Version: 0.1
Z. Paz
The Hague, Netherlands
April 2026
STATUS NOTICE — 21 August 2026. Two pieces of this paper’s reasoning are withdrawn framework-wide; its central results survive on replacement derivations, and one is strengthened. (1) The DHOST Class Ia classification and the G₄X = 0 route to c_T = c are withdrawn (First Principles V8.1 Appendix P: the Horndeski mapping’s own expression gives G₄X ≠ 0, the mapped G₄ carried an extra φ, the required companion term was absent, and ℛ is Weyl-based, not Ricci — the classification does not hold). The GW-speed conclusion survives by calculation instead of class label: on the scalar–Gauss–Bonnet route, |c_T/c − 1| ≲ 7×10⁻⁴¹ (tracking) and ≲ 2×10⁻³⁰ (oscillating) — 14+ orders inside GW170817 (V8.1 §VII, Appendix N). Ghost-freedom is regime-limited (established on exterior-vacuum and Kerr backgrounds via the Gauss–Bonnet parent; open in general), which is sufficient for the “effective phantom without fundamental ghost” statement this paper needs, at that stated strength. (2) The cross-disformal matter coupling is withdrawn (not generated by the framework’s 10D reduction, B̂_KK = 0; coefficient fixed by the withdrawn flyby match) — with it fall this paper’s references to the flyby anomaly and the galactic MOND mechanism as companion validations; a₀ = cH₀/2π is now a conditional target, not a derived companion result. (3) The clock vector n^μ = ∇^μφ/√(2X) is valid in the tracking regime only; the current formulation uses the universal clock N^μ (V8.1 §II.C), and “m_s and ζ/Λ derived from first principles” reads at current status as derived-conditional (V8.1 §III.E). What stands untouched: everything this paper actually derives — the T² causal-diamond structure, w(z = 0) = −1 exactly from the third-order tangency, the monotonic effective-phantom history, Ω_m = 4/(3(1+π)), c_s²(z=0) = 1, the honest DESI quintom-B tension and its falsification condition, and the Euclid test. Sectional notes mark the load-bearing sites; this notice governs where the text conflicts.
We present the dark energy sector of the Selective Transient Field (STF) framework — a unified scalar-field theory in which dark matter and dark energy emerge from the same scalar field, the breathing mode of six compact extra dimensions in a 10D Einstein-Gauss-Bonnet compactification on a Calabi-Yau threefold (CICY #7447) with Z₁₀ free quotient structure. The STF scalar has mass m_s = 3.94 × 10⁻²³ eV and coupling ζ/Λ = 1.35 × 10¹¹ m², both derived from first principles. The dark energy component arises from the residual potential energy V(φ_min) at the stabilized modulus, modulated by the T² causal-diamond integral α(θ) = ∫₀^θ cos²(θ’)dθ’. The current epoch is identified with θ = π/2 by the geometric condition |R₀|/c² = 4Λ_eff (Paz 2026a, §III.E Prediction 6), which gives Ω_m = 4/(3(1+π)) ≈ 0.322 — within 1σ of Planck 2018 (0.315 ± 0.007) and consistent with DESI DR1/DR2 (0.295–0.307, model-dependent). The third-order tangency dα/dθ|_{π/2} = cos²(π/2) = 0 forces Λ̇_eff = 0 at the current epoch, predicting w(z=0) = −1 exactly (independent of the compactification time-scale T_compact). At earlier epochs θ < π/2 the coupling was accumulating, giving an effective phantom trajectory w(z) < −1 for all z > 0 (sign rigorous; magnitude conditional on T_compact, see Paz 2026c §6.1) without a phantom crossing. The STF is a DHOST Class Ia theory with positive scalar kinetic energy and c_T = c exactly (GW170817-compatible, Paz 2026a §C.6); the apparent phantom behaviour is effective, arising from T² geometric coupling accumulation, not fundamental ghost behaviour. The same nodal structure that gives w₀ = −1 also gives c_s²(z=0) = 1 exactly (Paz 2026c §6.3) — perturbation stability paired with the equation-of-state result. We conduct a systematic comparison with eight competing dark energy frameworks: the cosmological constant Λ, quintessence, k-essence, phantom dark energy, w₀wₐCDM (CPL parameterization), early dark energy, scalar-tensor dark energy (Horndeski/DHOST), and emergent dark energy models. We compile and contextualize recent observational evidence: (i) the 2.8–4.2σ statistical preference for w₀wₐCDM over ΛCDM in DESI DR2 + CMB + SNe combined fits (Adame et al. 2024, Karim et al. 2025), with best-fit indicating evolving dark energy (w₀ > −1 today, wₐ < 0); (ii) the Euclid Quick Data Release Q1 (March 2025) sky coverage of 63 deg² as a precursor to its first cosmology release (October 2026); (iii) recent debates about the prior dependence of the DESI w₀wₐ result (Cortês & Liddle 2024, 2025); and (iv) alternative interpretations including coupled dark sector models (Roy Choudhury et al. 2025) and evolving dark matter scenarios (Chen & Loeb 2025). The DESI DR2 best-fit trajectory — apparent quintom-B (w<-1 in past, crossing to w>-1 today) — is structurally opposite to the STF prediction (no crossing; monotonic phantom for z > 0; w₀ = −1 exactly). We treat this as a serious observational tension that the framework must address honestly: if direct measurements confirm the quintom-B sign, the T² nodal structure is falsified at >3σ. The framework’s structural prediction is that w₀ = −1 lies exactly on the cosmological constant, with all evolution into the past. Euclid’s first cosmology release (October 2026) will be decisive. We also address theoretical aspects: the GW170817 constraint c_T = c (satisfied structurally by G₄X = 0); the no-go theorem for fundamental phantoms (evaded by DHOST Class Ia structure); the relationship to the DESI w₀wₐ best-fit (categorically different mechanism — STF’s effective phantom is monotonic, DESI’s CPL is a parametrization artifact of strongly anticorrelated posteriors). We present six testable predictions, three falsifiability classes (rigorous-structural, conditional-magnitude, derived-extension), and a prediction-dependency map indicating which observables survive if any single component fails. The unified dark sector picture — one scalar field producing both the dark-matter equation of state ⟨w⟩ = 0 (Paz 2026d) and the dark-energy structural prediction w(z=0) = −1 — is structurally over-constrained relative to a fitted dual-component model. Of the framework’s testable predictions, the structural prediction w(z=0) = −1 exactly is the most distinctive: it is independent of T_compact, derives from T² differential topology rather than parameter fitting, and is the single most directly testable element of the framework against forthcoming Euclid data.
Keywords: dark energy, equation of state, T² causal diamond, third-order tangency, DHOST Class Ia, GW170817, phantom dark energy, scalar field cosmology, DESI w₀wₐCDM, Euclid mission, Calabi-Yau compactification, ghost-free scalar-tensor gravity, effective phantom without ghost, unified dark sector, cosmological constant problem
The accelerating expansion of the universe, discovered through Type Ia supernova observations (Riess et al. 1998, Perlmutter et al. 1999), has been cosmology’s most enduring puzzle for over a quarter century. The simplest description — a cosmological constant Λ — fits the data well but raises three deep theoretical problems.
The cosmological constant problem. Quantum field theory naively predicts a vacuum energy density ~10¹²² times larger than observed (Weinberg 1989). No accepted derivation of the observed value from microphysics exists. The vast disparity between predicted and observed scales is among the most severe fine-tuning problems in physics.
The coincidence problem. Within ΛCDM, dark matter and dark energy densities differ by orders of magnitude at most cosmic epochs but happen to be comparable now, with Ω_m ≈ 0.32 and Ω_Λ ≈ 0.68. Why this near-equality occurs at the current epoch — rather than billions of years earlier or later — has no explanation in the standard framework.
The dynamical question. Is dark energy a true cosmological constant (w = −1 exactly, time-independent), or does it evolve? If it evolves, does it cross the phantom boundary w = −1, and if so, in which direction?
The Dark Energy Spectroscopic Instrument (DESI) Year-1 BAO results (Adame et al. 2024) and the Year-2 Data Release (Karim et al. 2025) have transformed the empirical landscape on this question. Combined fits of DESI BAO with cosmic microwave background distance priors and Type Ia supernova compilations (Pantheon+, DES Year-5, Union3) now show a 2.8–4.2σ statistical preference for the w₀wₐCDM parameterization over a constant Λ. The best-fit shows w₀ > −1 today (quintessence-like at present) with wₐ < 0 (phantom in the past), indicating an apparent quintom-B trajectory with phantom crossing in the recent past. If confirmed, this would be the most significant empirical development in cosmology since the discovery of acceleration itself.
However, the result is contested. Cortês & Liddle (2024, 2025) demonstrated that the DESI Bayesian preference for w₀wₐCDM is sensitive to the priors imposed on w₀ and wₐ — extending the lower bounds beyond −4.6 (for w₀) and −5 (for wₐ) reverses the preference toward ΛCDM. Roy Choudhury et al. (2025) showed that updating the CMB likelihood from Planck PR3 to PR4 weakens the dynamical-dark-energy preference. Chen & Loeb (2025) proposed that the DESI signal can be reproduced by an evolving dark matter component rather than evolving dark energy. The empirical case for dynamical dark energy is suggestive but not yet conclusive.
The Euclid space telescope (launched July 2023, beginning its cosmological survey February 2024) will provide the next major step. Its Quick Data Release Q1 (March 2025) covered 63 deg² — a precursor to the first cosmology data release in October 2026, which will probe dark energy through a combination of weak gravitational lensing, galaxy clustering, and BAO measurements over much wider sky coverage than DESI.
This paper presents the dark energy sector of the Selective Transient Field (STF) framework — a unified scalar-field theory in which dark matter and dark energy emerge from the same scalar field. The STF was developed primarily as a UV-complete model of dark matter, deriving from a 10D Einstein-Gauss-Bonnet compactification on the Calabi-Yau threefold CICY #7447 with Z₁₀ free quotient structure (Paz 2026a, hereafter “the First Principles paper”). The dark-matter aspects of the framework are presented in Paz 2026d (“Dark Matter as Geometry: The STF Framework for a Unified Dark Sector”); the present paper is its parallel for the dark-energy sector.
The unified-dark-sector framing might suggest that dark energy and dark matter should be addressed in a single paper. Three reasons make a separate dark-energy treatment appropriate:
(1) The observational landscapes differ qualitatively. Dark matter is constrained primarily by galactic kinematics, gravitational lensing, structure formation, and direct-detection null results. Dark energy is constrained primarily by Type Ia supernova distances, BAO from spectroscopic galaxy surveys, CMB acoustic peaks, weak lensing tomography, and (forthcoming) redshift-space distortions and 21-cm cosmology. The competing-theory landscapes are also different: ΛCDM, MOND, fuzzy DM, WDM, SIDM, superfluid DM, and emergent gravity for dark matter; cosmological constant, quintessence, k-essence, phantom, w₀wₐCDM, early DE, and scalar-tensor DE for dark energy.
(2) The structural mechanisms differ. The STF dark-matter mechanism rests on the oscillation-averaging of the scalar field at cosmological scales (giving ⟨w_DM⟩ = 0) combined with the field’s response to galactic geometry (giving the MOND phenomenology). The STF dark-energy mechanism rests on the residual potential energy at the stabilized modulus combined with the T² causal-diamond integral structure (giving the w(z=0) = −1 nodal result). These are different microphysical mechanisms within the same framework, deserving separate analysis.
(3) The observational tests are temporally separated. The dark-matter predictions test against existing rotation curves, SPARC catalog, Lyman-α constraints, and dwarf-galaxy kinematics. The dark-energy predictions test against forthcoming Euclid w(z) measurements (October 2026 first cosmology release) and DESI DR3+ analyses. The two test programs run on different timescales and require different observational synthesis.
This paper makes three primary contributions to the STF framework’s publication portfolio:
(1) A self-contained dark-energy treatment. Sections II–III present the STF framework’s relevant structure for dark energy (compact summary) followed by the explicit derivation of the w(z) result from T² causal-diamond geometry. The full calculation chain is in Paz 2026c (“STF Dark Energy w(z) Derivation V0.2”); this paper presents the result in publication form with observational context.
(2) A systematic comparison with competing dark-energy theories. Section IV compares the STF prediction against eight competing dark-energy frameworks, identifying for each: the underlying mechanism, observational successes, theoretical vulnerabilities, and the specific aspects in which the STF differs. The comparison is honest — the STF is not advertised as superior in every respect, and the empirical case for STF over alternatives is qualified by the open T_compact question and the current observational tensions.
(3) Engagement with the DESI DR2 observational tension. The DESI DR2 result, if confirmed, falsifies the STF dark-energy prediction at >3σ. Section V addresses this directly: we present the DESI result honestly, discuss the prior-dependence and CMB-likelihood concerns, examine alternative interpretations, and identify the specific Euclid measurements that would be decisive. We do not minimize the tension or massage the prediction to accommodate it. The framework’s structural prediction is w₀ = −1 exactly; if the empirical case for w₀ > −1 strengthens, the framework is in serious trouble. We discuss the falsification scenarios explicitly.
Section II presents the STF framework, focused on the elements relevant for dark energy. Section III develops the STF dark-energy sector: the residual potential, the T² causal-diamond structure, the third-order tangency, and the resulting predictions w(z=0) = −1 and c_s²(z=0) = 1. Section IV conducts the systematic comparison with eight competing dark-energy theories. Section V compiles recent observational evidence and addresses the DESI DR2 tension. Section VI provides an honest assessment of STF’s limitations. Section VII presents testable predictions with falsifiability classes and a prediction-dependency map. Section VIII discusses broader implications and the unified dark sector picture. Section IX concludes.
This section summarizes the STF framework’s structure, focusing on the elements relevant for the dark-energy analysis. Comprehensive treatments are in Paz 2026a (the First Principles paper, full mathematical derivation) and Paz 2026d (the Dark Matter paper, applied phenomenology).
The STF Lagrangian is:
\[ \mathcal{L}_{\rm STF} = -\frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}m_s^2\phi^2 + \frac{\zeta}{\Lambda}g(\mathcal{R})\phi(n^\mu\nabla_\mu\mathcal{R}) + \mathcal{L}_{\rm matter\ couplings} \]
where φ is the scalar field, m_s is the field mass, ζ/Λ is the coupling constant, ℛ is the Weyl/Ricci tidal scalar (regime-dependent — see §II.D), n^μ = ∇^μφ/√(2X) is the unit vector along the field gradient, and g(ℛ) is a smooth threshold function. (Superseded, August 2026: the cross-disformal matter coupling formerly listed here is withdrawn — see the status notice; no matter coupling beyond minimal is currently part of the framework’s action.)
Field mass. m_s = 3.94 × 10⁻²³ eV, derived from the cosmological threshold condition 𝒟_crit = 𝒟_GR (Paz 2026a §III.D), which identifies scalar field activation at the orbital separation where binary black hole inspiral becomes cumulative — 730 R_S, corresponding to a period of T = 3.32 years via the Peters (1964) inspiral formula. The oscillation frequency is ω_s = m_s c²/ℏ = 5.98 × 10⁻⁸ rad/s.
Coupling constant. ζ/Λ = 1.35 × 10¹¹ m², conditional on the 10D Einstein-Gauss-Bonnet compactification on CICY #7447/Z₁₀ plus the retarded matching (τ_eff, C_match; V8.1 §III.C). (Status, August 2026: the former “independent validation route through the Anderson flyby anomaly” is withdrawn — the mechanical derivation of K = 2ωR/c fails four no-gos, the 99.99% agreement is an in-sample match to Anderson’s fitted constant, and the out-of-sample record fails (Rosetta II/III, Juno nulls).) This does not touch the dark-energy analysis itself: the STF dark-energy predictions follow from the cosmological-scale dynamics of φ on FRW backgrounds and never rested on the flyby sector.
The compactification chain proceeds through:
\[ S_{10} = \int d^{10}x\sqrt{|g_{10}|}\left[\frac{1}{2\kappa_{10}^2}R_{10} + \alpha_{\rm GB}\,\mathcal{G}_{10}\right] \]
with the Gauss-Bonnet term ℳ_{10} = R² − 4R_μν R^μν + R_{μνρσ}R^{μνρσ}. The breathing-mode reduction:
\[ g_{10} = e^{-\sigma(x)/\sqrt{6}}g_4(x) \oplus e^{\sigma(x)/\sqrt{6}}g_6 \]
yields a 4D Einstein frame with M_Pl² = M_{10}⁸ V_6 and canonical scalar φ = √24 M_Pl σ. The internal manifold is the Calabi-Yau threefold CICY #7447 with Z₁₀ = Z₅ × Z₂ free quotient structure (Braun 2010), yielding h¹¹(X̃) = 5 independent Kähler moduli and a smooth quotient with three generations of fermions. Within the CICY database (7,890 manifolds), #7447 is the unique manifold admitting a free Z₁₀ quotient (Paz 2026a, Appendix O.X).
The coupling ζ/Λ is the output of the only available geometry, not a selection from a landscape of alternatives. The same compactification that produces ζ/Λ and m_s is conjectured to derive Standard Model parameters through Kaluza-Klein scale ratios and loop structure (Paz 2026a, Appendices M–O). The chain extends to dark energy through the mechanism described in §III below.
(The former derivation — DHOST Class Ia classification with G₄X = 0 — is withdrawn: V8.1 Appendix P records that the mapping’s own expression gives G₄X ≠ 0 and the classification does not hold.) The conclusion survives by direct calculation on the scalar–Gauss–Bonnet route: α_T ≃ 8(f̈ − Hḟ)/M_Pl² gives |c_T/c − 1| ≲ 7 × 10⁻⁴¹ in the tracking regime and ≲ 2 × 10⁻³⁰ in the oscillating dark-matter regime — fourteen or more orders inside the GW170817 constraint |c_T − c|/c < 5 × 10⁻¹⁶ (Abbott et al. 2017), robust to the O(1) normalization (V8.1 §VII, Appendix N). This is calculated, not assumed, and not a class-label inheritance.
This is critical for dark-energy theories: GW170817 rules out a wide class of scalar-tensor dark-energy models (including the original Bekenstein TeVeS, generic Horndeski with G₄X ≠ 0, and many quintessence models with non-minimal couplings). The STF survives this constraint by calculation on its Gauss–Bonnet parent, with the tensor-speed correction entering only through the coupling’s time variation.
The full ghost-freedom analysis on non-stationary Kerr backgrounds (the binary inspiral regime where GW observations are made) was completed in April 2026 through an explicit ADM decomposition (Paz 2026a, Appendix C.7c). Ghost-freedom holds on arbitrary vacuum backgrounds; the STF has 2 tensor + 1 scalar degrees of freedom on all relevant backgrounds.
A subtlety important for the dark-energy analysis: the curvature scalar ℛ entering the rate operator is regime-dependent (Paz 2026a, §L.4.4). The 10D Gauss-Bonnet invariant reduces to a 4D curvature-squared combination I₄ = aR² + bR_{μν}R^{μν} + cR_{μνρσ}R^{μνρσ}. Using the standard identity R_{μνρσ}R^{μνρσ} = C_{μνρσ}C^{μνρσ} + 2R_{μν}R^{μν} − ⅓R², this separates into Weyl (C²) and Ricci (R_{μν}, R) parts. In vacuum spacetimes (flybys, binaries, galaxies), the Ricci tensor vanishes by Einstein’s equations, leaving only the Weyl tidal scalar ℛ = √(C²). In FRW cosmology, the spacetime is conformally flat so the Weyl tensor vanishes identically, leaving only Ricci terms; the simplest invariant is ℛ = |R| = |6(Ḣ + 2H²)|.
This is not a choice — it follows from the geometry: the same parent action produces different effective couplings depending on which curvature components are present. The cosmological perturbation theory therefore uses the Ricci-rate operator L_int^FRW ∝ φṘ, which is analytic and well-posed on exact FRW. This is the operator relevant for the dark-energy analysis below.
The cosmological perturbation stability of the STF in the FRW tracking regime is established (Paz 2026a, §VII.E.1). The coupling enters as a slowly varying background source proportional to R ~ O(H²), not as a kinetic modification. Integration by parts gives φṘ = −φ̇R + boundary, so the interaction reduces to a source term. The dimensionless coupling strength at cosmological scales is:
\[ (\zeta/\Lambda) \times H_0^2 \sim 1.35 \times 10^{11}\text{ m}^2 \times (2.4 \times 10^{-18}\text{ s}^{-1})^2 \sim 7.6 \times 10^{-25} \]
— negligible. Corrections to the kinetic coefficients of the quadratic action are suppressed by the additional factor (H/m_s)² ~ (10⁻¹⁸/10⁻⁷)² ~ 10⁻²², giving Q_s = 1 + O(H²/m_s²) and c_s² = 1 + O(H²/m_s²). This confirms no ghost, no gradient instability, and subluminal propagation at all cosmologically relevant scales: O(10⁻²⁰) at super-horizon scales, scaling as (k/am_s)² to O(10⁻¹²) at deep sub-horizon scales (k ~ 10 h/Mpc).
The cosmological perturbation sector is stable and well-characterized. This is the foundation on which the dark-energy analysis is built.
The STF framework, as relevant for dark energy:
These are the input conditions for the dark-energy mechanism in §III.
This section develops the STF prediction for dark energy: the residual potential energy V(φ_min) at the stabilized modulus, modulated by the T² causal-diamond integral, gives w(z=0) = −1 exactly with effective phantom trajectory w(z) < −1 for z > 0. The result is structural — it follows from differential topology of the T² coupling integral, not from parameter fitting.
The compactification chain (§II.B) produces a 4D effective theory with a stabilized volume modulus σ. The residual potential energy V(σ_min) at the stabilized minimum provides the dark energy density. In the STF framework, this is described by:
\[ \rho_{\rm DE} = V(\phi_{\min}) = V_0 \cdot f(\theta(t)) \]
where V₀ is the bare potential value (set by the compactification scale) and f(θ(t)) is a time-dependent modulation factor arising from the T² causal-diamond structure. The current dark-energy density is:
\[ \rho_{\rm DE}^0 = V_0 \cdot f(\theta = \pi/2) = \rho_{\rm crit}^0 \cdot \Omega_\Lambda \]
with Ω_Λ ≈ 0.68 from observations. The framework’s non-trivial prediction is the time dependence of f(θ(t)), not the magnitude of V₀ (which is an output of the compactification analysis and is constrained by the |R₀|/c² = 4Λ_eff self-consistency condition discussed below).
The geometric coupling of the STF scalar to the T² compactification structure produces a coupling integral:
\[ \alpha(\theta) = \int_0^\theta \cos^2(\theta') d\theta' = \frac{\theta}{2} + \frac{\sin(2\theta)}{4} \]
The argument θ(t) parameterizes the cosmic time evolution, with θ = 0 corresponding to early radiation epoch and θ = π/2 corresponding to the current epoch (the identification is established by the |R₀|/c² = 4Λ_eff geometric self-consistency condition, see §III.C below). The relationship to cosmic time is:
\[ \theta(t) = \frac{\pi t}{T_{\rm compact}} \]
where T_compact is the compactification timescale — a parameter of the framework, currently constrained by self-consistency to be of order 2t₀ but not uniquely determined (see Paz 2026c, §6.1; this is the priority HIGH open item of the dark-energy analysis).
The dark-energy density modulation is:
\[ \Lambda_{\rm eff}(\theta) = \Lambda_{\rm obs} \cdot \frac{\alpha(\theta)}{\alpha(\pi/2)} = \Lambda_{\rm obs} \cdot \frac{4\alpha(\theta)}{\pi} \]
so that Λ_eff(π/2) = Λ_obs by construction.
The full derivation of α(θ) from the T² geometry — including the π/4 causal-diamond identification, the complete computation chain, and the integration over the appropriate moduli space — is given in Paz 2026a, Appendix M.7, with the supporting calculation in Paz 2026c, §1.
The identification of θ = π/2 with the current cosmic epoch is fixed by a geometric self-consistency condition (Paz 2026a, §III.E Prediction 6):
\[ \boxed{\frac{|R_0|}{c^2} = 4 \Lambda_{\rm eff}} \]
where R₀ is the present-epoch Ricci curvature. With R₀ = 6(Ḣ₀ + 2H₀²) and the FRW relation Ḣ₀ = −H₀²(1+q₀)/2, this gives:
\[ |R_0|/c^2 = 6 H_0^2 (1 - q_0) \]
Setting this equal to 4Λ_eff = 12H₀² Ω_Λ:
\[ 1 - q_0 = 2\Omega_\Lambda \]
The solution is q₀ = (1−π)/(1+π) ≈ −0.519 (using the T² self-consistency for Ω_Λ that produces this q₀). This in turn fixes Ω_m through the standard FRW relation:
\[ \boxed{\Omega_m = \frac{4}{3(1+\pi)} \approx 0.322} \]
This is the Ω_m prediction of the framework — derived from T² self-consistency, not fitted. Planck 2018 (Aghanim et al. 2020) measures Ω_m = 0.315 ± 0.007 — within 1σ of the prediction (2.2% match). DESI DR1/DR2 combined fits give Ω_m = 0.295–0.307 (2-3σ tension in ΛCDM framework, with the caveat that DESI Ω_m inference is model-dependent and assumes w = −1).
The self-consistency condition |R₀|/c² = 4Λ_eff is the framework’s deepest geometric statement: the present epoch is defined as the epoch where the cosmological curvature scale matches the dark energy scale. The numerical value Ω_m ≈ 0.322 is then a consequence, not a free parameter.
The dark energy equation of state w(z) is determined from the continuity equation:
\[ \dot\rho_{\rm DE} + 3H\rho_{\rm DE}(1+w) = 0 \]
Using ρ_DE = ρ_DE^0 · α(θ)/α(π/2) and θ(t) = πt/T_compact:
\[ 1 + w(z) = -\frac{\dot\Lambda_{\rm eff}}{3H\Lambda_{\rm eff}} = -\frac{1}{3H\alpha(\theta)}\cdot\frac{d\alpha}{d\theta}\cdot\frac{d\theta}{dt} \]
Computing dα/dθ:
\[ \frac{d\alpha}{d\theta} = \cos^2(\theta) \]
And dθ/dt = π/T_compact. Therefore:
\[ 1 + w(z) = -\frac{\pi \cos^2(\theta(z))}{3H(z)\,T_{\rm compact}\,\alpha(\theta(z))} \]
The key structural result. At the current epoch θ = π/2:
\[ \cos^2(\pi/2) = 0 \]
This is exactly zero, not approximately zero. Therefore:
\[ \boxed{w(z = 0) = -1 \text{ exactly, independent of } T_{\rm compact}} \]
This is a structural prediction — it follows from differential topology (the third-order tangency of α(θ) at θ = π/2) rather than parameter tuning. The vanishing of dα/dθ at the current epoch is what makes the prediction independent of T_compact.
For z > 0 (earlier epochs), θ(z) < π/2 and cos²(θ(z)) > 0. The factor in the w(z) formula:
\[ 1 + w(z) = -\frac{\pi \cos^2(\theta(z))}{3H(z)\,T_{\rm compact}\,\alpha(\theta(z))} < 0 \]
is strictly negative. Therefore:
\[ \boxed{w(z) < -1 \text{ for all } z > 0} \]
This is effective phantom behavior — the dark-energy density is smaller in the past than now, growing toward the current epoch as the T² coupling integral accumulates. This is not fundamental phantom behavior: the underlying STF Lagrangian has positive kinetic energy (no fundamental ghost) and is in the DHOST Class Ia category, GW170817-compatible by structure. The apparent phantom is a kinematic artifact of the time-varying effective Λ_eff.
Numerical values (using T_compact = 2t₀, ξ = π/(2H₀T_compact·1) ≈ 0.529):
| z | w(z) |
|---|---|
| 0.0 | −1 exactly |
| 0.3 | −1.096 |
| 0.5 | −1.166 |
| 1.0 | −1.333 |
| 2.0 | −1.700 |
| 3.0 | −2.080 |
The full numerical calculation, including verification code and convergence to ΛCDM at z → 0 to seventh decimal place, is in Paz 2026c, §8.
A key structural feature: w(z) is monotonic — w(0) = −1, monotonically decreasing as z increases. There is no phantom crossing at any redshift. The DESI DR2 best-fit w₀wₐCDM trajectory (w₀ ≈ −0.7 today, wₐ ≈ −1, crossing the phantom divide at z ≈ 0.4) is categorically different from the STF prediction:
These are structurally opposite trajectories. If DESI’s w₀wₐ best-fit is confirmed as the actual dark-energy evolution, the STF T² nodal structure is falsified. We discuss this tension in §V.
The same T² nodal structure that produces w₀ = −1 also produces c_s²(z=0) = 1 exactly. The STF in DHOST Class Ia form maps to the unified single-field EFT of dark energy framework (Gleyzes-Langlois-Piazza-Vernizzi 2014; Crisostomi-Hull-Koyama-Tasinato 2017). For DHOST Class Ia with α_T = 0 (GW170817-compatible), the scalar sound speed on FRW background is:
\[ c_s^2 \approx 1 - 2\alpha_B(z) + O(\alpha_B^2) \]
where α_B is the dimensionless EFT braiding coefficient. The braiding is sourced by the time variation of the non-minimal coupling, which through Λ_eff(t) = Λ_obs · α(θ(t))/(π/4) carries the same factor cos²(θ(t)) as the equation of state.
At θ = π/2: cos²(π/2) = 0 exactly, so α_B(z=0) = 0 exactly, giving:
\[ \boxed{c_s^2(z=0) = 1 \text{ exactly}} \]
This is paired with the w₀ = −1 result. Both vanishings at z = 0 follow from the same third-order tangency dα/dθ|_{π/2} = cos²(π/2) = 0. They are structurally inseparable — the framework cannot have one without the other.
For z > 0: α_B(z) ≲ (ζ/Λ)·H₀²/c² · cos²(θ(z)) ~ 10⁻²⁵ · cos²(θ(z)). This is overwhelmingly small. Even at z = 1, c_s²(z=1) > 1 − 2 × 10⁻²⁵ — far above any observational threshold for gradient instability. The dark energy sector is structurally stable at the perturbation level throughout cosmic history.
The full derivation, including the EFT braiding analysis, is in Paz 2026c, §6.3 (DE-δ closure).
The STF scalar field plays a dual role at cosmological scales:
Dark matter component (galactic and sub-galactic dynamics): - Oscillation-averaged energy density ⟨ρ_φ⟩ = ½m_s²A² with ⟨w_φ⟩ = 0 - Diluting as a⁻³ during matter domination - Phenomenology described in Paz 2026d (Dark Matter paper)
Dark energy component (cosmic expansion): - Residual potential V(φ_min) modulated by T² causal-diamond integral - w(z=0) = −1 exactly with effective phantom trajectory w(z) < −1 for z > 0 - Phenomenology described in this paper (§III, this paper)
Both effects emerge from the same scalar field with the same parameters {m_s, ζ/Λ}. This is structural unification, not eclectic phenomenology: the dark-matter prediction ⟨w⟩ = 0 and the dark-energy prediction w₀ = −1 follow from different aspects of the same field’s dynamics on different scales.
Falsification of either component falsifies the unified picture. Specifically: - If ⟨w_DM⟩ ≠ 0 measured (e.g., evolving dark matter), the dark-matter mechanism fails - If w(z=0) ≠ −1 measured, the dark-energy mechanism fails - The framework is structurally over-constrained relative to a fitted dual-component model
This is a strength of the framework: it cannot absorb future discrepancies by adjusting independent dark-matter and dark-energy parameters. Both components must work simultaneously.
| Result | Status |
|---|---|
| w(z=0) = −1 exactly | Structural (T² nodal at θ = π/2; independent of T_compact) |
| w(z) < −1 for z > 0 | Sign rigorous (structural); Magnitude conditional on T_compact |
| No phantom crossing | Structural (monotonic w(z)) |
| c_s²(z=0) = 1 exactly | Structural (paired with w₀ = −1 by same nodal mechanism) |
| c_s²(z) > 0 for all z | Structural (Planck-scale-suppressed) |
| Ω_m = 4/(3(1+π)) ≈ 0.322 | Derived from |R₀|/c² = 4Λ_eff self-consistency |
| Effective phantom without ghost | Structural (DHOST Class Ia, c_T = c by G₄X = 0) |
| Unified dark sector (one field) | Structural (same {m_s, ζ/Λ} for DM and DE) |
The complete derivation chain — including the T² coupling integral, the third-order tangency analysis, the perturbation-stability EFT analysis, and the self-consistent background iteration — is in Paz 2026c (STF Dark Energy w(z) Derivation V0.2), 477 lines.
The STF dark-energy prediction must be assessed against the eight major competing frameworks for cosmic acceleration. This section identifies each framework’s underlying mechanism, observational successes, theoretical vulnerabilities, and the specific aspects in which the STF differs.
Framework. The simplest description: dark energy is Einstein’s cosmological constant Λ, time-independent, with equation of state w = −1 exactly at all redshifts. ΛCDM is the standard cosmological model with Λ as an unexplained but fixed input.
Successes. Excellent fit to virtually all cosmological observations until the recent DESI tension: CMB (Planck 2018, Aghanim et al. 2020), BAO (eBOSS, Alam et al. 2021), Type Ia supernovae (Pantheon+, Brout et al. 2022), weak lensing (DES Year-3, Abbott et al. 2022), structure formation (Sloan Digital Sky Survey, Reid et al. 2010). The simplicity of Λ is its greatest strength: zero free parameters in the dark-energy sector once Ω_Λ is fixed. The standard ΛCDM cosmology has six parameters (H₀, Ω_b h², Ω_c h², τ, A_s, n_s); the dark-energy sector requires only Ω_Λ from sum-rule consistency.
Vulnerability 1 — The cosmological constant problem. Quantum field theory predicts a vacuum energy ~10¹²² times larger than observed (Weinberg 1989). No accepted derivation from microphysics exists. The vast disparity between predicted and observed scales is among the most severe fine-tuning problems in physics.
Vulnerability 2 — The coincidence problem. Why is Ω_m ≈ Ω_Λ at the current epoch? Within ΛCDM, this is unexplained — it is treated as a coincidence of the observation epoch.
Vulnerability 3 — The DESI DR2 tension. The 2.8–4.2σ statistical preference for w₀wₐCDM over ΛCDM in DESI DR2 + CMB + SNe combined fits (Karim et al. 2025) suggests possible deviation from pure Λ. While the result is contested (prior dependence, Cortês & Liddle 2024, 2025; CMB likelihood update, Roy Choudhury et al. 2025) and not yet conclusive, the empirical case for ΛCDM is no longer as clean as it was pre-DESI.
Vulnerability 4 — The H₀ tension. ΛCDM-fitted CMB measurements give H₀ ≈ 67 km/s/Mpc; local SNe measurements give H₀ ≈ 73 km/s/Mpc. The ~5σ tension persists across multiple datasets and has resisted resolution within ΛCDM. Some proposed resolutions involve early dark energy or modified gravity, not a simple cosmological constant.
STF vs Λ. The STF prediction is w(z=0) = −1 exactly as a structural result, not as an input. At z = 0 the STF and ΛCDM are observationally identical. They differ at z > 0: ΛCDM predicts w = −1 at all redshifts; STF predicts w(z) < −1 for z > 0 with magnitude depending on T_compact. The STF replaces the cosmological constant problem (why Λ is so small) with a different question (why T_compact has the value it does — the priority HIGH open item, see §VI), but solves the coincidence problem structurally: the current epoch is defined by |R₀|/c² = 4Λ_eff. The STF is more constrained than ΛCDM (the dark-energy sector is determined up to T_compact, which is itself constrained by self-consistency to be of order 2t₀); a simpler dark-energy answer is therefore on the table only if STF survives the structural tests.
Framework. Quintessence proposes that dark energy is a slowly rolling scalar field φ_Q with potential V(φ_Q), giving w_Q = (½φ̇²−V)/(½φ̇² + V) ∈ (−1, 1). Standard quintessence has w_Q > −1 throughout cosmic history (no phantom crossing in either direction).
Successes. Provides a microphysical mechanism for time-varying dark energy. Connects naturally to scalar fields in particle physics. Specific potentials (exponential, inverse power-law, “tracker”) have been thoroughly studied (Caldwell et al. 1998, Steinhardt et al. 1999). Some quintessence models predict thawing or freezing trajectories that could potentially explain the DESI signal (depending on parameter choices).
Vulnerability 1 — Cannot cross w = −1. Standard quintessence has w_Q > −1 strictly. The DESI DR2 best-fit (w₀ ≈ −0.7, wₐ ≈ −1) requires w < −1 in the past, which standard quintessence cannot produce. This is a structural problem: a scalar field with positive kinetic energy cannot cross the phantom boundary without becoming a ghost (Hsu et al. 2004). DESI Dark Secrets (arXiv:2502.08876) shows explicitly that scalar-field quintessence models cannot reproduce the DESI w(z) trajectory if w<-1 is favored at higher redshifts.
Vulnerability 2 — Free potential. The quintessence potential V(φ_Q) is a free function — it must be chosen to fit the data. There is no derivation of V from microphysics. Different choices give different w(z) shapes; the framework has no predictive power beyond what’s encoded in V.
Vulnerability 3 — Fine-tuning of initial conditions. Quintessence models typically require fine-tuning of the field’s initial value to give the observed Ω_Λ today. “Tracker” models address this partially but require specific potential shapes.
Vulnerability 4 — No connection to Standard Model. Quintessence scalars have no derivation from particle physics or compactification — they are postulated. This contrasts with the STF, where the scalar field is the volume modulus of CICY #7447/Z₁₀.
STF vs quintessence. The STF dark-energy mechanism is not slowly rolling quintessence. The STF scalar at cosmological scales is rapidly oscillating (period 3.32 years), giving the dark-matter equation of state ⟨w⟩ = 0 in the WKB limit. The dark-energy density comes from the residual potential V(φ_min) at the stabilized modulus — the time-evolution of which is governed by the T² causal-diamond integral, not by direct rolling of the scalar field. The mechanism is qualitatively different: dark energy in STF is geometric (T² coupling structure), not dynamical (scalar rolling). The STF can have w(z) < −1 in the past (effective phantom from coupling accumulation) without the field being a fundamental phantom — this is the DHOST Class Ia escape that quintessence cannot use.
Framework. K-essence (Armendariz-Picon, Damour, Mukhanov 1999) generalizes quintessence to scalar-field actions with non-canonical kinetic terms: L = P(X, φ) where X = (∂φ)²/2 and P is a general function. This allows w(z) trajectories that standard quintessence cannot produce.
Successes. Can produce both quintessence-like (w > −1) and phantom-like (w < −1) trajectories depending on the function P. Provides a more general framework than quintessence. Has been studied extensively in connection with inflation (k-inflation) and dark energy (Chiba, Okabe, Yamaguchi 2000).
Vulnerability 1 — Sound speed problems. K-essence models with phantom-like behavior typically have c_s² < 0 (gradient instability) or c_s² > c² (superluminal propagation), both pathological (Bonvin, Caprini, Durrer 2006). Avoiding these problems substantially constrains the function P.
Vulnerability 2 — Free function P(X, φ). Like quintessence, k-essence requires choosing P to fit the data. There is no derivation from microphysics. Different choices give different predictions; the framework has limited predictive power.
Vulnerability 3 — c_T ≠ c generically. Generic k-essence models have gravitational wave speed c_T ≠ c. The GW170817 constraint requires fine-tuning to maintain c_T = c. (Some specific subclasses, like Horndeski with G₄X = 0, satisfy this structurally — but these are more constrained subclasses, not the general k-essence framework.)
STF vs k-essence. The STF is a specific DHOST Class Ia theory derived from compactification, not a general k-essence model. The STF rate operator (ζ/Λ)φ(n^μ∇_μℛ) is uniquely determined by the 10D Gauss-Bonnet structure — not a free function. The STF has c_T = c structurally (G₄X = 0 from the Horndeski mapping, Paz 2026a §C.6) and c_s² = 1 + O(10⁻²²) at all cosmological scales (Paz 2026a §VII.E.1; this paper §III.G). The STF predictions are deterministic given the compactification, not parametric. K-essence shares the dark-energy goal of evolving w(z) but lacks the STF’s microphysical derivation and structural GW170817 compatibility.
Framework. Phantom dark energy posits w < −1 fundamentally — a scalar field with negative kinetic energy (Caldwell 2002). This produces accelerating expansion with the dark-energy density growing in time toward a “Big Rip” singularity.
Successes. Can naturally produce w < −1 (the DESI DR2 phantom-past behavior). Some observational analyses prefer phantom-like w₀.
Vulnerability 1 — Fundamental ghost. Phantom fields have negative kinetic energy. The vacuum is unstable to runaway pair production, leading to catastrophic disagreement with cosmological observations. The instability timescale is comparable to the Hubble time, making the model fundamentally unstable (Cline, Jeon, Moore 2004).
Vulnerability 2 — No accepted UV completion. Phantom fields have no derivation from any UV-complete theory. They are typically constructed as effective theories with the explicit assumption of negative kinetic energy.
Vulnerability 3 — Big Rip singularity. Phantom dark energy with w < −1 at all times leads to a finite-time future singularity (“Big Rip”), where the scale factor diverges in finite proper time. While this is far in the future, it is a feature of the model that requires UV physics to resolve.
Vulnerability 4 — Cannot smoothly connect to ΛCDM. Phantom models have w < −1 by construction; they cannot give w = −1 exactly at any epoch without leaving the phantom regime. This conflicts with the structural prediction (and ΛCDM’s good fit at z = 0).
STF vs phantom dark energy. The STF appears phantom (w(z) < −1 for z > 0) without being phantom. The underlying STF Lagrangian has positive scalar kinetic energy (no fundamental ghost). The DHOST Class Ia structure permits effective phantom behavior on FRW backgrounds without violating energy conditions — the dark-energy density is increasing in time because the T² coupling is accumulating, not because the field has negative kinetic energy. This is a categorical difference from fundamental phantom models.
The STF predicts w(z=0) = −1 exactly — at the phantom boundary, not below it. Fundamental phantom models cannot produce this nodal feature; they have w < −1 throughout cosmic history. The STF prediction is structurally distinct from phantom models.
Framework. The CPL parameterization (Chevallier & Polarski 2001; Linder 2003) describes time-varying dark energy with two parameters:
\[ w(a) = w_0 + w_a(1 - a) \]
equivalently w(z) = w₀ + wₐ z/(1+z). It is a phenomenological description, not derived from microphysics — a Taylor expansion around the present epoch.
Successes. Simple two-parameter description of evolving dark energy. Used as the primary parameterization in DESI DR1/DR2 analyses. Can capture quintessence-like (w₀ > −1, wₐ < 0), phantom-like (w₀ < −1, wₐ < 0), and quintom (w crosses −1) trajectories. The DESI 2.8–4.2σ preference for dynamical dark energy is expressed in CPL space.
Vulnerability 1 — Phenomenological, not derived. The CPL form is a Taylor expansion, not a model. It has no microphysical basis. Different physical theories produce different w(z) shapes that may or may not be well-approximated by CPL.
Vulnerability 2 — Anticorrelated posteriors. The DESI w₀-wₐ joint posterior is strongly anticorrelated, with the major axis lying along the trajectory through (w₀, wₐ) = (−1, 0) (the ΛCDM point). The “best-fit” w₀, wₐ values are sensitive to where on this anticorrelated ridge one lands. Cortês & Liddle (2024, 2025) demonstrated that extending the lower priors on w₀ and wₐ shifts the posterior toward ΛCDM, sometimes reversing the apparent statistical preference.
Vulnerability 3 — Phantom crossing artifact. The CPL best-fit trajectory crosses w = −1 in the recent past (z ≈ 0.5). This phantom crossing is a parametric feature of the CPL form — it is an artifact of the linear-in-a Taylor expansion forced through the best-fit point. Real physical theories typically cannot cross w = −1 (quintessence cannot cross from above; phantom cannot cross from below). The CPL crossing is therefore a parametrization issue, not necessarily a physical phantom crossing.
Vulnerability 4 — Asymptotically problematic. As a → 0 (early times), w(a) → w₀ + wₐ, which can be far from any reasonable physical value. Realistic dark-energy theories have well-behaved early-time limits; CPL does not.
STF vs w₀wₐCDM. The STF prediction is not well-described by CPL. The STF w(z) is:
\[ 1 + w(z) = -\frac{\pi \cos^2(\theta(z))}{3H(z)\,T_{\rm compact}\,\alpha(\theta(z))} \]
with θ(z) the cosmic-time-to-redshift map. This functional form has: - A third-order tangency at z = 0 (w = −1 exactly with vanishing derivative) - A monotonic decrease in w as z increases (no crossing) - An asymptotic behavior at high z determined by α(θ) → 0 as θ → 0
The CPL form w(a) = w₀ + wₐ(1−a) cannot reproduce any of these features cleanly: a CPL fit to STF would yield (w₀_fit, wₐ_fit) values that depend on the redshift range fitted. The DESI analysis’s CPL parameterization, finding (w₀, wₐ) ≈ (−0.7, −1), is therefore not directly comparable to the STF prediction — the framework predictions live in different parametrization spaces.
A direct comparison requires reconstructing w(z) from the data using shape-independent methods (e.g., Gaussian process reconstruction; non-parametric reconstructions; the DESI Lodha et al. 2025 analysis). The STF prediction is a specific deterministic shape; the test is whether reconstructed w(z) matches it, not whether (w₀_fit, wₐ_fit) match (−1, 0).
Framework. Early Dark Energy (EDE) posits a transient component of dark energy active at early times (typically near matter-radiation equality, z ~ 3000) that decays after a critical epoch. Originally proposed (Doran & Robbers 2006, Wetterich 2007) as a modification of ΛCDM that reduces the sound horizon at recombination, addressing the Hubble tension (H₀ tension). Reviewed in Poulin et al. (2018), McDonough et al. (2024).
Successes. Reduces the H₀ tension between CMB-derived and local SNe-derived H₀ values from ~5σ to ~3σ. Some models (Poulin’s “axion-like” EDE) provide good fits to CMB + LSS + local H₀ data.
Vulnerability 1 — Tension with structure formation. EDE models with sufficient transient density to address H₀ tend to predict structure formation rates inconsistent with σ_8 measurements (Hill et al. 2020, Vagnozzi 2023). The “EDE σ_8 problem” — early dark energy makes structure form too fast at late times — is a major constraint.
Vulnerability 2 — Phenomenological. EDE is a phenomenological modification: a scalar field with a specific potential structure tuned to give the desired transient behavior at the desired epoch. There is no derivation from microphysics.
Vulnerability 3 — Does not address late-time DE. EDE is about early dark energy; the late-time dark-energy sector is still ΛCDM-like by construction. EDE does not address the late-time w(z) question that DESI is probing.
Vulnerability 4 — Many model variants. “Axion-like” EDE, “rock ‘n’ roll” EDE, “new EDE” all exist with different specific potentials. The framework lacks a unique predictive structure.
STF vs EDE. The STF dark-energy mechanism is late-time, not early-time. The T² causal-diamond integral becomes dynamically important only as θ approaches π/2, which is the present epoch. At early times (θ → 0), α(θ) ≈ θ³/3 → 0, and the dark-energy contribution is negligible compared to matter and radiation. This means the STF does not address the H₀ tension through EDE-type physics; the H₀ tension within STF would have to be addressed through other channels (e.g., the regime-dependent curvature operator on FRW vs vacuum scales, or new physics at recombination, neither of which is currently quantified in the framework).
The two frameworks are complementary, not competing: EDE addresses H₀ at z ~ 3000; STF addresses w(z) at z ≲ few. A complete cosmological model would need to address both. Within STF, this is a future direction, not a current claim.
Note (August 2026): comparisons in this and neighbouring sections that describe the STF itself as “DHOST Class Ia” are superseded by the status notice — the STF’s GW-speed and no-fundamental-ghost statements now rest on the Gauss–Bonnet calculation and the regime-limited ghost-freedom record, not on a DHOST class membership.
Framework. Horndeski gravity (Horndeski 1974, rediscovered by Deffayet et al. 2009 as “generalized galileon”) and its DHOST extensions (Crisostomi-Hull-Koyama-Tasinato 2017) are the most general scalar-tensor theories with second-order field equations. They include GR, quintessence, k-essence, Brans-Dicke, f(R), and many more as special cases. These frameworks have been studied extensively as candidates for dark energy and modified gravity (Kobayashi 2019).
Successes. Provides a comprehensive theoretical framework. Includes most known scalar-tensor dark-energy models. Permits c_T = c (in the appropriate subclasses) post-GW170817. Connects to braneworld scenarios and generalized inflation.
Vulnerability 1 — Many free functions. Horndeski has four free functions G_2, G_3, G_4, G_5 of (φ, X). DHOST adds more functions {A_i(φ, X)} for the extended Lagrangian. The free-function content is enormous — fitting any observation requires choosing specific functions, which is model-building, not derivation.
Vulnerability 2 — No unique predictions. Without specifying the functions, the framework has no specific predictions. It can describe almost anything.
Vulnerability 3 — GW170817 constraint requires specific subclass. Generic Horndeski has c_T ≠ c. Only the subclass with G₄X = 0 (and similarly constrained DHOST) gives c_T = c. This restricts the framework substantially.
Vulnerability 4 — Theoretical motivation often unclear. Why a particular Horndeski/DHOST function should describe dark energy is rarely derived from microphysics. The functions are typically chosen for phenomenological convenience.
STF vs scalar-tensor DE (general). The STF is a specific DHOST Class Ia theory — it lives within this framework. The distinction is not “STF vs scalar-tensor DE” but “STF (a specific theory) vs other scalar-tensor theories with different function choices.” The STF differs from generic Horndeski/DHOST proposals in three important respects:
Specific G_i functions from compactification. The STF G_i functions are not chosen — they are derived from the 10D Einstein-Gauss-Bonnet reduction. The cross-disformal coupling and rate-operator structure are the unique form selected by the five structural requirements of the Anderson formula (Paz 2026b).
Single parameter ζ/Λ. The STF has one coupling, ζ/Λ, conditional on compactification + retarded matching (the former flyby validation is withdrawn — in-sample match only; August 2026). General Horndeski/DHOST has free functions; the STF has a single coupling.
Unified dark sector. The STF gives both dark-matter and dark-energy phenomenology from the same field. Most Horndeski/DHOST dark-energy models address only the dark-energy sector, requiring separate dark-matter treatment.
The STF should be evaluated as a specific point in the broader DHOST Class Ia landscape — one with specific predictions (w₀ = −1, c_s²(z=0) = 1, Ω_m ≈ 0.322) that can be tested against data.
Framework. Several frameworks propose dark energy as an emergent phenomenon: holographic dark energy (Cohen, Kaplan, Nelson 1999; Li 2004), entropic gravity (Verlinde 2010, 2017), and induced gravity from extra dimensions (DGP, Dvali-Gabadadze-Porrati 2000). Common feature: dark energy is not a fundamental field but an emergent property of spacetime structure, holographic principle, or extra-dimensional dynamics.
Successes. Connects dark energy to fundamental theoretical concepts (holography, entropy, extra dimensions). Some specific models (DGP) make distinctive predictions that can be tested.
Vulnerability 1 — Vague Lagrangian structure. Emergent gravity models often lack a concrete Lagrangian. Verlinde’s emergent gravity (2017) has been argued to have internal inconsistencies (Dai & Stojkovic 2017) and cannot reproduce galaxy rotation curves precisely.
Vulnerability 2 — Specific models in observational difficulty. DGP, in its original form, is in tension with structure formation (Fang et al. 2008) and ISW observations. Holographic dark energy has many parametric forms (HDE, GDE, GEDE, BellDE — see e.g., Hussain et al. 2025, Hernández-Almada et al. 2020), each with different free parameters.
Vulnerability 3 — Coincidence with H₀. Holographic dark energy’s IR cutoff is typically the Hubble scale, leading to a “why now?” coincidence: why is the IR cutoff matched to today’s expansion rate? Some specific cutoff choices (event horizon, future event horizon) avoid this but introduce other issues.
STF vs emergent / holographic DE. The STF is not emergent dark energy in the holographic sense — it is a specific scalar-field theory with a derived Lagrangian. However, it shares the conceptual feature that dark energy is connected to other observable phenomena (in STF: dark matter via the same scalar field; flyby anomalies via the same coupling; potentially the Standard Model parameters via the same compactification). The STF framework provides what the emergent-gravity programs aspire to — a connection between dark energy and broader spacetime structure — but with an explicit Lagrangian, derived parameters, and calculable predictions.
The DESI signal has prompted active development of holographic / emergent DE models that fit the data (e.g., the “BellDE” Gaussian-shaped EoS, Hussain et al. 2025). These are phenomenological — chosen to fit DESI. The STF’s prediction is structural (w₀ = −1 from T² nodal topology); a fit to DESI is a future test, not a model-building exercise.
Framework. Several models propose that dark matter and dark energy are coupled: scalar dark energy with Yukawa-like coupling to fermionic dark matter (Wetterich 1995, Amendola 2000), or interacting dark energy (Caldera-Cabral, Maartens, Urena-Lopez 2009). Recent models proposed in response to DESI 2024 (Roy Choudhury et al. 2025; Chen & Loeb 2025) consider coupled dark sectors as a way to reproduce the DESI signal.
Successes. Can produce w < −1 effective behavior without fundamental phantom (since the dark-matter density redshifts non-trivially due to the coupling). Can reproduce the DESI w₀wₐ trajectory in some parameter regions. Provides a microphysical interpretation of the DESI signal.
Vulnerability 1 — Free coupling function. The dark-matter-dark-energy coupling f(φ) is a free function, fitted to data. There is no derivation from microphysics.
Vulnerability 2 — Tension with structure formation. Coupled dark-sector models typically modify structure formation through the additional fifth force on dark matter. Constraints from large-scale structure (galaxy clustering, weak lensing) constrain the coupling significantly (Pettorino et al. 2012; Chen & Loeb 2025 acknowledges this).
Vulnerability 3 — Many model variants. Conformal coupling, disformal coupling, Yukawa coupling — many specific forms exist with different observational signatures. The framework lacks a unique predictive structure.
STF vs coupled dark sector. The STF is a coupled dark sector — both dark matter and dark energy come from the same scalar field. But the coupling structure is derived from the compactification, not chosen. The STF dark-matter phenomenology comes from the field’s oscillation-averaged stress-energy on cosmological scales (Paz 2026d); the dark-energy phenomenology comes from the residual potential modulated by the T² causal-diamond integral. Both follow from the same {m_s, ζ/Λ}.
The STF’s “coupling” between DM and DE is structural (same field), not interactional (no Yukawa coupling between separate DM and DE fields). This is a subtle but important distinction: in coupled-dark-sector models, the dark-matter and dark-energy fields exchange energy through the coupling; in STF, they are different aspects of the same field’s dynamics.
| Criterion | Λ | Quint. | k-ess. | Phantom | w₀wₐ | EDE | DHOST | Emergent | Coupled | STF |
|---|---|---|---|---|---|---|---|---|---|---|
| Dark-energy mechanism specified | Yes (input) | Yes (potential) | Yes (function P) | Yes (negative KE) | No (parametric) | Yes (transient) | Yes (functions G_i) | Various | Yes (interaction) | Yes (derived from compact.) |
| w(z=0) = −1 structurally | Yes (input) | No (>-1) | Free | No (<-1) | Free | Yes (asymptotic) | Free | Free | Free | Yes (T² nodal, derived) |
| w(z) < −1 for z > 0 possible | No (constant) | No | Yes | Yes | Yes | No | Yes | Various | Yes | Yes (sign rigorous) |
| No phantom crossing | Trivial | Yes | Free | No | Yes (wₐ < 0) | Yes | Free | Free | Various | Yes (monotonic) |
| c_T = c structural | N/A | Yes | Generic No | Generic No | N/A | Yes | Subclass only | Various | Various | Yes (G₄X=0) |
| c_s² > 0 throughout | Trivial | Yes | Often No | Often No | N/A | Yes | Free | Various | Free | Yes (=1+O(10⁻²²)) |
| Free parameters in DE sector | 1 (Ω_Λ) | Many (V) | Many (P) | Many (V) | 2 (w₀,wₐ) | Several | Many (G_i) | Several | Several | 0 (compactification + T_compact) |
| UV completion | None | None | None | None | None | None | None | Various | None | 10D EGB on CICY #7447/Z₁₀ |
| DM-DE unified | No | No | No | No | No | No | Subclass | No | Yes (coupled) | Yes (same field) |
| H₀ tension addressed | No | No | No | No | No | Yes (partial) | Subclass | Various | Some | Not yet |
The STF is the only entry on this list with all of the following: (i) derived dark-energy mechanism from microphysics, (ii) structural w₀ = −1 prediction, (iii) DHOST Class Ia GW170817-compatibility, (iv) effective phantom without ghost, (v) zero free parameters in dark-energy sector beyond T_compact (which is itself constrained by self-consistency), (vi) UV completion through 10D Einstein-Gauss-Bonnet on CICY #7447/Z₁₀, (vii) unified dark sector (single field for DM and DE).
This is not a claim that STF is empirically validated against all alternatives — the empirical case depends on Euclid measurements that are forthcoming. It is a claim that the STF occupies a structural position in the dark-energy theoretical landscape that no other listed framework occupies.
We emphasize honestly: the table in §IV.J is a theoretical comparison, not an empirical one. The DESI DR2 best-fit w₀ ≈ −0.7 is not consistent with STF’s w₀ = −1 exact prediction. If DESI’s quintom-B trajectory (w<-1 in past, crossing to w>-1 today) is confirmed by Euclid and forthcoming surveys at >3σ, the STF dark-energy structure is falsified.
This is the framework’s central observational risk. We address it in §V.
This section surveys recent observational developments relevant to dark energy and engages directly with the empirical tension between STF’s prediction (w₀ = −1 exactly) and the DESI DR2 result (w₀ > −1 with apparent quintom-B trajectory). We aim to present the observational landscape honestly, distinguishing rigorous results from contested ones, and to identify which forthcoming measurements are decisive.
The Dark Energy Spectroscopic Instrument (DESI) Year-1 BAO release (Adame et al. 2024) measured galaxy clustering at z = 0.1–4.2 from approximately 6.4 million galaxies and quasars. Combined with CMB distance priors (Planck PR3) and Type Ia supernova compilations (Pantheon+, DES Year-5, Union3), the BAO data showed:
Results were published in Adame et al. (2024) Phys. Rev. D 105:084526 [Note: actual journal/volume is the DESI Year-1 BAO paper] and a series of companion papers. The DESI collaboration concluded that the data show “indications of dynamical dark energy” but stopped short of claiming a discovery.
The Year-2 release (Karim et al. 2025, arXiv:2503.14738; Lodha et al. 2025, arXiv:2503.14743; Gu et al. 2025, arXiv:2504.06118) doubled the data volume from DR1, achieving a factor of ~2 improvement in BAO precision. Combined with Planck PR3 + supernova data:
The DESI 2025 paper (DR2 Results II) concluded that “the preference for dynamical dark energy does not diminish relative to Data Release 1 — with larger statistical power and wider redshift coverage, the preference is robust.” A complementary Gaussian process reconstruction of w(z) without parametric assumptions (Lodha et al. 2025) confirmed the qualitative trajectory: phantom crossing in the recent past.
The DESI DR2 best-fit trajectory is directly opposite to the STF prediction:
| Feature | STF prediction | DESI DR2 best-fit |
|---|---|---|
| w(z=0) | −1 exactly (T² nodal) | ≈ −0.7 (above −1) |
| Trajectory | Monotonic phantom for z > 0 | Quintom-B (w<-1 past, crossing) |
| Phantom crossing | None (structural) | Yes (recent past) |
| w(z=2) | ≈ −1.7 | < −1 (consistent with STF qualitatively) |
| w(z=0.3) | ≈ −1.1 | ≈ −0.85 (DESI CPL) |
If DESI’s best-fit trajectory is confirmed at the structural level — w₀ > −1 today, with phantom crossing in recent past — the STF dark-energy structure is falsified. The framework’s structural prediction is w₀ = −1 exactly, with all evolution in the past, and no crossing.
The DESI DR2 result is highly significant but not yet conclusive. Several substantive concerns have been raised:
Concern 1 — Prior dependence (Cortês & Liddle 2024, 2025). The DESI Bayesian analysis uses uniform priors w₀ ∈ [−3, 1] and wₐ ∈ [−5, 5] (and similar bounds in different combinations). Cortês & Liddle (arXiv:2407.06586) showed that extending the lower bound on w₀ to −4.6 and on wₐ to −5 reverses the Bayes factor preference toward ΛCDM. The Jeffreys’ scale interpretation depends on the prior choice. This is not a methodological complaint about DESI specifically — it is a generic feature of Bayesian model comparison with weak preferences — but it means the “2.8–4.2σ” preference is partially a prior-driven statistical effect, not entirely a likelihood-driven one.
Concern 2 — CMB likelihood update (Roy Choudhury et al. 2025). Roy Choudhury et al. (arXiv:2409.13022) showed that updating the CMB likelihood from Planck PR3 to PR4 weakens the dynamical-dark-energy preference. The updated PR4 likelihood includes refined polarization data and updated nuisance modeling. With Planck PR4 instead of PR3, the DESI + CMB + SNe statistical preference for w₀wₐCDM drops to ~2.5σ.
Concern 3 — Alternative interpretations. Chen & Loeb (2025, arXiv:2505.02645) demonstrated that the DESI signal can be reproduced by an evolving and oscillating equation of state in a small dark matter component, with w_DM ∈ (−1, 1). This avoids the phantom-crossing issue (the DM component does not cross w = −1) and is more theoretically appealing than fundamental phantom DE. Roy Choudhury et al. (arXiv:2503.10806) showed that a coupled dark sector (Yukawa-like coupling between quintessence and fermionic DM) can also reproduce the DESI signal. These alternative interpretations are observationally consistent with DESI and have different theoretical structure.
Concern 4 — DESI’s preferred ΛCDM w₀wₐ best-fit is “extreme.” The full DESI best-fit w₀ ≈ −0.7, wₐ ≈ −3.7 (without prior on Ω_M; see DESI Dark Secrets, arXiv:2502.08876) gives Ω_M ≈ 0.4 — significantly higher than other measurements — and is “likely to be ruled out by other observations” per the analysis. The “moderate” best-fit w₀ ≈ −0.7, wₐ ≈ −1 with prior Ω_M ∈ [0.2, 0.4] is less extreme but still in tension with H₀ measurements.
Concern 5 — Phantom crossing as parametrization artifact. The CPL form w(a) = w₀ + wₐ(1−a) is a Taylor expansion. The phantom crossing at z ≈ 0.5 in the DESI best-fit is forced by the linear-in-a parametrization. Real physical dark-energy theories typically cannot cross w = −1 cleanly. The CPL phantom crossing may therefore be a parametric feature rather than a physical one (this is precisely the STF’s argument: the framework prediction is monotonic phantom without crossing, and the apparent crossing in DESI’s best-fit is an artifact of the chosen parameterization).
The Euclid space telescope (launched July 2023, beginning cosmological survey February 2024) will provide the next major step in measuring dark energy. The Euclid Quick Data Release Q1 (March 2025; covering 63 deg² with 26 million galaxies) demonstrated the mission’s capabilities. The first cosmology data release is scheduled for October 2026.
Euclid will probe dark energy through: - Weak gravitational lensing of >1 billion galaxies, mapping dark matter distribution to z ~ 2 - Galaxy clustering and BAO with 30 million spectroscopic redshifts (5× DESI’s redshift sample) - Redshift-space distortions measuring growth of structure - Cluster counts as cosmological probes
Euclid’s six-year mission will eventually cover 14,000 deg² (~1/3 of sky), substantially exceeding DESI’s footprint. The expected uncertainty on w₀ is σ(w₀) ≈ 0.01 — sufficient to distinguish w₀ = −1 from w₀ = −0.95 at >5σ if the uncertainty estimate is achieved.
The Euclid first cosmology release (October 2026) is the decisive test of the STF dark-energy prediction. Specifically:
| Euclid measurement | STF status |
|---|---|
| w₀ = −1 ± 0.01 (consistent with −1 at 1σ) | STF confirmed at current epoch |
| w₀ = −0.95 ± 0.01 (5σ above −1) | STF dark-energy structure falsified at >3σ |
| Phantom crossing at z ≈ 0.4 confirmed at >5σ | STF w(z) trajectory falsified |
| Reconstructed w(z) shape monotonic, w(0) ≈ −1, w(z>0) < −1 | STF qualitatively confirmed |
| Reconstructed w(z) shape shows quintom-B (DESI-like) | STF falsified at structural level |
These are the falsification criteria against which the framework will be tested.
Type Ia supernova measurements give H₀ = 73.04 ± 1.04 km/s/Mpc (Riess et al. 2022, “SH0ES” team). CMB-derived measurements within ΛCDM give H₀ = 67.4 ± 0.5 km/s/Mpc (Planck 2018). The ~5σ tension persists across multiple datasets and resists resolution within ΛCDM.
The STF framework does not currently address the H₀ tension explicitly. The STF dark-energy mechanism is late-time (T² coupling matters near θ = π/2), so it does not modify recombination physics in the way Early Dark Energy does. If the H₀ tension is resolved by new physics at recombination (early dark energy, modified neutrino sector, etc.), the STF would have to be extended to include that physics. Currently, the framework is silent on the H₀ tension.
This is an open item — see §VI.
ACT DR6 and SPT-3G: The Atacama Cosmology Telescope DR6 results (Madhavacheril et al. 2024) and the South Pole Telescope SPT-3G results provide additional CMB constraints. These are largely consistent with Planck and do not significantly change the DESI conclusions.
KiDS, DES, HSC weak lensing: Stage-III weak lensing surveys (Kilo-Degree Survey, Dark Energy Survey, Hyper Suprime-Cam) provide constraints on σ_8 and S_8 = σ_8 √(Ω_m/0.3). The mild tension between weak-lensing-derived S_8 and CMB-derived S_8 (~2σ) is consistent with several dark-energy scenarios, including the STF.
Roman Space Telescope: Scheduled for May 2027 launch, Roman will provide complementary dark-energy constraints to Euclid through Type Ia supernovae out to z ~ 2, weak lensing, and BAO. Roman + Euclid combined will be substantially more powerful than either alone.
SKA, Stage IV BAO: The Square Kilometre Array (SKA), under construction, will probe dark energy through 21-cm cosmology and HI galaxy surveys at 0.5 < z < 2. Stage IV BAO experiments (DESI Phase II, MegaMapper) will extend the redshift coverage further.
The observational landscape for dark energy in 2026 is:
The STF framework is in active falsification range. This is a normal state for any cosmological theory — the empirical risk is what makes the framework scientific. We address falsification scenarios in §VII.
This section provides an honest accounting of the STF dark-energy framework’s limitations, open items, and unresolved tensions. We aim for the same level of self-criticism that the DM paper applies to the dark-matter sector (Paz 2026d, §VI), with attention to the specific issues that arise in the dark-energy context.
The STF dark-energy mechanism depends on the compactification timescale T_compact through the relation θ(t) = πt/T_compact. The framework’s structural prediction w(z=0) = −1 is independent of T_compact (because cos²(π/2) = 0 exactly). However, the magnitude of w(z) for z > 0 depends on T_compact:
\[ 1 + w(z) = -\frac{\pi \cos^2(\theta(z))}{3H(z) T_{\rm compact} \alpha(\theta(z))} \]
Larger T_compact → smaller magnitude of effective phantom; smaller T_compact → larger magnitude. The current self-consistency analysis (Paz 2026c §6.1, §6.2) constrains T_compact to be of order 2t₀, but the exact value is determined by the DHOST field equation on FRW + T² background, which has not yet been solved analytically.
Status: priority HIGH open item. The full T_compact derivation requires solving the DHOST equation of motion for the volume modulus σ on the FRW + T² background, accounting for the breathing-mode kinetic structure and the moduli-stabilization potential. Estimated effort: focused calculation on the order of a week of work, using standard EFT-of-DE machinery (GLPV, Crisostomi-Hull-Koyama-Tasinato).
If T_compact is determined favorably (consistent with self-consistent background analysis), the framework gains a fully predictive w(z) curve. If T_compact is determined unfavorably (inconsistent with cosmological observations of phantom magnitude), the framework needs revision.
The DESI DR2 best-fit (w₀ ≈ −0.7, quintom-B trajectory) is in structural tension with the STF prediction (w₀ = −1 exactly, monotonic phantom). The framework currently relies on:
If Euclid confirms DESI’s quintom-B trajectory at >3σ, the STF dark-energy structure is falsified. The framework would survive as a dark-matter theory (Paz 2026d) but the unified-dark-sector picture would fail in the dark-energy component.
Status: outstanding empirical risk. The framework cannot adjust its prediction to fit DESI without abandoning the structural T² nodal mechanism (which is the framework’s central dark-energy claim). The choice is binary: STF’s structural prediction stands or it is falsified.
The STF framework does not currently address the H₀ tension between CMB-derived and SNe-derived Hubble constants. The dark-energy mechanism is late-time, so it does not affect the recombination physics that would resolve H₀ tension through Early Dark Energy. The STF is currently silent on H₀.
If the H₀ tension is real and requires new physics at recombination, the STF would need to be extended. The most natural extension would be through the regime-dependent curvature operator (§II.D): on FRW, the Ricci-rate operator dominates; near recombination, both Ricci and Weyl contributions could matter. This has not been quantitatively analyzed in the framework.
Status: open item, lower priority than T_compact. The H₀ tension is a “nice-to-have” for a complete cosmological framework but not directly relevant to the structural dark-energy prediction.
Standard ΛCDM has a “why is Λ so small?” problem (vacuum energy 10¹²² times larger than observed). The STF framework partially addresses this through the residual potential V(φ_min) at the stabilized modulus — the “small” value of Λ_eff is the output of the moduli stabilization, not a free input. However, the specific value of V(φ_min) depends on details of the moduli potential that are not fully derived in the framework.
The structural achievement is that V(φ_min) is consistent with the |R₀|/c² = 4Λ_eff self-consistency condition, giving Ω_m = 4/(3(1+π)) ≈ 0.322 (matching Planck 2018 within 1σ). But the absolute scale V₀ is still set by the compactification volume and the breathing-mode dynamics — not derived from first principles in a strong sense.
Status: partial progress. The STF replaces “why is Λ so small?” with “why does the moduli potential have this structure?” — a different, perhaps more tractable question, but not a complete solution. The cosmological constant problem remains a real concern.
A general no-go theorem rules out dark-energy models with w < −1 implemented through fundamental fields with positive kinetic energy (Hsu et al. 2004, Cline et al. 2004). The STF evades this through DHOST Class Ia structure: the effective w(z) on FRW can be < −1 due to the time-varying coupling Λ_eff(t), without the underlying scalar having negative kinetic energy. This is the “effective phantom without ghost” mechanism.
The validity of this evasion has been verified at the level of: - Background dynamics (Paz 2026c §6.2: self-consistent Friedmann iteration converges) - Perturbation stability (Paz 2026c §6.3: c_s²(z=0) = 1 exactly, c_s²(z>0) > 0 with O(10⁻²²) suppression) - Tensor mode propagation (Paz 2026a §C.6: c_T = c structurally) - Ghost-freedom on Kerr backgrounds (Paz 2026a §C.7c: 2 tensor + 1 scalar dof on arbitrary vacuum backgrounds)
Status: established. The phantom problem evasion is rigorous within DHOST Class Ia. However, this is the theoretical guarantee; the empirical test is whether the framework’s specific predictions match observations (which is the DESI tension, §VI.B).
Recent literature has raised the question of whether dark energy is relevant at cluster scales — could the local dark-energy density influence cluster dynamics in observable ways? The STF framework has not addressed this question. The DM paper’s §III.G phase census places clusters in the “decoherent” regime (DM behaves CDM-like), but does not analyze the role of dark energy specifically at cluster scales.
Status: not yet addressed in framework. Likely a small effect (Λ_eff is small compared to cluster gravitational scales) but worth quantifying.
The STF dark-energy prediction depends on the rate operator (ζ/Λ)φ(n^μ∇_μℛ), specifically the FRW-Ricci form. This operator is part of the broader STF Lagrangian. *(Status, August 2026: the cross-disformal matter coupling formerly invoked for galactic phenomenology and flyby validation is withdrawn — B̂_KK = 0, and its coefficient came from the withdrawn flyby match; the galactic leg is an open programme and the flyby is an empirical target explained by a measurement theorem, V8.1 App. F–M.)*
For dark-energy purposes specifically, this is less critical than for dark-matter purposes: the dark-energy mechanism uses only the rate operator on FRW, not the cross-disformal matter coupling. So the dark-energy predictions are valid even if the cross-disformal UV origin remains unresolved. But the framework as a whole has this open item.
Status: acknowledged open item, not blocking dark-energy analysis.
The STF dark-energy analysis depends on the π/4 causal-diamond derivation in Paz 2026a, §M.7 (and the supporting calculation in Paz 2026c). Specifically:
DE-α (current epoch identification θ = π/2): Inherited from Energy V0.2 §A.3-A.4. This sets the angle θ = π/2 as the present epoch via the π/4 causal-diamond geometric argument. The argument is rigorous; θ = π/2 is structurally identified, not chosen.
DE-β (self-consistent background): CLOSED in Paz 2026c §6.2 (April 2026). Self-consistent Friedmann iteration converges to t₀H₀ = 0.957474 vs ΛCDM 0.944744 (1.35% shift); w(z) corrections ≲ 5% for T_compact = 2t₀.
DE-γ (T_compact determination): Open priority HIGH (Paz 2026c §6.1).
DE-δ (perturbation stability): CLOSED at leading order in Paz 2026c §6.3 (April 2026). c_s²(z=0) = 1 exactly; c_s²(z>0) > 1 − 2 × 10⁻²⁵ throughout cosmic history.
The current state of the dark-energy analysis, as of April 2026, has DE-α inherited from Energy V0.2 (rigorous), DE-β and DE-δ closed, and DE-γ open as the priority HIGH item. The framework is in a stable analytic state, with the empirical question (DESI tension, §VI.B) being the dominant remaining concern.
| Item | Status | Priority |
|---|---|---|
| T_compact determination | Open, ~1 week computation | HIGH |
| DESI DR2 empirical tension | Open, awaiting Euclid 2026 | CRITICAL (empirical) |
| H₀ tension | Not addressed | Medium |
| Cosmological constant problem | Partially addressed | Medium |
| Phantom problem evasion | Established (DHOST Class Ia) | Closed |
| Cluster-scale DE effects | Not addressed | Low |
| Cross-disformal UV origin | Not blocking DE | Low (for DE) |
| DE-α (θ = π/2 identification) | Closed (rigorous) | Closed |
| DE-β (self-consistent background) | Closed (V0.2) | Closed |
| DE-δ (perturbation stability) | Closed (V0.2 leading order) | Closed |
| DE-γ (T_compact) | Open | HIGH |
The framework’s dark-energy sector is in a state where the theoretical analysis is largely closed (DE-α, DE-β, DE-δ all resolved; DE-γ pending T_compact derivation), but the empirical test is in active falsification range against DESI. Euclid 2026 is the decisive measurement.
The STF dark-energy sector makes six quantitative predictions testable with current or near-future observational capabilities. We classify them by falsifiability tier and present a prediction-dependency map indicating which observables survive if individual components fail.
The structural prediction. The current-epoch dark-energy equation of state is exactly w₀ = −1, derived from the third-order tangency of the T² coupling integral α(θ) at θ = π/2.
\[\boxed{w(z=0) = -1 \quad \text{exactly, independent of } T_{\rm compact}}\]
Tier: Tier 1 structural (universal differential topology — vanishing of dα/dθ at θ = π/2). This is the most robust dark-energy prediction of the framework. It does not depend on T_compact, on details of the moduli stabilization, or on numerical fitting.
Test: Euclid first cosmology release (October 2026), with σ(w₀) ≈ 0.01 expected. Falsification: w₀ measured significantly above −1 at >3σ would falsify the T² nodal mechanism.
This is the prediction most directly under empirical pressure from DESI DR2 (which prefers w₀ ≈ −0.7). The framework’s structural claim is that DESI’s signal, if real, must reflect either parametrization artifacts (CPL crossing as Taylor-expansion feature) or alternative physics (evolving DM, coupled dark sector) rather than fundamental dark-energy evolution above −1 at the present epoch.
The sign prediction. At all redshifts z > 0, w(z) < −1, with magnitude conditional on T_compact:
\[\boxed{w(z) < -1 \quad \text{for all } z > 0}\]
Tier: Tier 1 sign (structural — follows from cos²(θ) > 0 for θ < π/2 and α(θ) > 0); Tier 4 magnitude (depends on T_compact, currently constrained to ≃ 2t₀ but not uniquely determined).
Test: Euclid + Roman combined w(z) reconstruction over 0 < z < 2. Falsification scenarios:
The framework’s distinctive feature is the combination of w₀ = −1 exact + w(z>0) < −1 monotonic. Either w₀ above −1 or w-trajectory crossing −1 falsifies the structure.
The paired structural result. The dark-energy sound speed at the current epoch is exactly c_s² = 1, derived from the same T² nodal mechanism that gives w₀ = −1.
\[\boxed{c_s^2(z=0) = 1 \quad \text{exactly}}\]
Tier: Tier 1 structural (paired with w₀ = −1 by same nodal mechanism).
Test: Indirect — c_s² constrains structure formation in the dark-energy sector. The Euclid clustering + weak lensing analyses can constrain c_s² to ~0.1 precision. Direct measurement is not currently feasible. Falsification: any measurement showing c_s² substantially different from 1 at z = 0 would falsify the paired structural prediction.
The pairing of w₀ = −1 ↔︎ c_s²(z=0) = 1 is unique to the STF among dark-energy models. Other models with w₀ = −1 (the cosmological constant) trivially have c_s² undefined; models with effective w₀ = −1 (specific quintessence trajectories, fine-tuned DHOST functions) do not generically also have c_s²(z=0) = 1 exactly.
The trajectory shape. The STF w(z) is monotonic — w(0) = −1, monotonically decreasing as z increases. There is no phantom crossing at any redshift.
\[\boxed{\text{No } z \text{ such that } w(z) = -1 \text{ for } z > 0}\]
Tier: Tier 1 structural (follows from monotonic decrease of α(θ) and increase of cos²(θ) as θ decreases from π/2).
Test: Reconstructed w(z) from Euclid + DESI + Roman. Falsification: any phantom crossing at >5σ falsifies the structural trajectory shape.
This distinguishes the STF from: - w₀wₐCDM (CPL): Has phantom crossing built into the Taylor expansion - Quintom-A models: Have phantom crossing by construction - Chen & Loeb evolving DM: Has oscillating w with multiple crossings
The matter density. The total matter density (including dark and baryonic) is:
\[\boxed{\Omega_m = \frac{4}{3(1+\pi)} \approx 0.3220 \pm 0.0050}\]
Tier: Tier 2 derived from |R₀|/c² = 4Λ_eff self-consistency (Paz 2026a §III.E).
Test: Direct comparison with cosmological measurements: - Planck 2018: Ω_m = 0.315 ± 0.007 — within 1σ ✓ - DESI DR2 (within ΛCDM): Ω_m = 0.295–0.307 — 2-3σ tension - Euclid first cosmology release (Oct 2026): expected σ(Ω_m) < 0.005
If Euclid measures Ω_m converging to 0.322 ± 0.005, the T² self-consistency is confirmed. If Ω_m is measured outside [0.31, 0.34] at >3σ, the curvature–dark energy link is falsified (LEVELS 0-2 of the framework survive — the dark-matter sector is independent).
The tensor mode prediction. The gravitational wave speed is exactly c, structurally:
\[\boxed{c_T = c \quad \text{exactly, by } G_{4X} = 0}\]
Tier: Tier 1 structural (DHOST Class Ia with specific Horndeski mapping).
Test: Already validated by GW170817 (|c_T − c|/c < 5 × 10⁻¹⁶, Abbott et al. 2017). Future GW + EM coincidences (LIGO O5 onwards, LISA) will tighten the constraint further. Falsification: any future GW + EM coincidence showing c_T ≠ c at >5σ would falsify a foundational element of the framework (but this would also falsify all DHOST Class Ia dark-energy theories simultaneously).
| Tier | Definition | STF Predictions |
|---|---|---|
| Tier 1: Structural | Rigorous from differential topology / EFT theorems | Pred. 1 (w₀=−1 exact); Pred. 2 sign (w<-1 past); Pred. 3 (c_s²=1 paired); Pred. 4 (no crossing); Pred. 6 (c_T=c) |
| Tier 2: Derived | Computed from framework parameters with no fitting | Pred. 5 (Ω_m = 4/(3(1+π))) |
| Tier 3: Closure-conditional | Internally constrained, depends on closure principles | (None for DE; this tier exists for galactic γ_eff in Paz 2026e) |
| Tier 4: Magnitude/scope-conditional | Sign rigorous, magnitude depends on open parameter | Pred. 2 magnitude (w<-1 magnitude depends on T_compact) |
Each advertised prediction depends on different subsets of the STF inputs. The following map clarifies which predictions are independent of T_compact and which would be affected if individual components fail:
| Observable | Depends on m_s? | Depends on ζ/Λ? | Depends on T² nodal? | Depends on T_compact? | Survives if DESI quintom-B confirmed? |
|---|---|---|---|---|---|
| w(z=0) = −1 exact | No | No | Yes | No | No (falsified if Euclid confirms DESI) |
| w(z) < −1 sign for z > 0 | No | No | Yes | No | No (paired with w₀=−1) |
| w(z) < −1 magnitude | No | No | Yes | Yes | N/A (sign component falsified) |
| No phantom crossing | No | No | Yes | No | No (paired with monotonicity) |
| c_s²(z=0) = 1 exact | No | Indirect (Λ_eff) | Yes | No | Decoupled — survives even if w-trajectory fails |
| c_s²(z) > 0 throughout | No | Yes | Indirect | No | Survives (Planck-suppressed) |
| Ω_m = 0.322 | No | Indirect (Λ_eff) | Indirect | No | Decoupled — depends on |R₀|=4Λ_eff |
| c_T = c | No | No | No | No | Survives (DHOST Class Ia structural) |
Key result: Five of the eight observables are independent of T_compact. Three of the eight (Ω_m, c_T, c_s²) are decoupled from the T² nodal w(z) prediction and would survive even if Euclid confirms DESI’s quintom-B trajectory. The framework’s dark-energy core would be falsified at the structural level by DESI confirmation, but the broader STF framework (dark matter condensate, Standard Model derivations, the flyby measurement theorem, |R₀|=4Λ_eff curvature link) would survive.
Conversely, what would Euclid measurements that strongly confirm the STF look like?
If all four hold, the framework would have its first major empirical victory in the dark-energy sector. The Tier 1 structural predictions (w₀ = −1 exact, no crossing, c_s² = 1, c_T = c) would all be vindicated; the Tier 2 derived prediction (Ω_m) would also be confirmed.
This is the empirical target. We do not advertise it as likely — the DESI DR2 result currently points the other way — but it is the clean test case for the framework.
Dark energy is conventionally framed as a phenomenon: “the universe is accelerating; what is responsible?” The standard model (cosmological constant) treats dark energy as an unexplained input. Alternative models (quintessence, k-essence, phantom, EDE) treat it as a dynamical phenomenon requiring scalar fields with specific potentials.
The STF approach is structural. The dark-energy mechanism is not a phenomenological fit but a consequence of the same compactification (CICY #7447/Z₁₀) that produces dark matter and (conjecturally) Standard Model parameters. The T² causal-diamond integral structure is geometric, not parametric. The current epoch identification through |R₀|/c² = 4Λ_eff is a self-consistency condition, not a free choice.
This is a different epistemological position than typical dark-energy models. The STF asks: “Given the compactification and the T² coupling structure, what does the framework predict?” The answer (w₀ = −1 exactly, w<-1 past, Ω_m ≈ 0.322) is then a derived consequence — testable but not adjustable. If observations falsify the prediction, the framework is wrong; it cannot be saved by parameter tuning.
The T² causal-diamond integral α(θ) = ∫₀^θ cos²(θ’)dθ’ is the geometric heart of the STF dark-energy mechanism. The third-order tangency at θ = π/2 (where dα/dθ = cos²(π/2) = 0) is what gives the structural prediction w(z=0) = −1 exactly. This is differential topology — a feature of the geometric integral, not a parametric tuning.
The framework’s Calabi-Yau compactification produces specifically a T² causal diamond structure (Paz 2026a, Appendix M.7). This is not a generic feature of any compactification — it is specific to the CICY #7447/Z₁₀ geometry. The structural prediction therefore reflects a deep geometric fact about the specific Calabi-Yau used.
If the T² structure were different (e.g., T³, or non-symmetric T²), the coupling integral would have different tangency properties, giving different w(z) predictions. The framework’s prediction is a fingerprint of the specific compactification geometry. This is what makes the dark-energy test discriminating: not all compactifications give w(z=0) = −1; only this one does.
The STF dark-matter and dark-energy mechanisms emerge from the same scalar field with the same parameters {m_s, ζ/Λ}. The dark-matter mechanism uses the field’s oscillation-averaged stress-energy at cosmological scales (giving ⟨w_DM⟩ = 0); the dark-energy mechanism uses the residual potential modulated by T² coupling (giving w(z=0) = −1 exactly).
This is structural unification, not eclectic phenomenology. The framework cannot adjust dark-matter and dark-energy parameters independently — they share m_s and ζ/Λ. The compactification chain that derives ζ/Λ also constrains the moduli potential structure that gives V(φ_min). The geometry that produces the dark-matter response to galactic curvature also produces the T² causal-diamond integral for dark energy.
The unified picture has falsifiability advantages. A fitted dual-component model (separate DM with parameters {a, b, c} and DE with parameters {d, e, f}) has six free parameters; a discrepancy in one sector can be absorbed by adjusting that sector’s parameters. The STF has zero free parameters in either sector beyond the compactification + T_compact. A discrepancy in one sector falsifies the unified structure; the framework cannot escape by parameter retuning.
In Paz 2026d (Dark Matter paper), the structural argument is:
The cross-disformal phase transition produces collective phenomenology at galactic scales. The X^{3/2} phonon exponent is universal from fold-catastrophe topology; the marginal-stability closure derives the force amplitude γ(M_b) from {ζ/Λ, M_b, a₀, T_compact} alone with one structural assumption.
In the present DE paper, the structural argument is:
The T² coupling integral’s third-order tangency at θ = π/2 produces w(z=0) = −1 exactly. The structural prediction follows from differential topology of α(θ), not parameter fitting; the magnitude of w(z) for z > 0 depends on T_compact through the cosmological self-consistency condition.
Both arguments rely on a topological / differential-topology feature (fold catastrophe in the DM case; third-order tangency in the DE case) producing a structural prediction. Both have a magnitude-conditional residual depending on a single open parameter. Both connect to broader cosmological parameters through self-consistency conditions.
This parallel structure is not coincidental — it reflects the framework’s underlying methodology: derive the qualitative structure from geometry, then use closure/self-consistency to fix magnitudes. The framework is uniformly structural rather than ad-hoc.
We summarize the falsification scenarios across the predictions:
Strong falsification (Tier 1 structural): - w₀ measured significantly above −1 at >3σ → T² nodal mechanism falsified - Phantom crossing at any redshift confirmed at >5σ → monotonic trajectory falsified - c_T ≠ c measured by future GW + EM coincidence → DHOST Class Ia framework falsified
Conditional falsification (Tier 2 / Tier 4): - Ω_m measured outside [0.31, 0.34] at >3σ → curvature–dark energy link falsified (DM sector survives) - T_compact derivation gives values inconsistent with self-consistent background → magnitude prediction fails
Empirical pressure (current): - DESI DR2 best-fit (w₀ ≈ −0.7, quintom-B): in tension with framework, awaiting Euclid 2026 to confirm or refute
The framework is falsifiable on multiple independent channels. This is the appropriate state for a scientific theory making structural claims.
The cosmological constant problem is sometimes addressed through anthropic reasoning: the observed value of Λ might be a selection effect of the observer’s existence in a multiverse. The STF framework rejects this approach: the observed Λ_eff is derived from the moduli stabilization potential and the T² self-consistency condition, both consequences of the specific Calabi-Yau compactification.
If the STF prediction Ω_m = 0.322 holds, this is evidence against anthropic explanations: the value is derived geometrically, not selected anthropically. Conversely, if the STF prediction fails and no other geometric explanation is found, anthropic reasoning becomes more credible by default.
The framework’s structural prediction is therefore not just an empirical claim but a philosophical claim about the nature of cosmological constants. We acknowledge this is a strong stance and defer further discussion to a separate paper.
The same compactification that produces dark matter and dark energy is conjectured to derive Standard Model parameters (Paz 2026a, Appendices M-O). Specific results from V7.9 include:
The dark-energy sector therefore connects to the broader STF framework’s claims about the Standard Model. If the dark-energy prediction holds (w₀ = −1 exact, Ω_m = 0.322), this is independent confirmation of the underlying compactification structure. If the dark-energy prediction fails, it does not directly falsify the SM derivations (which are independent), but it does cast doubt on the framework’s broader claim of unification.
Specific computational targets in the framework’s development:
T_compact derivation (priority HIGH, ~1 week): Solve the DHOST equation of motion for the volume modulus on FRW + T² background. Outcome: fully determined w(z) shape.
H₀ tension treatment (priority MEDIUM, multi-week): Investigate whether the regime-dependent curvature operator has effects at recombination that could resolve the H₀ tension within the framework.
Cluster-scale dark-energy effects (priority LOW, multi-week): Quantify whether local Λ_eff has observable effects on cluster dynamics.
Cross-correlation with galactic γ_eff(z) (priority MEDIUM, ongoing): The galactic dark-matter coupling γ_eff might depend on z through the cosmological background. Investigation in progress (Paz 2026e, Branch I-δ in V7.9 audit).
Direct Euclid forecast (priority MEDIUM, available data): Forecast specific Euclid measurements that would distinguish STF from competing models. Use Euclid data products as they become available in 2026-2028.
We have presented the dark-energy sector of the Selective Transient Field framework, a unified scalar-field theory in which dark matter and dark energy emerge from the same scalar field — the breathing mode of six compact extra dimensions in a 10D Einstein-Gauss-Bonnet compactification on the Calabi-Yau threefold CICY #7447 with Z₁₀ free quotient structure.
The dark-energy mechanism uses the residual potential V(φ_min) at the stabilized modulus, modulated by the T² causal-diamond integral α(θ) = ∫₀^θ cos²(θ’)dθ’. The current epoch is identified with θ = π/2 by the geometric self-consistency condition |R₀|/c² = 4Λ_eff, giving Ω_m = 4/(3(1+π)) ≈ 0.322 — within 1σ of Planck 2018 measurements. The third-order tangency dα/dθ|_{π/2} = cos²(π/2) = 0 exactly produces the structural prediction:
\[\boxed{w(z=0) = -1 \quad \text{exactly, independent of } T_{\rm compact}}\]
For z > 0, the coupling accumulates, giving an effective phantom trajectory w(z) < −1 with no phantom crossing. The DHOST Class Ia structure permits this effective phantom behavior without fundamental ghost. The same nodal structure that gives w₀ = −1 also gives c_s²(z=0) = 1 exactly — perturbation stability paired with the equation-of-state result. The gravitational wave speed satisfies c_T = c exactly (structural, GW170817-compatible).
We compared the STF framework with eight competing dark-energy theories (cosmological constant, quintessence, k-essence, phantom DE, w₀wₐCDM CPL, early dark energy, scalar-tensor DHOST, emergent/holographic, coupled dark sector) and identified the STF’s structural position: the only entry occupying all of {derived mechanism from microphysics, structural w₀ = −1, DHOST Class Ia GW170817-compatibility, effective phantom without ghost, zero free parameters beyond T_compact, UV completion via 10D EGB compactification, unified dark sector}.
We also engaged honestly with the DESI DR2 empirical tension: the apparent quintom-B trajectory (w<-1 past, crossing to w>-1 today) is structurally opposite to the STF prediction (w₀ = −1 exact, monotonic phantom past, no crossing). The DESI result is contested — prior dependence (Cortês & Liddle 2024-25), CMB likelihood updates (Roy Choudhury et al. 2025), and alternative interpretations (Chen & Loeb 2025: evolving dark matter, not dark energy) leave the empirical case unresolved. Euclid’s first cosmology release in October 2026 is the decisive test.
The framework’s structural prediction w(z=0) = −1 exact has the appropriate epistemological status: it is testable (via Euclid σ(w₀) ≈ 0.01) and falsifiable (any measurement of w₀ significantly above −1 at >3σ falsifies the T² nodal mechanism). The prediction is structural, not parametric — the framework cannot accommodate w₀ ≠ −1 by parameter tuning without abandoning the central T² nodal claim.
We have presented six testable predictions with falsifiability tiers and a prediction-dependency map showing which observables survive if individual components fail. Five of eight predictions are independent of the open parameter T_compact; three of eight are decoupled from the T² nodal w(z) prediction itself (Ω_m, c_T, c_s²) and would survive even if Euclid confirms DESI’s quintom-B trajectory. The unified dark-sector picture is structurally over-constrained relative to fitted dual-component models.
The framework is in active falsification range. The DM paper (Paz 2026d) presents the dark-matter case; the DE V0.2 supporting derivation (Paz 2026c) presents the calculation chain; the present paper presents the applied dark-energy treatment with observational synthesis and theoretical comparison. Together they form a coherent triple covering the framework’s dark-sector predictions and their empirical test.
If Euclid 2026 confirms w₀ ≈ −1 ± 0.01 with monotonic w(z) trajectory, the STF framework will have achieved its first major empirical confirmation in the dark-energy sector — a structural validation distinct from any fitted-parameter dark-energy model. If Euclid 2026 confirms DESI’s quintom-B trajectory at >3σ, the STF dark-energy structure is falsified, the unified-dark-sector picture fails, and the framework survives only as a dark-matter theory (Paz 2026d).
Either outcome is informative. The framework is structurally honest about the empirical risk. The next 12-24 months will be decisive.
This research was conducted independently, without institutional affiliation or external funding. The author thanks the open-source scientific community for the publicly available datasets and software tools that made this work possible.
The author acknowledges the use of Claude AI (Anthropic, 2024–2026) for assistance with mathematical formulation, statistical code implementation, and manuscript language editing. The Selective Transient Field theoretical framework, research hypothesis, experimental design, data analysis methodology, and all scientific interpretations are entirely the author’s original intellectual contributions. All decisions regarding data analysis, parameter selection, statistical methods, and conclusions represent the author’s independent scientific judgment. Claude was used as a research and writing assistant tool, not as a co-author or independent analyst. The current paper was developed in dialogue with similar AI collaboration. The author bears responsibility for all framework choices, claims, and conclusions.
The framework’s empirical predictions will be tested by data from the DESI collaboration (Karim et al. 2025), the Euclid Consortium (Q1 release 2025; first cosmology release October 2026), the Vera C. Rubin Observatory (LSST, beginning operations 2025), the Roman Space Telescope (launching 2027), the Square Kilometre Array (SKA, under construction), and other current and forthcoming surveys. We acknowledge the enormous observational efforts that make these tests possible.
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April 2026. Z. Paz, The Hague, Netherlands. Corresponding author: [email]
Companion papers in the framework: - STF First Principles V7.9 (Paz 2026a) — main framework derivation - STF Dark Matter Full v2 aligned (Paz 2026d) — applied dark-matter paper, parallel structure to this paper - STF Dark Energy w(z) Derivation V0.2 (Paz 2026c) — supporting calculation for w(z) result - STF Galactic Sector Marginal-Stability Closure V0.1 (Paz 2026e) — supporting calculation for γ_eff - STF Cross-Disformal Coupling (Paz 2026b) — UV motivation for cross-disformal matter coupling
@article{paz2026darkenergy,
author = {Paz, Z.},
title = {Dark Energy as Geometry: The STF Framework and the T² Causal Diamond},
year = {2026},
version = {V0.1},
url = {https://existshappens.com/papers/dark-energy/}
}