Dark Energy as Parent-Proton Merger
Definition
ODTOE identifies dark energy with the process of parent-proton mergers at recursion level d=12 in the matryoshka structure, regulated by a scalar field χ(x,t). The cosmological fractions Ω_Λ : Ω_DM : Ω_b = φ² : 1 : Z = 68.86% : 26.30% : 4.83% follow from π and φ with zero adjustable parameters and match Planck 2018 to within 0.54σ.
Formula
Related Terms
φ-Resonance (Golden Ratio)
φ-resonance is the role of the golden ratio φ = 1.618… as the unique stable resonance frequency in ODTOE, selected from the potentiality field by the KAM theorem because φ is the «most irrational» number. φ is the fixed point of the self-referential map f(x) = 1 + 1/x and appears in fundamental constants, nested φ-tori and recursive structure of reality.
Toroidal Topology of Reality
In ODTOE reality has the topology of nested φ-tori whose major-to-minor radius ratio R/r = φ — the maximally KAM-stable configuration. Continuous phase dynamics (π-rotation) and discrete quantum transitions (φ-jumps) are projections of one quasiperiodic trajectory on these tori; the photon is the bridge quantum of the spiral gap (π−3)².
Black Hole as Deconfiguration Operator
A black hole in ODTOE is not an object that curves spacetime but the ultimate deconfiguration operator D̂ — the inverse of Ô — that returns configurations back into the potentiality field H. The event horizon is the boundary where configuration inertia I(C) → ∞; the information paradox is resolved naturally because information returns to H, and Hawking radiation is spontaneous re-actualization.
Source Articles
A Geometric Resolution of the Hubble Tension: Dark-Energy and Dark-Matter Unification via Parent-Proton Mergers in the ODTOE Matryoshka
Geometric mechanism simultaneously resolving cosmological constant problem and H₀ tension within single one-parameter framework. Postulate of Geometric Primacy (GP) fixes asymptotic dark-sector attractor φ²:1:Z as topological invariant. Hubble tension between Planck 2018 H₀=67.4 and SH0ES H₀=73.04 at ~5σ level. Dark energy identified with merger process of parent-protons at level d=12. Merger rate regulated by scalar field χ(x,t). Three claims: χ-regimes classify expansion histories; anisotropic Δχ reproduces H₀ tension; dark sector unified with Ω_DM and Ω_Λ as two aspects of 2%-residue (π−3)². Merger limit N^{local}_{max}=Ω_DM/(π−3)²≈13.12. Single fitting parameter η.
Cosmological Fractions from Toroidal Architecture: Deriving Dark Energy, Dark Matter and Baryonic Matter from π and φ
Within the toroidal ODTOE model, the cosmological fractions of dark energy, dark matter and baryonic matter are derived from two structural invariants: π and φ. The φ-torus possesses three topological sectors: inter-level (R², gravitational inertia), intra-level (r²=1), and gap sector (Z=(π−3)/[1−(π−3)φ]). Normalized fractions: ΩΛ:ΩDM:Ωb = φ²:1:Z = 68.86%:26.30%:4.83%. Planck 2018 comparison: dark energy 0.54σ, dark matter 0.32σ, baryonic 1.64σ. Zero adjustable parameters.
Eternal Expansion: Transcendence of π as Proof of the Inexhaustibility of Reality
The mechanism of Universe expansion is formalized within the toroidal ODTOE model. The Lindemann theorem (1882) on the transcendence of π proves that the trajectory on the φ-torus never closes, making expansion infinite and inexhaustible. Potentiality pressure F=(π−3)²·|H|/|C| acts at every observation cycle. A scale factor a(n)=(1+ε/(2πφ))ⁿ describes exponential growth of the effective φ-torus radius. Accelerated expansion (ä>0) follows from (π−3)⁴>0 without invoking Λ as a free parameter. Dark energy fraction ΩΛ=φ²/(φ²+1+Z)=68.86% matches Planck 2018 within 0.54σ.