ODTOE Articles

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.

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.

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.

Closing the B-zero boundary topology marker of Article C. Topological structure of boundary ∂_B C of configuration space C at B→0. Criterion of finite-affine-parameter termination of Φ-iteration sequence (Theorem E.T2). Formal definition of trapped ODTOE-configuration via causal cone J⁺_O (Definition E.D1). Full ODTOE singularity theorem E.T1 as structural analog of Hawking–Penrose theorem. Five anti-circular proof steps: ODTOE Raychaudhuri inequality (E.L1), focusing along null directions (E.L2), finite-parameter focusing (E.L3), Φ-iteration behavior near ∂_B C (E.L4), Φ-iteration incompleteness as vanishing of causal future J⁺_O.

HD 140283 star age paradox. Different dating methods reflect different observation operators applied to unified field of potential states.

Reality as infinite recursion of self-observation cycle Φ=ι∘Ô. Each dimensionality level d contains 17 structural roles, two-level observer window spans 39 configurations. Life as recursion invariant inevitably emerging at every level d from −∞ to +∞. Time as strange-loop iteration flowing differently at each level. Five levels of inter-level navigation through field H. Threshold permeability P(d→d+1|S) with width δS=(π−3)². Observer trichotomy (B,A,H)=(spirit,soul,body). Cosmological proportions ΩΛ:ΩDM:Ωb=φ²:1:Z match Planck 2018 within 1.2σ. All formulas verified to 50+ significant digits.