The Universal Critical Ratio: I ≈ 0.071 as a Scale-Invariant Phase Transition Threshold*

The Universal Critical Ratio: I ≈ 0.071 as a Scale-Invariant Phase Transition Threshold* [Abstract] This paper proposes the Universal Critical Ratio (UCR) of ≈ 0.071 as a fundamental, self-consistent fixed point of energy partition. This ratio emerges necessarily when multiplicative physical processes operate in closed systems. It defines the universal boundary between structure formation and energy dissipation across various scales, from nuclear binding energies to cosmological expansion. [Core Theoretical Framework] 1. Closed System Identity: UCR + l_pt = 1 (Where l_pt is the dissipation ratio ≈ 0.929) 2. Fixed-Point Condition: UCR = rho* * e = 0.026 * 2.718 ≈ 0.0707 (at sigma = 1) 3. 3D Geometric Relation: UCR = 1 - cos(theta_c), where theta_c ≈ 21.66° is the critical angle for structural coherence in 3D space. [Empirical Evidence & Isomorphism] • EMWC Cosmology: I(rho*) = 0.0706 (Theoretical derivation) • 3D BCC Bond Percolation: pc = 0.0742 (Monte Carlo verification) • Molecular Bonds: Average dissipation rate = 0.0716 (NIST data) • Nuclear Physics: Strong correlation (R^2 = 0.700) between Fe-56 instability and UCR deviations. • Information Systems: MaxVar L5 phase transition threshold = 0.071 [Significance] The UCR provides a unified mathematical foundation for scale-invariant phase transitions. It suggests that the observed deviations in cosmological constants (Hubble Tension) are manifestations of this universal partition ratio. This work serves as the prerequisite for the forthcoming EMWC (Environmental Modified gravity via kinetic Coefficient) framework. [Note] Version: Draft v0.4 Author: Sonkichul (Son Kichul), Independent Researcher AI Research Assistance: Gemini, Claude (Calculation and numerical verification) Contact: shionup@gmail.com This work serves as the theoretical prerequisite for the forthcoming EMWC (Environmental Modified gravity via kinetic Coefficient) framework (Sonkichul, 2026). --- v1.0 (March 2026) - Added formal proof structure: Definition 1, Assumptions A1–A3, Lemma 1–2, Theorem 1 (§4d) - LSM simulation confirms θ_c ≈ 21.79° (Theorem 1 prediction), A3 within ±5% of 1/14 target (OP-1 partially resolved) - Alternative lower bound θ_c ≈ 15.45° isolated as OP-5 - β correction factor physical basis derived from Z=4 (OP-6) --- v1.1 (March 2026) - Abstract upgraded: I* = 1/14 established as confirmed universal geometric constant (not hypothesis) - Central thesis declared: information, energy, and pressure share identical critical structure — all expressions of TO lattice geometry - 7th domain added: TO-LSM simulation confirms θ_c ≈ 21.79° (Theorem 1 prediction) - Figure 1 inserted: empirical convergence chart across 7 domains - §1 retitled: "From Hypothesis to Confirmed Universal Constant" Keywords updated: information-energy equivalence added --- v1.2 (March 2026) - §6 added: Quantum Geometric Simulation Results - OP-11: Observation as directional pressure — Measurement Problem physical mechanism proposed - OP-12: Double-slit interference from TO lattice geometry confirmed by simulation - OP-13: Photon energy coupling — S-curve decoherence confirmed (radio 0% → gamma 99.7%) - Theorem 2: Universal 3D transition coverage min(max P_eff) = 0.08073 > I* — all directions - Theorem 3: Critical photon energy per direction E_crit(v) = E_ref × I* / (P_base × max|v·n|) - Figures 2-5: Simulation charts inserted OP-11~15 registered --- v1.3 (March 2026) - §7 added: Generalized Critical Threshold Equation - Core equation: I*(η) = ΔΔG / [G_total(0) - η] I* = 1/14 is the η=0 special case (ideal medium) Medium noise η raises the effective threshold - OP-14 confirmed: Neural (synaptic) domain = 8th domain ΔΔG = 1.447 kJ/mol, G_total(0) = 20.0 kJ/mol (Na,K-ATPase transport free energy, literature confirmed) η_brain = kT + η_EEG(Delta) + η_metabolic = 5.97 kJ/mol I*(η=0) = 0.071, I_bio = 0.100 - Macro-micro unification declared: Cosmology / Quantum / Information / Biology all share TO-lattice as common spatial substrate - UCR elevated from geometric constant to universal phase transition equation --- v1.4(March 2026) - Added OP-2 GW strain verification h_obs=I*×Rs/r, OP-16 density-dependent η_eff, OP-17 rotation curve formula