An Epoch-Dependent Acceleration Scale from Bounded Topology: Predictions for High-Redshift Galactic Dynamics

Abstract: Under the framework's local-epoch reading, we propose that the modified Newtonian dynamics (MOND) acceleration scale evolves with cosmic epoch as $\afid(z) = \afid(0)\,\Ez$, where $\Ez = \Hz/H_0$. This single input predicts three correlated direct dynamical observables at high redshift: baryonic Tully--Fisher relation (BTFR) normalization ($E^{-1}$), MOND transition radius ($E^{-1/2}$), and asymptotic velocity at fixed baryonic mass ($E^{+1/4}$), together with a supporting fixed-geometry collapse-time heuristic ($E^{-1/4}$). The redshift evolution is normalized to the local SPARC value. Existing intermediate-redshift measurements are mixed and not yet mutually consistent: a KMOS3D BTFR tension, radial-acceleration-relation evolution in the predicted direction (MUSE-DARK III), and a flat baryonic Tully--Fisher zero point in tension with the normalization prediction (MUSE-DARK II) are each treated in Section~\ref{sec:constraints}. A particularly clean near-term test is matched-systematics kinematic follow-up at the original {"U}bler redshifts; existing JWST/NIRSpec data partially cover the lower of the two redshifts, and the matched baryonic-mass campaign remains to be carried out at both. Extending the same evolution to gravitational lensing would require a relativistic completion not supplied here and is noted only as a conditional outlook. Under the stated sector and well assignments, the framework yields $\afid/(cH) = 0.1845$; the mechanism is presented in Section~\ref{sec:derivation} and Appendix~\ref{app:framework}.

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Introduction

The MOND acceleration scale $\afid \approx 1.20 \times 10^{-10}$~m/s$^2$ has governed the phenomenology of galactic rotation curves for four decades [Milgrom1983]. Fitted to the local SPARC sample, the relation has an observed scatter of $0.13$~dex [McGaugh2016,Lelli2016]. Later joint inference reduces the inferred intrinsic component to $0.034 \pm 0.001\,\text{(stat)} \pm 0.001\,\text{(sys)}$~dex, conditional on the assumed SPARC error model [Desmond2023underlying]. The acceleration scale also satisfies a numerical coincidence that lacks a generally accepted explanation: $\afid/(cH_0) \approx 0.18$, with the Hubble rate setting the dimensional scale of the MOND threshold [Milgrom1999,McGaugh2004]. Within standard MOND this ratio is an accident; within $\Lambda$CDM the ratio has no established physical counterpart, though the acceleration scale itself has proposed origins in baryonic feedback, discussed in the simulation comparison (Section [ref:sec:lcdmsims]).

Two current problems motivate examining this scale across cosmic time.

First, the coincidence itself. Using the standard SPARC normalization [Lelli2016] and Planck 2018 $H_0$ [Planck2018], $\afid/(cH_0) = 0.183$, numerically stable across four decades of improving measurements.

Second, JWST has revealed candidate galaxies at $z \sim 7$--$10$ with stellar masses $M_\star \gtrsim 10^{10}\,\Msun$ assembled within 600~Myr of the Big Bang [Labbe2023,BoylanKolchin2023], discussed as a potential early-structure tension under standard halo-growth assumptions. Spectroscopic follow-up has since reclassified some similarly red, compact candidates as dust-reddened broad-line active galactic nuclei (AGN) rather than pure stellar populations -- the “little red dots” [Kocevski2023,Matthee2024,Kocevski2025rise] -- and revised some Labb{'e}-type masses downward once an AGN continuum contribution is included [Wang2024rubies]; the extent to which this affects the specific sample used below is not established here, but it tempers the “impossible galaxy” framing for the population as a whole. If $\afid$ were larger at early times, gravitational collapse of galactic-scale systems would proceed faster, easing this tension as a fixed-geometry heuristic with the scope limits set out in Section [ref:sec:collapse]. A related application to the DESI DR2 dark-energy signal [DESI2025] is treated in a separate preprint; this paper is limited to the galactic-dynamics consequence of an epoch-dependent $\afid$.

This paper develops one hypothesis and its consequences. Within a bounded-topology measurement framework on $S^1 = \partial(\Mob) \hookrightarrow S^3$ (Appendix [ref:app:framework]), the Milgrom ratio resolves to the ratio of two phase-operator values at Fibonacci wells on the 120-domain native to $S^3/2I$:

$$\frac{\afid}{cH} = \frac{\Cval{13}}{\Cval{34}} = 0.1845, \label{eq:milgrom-ratio}$$

agreeing with the observed $0.1833$ at the $0.7\%$ level. This is a central-value match under the adopted SPARC and Planck calibrations, to which no statistical significance is assigned: the local $\afid$ carries a $\sim 20\%$ systematic calibration range and the ratio depends further on the adopted $H_0$. What is nontrivial is that this central-value match falls at a combinatorially sparse pairing (Section [ref:sec:derivation]). Among the framework's six Fibonacci-well pairs, $(13, 34)$ is the unique pair that reproduces the observed ratio within $1\%$ (Section [ref:sec:derivation]).

Because both $\afid$ and $H$ reference the same epoch-dependent hierarchy in the framework's scaling law, the ratio (eq:eq:milgrom-ratio) holds at every cosmic epoch under the framework's local-epoch reading (Section [ref:sec:derivation]). The evolution follows:

$$\afid(z) = \afid(0)\,\Ez, \qquad \Ez \equiv \Hz/H_0. \label{eq:evolution}$$

The bare possibility $\afid \propto H$ has appeared previously in MOND-motivated discussions of the Milgrom coincidence [Milgrom1999,Limbach2008,HossenfelderMistele2018,Milgrom2020,Maeder2023,DelPopoloChan2024,Gueorguiev2024]. Limbach etal.\ did more than note the coincidence: they derived the BTFR-evolution channel under $\afid \propto cH$ and tested it against Tully–Fisher data to $z = 1.2$, finding the coupling disfavored at the precision then available, though their systematics-adjusted reading marginally favored a competing coupling to the dark-energy density over the Hubble-rate coupling tested here. A rising acceleration scale with redshift also has a direct low-redshift observational precedent: V{\u a}r{\u a}{\c s}teanu etal.\ [Varasteanu2025] report tentative evidence ($\sim 2.4\sigma$) for a rising acceleration scale to $z \sim 0.08$ using MIGHTEE-HI. The novelty here is not the dimensional guess, nor the BTFR-evolution channel itself, which follows from deep-MOND relations once $\afid(z)$ is specified and which Limbach et~al.\ already derived and tested; it is the specific framework mechanism fixing the ratio at $\Cval{13}/\Cval{34}$ and the evolution linear in $\Ez$, from which the observable exponents follow together rather than as independent choices.

Three direct dynamical predictions, in two independent observable families, follow as different powers of $\Ez$, with no additional free parameter beyond the local SPARC calibration. A supporting collapse-time heuristic follows as well. A particularly clean near-term test requires a dedicated matched-systematics campaign at the original {"U}bler redshifts, using JWST NIRSpec integral-field-unit (IFU) or ground-based IFU observations (Section [ref:sec:euclid]). The framework's monotonic BTFR prediction is currently in tension with {"U}bler et~al.'s two-redshift trend-shape measurement [Ubler2017], the only internally homogeneous two-bin measurement of its kind; a second, single-redshift zero-point measurement (MUSE-DARK II) disagrees more strongly with the {"U}bler measurement than with the prediction. Both tensions are quantified in Section [ref:sec:constraints].

Section [ref:sec:derivation] gives the conditional derivation of (eq:eq:milgrom-ratio) and (eq:eq:evolution). Section [ref:sec:channels] propagates the evolution into three direct dynamical predictions and one supporting heuristic. Section [ref:sec:constraints] treats existing constraints. Section [ref:sec:euclid] specifies the near-term observational program. Section [ref:sec:conclusions] closes.

Derivation

The framework's measurement postulate (Appendix [ref:app:framework]) maps any Planck-normalized observable to a phase position on the 120-domain native to $S^3/2I$:

$$\frac{A}{A_P} = C(\Theta) \cdot N^n, \qquad C(\Theta) = 2\sin^2(\pi\Theta), \label{eq:scaling-law}$$

where $A_P$ is the Planck-scale reference matching the dimension of $A$, built from $c$, $\hbar$, and $G$ (for a time, $t_P = \sqrt{\hbar G/c^5}$; for an acceleration, $a_P = c/t_P = \sqrt{c^7/(\hbar G)}$), $\Theta \in \{k/120\}$ is the phase position, $n$ is the manifold-mode index set by the embedding hierarchy $S^1 \subset \Mob \subset S^3$, and $N$ is a dimensionless hierarchy normalization proportional to $(\sqrt{\Omega})^{-1}$ within each mode class, its exact value (the $O(1)$ proportionality constant) fixed by calibrating one reference observable per sector. The phase operator $C(\Theta)$ is the unit-mean squared intensity of the node-centered representative of the lowest anti-periodic mode on the \Mob{} surface (Appendix [ref:app:120-domain]). A selection rule assigns edge-mode observables ($n = 1$) to the kinematic hierarchy $\Omega_H$ and surface-mode observables ($n = 2$) to the eigenvalue hierarchy $\Omega_\Lambda$ (Appendix [ref:app:selection]). The normalization $N$ is associated with the embedding hierarchy: for edge modes ($n = 1$), the relevant kinematic hierarchy is $\Omega_H(z) = (c/(H(z)\ell_P))^2$, so $\Omega_H(z)^{-1/2} = H(z)\,t_P$. The normalization used throughout this paper, denoted $N_H(z)$ without ambiguity, is fixed by calibrating on $H$ (Eq. (eq:eq:H-calibration)): $N_H(z) = H(z)\,t_P/\Cval{34} = \Omega_H(z)^{-1/2}/\Cval{34}$, so the calibration constant for this sector is the reciprocal phase factor $\Cval{34}^{-1}$. For surface and space modes, the corresponding normalization $N_\Lambda \propto (\sqrt{\Omega_\Lambda})^{-1}$ is epoch-independent; its magnitude is set by the surface sector's calibrating observable $\Lambda$, whose value is fixed independently by the surface eigenvalue (Appendix [ref:app:selection]) rather than by evaluating the scaling law forward. The postulate thus places every observable at a phase position, while one calibrating observable per sector fixes that sector's $N$.

Calibration and prediction.

The eligibility rules assign $\afid$ to $\Theta = 13/120$ and $H$ to $\Theta = 34/120$, both edge modes (Appendix [ref:app:wells]). The scaling law gives, at any epoch $z$:

$$\begin{aligned} \frac{\afid(z)}{a_P} &= \Cval{13} \cdot N_H(z), \label{eq:a0-prediction}\\ H(z)\,t_P &= \Cval{34} \cdot N_H(z). \label{eq:H-calibration} \end{aligned}$$

Equation (eq:eq:H-calibration) is the calibration: $H$ is the measured input that fixes $N_H(z) = H(z)\,t_P/\Cval{34}$ at every epoch, with the same role as the QCD coupling measured at a reference process. Equation (eq:eq:a0-prediction) is the prediction: given $N_H(z)$ from (eq:eq:H-calibration), the well at $\Theta = 13/120$ produces $\afid(z)$. Substituting:

$$\boxed{\;\frac{\afid(z)}{c\,H(z)} = \frac{\Cval{13}}{\Cval{34}} = 0.1845\;} \label{eq:milgrom-ratio-derived}$$

The two well positions enter differently: $H$ calibrates the shared normalization $N_H(z)$ from its assigned position at $\Theta = 34/120$, while $\afid$'s assigned position at $\Theta = 13/120$ is not itself calibrated against any observed value. Both share the same $N_H(z)$, so (eq:eq:milgrom-ratio-derived) follows algebraically once the assignments are made. $H$'s specific position enters observably only through this resulting ratio: Eq. (eq:eq:H-calibration) would calibrate $N_H(z)$ identically for any candidate position in place of $34/120$, so it is the substitution into Eq. (eq:eq:a0-prediction), not the calibration step itself, that gives $34$ its observable consequence. The content is not the algebra but the output: the ratio lands at $0.1845$ against an observed central value $0.1833$, a $0.7\%$ central-value match under the adopted SPARC and Planck calibrations, to which no statistical significance is assigned (the local $\afid$ alone carries a $\sim 20\%$ systematic range). Numerically, $\afid^\text{obs}/(cH_0^\text{obs}) = 1.20 \times 10^{-10} / 6.548 \times 10^{-10} = 0.1833$. Of the 7,021 unordered distinct nonzero phase-position pairs on the 120-domain, 24 reproduce this ratio within $1\%$; the reflection symmetry $C(k) = C(120-k)$ collapses them to 6 unique phase-operator value pairs. Among the framework's six Fibonacci-well pairs, $(13, 34)$ is the unique match (Figure [ref:fig:milgrom-sparsity]). The agreement is combinatorially sparse on the unrestricted domain and unique within the declared Fibonacci candidate set; these counts are descriptive, not statistical significances, since neither the unrestricted domain nor the six-pair candidate set has an independently justified uniform selection measure. The $1\%$ tolerance is itself a round-number choice made with the achieved $0.68\%$ miss already known: a $0.5\%$ tolerance gives only $12$ pairs and excludes the framework's own match entirely.

figure1 Milgrom-ratio sparsity on the 120-domain. Of 7,021 unordered nonzero phase-position pairs, 24 reproduce the observed $\afid/(cH_0) = 0.1833$ within $1\%$; reflection symmetry $C(k) = C(120-k)$ collapses these to 6 unique phase-operator value pairs. Within the framework's Fibonacci-well subset $\{13, 21, 34, 55\}/120$, only the pair $(13, 34)$ matches, marked with a star.

Local-epoch reading.

Equation (eq:eq:milgrom-ratio-derived) holds at every $z$ because $N_H(z)$ is calibrated through the local $H(z)$ at the observer's epoch, not frozen at $N_H(0)$. The framework adopts this local-epoch reading as a distinct commitment: the topology alone does not presently provide a dynamical derivation selecting it over an $N_H(0)$-anchored alternative, although the absence of a preferred epoch in the bounded domain $S^3$, with $\partial S^3 = \emptyset$, and the standing wave's explicit $t$-dependence in $\Psi(t) = \cos(t/2)$ motivate it. The two readings are observably distinct at every $z > 0$: at $z = 2$ the BTFR normalization differs by a factor of $3$ between them, and matched-systematics kinematic testing (Section [ref:sec:euclid]) will distinguish them.

Why $\Lambda$ does not evolve.

The same scaling law places $\Lambda$ at the antinode $\Theta = 60/120$ with $n = 2$ (surface mode), referenced to $\Omega_\Lambda$: the fixed eigenvalue hierarchy associated with the $\Lambda$ eigenvalue, not the redshift-dependent fractional density parameter of standard cosmological notation. Under the local-epoch reading, this eigenvalue hierarchy is the same at every $z$. The phase position $60/120$ is the fixed point of the reflection symmetry $C(k) = C(120-k)$ (Appendix [ref:app:wells]), so $d\ln C/d\Theta = 0$ there by construction: $C$ is symmetry-protected against first-order phase displacement at the antinode, not protected against arbitrary perturbations of the operator or geometry. The two predictions are structurally inverse: $\afid$ evolves because it references $\Omega_H$; $\Lambda$ does not because it references $\Omega_\Lambda$. This inversion follows from the framework's selection rule; a first-principles derivation of that rule remains open (Appendix [ref:app:selection]).

Thus the result is a conditional prediction of the framework: given the scaling law, selection rule, assigned well positions, $H$-calibrated normalization, and local-epoch reading, the redshift evolution follows by substitution, with no additional evolution parameter.

Observable channels

The evolution $\afid(z) = \afid(0)\,\Ez$ from Section [ref:sec:derivation] propagates into galactic observables through different powers of $\Ez$. We adopt the flat $\Lambda$CDM Friedmann form, normalized to the displayed parameters' own sum so that $\Ez = 1$ exactly at $z=0$ (the rounded Planck values below sum to $1.000092$, not $1$; the normalization imposes flatness exactly rather than carrying that rounding residual through $\Ez$):

$$\Ez = \sqrt{\frac{\Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_{\Lambda,\mathrm{dens}}}{\Omega_m + \Omega_r + \Omega_{\Lambda,\mathrm{dens}}}}$$

with Planck 2018 parameters [Planck2018] ($\Omega_m = 0.315$, $\Omega_r = 9.2 \times 10^{-5}$, $\Omega_{\Lambda,\mathrm{dens}} = 0.685$, $H_0 = 67.4$~km/s/Mpc; here $\Omega_{\Lambda,\mathrm{dens}}$ denotes the standard cosmological density fraction, distinct from the framework's eigenvalue hierarchy $\Omega_\Lambda$ of Section [ref:sec:derivation]). The prediction $\afid(z) \propto H(z)$ holds for any expansion history; adopting $\Lambda$CDM is a transparency choice that allows direct comparison with published measurements. Table [ref:tab:cosmology] provides the reference values consumed by the channels below.

Reference cosmological quantities used throughout this paper, across the redshift range relevant here: the normalized expansion rate $\Ez \equiv \Hz/H_0$, the Hubble rate $\Hz$, the predicted acceleration scale $\afid(z)$, the deep-MOND enhancement factor $\sqrt{\Ez}$, and the cosmic age $t_\text{age}$ at each $z$.

Tully–Fisher normalization: \texorpdfstring{$\Ez^{-1}$}{E(z) to the minus 1}

In the deep-MOND limit, the baryonic Tully–Fisher relation (BTFR) is $\vflat^4 = G\,\Mb\,\afid$ [Famaey2012], empirically confirmed locally as the small-scatter BTFR [Lelli2016b]. With evolving $\afid$:

$$\boxed{\;\frac{A_\text{BTFR}(z)}{A_\text{BTFR}(0)} = \frac{1}{\Ez}\;}, \qquad A_\text{BTFR} \equiv \frac{1}{G\,\afid}. \label{eq:btfr}$$

The interpolating-function dependence cancels in the ratio: any $\mu(x)$ that fits the local SPARC sample inherits the same fractional evolution. At fixed baryonic mass, $\vflat(z) = \vflat(0)\,\Ez^{1/4}$; at fixed velocity, $\Mb(z) = \Mb(0)/\Ez$.

BTFR evolution at six reference redshifts.

An $L^*$ disk ($\Mb = 6 \times 10^{10}\,\Msun$, $\vflat(0) = 176$~km/s) is predicted to have $\vflat(z=2) = 232$~km/s, a $32\%$ increase at fixed baryonic mass. Existing tests of this channel are treated in Section [ref:sec:constraints].

MOND transition radius and asymptotic velocity: \texorpdfstring{$\Ez^{-1/2}$, $\Ez^{+1/4}$}{E(z) powers}

The radius where Newtonian gravity equals $\afid(z)$ is $\rM = \sqrt{G \Mb / \afid(z)}$, giving

$$\boxed{\;\frac{\rM(z)}{\rM(0)} = \frac{1}{\sqrt{\Ez}}\;}. \label{eq:rM}$$

At $z = 2$, the transition happens at $57\%$ of the local radius. The asymptotic velocity rises as $\Ez^{1/4}$ (from Section [ref:sec:btfr]). Both scalings are mass-independent in the deep-MOND limit.

Predicted MOND transition radius and asymptotic velocity at $z = 0$ and $z = 2$ for four SPARC archetypes. The fractional shifts $\rM(2)/\rM(0) = 0.574$ and $\vflat(2)/\vflat(0) = 1.320$ apply identically across the table.

The comparison isolates the effect of $\afid(z)$ at fixed baryonic distribution. Real galaxies at $z = 2$ differ in structural properties; the observer-facing formulation is: given an observed galaxy at redshift $z$ with measured $\Mb$ and $\vflat^\text{obs}$, compare $\vflat^\text{obs}/\Ez^{1/4}$ to the local SPARC value for galaxies of the same $\Mb$.

Collapse time: \texorpdfstring{$\Ez^{-1/4}$}{E(z) to the minus 1/4} (supporting evidence)

In the deep-MOND regime, $\geff = \sqrt{\gN \cdot \afid(z)}$ scales as $\sqrt{\Ez}$ at fixed source, and the free-fall time $\tff \propto 1/\sqrt{\geff}$ shortens as $\Ez^{-1/4}$. This channel builds on a longer-standing result: fixed-$a_0$ MOND already predicts faster gravitational collapse than $\Lambda$CDM, independent of any redshift dependence in $a_0$ itself [Sanders1998,Sanders2008], a mechanism McGaugh etal. [McGaugh2024structure] connect explicitly to the JWST early-massive-galaxy tension. The $\Ez^{-1/4}$ correction developed here is a distinct, additional effect specific to this framework's epoch-dependent $\afid(z)$, layered on top of whatever baseline speedup fixed-$a_0$ MOND already supplies; it is not a restatement of the Sanders mechanism, which contains no $a_0(z)$ term. At $z \sim 8$, the factor $\Ez^{1/4} \approx 2$ shortens the characteristic deep-MOND collapse time by approximately a factor of two. Once in the edge regime, this could ease, but would not by itself resolve, the star-formation-efficiency pressure for the Labb{'e} etal.\ [Labbe2023] massive candidates. Whether this resolves the impossibility constraint for any specific candidate depends on halo-mass priors the framework does not modify; the per-object analysis is reserved for future work. The fourth-root dependence makes this correction weakly sensitive to $\Ez$: a $10\%$ shift in $\Ez$ changes the correction by less than $3\%$. This is a fixed-geometry free-fall timescale at fixed source properties; it does not incorporate perturbation growth, halo abundance, gas cooling, angular momentum, or feedback, so a measured stellar age alone cannot cleanly isolate this specific scaling from the rest of high-redshift structure formation. It further presumes a system already in the deep-MOND (edge) regime, where $\afid(z)$ applies: the framework assigns cosmological perturbations to the space sector, which does not inherit the evolving $\afid$ (Section [ref:sec:cmb]), and it does not specify when a growing perturbation crosses into an edge-sector, galactic-scale object. The channel therefore bounds the collapse of an already-galactic overdensity, not the assembly of a galaxy from a linear perturbation. It is supporting, qualitative evidence for the direction and rough magnitude of the effect, not an independent quantitative test on the same footing as the two independent families above.

Summary: predictions from one input

Summary of the direct predictions and the supporting collapse-time heuristic of the evolving acceleration scale, evaluated at $z = 2$.

The three direct exponents are correlated manifestations of one input: any single channel tests $\afid(z) = \afid(0)\,\Ez$, and agreement among channels at one redshift tests whether the deep-MOND scaling is universal.

Existing constraints

The comparisons in this section share one methodological condition, stated here once. The published measurements differ in sample selection, velocity statistic, baryonic-mass inference, and local calibration, and no covariance model connects them. Gap-to-quoted-error ratios below are therefore descriptive, not formal significances, whether between surveys or between any survey and this paper's prediction. Tiley et~al.\ [Tiley2019] report a comparable pattern independently: KROSS and SAMI stellar-mass Tully–Fisher zero points differ by as much across matched-baseline choices as across the redshift range itself. This provides evidence from outside the present comparison that survey-to-survey systematics of this kind are common in this literature, not unique to the measurements below. The falsification thresholds of Table [ref:tab:falsification] apply only under the matched-systematics condition of Section [ref:sec:euclid], which no existing cross-survey comparison meets. Each subsection below states only its specific systematics beyond this shared condition.

Falsification criteria.

Falsification criteria for the direct Section [ref:sec:channels] predictions and the collapse-time heuristic. A matched-systematics discrepancy at $\geq 2\sigma$ is treated as a reportable tension; the pre-committed decision threshold for retiring the prediction, not a formally derived joint-likelihood criterion, is either $\geq 3\sigma$ in one channel or mutually consistent $\geq 2\sigma$ failures across independent redshift bins or observables.

The collapse-time row is the weakest: it is a fixed-geometry heuristic (Section [ref:sec:collapse]), not an independent structure-formation calculation, so a reportable tension there constrains the heuristic's applicability rather than the underlying $\afid(z)$ relation directly. The BTFR normalization and MOND transition radius rows probe the same evolving $\afid(z)$ through the same baryonic mass model rather than independent physics; concordant failures in both should not by themselves be read as two independent channels under the mutually-consistent-failures criterion above. The trend-shape row exceeds this threshold at face value (Section [ref:sec:ubler]); a formal test awaits the matched-systematics program of Section [ref:sec:euclid].

The {"U}bler tension

The KMOS3D analysis of {"U}bler etal.\ [Ubler2017] is the only internally homogeneous, fixed-slope, two-redshift trend-shape test of the Section [ref:sec:btfr] prediction using a single survey and velocity statistic. Other intermediate-redshift BTFR studies exist with different redshift coverage, velocity definitions, and gas treatments -- Puech etal. [Puech2010] find no BTFR zero-point evolution to $z\sim0.6$ within $\pm0.08$dex, while Sharma etal. [Sharma2024] report a subtle deviation across $0.6<z<2.5$ that they attribute to evolutionary processes in disk formation -- but neither matches {"U}bler's specific two-bin, fixed-slope, single-instrument construction. Section [ref:sec:musedark2] treats a second, single-redshift zero-point measurement. {"U}bler etal.'s fixed-slope BTFR zero-point offsets relative to the local Lelli etal.\ [Lelli2016b] baseline are

$$\Delta b^{\text{obs}}(z{=}0.9) = -0.44 \pm 0.04~\text{dex}, \qquad \Delta b^{\text{obs}}(z{=}2.3) = -0.27 \pm 0.05~\text{dex}.$$

The framework predicts

$$\Delta b^{\text{pred}}(z{=}0.9) = -0.227~\text{dex}, \qquad \Delta b^{\text{pred}}(z{=}2.3) = -0.540~\text{dex}.$$

The magnitudes overlap: both the prediction and the data place the high-$z$ BTFR substantially below the local relation. The trend shapes do not: the data are non-monotonic ($-0.44 \to -0.27$), the prediction strictly monotonic ($-0.227 \to -0.540$) (Figure [ref:fig:ubler]).

This comparison also carries an unaddressed velocity-definition gap: {"U}bler et~al.\ report the finite-radius circular velocity $v_{\rm circ,max}$, while the framework's deep-MOND relation $\vflat^4 = G\Mb\afid$ is defined at the asymptotic $\vflat$. The two converge for galaxies with well-resolved flat outer rotation curves but are not automatically interchangeable in general; this paper does not independently establish that equivalence for the {"U}bler sample, so the comparison above is indicative rather than a fully closed test.

figure2 KMOS3D BTFR comparison. Vertical axis is the BTFR zero-point offset $\Delta b$ relative to the local ($z=0$) baseline, at the two {"U}bler redshifts $z = 0.9$ and $z = 2.3$; error bars are the quoted $1\sigma$ statistical uncertainties. The framework predicts a strictly monotonic decrease with $z$ (solid curve); the {"U}bler et al.\ [Ubler2017] data (points) are non-monotonic. Statistical treatment in Section [ref:sec:ubler] and Appendix [ref:app:stats].

The tension is quantified across three uncertainty budgets:

Per-bin and joint $\sigma$-tension for the KMOS3D BTFR comparison under three uncertainty budgets. Entries are computed from the unrounded zero-point offsets, $-0.443$ and $-0.270$ dex, which the text above quotes rounded to two decimals.

Even under the most conservative budget (Table [ref:tab:sigma-tension]), the total two-bin discrepancy is $\chi^2 = 8.16$ on two degrees of freedom, $2.4\sigma$ (Appendix [ref:app:stats]): a reportable tension under Table [ref:tab:falsification]'s $\geq 2\sigma$ criterion, but below its $\geq 3\sigma$ retirement threshold. The shape-isolated statistic differences the two bins so the shared local baseline cancels, giving a more direct test of the monotonic prediction. Treating the velocity-correction systematic as independent between bins at the $0.10$-dex-per-bin level gives a sensitivity case of $3.1\sigma$; this treatment assumes the two bins' systematic responses are uncorrelated, not anti-correlated, an assumption not independently tested here. Propagated instead through this paper's own forward-modeled bias-correlation structure, which returns a coherent survey-level effect rather than an independent one, the sensitivity case is $6.8\sigma$ (Appendix [ref:app:stats]). Both sensitivity cases would read as formal falsification under Table [ref:tab:falsification]'s criteria taken at face value, but neither is compared against that threshold here: both depend on this paper's own assumed covariance structure, not an independently validated one (Appendix [ref:app:stats]).

A four-bias forward model (radial-position uncertainty, beam-smearing of $\sigma_0$, non-asymptotic rotation curves, selection-induced mass shifts) tested whether the {"U}bler pressure-support correction can produce the observed non-monotonic pattern from a monotonic underlying BTFR (Appendix [ref:app:stats]). No combination reproduces the observed pattern; the joint $L_2$ best fit leaves residuals of $(+0.27, -0.16)$dex. The model does not test the neutral-gas systematic Jeanneau etal.\ propose (Section [ref:sec:musedark2]). Resolution will come from follow-up at the original {"U}bler redshifts matched in tracer and in baryonic-mass definition, not from a single-instrument remeasurement alone (Section [ref:sec:euclid]).

MUSE-DARK III

Ciocan etal.\ [Ciocan2026] measure the radial acceleration relation directly for 79 star-forming galaxies at $0.33 < z < 1.44$ (MUSE Hubble Ultra Deep Field), using the disk–halo decomposition of the parent sample performed in PaperI [Ciocan2026a]. MUSE-DARK III is the first dedicated intermediate-redshift RAR measurement from resolved dynamical modeling; unlike the Section [ref:sec:ubler] BTFR comparison, which infers evolution from a zero-point offset, MUSE-DARK III fits the characteristic acceleration scale itself. They find statistically significant evolution relative to the local RAR and report $\afid(z{\sim}1) = 2.38^{+0.12}_{-0.10} \times 10^{-10}$~m/s$^2$, with the quoted interval a $95\%$ confidence interval in their stated convention, not the $1\sigma$ bar this paper's own uncertainties elsewhere denote.

This agrees with the predicted direction of $\afid(z) \propto H(z)$. The magnitude is not an exact match: at $z = 1$ this paper predicts $\afid(1) = 2.15 \times 10^{-10}$m/s$^2$ (Table [ref:tab:cosmology]), about $10\%$ below the MUSE-DARK III central value. Reading McGaugh etal.'s published systematic uncertainty on the local acceleration scale ($\afid(0) = 1.20 \pm 0.24 \times 10^{-10}$~m/s$^2$ [McGaugh2016], one calibration treatment rather than a universal uncertainty) as a conservative envelope and propagating it through $\afid(1) = \afid(0)\,E(1)$ gives approximately $1.72$--$2.58 \times 10^{-10}$m/s$^2$ at $z = 1$. This range contains the MUSE-DARK III central value, although its $95\%$-CI convention makes this containment check looser than a nominal $1\sigma$ comparison would be (Section [ref:sec:constraints]). Ciocan etal.\ do not state the corresponding $1\sigma$ width, so a directly comparable statistical statement is not constructed here.

Ciocan et~al.\ also report a fitted linear rate, $\afid(z) = \afid(0) + a_1 z$, with $a_1 = 1.59^{+0.11}_{-0.10} \times 10^{-10}$~m/s$^2$ per unit $z$ under their preferred DC14 decomposition -- appreciably steeper than a free fit of this paper's own $\afid(0)\,E(z)$ curve over the same range ($\approx 1.20 \times 10^{-10}$~m/s$^2$ per unit $z$). Two further robustness fits span this same range: a galaxy-by-galaxy $\Lambda$CDM decomposition (their Appendix D, Eq.~D.2) gives $a_1 = 1.63^{+0.13}_{-0.12} \times 10^{-10}$~m/s$^2$, steeper still, while a MOND-framework fit (their Appendix E, Eq.E.5) gives $a_1 = 1.20^{+0.10}_{-0.10} \times 10^{-10}$m/s$^2$, numerically matching this paper's free fit. Ciocan etal.\ treat all of these as reinforcing one conclusion -- in their words, the results “reinforce the presence of a systematic increase of the characteristic acceleration scale with redshift,” though the MOND fits underperform the dark-matter-based models for over $60\%$ of the sample by their own Bayesian model comparison. The numerical match between their MOND-derived rate and this paper's free fit is therefore not evidence that the two analyses agree: Ciocan etal.'s preferred decomposition remains DC14, appreciably steeper than the framework's curve, and they themselves state that, in the context of modified gravity, their measured $\afid(z)$ “is faster than that of $H(z)$.” Whether this becomes a formal tension once calibration systematics, effective redshifts, and covariances are matched between the two analyses is not yet established (Figure [ref:fig:musedark3]).

figure5 MUSE-DARK III radial acceleration relation. The framework curve $\afid(z) = 1.20\,E(z)$ against the reported central value $\afid(z{\sim}1) = 2.38^{+0.12}_{-0.10}$ (Ciocan et al.\ [Ciocan2026]); shaded band is the McGaugh et al.\ [McGaugh2016] local-calibration envelope propagated through $E(z)$, read as descriptive rather than a formal goodness-of-fit region (Section [ref:sec:constraints]).

\FloatBarrier

MUSE-DARK II: a flat BTFR zero point at \texorpdfstring{$z\sim1$}{z 1}

Jeanneau etal.\ [Jeanneau2026] analyze 95 rotationally supported, strongly lensed star-forming galaxies at $0.5 < z < 1.5$ from the MUSE Atlas of Lensing Clusters (Abell370, Abell2744, MACS0416, AbellS1063), a sample independent of the MUSE-DARK III blank-field sample of Section [ref:sec:musedark]. They report no detectable evolution of the baryonic Tully–Fisher zero point relative to $z \approx 0$,

$$\Delta b_\text{bTFR} = 0.00^{+0.06}_{-0.06}~\text{dex},$$

alongside a significant stellar-mass Tully–Fisher shift ($\Delta b_\text{sTFR} = -0.42^{+0.05}_{-0.05}$~dex); they attribute the flat baryonic result to increasing cold-gas mass at higher redshift compensating the stellar-mass evolution.

This is a direct comparison with the normalization channel of Section [ref:sec:btfr]. At $z = 1$, the framework predicts $\Delta b_\text{pred} = -\log_{10}E(1) = -0.253$dex, roughly $4.2$ times Jeanneau etal.'s own $0.06$-dex sample uncertainty considered in isolation. Jeanneau et~al.\ themselves note that a fair comparison should also include the $\pm 0.16$-dex statistical uncertainty in the local BTFR zero point they adopt; combined with their sample uncertainty in quadrature, the same $-0.253$~dex shift is $1.5$ times the total, a substantially weaker nominal discrepancy than the isolated figure suggests (Section [ref:sec:constraints]).

{"U}bler et~al.'s $\Delta b^\text{obs}(z{=}0.9) = -0.44 \pm 0.04$dex and Jeanneau etal.'s $0.00 \pm 0.06$dex sit $0.44$dex apart, nominally $6\times$ their combined quoted errors, on the same descriptive basis as the $4.2$ figure above. Thus, two kinematically measured BTFR results disagree with each other by a larger nominal margin than either disagrees with the prediction, consistent with unmodeled cross-survey systematics though not itself proof of them. Jeanneau etal.\ attribute part of the gap with {"U}bler to neutral gas neglected in {"U}bler's baryonic mass estimates, a systematic the Section [ref:sec:ubler] forward model does not test; Jeanneau etal.'s own gas masses are themselves scaling-relation estimates rather than direct detections, which they describe as a limitation of their own analysis, so this is not a clean direct-versus-indirect contrast between the two surveys.

Genuinely direct, neutral-hydrogen (H,i)-based BTFR measurements exist at lower redshift: Ponomareva etal. [Ponomareva2021], Gogate etal. [Gogate2023], and Jarvis etal. [Jarvis2025] find no evolution (or, in Jarvis etal.'s case, only tentative high-mass flattening they do not themselves rule out as evolution) in the MIGHTEE-HI/BUDHIES BTFR from $z \sim 0.08$ to $0.35$. The framework predicts shifts of only $-0.017$ to $-0.082$~dex over this range, small enough that these results are consistent with it because the studies are confined to low redshift, not an independent test of it.

The nominal $4.2$ figure would read as a formal falsification under Table [ref:tab:falsification]'s criteria taken at face value, but this comparison's cross-survey, unmatched-baseline character means that threshold does not apply (Section [ref:sec:constraints]). A matched-systematics reconciliation of all three intermediate-$z$ results is not available on the published data: they differ in sample selection, velocity statistic, and baryonic-mass treatment, which leaves them systematics-limited rather than reconcilable at present; a companion analysis develops the decomposition dependence of the measured rate and the per-galaxy universality estimator further.

RC100: no detected BTFR evolution across a larger sample

McGaugh, Schombert, Lelli & Franck [McGaugh2024structure] analyze the baryonic Tully–Fisher and dark-matter-fraction–surface-brightness relations using the Nestor Shachar et~al. [NestorShachar2023] RC100 sample, 100 rotation curves at $0.6 < z < 2.5$ drawn from KMOS3D and SINS/zC-SINF, binned as $0.6 < z < 1.22$, $1.22 < z < 2.14$, and $2.14 < z < 2.53$. They report both relations persisting to the highest bin with, in their words, “little if any indication of evolution in either relation up to $z \approx 2.5$.”

The comparison is qualitative on their side: no fitted zero-point or slope against redshift is reported, only that the plotted relations show no visible trend across the three bins. McGaugh et~al.
themselves note they “made no attempt to reconcile the choice of circular velocity measure or the precise definition of baryonic mass” across the low- and high-$z$ subsamples, underscoring the systematics-limited character of this comparison. This framework predicts $\Delta b_\text{pred} = -\log_{10}\Ez$ of $-0.148$, $-0.308$, and $-0.509$~dex at the bin edges $z = 0.6$, $1.22$, and $2.14$, a $-0.434$dex shift across the sampled range, comparable in size to the Section [ref:sec:ubler] {"U}bler gap. A sample an order of magnitude larger than the 6–95-galaxy samples used in the comparisons elsewhere in this paper shows no such trend at face value, though RC100 is not a fully independent check: it draws from the same KMOS3D and SINS/zC-SINF parent surveys as {"U}bler (Section [ref:sec:ubler]) and Genzel etal.
(Section [ref:sec:other-regimes]). Whether a fitted zero-point extracted from the RC100 data under this paper's own velocity and mass conventions would recover the predicted shift is not established here; absent that fit, this is a qualitative indication against the predicted evolution, not a formal one.

Del Popolo & Chan [DelPopoloChan2024] report a related exercise on this same RC100 sample: fitting $a_0(z)$ via Bayesian inference to a 17-galaxy high-$z$ subset selected by an acceleration cut not applied symmetrically to their low-$z$ SPARC comparison, they find a nominal anti-correlation with redshift, opposite in sign to the evolution predicted here. Gueorguiev's [Gueorguiev2024] reanalysis using the complete 100-galaxy sample finds this signal does not survive: the slope is consistent with zero ($0.01 \pm 0.2$), matching the qualitative null McGaugh et~al.\ report above rather than confirming either sign of evolution.

\texorpdfstring{$\Lambda$CDM}{Lambda-CDM} simulation predictions

$\Lambda$CDM hydrodynamical simulations divide on whether the radial acceleration relation evolves with redshift. EAGLE reports the relation essentially invariant: tracing $z = 0$ progenitors to higher redshifts leaves the mean relation closely similar, though the same discussion notes the residuals carry a slight but systematic trend with redshift, and EAGLE's own preferred $g^\dagger \approx 2.6 \times 10^{-10}$~m/s$^2$ runs $\approx 2.2\times$ above the SPARC-calibrated value it is compared against [Ludlow2017]. MUGS2 reports evolution instead, a low-$g_\text{bar}$ slope that flattens with increasing redshift as disks become baryon-depleted, and notes that describing all epochs with a single functional form would require a significantly redshift-dependent $g^\dagger$ [KellerWadsley2017]. Magneticum recovers a rising characteristic acceleration, larger by a factor $\simeq 3$ between $z = 0$ and $z = 2.3$, with the increase slower than an $\afid \propto \Hz$ scaling gives [Mayer2023]. The direction of (eq:eq:evolution) is reproduced by part of this literature and contradicted by another part, and where it is reproduced, the $\Ez$ form adopted here grows faster than the simulation does (Figure [ref:fig:lcdmsims]). This disagreement within the simulation literature remains unresolved rather than a settled $\Lambda$CDM expectation, and the matched-systematics program of Section [ref:sec:euclid] is what would separate the cases.

This same tension is recognized within the simulation literature itself: Ciocan etal.\ (Section [ref:sec:musedark]) cite this MUGS2/EAGLE disagreement, alongside Garaldi etal. [Garaldi2018] and Tenneti etal. [Tenneti2018], as leaving the degree of $z$-evolution of $a_0$ in hydrodynamical simulations uncertain, attributing it to differences in feedback prescriptions. Garaldi etal.\ find $\Lambda$CDM satellite galaxies follow the same RAR as brighter systems but with much larger scatter; Tenneti et~al.\ measure the RAR in MassiveBlack-II at $z \approx 0$ only, citing the MUGS2/EAGLE tension as open motivation rather than resolving it.

SIMBA [Glowacki2021] addresses the BTFR-normalization channel directly (a different observable from the RAR-shape comparisons above), predicting zero-point evolution to $z=1$ that differs in sign and magnitude by velocity statistic ($\vflat$ vs.\ the HI line widths $W_{20}$/$W_{50}$) -- independently illustrating, within a single simulation, the same velocity-definition sensitivity Sections [ref:sec:ubler] and [ref:sec:euclid] flag as an observational systematic, though it does not itself test the $\afid(z) \propto \Hz$ direction.

figure6 Radial acceleration relation across $\Lambda$CDM hydrodynamical simulations against the framework curve $\afid(z)/\afid(0) = \Ez$, where $\Ez \equiv \Hz/H_0$. EAGLE [Ludlow2017] reports the relation essentially invariant with redshift (no numeric slope stated, shown schematically; its own preferred normalization runs $\approx 2.2\times$ high against the observed value); Magneticum [Mayer2023] recovers a rise of $\simeq 3\times$ by $z = 2.3$, slower than the framework's $E(2.3) \approx 3.47$; MUGS2 [KellerWadsley2017] reports evolution expressed as a flattening RAR slope rather than a single $\afid(z)$ curve and is not plotted on these axes.

Galaxy clusters

MOND analyses of local galaxy clusters [Pointecouteau2005] report a residual mass discrepancy of roughly a factor of $5$ at $r \approx 0.5\,\Rvir$. The framework's evolving $\afid$ does not address this: the relevant clusters sit at $z \lesssim 0.15$, where $\Ez \leq 1.08$ and the framework correction is less than $8\%$. A factor-of-$5$ discrepancy is not materially reduced by an $8\%$ correction. This discrepancy is an open problem shared by all MOND-class theories [Sanders2003,Angus2008,Famaey2012] and predates the present framework, which neither introduces nor resolves it.

The CMB at recombination

A naive application of $\afid(z)$ to cosmological perturbations at $z = 1090$ gives $\afid \approx 23{,}000 \times \afid(0)$, placing recombination-scale perturbations deep in the MOND regime and disrupting the acoustic peak structure that Planck fits at sub-percent precision. This is not a problem unique to this framework: relativistic completions of MOND generically face it, and Skordis & Z{\l}o{'s}nik's [SkordisZlosnik2021] Aether-Scalar-Tensor (AeST) theory demonstrates it can be overcome for a constant $\afid$, fitting the CMB and matter power spectra while retaining a MOND limit in galaxies, though its free-function choice is under active scrutiny and may need to be more elaborate than the original [Verwayen2024,RosaZlosnik2024]. The burden here is therefore comparative -- showing this framework's evolving $\afid(z)$ is compatible with the CMB -- rather than showing compatibility is unprecedented in principle.

The framework's selection rule (Section [ref:sec:derivation], Appendix [ref:app:selection]) separates this channel structurally. The rule assigns $\afid$ to the edge sector ($n = 1$, $\Omega_H$) and cosmological perturbations to the space sector ($n = 3$, $\Omega_\Lambda$). Under this assignment, the $\afid(z)$ scaling governs galactic dynamics and does not propagate to perturbation evolution. The edge and surface clauses of this assignment are exercised elsewhere: the edge clause by the Section [ref:sec:derivation] Milgrom-ratio $0.7\%$ calibration-dependent central-value match, the surface clause by the $\Lambda$-constant prediction. The space clause invoked here is exercised only in this consistency argument; it is an independent assumption, not corroborated by either of the other two.

The consistency condition is a compatibility requirement, not yet a demonstrated result: writing a minimal leakage coupling $\geff = \gN + \varepsilon\sqrt{\gN\,\afid(z)}$ and translating Planck's first-peak precision through it gives $\varepsilon \leq 1.2 \times 10^{-5}$, a conditional bound resting on an adopted $\gN$ and an unchecked linear-leakage assumption rather than a full Boltzmann-hierarchy calculation. Exact edge/space decoupling ($\varepsilon = 0$) would satisfy it trivially; a first-principles perturbation-theory derivation of that decoupling, rather than its structural assumption, is the principal open task for the framework's CMB-scale predictions.

Other regimes

Local SPARC and THINGS measurements are consistent with the Section [ref:sec:derivation] $0.7\%$ calibration-dependent central-value ratio match: $H_0$ calibrates $N_H(0)$, and $\afid(0)$ follows from the coprimality-assigned well at $\Theta = 13/120$ once $N_H(0)$ is fixed, giving $\afid(0) = 1.208 \times 10^{-10}$m/s$^2$ against the adopted $1.20 \times 10^{-10}$. The Genzel etal.\ [Genzel2017] SINS/zC-SINF sample of six massive star-forming disks at $z = 0.9$--$2.4$ reports declining outer rotation curves with low inferred dark-matter fractions. No fixed-slope BTFR zero point is published, so the sample does not test the Section [ref:sec:btfr] normalization prediction in either direction. It does bear on Section [ref:sec:rotcurves]: Genzel etal.\ determine the curves out to $\approx 3.1$--$3.6$ half-light radii; with the fitted, gas-and-bulge-inclusive baryonic mass Genzel etal.\ report for each galaxy, and the low end of that extent range ($3.12\,R_{1/2}$), the observed curves are measured $1.6$ to $2.4$ times each galaxy's predicted $\rM(z)$; at the high end of the extent range ($3.60\,R_{1/2}$) the ratios run $1.9$ to $2.8$. This places the measured extent well past the radius where the local acceleration crosses $\afid(z)$; $\rM(z)$ marks that acceleration threshold but not, by itself, the radius by which an extended, non-point-mass disk must already have reached its asymptotic, deep-MOND velocity, which also depends on how much baryonic mass remains enclosed beyond $R_\text{out}$.

This comparison should be read alongside Milgrom's [Milgrom2017] own analysis of the same six galaxies and a larger, stacked sample from the same collaboration. Milgrom finds accelerations at $R_{1/2}$ of $(3$--$11)\,a_0$ and MOND-predicted asymptotic speeds at $0.55$--$0.75$ of the observed rotation-curve maxima: a decline beyond the peak is the expected constant-$a_0$ MOND signature for these baryon-dominated systems, so the $R_\text{out}/\rM(z) > 1$ figures above cannot by themselves be read as an unfavorable indication for the evolving-$a_0$ law specifically, absent a forward model of each galaxy's resolved baryonic profile. Lang etal. [Lang2017] independently corroborate this from an entirely different mechanism, averaging rotation curves for 101 KMOS3D and SINS/zC-SINF galaxies at $0.6 \lesssim z \lesssim 2.6$ (four of the six Genzel etal.\ galaxies among them) and finding a significant outer decline explicable by baryon-dominated, pressure-supported disks under standard gravity, with no MOND content in their analysis. Milgrom further argues the data “all but exclude” a MOND constant of $\sim 4a_0$ at $z \sim 2$ under a power law such as $a_0 \propto (1+z)^{3/2}$; this framework's own $E(2.3) \approx 3.47$ sits close to that excluded region, though reconciling the two bounds is beyond the present scope. Per-galaxy inputs and the transition-radius ratios underlying the ranges above are given in Table [ref:tab:genzel] (Appendix [ref:app:stats]).

The low dark-matter fractions are not themselves adverse: they are quoted at $R_{1/2}$, which lies inside the predicted transition radius for the sample, where the framework expects Newtonian behavior. This per-galaxy comparison covers the transition-radius channel only; no fixed-slope BTFR zero point is published for this sample, so it remains inconclusive on the transition-radius channel absent the forward model discussed above (Section [ref:sec:other-regimes]), and in any case not a formal falsification (Section [ref:sec:constraints]).

Strong-lensing time-delay cosmography (TDCOSMO [Millon2020], H0LiCOW [Wong2020]) is not presently predicted by this framework: like galaxy–galaxy lensing (Section [ref:sec:euclid]), time-delay cosmography depends on the lensing potential. Neither this regime nor the SINS/zC-SINF sample above tests the framework's normalization prediction independently: the Genzel et~al.\ sample has no published BTFR zero point to compare against, and strong-lens time delays are not yet predicted at all.

Summary

Constraint status across ten regimes.

Near-term observational tests

A particularly clean near-term test of $\afid(z) \propto H(z)$ is matched-systematics kinematic follow-up at the {"U}bler redshifts, where the framework's strictly monotonic BTFR prediction already stands in tension with existing data (Section [ref:sec:ubler]). Both {"U}bler bins come from KMOS3D, so the trend-shape statistic is internal to one survey and its shape is not a cross-instrument artifact. What it remains exposed to are the two systematics Section [ref:sec:ubler] identifies: the finite-radius $v_{\rm circ,max}$ standing in for the asymptotic $\vflat$ at which the deep-MOND relation is defined, and neutral gas omitted from the baryonic masses. Addressing them requires kinematics extended far enough out to reach the flat part of the curve, together with gas-inclusive baryonic masses at both redshifts, the latter drawing on radio or millimeter facilities rather than on the same instrument. The decisive test is therefore matched in tracer and in mass definition across the two bins, which is a stronger and different requirement than matched hardware.

What the kinematic program measures.

At $z \sim 0.9$, JWST/NIRSpec data already exist: MSA-3D delivers resolved rotation curves and the stellar-mass Tully–Fisher relation over $0.5 < z < 1.7$ using MSA slit-stepping, not standard IFU mode, and reports the stellar-mass relation rather than the baryonic-mass, matched-systematics comparison to the {"U}bler baseline (Section [ref:sec:ubler]) that would test the trend-shape prediction directly. The same MSA-3D data extend to the transition-radius and asymptotic-velocity predictions of Section [ref:sec:rotcurves] at $z \sim 0.9$, with the same stellar-mass caveat. Among the 19 MSA-3D galaxies with radial coverage reaching $\gtrsim 2R_e$, six show rising, six approximately flat, and seven declining rotation curves; a simplified seeing-degradation test shifts the inferred stellar-mass normalization by $\approx +0.22$~dex at fixed circular velocity, illustrating the resolution-driven systematic that this paper's own matched-tracer requirement is designed to control.

Mancera Pi{~n}a etal. [ManceraPina2026] report moderate evidence for stellar Tully–Fisher evolution at $z \sim 0.9$ using flat-part velocities -- the deep-MOND-relevant statistic -- relative to the Marasco etal. [Marasco2025] local relation, though Jeanneau et~al. [Jeanneau2026] attribute the apparent slope change to the choice of reference relation rather than real evolution, and MSA-3D's own comparison shows a similarly sized cross-survey spread its team cautions against over-reading. Both remain stellar-mass, not baryonic-mass, measurements, so neither yet tests the Section [ref:sec:btfr] normalization prediction directly.

At $z \approx 2.3$, no matched program is yet identified. The decisive test requires a dedicated future campaign applying that prescription at both {"U}bler redshifts, not a program already executing (Table [ref:tab:schedule]).

Euclid contributes photometry, stellar-mass estimates, and redshift and sample selection to this broader observational landscape [Euclid2024], but its Near-Infrared Spectrometer and Photometer (NISP) channel is a slitless spectrograph designed to deliver redshifts, not spatially resolved rotation curves, so it does not supply the velocity measurements these predictions require.

Higher-leverage kinematic data exist at redshifts this paper does not currently engage: KURVS resolves outer rotation-curve shapes for 22 galaxies at $z \sim 1.5$ [Puglisi2023]; ALMA has delivered dynamically cold, resolved curves at $z \sim 4$--$5$ and a larger archival sample across $z = 0.5$--$3.5$ via ALPAKA [Rizzo2020,RomanOliveira2023,Rizzo2023alpaka]; and JWST/NIRSpec and NIRCam are beginning to add disk kinematics and stellar-mass Tully–Fisher measurements at $z \sim 4$--$6$, including a reported zero-point offset of $\approx -1.3$dex relative to the $z \sim 0.9$--$2.3$ cosmic-noon relation of {"U}bler etal. [Ubler2017], suggesting the stellar relation may already be emerging there [deGraaff2024,Danhaive2025geko]. At $z = 4.2$, for example, the framework predicts $\rM(4.2)/\rM(0) = 0.386$ and $\vflat(4.2)/\vflat(0) = 1.610$, substantially larger leverage than at the {"U}bler redshifts. None of these works were designed as a baryonic Tully–Fisher or transition-radius test, and whether they meet this paper's matched-systematics condition (gas-inclusive baryonic masses are harder to determine at these redshifts) is not established here; they are noted as a higher-leverage opportunity, not engaged as a comparison.

Near-term observational schedule, distinguishing existing data from the still-required matched-systematics, baryonic-mass follow-up.

A conditional lensing outlook.

The framework supplies a Newtonian-equivalent dynamical prediction but not a relativistic lensing law: outside general relativity, the potential governing photon deflection and the potential governing non-relativistic dynamics need not coincide, a distinction long recognized in the MOND literature and the reason relativistic completions such as TeVeS [Bekenstein2004] were developed for MOND itself. Supplying that completion for the present framework is future work. If a future relativistic completion shows the lensing potential inherits the same $\Ez$ scaling as $\afid(z)$, stacked galaxy–galaxy lensing, including from Euclid, would provide a further test of the redshift evolution; if it does not, the dynamical predictions above stand or fall independently of any lensing measurement. This manuscript carries no numerical lensing forecast; an exploratory calculation under an assumed completion is archived separately (Data availability) as a conditional forecast, not a manuscript claim.

For the kinematic test, no new instrument or observational capability is required, only the specific matched-tracer campaign at both {"U}bler redshifts. This paper's direct predictions are pre-registered on Zenodo at a fixed version ahead of that follow-up (DOI: 10.5281/zenodo.20045115; Data availability).

Conclusions

The observed Milgrom ratio $\afid/(cH_0) \approx 0.18$ is numerically reproduced, conditional on the bounded-topology framework's stated sector and well assignments (Appendix [ref:app:framework]): a fixed ratio of two phase-operator values at Fibonacci wells, $\afid/(cH) = \Cval{13}/\Cval{34} = 0.1845$, agreeing with the observed $0.1833$ at $0.7\%$. The $\afid$ and $H$ wells are specified by separately assigned positions at $z = 0$ sharing the same edge-mode hierarchy; no formula in that assignment directly consults the observed ratio, and $H_0$ calibrates their shared normalization; the ratio follows algebraically and is unique among the framework's Fibonacci-well pairs within the stipulated $F_7$--$F_{10}$ window (Section [ref:sec:derivation]).

Under the framework's local-epoch reading, because both observables reference the same epoch-dependent hierarchy, the ratio holds at every cosmic epoch: $\afid(z) = \afid(0)\,\Ez$. Three direct predictions, in two independent observable families, follow as different powers of $\Ez$, with no additional free parameter. The BTFR normalization shifts as $\Ez^{-1}$ and the asymptotic velocity at fixed baryonic mass rises as $\Ez^{+1/4}$ -- the same deep-MOND relation solved for two variables, not independent tests of it -- while the MOND transition radius contracts as $\Ez^{-1/2}$, a genuinely separate, radially resolved test. A supporting fixed-geometry heuristic gives the gravitational collapse time shortening as $\Ez^{-1/4}$. The exponents move together, set by one input; agreement between the two independent families at a fixed redshift is the test of the deep-MOND scaling.

The prediction lives in a constraint landscape summarized in Table [ref:tab:constraint-status]. Two measurements currently present tensions: {"U}bler's KMOS3D BTFR trend shape (an absolute two-bin model–data discrepancy of $2.4\sigma$ under the conservative uncertainty budget, with a separate shape-only statistic of $3.1$--$6.8\sigma$ depending on the assumed inter-bin covariance, Section [ref:sec:ubler]), and MUSE-DARK II's flat BTFR zero point at $z \sim 1$ (nominally $4.2\times$ falling to $1.5\times$ with local-baseline uncertainty included, Section [ref:sec:musedark2]). The remaining entries comprise one qualitative echo (RC100's null trend across a much larger sample), one inherited problem (the cluster-scale MOND discrepancy), one inconclusive per-galaxy comparison (SINS/zC-SINF outer-curve declines, confounded with the generic decline standard MOND already predicts for these systems), and one assumed decoupling awaiting derivation (the CMB channel). MUSE-DARK III independently measures $\afid(z{\sim}1) = 2.38^{+0.12}_{-0.10} \times 10^{-10}$~m/s$^2$ via the RAR [Ciocan2026], agreeing with the predicted direction and, at ${\sim}11\%$ above this paper's own prediction, inside a conservative reading of the SPARC systematic envelope (Section [ref:sec:musedark]). A separate preprint applies the same selection rule to the dark-energy sector (DOI: 10.5281/zenodo.19798852).

A particularly clean test is matched-systematics kinematic follow-up at the {"U}bler redshifts (Section [ref:sec:euclid]), which would determine whether the observed non-monotonic BTFR trend survives matched systematics, directly testing the prediction that already stands in tension with existing data. Rotation-curve data at $z \sim 0.9$ already exist (Section [ref:sec:euclid]) but deliver a stellar-mass relation rather than the baryonic-mass comparison, so the matched campaign is still needed at both redshifts.

---

Appendix

} } } }

Framework foundations

This appendix supplies the structural detail underlying the derivation in Section [ref:sec:derivation], where the bounded topology, phase operator, scaling law, and selection rule are stated. It develops the 120-domain (Appendix [ref:app:120-domain]), the selection rule's physical basis (Appendix [ref:app:selection]), and the well-assignment eligibility conditions (Appendix [ref:app:wells]).

The 120-domain

The physical observable space is the quotient $S^3/2I$, where $2I$ is the binary icosahedral group with $|2I| = 120$. The discrete subgroups of $\text{SU}(2) \cong S^3$ are classified: cyclic groups $\mathbb{Z}_n$, binary dihedral groups $2D_n$, and three exceptional groups (binary tetrahedral $|2T| = 24$, binary octahedral $|2O| = 48$, binary icosahedral $|2I| = 120$). Three constraints select $2I$.

First, the cyclic and binary dihedral families are open: each requires an external choice of $n$, is unbounded in principle, and is excluded by the framework's input-minimization. Second, among the three exceptional groups, which are parameter-free and fixed by the classification itself, $2I$ is the largest, giving the maximum spectral resolution compatible with $S^3$ among the cases the framework admits. Third, the icosahedral group is the unique exceptional case whose branch orders $(2,3,5)$ are consecutive Fibonacci numbers satisfying $2 + 3 = 5$; the tetrahedral $(2,3,3)$ and octahedral $(2,3,4)$ branch data do not have this property, so this condition discriminates among the exceptional cases on its own. In the framework, the quotient order $|2I| = 120$ defines the discrete phase domain used to label observable positions, with spacing $\Delta\Theta = 1/120$; the Fibonacci positions on that domain form the framework's declared sampling lattice, whose candidate set and bounds are developed in Appendix [ref:app:wells].

The quotient $S^3/2I$ is the Poincar{'e} homology sphere: the classical example of a closed 3-manifold with the integral homology of $S^3$ but nontrivial fundamental group, obtained as the quotient of $S^3$ by the binary icosahedral group $2I$ [Saveliev2012]. The same group occupies the $E_8$ position in the ADE classification of finite subgroups of $\text{SU}(2)$, and its representation theory maps to the $E_8$ Dynkin diagram through the McKay correspondence [McKay1980]. More generally, quotients $S^3/\Gamma$ by finite freely acting subgroups are spherical space forms, classified in the standard space-form literature [Wolf2011]. This is the classification within which the three constraints above operate; the $E_8$/McKay position of $2I$ recurs in the candidate-well analysis of Appendix [ref:app:wells].

The phase position is quantized: $\Theta \in \{k/120 : k = 0, 1, \ldots, 119\}$. For photon-mediated observables, the bosonic projection $|\psi|^2$ erases the anti-periodic sign flip; the framework takes this to restrict the accessible grid to even numerators, a 60-position quotient, though sign-erasure alone does not by itself derive this specific quotient construction, and the even-numerator restriction should be read as a framework postulate pending an explicit derivation. Dynamical observables access the full 120-domain. Which physical observables count as photon-mediated versus dynamical is a classification the framework assigns, not a property derived from the observable's own definition; for $H_0$ in particular, the photon-mediated assignment is a framework rule (Appendix [ref:app:wells]), not a consequence of what the Hubble parameter measures.

The chronon, the smallest phase advance the domain can register, is $\Delta t_\text{min} = 4\pi/120 = \pi/30$ in the dimensionless phase argument $t$ of $\Psi(t)=\cos(t/2)$, not yet a physical time interval; translating it to proper time requires a phase-to-time mapping, which this paper does not supply. This material is not used elsewhere in the $\afid(z)$ derivation; a corresponding minimum-action claim is not established here and is left to the foundational deposit.

The phase dependence of $C(\Theta)$ follows from the lowest anti-periodic mode, up to the choice of phase origin. Let $\Theta \in [0,1]$ parameterize one traversal and consider

$$-\partial_\Theta^2\psi = \lambda\psi, \qquad \psi(\Theta+1)=-\psi(\Theta).$$

For an eigenfunction $\psi(\Theta)=e^{iq\Theta}$, the anti-periodic condition requires

$$e^{iq}=-1, \qquad q=(2m+1)\pi, \qquad m\in\mathbb{Z}.$$

Thus the real anti-periodic modes are $\sin[(2m+1)\pi\Theta]$ and $\cos[(2m+1)\pi\Theta]$, and the lowest eigenspace has wavenumber $\pi$. With the framework's phase-origin convention that $\Theta=0$ is a node, a representative of this eigenspace is

$$\psi_0(\Theta)=\sin(\pi\Theta).$$

The observable phase weight is the squared amplitude. Since

$$\int_0^1 |\psi_0(\Theta)|^2\,d\Theta = \int_0^1 \sin^2(\pi\Theta)\,d\Theta = \frac{1}{2},$$

normalization to unit mean gives

$$C(\Theta)=2\sin^2(\pi\Theta).$$

Anti-periodicity therefore fixes the odd, or half-integer, mode spectrum and its fundamental harmonic; the node convention fixes the phase origin. A fuller framework-level treatment is given in the foundational deposit [Shatto2025].

The selection rule

The scaling law (Section [ref:sec:derivation], Eq. [ref:eq:scaling-law]) assigns each observable a manifold-mode index $n$ from the embedding hierarchy $S^1 \subset \Mob \subset S^3$ and a corresponding hierarchy normalization $N$. The physical basis for the $(n, \Omega)$ assignment is:

Edge modes ($n = 1$, $\Omega_H$).

The boundary $S^1$ is a kinematic locus: it carries no intrinsic eigenvalue (the surface eigenvalue lives one dimension up, on the \Mob{} strip, where $\Lambda$ is placed). What an edge-mode observable references is the embedding scale of $S^1$ in the ambient cosmological geometry. The natural such scale is the Hubble horizon $R_H = c/H$, and its dimensionless Planck ratio is $\Omega_H = (c/(H\ell_P))^2$. Because $H$ evolves with cosmic epoch, edge-mode observables inherit that evolution through the calibrated normalization $N_H(z) = H(z)\,t_P/\Cval{34}$.

Within a given manifold-mode sector, the hierarchy normalization is sector-wide rather than observable-specific. Observable-to-observable variation within the sector is carried by the phase factor $C(\Theta)$; the common normalization is fixed by the embedding scale. Consequently, all edge-mode observables with $n = 1$ share the same $N_H(z)$, while their relative magnitudes are determined by their distinct phase positions. This sector-wide normalization is part of the framework's measurement rule; it does not follow from the edge-mode classification alone.

Surface modes ($n = 2$, $\Omega_\Lambda$).

The \Mob{} band carries the surface eigenproblem. It is the edge-identified quotient of a spherical band on a totally geodesic great $S^2 \subset S^3$, inheriting that band's constant-curvature metric; a Möbius band is non-orientable and therefore cannot itself be totally geodesic in $S^3$ (every totally geodesic surface there lies in an orientable great $S^2$), which is precisely why its twist is geometric rather than externally imposed. The orientation twist excludes a continuous global constant section. Because the collapsed fiber is a limit-circle cone point, however, the spectral bottom depends on the self-adjoint realization: the Friedrichs realization has a discontinuous zero mode, while the sector-regular bridging family has a negative defect-bound state. The quantity used here is therefore the first positive eigenvalue.

For the Friedrichs realization, and for the sector-regular bridging family with $\delta_0 > 2R/e$, the companion spectral analysis gives

$$\lambda_{+}(W)= \begin{cases} 2/R^2, & 0 < W \leq \pi R/2,\\[2pt] \alpha_0(\alpha_0+1)/R^2, & \pi R/2 < W < \pi R, \end{cases} \qquad \alpha_0=\frac{\pi R}{2W}.$$

The framework adopts the narrow-band condition $0 < W \leq \pi R/2$ as an input, so the applicable first positive eigenvalue is $\lambda_{+}=2/R^2=R_\Sigma$ [Shatto2026eigen]. An exact scalar spectral calculation supplies this value. Separately, the Weitzenb"ock identity on $S^3$ gives the compatible lower bound $\lambda \geq 2/R^2$ for the relevant $1$-form fluctuations; this is a consistency bound, not a second equality derivation.

This is a distinct spectral problem from the dimensionless $\Theta$-domain eigenvalue of Appendix [ref:app:120-domain]: that problem fixes the functional shape of $C(\Theta)$ on the unit interval, while $\lambda_{+} = 2/R^2$ fixes the absolute curvature scale carried into $\Lambda$. The Gauss–Codazzi conversion, applied to the totally geodesic covering great-$S^2$ band ($K_{ij} = 0$, so the Codazzi equation is trivial and only the Gauss equation contributes) under an isotropic embedding (motivated by, though not equivalent to, the CMB's observed temperature isotropy to $10^{-5}$) and a de Sitter vacuum, gives $\Lambda_\text{obs} = (3/2)\lambda_{+} = 3/R^2$; the factor $3/2$ is the isotropic spatial Ricci trace and the de Sitter normalization is imported from general relativity. The hierarchy ratio $\Omega_\Lambda = (\Lambda\ell_P^2)^{-1}$ is set by this eigenvalue and does not evolve. $\Omega_\Lambda$ and $N_\Lambda$ are defined from this independently established $\Lambda_\text{obs}$, not solved for by numerically evaluating the general scaling law of Section [ref:sec:derivation] on the surface sector; that law is used here only to place $\Lambda$ in the epoch-independent hierarchy, not to fix its magnitude.

Space modes ($n = 3$, $\Omega_\Lambda$).

Cosmological perturbations propagate on the spatial slice $S^3$ and reference the same fixed eigenvalue hierarchy as the surface. Under this assignment, perturbation evolution is governed by $\Omega_\Lambda$, not $\Omega_H$, and inherits no $\afid(z)$ modification. This is the structural basis for the CMB consistency argument in Section [ref:sec:cmb].

Well assignments

Well assignment proceeds in two layers. Three eligibility conditions first determine which grid, and for $\Lambda$ which specific position, is consistent with each observable's classification; separate assignment rules, coprimality for dynamical observables and the even-numerator requirement for photon-mediated ones, then select the unique well from within the eligible candidates. The three eligibility conditions are:

[label=\arabic*., leftmargin=*]

• Manifold index separates edge modes ($n = 1$: $H_0$, $\afid$) from surface modes ($n = 2$: $\Lambda$).

• Bosonic projection. Photon-mediated observables access only the 60-grid (even numerators); dynamical observables access the full 120-grid.

• Antinode requirement. $\Lambda$ is assigned to the reflection-fixed antinode $\Theta = 60/120$, where the reflection symmetry $C(k) = C(120-k)$ forces $d\ln C/d\Theta = 0$: the phase factor is stationary to first order there, not protected against arbitrary perturbations.

The manifold index and the bosonic projection are independent axes: the former (edge vs.\ surface) fixes the sector and its normalization $N$, while the latter (photon-mediated vs.\ dynamical) fixes only which grid positions are accessible. An observable is classified on each axis separately; $H_0$ is an edge mode ($n = 1$, sharing $N_H(z)$) and photon-mediated (even-numerator grid), with no tension between the two, and its shared normalization with $\afid$ rests on the edge-mode classification alone.

The candidate set $\{13, 21, 34, 55\}/120$ is the framework's Fibonacci-well restriction, corresponding to $F_7$ through $F_{10}$. The lower bound follows from a divisibility seam together with one interpretive premise: $1, 2, 3, 5, 8$ all divide $120$ and have $\mathrm{lcm} = 120$ exactly, so each reduces to a unit fraction $1/m$ -- the generator of a coarser sub-lattice that $120$ refines -- and the framework treats such positions as constituents of the domain's construction rather than new sampling positions at resolution $120$. This premise is narrower than mere reducibility: $21/120 = 7/40$, $34/120 = 17/60$, and $55/120 = 11/24$ also sit on coarser sub-lattices but do not reduce to unit fractions, so the premise does not absorb them. Under that premise, $13$ is the first Fibonacci number that does not divide $120$ and so is the first contributing genuinely new resolution. The upper bound is a stipulated window rather than a derived one: the reflection symmetry $C(k) = C(120-k)$ prevents double-counting only when a reflected value lands on another member of the candidate set, and it does not here -- none of $\{13,21,34,55\}$'s four reflections ($107$, $99$, $86$, $65$) coincide with another member, so reflection performs no exclusionary work inside this restricted set, unlike its role in the whole-domain sparsity count of Section [ref:sec:derivation]. $F_{11} = 89$ is excluded by the lower-half labeling convention (candidate wells are drawn from $k \le 60$ and must themselves be Fibonacci), not by an arithmetic identity. That convention does not affect the ratio-match result: admitting $C(89) = C(31) = 1.0523$ as a fifth candidate value creates no second match to the observed ratio, the closest being $C(13)/C(31) = 0.212$, off by $15.5\%$. The convention does have consequences elsewhere, however: $89$ is itself coprime to $120$ ($\gcd(89,120)=1$), so admitting it would break the coprime-rule uniqueness of assignment (a) below, giving $\{13,89\}$ rather than a unique coprime well; the lower-half convention is therefore load-bearing for that assignment, not only for the ratio match. What remains open is not the extent of the window but its principle: why a divisibility seam and the icosahedral Fibonacci recurrence, rather than a variational criterion, should fix the sampling positions at all.

Under the eligibility conditions:

[label=(\alph*), leftmargin=*]

• $\afid$ is assigned $\Theta = 13/120$, the unique coprime Fibonacci well within the stipulated $F_7$--$F_{10}$ window. This position lies on the full 120-grid and is therefore compatible with the classification of $\afid$ as a dynamical, non-photon-mediated observable. Coprimality supplies the assignment rule; full-grid access supplies the eligibility condition.

• $H_0$ is assigned $\Theta = 34/120$: the unique even-numerator Fibonacci well among $13/120$, $21/120$, $34/120$, and $55/120$, consistent with the bosonic-projection condition for a photon-mediated observation.

• $\Lambda$ is assigned $\Theta = 60/120$: the antinode, where the phase operator takes its maximum and the log-slope vanishes.

The wells $21/120$ and $55/120$ remain unassigned in this paper.

The remaining question is whether this selection could be an artifact of already knowing the target ratio. Three independent checks bear on it: the arithmetic conditionality of the pick itself, the existence of an independent mathematical criterion that reproduces it, and the chronology of when the selection rules were fixed relative to the comparison.

The status of the $(13,34)$ pair is conditional: it is mechanically unique given the framework's stipulated rules, as verified below, but it is not independently shown to be free of the target's influence on those rules. Within the candidate Fibonacci set $\{13,21,34,55\}/120$, the numerator $13$ is uniquely coprime to $120$, while $34$ is the unique even numerator. Thus, once the framework's coprime-well rule for $\afid$ and its bosonic even-numerator rule for $H_0$ are imposed, the pair is arithmetically unique conditional on the stipulated $F_7$--$F_{10}$ window. No formula in this selection directly consults the observed value $\afid/(cH_0) = 0.1833$; whether the rules generating the candidate window and the assignment criteria were themselves shaped by the four-decade-old public knowledge of that target is not something this manuscript establishes, and is not claimed.

Two qualifications keep that statement at its correct strength. First, the coprimality of $13$ is not independent of the lower bound: $1$ is also coprime to $120$, and it is the divisibility seam, not coprimality, that removes it. The seam is therefore load-bearing, and the coprimality rule is unique only on the set already fixed by the seam. Second, the $E_8$ structure is not independent corroboration. The exponents of $E_8$ are the totatives of its Coxeter number $h = 30 = 2\cdot3\cdot5$, and $30$ and $120$ carry the same prime radical, so coprimality to $30$ and coprimality to $120$ are the same condition for every integer. That $13$ is the only nontrivial Fibonacci $E_8$ exponent and that it is the only candidate coprime to $120$ are one fact stated twice. The $E_8$ framing places the seam inside active representation theory; it does not test it a second time.

A variational route to the same selection was tested against eight interference functionals, scoring all $\binom{51}{4} = 249{,}900$ four-subsets of $\{9,\ldots,59\}$ against the anti-periodic boundary modes $\varphi_m(k) = e^{i\pi(2m+1)k/120}$ under differing mode bands, weights, and normalizations (full construction in the foundational deposit); $\{13,21,34,55\}$ is extremal in none of them, since the well gaps are golden self-similar while these functionals favor uniform or maximally spread configurations. A ninth, combinatorial diagnostic -- the additive gap condition $g_3 = g_1+g_2$ on a four-subset's sorted gaps, satisfied by the wells' own gaps $(8,13,21)$ -- fares no better: $4{,}900$ of the $249{,}900$ four-subsets satisfy it non-uniquely, since the condition leaves one of the four positions unconstrained. Neither route selects the wells uniquely. A functional built specifically around $\varphi$-scaled spacing could in principle do so, but only by encoding the answer rather than deriving it, which is why none of the nine tested here was constructed that way.

This uniqueness should not be confused with a first-principles derivation of the candidate wells or of the observable-to-well eligibility rules. In particular, access to the full 120-grid alone does not select $13/120$, because $21/120$ and $55/120$ are also available on that grid. The manifold-mode classification, bosonic-projection filter, selection rule, and the prediction $\afid(z)\propto H(z)$ were specified in the framework's foundational deposit at the version current on 2026-06-24, the date of the present numerical comparison (DOI: 10.5281/zenodo.18729503), prior to that comparison. That precedence bears only on this specific 2026 comparison; it does not extend to the target ratio itself, which Section [ref:sec:derivation] already notes has been public since Milgrom (1983) and numerically stable across four decades of measurement, so the deposit date cannot certify that the rules were chosen without that long-known value in view. Conditional on those framework inputs, the $(13,34)$ split and its phase-factor ratio are fixed; a first-principles derivation of the Fibonacci-well restriction and the observable-to-well eligibility rules remains future work.

Statistical treatment of the intermediate-redshift comparisons

The {"U}bler two-bin discrepancy and shape statistic

The per-bin residuals under the most conservative uncertainty budget (Table [ref:tab:sigma-tension]) combine as $\chi^2 = T(0.9)^2 + T(2.3)^2 = 8.16$ on two degrees of freedom, corresponding to $p \approx 0.017$ and an equivalent two-sided Gaussian significance of $2.4\sigma$. This is a total two-bin model-data discrepancy: it sums each bin's absolute deviation from the model and does not isolate whether the direction of change between bins matches the prediction.

A shape-isolated test requires differencing the two bins, which cancels the local-baseline term shared between them: the observed change is $+0.17$~dex against a predicted change of $-0.313$~dex, a $0.486$~dex gap. The uncertainty on this gap has two contributions. The per-bin statistical errors ($0.04$ and $0.05$~dex) are independent between bins and add in quadrature. The velocity-correction systematic is a single-instrument effect: the same thick-disk prescription is applied at both redshifts, so a coherent survey-level bias amplitude, scaled by each effect's resolution-driven redshift dependence, is propagated through both bins together. Under that coherent treatment the forward model returns a differenced-systematic scatter of $0.031$~dex, giving a total gap uncertainty of $0.071$~dex and a shape-isolated significance of $6.8\sigma$. Treating the velocity-correction term as independent between bins at the $0.10$~dex-per-bin level instead gives $3.1\sigma$; this independent-bin treatment assumes the two bins' systematic responses are uncorrelated rather than anti-correlated, an assumption not independently tested here. Both figures exceed the $2.4\sigma$ total-offset figure, so the shape-isolated test strengthens the tension. The bias parameterization, redshift scalings, and random seed are those of the forward-model configuration in the following subsection.

Forward-model methodology

Mock galaxies at $z = 0.9$ and $z = 2.3$ were generated under the framework's prediction with literature-realistic distributions of baryonic mass, scale length, and velocity dispersion [Wisnioski2015]. The {"U}bler thick-disk correction $v_\text{circ}^2(r) = v_\text{rot}^2(r) + 2\sigma_0^2\,r/R_d$ was applied with the published selection cut $v_\text{rot,max}/\sigma_0 > 4.4$. Four bias models were swept over literature-plausible ranges. Each enters at a literature-sourced maximum amplitude at $z = 0.9$, scaled to $z = 2.3$ by a redshift-dependent multiplier: radial-position uncertainty (RPU, up to $5\%$ reduction in recovered $v_\text{rot}$, scaling $\times 1.7$; Genzel etal.\ 2014, {"U}bler etal.\ 2017 \S 3.2), beam-smearing inflation of $\sigma_0$ (BS, up to $30\%$, scaling $\times 1.9$; Wisnioski etal.\ 2015 \S 3, Burkert etal.\ 2016 Fig.4), non-asymptotic rotation-curve suppression of $v_\text{rot,max}$ below $\vflat$ (NA, up to $7\%$, scaling $\times 1.7$; Genzel etal.\ 2017 \S 2.2), and a signed shift in the $v/\sigma > 4.4$ selection threshold (SI, up to $\pm 15\%$, not redshift-scaled, a proxy for heterogeneous selection across the two bins). $N = 5000$ mock galaxies per redshift were drawn with $\log_{10}(\Mb/\Msun) \sim \mathcal{N}(10.5, 0.40)$, seeded deterministically (base seed $42$). The four bias amplitudes and the SI sign were swept on an $11$-point grid per dimension over $[0,1]^4 \times \{-1,+1\}$, with local refinement (half-width $0.10$, $11$-point grid) around the coarse-grid optimum. Optimizing the bias parameters for the $z = 0.9$ fit gives $\Delta b(0.9) = -0.29$~dex, leaving the $z = 0.9$ point $0.15$~dex above observed; at the same parameters $\Delta b(2.3) = -0.65$~dex, $0.38$~dex below observed. The joint $L_2$ best fit gives $(\Delta b(0.9), \Delta b(2.3)) = (-0.17, -0.43)$ at residuals $(+0.27, -0.16)$~dex. These are computational outputs of the archived pipeline's ubler_forward_model.py, and are not otherwise derivable from this manuscript. The mock-generation script, bias parameterization, and random seeds are archived with the rest of the analysis pipeline at the Zenodo deposit cited in the Data availability section.

SINS/zC-SINF per-galaxy transition-radius comparison

Table [ref:tab:genzel] gives the per-galaxy inputs underlying the $R_\text{out}/\rM(z)$ ranges quoted in Section [ref:sec:other-regimes].

SINS/zC-SINF per-galaxy inputs and the transition-radius comparison. $M_b$ is Genzel et al.'s fitted gas-and-bulge-inclusive baryonic mass; $R_{1/2}$ is the H-band half-light radius; $R_\text{out}$ is the low–high end of the $3.12$--$3.60\,R_{1/2}$ measured-extent range they report; $\rM(z)$ follows from Eq. (eq:eq:rM). Ratio cells are computed from unrounded $\rM(z)$ and $R_\text{out}$; dividing the printed (rounded) columns directly can differ by up to $0.01$.

Acknowledgments

The author thanks the observational teams whose published data made this work possible: the SPARC collaboration (Lelli, McGaugh, Schombert) for the local rotation-curve sample [Lelli2016,McGaugh2016]; {"U}bler and the KMOS3D team for the intermediate-redshift Tully–Fisher measurements [Ubler2017]; Wisnioski and the KMOS3D team for the kinematic-dispersion data used in the Section [ref:sec:ubler] forward model [Wisnioski2015]; Genzel and the SINS/zC-SINF team for the high-redshift rotation curves [Genzel2017]; Pointecouteau and Silk for the X-ray cluster analysis [Pointecouteau2005]; the JWST observing teams and the Labb{'e} group for the early-galaxy catalog [Labbe2023]; the Planck collaboration for the cosmological parameters and CMB amplitudes [Planck2018]; and the DESI collaboration for the dark-energy survey results [DESI2025]. No funding was received for this study, and the author has no competing interests to declare. During the preparation of this manuscript the author used large language models to assist with drafting and editing the exposition and to check symbolic and character-theoretic computations; the author conceived the results, reviewed and edited all text, and takes full responsibility for the content of the publication.

Data availability

All numerical predictions, tabulated framework outputs, and source code for the analysis pipelines and figure-generation scripts are archived at Zenodo (concept DOI: 10.5281/zenodo.19980665), mirrored from https://github.com/dmobius3/a0z. The archive includes the combinatorial baseline table (supporting Section [ref:sec:derivation]); the Section [ref:sec:channels] prediction tables (Tables [ref:tab:cosmology]--[ref:tab:five-exponents]) in CSV format; the {"U}bler $\sigma$-tension table and the four-bias forward-model script (mock-generation, bias parameterization, and random seeds; supporting Section [ref:sec:ubler]); the CMB leakage-bound table (supporting Section [ref:sec:cmb]); the Labb{'e} candidate speedup factors (supporting Section [ref:sec:collapse]); and an exploratory conditional lensing forecast (Newtonian-equivalent $\Mdyn/\Mb$ enhancement under an assumed relativistic completion, with a Navarro–Frenk–White (NFW)-based $\Lambda$CDM comparison), discussed as an outlook in Section [ref:sec:euclid] but archived separately and not carried as a claim in this manuscript. The Section [ref:sec:musedark2] comparison arithmetic can be reproduced directly from the quoted published values in [Ubler2017,Jeanneau2026] and this paper's own $E(z)$ and is not separately archived. The direct predictions of Section [ref:sec:channels] are additionally pre-registered at a fixed Zenodo version (DOI: 10.5281/zenodo.20045115) ahead of the matched-tracer follow-up of Section [ref:sec:euclid]; the version is cited rather than the concept DOI so the deposit cannot silently advance to a later release before the confirming or disconfirming data arrive.

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