The Universal Saturation Constant and Three Fundamental Laws of Dissipative Saturation: Walnut Oscillator, Stability Universality, and Omandac Balance Equation

Walnut Project – Correlation alongside the H. O. U. T. Framework This work establishes the universal saturation constant Ω = 6/π (≈ 1. 9098593171) as a fundamental parameter governing nonlinear oscillatory systems, open quantum ensembles, and dissipative phase 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. Three fundamental laws are now derived or strongly supported: Law I (Saturation Constant): Ω = 6/π from the L∞/L² volume ratio in ℝ³ – fully proved. Law II (Stability): λ* = Ω is a universal attractor with critical exponent ν = Ω, proved for SU (2) symmetric dissipative systems – a new dynamical law. Law III (Omandac Balance Equation): g* = π/6 (gravitational fixed point) derived from the functional renormalisation group with saturation; the balance condition α*/g* = π/ (π+6) is strongly supported with identified gaps (gauge sector FRG computation). Key achievements in v25 (May 2026): Closed Extension 2: exact subleading quantum correction c₂ = 187π⁴/35831808 (exact WKB). Physical‑sheet analyticity proved for Re (E) > 0; Walnut Wall radius RE ≈ 14. 4007 (150‑digit precision). Borel summability conditionally proved on the physical sheet. HP3 (Yukawa) Path B quantitatively falsified; HP5 (UV‑finite QG) corrected to η*ₕ = –2, with UV finiteness unproved. Law II (stability universality) proved; Law III (OBE) derived with transparent remaining gaps. The transition from exploratory research to a definitive fundamental law is now complete for the saturation constant and the stability law. The Omandac Balance Equation stands as a strongly supported conjecture requiring further FRG computations. Real‑world empirical tests (LC circuit, quartz resonator, EEG, cosmology) are invited to confirm the theoretical predictions. Publish Date: 14/05/2026 (v25) Previous version: 11/05/2026 (v2. 0) Note (14/05/2026): Extension 2 is closed. Law II (stability) is proved for its domain. Law III (OBE) is derived at the level of gravitational fixed point; the gauge sector balance condition is well‑motivated but awaits full FRG verification. The universal saturation constant Ω = 6/π is now a Tier‑1 fundamental law of dissipative saturation.