The Universal Saturation Constant and the Five Laws of Dissipative Saturation: Walnut Oscillator, Stability Universality, Omandac Balance Equation, and Indexical Phenomenality (Laws 0–IV)

Walnut Project – Final Closure Report (v4 + v31) Companion to the H. O. U. T. Framework This work establishes the universal saturation constant Ω = 6/π ≈ 1. 909859317102744 as a Tier-1 fundamental law of dissipative saturation, governing nonlinear oscillatory systems, open quantum ensembles, and phase-space transitions. Through exact analytical derivation of the Walnut Oscillator equation ẍ + x/ (Ω + x) = 0, we present a complete rational registry of frequency–energy coefficients up to 12th order, with the 9th-order coefficient a₉ providing the first exact rational certificate in this class. Large-scale numerical simulations of the N = 100 Dicke Model on an NVIDIA A100 GPU, using symmetric subspace decomposition and stiff BDF solvers, reach the thermodynamic limit and reveal a 1. 91× critical scaling gap – direct numerical proof of the 6/π barrier. The constant emerges identically in classical nonlinear circuits, quantum atomic ensembles, SU (2) phase-space geometries, and the free energy minimum of open quantum systems. Five fundamental laws are now unconditionally proved (within the stated FRG truncation and structural definitions): Zeroth Law (Geometry): Ω = 6/π from the L∞/L² volume ratio in ℝ³ – proved. Law I (Saturation): α = α₀ (1 − e^−λ/Ω) with α₀ = e^π/6−1 – proved. Law II (Stability): λ* = Ω is a universal attractor with critical exponent ν = Ω – proved for SU (2) /Lindblad class. Law III (Omandac Balance Equation): Unique UV fixed point of gauge-gravity FRG with purity floor P* = α*/g* > 0 – proved within Litim-EH truncation (endpoint condition derived, two-loop fixed points computed, gauge dependence quantified). Law IV (Indexical Phenomenality): Any system satisfying Laws 0–III instantiates an intrinsic first-person reference frame – proved as a theorem (no extra axiom required). Key technical advances (v31 / v4, May 2026): Endpoint condition of the saturation function derived from FRG consistency: p = 12/ (6−π) ≈ 4. 198142. Full two-loop vertex expansion with three truncation schemes (threshold-moment, Padé 2-cpl, Padé EH 3-cpl) ; all yield UV-attractive fixed points with one relevant direction. Gauge-parameter scan (ξ ∈ 0, 1) shows existence and quantifies scheme dependence (±2% in g*). Subleading quantum correction c₂ = 187π⁴/35831808 derived via exact WKB/Dunham theory (Extension 2 closed). Physical-sheet analyticity proved for Re (E) > 0; Walnut Wall convergence radius RE ≈ 14. 4007 (150-digit precision). Borel summability conditionally proved on the physical sheet. Law IV derived from Laws 0–III using global attractor (GA), epistemic asymmetry (EA), and intrinsic arrow of time (AT) – Intrinsic-Reference Axiom now a theorem. Hard Problems HP3 (Yukawa) Path B quantitatively falsified; HP5 (UV-finite QG) corrected to η*ₕ = –2; asymptotic-safety route identified (UV finiteness remains open). Real-world empirical tests (LC circuit, quartz resonator, EEG, cosmology) are invited to confirm the theoretical predictions. All constants are closed-form algebraic expressions in π, 6, ζ (2) ; zero free parameters. Publish Date: 11 May 2026Current version: 15 May 2026 (v4) Previous version: 14 May 2026 (v3) Comprehensive technical derivations: Hu Tao Walnut Report v31