Dark Energy as a Disequilibrium Residual

DESI DR2, the Volovik residual, and two readings the data retired

In the substrate the vacuum self-tunes to zero energy at equilibrium, so all of today’s dark energy is a residual left by expansion outrunning relaxation. Its scale is the substrate’s own, \rho_\Lambda^{1/4} \approx m_1c^2. That part of this chapter stands. What the chapter does not have is a dark-energy history the data can tell apart from a cosmological constant. Two attempts to supply one were tested and retired. The first read a spatial structure into the DESI data: a “moraine crust” left by the previous bubble, crossed at a transcritical point M(z) = 1 at z = 1.588, ringing as an undular bore. It is retracted (§ What was retracted). The M = 1 condition is the turnover of the angular-diameter distance, a property of every observer’s past light cone, and the spline structure it was meant to explain had about three degrees of freedom. The second was homogeneous: dark energy switches on as the diluting mean density approaches the lattice’s marginal density n_\text{tr} (§ The n_\text{tr} clock). Fitted to DESI DR2 BAO, Pantheon+ and Planck distance priors, it fails its own falsifier (WIP-38). Critical slowing at the marginal point produces a phantom drift the data reject, and the only switch-on the data allow comes while the mean density is still at least 40 times n_\text{tr}, too early to see. The prediction C = 1, no dark energy in the deep past, is therefore allowed but untested, and the dark-energy statement reverts to “Volovik form, value open”. The proposed S_8 suppression does no better: keyed to any history the data allow, it over-suppresses growth (§ Structure growth).

The DESI Anomaly: A Changing Dark Energy

The current standard model \LambdaCDM assumes dark energy is a cosmological constant: w = -1 at all times, with no evolution (w_0 = -1, w_a = 0 in the CPL parameterization w(a) = w_0 + w_a(1-a)).

The DESI data put pressure on this.¹ The combined DESI DR2 + CMB fit prefers w_0 = -0.42 \pm 0.21 and w_a = -1.75 \pm 0.58. Read literally, dark energy today behaves like quintessence (w > -1); tracing back in time it crossed w = -1 around z \approx 0.5, and before that it sat in a phantom regime (w < -1). How strong the preference is depends on the supernova sample: the 2025–2026 host-mass recalibrations generally weaken it.

A fundamental phantom fluid is theoretically toxic: w < -1 with positive energy density violates the null energy condition, and semiclassical realizations generically develop ghosts, gradient instabilities, or superluminality. The escape route the field has converged on is to make the crossing apparent. No physical component ever has w < -1; a \LambdaCDM observer’s bookkeeping merely reads one. The sharpest published construction is the dark axion–dark baryon model of Khoury, Lin & Trodden (below). The substrate shares that diagnosis. Its version was that dark energy is a relaxation residual whose density was smaller in the past, read through \LambdaCDM glasses. The fit below shows why that version gets only half the signal: a residual that only builds reads as phantom at every epoch and never reaches w_0 > -1.

The Volovik residual

From Gravity §G4–G5: in thermodynamic equilibrium, the substrate’s vacuum energy is exactly zero. The Gibbs-Duhem relation at T = 0 gives \varepsilon + P = 0 with \varepsilon = 0, so the superfluid self-tunes. The observed dark energy is entirely a residual from cosmic expansion preventing full relaxation:

\rho_\Lambda = \rho_\text{substrate} \cdot (\delta T / T_c)^2

Measured against the substrate’s own ground-state density \rho_\text{substrate}\approx\rho_\text{DM}, this disequilibrium is order unity: \delta T/T_c = \sqrt{\rho_\Lambda/\rho_\text{DM}}\approx1.6. The famous 10^{-61.5} appears only when the reference is the gravitational Planck density (see Gravity § The residual; the two differ by exactly the hierarchy factor (m_1/M_\text{Pl})^2). The cosmological constant’s nonzero value today is not a fine-tuned residual. It is a disequilibrium not yet drained.

Write the dark energy density relative to its present base value as f(z) \equiv \rho_\Lambda(z)/\rho_{\Lambda,\text{base}}, with a bulk-deficit term

f(z) = 1 - C\,\frac{z^2}{z^2 + z_b^2} + \ldots, \qquad f(z\to\infty) = 1 - C.

C = 1 is the Volovik prediction. Setting C = 1 gives f \to 0: dark energy was zero in the deep past, when the substrate was close to equilibrium. If fits had preferred C = 0.3 or 1.7, the model would have a fit but not a prediction. C = 1 is fixed by the framework independently of DESI. What the data say about it is in § The n_\text{tr} clock: they allow C = 1 only if the switch-on is early, with dark energy at half its present density by z \gtrsim 2.4 for the shape tested.

In the equation of state,

w(z) = -1 + \frac{(1+z) \cdot f'(z)}{3\,f(z)},

a dark energy density that was smaller in the past (f' < 0 at high z) reads as w < -1. The phantom regime is not exotic physics. It is the signature of a residual that has been building up.

What would set the history is the relaxation dynamics, written as a tracker ODE, \dot{(\delta T)} = -\delta T/\tau_\text{relax} + \alpha H (Gravity § The residual; WIP-16). Two forms of \tau_\text{relax} have been tried. The honest-stack fit (WIP-38) rejects both wherever their relaxation acts within the observed epoch:

• Critical slowing as the residual shrinks (\tau \propto \delta T^{-2}, the gravity chapter’s form). Its quasi-static solution is \delta T \propto H^{1/3}, so \rho_\Lambda \propto H^{2/3} grows into the past; only the fraction \Omega_\Lambda goes to zero. That is not C = 1, and on its attractor branch the fit rejects it at \Delta\chi^2 = +79.
• Critical slowing at the marginal point (\tau \propto |\bar n/n_\text{tr} - 1|^{-1}, the clock below). This one is C = 1, but it stops relaxing just as the residual should settle, so \delta T keeps growing until today.

Each fits only when its relaxation is frozen or nearly so, so that the residual is an inherited near-constant. In the frozen limit that is a cosmological constant. The gravity form’s best fit is a slow drain of the inherited value (w_0 = -0.96, \Delta\chi^2 = -1.9, not significant), the opposite of C = 1. The tracker never selected the present value anyway: \rho_\Lambda/\rho_\text{DM} is an inherited initial condition, set by how far \mathcal{B}^{-1} had relaxed.

One consequence needs saying plainly. A residual that only grows has \dot\rho_\Lambda > 0, which means w \le -1 at every epoch. That reproduces the phantom side of the DESI result but not a present-day w_0 > -1. To get the crossing, f(z) must peak in the recent past and turn over before today. On the honest stack the preference for w_0 > -1 is present: CPL improves on \LambdaCDM by \Delta\chi^2 = 7.5, and its best fit is exactly that shape. Dark energy rises from near zero in the deep past, peaks about 7\% above today’s density at z \approx 0.4, and has been falling since. That is the Volovik shape with an overshoot. No build-up form tested supplies the overshoot. Whether the preference survives the supernova recalibrations no longer decides anything for the substrate. A stronger preference cannot be met by a residual that only builds, and a weaker one leaves an early C = 1 switch-on indistinguishable from \Lambda.

The n_\text{tr} clock: tested and excluded

The Volovik residual says what dark energy is. It does not say when it switches on. The framework offered a candidate, stated as a conjecture in Galactic Dynamics § Why now?.

Above the marginal density n_\text{tr}, a uniform substrate is tachyonically unstable and must clump (Substrate Particles § The Marginal Point; The Winding Down). Clumping is a relaxation channel: it lets the medium shed super-criticality and stay near equilibrium, with \rho_\Lambda \approx 0. The diluting cosmic mean falls as

\frac{\bar n(z)}{n_\text{tr}} = \frac{\bar n_0}{n_\text{tr}}\,(1+z)^3,

and the framework places today’s mean at the marginal value, \bar n_0 \approx n_\text{tr}. The conjecture was that as \bar n approaches n_\text{tr}, the clumping engine winds down, relaxation slows, and the residual starts to build. Dark energy would then emerge because the medium is reaching its marginal point, and three “why now” coincidences would become one event:

• \rho_\Lambda \sim \rho_\text{DM} today;
• Milgrom’s a_0 \approx cH_0/2\pi (the diluting mean currently equals the fixed cell density);
• the bridge equation’s \rho_\text{DM}, fixed by \sin^2\theta_W and m_e, matching Planck’s \Omega_c.

Unlike the transcritical crossing, \bar n/n_\text{tr} is a function of cosmic time alone. Every comoving observer sees the same value at the same epoch, so the trigger is homogeneous and belongs in the Friedmann equation. Writing x \equiv \bar n/n_\text{tr} = (1+z)^3, the model is

f(z) = g(x), \qquad g(1) = 1, \qquad g(x) \to 0 \text{ as } x \to \infty \quad (C = 1).

One wrinkle was clear from the start: \bar n > n_\text{tr} at every z > 0, so the onset must precede the crossing.

What the fit found (WIP-38). The stack is DESI DR2 BAO, Pantheon+ without SH0ES, and Planck 2018 compressed distance priors, with h, \omega_b and \omega_c free in every model. The pipeline reproduces the standard results: \LambdaCDM at \Omega_m = 0.301, and CPL at w_0 = -0.85, w_a = -0.51.

Model	extra parameters	\Delta\chi^2 vs \LambdaCDM	What it shows
CPL (benchmark)	2	-7.5	the data’s preferred shape: Volovik plus an overshoot
clock as a switch-on, g = e^{-(x-1)/\Delta}	1	-1.6	not significant; half-on at z \approx 4, when \bar n \approx 135\,n_\text{tr}
clock as a tracker, relaxation slowing as x \to 1	1	+87 with freeze-out at z \approx 30	fits only when frozen; \LambdaCDM in the limit
same, crossing allowed in the past	2	\ge +52 while the crossing shapes g	its overshoot comes as a spike
the gravity chapter’s tracker, \tau \propto \delta T^{-2}	1	+79 on its attractor	not C = 1; its best fit (-1.9) is a residual slowly draining

Three things follow.

1. The clock’s mechanism is excluded. Critical slowing at the marginal point freezes the residual and then lets it grow by a fixed amount per e-fold, all the way to today. That is a phantom drift (w_0 = -1.20 for a freeze-out at z \approx 30), and the data reject it. The tracker gets within 2\sigma of \LambdaCDM only if its relaxation stopped by z \sim 10^{10}, when \bar n/n_\text{tr} \sim 10^{30}. Nothing about that involves the marginal point.
2. The surviving switch-on is too early to see. For the exponential shape, dark energy must already be at half its present density by z_{1/2} \gtrsim 2.4 (2\sigma), when \bar n/n_\text{tr} \gtrsim 40. So C = 1 is allowed, but only as a switch-on early enough to be dynamically invisible. That does not test it.
3. The sign problem is generic. A residual that only builds has w \le -1, and the stack prefers w_0 > -1. No form tested supplies the overshoot that would bridge the two.

By the conjecture’s own falsifier, the dark-energy statement reverts to “Volovik form, value open” (Gravity § The residual).

What survives. The fit tests whether dark energy switches on near n_\text{tr}, not whether today’s mean sits at n_\text{tr}. The crossing itself is untouched, and so are the two coincidences that are literally statements about it: Milgrom’s a_0 \sim cH_0 and the bridge \rho_\text{DM} matching Planck’s \Omega_c. The third coincidence, \rho_\Lambda \sim \rho_\text{DM}, is no longer an event at the crossing, because dark energy was in place long before it. What still ties it to the other two is the scale identity \rho_\Lambda^{1/4} \approx m_1c^2. A residual of order the cell density matches the diluting mean at whatever epoch the mean passes the cell density. That reduces three coincidences to one, why we live at the crossing, which is a smaller problem but not an explanation.

What was retracted: the transcritical moraine

Earlier versions of this chapter built a much richer structure on top of the Volovik residual. Our bubble’s wall was said to have decelerated through a shell of remnant material from the previous cycle (the “moraine crust”). The flow was supposed to cross criticality at M(z) = H(z)\,d_\text{proper}(z)/c = 1, which Planck parameters place at z = 1.588. The DESI data were read as a transcritical undular bore: a soliton at z \approx 2.3, a wake at 2.0, a recovery-zone node at 1.6, and a chirped downstream wave train. z = 1.588 was billed as a zero-parameter prediction. None of that survives review:

1. M = 1 is the angular-diameter-distance turnover. H\,d_p = c is algebraically the condition d(d_p)/dz = 0, the maximum of d_A, which sits at z \approx 1.59 for Planck parameters. It is a well-known property of our past light cone. Every comoving observer finds it in their own sky. It describes the geometry of looking, not anything in the medium.
2. It is not a Mach number. The Hubble flow is the substrate’s own bulk flow, not motion through it (Spacetime Dynamics § Friedmann setup). Material at the supposed moraine is at rest relative to its local substrate. Its recession speed relative to us is observer-dependent, and a Mach number has to be local.
3. A physical shell has one centre. It would sit on our Hubble sphere only if we were at the nucleation point. Otherwise it would appear at different redshifts in different directions.
4. It clashed with inflation. In the inflation chapter the bubble wall exits our past light cone when inflation ends (§ The number of e-foldings). There is no wall left inside our observable volume to strike a remnant at z \approx 2.2.

The data side failed too (§ What the fits show). The structure the bore was meant to explain came from a 15-knot spline fitted to 18 data points, with its key knots in a redshift gap.

What a consistent previous-cycle remnant would look like is a separate question, now in Speculation: A Remnant Inside the Bubble. There, remnant material swallowed and inflated along with our patch sits inside our volume as a local feature. Its signatures are anisotropic (a disk or dipole on the sky, a local void), not an isotropic shell in redshift.

What the fits show

The moraine model was fitted with a freeform spline. That record is kept here because it explains how the earlier claim arose.

The data. DESI DR2 BAO supplies 13 numbers at seven effective redshifts: z = 0.295 (D_V only), 0.51, 0.706, 0.934, 1.321, 1.484 and 2.33. The Jia et al. reconstruction adds five H_0(z) bins:²

Bin	z_\text{mid}	H_0 (km/s/Mpc)
1	0.1	72.20 \pm 0.19
2	0.3	71.62 \pm 0.36
3	0.5	69.78 \pm 0.51
4	0.7	68.13 \pm 0.67
5	2.5	67.23 \pm 0.84

Why the spline structure is not evidence. A 15-knot freeform spline in f(z) fitted to these 18 points reached \chi^2 \approx 9, against 68 for \LambdaCDM with free H_0. But:

• Almost no degrees of freedom. Fifteen knots on eighteen data points leaves about three. A spline with that freedom will fit almost anything.
• The “features” sit where there is no data. There is no BAO point between z = 1.484 and 2.33, so the knots at 1.6 and 2.0 were set by spline interpolation, a single Lyα pair and the Jia z = 2.5 bin. The four knots below the first BAO point (z = 0.295) rest mostly on the single Jia z = 0.1 bin. The alternating ridges and voids, the chirp and the rank-ordered carrier were all read off this knot pattern.
• The two data sets are not independent. The Jia H_0(z) bins are built from DESI BAO with Pantheon+ calibration, so a joint DESI + Jia fit counts DESI twice.
• BAO alone prefers \LambdaCDM. Fitted self-consistently, DESI BAO by itself drives the crust amplitude to zero (\chi^2 \approx 11 at H_0 \approx 69), consistent with the literature.
• A physical profile does much worse. The 3–4 parameter dispersive-shock profile reached only \chi^2 \approx 33–48.

So the joint fit showed that DESI + Jia can accommodate a present-epoch excess f(0) \approx 1.15–1.25 above a lower base density. It did not show oscillatory structure, and its \chi^2 against \LambdaCDM was not a model comparison. The real test needed independent data, used directly, with a model of a few parameters at most. That test is the n_\text{tr}-clock fit above. On independent data no clocked form carries a present-epoch excess, and the one departure from \LambdaCDM the stack prefers is CPL’s overshoot.

Full Friedmann self-consistency

One piece of the earlier work holds for any f(z) model: the bookkeeping that keeps the background flat (WIP-18). The expansion law must be normalized so H(0)\equiv H_0:

\frac{H^2(z)}{H_0^2}=\Omega_m(1+z)^3+\Omega_r(1+z)^4+\Omega_{\Lambda,\text{base}}\,f(z),\qquad \Omega_{\Lambda,\text{base}}=\frac{1-\Omega_m-\Omega_r}{f(0)}.

The Volovik base density sits a factor f(0) below today’s observed \Omega_\Lambda. The original bootstrap held \Omega_\Lambda=1-\Omega_m-\Omega_r fixed and plugged in f(z). With a present-epoch excess f(0)\approx1.25, that ran the model at H(0)/H_0\approx1.08, an 8% flatness violation. Closing this leak (friedmann_joint_selfconsistent.py, fit_freeform_selfconsistent.py) gave two results that carry over:

• The true H_0 stays Planck-consistent. The bootstrap’s “H_0(\text{local})=71.8 versus background 67.4” split was the flatness leak. With flatness enforced, H_0(\text{local})=H_0(\text{background}) by construction, and the fit keeps the true H_0 \approx 67–70 with \Omega_m h^2 at the Planck value (0.143).
• A present-epoch excess can mimic a high local H_0, but the data do not supply one. If f(0) > 1, a \LambdaCDM observer reconstructing H_0(z) from comoving distances reads a descending curve: higher at low z, Planck-like at high z. That is a way the Hubble tension could be a reconstruction artifact. The honest-stack fit (WIP-38) does not support it. No clocked form has an excess, because each rises monotonically to today. The clocked forms that do reproduce the Jia descent are the phantom trackers, and they do it by raising the true H_0 (to 72.6 for a freeze-out at z \approx 30), not by an artifact. The same stack rejects them at \Delta\chi^2 = +87. The candle side of the question is WIP-35.

Structure growth and S_8

The S_8 tension

The parameter S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3} measures the amplitude of matter density fluctuations at 8 h^{-1} Mpc, weighted by \Omega_m. Planck CMB observations predict S_8 = 0.832 \pm 0.013 by evolving the primordial power spectrum forward through the standard growth equation. Weak lensing surveys, which measure the actual clumping of matter at low redshift, found less:

Survey	S_8
Planck CMB	0.832 \pm 0.013
KiDS-1000	0.759^{+0.024}_{-0.021}
DES Y3	0.776 \pm 0.017
HSC Y3	0.769^{+0.031}_{-0.034}
KiDS-Legacy (2025)	0.815^{+0.016}_{-0.021}
DES Y6 3×2pt (2026)	0.789 \pm 0.012

Through 2024 the gap was 2–3\sigma and persistent across independent surveys. It has since narrowed. The final KiDS analysis, with improved redshift calibration, moved to 0.815, 0.73\sigma from Planck.³ DES Y6 sits 2.6\sigma low in the S_8 projection and 1.8\sigma in the full parameter space.⁴ What remains is a question of a few percent, not of ten.

Two channels of suppression

A late-time change in the substrate would act on growth through two physically distinct routes.

The background route (Hubble friction). The dark-energy history f(z) changes the Hubble friction on growing modes. On the only history the data allow (§ The n_\text{tr} clock), dark energy is close to constant over the growth epoch, and this route is small: S_8 = 0.816, against 0.804 for \LambdaCDM on the same stack.

The coupling route (boundary disruption). This is the channel unique to the substrate. Gravity’s coherent, low-acceleration response runs through the quadratic current-phase relation of the counter-rotating boundary (Galactic Dynamics). Anything that disrupts that boundary coherence lowers the effective coupling to perturbations, G_\text{eff}, without touching the homogeneous energy content.

The efficiency of that disruption was anchored as follows. The disturbance couples into the boundary through the HVBK mutual-friction interface, efficiency \alpha_{mf}, and the coupled energy disrupts the coherent response through the same interface, \alpha_{mf} again. Both steps must succeed, giving \alpha_{mf}^2. The factor of 2 comes from the dissipative and reactive components contributing comparably at the substrate’s operating point:⁵

\eta = 2\alpha_{mf}^2 = 2\left(\frac{\sin^2\theta_W}{1 - \sin^2\theta_W}\right)^2 = 0.181, \qquad \eta' = 2\alpha_{mf}^2(1-\sin^2\theta_W) = 0.139.

The second value is the same two-step result reduced by one further power of the angle (since 1/(1+\alpha_{mf}) = 1 - \sin^2\theta_W). The argument is heuristic: natural for a two-step mutual-friction process, but not derived from a full turbulence calculation.

What the test found. The earlier calculation placed the \eta = 0.181 feature at the transcritical crossing z = 1.588 and the \eta' = 0.139 feature at a “downstream exit” z \approx 0.52, both read off the moraine geometry, and reached S_8 = 0.816 on the self-consistent background. That placement went with the moraine. WIP-38 keyed the suppression to the clock instead, G_\text{eff}/G = 1 - \eta W(z), with the window bracketed two ways: W \propto g (tracking the built residual) and W \propto \mathrm{d}g/\mathrm{d}N (acting only while it builds). Because the surviving switch-on is at z \sim 4, both windows are open across the whole growth epoch:

	W \propto g	W \propto \mathrm{d}g/\mathrm{d}N	eBOSS f\sigma_8 \chi^2 (5 points; 7.3 without suppression)
\eta' = 0.139	0.752	0.761	14.2 / 11.5
\eta = 0.181	0.733	0.744	18.4 / 13.8

The anchored suppression drives S_8 below every current lensing value. The growth-rate data, which were not fitted, disfavour it at \Delta\chi^2 = +4 to +11. Meanwhile the tension it was built to ease has narrowed to a few percent. The coupling as anchored is not supported, and the framework makes no S_8 prediction. A version that survived would need a narrow window, which nothing in the framework now supplies, or an efficiency several times smaller. The 2\alpha_{mf}^2 efficiency itself remains an underived heuristic (HVBK Mutual Friction). The test bears on its use here, not on the number.

One finding from the earlier work carries over because it does not depend on placement. The nonlinear corrections do not lower S_8 (WIP-20). On 8\,h^{-1}Mpc scales the modes are deep in the MOND regime, where the boost enhances growth. Taken literally, MOND pushes S_8 up.

Why G_\text{eff} stays out of Friedmann

The suppression is a modification of the linearized growth equation with G \to G_\text{eff}, applied to perturbation growth only, not to the background. The same symbol G appears in Friedmann, H^2 = (8\pi G/3)\rho, but there it multiplies the homogeneous energy density that sources expansion. Boundary disruption removes no energy from the background. It weakens the boundary’s coherent gravitational response to perturbations. Force coupling and energy content are different physical quantities, and only the latter belongs in Friedmann. The genuine background modification is the dark-energy history itself, carried by f(z). (Applying G_\text{eff} to both sides was an early bookkeeping error.)

Treating the two appearances of G separately is not peculiar to the substrate. The “cosmic glitch in gravity” model (below) does it in the opposite direction. It lets the coupling in Friedmann, G_\text{cosmo}, differ from the Newtonian coupling in the Poisson equation, G_\text{N}, and Planck+ACT+SPT with DESI DR2 prefer G_\text{cosmo}/G_\text{N} = 0.9920 \pm 0.0025 at 3.3\sigma. In matter domination the linear growth equation is \ddot\delta + 2H\dot\delta = 4\pi G_\text{Poisson}\,\bar\rho_m\,\delta = \tfrac32\,\frac{G_\text{Poisson}}{G_\text{Friedmann}}\,H^2\delta, so growth sees only the ratio of the two couplings. The substrate’s proposed coupling would lower the numerator while the residual builds, by up to 14–18\%; the test above finds that amount over-suppresses growth. The glitch lowers the denominator at all epochs, giving a constant ratio of 1.008, which enhances growth. Both leave local, high-acceleration gravity at Newton’s value, so both pass the supernova test of the next section in the same way.

Type Ia supernovae as a test of G_\text{eff}

Type Ia supernovae, the standard candles, constrain any such coupling. Direct host-galaxy age measurements find a 5.5\sigma correlation between standardized SN Ia magnitude and progenitor age, at -0.030 \pm 0.004 mag/Gyr. Over 0 < z < 1, where the mean progenitor age drifts by \sim 5 Gyr, that is a \sim 0.16 mag drift that mimics acceleration. Corrected for it, the SN Hubble diagram aligns with DESI’s w_0w_a and reads q_0 = +0.18 \pm 0.06, a non-accelerating universe.⁶ The DES/SH0ES rebuttal holds that the standard host-mass correction already absorbs the age dependence, and that the measured evolution of the mass step, -0.028 \pm 0.034 mag per unit z, is consistent with zero. It adds that the claimed age gap conflates host age with progenitor age by a factor of three to five.⁷ The exchange is not settled. The framework does not need it settled to state its own position. That position has two halves, one about gravity and one about astrophysics, and they must be kept apart.

The gravity half: the candle is untouched. “Gravity was stronger earlier” in this framework means the MOND scale, a_0(z) = a_0(0)(1+z)^{3/2} (Early Structure Formation). A white dwarf’s interior sits at \sim 10^{6}–10^{8} m/s^2, some sixteen orders of magnitude above a_0. That is the incoherent high-acceleration regime, where each boundary scatters the ebbing current independently and gravity is exactly Newtonian (Galactic Dynamics). The evolving a_0 therefore never enters the Chandrasekhar mass, the ^{56}Ni yield, or the light curve, at any redshift. The framework predicts no intrinsic, gravitational evolution of SN Ia luminosity. Whatever evolution the data contain is astrophysical.

And the SN data are what force G_\text{eff} to stay out of local physics. Suppose the disruption did act on Newton’s constant inside stars. The Chandrasekhar mass scales as M_\text{Ch} \propto G^{-3/2}, and the peak luminosity follows it. On the raw Chandrasekhar scaling weaker gravity means brighter supernovae. Once light curves are shape-matched the sign reverses and weaker gravity means fainter ones, L_\text{std} \propto G^{+1.46}.⁸ Either way, an anchored suppression would print itself on the Hubble diagram:

Suppression	G_\text{eff}/G	\lvert\Delta M\rvert, Chandrasekhar (G^{-3/2})	\lvert\Delta M\rvert, standardized (G^{+1.46})
\eta' = 0.139	0.861	0.24 mag	0.24 mag
\eta = 0.181	0.819	0.33 mag	0.32 mag

A quarter-magnitude feature anywhere in 0.15 \lesssim z \lesssim 0.6, the best-sampled stretch of Pantheon+ and DES-SN5YR, is excluded by an order of magnitude. The binned residuals there are flat to a few hundredths of a magnitude, and the in-repo mass-step search (Galactic Dynamics, prediction 7) returns steps consistent with zero at \pm 25–40 mmag. So the statement in the section above is not a modeling convenience. The supernovae measure Newton’s G through the late-time epoch and find it constant at the few-percent level, while the anchored coupling would be a 14–18\% suppression over the same redshifts. Only a coupling that lives in the coherent, low-acceleration channel, and is invisible in the Newtonian regime, can satisfy both. The Weinberg-anchored \eta values were a proposal about S_8 and about a_0(z), now unsupported (§ Structure growth). They never predicted anything about white dwarfs, and the SN Ia record is what says so.

The same logic separates the substrate from the varying-G routes to the Hubble tension. Marra & Perivolaropoulos⁹ resolve it by making SNe Ia at z \gtrsim 0.01 intrinsically \approx 0.2 mag brighter through a 10\% weaker G_\text{eff} in the past. The substrate does not use the candle at all: its candidate Hubble-tension mechanism was a distance artifact (§ Full Friedmann self-consistency), and by the argument just given its G_\text{eff} cannot reach a white dwarf. The two are distinguishable by exactly the test above. A G-step model predicts a step in intrinsic luminosity at z_t; the substrate predicts none anywhere.

The astrophysical half: where a real age effect would come from. If the Yonsei correlation survives, it is a progenitor-population effect. The erratics chapter discusses a speculative transit-triggered SN Ia channel that needs only a white dwarf and so would preferentially fire in old stellar populations. Whether it has the right sign and a 0.03 mag/Gyr amplitude is not computed; Leung et al. find transit-triggered explosions close to standard [R144]. What the two halves fix together is the division of labour: gravity contributes zero, and any evolution in the candle is an astrophysical statement.

DESI reference data

The DESI DR2 + CMB combined constraints used for comparison:¹⁰

Quantity	Value
w_0	-0.42 \pm 0.21
w_a	-1.75 \pm 0.58
Correlation \rho_{w_0, w_a}	-0.85
\Omega_m	0.315

The error ellipse in the w_0–w_a plane is rotated by \sim 17° from the w_0 axis, with semi-axes (0.160, 0.920) and (0.262, 1.508) for the 1\sigma and 2\sigma contours respectively.

With Pantheon+ added, DESI DR2 finds w_0 = -0.838 \pm 0.055, w_a = -0.62^{+0.22}_{-0.19}, a 2.8\sigma preference (\Delta\chi^2 = -10.7) over \LambdaCDM. The WIP-38 pipeline, with compressed CMB priors in place of the full likelihood, finds w_0 = -0.85, w_a = -0.51 at \Delta\chi^2 = -7.5.

The logotropic precedent: Chavanis’s dark fluid

The substrate’s dark sector follows the log-EOS unification of dark matter and dark energy from Chavanis’s logotropic dark fluid:¹¹ a single dark fluid whose rest-mass energy plays dark matter and whose internal energy plays dark energy, governed by a logarithmic equation of state P = A\ln(\rho/\rho_P). Both model the logarithmic self-interaction of an elemental particle, whose rest-mass close-packing is the dark matter and whose log “internal energy” is the dark-energy residual. The substrate adds a microphysics (a Zloshchastiev condensate at its marginal point) and a cyclic cosmology underneath the same equation of state. The two models compare as follows.

The structural map:

	Chavanis logotropic	Substrate (dc1)
EOS	P = A\ln(\rho/\rho_P)	log-NLSE, V = -b\,n[\ln(na^3)-1]
Log coupling	A = B\,\rho_\Lambda c^2 (logotropic temperature)	b = m_1c^2 (per-particle energy)
Dark matter	rest-mass \rho c^2	rest-mass at close-packing, n_1\xi^3\approx1
Dark energy	internal energy u = -A[1+\ln(\rho/\rho_P)]	internal energy b\,n = \rho_\text{DM}c^2 at close-packing
DM abundance	not set (misalignment / dark magic)	\rho_\text{DM}c^2 = (m_1c^2)^4/(\hbar c)^3 (close-packing)

The substrate is a logotropic dark fluid parametrized by a per-particle energy b = m_1c^2 = \rho_\Lambda^{1/4} rather than by a pressure scale A. The two forms are the same physics expanded about different variables. Compared quantitatively, they differ on three points.

1. The \Omega ratio

Chavanis’s prediction is the present ratio of dark energy to dark matter \frac{\Omega_\text{de,0}}{\Omega_\text{dm,0}} = e = 2.71828\ldots \quad\text{vs. observed } 2.669\pm0.08\ (1.8\%). But it uses a “dark magic” relation that pins the present DE density \rho_\Lambda to the fundamental constant A with infinite precision, giving our epoch a central place in cosmic history. He suggests that the e-relation may belong to another theory.

The substrate has an equilibrium ratio that is not e but 1: close-packing sets \rho_\Lambda = \rho_\text{DM}. The observed \Omega_\Lambda/\Omega_\text{DM}\approx2.6 is a transient disequilibrium inherited from the previous bubble and not yet drained (§ The Volovik residual; Gravity § The residual). The substrate-referenced disequilibrium is its square root, \delta T/T_c = \sqrt{\rho_\Lambda/\rho_\text{DM}} = 1.61. The substrate once claimed more than this: a relaxation tracker whose attractor forgets its initial condition and reproduces the dark-energy history, so that “now” would be a generic point on a universal freeze-out rather than a tuned instant. The honest-stack fit (WIP-38) rejects every tracker form wherever its relaxation acts within the observed epoch. The data accept the tracker only frozen, with the residual an inherited constant. The n_\text{tr} clock, offered as the reason the freeze-out happens now, is excluded with it.

So Chavanis has a precise number with an admitted coincidence, and the substrate has neither the number nor a tested mechanism that selects it. What it has is the self-tuning that sets the equilibrium value to zero and the scale identity that puts \rho_\Lambda near \rho_\text{DM}. The frozen value \rho_\Lambda/\rho_\text{DM} is the inherited relaxation depth of \mathcal{B}^{-1}, so nothing the framework has tested outputs 2.6 the way Chavanis’s coincidence outputs e. He pays “dark magic” for a sharp number. The substrate pays with the number and, for now, with the mechanism too. Deriving the ratio, for instance from the nucleation dynamics of \mathcal{B}^{-1}, is still the single calculation that would let the substrate rival e = 2.718 on its own terms.

2. The universal halo surface density: parity, reached more naturally

Chavanis’s other prediction is astrophysical: logotropic halos have a universal surface density \Sigma_0 = 0.01955\,\frac{c\sqrt{\Lambda}}{G} = 133\ M_\odot/\text{pc}^2 \quad\text{(observed } 141^{+83}_{-52}\text{)}, with no free parameter. This is the same c\sqrt{\Lambda}/G \sim 135\ M_\odot/\text{pc}^2 scale as the MOND surface density a_0/(2\pi G) = 137\ M_\odot/\text{pc}^2. The substrate already predicts that, because a_0 = c\sqrt{G\rho_\text{DM}} is central to it, coming from the quadratic current-phase relation of the counter-rotating boundary. So this is a point of parity the substrate can claim, arguably reached more naturally. Chavanis must interpret his logotropic constant A as a fundamental constant to make the surface density universal. In the substrate, the halo surface-density scale, the MOND acceleration scale and the dark-energy scale are three faces of the single density \rho_\text{DM}. The DE-scale ↔︎ MOND-scale tie the substrate wants is exactly what Chavanis’s coincidence between A (cosmological) and \Sigma_0 (galactic) reports independently.

3. Chavanis’s B is the substrate’s weak-gravity hierarchy

The logotropic parameter is B = \frac{A}{\rho_\Lambda c^2} = \frac{1}{\ln(\rho_P/\rho_\Lambda)} = \frac{1}{283} = 3.53\times10^{-3}. He notes it is “essentially the inverse of the famous number 123”, the cosmological-constant hierarchy, and that it is nonzero only because \hbar\neq0: in the classical limit \rho_P\to\infty, B\to0 and the model reduces to \LambdaCDM. The argument \ln(\rho_P/\rho_\Lambda) = 283 is, identically, the substrate’s weak-gravity number: \ln(\rho_P/\rho_\Lambda) = 2\,\bigl|\ln(m_1/M_\text{Pl})^2\bigr| = 283.4, \qquad\Longrightarrow\qquad B_\text{Chavanis} = \frac{1}{2\,|\ln(m_1/M_\text{Pl})^2|}. So the single hand-tuned constant that makes Chavanis’s logarithm cosmological is the substrate’s “small \Lambda = weak gravity” ratio (m_1/M_\text{Pl})^2 (Gravity § Small \Lambda is the same number as weak gravity). Chavanis inputs the 123-orders hierarchy through B and correctly identifies its quantum origin (\hbar\neq0). The substrate routes it through the marginal-point critical slowing: the frozen photon mass at the \mu\to0 edge of stability, where (m_1/M_\text{Pl})^2 = (\ell_\text{Pl}/\xi)^2 is the squared ratio of the Planck length to the substrate cell. Same hierarchy, same recognition that it is quantum in origin, two mechanisms for its value. The substrate claims to explain the number Chavanis assumes.

What the substrate adds beyond Chavanis

Chavanis’s background is, up to the present epoch, indistinguishable from \LambdaCDM. His logotropic model departs only \sim27 Gyr in the future, in a phantom, super-de-Sitter era. After WIP-38 the substrate’s background is indistinguishable from \LambdaCDM too. Its one departure, C = 1 (\rho_\Lambda \to 0 in the deep past), is allowed only with a switch-on at z \gtrsim 2.4, where it does not show. What the substrate adds is the microphysics behind the equation of state and the identification of Chavanis’s B with weak gravity. Where Chavanis is ahead is the number: his e and his \Sigma_0 = 133 are sharp, parameter-free predictions, and the substrate’s honest answer on the ratio is still a mechanism in search of its value. Honoring the precedent means saying both plainly.

The interacting-dark-sector alternative: the DADB model

The strongest physically motivated competitor in reading the DESI anomaly as apparent is the dark axion–dark baryon (DADB) model of Khoury, Lin & Trodden.¹² It is a separate program from the superfluid dark matter work of the same author that the galactic chapters engage. Dark matter is composed of dark baryons, and the associated dark-QCD axion plays the role of dark energy. Finite-density corrections to the dark quark condensate make the dark-baryon mass axion-dependent, so an observer who assumes constant-mass dark matter attributes the mass evolution to an effective dark energy with w_{\rm eff} < -1. Implemented in a Boltzmann code and fit to Planck + DESI DR2 BAO + SNe, the model beats \LambdaCDM by \Delta\chi^2 = -14.48 with three extra parameters: a canonical scalar field, no ghosts, no fine-tuning.

Where the frameworks agree

• There is no fundamental phantom. No physical component ever has w < -1. In DADB the misreading sits in the dark matter column of the ledger, a mass evolution booked as dark energy, and it reproduces the full apparent crossing. In the substrate it would sit in the dark-energy history itself, a residual that was smaller in the past, read through \LambdaCDM glasses. That reading gets the phantom side and not the crossing, because a residual that only builds never reaches w_0 > -1 (§ The Volovik residual).
• The true H_0 stays Planck-consistent. DADB’s best fit gives H_0 = 67.65 \pm 0.68 km/s/Mpc, still 4.4\sigma from SH0ES. The model’s built-in early-dark-energy component peaks at f_{\rm EDE} \simeq 0.008, an order of magnitude too small to close the gap. The substrate’s flatness-corrected bookkeeping likewise keeps the true H_0 at \sim 67–70 (§ Full Friedmann self-consistency). Neither framework buys a genuinely high local expansion rate.
• The CMB constrains the high-z history. Paper II’s geometric argument is that the CMB anchors the acoustic scale \theta_* and the equality-era matter density, so the late-time trend cannot be extrapolated monotonically into the past. It supplies ready-made machinery for the substrate’s own CMB check (open calculation 3).

An honest benchmark

The DADB \chi^2 table is the right external yardstick on the standard likelihood stack: \Delta\chi^2 = -14.48 relative to \LambdaCDM (CMB + DESI DR2 + DES-Dovekie SNe, 3 parameters), against -12.43 for the two-parameter CPL parameterization on the same data. Fit to CMB + DESI without any SNe, DADB still prefers its coupling at \Delta\chi^2 = -8.94. The substrate now has a number on a comparable stack, with compressed Planck priors and Pantheon+ in place of DES-Dovekie. Its one-parameter clock reaches \Delta\chi^2 = -1.6, against -7.5 for CPL on the same data (WIP-38). That puts it on the table, at the \LambdaCDM end.

Where the frameworks part

Observable	DADB prediction	Substrate prediction
High-z dark energy	Axion potential energy present throughout	C = 1: \rho_\Lambda \to 0 at high z, allowed only with a switch-on at z \gtrsim 2.4 (untested)
Structure growth	Scale-dependent: small-scale P(k) enhanced, largest scales suppressed; net growth slightly up	No S_8 prediction: the anchored suppression over-suppresses on every history the data allow
Equality era	EDE-like bump, f_{\rm EDE} \simeq 0.008 at z \simeq 3500	Nothing localized at equality; only the smooth C = 1 deficit

Notably, DADB’s data-preferred solution migrated: Paper I’s regime gave a lower S_8, while Paper II’s best fit predicts slightly enhanced net growth. With the substrate’s coupling withdrawn, the growth row no longer discriminates between the two. With KiDS-Legacy at 0.815, the lensing data also press less hard on DADB’s slight enhancement than they once would have. The equality row is a discriminator that Planck already constrains and CMB-S4 will sharpen.

The SVT-internal alternative: Zloshchastiev’s quintom

There is a third comparison, and it comes from inside superfluid vacuum theory itself. Zloshchastiev, whose logarithmic EOS is the substrate’s defining nonlinearity, derives an inflation-to-dark-energy cosmology from the same equation ([R150]). The laminar-flow dilaton of the log superfluid, perturbed, yields a non-minimally coupled quintom: quintessence plus a tachyonic phantom with exponential kinetic coupling, “not postulated but derived”. He offers it as a resolver of the Hubble tension.

Where the frameworks agree, and it is the deep point: neither has a fundamental phantom. His phantom is a projection of density fluctuations; the substrate’s is a \LambdaCDM bookkeeping reading of a growing residual. Both say DESI’s w_0 = -0.42, w_a = -1.75 is emergent, the same diagnosis DADB reaches by yet another route.

Where they divide: his dark-energy potential \Delta V(\phi,\sigma) is “chosen ad hoc, as is common in cosmological models” (his words), and the coupling constant is free. Both transitions are homogeneous. The substrate’s candidate for what sets the onset was the approach of the mean density to the marginal point, which is the same marginal point his log EOS defines. That clock did not survive its fit (§ The n_\text{tr} clock), so neither framework fixes when the dark-energy term turns on.

His inflation (dilaton-driven, \rho \propto \tau^{-2} laminar de Sitter) and the framework’s (latent heat of the boil) are likewise different engines for the same epoch. His needs a reason the laminar phase exists and ends, which the boil provides.

The one-parameter alternative: a cosmic glitch in gravity

The most economical competitor to date changes no component of the universe, only how strongly the homogeneous universe gravitates. Wen, Hergt, Afshordi & Scott¹³ define a single parameter, \Omega_\text{g} \equiv 1 - G_\text{N}/G_\text{cosmo}. Here G_\text{cosmo} enters the Friedmann equation and G_\text{N}, the locally measured constant, enters the Poisson equation. Fitting Planck+ACT+SPT with DESI DR2 gives \Omega_\text{g} = -0.0081 \pm 0.0025, so the background gravitates 0.8\% more weakly than local systems. The minimum \chi^2 improves by 8.6 over \LambdaCDM, against 10.9 for w_0w_a with twice the parameters, and the Bayesian evidence ties (\Delta\ln\mathcal Z = +3.2 for each). The strongest result is that the glitch fitted to the CMB alone predicts the 1–2\% downward shift of the DESI distances at z \lesssim 1, while w_0w_a fitted to the CMB alone predicts nothing.

Where the frameworks agree

• Gravity, not a new fluid. Both read DESI’s evolving w as a misreading of something else and not as a phantom component. In the glitch the misreading is a coupling mismatch booked as dark energy. Its effective dark-energy density is negative at early times and crosses zero, so w_\text{DE} formally diverges. Both carry less dark energy than \LambdaCDM at high redshift: the glitch’s early negative term and the substrate’s Volovik deficit (f \to 0).
• The Friedmann and Poisson couplings are distinct. This is the split above, now with a 3.3\sigma empirical preference that it is real. The two frameworks put the deviation on opposite sides of the ratio.
• Same corner of theory space. A G_\text{cosmo} \ne G_\text{N} mismatch is the generic low-energy signature of gravity with a preferred foliation: Hořava–Lifshitz, Einstein-aether, cuscuton. Wen et al. conclude that the only viable versions have luminal gravitons, a scalar mode that local experiments cannot see, and a deviation visible only cosmologically. A superfluid substrate has a preferred rest frame by construction, gives c_\text{GW} = c exactly, and has a breathing mode invisible to co-moving detectors (Gravitational Wave Polarization). The map is not one-to-one: their khronon must be nearly incompressible (c_s^2 \gtrsim 10^4), whereas the substrate’s scalar branch travels at c.

An honest benchmark

The glitch achieves with one parameter what the n_\text{tr} clock was meant to. On the standard likelihood stack it resolves the CMB–BAO tension in the (r_d h, \Omega_m) plane from 2.1\sigma to 0.2\sigma. Its test, a CMB-calibrated prediction confirmed by DESI, is the hard kind to pass. Its preference weakens when CMB lensing and supernovae are added (\Delta\ln\mathcal Z falls to +1.1). The clock, fitted (WIP-38), gains only 1.6 on a compressed-prior stack where CPL gains 7.5. The two have not been compared on one stack. Compressed priors cannot test the glitch, because it changes the early-time expansion they assume, so that comparison needs the full CMB likelihood.

Where the frameworks part

Observable	Cosmic glitch prediction	Substrate prediction
Growth amplitude	Enhanced: \sigma_8 up 2.8\%, S_8 = 0.835; f\sigma_8 up a nearly constant \approx 3\% over 0.3 < z < 1.5	No prediction; the anchored suppression is not supported, so \LambdaCDM-like growth
Shape of w_\text{eff}(z)	Smooth, one parameter, tracks the dominant species	Indistinguishable from w = -1 where measured; C = 1 allowed only with a switch-on at z \gtrsim 2.4
Potentials in matter era	Slowly growing, \Phi \propto a^{-3\Omega_\text{g}/5}, giving an early ISW signal absent in \LambdaCDM; no slip, \Phi = \Psi	Constant, as in \LambdaCDM

The growth row is still the sharpest test of the glitch. DESI and Euclid full-shape RSD will measure f\sigma_8(z) at the percent level, and a flat +3\% offset would favour it. A null result would not separate the substrate from \LambdaCDM. The glitch also worsens the S_8 tension, though with KiDS-Legacy and DES Y6 at 0.79–0.82 that tension is now small.

Where this leaves dark energy

Three results stand.

• Self-tuning. The equilibrium vacuum energy is zero (Gibbs–Duhem), and today’s \rho_\Lambda is a disequilibrium residual, order unity in the substrate’s own units (Gravity § The residual).
• The scale. \rho_\Lambda^{1/4} = 2.24 meV \approx m_1c^2. Dark energy and dark matter are two fourth roots of one substrate density, with the ratio 2.59 measured, not derived.
• The history is \LambdaCDM’s, as far as the data reach. C = 1 is allowed only with a switch-on at z \gtrsim 2.4. Every relaxation tracker tried fits only when frozen or nearly so. The departure the data prefer needs an overshoot the substrate does not have.

What would change this:

1. A derived switch-on. A \tau_\text{relax} computed from the substrate’s viscous response (WIP-16) must place the switch-on at z \gtrsim 2.4, or C = 1 fails outright. One that lands near the bound would make C = 1 a target for the full CMB likelihood and for z \approx 2–3 probes.
2. A mechanism for an overshoot. Something that resumes relaxation after the residual has built, smoothly, without the spike the past-crossing tracker gave. CPL’s best fit says what shape it would need: a peak about 7\% above today’s density near z \approx 0.4.
3. A selector for the value. The ratio 2.59 is the inherited relaxation depth of \mathcal{B}^{-1}. Deriving it from the nucleation dynamics (A Universe That Boils, breadcrumb 1) would give the substrate a number to set beside Chavanis’s e.

Failing those, the statement is “Volovik form, value open”, which is where WIP-38’s falsifier left it.

Open calculations

1. Fit the n_\text{tr} clock. Done (2026-11-03), negative (WIP-38). The clocked residual fails its own falsifier (§ The n_\text{tr} clock).

2. Derive the onset scale from \tau_\text{relax} (WIP-16). Compute \tau_\text{relax} from C7 and the substrate viscous parameters, then solve H(z_b)\,\tau_\text{relax} = 1. The target is now a bound: the switch-on must come at z \gtrsim 2.4.

3. Run the full CMB likelihood. Compressed priors served for the late-time models tested so far. An early switch-on at z \sim 2–5 is where they lose information: most of the exponential clock’s small gain came from the CMB term. The cosmic-glitch comparison needs the full likelihood anyway. The first-order geometric machinery in Khoury–Lin–Trodden Paper II (§ DADB) is a shortcut for the distance part. Compute \delta\ln H(z) for the switch-on profile, integrate \delta D_M(z_*) = -\int \mathrm{d}a\, \delta\ln H / (a^2 H^{(\Lambda)}), and check it against the Planck \theta_* precision.

4. Recompute S_8 with the coupling keyed to the clock. Done (2026-11-03), negative. The anchored suppression over-suppresses (§ Structure growth). A full HVBK derivation of 2\alpha_{mf}^2, in particular whether the reactive coefficient \alpha_{mf}' really contributes comparably to the dissipative \alpha_{mf}, remains open but no longer carries a cosmological prediction.

5. Connect to a_0(z). Withdrawn. The proposed 7–9\% late-time dip in a_0, a_0 \to a_0\sqrt{1-\eta(z)}, was the same coupling read at galactic scale, and it goes with the coupling. The baseline a_0(z) = a_0(0)(1+z)^{3/2} of Galactic Dynamics does not depend on it.

6. Test the robustness of the verdict against the 2025–2026 supernova corrections (WIP-35). The host-mass recalibrations (DES-Dovekie, corrected Pantheon+ from Hoyt et al. 2026, Union3.1) shift \sim 0.03–0.05 mag on a subset of the sample and generally weaken the dynamical-dark-energy preference. The progenitor-age correction of Son et al. [R175] is a \sim 0.16 mag monotonic drift over 0 < z < 1, five times larger, and is contested [R176]. Rerunning the WIP-38 fit under each is worth doing, but it cannot revive the clock. A stronger w_0 > -1 preference is what a build-up residual cannot produce, and a weaker one leaves C = 1 invisible. Why the candle itself is gravitationally untouched, whichever way the correction goes, is § Type Ia supernovae as a test of G_\text{eff}.

7. Look for a remnant the honest way (WIP-39). If a previous-cycle remnant sits inside our volume, it shows up as anisotropy, not as an isotropic feature in z. Splitting DESI BAO and Pantheon+ by hemisphere and cross-checking against the quasar dipole excess (The Two Dipoles) is the test; see A Remnant Inside the Bubble.

Footnotes

1. DESI Collaboration, “DESI 2024 VI: Cosmological Constraints from Baryon Acoustic Oscillations,” arXiv:2404.03002, 2024. Updated in DR2, 2025.

2. Jia et al. (2025), ApJL 994 L22, Table 2. Using DESI+PP calibration.

3. Wright, A.H. et al., “KiDS-Legacy: Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey,” A&A 703, A158 (2025), arXiv:2503.19441.

4. DES Collaboration (Abbott, T.M.C. et al.), “Dark Energy Survey Year 6 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” arXiv:2601.14559 (2026).

5. In the HVBK formalism, the mutual friction force has dissipative (\alpha_{mf}) and reactive (\alpha_{mf}') components. At the substrate’s operating temperature, both channels are active, giving a total efficiency 2\alpha_{mf}^2. See Hall & Vinen (1956), Bekarevich & Khalatnikov (1961).

6. Son, J., Lee, Y.-W., Chung, C., Park, S. & Cho, H., “Strong progenitor age bias in supernova cosmology. II. Alignment with DESI BAO and signs of a non-accelerating universe,” MNRAS 544, 975, 2025, arXiv:2510.13121 [R175].

7. Wiseman, P., Popovic, B., Sullivan, M., Riess, A.G., Scolnic, D., et al., “Still accelerating: type Ia supernova cosmology is robust to host galaxy age evolution,” MNRAS 549, 2026, arXiv:2601.13785 [R176]; the Yonsei reply is Chung et al., “Still non-accelerating,” MNRAS 551, 2026, arXiv:2605.21586.

8. Wright, B.S. & Li, B., “Type Ia supernovae, standardizable candles, and gravity,” Phys. Rev. D 97, 083505, 2018, arXiv:1710.07018 [R177].

9. Marra, V. & Perivolaropoulos, L., “Rapid transition of G_\text{eff} at z_t \simeq 0.01 as a possible solution of the Hubble and growth tensions,” Phys. Rev. D 104, L021303, 2021, arXiv:2102.06012.

10. DESI Collaboration (2025), DR2 combined analysis.

11. Chavanis, P.-H., “Predictions from the logotropic model: the universal surface density of dark matter halos and the present proportion of dark matter and dark energy,” Eur. Phys. J. Plus 137, 525 (2022); arXiv:2201.05903. Building on Chavanis, Eur. Phys. J. Plus 130, 130 (2015).

12. Khoury, J., Lin, M.-X. & Trodden, M., “Apparent w<-1 and a Lower S_8 from Dark Axion and Dark Baryons Interactions,” Phys. Rev. Lett. 135, 181001 (2025), arXiv:2503.16415 (Paper I); “Cosmological Evidence for Dark Axion–Dark Baryon Interactions from Apparent Phantom Crossing,” arXiv:2607.16191 (2026) (Paper II, the full likelihood analysis).

13. Wen, R.Y., Hergt, L.T., Afshordi, N. & Scott, D., “Is DESI Seeing Dynamical Dark Energy, or a Cosmic Glitch in Gravity?” arXiv:2609.40176 (2026); the model is introduced in Wen et al., JCAP 2024, 045, arXiv:2311.03028.