The Selective Transient Field (STF) produces three structurally distinct effects from a single scalar field with two derived parameters — the mass m_s = 3.94 × 10⁻²³ eV and the coupling ζ/Λ = 1.35 × 10¹¹ m². In the cosmological background, the mass term produces oscillation-averaged cold-dark-matter-like behavior (⟨w⟩ = 0, ρ ∝ a⁻³) through the standard fuzzy-scalar mechanism; the Ricci-rate coupling sources dark energy through the T² topological pump with equation of state w(z=0) = −1 exactly and ghost-free effective phantom history w(z) < −1 for z > 0. At galactic scales, the same coupling produces MOND phenomenology through the nonperturbative cross-disformal regime. This paper derives the unified cosmological sector: the WKB mass-oscillation mechanism, the Friedmann budget check Ω_b + Ω_DM,osc + Ω_Λ,T² ≈ 1.00 from a single field, and the structural independence of the three mechanisms by their controlling parameters. The dark-energy equation of state w(z) is derived in [Paz 2026f] and its structural results are integrated here. Falsifiable predictions: w(z) trajectory distinct from DESI CPL best-fit (Euclid σ(w₀) ∼ 0.01–0.02); Ω_m = 4/(3(1+π)) = 0.3219 from T² self-consistency; a₀(z) = cH(z)/2π cross-epoch evolution distinguishing STF from Milgromian MOND.
The STF Lagrangian in the curvature-coupling sector is [Paz 2026c, §III.A]:
$$\mathcal{L}_{\rm STF} = \frac{1}{2}(\nabla_\mu\phi)^2 - \frac{1}{2}m_s^2(\phi-\phi_0)^2 + \frac{\zeta}{\Lambda}\phi\,(n^\mu\nabla_\mu\mathcal{R})$$
with m_s and ζ/Λ derived from independent first-principles routes — m_s from the cosmological threshold condition 𝒟_crit = 𝒟_GR at the 730 R_S binary-inspiral activation [Paz 2026c, §III.D], ζ/Λ from 10D compactification over CICY #7447/Z₁₀ [Paz 2026c, Appendix O]. The field produces three distinct physical effects:
| Mechanism | Lagrangian term | Scale | Observable | Controlling parameter |
|---|---|---|---|---|
| Mass oscillation around potential minimum | (1/2)m_s²(φ − φ₀)² | Cosmological | ⟨w⟩ = 0, ρ ∝ a⁻³ (dark matter) | m_s |
| Ricci-rate coupling to curvature gradient | (ζ/Λ)φ(n^μ∇_μℛ) | Cosmological | w(z) ≤ −1, Λ_eff via T² pump (dark energy) | ζ/Λ, T² topology |
| Cross-disformal matter coupling, nonperturbative regime | g̃_μν = g_μν + B̂(∂φ∂ℛ + ∂ℛ∂φ) | Galactic (h ≫ 1 at r ≳ kpc) | a₀ = cH₀/2π, flat rotation curves | ζ/Λ |
Structural independence of the controlling parameters. The cosmological perturbation spectrum — and thus any Lyman-α or CMB-based constraint on the dark-matter sector — depends on m_s alone. The galactic phenomenology and the cosmological dark-energy contribution depend on ζ/Λ (and, for dark energy, the T² topology). The two parameters have independent derivation routes and independent observational probes. Neither is fitted to the other’s domain.
This structural independence has a consequence that organizes the rest of the paper: each mechanism stands or falls on its own evidence. Observational challenges to the mass-oscillation dark-matter role (controlled by m_s) do not threaten the galactic-sector or dark-energy predictions (controlled by ζ/Λ and T² structure).
The galactic MOND sector is developed in [Paz 2026h]. The present paper derives the two cosmological mechanisms and their combined consequences.
In FRW, the STF field equation [Paz 2026c, §VI.C.1]:
$$\ddot\phi + 3H\dot\phi + m_s^2(\phi - \phi_0) = \kappa\dot{R}$$
admits two distinguished configurations that coexist:
The general cosmological solution is a superposition. At the current epoch, m_s/H₀ ≈ 2.5 × 10¹⁰: the oscillator is vastly under-damped in its own frame (many oscillation periods per Hubble time) and over-damped in the cosmic-expansion frame. The tracker is adiabatic; the oscillating part δφ evolves independently around it.
The tracker contribution ρ_tracker = V(φ_tracker) = (κṘ)²/(2m_s²) evaluates to ~10⁻⁹² of the critical density today [Paz 2026c, Appendix M] — entirely negligible. The cosmologically relevant contribution from the mass term is the oscillation.
For δφ in the adiabatic regime m_s ≫ H, the WKB solution is:
δϕ(t) ≈ Φ(t)cos (mst+θ), Φ(t) ∝ a(t)−3/2
The amplitude decreases as a⁻³/² from Hubble friction [Turner 1983]. Energy density and pressure:
$$\rho_{\rm osc} = \frac{1}{2}\dot{\delta\phi}^2 + \frac{1}{2}m_s^2\delta\phi^2,\qquad p_{\rm osc} = \frac{1}{2}\dot{\delta\phi}^2 - \frac{1}{2}m_s^2\delta\phi^2$$
Averaging over one oscillation period τ_osc = 2π/m_s ≈ 3.32 yr — shorter than the Hubble time by a factor of ~10¹⁰ — kinetic and potential energies equilibrate:
$$\langle\dot{\delta\phi}^2\rangle_{\rm osc} = \langle m_s^2\delta\phi^2\rangle_{\rm osc} = \frac{1}{2}m_s^2\Phi^2$$
giving:
$$\boxed{\langle\rho_{\rm osc}\rangle = \frac{1}{2}m_s^2\Phi^2 \propto a^{-3},\qquad \langle p_{\rm osc}\rangle = 0,\qquad \langle w\rangle = 0}$$
This is the standard fuzzy-scalar result [Hu, Barkana & Gruzinov 2000; Marsh 2016]. The STF at m_s = 3.94 × 10⁻²³ eV, for modes inside the Hubble radius, behaves as a homogeneous pressureless cold-dark-matter fluid at the background level.
For a scalar Hubble-frozen until m_s = H and oscillating thereafter [Marsh 2016], the current energy density:
$$\rho_{\rm osc,0} \sim m_s^2 \Phi_{\rm i}^2 \left(\frac{a_{\rm osc}}{a_0}\right)^3$$
where Φ_i is the field amplitude at oscillation onset and a_osc the scale factor at that epoch. For m_s = 3.94 × 10⁻²³ eV, oscillation begins when H = m_s: in the radiation-dominated era that applies at these epochs, z_osc ≈ 1.6 × 10⁶. For Φ_i such that ρ_osc,0 matches observations, Ω_DM,osc ≈ 0.27.
Status. The STF mass-oscillation mechanism is sufficient in form to produce the observed dark-matter density when Φ_i is chosen appropriately. The mechanism is identical to standard fuzzy dark matter; the STF adds no new physics to the cosmological dark-matter production beyond the identification of m_s with the specific value derived from BBH-threshold physics. A first-principles derivation of Φ_i from compactification dynamics or the Cascade initial-condition structure [Paz 2026b] is an open item (§6.2).
The Ricci-rate coupling (ζ/Λ)φ(n^μ∇_μℛ) sources two distinct contributions to dark energy — the driven-minimum potential and the T² topological pump — separated by 92 orders of magnitude at the current epoch. Only the T² contribution is cosmologically observable.
The tracker potential V(φ_tracker) = (κṘ)²/(2m_s²) gives an energy density ρ_UV ∼ 10⁻¹⁵⁸ eV² at current Ṙ [Paz 2026c, Appendix M]. This is 92 orders of magnitude below the observed dark-energy density. Its equation of state at the current epoch is w = −1 + 2(H₀/m_s)² ≈ −1 + 3 × 10⁻²¹ [Paz 2026c, Theorem M.5.7] — effectively a cosmological constant at the 10⁻²¹ precision level, but at a density too small to be observable.
The Ricci-rate coupling, integrated over the T² compact-time topology with the fundamental mode φ(θ) = cos(θ), θ = πt/T_compact, produces a topological contribution to Λ_eff through the coupling-accumulation integral:
$$\alpha(\theta) = \int_0^\theta \cos^2(\theta')\,d\theta' = \frac{\theta}{2} + \frac{\sin 2\theta}{4}$$
with current epoch at θ = π/2 (the causal-diamond boundary fixed by full-period cancellation). This gives α(π/2) = π/4 exactly, and:
$$\Lambda_{\rm eff} = \frac{\pi}{4}\cdot\frac{|\dot R|}{H_0 c^2} = 1.124 \times 10^{-52}\,{\rm m}^{-2}$$
matching Λ_obs = 1.100 × 10⁻⁵² m⁻² to 2.2% with zero free parameters [Paz 2026e, §II.2].
The full five-step derivation of the causal-diamond boundary identification and the π/4 result is given in [Paz 2026e, §II.2]. The full z-evolution of Λ_eff(z), the continuity-equation derivation of w(z), the numerical trajectory, and the Python verification code are given in [Paz 2026f]. The present paper integrates these results into the unified cosmological architecture without reproducing them.
Three results from [Paz 2026f] are used in §4 and §5:
(i) At z = 0: w(z=0) = −1 exactly, from third-order tangency dα/dθ|_{π/2} = cos²(π/2) = 0. Independent of T_compact.
(ii) For all z > 0: w(z) < −1 monotonically, no phantom crossing from above. Effective phantom trajectory; ghost-free (DHOST Class Ia, α_T = 0, GW170817-compatible). See [Paz 2026g] for why this evades the standard phantom no-go theorem.
(iii) At z = 3: Ω_Λ,STF(z=3) ≈ 0.01 — approximately 1% of critical density — negligible at matter-dominated epochs. (Computed from α(θ(3))/(π/4) × Ω_Λ,0 / E(3)² using the w(z) V0.1 trajectory at T_compact = 2t₀.)
The two cosmological contributions from §§2–3 are structurally distinct:
These are not two estimates of the same energy density. They project onto different aspects of the field configuration. The 92-order-of-magnitude gap between the UV tracker potential and the T² pump (§3.1–3.2) is the quantitative measure of how distinct the two cosmological sources are.
At z = 0:
$$3M_{\rm Pl}^2 H_0^2 = \rho_b + \rho_r + \rho_{\rm osc} + \rho_{\rm tracker} + \rho_{\Lambda,T^2}$$
with:
Summing the non-negligible contributions:
$$\Omega_b + \Omega_{\rm DM,osc} + \Omega_{\Lambda,T^2} = 0.05 + 0.27 + 0.68 = 1.00$$
The same scalar field carries both the dark-matter role (through its mass term) and the dark-energy role (through its coupling term via the T² topology). Together they account for ≈ 95% of the cosmic energy budget. Baryons supply the remaining ≈ 5%. No additional dark-sector species are invoked.
The structural independence argued in §1 reflects the mathematical orthogonality of the three mechanisms:
The three probes — CMB/LSS for ρ_osc, Euclid/supernovae for w(z) via ρ_Λ,T², SPARC/Gaia for galactic a₀ — do not inform each other through the field equation. This is what allows a failure in one sector (e.g., cosmological perturbation constraints on m_s, addressed in [Paz 2026h]) to be structurally quarantined from the other two sectors.
The T² pump mechanism imposes an additional structural relation on the cosmological budget through the requirement |R₀|/c² = 4Λ_eff. This fixes a specific matter fraction Ω_m = 4/(3(1+π)) = 0.3219, derived in [Paz 2026e, §II.3]. The derivation and observational comparison to Planck 2018 (consistent at +1σ) and DESI (2–3σ tension, model-dependent) are given there; the present paper imports this constraint as an additional prediction from the same T² architecture underlying the w(z) trajectory.
Prediction (§3.3): w(z=0) = −1 exactly (third-order tangency at θ = π/2); w(z) < −1 monotonically for all z > 0; no phantom crossing from above.
Distinguishes from:
Test: Euclid (operating) will constrain w₀ to σ ≈ 0.01–0.02 by 2027. STF is falsified if w₀ is measured significantly above −1 at > 3σ robust to systematics.
See §4.4; derivation in [Paz 2026e, §II.3]. Prediction: Ω_m = 4/(3(1+π)) = 0.3219. Euclid projected σ(Ω_m) ≈ 0.002–0.003 will test this decisively. Falsified if Ω_m < 0.31 or > 0.34 at > 5σ. The falsification isolates to the T² curvature–dark-energy link; the rest of the STF framework survives.
Prediction: a₀(z) = cH(z)/2π. The MOND acceleration scale evolves with the Hubble parameter because its derivation from the cross-disformal coupling structure ties it to the current cosmological expansion rate, not to a fundamental constant.
Distinguishes from Milgromian MOND: Standard MOND treats a₀ as a constant with no predicted z-dependence. STF predicts a specific H(z) tracking.
Test: High-redshift galaxy rotation curves (JWST, next-generation telescopes). If a₀ is observed constant across redshift contrary to STF’s cH(z)/2π prediction, the cross-disformal coupling structure must be revised.
A consequence of the structural independence (§1, §4.3):
| Observation | Falsifies | Survives |
|---|---|---|
| w₀ significantly > −1 at > 3σ | T² pump mechanism | Mass-oscillation DM, galactic sector |
| Ω_m outside [0.31, 0.34] at > 5σ | T² self-consistency (curvature–DE link) | Core STF framework |
| a₀ observed constant across z | Cross-disformal cosmological scaling | Mass-oscillation DM, T² pump |
| Lyman-α tightening excludes m_s ≤ 10⁻²⁰ eV | Mass-oscillation DM (as formulated) | T² pump, galactic sector |
The framework is falsifiable piece-by-piece. No single observation collapses the architecture; each sector carries its own falsifiers tied to its own controlling parameters.
The magnitude of |1+w(z)| at z > 0 scales as ξ = 1/(H₀T_compact). The θ_now = π/2 identification gives T_compact = 2t₀ ≈ 27.6 Gyr, yielding |1+w(z=0.3)| ≈ 0.095. Larger T_compact (up to ∼ T_depart ≈ 2.4 × 10¹⁴ yr) shrinks the deviations toward observationally indistinguishable from Λ. The full DHOST field-equation solution is needed to fix T_compact from first principles; the full sensitivity analysis across T_compact ∈ {2t₀, 20t₀, 200t₀, T_depart/H₀} is given in [Paz 2026f, §6.1]. The structural results — w(z=0) = −1 exactly, no crossing, monotonic phantom history — hold regardless of T_compact.
The cosmological Ω_DM,osc depends on Φ_i, the field amplitude at the onset of oscillation. The present paper takes Φ_i as a free parameter fitting observed Ω_DM ≈ 0.27. A first-principles derivation from compactification dynamics (Appendix J-class inflation analysis [Paz 2026c]) or from the Cascade initial-condition structure [Paz 2026b] is deferred to future work.
The 10D Gauss-Bonnet reduction produces two distinct descendants of the e^{κσ} I₄(g) term, where I₄ is the curvature-squared invariant — reducing to C_μνρσ C^μνρσ in vacuum (Ricci-flat, only Weyl survives) and to a R² + b R_μν R^μν in FRW (conformally flat, only Ricci terms survive), with coefficients a, b set by the compactification ([Paz 2026c, §L.4, §L.5.1]). The first descendant, γφ I₄ (linear in φ, obtained in §O.3.3 by Taylor-expanding e^{6σ} around σ₀), becomes the STF rate coupling through EFT matching and is the origin of the (ζ/Λ)φ(n^μ∇_μℛ) operator in the Lagrangian. The second descendant, extracted at quadratic order in δσ in [Paz 2026c, §O.3.4]:
$$\Delta\mathcal{L}^{(2)} = \frac{3}{8}\,\lambda_{\rm GB}\,e^{6\sigma_0}\,\phi^2\,I_4(g_{\rm bg})$$
shifts the effective scalar mass on non-vacuum backgrounds:
$$m_{s,{\rm eff}}^2 = m_s^2 + \frac{3}{4}\,\lambda_{\rm GB}\,e^{6\sigma_0}\,I_4(g_{\rm bg})$$
In vacuum, I₄ = 0 on the homogeneous Minkowski background, so V_stab alone determines m_s,vac = 3.94 × 10⁻²³ eV through V’’(σ₀)/(24M_Pl²) [Paz 2026c, §L.3]. The second-order descendant contributes only on non-vacuum backgrounds. In FRW, with I₄ scaling as H⁴, the present-day correction is I₄ ∼ H_0⁴ ∼ 10⁻¹³² eV⁴ and Δm²_eff ∼ 10⁻¹¹² eV² — utterly negligible compared to m_s,vac² ∼ 10⁻⁴⁵ eV². The correction was significant in the early universe (H ∼ M_Pl) and asymptotes to the vacuum value as H → 0 [Paz 2026c, §L.3 note, §O.3.4]. The sign of the correction depends on λ_GB and the I₄ coefficients (a, b in a R² + b R_μν R^μν), and has not been determined.
The open question is whether the regime-dependent Ricci decomposition of I₄ in FRW — the specific combination of R² and R_μν R^μν that emerges from the compactification — contains terms whose H-scaling at intermediate epochs (relevant for structure formation at z ∼ 1–5) differs from the full Gauss-Bonnet invariant’s H⁴ scaling. The updated first-principles paper explicitly flags this question [Paz 2026c, §O.3.4, final paragraph]: “Whether the correction is significant at intermediate epochs relevant for structure formation depends on the precise coefficients c₁, c₂ from the compactification and on whether the regime-dependent Ricci decomposition introduces terms with lower H-scaling than the full Gauss-Bonnet invariant.” Every other route to modifying the effective perturbation-spectrum mass has been checked and closed: coupling corrections at cosmological scales are 10⁻²⁰-suppressed [Paz 2026c, §VII.E.1]; threshold inputs to m_s,vac are locked (M_c is fixed by the 10D formula and measured via LIGO chirp masses, the 4π² is a topological invariant, H₀ is measured); no channel-dependent threshold splitting is possible because the curvature channel in FRW is still metric curvature and M_Pl remains the reduction scale.
This is the primary open target. Whether the mass-oscillation perturbation spectrum can accommodate Lyman-α constraints without modifying the framework rests on this single question. Resolution requires the explicit regime-dependent Ricci decomposition of I₄ in FRW and a check for any descendant scaling as H^n with n < 4.
The cross-disformal coefficient B̂ = (27/8) μ² ℛ / ((ζ/Λ) Y^{3/2} c) is derived in Schwarzschild geometry with ℛ the Kretschmann scalar [Paz 2026d, §3.2]. The cosmological perturbation analysis [Paz 2026c, §VII.E.1] establishes that the scalar kinetic and sound-speed functions in the FRW tracking regime are Q_s = 1 + O((H/m_s)²) and c_s² = 1 + O((H/m_s)²), with (H₀/m_s)² ∼ 10⁻²¹ at late times — so coupling corrections to the matter transfer function are negligible at all cosmologically relevant wavenumbers. This settles the question: cross-disformal is negligible cosmologically. The CMB’s agreement with linear perturbation theory at sub-percent precision [Planck 2018] is independent observational confirmation.
Boltzmann caveat. The matter-era suppression estimate is not the full story for Lyman-α specifically. A complete Boltzmann computation including radiation-era evolution would generally shift the fuzzy-scalar suppression onset to lower k than the matter-era estimate, making the Lyman-α tension in the mass-oscillation sector somewhat worse, not better. This reinforces rather than weakens the importance of the §6.3 open item.
The STF cosmological sector is a unified dark-sector architecture from a single scalar field. Two derived parameters — m_s from vacuum (BBH threshold) physics, ζ/Λ from 10D compactification — control three structurally independent mechanisms:
The Friedmann budget closes: Ω_b + Ω_DM,osc + Ω_Λ,T² = 0.05 + 0.27 + 0.68 ≈ 1.00. The three mechanisms are independently falsifiable, with distinct observational probes and controlling parameters. A failure in any single sector does not cascade.
Primary falsifiers: Euclid w(z) for the T² pump; Euclid Ω_m for T² self-consistency; JWST-era high-z rotation curves for cross-epoch a₀(z); Lyman-α tightening for the mass-oscillation mechanism. Primary open theoretical target: whether the regime-dependent I₄ decomposition in FRW contains descendant terms with H-scaling lower than H⁴, which is the one remaining route by which the cosmological perturbation spectrum could accommodate Lyman-α constraints without modifying the framework.
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.
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[Paz 2026e] Z. Paz, “STF Energy V0.4 — The Energy Problem,” (2026).
[Paz 2026f] Z. Paz, “STF Dark Energy Equation of State: w(z) Derivation,” V0.1 (2026).
[Paz 2026g] Z. Paz, “The Phantom Problem: Theoretical Pathologies in DESI DR2 Dark Energy Claims,” V0.2 (2026).
[Paz 2026h] Z. Paz, “Dark Matter as Geometry,” V2.0 (2026).
[Hu, Barkana & Gruzinov 2000] W. Hu, R. Barkana, A. Gruzinov, Fuzzy Cold Dark Matter, Phys. Rev. Lett. 85, 1158.
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End of draft.
@article{paz2026cosmologicalsector,
author = {Paz, Z.},
title = {The STF Cosmological Sector: Unified Dark Matter and Dark Energy from a Single Scalar Field},
year = {2026},
version = {V0.1},
url = {https://existshappens.com/papers/cosmological-sector/}
}