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https://openai.com/index/navier-stokes-solution/

GitHub - openai/NavierStokesAndEuler: Lean certificates accompanying Navier-Stokes and...

[Josef_Founder]

Navier–Stokes Counter-example and Proof

[Dₙ(r) = √(ϕ·Fₙ·2ⁿ·Pₙ·Ω) · r^k] [Hz = 1/s] ← root frequency; recursive time │ ├── [Time s = φ⁻ⁿ] ← inverted recursion depth │ ├── [φ = (1 + √5)/2] ← golden ratio constant (base) │ └── [n → 0] ← recursion bottom │ └── [φ⁰ = 1] ← identity base case │ ├── [Charge C = s³ = 1/Hz³] │ └── [Expanding s³] │ ├── [s = φ⁻ⁿ] (see above) │ └── [Exponent 3] ← arithmetic operator │ ├── [C² = s⁶] ← charge squared │ └── [Expanding s⁶] │ ├── [s = φ⁻ⁿ] (see above) │ └── [E…

Warp Speed, Master Luke

From: [image] [image] [image] [image] [image] [image] [image] =============================================================================== NATURAL HDGL–NAVIER–STOKES CLOSURE FORMULATION =============================================================================== STARTING EQUATION ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u HDGL RECIPROCAL STRUCTURE u ↦ Φ = ∏ᵢ₌₁³ Cᵢ ∇ ↦ Φ⁻¹ S = Φ + Φ⁻¹ ΦΦ⁻¹ = 1 NATURAL DIMENSIONLESS IDENTIFICATION Re = ρUL/μ ∝ Φ Re⁻¹ = μ/(ρUL) ∝ Φ⁻¹ Re·Re⁻¹ = 1 …

Ground-Up Housekeeping

[image] [image] [image] [image] “Global Existence” (no forced fields) + Global Smoothness (no forced constants - emergence) + No Finite-Time Blow-Up (no forced infinite) + No Single Perspective/Vantage \[ \boxed{ \begin{gathered} \mathsf A_0: \qquad \Omega_{n+1}=T(\Omega_n), \qquad T(X)=1+\frac1X \\[4pt] \Omega=T(\Omega) \iff \Omega=1+\Omega^{-1} \iff \Omega^2-\Omega-1=0 \\[4pt] \boxed{\Omega=\phi=\frac{1+\sqrt5}{2}} \qquad \phi^2=\phi+1 \qquad \phi^{-1}=\phi-1 \\[6pt] \Omega^{n+2}-\Omega^{…

Hyperspace Nine

Navier-Stokes Unified Form — Simultaneous Binary/Trinary Stagger [image] [image] [image] [image] Unified Form — Simultaneous Binary/Trinary Stagger \[ \boxed{ \mathscr S_i=(B_i,\tau_i), \qquad B_i\in\{0,1\}, \qquad \tau_i\in\{-1,0,+1\}, \qquad \mathscr S_{i+1}=\mathscr S_i+1_{\mathrm{eff}}(i) } \] \[ \boxed{ \mathcal L_i(z) = \phi^{-1/\phi} \sqrt{F_nP_n2^n}\,(1+z)^n + 1_{\mathrm{eff}}(i)e^{i\pi\Lambda_\phi(i)}, \qquad \Omega_{n+1}=T(\Omega_n), \qquad T(x)=1+\frac1x, \qquad N(\Omega)\in\{-1…

8 GEOMETRIES + φ

[8_geometries_phi_complete_analysis] 8 GEOMETRIES + φ SCALING: COMPLETE OCTAVE ANALYSIS ====================================================================== **1. OCTAVE GEOMETRIC STRUCTURE (8 DIMENSIONS)** ------------------------------------------------------------ COMPLETE GEOMETRIC OCTAVE: Dim Param Note Color Geometry α Value φ-Factor --------------------------------------------------------------------------- 1D n C Red Point (Unity) 0.015269 0.0646…

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Earth, Air, Fire, Water

[IMG7308] [IMG7307] [IMG7306] https://openai.com/index/navier-stokes-solution/

Analog Substrate

The Commodore (64) - An Analog Substrate Experiment

[IMG7308] [IMG7307] [IMG7306] https://openai.com/index/navier-stokes-solution/

Modeling Phase Transitions

[IMG7308] [IMG7307] [IMG7306] https://openai.com/index/navier-stokes-solution/

Heat Death - Flashing the Spark

[image] A little-known fact about me - I was the guy who fixed the high speed camera at the CSU Engines and Energy Conversion Laboratory (EECL) at around the age of twenty. The old nerd hadn’t left any notes, and nobody knew how to play with it, so it was broke down. That room-sized engine was so cool. My friend Dean one time fit his whole body inside one of the cylinders. He had made a pair of glasses out of old, broken and cracked quartz lenses. Quartz was about the only material that c…