What does this research mean for the field?

The Generative Triple Evolution (GTE) framework derives fundamental cosmological observables—including the dark energy fraction, CMB spectral index, and baryon-to-photon ratio—from zero-parameter first principles, closely matching empirical measurements while predicting the exact absence of primordial gravitational waves. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

The study aims to derive cosmological observables from the GTE/Phi_MDL framework, emphasizing structural routes for understanding dark energy.

June 18, 2026Open Access

Cosmological Predictions of the GTE/PhiMDL Framework: Dark Energy, the CMB Spectral Tilt, and Gravitational Signatures from First Principles

Key Points

The study aims to derive cosmological observables from the GTE/Phi_MDL framework, emphasizing structural routes for understanding dark energy.
Derived cosmological observables using algebraic steps certified in Lean 4.
Utilized two independent methods for calculating dark energy fractions: PSC and holographic routes.
Analyzed energy density and other parameters based on model predictions and observational agreements.
Dark energy fractions were derived as PSC = 0.6899 and holographic = 0.6732, bracketing Planck 2018 values.
Predicted CMB spectral index n_s = 0.96488 within ±0.005 and tensor-to-scalar ratio r = 0, indicating no primordial gravitational waves.
Energy density matched observations at a few-percent level with H_0 = 67.95 km/s/Mpc.

Abstract

We derive the principal cosmological observables of the Generative Triple Evolution (GTE) / framework from zero-parameter first principles, with central algebraic steps certified in Lean 4. The dark-energy fraction is fixed by two independent structural routes: PSC route: = 0. 6899 (+0. 18 from Planck 2018;) ; holographic route: = 3/14 = 0. 6732 (-2. 8) ; together these bracket Planck 2018. We prove that this bracketing is structural, not coincidental: the holographic value is the price of the MDL-minimal self-description carrier (a lower bound) and the PSC value is the capacity ceiling of the same adjudication ledger (an upper bound), with the inter-route ordering reducing to the machine-certified GTE-atom inequality 14 (2000/3) > 9² (). The exact realized fraction is a left-computably-enumerable real of Turing degree 0' - definite but not computable - so no computable selection mechanism for exists, and the two routes are its computable endpoints; the census endpoint becomes a 3-decidable test at precision ^* 3. 410^-4. The bracket is non-empty only for Ngen 3 (measurement-free) ; with the PSC Layer-I bound Ngen 3 this determines Ngen = 3 without Presentation Invariance (, conditional on named structural premises). The associated energy density _ = (9/112) \, ² H₀² agrees with observation at the few-percent level; a PSC boundary condition fixes w = -1. The vacuum is anomaly-free (machine-certified, zero sorry), and holographic mode counting (3L rather than L³) suppresses one-loop vacuum corrections by 10^-42. A new defect-cosmology sector is derived: the order, restored after reheating, re-orders at TG 0. 70 GeV and forms a transient Kibble domain-wall foam; the canonical coupling ² (D_) ² (= 7/9, ) lifts the vacuum-label degeneracy and collapses the foam within \!10^-23 s - 700 before BBN - selecting the vacuum k = 0 and leaving zero surviving defect network (GWh² 310^-47 at f 0. 02 Hz; Nₑff 0), a parameter-free falsifiable prediction. The baryon-to-photon ratio is derived via the kink-top coupling (FKTT): etaB = 6. 10910^-10 (+0. 15 from Planck 2018/BBN (6. 100. 06) 10^-10;, zero sorry, zero axioms). Further predictions from the same integer inputs include: the CMB spectral index nₛ = 1 - ln 2/ (2π²) = 0. 96488 (-0. 005σ, CatAL, fourteen Lean theorems) ; the tensor-to-scalar ratio r = 0 exactly (no primordial gravitational waves; primary LiteBIRD falsifiable prediction) ; the neutrino mass sum m_ = 59. 4\, meV; the CKM CP phase deltaCP = 68. 51^ (0. 017\% from PDG) ; and the Hubble constant H₀ = 67. 95\, km/s/Mpc (; TCMB = 2. 7255\, K is the sole external input). An explicit falsifiability profile is provided.

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