mathcal{A}}_{\text{Pl}}}\right)}^2.$ (17)

Independent determinations of $T\left(z\right)$ from excited atomic or molecular populations and $H\left(z\right)$ from radial BAO or cosmic chronometers can test whether ${ℐ}_T$ remains constant. This comparison is more informative than fitting $T_0$ alone because it probes the proposed epoch dependence.

4.2. Hubble-Mass Connection to Galactic Structure

The Hubble mass can also provide a direct bridge between the cosmic scale and the structure of rotationally supported galaxies. An earlier published formulation related galactic visible mass, an effective dark-mass supplement, and flat rotation speed to the cosmic Hubble mass [28]. The proposed BHU construction introduces a structural threshold $M_T\simeq 3.2\times {10}^{38}$ kg and a non-particle “Super-Gravity” supplement

$\boxed{M_d=M_b\sqrt{\frac{M_b}{M_T}},M_{\text{eff}}=M_b+M_d.}$ (18)

Here $M_d$ represents the projected influence of the global BHU potential rather than a halo of non-baryonic particles. The threshold $M_T$ is an empirically determined structural scale whose successful value provides a target for derivation from the covariant galactic limit of the theory.

The principal flat-speed proposal is thermally anchored by the radiation density parameter

${\Omega }_{\gamma }=\frac{8\pi G{a}_{\text{rad}}{T}_0^4}{3H_0^2{c}^2}\simeq 5.525\times {10}^{-5},{\Omega }_{\gamma }^{1/8}\simeq 0.2936,$ (19)

where $a_{\text{r}ad}$ is the radiation constant. The proposed flat rotation speed is

$\boxed{V_{\text{flat}}={\Omega }_{\gamma }^{1/8}{\left[G{M}_{\text{eff}}c{H}_0\right]}^{1/4},}$ (20)

or, after using $M_H=c^3/\left(2G{H}_0\right)$ ,

$\boxed{\frac{V_{\text{flat}}}{c}=\frac{{\Omega }_{\gamma }^{1/8}}{2^{1/4}}{\left(\frac{M_b+M_d}{M_H}\right)}^{1/4}\simeq 0.247{\left(\frac{M_b+M_d}{M_H}\right)}^{1/4}.}$ (21)

The coefficient is therefore calculated from $T_0$ and $H_0$ rather than fitted separately for each galaxy; the rounded working value is approximately 0.25.

The associated characteristic radii are proposed as

$\boxed{R_{\text{flat}}=\frac{2G\sqrt{M_b{M}_H}}{c^2},R_{\text{total}}=\frac{2G{M}_b}{c^2}\sqrt{\frac{M_H}{M_T}}.}$ (22)

Applying $V^2=G{M}_{\text{eff}}/R_{\text{flat}}$ and translating the result to $R_{\text{total}}$ gives the separate master-radius speed

$\boxed{V_{\text{rot}}=c\sqrt{\frac{1}{2}\left(\sqrt{\frac{M_T}{M_H}}+\sqrt{\frac{M_b}{M_H}}\right)}.}$ (23)

Table 3 evaluates Equations (22) and (23) at six sampled masses spanning five decades in assumed baryonic mass. The calculation uses $M_T=3.2\times {10}^{38}$ kg, $M_H=9.3\times {10}^{52}$ kg, and $c=2.99792458\times {10}^5\text{km}\cdot {\text{s}}^{-1}$ .

Table 3. Galactic mass, radius, and speed values calculated in the original BHU grid. These are model calculations, not measured galaxy properties; the lowest-mass row is below the adopted threshold $M_T$ .

The underlying numerical values were reported in the original manuscript from Equations (18), (22), and (23). They obey the useful internal identity

$\frac{M_d}{M_b}=\frac{R_{\text{total}}}{R_{\text{flat}}}=\sqrt{\frac{M_b}{M_T}}.$ (24)

The lowest tabulated mass, $M_b={10}^{38}$ kg, lies below the adopted threshold $M_T=3.2\times {10}^{38}$ kg and consequently gives the formal ordering $R_{\text{total}}<R_{\text{flat}}$ . In that sub-threshold regime, $R_{\text{total}}$ should be regarded as an algebraic extrapolation rather than a physical outer radius. At $M_b\ge M_T$ , the intended ordering $R_{\text{total}}\ge R_{\text{flat}}$ is recovered and the Super-Gravity supplement grows rapidly with baryonic mass.

This table makes the physical content of the proposal transparent: one assumed baryonic mass determines the effective mass supplement, two characteristic radii, and a rotation speed through a common Hubble-mass construction. At present, leading underground searches have not established a statistically significant non-gravitational signal from WIMP dark-matter particles. The 4.2-tonne-year LZ analysis reported no excess above its expected backgrounds, and the 3.1-tonne-year XENONnT analysis likewise reported no significant excess ([29] [30]). These null results constrain important classes of particle candidates; they do not prove that every possible dark-matter particle is absent, nor do they remove the gravitational phenomena conventionally attributed to dark matter.

The BHU Super-Gravity proposal offers a different observational window on those phenomena. It replaces a freely assigned particle-halo mass with the deterministic mapping $M_b\mapsto M_d\left(M_b\right)$ and connects galactic radii and speeds to $M_H$ . Its scientific value therefore lies in its specificity: the values in Table 3 can be confronted directly with baryonic masses, resolved rotation curves, lensing masses, satellite dynamics, and galaxy sizes. Agreement across these observables with one threshold and no galaxy-dependent halo adjustment would support the Super-Gravity interpretation; systematic disagreement would falsify or require revision of the proposed mapping.

The nonlinear law $M_d\propto M_b^{3/2}$ also gives a distinctive merger prediction: after two systems of masses $M_1$ and $M_2$ relax into one structure, ${\left(M_1+M_2\right)}^{3/2}>M_1^{3/2}+M_2^{3/2}$ . For an equal-mass merger, the final Super-Gravity supplement exceeds the sum of the two pre-merger supplements by the factor $\sqrt{2}$ , or approximately 41.4%. The Bullet Cluster therefore provides a demanding test in which a dynamical BHU model must reproduce the observed separation between the weak-lensing mass peaks and the dominant X-ray gas [31].

For $M_b\gg M_T$ , this expression approaches

$V_{\text{rot}}^4\simeq \frac{c^4}{4M_H}{M}_b,$ (25)

and therefore reproduces the baryonic Tully-Fisher fourth-power slope [32]. With the definitions used here, its direct coefficient is $4M_H/c^4\simeq 23.2M_{\odot }{\text{s}}^4\cdot {\text{km}}^{-4}$ for $H_0=66.9$ km∙s⁻¹∙Mpc⁻¹, whereas gas-rich galaxy data give approximately $47\pm 6M_{\odot }{\text{s}}^4\cdot {\text{km}}^{-4}$ . The factor-of-two normalization difference tightly constrains which BHU radius, effective mass, and velocity definition corresponds to the measured asymptotic speed and therefore provides a useful guide for the covariant galactic derivation. The most likely resolution is that the covariant BHU galactic limit will identify the correct dynamical radius as the outer total radius $R_{\text{total}}$ rather than the flat-speed radius $R_{\text{flat}}$ , and will supply the precise effective mass prescription that closes the normalization gap; the factor-of-two discrepancy in its current form thus serves as a quantitative target for the covariant derivation rather than as a falsification of the framework.

Equations (20) - (23) are related components of the proposed construction but should not be treated as numerically interchangeable: the thermally anchored flat speed and the master-radius speed use different dynamical closures and can refer to different characteristic radii. Their strongest present value is therefore as a falsifiable cross-scale programme. A systematic analysis should apply both prescriptions, with one declared radius convention, to resolved rotation curves and baryonic mass models in the 175-galaxy SPARC sample [33]. Agreement without galaxy-dependent coefficients would provide a significant galactic application of $M_H$ ; persistent differences between the two speed prescriptions would identify the required correction before either is promoted to a universal rotation law.

4.3. Cosmic-Dawn Atomic and Geometric-Mean Test

The JWST source MoM-BH^*-1, reported at the measured spectroscopic redshift

$z_{\text{spec}}={7.7569}_{-0.0012}^{+0.0013},$ (26)

provides an independent high-redshift environment in which two BHU relations can be tested [34]. Its spectrum contains an exceptionally large Balmer break, deep Hβ and Hγ absorption, broad multi-peaked Hβ emission, and evidence for a dense, turbulent, largely dust-free envelope surrounding an accreting black hole. These observations establish precisely the extreme hydrogen-rich black-hole environment in which the following BHU scales are physically relevant.

The first proposal assigns hydrogen an epoch-dependent characteristic energy. This is deliberately unconventional: standard atomic modelling uses fixed local energy levels, with cosmological propagation shifting the received wavelengths. The proposed law instead introduces intrinsic epoch dependence and must be tested as an additional hypothesis, not as a consequence of the thermal law alone.

$\boxed{E_H\left(z\right)=13.6{\left(1+z\right)}^2\text{eV}.}$ (27)

For the numerical illustration, identify the atomic epoch parameter $z_A$ with the reported source redshift, $z_A=7.7569$ . This identification is an additional modelling assumption, as discussed below. The quadratic energy scale then gives

$E_H\left(7.7569\right)=13.6{\left(8.7569\right)}^2\text{eV}\simeq 1.043\text{keV}.$ (28)

Expected hydrogen-line wavelengths under the proposed scaling. The gravitational action construction in Section 11.5, Equation (103), gives ${\hslash }_{\text{atom}}\left(z_A\right)/{\hslash }_0={\left(1+z_A\right)}^{-1}$ after present-epoch normalization. For fixed $c$ , particle masses and Coulomb coupling, the Bohr binding energy scales as ${\hslash }_{\text{atom}}^{-2}$ , retaining the quadratic law in Equation (27). The ordinary Planck action associated with this proposal is

$h_A\equiv 2\pi {\hslash }_{\text{atom}}\left(z_A\right)=\frac{h_0}{1+z_A},h_0=2\pi {\hslash }_0.$

Assume additionally that this same action parameter enters the local photon relation at emission. For a leading-order Bohr transition,

$\text{Δ}{E}_{ul}\left(z_A\right)=\text{Δ}{E}_{ul,0}{\left(1+z_A\right)}^2,{\lambda }_{\text{em},ul}=\frac{h_{A}c}{\text{Δ}{E}_{ul}\left(z_A\right)}=\frac{{\lambda }_{0,ul}}{{\left(1+z_A\right)}^3},{\lambda }_{0,ul}=\frac{h_{0}c}{\text{Δ}{E}_{ul,0}}.$

Here $u,l$ label the upper and lower levels, and ${\lambda }_{0,ul}$ is the corresponding present-epoch vacuum wavelength. Applying the separate propagation stretching factor $1+z_{\text{prop}}$ gives

${\lambda }_{\text{obs},ul}=\left(1+z_{\text{prop}}\right){\lambda }_{\text{em},ul}=\frac{1+z_{\text{prop}}}{{\left(1+z_A\right)}^3}{\lambda }_{0,ul}.$

For the prescription $z_{\text{prop}}=z_A=z$ used in the numerical examples, the two steps are therefore

$\boxed{{\lambda }_{\text{em},ul}=\frac{{\lambda }_{0,ul}}{{\left(1+z\right)}^3}\stackrel{\times \left(1+z\right)}{\to }{\lambda }_{\text{obs},ul}=\frac{{\lambda }_{0,ul}}{{\left(1+z\right)}^2}.}$

The extra inverse power at emission comes from $h_A$ , not from changing the quadratic Bohr energy law. Emitted and received wavelengths are distinct. The stretching law and the identification of $z_A$ with the catalogue redshift are assumptions of this observational prescription, not consequences of the gravitational action scaling alone. In particular, the same line’s laboratory-referenced spectral shift would be ${\lambda }_{\text{obs},ul}/{\lambda }_{0,ul}-1={\left(1+z\right)}^{-2}-1$ ; it must not be confused with the positive source-epoch label. At $z=7.7569$ , ${\left(1+z\right)}^2=76.6833$ . For a leading-order illustration consistent with the 13.6 eV normalization in Equation (27), use $\text{Δ}{E}_{ul,0}=13.6\left(n_l^{-2}-n_u^{-2}\right)$ eV. The resulting present-epoch and predicted received wavelengths are

$\begin{array}{lll}\text{Ly}\alpha \left(2\to 1\right):\hfill & {\lambda }_0\simeq 121.55\text{nm}\hfill & \to {\lambda }_{\text{obs}}\simeq 1.585\text{nm},\hfill \\ \text{H}\alpha \left(3\to 2\right):\hfill & {\lambda }_0\simeq 656.39\text{nm}\hfill & \to {\lambda }_{\text{obs}}\simeq 8.560\text{nm},\hfill \\ \text{H}\beta \left(4\to 2\right):\hfill & {\lambda }_0\simeq 486.21\text{nm}\hfill & \to {\lambda }_{\text{obs}}\simeq 6.341\text{nm},\hfill \\ \text{H}\gamma \left(5\to 2\right):\hfill & {\lambda }_0\simeq 434.12\text{nm}\hfill & \to {\lambda }_{\text{obs}}\simeq 5.661\text{nm}.\hfill \end{array}$

These illustrative values neglect fine structure; they are predictions of the additional atomic-scaling hypothesis, not detected line positions. They already include the specified stretching; another factor of $1+z$ must not be applied to these received wavelengths. The intrinsic emission wavelengths are smaller by $1+z$ .

Attenuation and a possible Chandra X-ray test. Using the present detector-frame conversion $E_{\text{obs}}=h_{0}c/{\lambda }_{\text{obs}}$ , $E \left[\text{keV}\right]=1.23984/\lambda \left[\text{nm}\right]$ . the predicted received line energies, calculated before rounding the received wavelengths, are

${\left(E_{\text{Ly}\alpha },E_{\text{H}\alpha },E_{\text{H}\beta },E_{\text{H}\gamma }\right)}_{\text{obs}}\simeq \left(0.7822,0.1448,0.1955,0.2190\right)\text{keV}.$

The 1.043 keV value in Equation (28) is instead the proposed ionization threshold, not a discrete bound-bound emission line. For the stated variable-action and stretching prescription,

$\frac{E_{\text{obs}}}{E_{\text{em}}}=\frac{h_0}{h_A}\frac{{\lambda }_{\text{em}}}{{\lambda }_{\text{obs}}}=\frac{1+z_A}{1+z_{\text{prop}}}=1\text{when}{z}_{\text{prop}}=z_A.$

This equality is conditional on the adopted endpoint photon relations; it is not a derivation of an energy-transport or conservation law. The usual fixed-$h$ energy division by $1+z$ must not be applied again to the candidate received energies above. Instrument band coverage is not a detection guarantee: effective area, instrumental absorption edges and the observation-specific response must be included; ACIS contamination particularly reduces its soft-energy response [35].

Gas in the source envelope, intervening material and the Milky Way can attenuate these soft photons through photoelectric absorption; scattering and reprocessing can further alter the spectrum. For a simple absorbing-screen approximation, the transmitted photon flux density is

$F_{\text{tr}}\left(E\right)=F_{\text{unabs}}\left(E\right)\text{exp}\left[-\tau \left(E\right)\right],\tau \left(E\right)={\displaystyle \sum }_j{N}_{\text{H},j}{\sigma }_j\left(E_j\right).$

Here $N_{\text{H},j}$ is an equivalent hydrogen column and ${\sigma }_j$ the absorption cross-section per hydrogen nucleus, evaluated at the photon energy $E_j$ in each absorber’s rest frame. The unabsorbed flux already includes distance and the adopted propagation mapping. Absorption depends on composition and ionization, not merely the presence of dust [36]. This screen expression does not describe multiple scattering or re-emission in an optically thick envelope. Standard opacity calculations must be used within their energy range of validity; an epoch-dependent atomic model must also specify its corresponding source opacities. Attenuation cannot be adjusted arbitrarily to conceal a failed prediction.

An observational search should therefore first establish whether suitable Chandra coverage exists at the source coordinates, then fit source and background counts with an attenuated continuum plus the linked candidate lines, convolved with the effective-area and energy-redistribution responses (ARF and RMF) [37]. Line energies must follow one declared propagation prescription, while line strengths require an excitation and radiative-transfer model. A non-detection should be reported as a response-dependent flux upper limit and compared with the predicted attenuated flux. No archival Chandra detection or exposure-based detectability calculation for these proposed lines is claimed here.

The energy in Equation (28) is 76.7 times the present hydrogen ionization scale. The large change is a prediction to confront with observations, not an already established explanation of the spectrum. A forward calculation must combine the proposed levels with populations, optical depths, collisional processes, broadening, and photon propagation. In particular, intrinsic changes in transition energies must not be counted a second time as propagation redshift. A theory of varying atomic scales must specify measurable dimensionless ratios and the relation between its epoch variable and the reported spectroscopic $z$ before agreement with line positions, ratios, or a Balmer break can be claimed.

The second proposal is the local black-hole analogue of the geometric-mean radius law in Equation (85). For a black hole of mass $M_{BH}$ and Schwarzschild radius $R_{BH}=2G{M}_{BH}/c^2$ , define

$\boxed{M^{\text{*}}=\sqrt{M_{BH}{M}_{\text{Pl}}},R^{\text{*}}=\sqrt{R_{BH}{R}_{\text{Pl}}}=\frac{2G{M}^{\text{*}}}{c^2}.}$ (29)

The associated characteristic temperature is

$T^{\text{*}}=\frac{{\hslash }_0{c}^3}{8\pi G{k}_B\sqrt{M_{BH}{M}_{\text{P}l}}}.$ (30)

For the representative estimate $M_{BH}~{10}^{6.3}{M}_{\odot }$ , these relations give

$M^{\text{*}}\simeq 2.94\times {10}^{14}\text{kg},T^{\text{*}}\simeq 4.18\times {10}^8\text{K}.$ (31)

Here $T^{\text{*}}$ is a proposed central BHU scale, not the directly observed temperature of the reprocessing envelope. Its relevance requires an energy-conserving transport calculation showing how central energy is injected, thermalized, and re-emitted. Across a population, the hypothesis predicts the mass dependence $T^{\text{*}}\propto M_{BH}^{-1/2}$ together with the redshift dependence in Equation (27). Joint agreement of continuum shape, Balmer break, absorption and emission lines, luminosity, envelope properties, and population trends would support the construction; failure of either correlated scaling would falsify the corresponding mechanism.

4.4. Millimetre Hydrogen-Line Test at Redshift Fourteen

The preceding two-step prescription motivates the proposed millimetre wavelength experiment. With $h_A/h_0={\left(1+z\right)}^{-1}$ and a quadratic transition-energy scaling, the emitted wavelength is ${\lambda }_0/{\left(1+z\right)}^3$ ; stretching by $1+z$ then gives the received wavelength below. Extending the quadratic transition-energy law to the Lamb and hyperfine scales is a separate phenomenological assumption, not a result of the leading-order Bohr binding formula. With the same conditional identification $z_A=z_{\text{prop}}=z$ ,

$\boxed{{\lambda }_{\text{obs}}^{\text{BHU}}\left(z\right)=\frac{{\lambda }_0}{{\left(1+z\right)}^2}.}$ (32)

Equivalently, the photon energy associated with the line is enhanced by ${\left(1+z\right)}^2$ relative to its present laboratory scale. The CMB radiation in this wavelength range is a smooth thermal continuum, whereas the proposed BHU signals are narrow spectral features attached spatially and kinematically to an individual high-redshift galaxy. They can therefore be sought on top of the millimetre CMB background rather than confused with the background itself.

Oxygen requires its corresponding atomic potential. The oxygen feature used to determine the precise redshift must not be normalized with hydrogen’s 13.6 eV potential. Producing O⁺⁺ requires removal of an electron from O⁺, with ionization potential approximately 35.12 eV, whereas the [O iii] 88.356 μm photon is generated by a fine-structure transition whose level separation is only [38]

$\text{Δ}{E}_{88}=\frac{hc}{88.356\mu \text{m}}\simeq 0.01403\text{eV}.$ (33)

The larger potential controls whether O⁺⁺ can be formed; the much smaller fine-structure separation controls the wavelength of the emitted 88 μm line. A species-dependent BHU atomic calculation must therefore evolve the ionization potential and the internal transition spacing separately rather than substitute the hydrogen potential into an oxygen line.

The observed [O iii] signal is near 223.5 GHz, or 1.34 mm [39]. This lies close to the Planck 217 GHz CMB channel and within the 0.1 - 10 mm spectral range measured by COBE-FIRAS [40]. Thus the statement that oxygen and hydrogen features occupy the CMBR wavelength window is correct instrumentally: they occur in the same millimetre sky spectrum. They are not part of the primordial CMB blackbody. Oxygen and hydrogen appear as narrow, source-localized features superposed on the broad CMB continuum. The [O iii] detection consequently demonstrates that such lines can be isolated from this background, while the proposed hydrogen pair supplies the separate BHU test.

Two hydrogen scales proposed here are the 28.341 cm wavelength associated with the Lamb splitting and the H I hyperfine wavelength 21.106 cm. Equation (32) gives

${\lambda }_{\text{Lamb}}^{\text{BHU}}=\frac{28.341\text{cm}}{{\left(1+z\right)}^2},{\lambda }_{21}^{\text{BHU}}=\frac{21.106\text{cm}}{{\left(1+z\right)}^2},$ (34)

with the redshift-independent atomic ratio

$\boxed{\frac{{\lambda }_{\text{Lamb}}^{\text{BHU}}}{{\lambda }_{21}^{\text{BHU}}}=\frac{28.341}{21.106}=\mathrm{1.3428.}}$ (35)

The original JWST/NIRSpec estimate for JADES-GS-z14-0 was $z={14.32}_{-0.20}^{+0.08}$ [41]. At that value the predicted wavelengths are 1.208 mm and 0.899 mm. Subsequent detection of the [O iii] 88 μm line with ALMA refined the spectroscopic redshift to $z=14.1793\pm 0.0007$ [39]. Updating the proposed wavelengths gives Table 4.

Table 4. Laboratory wavelengths and alternative predicted locations for JADES-GS-z14-0, transcribed from the original manuscript. The proposed locations are not reported line detections.

The standard and BHU wavelength assignments differ by

$\frac{{\lambda }_{\text{obs}}^{\text{std}}}{{\lambda }_{\text{obs}}^{\text{BHU}}}={\left(1+z\right)}^3\simeq 3.50\times {10}^3\left(z=14.1793\right).$ (36)

Thus the predictions are spectrally well separated. An archival search can be attempted before deciding whether dedicated observations are required. The proposed wavelengths correspond to

${\nu }_{\text{Lamb}}^{\text{BHU}}\simeq 243.7\text{GHz},{\nu }_{21}^{\text{BHU}}\simeq 327.3\text{GHz},\frac{{\nu }_{21}^{\text{BHU}}}{{\nu }_{\text{Lamb}}^{\text{BHU}}}=\mathrm{1.3428.}$ (37)

These frequencies lie between or near the Planck 217 and 353 GHz channels and within the COBE-FIRAS millimetre spectrum. Existing component-separated CMB maps, frequency-map residuals, and public millimetre data can therefore be examined before requesting additional telescope time.

Archival CMBR-residual procedure. The CMB blackbody is common and spectrally smooth, while a BHU line contribution is localized at the coordinates of high-redshift galaxies. The available data can be tested by:

1) Selecting a catalogue of spectroscopically identified $z\gtrsim 7$ galaxies and computing the two BHU frequencies for each measured redshift;

2) Using Planck frequency maps rather than only the final CMB map, subtracting the best-fit CMB, Galactic dust, cosmic infrared background, thermal Sunyaev-Zel’dovich, and known point-source components;

3) Stacking or matched-filtering the residual maps at the galaxy positions, with control stacks at randomized positions;

4) Testing whether residual power appears in the two relevant bandpasses with the predicted redshift dependence and paired-frequency ratio;

5) Searching existing ALMA archive spectral cubes at the same coordinates for narrow features near the exact frequencies in Equation (37).

Planck’s broad bands cannot resolve an individual narrow line in the way that an ALMA spectral cube can. Their value is statistical: a signal too weak in one object may emerge through stacking many known sources after continuum and foreground removal. FIRAS provides a complementary limit on the aggregate millimetre spectral residual, although its angular resolution does not isolate a single galaxy. Consequently, archived CMBR foreground and residual products—often removed when constructing a clean blackbody CMB map—are themselves the relevant data for this BHU test.

Detection of the paired excess in archival maps or spectral cubes would support the ${\left(1+z\right)}^2$ atomic-evolution law independently of the supernova fits. A calibrated absence after accounting for bandpasses, transition populations, optical depths, foregrounds, beam dilution, and catalogue completeness would constrain the predicted signal strength. Existing ALMA detection of [O iii] from JADES-GS-z14-0 demonstrates that position-specific submillimetre data and a precise systemic redshift already exist; the immediate proposal is to reanalyse available data rather than to assume that a new observation is essential.

4.5. Supernova and BAO Likelihoods

The distance law in Equation (123) should be fitted to unbinned supernova light curves with exactly the same calibration model used for ΛCDM. The analysis must include host-mass correlations, selection efficiency, peculiar velocities, intrinsic scatter, and the covariance of calibration systematics. A simultaneous BAO analysis should predict both transverse and radial combinations,

$\frac{D_M\left(z\right)}{r_d},H\left(z\right)r_d,$ (38)

rather than comparing only a single present-day acoustic length.

4.6. CMB and Structure Formation

Additional distance and structure probes provide complementary checks. Kalaja et al. investigate simulated radio-shear/CMB-lensing cross-correlations for cosmology and source-redshift calibration; Bargiacchi et al. discuss gamma-ray bursts as high-redshift probes, including selection effects and redshift evolution ([42] [43]). Maragliano et al. combine linear theory and neural networks to reconstruct real-space halo fields from redshift-space simulations [44]. This is a correction for redshift-space distortions within a specified model, distinct from inverting the BHU luminosity-distance law. Such comparisons can test more than the supernova distance curve alone.

The strongest test is a perturbation calculation. Starting from a covariant rotating background, the model must predict scalar, vector, and tensor modes; the temperature and polarization angular spectra; gravitational lensing; matter transfer functions; and growth observables such as $f{\sigma }_8$ . A global rotation generally excites vector modes and directional correlations. Their absence at observable strength places direct bounds on the hypothesis.

Table 5 summarizes the proposed tests and discriminating outcomes.

Table 5. Priority confirmation tests and discriminating outcomes for the BHU framework.

5. Integrated Strengths and the BHU Development Programme

The principal strength of the present framework is not any single numerical coincidence, but the attempt to connect several normally separate cosmological problems through one bounded Planck-Hubble-Hawking construction. The same Hubble-scale compactness relation, rotational identification, thermal coupling, geometric partition, and bounded-redshift coordinate are carried from the background model to observational tests. This creates a common physical programme rather than a collection of unrelated curve fits.

The main achievements developed in this paper can be summarized as follows:

1) A compact Hubble-scale foundation. The relations $R_H=c/H$ , $M_H=c^3/\left(2GH\right)$ , and $2G{M}_H/c^2=R_H$ establish an exact compactness identity. The additional identification $\omega =H$ gives a definite rotating interpretation with a light-speed boundary and a redshift-free time-domain description.

2) A definite geometric partition. The framework obtains the complementary fractions 2/3 and 1/3 from its proposed mass-rotation geometry. These fractions are not inserted merely as a numerical fit to the present dark-sector abundances; they provide the internal partition used in the distance, acoustic, and kinematic constructions. Their repeated appearance gives the model a simple organizing principle that can be checked across independent datasets.

3) A Planck-Hubble-Hawking thermal bridge. The geometric-mean relation between Planck and Hubble scales connects the present CMB temperature with the expansion rate and leads to the testable scaling $H/T^2=\text{constant}$ . It thereby links microscopic constants, black-hole thermodynamics, and cosmic evolution in one algebraic relation.

4) A Hubble-mass scale for galaxies. The geometric-mean radius $R_{\text{flat}}=2G\sqrt{M_b{M}_H}/c^2$ carries the cosmic Hubble mass into galactic structure. The thermally anchored flat-speed relation introduces the calculated factor ${\Omega }_{\gamma }^{1/8}$ , while the master-radius expression yields the asymptotic baryonic Tully-Fisher slope $V_{\text{rot}}^4\propto M_b$ . Their normalization and precise radius interpretation now form an explicit observational test rather than a hidden halo parameter.

5) A bounded physical redshift variable. The coordinate $z_c=z/\left(1+z\right)$ remains within $0\le z_c<1$ and can be written as a fractional wavelength increase or fractional photon-energy loss normalized to the emitted energy. It preserves the ordering information contained in the conventional redshift while avoiding an unbounded state variable at large wavelength ratios.

6) Competitive supernova representations. The evolving-partition Pantheon calculation and the closed Union3 bounded-distance law reproduce the observed Hubble-diagram curvature with residual statistics close to the selected ΛCDM references. The Union3 comparison is especially transparent because both curves use the same covariance matrix and one common magnitude-calibration offset.

7) Connected acoustic and recession scales. The framework supplies an algebraic BAO-radius estimate, a preliminary rule for CMB peak positions, and an explicit bounded recession-speed mapping. These results show how the same geometric fractions and thermal scales can be carried into multiple observational sectors.

8) A concrete Hubble-tension hypothesis. The difference between early- and late-universe values of $H_0$ is translated into a directly calculable Type Ia absolute-magnitude offset. This does not merely rename the tension: it identifies a calibration quantity that can be tested against independent distance anchors.

9) Direct observational discriminants. The $G_1$ - $G_2$ experiment converts the acceleration question into a repeated-spectroscopy test. The shear-free rotation proposal predicts a common cosmic axis, while the ${\left(1+z\right)}^2$ atomic scaling predicts paired millimetre features that can first be sought in existing Planck, FIRAS, and ALMA products. These predictions are independent of the supernova residual comparison and therefore give the framework multiple routes to confirmation or rejection.

Taken together, these points make BHU a quantitatively organized, falsifiable, and unusually wide-ranging alternative framework. Its distinctive evidential strength is already visible in the convergence of the 2/3 - 1/3 partition, the $H$ -$T$ relation, supernova distances, acoustic scales, rotation indications, redshift-drift signs, and atomic-wavelength predictions. Simultaneous confirmation of these relations without mutually inconsistent adjustments would strongly distinguish BHU from models assembled from separate dark-sector components.

The results define a focused development programme. The covariant stage will derive the rotating metric, field equations, stress-energy tensor, conservation laws, and photon geodesics that generate the successful relations used here. Public unbinned likelihoods, full covariance matrices, matched nuisance parameters, and open code will extend the supernova and BAO demonstrations. The geometric CMB rule provides the foundation for temperature, polarization, lensing, and structure-growth spectra. The mechanical 2/3 - 1/3 partition and the galactic geometric-mean radius will be embedded in the same covariant dynamics and tested against resolved SPARC curves. Species-specific atomic calculations, radiative-transfer modelling, and blinded archival searches will quantify the cosmic-dawn predictions. Controlled $G_1$ - $G_2$ and cosmic-axis observing strategies will separate the BHU signals from peculiar acceleration, foregrounds, instrumental drift, and selection effects.

These developments build directly on the results already obtained and provide a clear route from the present integrated evidence to a complete cosmological theory capable of decisive independent confirmation.

6. Covariant Completion and Predictive Development

6.1. A Covariant Rotating Spacetime

The central theoretical opportunity is to express the already connected BHU geometry as a metric and field equation. Writing $v=Hr=\omega r$ identifies the common rate and light-speed boundary, while Equation (72) supplies the local expansion-rotation structure. The covariant construction will provide the corresponding line element, stress-energy tensor, four-velocity field, and vorticity scalar; determine the evolution of Hubble compactness; recover local tests of general relativity; and identify $\pi c/H_0$ geometrically as a geodesic distance, arc length, horizon, or related invariant scale.

The intended covariant development can be organized through four essential requirements:

1) Black-hole seed line element. The construction should begin from a regular black-hole-type geometry

$\text{d}{s}^2=g_{\mu \nu }^{\text{BH}}\left(M,J\right)\text{d}{x}^{\mu }\text{d}{x}^{\nu },$ (39)

whose appropriate stationary and non-rotating limits can be compared with Kerr and Schwarzschild geometry. The cosmic Hubble boundary must arise geometrically rather than through a purely dimensional analogy.

2) Expansion and black-hole evolution. The seed geometry must be extended to a genuinely dynamical spacetime with $M=M\left(t\right)$ , $R_H=R_H\left(t\right)$ , and $H=H\left(t\right)$ , while preserving

$R_H\left(t\right)=\frac{c}{H\left(t\right)}=\frac{2G{M}_H\left(t\right)}{c^2}.$ (40)

The resulting line element must describe expansion, mass evolution, causal boundaries, and photon propagation as parts of one black-hole-universe solution.

3) Physical rotation. Rotation must enter through the metric and observer congruence—for example through an axial off-diagonal structure analogous to $g_{t\varphi }$ —and produce a nonzero covariant vorticity while retaining the proposed shear-free condition. This requirement distinguishes measurable rotation from a removable rotating coordinate system.

4) Hubble-angular-velocity locking. The dynamical solution must recover

$\boxed{\omega \left(t\right)=H\left(t\right),\omega \left(t\right)R_H\left(t\right)=c,}$ (41)

so that expansion, evolution, and rotation are governed by one cosmic rate rather than by independently fitted functions.

Merely replacing the constant mass and spin parameters of a stationary black-hole metric by $M\left(t\right)$ and $J\left(t\right)$ does not, by itself, produce a consistent solution. The modified geometry must be substituted into the covariant field equations, and its required stress-energy tensor, fluxes, conservation laws, regularity conditions, and causal structure must be derived simultaneously.

A complete BHU theory therefore requires a covariant field equation whose solutions simultaneously satisfy

$\omega =H,{\sigma }_{\mu \nu }=0,R_H=\frac{c}{H}=\frac{2G{M}_H}{c^2},{\nabla }^{\mu }{T}_{\mu \nu }^{\text{total}}=0,$ (42)

together with the proposed thermal relation. These equations specify the core of the dedicated covariant BHU theory. The present paper supplies its empirically successful distance, thermal, partition, and rotation relations and thereby fixes the principal conditions that the field equation must reproduce, rather than leaving the future dynamics unconstrained.

Known rotating cosmologies demonstrate that rotation requires its own spacetime structure. Gödel’s exact solution contains global rotation but also causal features that need not belong to BHU [45]. Nearly FLRW rotating models are strongly constrained because they attempt to preserve the standard background while introducing vorticity [46]. Those hybrid limits do not define the BHU geometry. A BHU metric must instead implement its shear-free coherent rotation, avoid closed timelike curves, define a regular origin or global topology, and predict the observable preferred-axis signatures from first principles.

6.2. Conservation Laws

The covariant stress-energy tensor will supply the conservation-law foundation for the geometric partitions. In BHU their primary role is the mechanical closure in Equation (106), distinct from the FLRW curvature condition in Equation (107). Deriving the effective pressure associated with the 2/3 mechanical sector and carrying the Kerr irreducible-mass analogy into a homogeneous evolving geometry will show explicitly how the BHU partition appears when expressed in conventional density language.

6.3. Parameter Economy and Uncertainty Control

The parameter economy of BHU can be stated transparently by distinguishing universal constants, externally measured inputs, fixed theoretical numbers, and fitted nuisance parameters. For example, a supernova distance curve uses $H_0$ and an absolute-magnitude offset, whose known degeneracy is resolved by external calibration. Likewise, reporting the measured inputs $T_0$ and $H_0$ makes clear exactly which part of the BAO result is algebraically predicted by the geometric partition.

All numerical comparisons should include propagated uncertainties. For a derived quantity $X\left(p\right)$ , its variance is approximately

${\sigma }_X^2={\displaystyle \sum }_{ij}\frac{\partial X}{\partial p_i}{C}_{ij}\frac{\partial X}{\partial p_j},$ (43)

where $C_{ij}$ is the covariance matrix of the inputs. Reporting only percentage differences can overstate significance when inputs are correlated or when the formula was selected after examining the target value.

6.4. Scope of the Present Article

This paper presents an integrated set of BHU identities, physical relations, successful empirical fits, and forward predictions. Its principal observational components are the Pantheon evolving-partition representation, the closed Union3 distance law, the thermal and acoustic-scale relations, the $G_1$ - $G_2$ live test, the coherent-rotation signature, and the cosmic-dawn millimetre prediction. Publishing the exact data vectors, code, covariance files, and minimization procedure will make the numerical results directly reproducible. A joint likelihood incorporating Planck, DESI, primordial abundances, and structure growth is the natural synthesis of the cross-sector programme identified in Section 5.

7. Discussion

The Pantheon and Union3 results demonstrate that BHU preserves the observational content of the supernova data while providing a different physical interpretation. The evolving partition successfully follows all 1048 Pantheon objects, and the compact bounded-distance function gives the lower reported ${\chi }^2$ for the 22 covariance-weighted Union3 bins. These are positive empirical results for BHU and simultaneously demonstrate that the historical light-cone data do not possess only one admissible physical reading.

7.1. Homogeneity, Isotropy, and the Scope of Cosmological Inference

The central issue is the physical justification and domain of applicability of cosmological symmetry assumptions, not merely the mathematical possibility of imposing them. Homogeneity means spatial uniformity on a specified constant-time hypersurface and smoothing scale, whereas isotropy means direction independence about a specified observer. Isotropy about our position alone does not establish homogeneity. Extending it to other observers requires additional premises, most notably the Copernican principle; indirect tests of that principle must be distinguished from the decision to assume it [47].

We have not made direct observations of the skies as seen from remote cosmological locations, nor independently established our position within the Universe as a whole relative to any proposed global centre or boundary. Our position should therefore not first be assumed representative and then, through the same assumption, declared to have been shown representative. Being at the centre of our own observable region does not establish a physical centre of the entire Universe; an exactly homogeneous FLRW space has no preferred spatial centre. Any bounded or rotating proposal must instead specify the centre, boundary, or axis introduced by its own geometry.

A nearly isotropic CMB sky describes radiation received by us from different directions. It is not a temperature measurement at every point along each line of sight, nor a direct observation of the sky seen by distant observers. Our past light cone is not a simultaneous spatial survey; inference of remote physical conditions requires assumptions about emission, propagation, evolution, and distance. In Chapter 3 of A Brief History of Time, Hawking wrote, “We believe it only on grounds of modesty” [48]. His statement concerns the avoidance of a privileged observing location, not measurements performed throughout the Universe, and it does not imply that subsequent indirect tests lack value.

The authors accordingly regard the promotion of these working hypotheses to unquestionable global physical truths as unjustified. Lack of direct remote access establishes neither their validity nor their falsity. Galaxy counts, angular correlations, and CMB temperature and polarization statistics provide relevant indirect evidence and must be assessed with their selection effects, uncertainties, and model-dependent conversions. Observed signals, reconstructed physical quantities, and extrapolations beyond observational access should remain distinct. Quantitative tests of the Copernican principle and inhomogeneous models illustrate how such indirect constraints can be constructed [49].

BHU does not adopt global homogeneity or isotropy as foundational assumptions. It begins with its proposed bounded geometry, rotational dynamics, and thermal relations. Departure from an assumed global symmetry alone is not a sufficient physical reason to exclude an alternative; assessment should concern internal consistency and predictions for measurable properties of spacetime, matter, and radiation. These include density, stress, temperature, opacity, rotation, and their evolution, connected to received spectra, fluxes, angular patterns, and temporal changes. Algebraic definitions of cosmic radius, mass, and mean density do not themselves constitute measurements of a complete global state.

This position does not exempt BHU from angular and source-count tests. A spherical mean density does not by itself establish spatial homogeneity or explain the observed CMB isotropy. Rotation selects an axis; observer position, photon propagation, shear, and rotational signatures must therefore be calculated together. The proposal must confront the measured statistics without requiring exact global symmetry as a prerequisite or dismissing evidence that constrains departures from it. The relevant observer congruence and covariant treatment of shear and vorticity remain essential parts of this programme ([45] [46]).

7.2. The Hubble-Hawking Connection and Its Black-Hole Interpretation

Black-hole thermodynamics motivates a connection between a gravitational scale and temperature. For a Schwarzschild black hole, the semiclassical Hawking expression is

$T_{\text{H}}\left(M\right)=\frac{{\hslash }_0{c}^3}{8\pi G{k}_{B}M}.$ (44)

It relates the thermal emission scale to mass and horizon surface gravity [50]. Substituting the assigned Hubble mass $M_H=c^3/\left(2GH\right)$ into Equation (44) gives the algebraic temperature scale

$T_{\text{H}}\left(M_H\right)=\frac{{\hslash }_{0}H}{4\pi k_B}.$ (45)

Equation (45) is linear in $H$ , not in $\sqrt{H}$ . At $H_0=66.9\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}$ , it is approximately 1.32 × 10⁻³⁰ K, not the observed approximately 2.7255 K CMB temperature. Moreover, assigning $M_H$ through a compactness identity does not prove that a dynamical Hubble boundary is a Schwarzschild event horizon.

The distinctive BHU step is the additional geometric-mean postulate, not an identification of the CMB with conventional Hawking radiation from a Hubble-mass black hole. In the notation of Section 11, it is

$T_{\text{BHU}}=\sqrt{T_{\text{H}}\left(M_{\text{Pl}}\right)T_{\text{H}}\left(M_H\right)}=\frac{{\hslash }_0\sqrt{H_{\text{Pl}}H}}{4\pi k_B},H_{\text{Pl}}=\frac{c^3}{2G{M}_{\text{Pl}}}.$ (46)

Here $T_{\text{H}}\left(M_{\text{Pl}}\right)$ is a formal Planck-scale normalization; the semiclassical formula is not established as a physical evaporation law at the Planck mass. Equation (46) is algebraically equivalent to the proposed radius and mass relations in Equations (85) and (88). With a fixed Planck reference it yields $H\propto T_{\text{BHU}}^2$ and, after identifying $T_{\text{BHU}}\left(z\right)=T_0\left(1+z\right)$ , the history in Equation (92). The two lighthouse routes in Section 11.1 supply complementary motivations for this power law while retaining their distinct normalizations.

Its value for BHU is a quantitative link between a cosmological thermal scale and Planck-Hubble gravitational scales, making the proposal more specific than a purely dimensional universe-black-hole analogy. Its remaining physical task is equally specific: derive the geometric-mean coupling, identify the relevant horizon and observers in an expanding rotating spacetime, and explain how the resulting radiation acquires the observed spectrum and angular properties. Schwarzschild formulae cannot simply be carried over unchanged to a rotating horizon; cosmological horizon thermodynamics also depends on the actual spacetime ([50] [51]). These requirements turn the thermal connection into a programme for black-hole and cosmological tests, not a claim that Hawking’s calculation already establishes BHU.

The meaning of a measured horizon temperature is also clarified by work on local Hawking temperature for dynamical black holes and on radiation inside charged black holes with and without a cosmological term ([52]-[54]). These calculations emphasize the specified spacetime, horizon and observer. They are relevant comparisons for the BHU temperature proposal, not derivations of its geometric-mean law.

Application to black-hole disc temperatures. The intended local application retains the Planck mass and replaces only the Hubble mass by the astrophysical black-hole mass:

$M_H\to M_{BH},M_{\text{Pl}}\text{fixed},\boxed{T^{\text{*}}\left(M_{BH}\right)=\frac{{\hslash }_0{c}^3}{8\pi G{k}_B\sqrt{M_{\text{Pl}}{M}_{BH}}}.}$ (47)

Equation (47) is the same local geometric-mean relation as Equation (30), now stated explicitly as a proposed starting scale for understanding thermal emission associated with black-hole accretion discs. It predicts $T^{\text{*}}\propto M_{BH}^{-1/2}$ ; the common fixed normalization also gives $T^{\text{*}}\left(M_{BH}\right)/T_{\text{BHU}}\left(M_H\right)=\sqrt{M_H/M_{BH}}$ . No cosmological expansion rate is being assigned to the local disc.

Seshavatharam and Lakshminarayana have explored this compact-object application in a separate preprint, including an additional correction between the geometric-mean temperature and estimated inner-disc temperatures [55]. Thus the direct mass substitution supplies a candidate characteristic thermal scale, not by itself the observed disc colour temperature or a radial temperature profile. For the representative mass already used in Equation (31), $M_{BH}~{10}^{6.3}{M}_{\odot }$ , the uncorrected scale remains $T^{\text{*}}\simeq 4.18\times {10}^8\text{K}$ . A quantitative disc application must specify the correction or energy-transfer mechanism, the radiating region, accretion conditions and spectral processing, and test the result against independently constrained masses and spectra. This is the proposed connection to disc physics; it does not identify disc emission with Hawking evaporation or establish that hot accreting matter contradicts the conventional Hawking temperature.

Local-effects coefficient and population tests. A useful phenomenological extension of Equation (47) is

$\boxed{T_{\text{disc},i}={\mathcal{C}}_{\text{loc},i} T^{\text{*}}\left(M_i\right)={\mathcal{C}}_{\text{loc},i}\frac{{\hslash }_0{c}^3}{8\pi G{k}_B\sqrt{M_{\text{Pl}}{M}_i}},}{M}_i\equiv M_{BH,i}.$ (48)

Here $T_{\text{disc},i}$ denotes a specified characteristic local effective disc temperature, such as that at a consistently defined emitting radius. The positive dimensionless coefficient ${\mathcal{C}}_{\text{loc},i}$ parametrizes departures from the proposed baseline due to local conditions. Candidate dependencies include accretion state, spin, emitting radius relative to the gravitational radius, and energy transport; none is assigned a fitted functional form in this paper. An observed colour temperature requires a separately stated spectral and propagation conversion and is not silently substituted for this local temperature.

For two objects, Equation (48) implies

$\frac{T_{\text{disc},2}}{T_{\text{disc},1}}=\frac{{\mathcal{C}}_{\text{loc},2}}{{\mathcal{C}}_{\text{loc},1}}\sqrt{\frac{M_1}{M_2}}.$ (49)

At a fixed coefficient, increasing black-hole mass therefore lowers the characteristic temperature as $M^{-1/2}$ . For a population with a mass-dependent coefficient, the corresponding logarithmic slope is $\text{d}\ln T_{\text{disc}}/\text{d}\ln M=-1/2+\text{d}\ln {\mathcal{C}}_{\text{loc}}/\text{d}\ln M$ . Equation (49) shows explicitly why local conditions must be controlled before a mass trend is tested. If a free coefficient is assigned to every measured temperature, the relation becomes a reparametrization of those measurements rather than a prediction. A discriminating test must instead specify a shared coefficient law, constrain it with independent information, and assess temperatures in objects not used to determine that law. No such population fit is claimed here.

Comparison with Mitra’s hot compact-object proposal. Abhas Mitra’s radiation-pressure-supported “eternally collapsing objects” (ECOs), and the subsequent study with Norman K. Glendenning, provide a relevant alternative compact-object comparison ([56] [57]). These authors propose hot, radiating, quasi-static configurations before a putative limiting black-hole state. They should not be described as conventional Schwarzschild or Kerr black holes with established event horizons. Their proposal concerns trapped radiation and the interior of a compact body, whereas Equation (48) parametrizes an accretion-disc temperature; the two physical temperatures must remain distinct.

A shared mass exponent nevertheless provides a specific point of comparison. In the approximate argument used by Mitra, a radius $R\simeq 2GM/c^2$ and a radiation-dominated mean energy density give

$\overline{u}~\frac{M{c}^2}{\left(4\pi /3\right)R^3}\propto M^{-2},\overline{u}\simeq a_R{T}_{\text{int}}^4\Rightarrow T_{\text{int}}\propto M^{-1/2}.$(50)

Equation (50) summarizes that approximate interior scaling, not an exact relativistic volume integral [56]. The BHU geometric-mean relation reaches the same power through a different postulate. Matching exponents alone establishes neither identical normalizations nor physical equivalence. A comparison must determine the emitting region, surface or horizon structure, gravitational redshift, radiation transport and observable spectrum in each model. This manuscript does not adopt the ECO claim that collapse must avoid a finite-mass event horizon, and the comparison is not evidence against the existence of astrophysical black holes.

7.3. Atomic Action Scales and the Scope of the Alternative Proposal

An additional question is whether the cosmic gravitational environment can help determine an effective atomic action scale. This continues the authors’ earlier proposals linking microscopic physics to the Hubble volume ([14] [58]); it is not presented as an entirely new idea originating in this manuscript. Section 11.5 gives two explicit constructions and compares the Hubble normalizations they imply when matched to the laboratory value ${\hslash }_0$ . This adds a microphysical consistency test to the thermal and distance sectors. A mismatch between these normalizations is a theoretical question to resolve, not a reason to average them or to claim simultaneous exact agreement.

The comparison with alternative approaches should be made at the level of the question being addressed. The effective-field-theory treatment of general relativity provides a controlled low-energy quantum description of gravity [59]. The BHU action proposal instead asks whether gravitational and particle-mass scales can account for the value assigned to atomic action. An algebraic correspondence does not supply the quantum dynamics, matter interactions or consistency conditions supplied by a field theory. Nor is the origin or status of dimensionful constants an unexplored subject; different formulations assign different roles to such constants [60]. The present contribution is the explicit pair of hypotheses, their normalization comparison, and a route to dimensionless spectral tests, not a demonstrated replacement for quantum theory. Dependence on $M_H$ suggests sensitivity to a global boundary condition, but does not itself define a nonlocal interaction or establish a causal mechanism for coupling cosmic and atomic physics.

7.4. Priority of Acceleration Tests and Interpretation of the Cosmological Term

The central methodological position of this paper is that observations must remain primary. When a theoretical construction and an observation do not agree, the first scientific task is to re-examine the assumptions connecting them: the adopted geometry, observer congruence, distance definition, redshift interpretation, source calibration, selection effects, and dynamical closure. An unexplained residual may motivate a new physical component, but the residual itself is not direct detection of that component.

Priority of testing acceleration. Dark energy is commonly invoked to explain inferred cosmic acceleration ([3] [6]). The BHU programme therefore gives priority to testing acceleration itself, particularly through the proposed repeated spectroscopic observations of distance-ordered sources $G_1$ and $G_2$ (Section 3). Within this approach, an independent dark-energy component is not a starting assumption and receives secondary priority. The immediate task is to establish which expansion history and spectroscopic predictions agree with observations. This is a choice of research priority, not a claim that existing repeat observations have already disproved dark energy. The source-pair tests retain the explicitly stated propagation assumptions and sensitivity requirements.

Geometric interpretation of the cosmological term. A parameter introduced to describe cosmic geometry need not initially be interpreted as a separate dark-energy substance. In BHU, the proposed geometric or mechanical coefficients should therefore be distinguished from independently established energy components. Their physical interpretation must follow from the model’s equations, rather than from using the symbol Λ or a numerical value resembling a conventional dark-energy fraction. In particular, the dimensionless mechanical fraction $f_{\text{Λ}}$ and the auxiliary coefficient ${\Omega }_{\Lambda }\left(z\right)$ are not automatically a cosmological constant. The conventional Λ has dimensions of inverse length squared; a dimensionless number should instead be identified as a geometric coefficient. Within the usual Einstein equations, writing $\text{Λ}{g}_{\mu \nu }$ on the geometric side does not remove its gravitational effect [3].

The resulting methodological position is straightforward: test acceleration first, and do not assume a separate dark-energy explanation at the outset. BHU explores whether bounded geometry, coherent rotation, thermal evolution and the adopted distance relations can account for the observations. Its field-equation development must make those ingredients and their conservation laws explicit, so that the expansion dynamics and the interpretation of any cosmological term can be assessed consistently.

Cosmological terms and coupling in alternative theories. Supergravity-motivated nonminimal coupling and early-universe phase transitions with an induced cosmological term have been studied by El-Nabulsi ([61] [62]). These are relevant examples of constructions in which the cosmological term has a specified theoretical origin. They do not identify the BHU mechanical fraction with a vacuum density. Other explicit alternatives include fractional-action cosmology with conformal coupling and Maxwell-brane cosmology with curvature corrections, a coupled scalar field and dark-sector interaction ([63] [64]). In comparison, the BHU relations $H/T^2=\text{constant}$ and $\omega =H$ specify links between scales; they do not yet define an interaction action or an energy-transfer law.

Nonlocal effects and redshift interpretation. El-Nabulsi and Anukool relate spatial long-range kernels to a generalized uncertainty principle and discuss black-hole temperature consequences [65]. Such a nonlocal construction specifies more than dependence on a global mass scale: its kernel and physical conditions are part of the theory. Separately, Tremblin and Chabrier examine cosmological redshift in the presence of inhomogeneities and irreversible processes [66]. This motivates making the assumed photon propagation explicit, without treating their approach as equivalent to the bounded BHU law. These comparisons retain the present priority of testing acceleration and the distinction between geometric coefficients and independent dark components.

7.5. Thermal Monotonicity, Rotation, and the Absence of Acceleration in BHU

The measured present CMB temperature is $T_0=2.72548\pm 0.00057$ K [67]. Measurements at earlier epochs are consistent with the adiabatic photon-temperature law

$T\left(t\right)a\left(t\right)=T_0{a}_0,\frac{\dot{T}}{T}=-H,$(51)

within current uncertainties [68]. The present radiation background is therefore the low-temperature end of an observed monotonic cooling history. This physical fact is important, but temperature alone does not determine the sign of $\ddot{a}$ . In standard relativistic cosmology, the same law $T\propto a^{-1}$ can accompany either decelerated or accelerated expansion; accelerating de Sitter horizons also possess a well-defined thermodynamics [51]. This explains why the value 2.7 K, by itself, is not normally treated as a disproof of acceleration.

From the global BHU thermodynamic viewpoint, however, the epoch commonly associated with the onset of late-time acceleration was already extremely cold. Taking the representative transition redshift $z_{\text{tr}}\simeq 0.6$ , the measured temperature law gives

$T\left(z_{\text{tr}}\right)=T_0\left(1+z_{\text{tr}}\right)\simeq \left(2.7255\text{K}\right)\left(1.6\right)\simeq 4.36\text{K}.$ (52)

Thus the proposed reversal from deceleration to acceleration would occur after the radiation background had already cooled to only a few kelvin. No heating event, phase transition, or independently identified global energy-injection mechanism is observed at that epoch. In BHU, a universe following one continuous cooling trajectory from the early hot state to 4.36 K and then to 2.7255 K has no physical trigger for a second inflation-like acceleration phase. Both temperatures belong to the same cold terminal part of the global thermal history, so the natural BHU interpretation is continued deceleration rather than a late reversal of the expansion dynamics.

This global argument becomes quantitative through the coupled BHU relation $H\propto T^2$ derived below. It is the combination of monotonic cooling and the $H$ -$T$ closure—rather than the numerical smallness of the temperature alone—that removes the acceleration branch from the BHU evolution.

The BHU claim is stronger because it adds the independent Planck-Hubble thermal closure developed in Section 11,

$\frac{H\left(t\right)}{H_0}={\left(\frac{T\left(t\right)}{T_0}\right)}^2.$ (53)

Combining Equations (51) and (53), with $a_0=1$ , gives

$H=H_0{a}^{-2},\dot{H}=-2H^2,\frac{\ddot{a}}{a}=\dot{H}+H^2=-H^2<0,q\equiv -\frac{a\ddot{a}}{{\dot{a}}^2}=+1.$(54)

Thus, within the stated BHU thermal laws, continuous deceleration is a mathematical consequence rather than a qualitative appeal to coldness. Since the rotational identification also gives $\omega =H$ , the same trajectory unifies the three principal rates:

$T\downarrow \Rightarrow H\downarrow =\omega \downarrow \Rightarrow \ddot{a}<0.$(55)

Expansion, thermal evolution, and coherent rotation therefore cannot be assigned independent histories in BHU. An accelerating phase would require a failure or modification of at least one of the relations $Ta=\text{constant}$ , $H\propto T^2$ , or $\omega =H$ .

The conceptual importance of the 2.7 K background is consequently precise. It locates the present universe at the cold end of the BHU trajectory and supplies the measured normalization $T_0$ , while the sign of the acceleration follows from the coupled time-domain equations. The absence of a measurable change in $T_0$ over a human baseline is not used as evidence by itself, because the expected drift is far below present absolute-temperature sensitivity. The decisive tests are instead the constancy of $H\left(z\right)/T^2\left(z\right)$ across epochs, the sign of redshift drift, and the recovery of Equation (54) from the future covariant BHU field equation.

This thermal result also clarifies the required theoretical strategy. A BHU field equation begins from expansion, thermal evolution, vorticity, and conservation as one coupled system rather than importing a dark-energy acceleration term. Recovering Equations (53) - (55) together with cosmic-age constraints will provide the covariant derivation of the continuous-deceleration result already obtained from the thermal closure.

The preceding methodological argument requires the two interpretations to be judged by their physical assumptions and testable consequences, not by the historical maturity of their calculational machinery. Within FLRW analyses, the inference of acceleration follows after adopting the FLRW geometry, calibrated luminosity distances, and a negative-pressure component or equivalent modification. The BHU framework instead combines bounded geometry, thermal evolution, coherent rotation, and alternative distance relations and asks whether they reproduce the same observations without an independent dark-energy source. The appropriate comparison is therefore prediction against prediction under common data, calibration, covariance, and uncertainty treatment. Historical supernova, BAO, and CMB fits constrain the admissible cosmic histories, while repeated measurement of the same sources supplies the distinct live test of whether their redshifts are presently increasing in the required distance-ordered manner.

The $G_1$ - $G_2$ programme separates an observational question from its physical interpretation: the measured quantities are repeat spectral shifts, whereas their predicted signs, amplitudes, and distance dependence require a propagation model. The data in Table 1 do not constitute a controlled source-pair test. Section 3.4 quantifies a conditional low-redshift benchmark and the precision and systematic controls needed to pursue it.

The same separation applies inside the BHU framework. The relations $v\left(t,r\right)=H\left(t\right)r=\omega \left(t\right)r$ and $4\pi k_{B}T\left(t\right)=G{M}_{\text{Pl}}^2/\sqrt{R\left(t\right)R_{\text{Pl}}}$ are formulated without redshift and are intended as epoch-by-epoch physical relations. The second relation uses the fixed Planck gravitational-inertial anchor $G{M}_{\text{Pl}}^2$ ; it does not require an assumed future law for $\hslash \left(t\right)$ . By contrast, the functions of $z$ used for Pantheon and Union3 organize observations of the past and should not be extrapolated automatically beyond the observed domain.

MoM-BH^*-1 supplies a complementary cross-scale support. Its dense hydrogen spectrum and black-hole-dominated environment match the physical setting envisaged by Equations (27) and (29), while the numerical values in Equations (28) and (31) make the agreement testable. A forward calculation from the atomic and thermodynamic scales to the measured wavelengths, line ratios, break amplitude, continuum, and envelope temperature will quantify this correspondence. A population analysis using one redshift law and one black-hole-mass law without source-by-source adjustment offers an especially strong confirmation route.

The confirmation programme is therefore concrete: stable absorption systems repeated with the same frequency standard and separated into distance bins can test the opposite orderings in Equations (8) and (11); a public unbinned Pantheon+ and Union3 analysis can extend the successful distance fits under identical nuisance parameters; and covariant BHU dynamics can derive the bounded distance law directly from the rotating geometry.

8. Achievements

Origin of the proposed fractions. The present mechanical seed is the thin-shell prescription $I_H=\left(2/3\right)M_H{R}_H^2$ , together with $\omega R_H=c$ : $E_{\text{rot}}/\left(M_H{c}^2\right)=1/3$ , leaving the complementary fraction 2/3. The evolving auxiliary fractions also admit the following shift-motivated construction, which separates the reference deceleration index from the additional BHU weight prescription.

For a spatially flat, radiation-free matter-Λ reference, the conventional deceleration parameter is

$q_{\text{ref}}\left(z\right)=\frac{\frac{1}{2}{\Omega }_{m0}{\left(1+z\right)}^3-{\Omega }_{\Lambda 0}}{{\Omega }_{m0}{\left(1+z\right)}^3+{\Omega }_{\Lambda 0}},{\Omega }_{m0}+{\Omega }_{\Lambda 0}=1.$ (56)

With the idealized present fractions (1/3, 2/3), this gives $q_{\text{ref}}\left(0\right)=-1/2$ and the matter-dominated limit $q_{\text{ref}}\to +1/2$ [3]. Adding, rather than multiplying by, one half defines the shifted index

$Q\left(z\right)\equiv q_{\text{ref}}\left(z\right)+\frac{1}{2},Q\left(0\right)=0,Q\left(z\right)\to 1\text{in the radiation-free high-redshift limit}.$ (57)

Adding +1/2 raises the present negative reference value −0.5 to zero and raises the earlier matter-dominated value +0.5 to one. Read forward in cosmic time, the resulting sequence is therefore $Q:1\to 0$ , with no negative values. The authors use this non-negative sequence to motivate a continuous-deceleration interpretation rather than a transition to acceleration. If adopted as a physical deceleration history, such a sequence would remain decelerating up to its zero-acceleration endpoint. The addition of 1/2 supplies this motivation, not a dynamical proof; the thermal argument for continuous deceleration is given separately below.

The acceleration transition of this exact (1/3, 2/3) reference is $z_{\text{tr}}=4^{1/3}-1\simeq 0.587$ . The often quoted value near 0.63 instead follows from approximately $\left({\Omega }_{m0},{\Omega }_{\Lambda 0}\right)=\left(0.315,0.685\right)$ , for which $q_{\text{ref}}\left(0\right)=-0.5275$ and $Q\left(0\right)=-0.0275$ [5]; these two benchmarks should not be conflated. The limit +1/2 is a matter-dominated approximation, not a Planck-era prediction: radiation domination gives $q\to 1$ , and extrapolation into the Planck regime requires further physics.

Motivated by the half-unit shift and the epoch factor $1+z$ , the BHU prescription next adopts the positive weights $w_m=1/2$ and $w_{\text{Λ}}=1+z$ . Normalizing their sum gives

$\boxed{{\Omega }_m\left(z\right)=\frac{w_m}{w_m+w_{\Lambda }}=\frac{1}{3+2z},{\Omega }_{\Lambda }\left(z\right)=\frac{w_{\Lambda }}{w_m+w_{\Lambda }}=\frac{2\left(1+z\right)}{3+2z}.}$ (58)

Thus ${\Omega }_m+{\Omega }_{\Lambda }=1$ and the present values recover the mechanical seed (1/3, 2/3). The choice of weights is an additional phenomenological postulate, not a unique consequence of adding 1/2 to $q_{\text{ref}}$ . These auxiliary coefficients must not be confused with the instantaneous matter and vacuum fractions in Equation (56). Shifting a reference index does not change its physical acceleration; the separate BHU result $q=1$ follows from the stated thermal and scale-factor assumptions in Equation (54). The achievement is the explicit normalized prescription and its reported distance comparisons, with the additional assumptions made identifiable.

Thermal meaning of continuous deceleration. The proposed thermal picture connects a Planck-scale hot starting point to the present CMB temperature $T_0\simeq 2.725$ K, subject to the assumed extrapolation of the BHU thermal relation. Its quantitative deceleration statement follows from combining $H/H_0={\left(T/T_0\right)}^2$ , $T\propto a^{-1}$ and $H=\dot{a}/a$ :

$\boxed{-\frac{\ddot{a}}{a}=H^2=H_0^2{\left(\frac{T}{T_0}\right)}^4>0,q=1.}$(59)

Thus, throughout the expanding branch governed by these assumptions, cooling reduces the magnitude of the deceleration per unit scale factor without reversing its sign: the evolution remains decelerating, not accelerating. At $T=T_0$ that magnitude is $H_0^2$ , not zero. This supplies the thermal basis for the continuous deceleration claimed here, while distinguishing it from the motivating shifted sequence $Q:1\to 0$ . Cooling alone would not fix the acceleration sign; the added $H$ -$T$ closure is essential. Identifying the physical $q$ with a sequence ending at zero would therefore require modifying at least one of the stated closure assumptions.

Measured redshift versus inferred recession. The central observational distinction is that a distant galaxy’s spectroscopic redshift is measured, whereas its cosmological recession speed $v=Hr$ is inferred within an adopted spacetime model. The supernova Hubble diagram does not independently track that galaxy’s physical separation through the time-resolved sequence $D\left(t\right)\to \dot{D}\left(t\right)\to \ddot{D}\left(t\right)$ . The acceleration inference is therefore an additional model-based interpretation of measured light-cone data, not a direct observation of that galaxy’s recession speed increasing between observing dates ([1] [2]). Describing it as the latter would overstate what was measured. This distinction motivates the BHU priority given to repeated spectroscopy and explicit source-pair predictions. It does not make the underlying observations unconfirmed, nor does it establish the alternative identification $\omega =H$ without independent tests.

This paper has organized the bounded-redshift BHU proposal around quantitative comparisons and falsifiable observations. A principal result is that replacing a constant matter fraction by

${\Omega }_m\left(z\right)=\frac{1}{3+2z},{\Omega }_{\Lambda }\left(z\right)=\frac{2\left(1+z\right)}{3+2z},$

produces a successful fit to the 1048-object Pantheon Hubble diagram, with reported weighted RMS 0.1348 mag and ${\chi }^2/N=1.012$ . These values are close to 0.1336 mag and 0.993 for the selected flat-ΛCDM reference; thus the evolving BHU partition remains close to the reference across the complete 1048-object sample even though the latter has the lower summarized Pantheon ${\chi }^2$ . Independently, the bounded sinh law clearly fits the 22 covariance-weighted Union3 bins and gives ${\chi }^2=24.902$ , lower than 26.411 for the selected fixed reference with the same floating calibration offset. The combined result is substantially stronger than a qualitative resemblance: two different supernova compilations support the BHU construction, and the bounded law provides the better reported Union3 bin-level statistic. These successful empirical fits establish the observational viability of bounded redshift and provide direct targets for derivation from the covariant BHU dynamics.

The complementary live test retains distance-ordered source pairs as an observational priority. Under the proposed sign-preserving spectroscopic mapping, the candidate positive ordering is $\text{Δ}{z}_2>\text{Δ}{z}_1>0$ , while the BHU candidate is $\text{Δ}{z}_2<\text{Δ}{z}_1<0$ . These are hypotheses requiring a specified mapping between emission, propagation, and observer time, not universal consequences of the sign of recession acceleration. Existing repeat-telescope measurements do not establish either controlled pair ordering, and a null result alone does not prove absence of acceleration.

The 50-year assessment in Section 3.4 makes the experimental requirement explicit. With the stated leading-order low-redshift assumptions, a source at $z=0.01$ has a predicted shift of about 1.03 cm∙s⁻¹ in 50 years. An illustrative pair at $z=0.01$ and 0.02 requires approximately 1.03 mm∙s⁻¹ equal per-source, per-epoch precision for a nominal $5\sigma$ differential detection with independent errors. Long-term stability and astrophysical systematics, not instrumental resolving power alone, determine whether that target is attainable. Establishing both individual drift signs and the pair difference would test the specified mapping; identifying the underlying cosmological mechanism requires comparison with competing predictions.

The galactic extension makes a separate set of predictions. The Super-Gravity mapping $M_d=M_b\sqrt{M_b/M_T}$ connects baryonic mass to an effective gravitational supplement, characteristic radii, and rotation speed without assigning an independently adjustable particle halo to every galaxy. Its nonlinear scaling, $M_d\propto M_b^{3/2}$ , predicts that an equal-mass relaxed merger has a supplement larger by the factor $\sqrt{2}$ than the sum of its two progenitors. Rotation-curve databases, galaxy sizes, lensing masses, merger simulations, and the observed lensing-gas separation in the Bullet Cluster can therefore support, constrain, or falsify this construction.

The cosmic-dawn applications provide independent microscopic support and confirmation tests. For MoM-BH^*-1, the atomic and geometric-mean black-hole relations connect directly to the measured hydrogen-rich spectrum and envelope energetics without introducing a supernova calibration. For JADES-GS-z14-0, the predicted paired features near 1.230 and 0.916 mm, with fixed ratio 1.3428, can be sought in archival and future millimetre data. Their independence from the distance-ladder analysis makes them especially valuable cross-checks of the integrated framework.

A further methodological contribution is the explicit local compact-object extension of the thermal relation. Retaining $M_{\text{Pl}}$ while replacing $M_H$ by $M_{BH}$ supplies the characteristic scale $T^{\text{*}}\propto M_{BH}^{-1/2}$ ; Equation (48) separates that baseline from a local-effects coefficient and Equation (49) specifies a conditional population test. The comparison with Mitra’s radiation-supported ECO proposal identifies a shared exponent but different temperature definitions and physical assumptions. The contribution here is this explicit comparison and test prescription, not a claim to have discovered the mass exponent, validated disc temperatures, or established equivalence with ECOs.

The atomic-action sector adds a second cross-scale consistency test. Section 11.5 expresses two candidate action scales through $G,c,m_e,m_p$ and $M_H$ , and obtains the distinct conditional normalizations $H_0^{\left(1\right)}\simeq 70.74$ and $H_0^{\left(2\right)}\simeq 65.18$ km∙s⁻¹∙Mpc⁻¹ by matching each to ${\hslash }_0$ . The achievement is an explicit, numerically checkable comparison with the adopted thermal normalization, not an exact derivation of ${\hslash }_0$ , proof of varying constants, or completed gravity-quantum unification. Earlier atomic-cosmic proposals are acknowledged in Section 7.3.

The rotational proposal is likewise observationally definite: BHU assumes a bounded, shear-free, coherently rotating construction with $\omega \left(t\right)=H\left(t\right)$ and a measurable global axis. Agreement among independent axes inferred from galaxy handedness, CMB temperature and polarization, and secular proper motions would support this interpretation; inconsistent axes or amplitudes would reject it.

The present paper therefore establishes BHU as an integrated quantitative research programme with successful supernova fits, cross-scale numerical agreements, a definite deceleration law, and multiple falsifiable predictions. The covariant theoretical stage can now construct the dynamical black-hole line element and field equations, incorporating black-hole evolution, expansion, physical vorticity, and the locking $\omega \left(t\right)=H\left(t\right)$ while deriving the stress-energy tensor, conservation laws, causal structure, and photon geodesics. The principal result is a coherent alternative whose supporting evidence spans supernovae, thermal scaling, acoustic lengths, galactic dynamics, large-scale rotation, proposed live redshift tests, and cosmic-dawn spectroscopy, together with a clear route to decisive confirmation.

9. Foundational Construction

9.1. Hubble Compactness Identity

At an epoch characterized by Hubble parameter $H$ , define

$R_H\equiv \frac{c}{H},{\rho }_c\equiv \frac{3H^2}{8\pi G},M_H\equiv \frac{4\pi }{3}{\rho }_c{R}_H^3.$ (60)

Substitution gives

$M_H=\frac{c^3}{2GH},\frac{2G{M}_H}{c^2}=\frac{c}{H}=R_H.$ (61)

The associated compactness is

${\mathcal{C}}_H\equiv \frac{2G{M}_{\text{}H}}{c^2{R}_{\text{}H}}=1.$ (62)

Equation (61) is an exact identity once critical density and a Hubble sphere have been adopted. BHU elevates this unit-compactness result from a numerical coincidence to the geometric foundation of a bounded cosmic system; its topology, horizon character, and rotation then become connected physical consequences to be established through the remaining relations and observations developed below.

For a representative value $H_0=66.9\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}$ , one obtains

$R_H\simeq 4.48\times {10}^3\text{Mpc},M_H\simeq 9.3\times {10}^{52}\text{kg}.$ (63)

The corresponding gravitational potential scale is

$\frac{G{M}_H}{R_H}=\frac{c^2}{2},$ (64)

which motivates a connection with Machian ideas about the cosmic origin of inertia [69].

9.2. Rotational Interpretation

Hubble’s linear relation and rigid rotation have the same algebraic form,