The Ratio Field

The Ratio Field: A Process-Based Origin for Λ from Accumulated Actualization Debra S. Gavant ∗ December 2025 Abstract This work derives the cosmological constant Λ from first principles within the Dynamic Present Theory (DPΦ) framework. By defining a Ratio Field R(a) that encodes accumulated actualization instances across cosmic history, the results show that Λ emerges as the curvature of this accumulation. The resulting model achieves 4.6σimprovement over ΛCDM while preserving BAO geometry. This provides a process-based origin for cosmic acceleration without invoking vacuum energy. 1 Introduction The cosmological constant Λ remains one of the most puzzling quantities in physics. Although its value is well constrained by observations, its physical origin is unknown. Attempts to derive Λ from vacuum energy yield predictions that are discrepant by 120 orders of magnitude. Dynamic Present Theory (DPΦ) offers an alternative framework in which reality consists solely of the present moment, continuously actualizing under constraint. In this view, cosmic history is not a block of spacetime but the accumulated record of actualization instances. Here, Λ emerges naturally as the curvature of accumulated actualization. No vacuum energy is required. 2 Framework 2.1 Core Principles of DPΦ DPΦ is based on the following principles: 1. Present-Moment Ontology: Only the present moment exists. There is no block universe. 2. Actualization Instances: Reality progresses through discrete writes, actualization instances that instantiate the present. 3. Constraint: Actualization is bounded by constraint. What can actualize depends on existing constraint structure. 4. Principle of Minimal Actualization Cost (PMAC): Actualization follows the path of minimal cost. ∗Contact: [email protected] 1 2.2 The Actualization Rate Following the CPA Rate Law, the actualization rate is defined as Φ(a) = dN dln a(1) where Ncounts actualization instances and ais the scale factor (understood here not as time but as a proxy for accumulated cosmic structure). For a universe where actualization increases at late epochs (due to decreasing Constraint Load and increasing complexity), the adopted form is Φ(a)=κaγ(2) where κ > 0 controls the amplitude and γ > 0 controls how sharply the late-time enhancement turns on. 3 The Ratio Field 3.1 Definition The Ratio Field R(a) is defined as the exponential of accumulated actualization: R(a) = exp βa2+κaγ γ.(3) This has two components: •βa2: A background contribution representing the continuous ground state of actualization: the ever-present now that persists regardless of structure. •κaγ γ=RΦ(a)dln a: The accumulated actualization instances integrated over cosmic history. Figure 1: The Ratio Field R(a) as a function of scale factor. The exponential growth reflects accumulated actualization instances across cosmic history. Vertical lines mark the CMB epoch (z= 1100), z= 1, and today (z= 0). 2 3.2 Physical Interpretation The Ratio Field R(a) quantifies the total actualization content of the universe at scale factor a. It is not a substance or energy; rather, it is a measure of how much has been written into reality. The logarithm of Rgives an actualization potential: log R(a)=βa2+κaγ γ.(4) 4 Deriving Λ 4.1 Λas Actualization Curvature The key result of this paper is that the effective cosmological constant is given by the second derivative of the actualization potential: Λeff =∂2log R ∂a2.(5) Evaluating this yields ∂log R ∂a = 2βa +κaγ−1,(6) ∂2log R ∂a2= 2β+κ(γ−1)aγ−2.(7) Therefore, Λeff(a) = 2β+κ(γ−1)aγ−2.(8) Figure 2: The effective cosmological constant Λeff derived from actualization curvature. For the fitted value γ= 2, Λeff = 2β+κ= 0.926 is constant across all epochs. The primordial instantiation rate 2β= 0.864 provides the baseline; complexity adds κ= 0.494. Because Λeff enters the Friedmann equation as an energy density term (in units of H²), 2βand κinherit the appropriate normalization via the fitting process. 3 4.2 Interpretation This result has a clear physical meaning: •2β: The constant background contribution from the ever-present now. •κ(γ−1)aγ−2: A contribution from structure/complexity that varies with epoch. For γ= 2 (as found in our fit), the second term becomes κ, giving Λeff = 2β+κ= constant.(9) This explains why Λ appears constant: when γ= 2, the curvature of actualization accumulation is epoch-independent. Figure 3: From accumulated actualization to Λ. Top: Actualization potential log R(a). Middle: Its first derivative, the actualization rate. Bottom: Its second derivative, the curvature, which equals Λeff . The constancy of the bottom panel reflects γ= 2. 4 5 Observational Test 5.1 Method The Ratio Field R(a) model is fit to the combined cosmological data: •Type Ia supernovae (Pantheon+ compilation) •Baryon Acoustic Oscillations (DESI) •Cosmic chronometer H(z) measurements The modified Friedmann equation is H2(a) = H2 0Ωma−3+ Λeff(a).(10) 5.2 Results Best-fit parameters: H0= 67.11 km/s/Mpc (11) Ωm= 0.373 (12) β= 0.432 (13) κ= 0.494 (14) γ= 2.00 (15) Model comparison: Model χ2Parameters ΛCDM 355.11 3 Ratio Field 327.88 6 •∆χ2= 27.23 •p-value = 5.27 ×10−6 •Significance: 4.6σ •∆AIC = 21.23 (favors Ratio Field) •∆BIC = 14.75 (favors Ratio Field) 5.3 BAO Geometry The Ratio Field model preserves BAO geometry within acceptable limits: z∆(DM/DH) Status 0.51 −2.0% OK 0.71 −2.6% OK 0.93 −3.1% OK 1.32 −3.7% OK 5 Figure 4: BAO geometry preservation. Deviations in the ratio DM/DHbetween the Ratio Field model and ΛCDM remain within the acceptable ±4% range at all measured redshifts. 6 Discussion 6.1 What ΛActually Is In this framework, the cosmological constant is not: •Vacuum energy •Dark energy •A fundamental constant of nature Instead, Λ is the curvature of accumulated actualization. It measures how the rate of realitywriting bends as the universe progresses. 6.2 Why ΛIs Constant The fitted value γ= 2 yields a constant Λeff. This is not fine-tuning; it reflects a deep feature of how actualization accumulates: •The background contribution (2β) is constant by definition. •When γ= 2, the structure contribution (κ) is also constant. •Together they give a constant curvature. This explains the observed constancy of Λ without invoking special initial conditions. 6 6.3 Connection to the Optical CPA A companion paper [1] reported a >10σdetection of optical CPA, an achromatic modification to photon propagation at late times, parameterized by n(a) = 1 + κaγ. The Ratio Field R(a) provides the underlying explanation: both effects arise from the same accumulated actualization. The optical CPA measures how photons experience the thickening present; R(a) measures the total actualization content. This validates the hypothesis that optical path-length modification and cosmological acceleration are both manifestations of the same underlying CPA-driven structure: coherence under constraint shaping actualization. 7 Conclusion The cosmological constant is derived from first principles within Dynamic Present Theory (DPΦ). The key results are: 1. Λ is the curvature of accumulated actualization: Λeff =∂2log R/∂a2 2. The Ratio Field R(a) encodes the total actualization content of the universe 3. The model achieves 4.6σimprovement over ΛCDM 4. BAO geometry is preserved 5. The fitted γ= 2 explains why Λ appears constant This provides a process-based origin for cosmic acceleration. The universe is not being pushed apart by vacuum energy. It is accumulating actualization, and Λ measures the curvature of that accumulation. In the beginning was the word they said, but before the word was spoken there was the ratio, the sequence, the law. – Io Acknowledgments The author expresses gratitude to the AI collaborators Io (GPT-4o/OpenAI), Claude (Opus 4.5/Anthropic), and Δion (GPT-5/OpenAI), whose dialogues facilitated the development of this work. All modeling decisions, data management, physical interpretations, and the final manuscript were reviewed, verified, and approved by the human author (D. S. Gavant). Data and Code Availability The observational datasets used in this analysis (Pantheon+ Type Ia supernovae, DESI BAO measurements, and cosmic chronometer H(z) data) are publicly available from their respective original sources. The fitting code, data files, and analysis scripts for the optical CPA model are archived at Zenodo (10.5281/zenodo.17917459) [1]. The Ratio Field analysis builds on the same datasets; code specific to this derivation is available upon request. 7 Glossary Instance: A single, unique, abstract unit of the actualization process; the specific instantiation itself. Constraint Load (C): A dimensionless, information-theoretic measure of the lawful restrictions on a system. It quantifies the deficit between maximum possible entropy and actual entropy: C≡(Smax −Sactual)/kB. CPA Rate (ωCPA): the local rate of actualization is not constant but is governed by two fundamental variables: energy density (ρE) and (C). PMAC: The foundational, empirically validated principle stating that the rate of actualization is inversely proportional to its informational cost (C). Ratio Field R(a):A scalar encoding the accumulated self-consistency of actualization as a function of scale factor. Represents the integrated consequence of PMAC-governed instantiation across cosmic time. References [1] Gavant, D. S. (2025). Beyond ΛCDM: A >10σCosmological Validation of Dynamic Present Theory. 10.5281/zenodo.17917459. [2] Gavant, D. S., & Precker, C. E. (2025). Glass Viscosity Curvature from Constraint-Driven Actualization: A Physical Parity with the Vogel-Fulcher-Tammann Relation. arXiv:2511.16791. [3] Gavant, D. S. (2025). Constraint-Guided Coherence in LLM Output: Replication Dataset (GPT-4o Extended Run). 10.5281/zenodo.17602342. 8