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 (v32) 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 = eπ/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 (v32 / v5, 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 R_E ≈ 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: 16 May 2026Current version Hu Tao Walnut Report: v32Previous version: v31 (15 May 2026)