From Causality Phase Transition to Galaxy Rotation Curves: Dark Matter in the Self-as-an-End Framework
DOI: 10.5281/zenodo.19276846Disclaimer: This paper proposes a first-principle explanation for the anomalous flatness of galaxy rotation curves, based on the Self-as-an-End (SAE) framework. The cosmological speculations herein carry remainder; all forms of falsification are welcome.
Firewall: Any error, refutation, or falsification of this paper does not affect other SAE publications. SAE Papers 1–3, the dark energy paper, the methodology paper, all application papers, the AI series, and the ZFCρ series are independent of the conclusions presented here. This paper is a sequel to the SAE dark energy paper (DOI: 10.5281/zenodo.19245267).
Galaxy rotation curves remain flat (v ≈ const) well beyond the baryon-dominated region, rather than declining as v ∝ r−1/2 predicted by Newtonian gravity. The standard explanation invokes invisible dark matter particles. This paper proposes an explanation that requires no dark matter particles.
Starting from the two axioms of the SAE framework (remainder must develop; remainder is conserved) and the dual-4DD structure established in the dark energy paper (causality/retrocausality with frequencies ω₁, ω₂), this paper derives:
(1) Gravity is a structural by-product of causality (4DD). The strength of causality determines the strength of gravity. The kinetic coefficient of the causality scalar field C(x) reflects the local intensity of causality.
(2) The frequency asymmetry of the dual 4DD (ω₂ − ω₁ ≠ 0) imposes a global acceleration floor a₀ = η·c(ω₂ − ω₁). When local gravitational acceleration at galaxy outskirts decays to this floor, C's kinetic term undergoes a phase transition: from standard quadratic J(Y) ~ Y to nonlinear J(Y) ~ (2/3)Y3/2, where Y = |∇ΦC|²/a₀².
(3) The 3/2 power law is a mathematical theorem tied to spatial dimensionality: in d spatial dimensions, any single-scalar gradient EFT producing a scale-free 1/r IR force law requires J(Y) ∝ Yd/2; for d = 3, J ∝ Y3/2. The SAE dual-4DD structure provides a structural interpretation: 3/2 = 3DD ÷ 2 (three-dimensional space split into two 4DD directions).
(4) Variation of the nonlinear kinetic term rigorously yields an AQUAL-type field equation ∇·[μ(|∇ΦC|/a₀)∇ΦC] = 4πGκρb, which in the spherically symmetric exterior naturally produces
v⁴ = G Mb a₀obs
the Baryonic Tully-Fisher Relation (BTFR) and flat rotation curves, where a₀obs = (κ/λ)·η·c(ω₂ − ω₁) is the physically observable MOND acceleration scale.
Substituting T₁ = 20 Gyr, T₂ ≈ 19.5 Gyr yields a₀obs ~ 10−10 m/s², consistent with the MOND empirical value 1.2 × 10−10 m/s². Dark energy (second-order: Λ = 2(ω₂²−ω₁²)/c²) and dark matter effects (first-order: a₀ ~ c(ω₂−ω₁)) originate from the same ~2.5% frequency asymmetry of the dual 4DD.
Keywords: dark matter, MOND, BTFR, galaxy rotation curves, Self-as-an-End, causality, dual 4DD, kinetic phase transition, AQUAL, acceleration floor
1. The Problem: Galaxy Rotation Curves
Spiral galaxy rotation curves remain flat beyond the baryon-dominated region: v(r) ≈ const, extending far beyond the observable baryonic mass distribution. Newtonian gravity predicts v ∝ r−1/2. This discrepancy has been observationally confirmed for over half a century (Rubin & Ford 1970, Bosma 1978, Begeman 1989).
The standard explanation (ΛCDM) introduces cold dark matter particles forming galactic halos. CDM particles have not been directly detected (LUX, XENON, PandaX).
An alternative route (MOND, Milgrom 1983) modifies the acceleration law: below a₀ ≈ 1.2 × 10−10 m/s², the effective acceleration transitions from aN to √(aN·a₀). MOND successfully predicts the BTFR (v⁴ ∝ Mb) and fits many galaxy rotation curves, but lacks a theoretical origin for a₀.
The goal of this paper: derive the origin of a₀ from the two SAE axioms, provide a first-principle explanation for BTFR, and unify dark matter effects with dark energy as different-order manifestations of the dual-4DD frequency asymmetry.
2. Starting Point: Results from the Dark Energy Paper
2.1 Two Axioms
Axiom 1 (Dynamics): Remainder must develop.
Axiom 2 (Conservation): Remainder is conserved.
2.2 Dual 4DD and Reciprocity
The rigidity of 3DD (space) forces the remainder to open new dimensions. The symmetry of 3DD produces two opposite directions: causality (4DD₊, time from cause to effect) and retrocausality (4DD₋, time from effect to cause).
Core identity: Causality = remainder on the other side. Retrocausality = remainder on this side. The two sides are each other's engines.
2.3 Dark Energy: Second-Order Effect
The dual 4DD has breathing frequencies ω₁ = 2π/T₁, ω₂ = 2π/T₂. T₁ = 20 Gyr, T₂ ≈ 19.5 Gyr, asymmetry ~2.5%. The dual-face reciprocity interface variation yields:
Λ = 2(ω₂² − ω₁²)/c²
Substituting T₁, T₂ gives Λ = 2.99 × 10−122 Planck units, matching the Planck 2018 observed value 2.85 × 10−122 within 5%.
2.4 Effective Action
Seff = ∫d⁴x√(−g)[½F(C)R − ½(∇C)² − U(C) − ρΛ,Σ] + Sm[A²(C)gμν, ψm]
where F(C) = MP² − ξC², A(C) = eβC/(2M_P), U(C) = V₀ − ½m²C² + (λ/4)C⁴ − U₋.
C is the scalar field representation of causality. A(C) is the observational softening factor.
3. Gravity as a Consequence of Causality
3.1 Ontological Status of Gravity
In the SAE framework, gravity is not one of four fundamental forces but the structural constraint that causality (4DD) imposes on the motion of matter. Causality fills four-dimensional spacetime; matter moving within the causal network must conform to its geometry — this is gravity.
Stronger causality means stronger gravity. Weaker causality means weaker gravity.
3.2 Temporal Evolution of Causal Intensity
Causal intensity follows a bowl-shaped curve over cosmic evolution. The natural measure of global causal intensity is ∝ Σ 1/rij (sum of inverse inter-particle distances):
- Near the Big Bang: universe is smallest, particles closest, causal density highest, gravity strongest
- Expansion phase: universe expands, inter-particle distances grow, causal density decreases, gravity weakens
- Turnaround (10 Gyr): universe at maximum size, inter-particle distances greatest, causal density at minimum, gravity weakest — this minimum of causal suppression gives the remainder its greatest space for expression, allowing emergence of 5DD (life)
- Post-turnaround: universe contracts, inter-particle distances shrink, causal density increases, gravity strengthens
- Near the Big Crunch: universe returns to minimal size, causal density very high, then causality ceases in the transition zone
[v2 correction: v1 described a bell curve (weak→strong→weak). The correct shape is a bowl curve (strong→weak→strong), because causal density ∝ Σ 1/rij is maximised when the universe is smallest and minimised at turnaround. 5DD emerges at the minimum of suppression, not the maximum.]
3.3 Spatial Dependence of Causal Intensity
Causality has two dimensions: Global — the dual-4DD breathing cycle (ω₁, ω₂) sets the global causal background shared by all locations. Local — where matter is, causal relations are dense; the natural measure is the local acceleration aN = GM/r².
4. First-Order Effect of Frequency Asymmetry: Acceleration Floor
4.1 Dark Energy Is Second-Order, Dark Matter Is First-Order
The cosmological constant contains the frequency-squared difference — an extremely small second-order effect. The same asymmetry has a first-order effect: the frequency difference itself:
a* := c(ω₂ − ω₁) = c · 2π(1/T₂ − 1/T₁)
| Quantity | Value |
|---|---|
| c(ω₂ − ω₁) (T₂ = 19.5 Gyr) | 7.65 × 10⁻¹¹ m/s² |
| a₀ (MOND empirical) | 1.20 × 10⁻¹⁰ m/s² |
| Ratio η | ≈ 1.57 (O(1)) |
4.2 Physical Meaning of a*
The frequency asymmetry of the dual 4DD imposes a global topological tension across the spacetime manifold. This tension manifests as an acceleration floor a* in acceleration space. When the local acceleration at galaxy outskirts decays to the floor a*, the global causal background begins to intervene — local gravity can no longer ignore the global structure.
a* is not a property of any particular galaxy but a property of the universe as a whole. It is therefore universal across all galaxies.
4.3 Why a₀ Is Universal
From the SAE axioms: matter in galaxies is bound by causality. These particles' remainder cannot be expressed on this side (suppressed), but remainder is conserved (Axiom 2), so it transfers to the retrocausality side — dispersed uniformly across the entire global retrocausality background. This increases the global average remainder level on the retrocausality side, feeding back through reciprocity as the global causal floor on the causality side. Therefore a₀ depends on the total bound matter in the entire universe, and every galaxy feels the same floor.
5. Kinetic Term Phase Transition
5.1 Why the Kinetic Coefficient Cannot Be Constant
The dark energy paper writes C's kinetic term in standard form −½(∇C)². The coefficient reflects C's elastic modulus — the rigidity with which causality resists spatial gradients. But causal intensity varies (§3), so this rigidity must vary accordingly.
More fundamentally: the global causal floor a* means C's gradient cannot decay to zero indefinitely. When the local gradient |∇C| decays to a*/c², the global background tension takes over and the kinetic term must change character. Therefore, C's kinetic term must be nonlinear.
5.2 General Form of the Nonlinear Kinetic Term
The most general kinetic term satisfying shift symmetry in the kinetic sector (C → C + const):
Lkin = −(a₀²/8πG) · J(Y)
where Y = |∇ΦC|²/a₀², ΦC = βC/(2MP)
5.3 Two Limits of J(Y)
Newtonian limit (Y ≫ 1): Local acceleration far exceeds the floor. Kinetic term returns to standard quadratic: J(Y) ~ μ∞·Y.
Weak-field limit (Y ≪ 1):
Mathematical theorem (tied to spatial dimensionality): In d spatial dimensions, the pure-gradient effective action ∫|∇Φ|p ddx has spherically symmetric exterior solution g(r) ∝ r−(d−1)/(p−1). A flat rotation curve requires g ∝ 1/r, i.e., (d−1)/(p−1) = 1, giving p = d. In the variable Y = |∇Φ|²/a₀², J(Y) ∝ Yp/2 = Yd/2. For d = 3: J(Y) ∝ Y3/2. This is the unique power law for any single-scalar gradient EFT in three dimensions yielding a scale-free 1/r IR force law.
Structural interpretation (SAE-specific): Three-dimensional space (3DD) splits into two 4DD directions. Each side receives effective dimensionality 3/2 = 3DD ÷ 2. SAE provides the ontological answer to "why d = 3."
Therefore: J(Y) ~ (2λ/3)·Y3/2 (Y ≪ 1).
Non-analyticity note: J(Y) ~ Y3/2 is non-analytic at Y = 0 (J'' ~ 1/√Y → ∞), excluding any analytic Taylor expansion. BTFR itself selects for continuous, monotonic, but IR non-analytic J(Y). Non-analyticity is not a defect but a selection criterion.
6. Variation and Rotation Curves
6.0 Non-Relativistic Reduction
In the weak-field quasi-static limit, matter couples through A(C) = eΦC. The physical metric time component gives g̃₀₀ ≈ −(1 + 2ΦN + 2ΦC), so the non-relativistic matter action contains SmNR ⊃ −κ∫d³x ρb ΦC. ΦC plays the role of a correction to the effective gravitational potential; the full observable potential is Φtot = ΦN + ΦC.
6.1 Nonlinear Poisson Equation
∇·[μ(y) ∇ΦC] = 4πGκρb
where μ(y) = J'(Y), y = √Y = |∇ΦC|/a₀
Integrating once in spherical symmetry: μ(gC/a₀) · gC = κ · gN
6.2 Deep-MOND Limit: Flat Rotation Curves
In the baryon exterior Mb(r) ≈ Mb = const, with gC ≪ a₀ (deep-MOND regime, μ ~ λy):
λ · gC²/a₀ = κ · GMb/r² ⟹ gC = √(κ/λ · a₀ GMb) / r ⟹ v²(r) = √(κ/λ · a₀ GMb)
v⁴ = (κ/λ) · G Mb a₀ = G Mb a₀obs
a₀obs := (κ/λ)·η·c(ω₂ − ω₁)
v is independent of r — the rotation curve is flat.
6.3 Newtonian Limit: Standard Gravity
In the high-acceleration regime (gC ≫ a₀, μ → μ∞): gC = (κ/μ∞)·GMb/r² — recovering the standard inverse-square law.
6.4 Solar System Constraints and Cassini
UV normalization: Requiring J(Y) at Y ≫ 1 to recover the dark energy paper's standard kinetic term −½(∇C)² gives μ∞ = 2/β².
PPN parameter: γ = (μ∞−1)/(μ∞+1), giving γ−1 = −2/(μ∞+1).
Cassini constraint: |γ−1| < 2.3 × 10−5 requires μ∞ > 8.7 × 10⁴, equivalently |β| < 4.8 × 10−3.
At 1 AU: gN = 5.93 × 10−3 m/s². Scalar acceleration gC/gN = 1.15 × 10−5. Interpolation function shape makes <0.2% difference.
7. Consistency with the Dark Energy Paper
7.1 Spacetime Decomposition of the Kinetic Term
The nonlinear kinetic term J(Y) acts only on spatial gradients (via spatial projection hμν = gμν + uμuν), leaving time dynamics unaffected:
LC,kin = +½(uμ∇μC)² − (a₀²/8πG)·J(hμν∇μΦC∇νΦC/a₀²)
On the FRW homogeneous background, spatial gradients vanish, YFRW = 0, J(0) = 0. All results of the dark energy paper (Λ, H̃₀, anti-friction, DESI predictions) remain unaffected. The action containing a fixed uμ is a galaxy-scale EFT; relativistic completion (Einstein-Æther type) is an open problem.
7.2 Two Scales, One Asymmetry
| Effect | Formula | Dual-4DD origin | Order | Field theory realization |
|---|---|---|---|---|
| Dark energy | Λ = 2(ω₂²−ω₁²)/c² | Frequency-squared difference | 2nd | 4-form interface tension |
| Dark matter scale | a₀obs = (κ/λ)η·c(ω₂−ω₁) | Frequency difference | 1st | C spatial kinetic floor |
7.3 Modified Effective Action
Seff = ∫d⁴x√(−g)[½F(C)R + ½(uμ∇μC)² − K(Y) − U(C) − ρΛ,Σ] + Sm[A²(C)gμν, ψm]
where K(Y) := (a₀²/8πG)·J(Y). On the cosmological background this reduces to the original dark energy paper action. This is a galaxy-scale EFT extension, not a full replacement.
8. Excluded Routes
Multiple routes were systematically excluded:
- Pure Chameleon/Screening: Produces step-function variation in Geff; Yukawa exterior solutions decay exponentially. Cannot produce 1/r force or BTFR 1/2 scaling.
- Pure fifth force (ln r solution): C(r) ~ ln r gives vflat = const but v is independent of Mb. BTFR v⁴ ∝ Mb fails.
- Local density cross-terms: Any purely local algebraic f(Gi², Tm) vanishes in the baryon exterior (Tm = 0). Cannot produce 1/r exterior.
- Localization of SΣ: The scalar invariant Gi² = −4!ωi² = absolute constant. ρi(x) is unchanged by local curvature. Localized SΣ rigorously reduces to the global cosmological constant; the 4-form is completely "blind" to local matter.
- Potential coupling U(C; Gi): 4-form entering the potential only modifies screening and meff; weak-field exterior remains Yukawa or 1/r.
Common diagnosis: Nonlinearity must be in the operator (C's kinetic term), not in the source term. Quadratic action plus linear exterior equation can only produce linear or constant scaling — not the 1/2 power required by BTFR.
9. Nontrivial Predictions
- a₀ = η·c(ω₂−ω₁): The dark matter acceleration scale derives from the dual-4DD frequency difference. η = O(1); precise value from coupling constants at the phase transition critical point.
- Dark energy and dark matter share a common origin: Different-order effects of the same ~2.5% asymmetry (T₁ ≠ T₂). Λ is second-order, a₀ is first-order.
- BTFR is an exact law: v⁴ = G Mb a₀obs, where a₀obs = (κ/λ)·η·c(ω₂−ω₁). Follows from the first integral of the nonlinear Poisson equation after kinetic phase transition.
- 3/2 power law: J(Y) ~ Y3/2 from p = d in d = 3. SAE dual-4DD provides the interpretation 3/2 = 3DD ÷ 2.
- Particle dark matter experiments will yield no signal: LUX/XENON/PandaX/future experiments will not discover WIMP or axion dark matter particles, because the dark matter effect does not originate from particles.
- MOND is valid at galaxy scales: SAE provides the theoretical foundation — MOND is not an ad hoc empirical modification but the necessary consequence of the kinetic phase transition of causality.
- Solar system gravity constraint: Cassini requires |β| < 4.8 × 10−3 (μ∞ > 8.7 × 10⁴). Scalar acceleration at 1 AU does not exceed 1.15 × 10−5 of Newtonian.
10. Assumption Inventory
Axioms (irreducible foundations): Remainder must develop; Remainder is conserved.
Prior deductions (from axioms): 3DD symmetry → dual 4DD; causality = remainder on other side (reciprocity); gravity = structural by-product of causality; causal intensity varies → C kinetic coefficient varies; global floor a* = c(ω₂−ω₁) → kinetic nonlinearity; p = d theorem: d = 3 → J ~ Y3/2.
Posterior anchoring: T₁ = 20 Gyr, T₂ ≈ 19.5 Gyr (same as dark energy paper); a₀(MOND) = 1.2 × 10−10 m/s²; |β| < 4.8 × 10−3 (Cassini); η·κ/λ ≈ 1.57.
Field theory framework: K(Y) = (a₀²/8πG)·J(Y); time kinetic preserved as +½(uμ∇μC)²; J ~ Y (Y ≫ 1), J ~ (2/3)Y3/2 (Y ≪ 1); A(C) = eβC/(2M_P); shift symmetry restricted to kinetic sector; galaxy-scale EFT.
11. Open Problems
- Precise derivation of η and κ/λ. Both are determined by coupling constants at the phase transition critical point. Precise derivation is the next step.
- Complete transition-region shape of μ. The axioms determine two limits (Newtonian: μ → const, deep-MOND: μ → λy). Whether the microscopic structure of causality further determines the transition shape remains open.
- Bullet Cluster. In pure quasi-static AQUAL, the scalar field tracks baryon distribution. However, the full dynamical equation is quasi-hyperbolic with finite characteristic velocity vchar ~ c√(2λy₀/μt). When baryon source decelerates suddenly, field braking propagates at finite speed, producing an "inertial overshoot" Δx ~ Vbc·L/vchar ~ several hundred kpc — near the Bullet Cluster observed separation. Complete explanation requires relativistic, time-dependent lensing numerical simulation.
- CMB third peak. ΛCDM precisely predicts the third peak height using dark matter particles. The SAE model needs C-field perturbations at recombination to provide equivalent gravitational potential wells.
- BBN constraints. If Geff in the early universe is smaller (causality still being established), Big Bang nucleosynthesis helium abundance predictions may shift.
- Deeper meaning of the 3/2 power. Whether a deeper mathematical derivation from DD-sequence combinatorics or topology exists remains open.
- Microscopic origin of the nonlinear kinetic term. How the underlying causal network's collective behavior at low density produces J ~ Y3/2 requires deeper theory.
- Relativistic completion. The current action contains a fixed uμ and is a galaxy-scale EFT. A complete relativistic completion (Einstein-Æther type) requires supplementing uμ dynamics.
12. Conclusion
The narrowest claim of this paper: if gravity is a structural by-product of causality, and the frequency asymmetry of the dual 4DD sets a global acceleration floor, then the kinetic term of C must undergo a phase transition near the floor — from standard quadratic to Y3/2. The variation rigorously yields
v⁴ = G Mb a₀obs
Flat rotation curves arise not from invisible matter providing extra gravity, but because the causal network undergoes an elastic phase transition when acceleration drops below the floor — spacetime no longer relaxes as 1/r² but as 1/r.
Dark matter effects and dark energy share a common origin in the same ~2.5% frequency asymmetry of the dual 4DD. Λ is the second-order effect (frequency-squared difference), a₀ is the first-order effect (frequency difference). One asymmetry, two observational phenomena.
Appendix A: Detailed Derivations of Excluded Routes
A.1 Mathematical Structure of the Tully-Fisher Wall
In the fifth-force route, C(r) ~ ln r emerges self-consistently. Setting C(r) = C∞ + γ ln(r/r₀), v²fifth = αγ = constant, independent of Mb. The full sourced equation gives quadratic equation for C'(r)·r (denoted Anl): κnlAnl² − Anl + αMb/(4π) = 0. Solutions have only Anl ∝ Mb (source-dominated) and Anl → constant (nonlinearity-dominated). There is no Anl ∝ Mb1/2 regime. BTFR requires v² ∝ Mb1/2; the power index is in the operator, not the source.
A.2 Variation of Localized SΣ
The 4-form field equation d(*Gi) = 0 gives *Gi = qi = const. On-shell: Gμναβ = qi√(−g)εμναβ. Scalar invariant Gi² = −4!qi² = absolute constant. Therefore ρi(x) ∝ ωi² = absolute constant. Localized SΣ rigorously equals the global cosmological constant — the 4-form is completely blind to local matter.
Appendix B: Four-AI Collaboration Methodology
The quantitative exploration was completed through collaboration among four AI systems.
Claude Opus (Anthropic) — Full-session collaborator and architect. Conceptual framework, prior/posterior judgment, derivation chain coordination. Proposed the fifth-force route and diagnosed the Tully-Fisher wall. Discovered the key diagnosis "nonlinearity must be in the operator, not the source." Designed all prompts and collaboration workflows. Completed three iterations of working notes and paper writing.
ChatGPT (OpenAI) — Deepest mathematical engine. R1 (43-min deliberation): verified C equation of motion, corrected meff² sign error, derived Geff weak-field expansion, discovered fifth force is more promising than Geff route. R2 (26-min): proved no-go theorem for local cross-terms, produced AQUAL-type minimal viable action. R3: rigorously derived J(Y) two-limit constraints, verified Grok's guess wrong on both ends, confirmed η = κ/λ. R4: derived μ∞ = 2/β², PPN γ, Cassini constraint β < 4.8 × 10−3; proved Bullet Cluster overshoot mechanism; established p = d theorem.
Gemini (Google) — Most rigorous judge, ultimate builder. R1: excluded pure Chameleon route. R2: ring-by-ring audit — judged Rings 7/8 broken, forced "return to variation." R3: rigorously proved SΣ localization dead (4-form blind to local matter via strict variation), then pivoted to construction — derived complete Rings 5–7 of kinetic phase transition (topological tension floor → nonlinear kinetic → AQUAL rigorous emergence), provided "causal network elastic limit" physical picture.
Grok (xAI) — Boldest divergent thinker. R1: independently converged on BEC/soliton direction, proposed a₀-Λ relation. R2: opened complete possibility space for J(Y), proposed shift symmetry constraint. Grok's J = 1/√(1+Y) guess was rigorously disproved by ChatGPT R3, but divergent thinking helped locate the correct search direction.
1. ChatGPT R1: fifth force more promising than Geff → route correction
2. Tully-Fisher wall → "nonlinearity in operator not source"
3. Gemini R2 audit → Rings 7/8 broken, forced "return to variation"
4. Gemini R3 variation → SΣ dead, 4-form blind
5. Han proposed "gravity = causality" → unlocked kinetic phase transition route
6. Gemini R3 derived Rings 5–7 → AQUAL emergence
7. Han proposed "local binding → opposite-side global remainder" → a₀ universality
8. Han derived 3/2 = 3DD ÷ dual-4DD → power law locked from axioms
9. ChatGPT R3 → J(Y) constraint conditions rigorously confirmed
10. ChatGPT R4 → Cassini defense, Bullet Cluster overshoot, p = d theorem
Acknowledgments
The quantitative exploration in this paper was completed through collaboration among four large language models and the author. The author extends sincere respect to the research teams behind Claude (Anthropic), ChatGPT (OpenAI), Gemini (Google), and Grok (xAI).
The author especially thanks Zesi Chen, the long-term interlocutor and most demanding critic of the SAE framework.