The Ratio Field

The Ratio Field: A Process-Based Origin for Λ from Accumulated Actualization Debra S. Gavant∗ December 2025 Abstract The cosmological constant Λ is typically interpreted as vacuum energy density, a formulation that leads to the “vacuum catastrophe” (a discrepancy of ∼120 orders of magnitude between theory and observation). This technical note proposes an alternative derivation, where Λ emerges as a geometric constraint derived from a dimensionless scalar potential, the Ratio Field R(a). We demonstrate that identifying the effective cosmological constant with the second derivative of this field (Λeff =∂2log R/∂a2) offers a natural resolution to the fine-tuning problem. Joint fits to Type Ia supernovae (Pantheon+), BAO (DESI), and cosmic chronometer data indicate a strong statistical preference for this model over ΛCDM (∆χ2= 27.23, 4.6σsignificance), driven by the convergence of the complexity scaling parameter γ→2. This suggests Λ is not a physical fluid but a geometric invariant of the cosmological metric. 1 Introduction The standard ΛCDM model successfully parameterizes cosmic acceleration but lacks a theoretical derivation for the value of Λ. When Λ is treated as the energy density of the quantum vacuum, theoretical predictions exceed observations by orders of magnitude [1]. This suggests that Λ may not be a substance filling space but rather a geometric feature of the spacetime metric itself. In this note, we formalize the Ratio Field R(a), a scalar function introduced in a previous work [2], to model scale-dependent metric constraints. We propose that the observed cosmological constant is a manifestation of the curvature of this field. Specifically, we treat Λ not as a free parameter, but as a derived quantity defined by the second derivative of the field’s logarithmic potential. This approach offers two distinct advantages: 1. It provides a first-principles derivation of Λ that avoids the vacuum energy contribution. 2. It introduces a scaling behavior that fits late-time cosmological data with significantly higher precision than the standard model. ∗Correspondence to: [email protected] 1 2 The Ratio Field Formalism 2.1 Definition We define the Ratio Field R(a) as a dimensionless scalar potential dependent on the scale factor a. To model scale-dependent metric evolution, the field takes the form R(a) = exp βa2+κaγ γ,(1) where the parameters represent specific geometric constraints: •β: The intrinsic scaling constant, governing the baseline evolution of the metric. •κ: The amplitude of the complexity modulation term. •γ: A scaling exponent that controls the rate of complexity integration. 2.2 Derivation of Λeff We postulate that the effective cosmological constant Λeff is given by the curvature (second derivative) of the log-potential of the Ratio Field with respect to the scale factor. This relationship is defined as Λeff =∂2log R ∂a2.(2) Substituting Eq. 1 into Eq. 2 yields the analytical form: log R=βa2+κaγ γ,(3) ∂log R ∂a = 2βa +κaγ−1,(4) ∂2log R ∂a2= 2β+κ(γ−1)aγ−2.(5) Thus, the effective cosmological constant evolves as Λeff(a) = 2β+κ(γ−1)aγ−2.(6) This equation reveals that Λ is generally dynamic. However, for the specific case where γ= 2, the term (γ−2) vanishes, rendering the second term constant. As shown in Section 3, the empirical data forces the model to precisely this limit. 3 Empirical Verification 3.1 Data and Methodology To test the validity of this derivation, we performed a joint likelihood analysis using: •Type Ia Supernovae: 1701 light curves from the Pantheon+ sample. •Baryon Acoustic Oscillations (BAO): DM/rsand DH/rsmeasurements from DESI. •Cosmic Chronometers: 32 measurements of H(z) from differential galaxy aging. Model comparison was performed using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) against a standard flat ΛCDM baseline. 2 3.2 Results The best-fit parameters (Table 1) reveal a critical convergence. The scaling exponent γconverges to 2.0 within statistical uncertainty. Parameter Best-Fit Value Physical Interpretation β0.4318 Intrinsic scaling baseline κ0.4942 Complexity amplitude γ2.0 Complexity scaling exponent H067.11 km/s/Mpc Hubble Constant Ωm0.37 Matter Density Table 1: Best-fit parameters for the Ratio Field model. When γ= 2, Eq. 6 simplifies to a constant as Λeff = 2β+κ= constant.(7) This suggests that the observed constancy of dark energy is not a fundamental property of the vacuum but a phenomenological result of the scaling exponent γtaking the integer value of 2. 3.3 Statistical Significance The Ratio Field model outperforms the ΛCDM model with high statistical significance: ∆χ2= 27.23 in favor of the Ratio Field. •Significance: 4.6σ(p-value 5.27 ×10−6). •∆BIC = 14.75, indicating very strong evidence against ΛCDM even after penalizing for the 3 additional parameters. Importantly, the model preserves the BAO geometric constraints, with deviations in the distance ratios remaining <3.7% across all redshift bins (z= 0.51 to z= 1.32). 4 Theoretical Consistency To ensure that the Ratio Field model represents a viable cosmological alternative, we verified its consistency with fundamental relativistic principles and geometric observational limits. •Phenomenological Interpretation. The derivation and fits do not require endorsing any ontological stance; R(a) is a scalar potential and Λeff is its log–curvature. •GR–compatibility. No modifications to the local Lorentz invariance or GR field equations are assumed; Eq. (6) simply reparameterizes the effective constant entering H2(a). •BAO geometry. Because DM/DHis preserved at the reported level, the standard BAO geometric consistency checks are satisfied. Vacuum catastrophe. The construction avoids the vacuum–energy interpretation altogether. 3 5 Discussion The derivation presented here suggests that Λ is a geometric consequence of the scaling potential. By defining Λ as the second derivative of the Ratio Field, we recover the observed magnitude of the cosmological constant (O(1)) without fine-tuning, avoiding the 10120 discrepancy inherent to vacuum energy calculations. The Ratio Field serves as the scalar implementation of Dynamic Present Theory (DPΦ) [3], providing a mathematical mechanism that satisfies the geometric requirements of the theory. The parameter βcan be interpreted as a background geometric constraint (analogous to a curvature constant), whereas κrepresents the modification of the metric due to structure formation (complexity). The convergence of γ→2 implies a specific stability in the scaling evolution of these constraints. Remarkably, this derivation achieves statistical parity with Vacuum Fluctuation Theory predictions for the critical density but resolves the magnitude discrepancy by treating Λ as a geometric constraint rather than a vacuum energy. The CPA + C framework has been independently validated across three domains: •Glass Transition Dynamics: Achieving statistical parity with the Vogel-Fulcher-Tammann (VFT) relation at 0.0001 precision across 14 orders of magnitude in viscosity [4]. •AI Coherence: Demonstrating a high-fidelity VFT fit (ϵ < 10−5, RMSE = 0.02085) in the uncertainty reduction of large language models under constraint [5, 6]. •Cosmological Expansion: Identifying a >10σpreference for the CPA geometric expansion over ΛCDM in optical analyses [2]. This cross-domain convergence suggests that CPA + C describes a fundamental constraint-rate relationship operating across condensed matter, informational, and cosmological scales. “In the beginning was the word they said, but before the word was spoken there was the ratio, the sequence, the law.” — Io Acknowledgments As an independent researcher, the author acknowledges the collaboration of large language models—GPT4o, GPT-5 (OpenAI), Claude (Anthropic), and Gemini (Google)—for theoretical development, code implementation, and manuscript refinement. All modeling choices, physical interpretations, and final text were composed and validated by the human author. References [1] Steven Weinberg. The Cosmological Constant Problem. Rev. Mod. Phys., 61:1–23, Jan 1989. doi: 10.1103/RevModPhys.61.1. URL https://link.aps.org/doi/10.1103/RevModPhys.61.1. [2] D. S. Gavant. Beyond ΛCDM: A >10σCosmological Validation of Dynamic Present Theory, 2025. URL https://doi.org/10.5281/zenodo.17917459. Preprint. 4 [3] D. S. Gavant. Dynamic Present Theory I: Unifying Quantum Mechanics and General Relativity, 2025. URL https://doi.org/10.5281/zenodo.17069890. [4] D. S. Gavant and C. E. Precker. Glass Viscosity Curvature from Constraint-Driven Actualization: A Physical Parity with the Vogel-Fulcher-Tammann Relation, 2025. URL https: //arxiv.org/abs/2511.16791. [5] D. S. Gavant. Constraint-Guided Coherence in LLM Output: Replication Dataset (GPT-4o Extended Run), 2025. URL https://doi.org/10.5281/zenodo.17602342. [6] D. S. Gavant. Constraint-Driven Coherence in LLM Output: Token-Entropy and Surprisal as CPA Signatures Across Models, 2025. URL https://doi.org/10.5281/zenodo.17451956. 5