From 4f6ca74c2bec7aad2f70b627a566038de4882f21 Mon Sep 17 00:00:00 2001 From: Brandon Schneider Date: Thu, 21 May 2026 00:47:07 -0500 Subject: [PATCH] =?UTF-8?q?feat(lean):=20add=20Burgers-PhiNUVMAP=20bridge?= =?UTF-8?q?=20=E2=80=94=2016D=20golden-ratio=20projection=20for=20viscous?= =?UTF-8?q?=20shock=20fields?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - `burgersStateToSpectralWindow`: extracts inner lattice points as 8-bin PIST window - `burgersStateToRegime`: classifies velocity profile via spectral discriminant - `burgersFieldToPhiNUVMAP`: projects BurgersState into 16D φ-NUVMAP space (dims 0-7: velocity samples, 8: ν, 9: t, 10: max|u|, 11: KE, 12: dissipation, 13: CFL, 14-15: reserved) - `burgersPhiDissipationStep`: golden contraction s' = c + φ⁻¹·(s-c) as viscous dissipation operator, using 3-point moving average as attractor center - Eval witnesses for smooth parabola and shock-step fixtures Generated with [Devin](https://cli.devin.ai/docs) Co-Authored-By: Devin <158243242+devin-ai-integration[bot]@users.noreply.github.com> --- .../Semantics/Semantics/PistSimulation.lean | 148 ++++++++++++++++++ 1 file changed, 148 insertions(+) diff --git a/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean b/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean index 92ccc287..870b485f 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean @@ -1137,4 +1137,152 @@ def phiNUVMAPChaosRun (initial anchor : Array Q16_16) (epsilon : Array Q16_16) /- Scale level witness: stringy tree at scale level 3 (zoomed in). -/ #eval! treeDIATToPhiNUVMAPState (treeToDIAT fixtureStringyTree) vec16Zero 3 PhiSpectralMode.transient +-- ════════════════════════════════════════════════════════════ +-- §9 Burgers-PhiNUVMAP Bridge +-- ════════════════════════════════════════════════════════════ +-- +-- The Burgers equation u_t + u·u_x = ν·u_xx exhibits two regimes: +-- • ν large → smooth, diffusive, oscillatory (NÉEL / Δ < 0) +-- • ν → 0 → shock formation, discontinuous (BLOCH / Δ ≥ 0) +-- +-- The spectral pipeline in PistSimulation.lean (8-bin window, quadratic +-- fit, discriminant gate) classifies Burgers solutions directly. +-- +-- PhiNUVMAP adds a 16D golden-ratio fractal parameter space where: +-- • Each BurgersState maps to a 16D vector +-- • Golden contraction s' = c + φ⁻¹·(s-c) models viscous dissipation +-- • Shock detection = regime classification on the spectral window + +-- ── 9a. Burgers state → spectral window ──────────────────── + +/-- Extract the inner N-2 lattice points of a Burgers velocity field + as a spectral window for PIST quadratic fitting. + Drops boundary points (assumed zero or fixed). -/ +def burgersStateToSpectralWindow (N : Nat) (u : Array Q16_16) : List Q16_16 := + if N <= 2 then [] + else + let inner := u.extract 1 (N - 1) + -- Pad or truncate to exactly 8 bins for the fixture + if inner.size >= 8 then (inner.extract 0 8).toList + else + let pad := 8 - inner.size + inner.toList ++ List.replicate pad Q16_16.zero + +/-- Classify a Burgers velocity profile via spectral discriminant. + Fits quadratic to the velocity field; Δ < 0 → smooth (NÉEL), + Δ ≥ 0 → shock-prone (BLOCH). -/ +def burgersStateToRegime (N : Nat) (u : Array Q16_16) : MagneticRegime := + let window := burgersStateToSpectralWindow N u + if window.length < 3 then MagneticRegime.uglyAsymmetricPruning + else spectralWindowToRegime window + +-- ── 9b. Burgers state → 16D φ-NUVMAP projection ─────────── + +/-- Project a Burgers velocity field into the 16D φ-NUVMAP space. + Dimensions: + 0-7 : velocity field samples (8 bins, spectral coefficients) + 8 : viscosity ν + 9 : time t + 10 : max |u| (shock strength proxy) + 11 : kinetic energy Σu²/2 + 12 : energy dissipation rate (heuristic) + 13 : CFL-like number = max|u|·dt/dx + 14-15: reserved (boundary condition flags) -/ +def burgersFieldToPhiNUVMAP (N : Nat) (u : Array Q16_16) (ν t dx dt : Q16_16) + : Array Q16_16 := + let window := burgersStateToSpectralWindow N u + let padded := window ++ List.replicate (8 - window.length) Q16_16.zero + let w8 := padded.take 8 + let maxU := u.foldl (λ acc ui => + let abs_ui := if Q16_16.lt ui Q16_16.zero then Q16_16.neg ui else ui + if Q16_16.gt abs_ui acc then abs_ui else acc) Q16_16.zero + let ke := Q16_16.div (u.foldl (λ acc ui => Q16_16.add acc (Q16_16.mul ui ui)) Q16_16.zero) (Q16_16.ofNat 2) + let diss := Q16_16.mul ν ke -- heuristic: dissipation ∝ ν·E + let cfl := if dx = Q16_16.zero then Q16_16.zero + else Q16_16.div (Q16_16.mul maxU dt) dx + -- Build 16D vector from components + let base := w8 ++ [ν, t, maxU, ke, diss, cfl, Q16_16.zero, Q16_16.zero] + -- Ensure exactly 16 elements + let base16 := if base.length >= 16 then List.take 16 base else + base ++ List.replicate (16 - base.length) Q16_16.zero + base16.toArray + +-- ── 9c. Golden contraction as viscous dissipation ────────── + +/-- Apply one golden-contraction dissipation step to a Burgers field. + Conceptually: u' = u_smooth + φ⁻¹·(u - u_smooth) + where u_smooth is a low-pass filtered version (the "center"). + For this witness, the center is a parabolic fit to the field. -/ +def burgersPhiDissipationStep (N : Nat) (u : Array Q16_16) (ν dx dt : Q16_16) + : Array Q16_16 := + let state16 := burgersFieldToPhiNUVMAP N u ν Q16_16.zero dx dt + -- The "center" is a smoothed version: for this witness, we use + -- a simple moving average (3-point stencil) as the attractor. + let smooth i := + if i > 0 ∧ i + 1 < u.size then + Q16_16.div (Q16_16.add (Q16_16.add u[i-1]! u[i]!) u[i+1]!) (Q16_16.ofNat 3) + else u[i]! + let center := Array.ofFn (n := N) (fun i : Fin N => smooth i.val) + let center16 := burgersFieldToPhiNUVMAP N center ν Q16_16.zero dx dx + phiContract state16 center16 + +-- ── 9d. Verification witnesses ─────────────────────────── + +/- Smooth velocity field: parabola u(x) = x·(4-x) on [0,4]. + Quadratic, symmetric, no shock. Should classify as BLOCH + (real discriminant, single smooth basin). -/ +def fixtureBurgersSmooth : Array Q16_16 := #[ + Q16_16.zero, -- u[0] = 0 + Q16_16.ofNat 3, -- u[1] = 3 + Q16_16.ofNat 4, -- u[2] = 4 + Q16_16.ofNat 3, -- u[3] = 3 + Q16_16.zero -- u[4] = 0 +] + +/- Shock-like velocity field: step function u = [0,0,2,2,0]. + Sharp discontinuity, high gradient. Should classify as NÉEL + (complex discriminant, oscillatory/underresolved). -/ +def fixtureBurgersShock : Array Q16_16 := #[ + Q16_16.zero, -- u[0] = 0 + Q16_16.zero, -- u[1] = 0 + Q16_16.ofNat 2, -- u[2] = 2 + Q16_16.ofNat 2, -- u[3] = 2 + Q16_16.zero -- u[4] = 0 +] + +/- Spectral window extraction from smooth field. -/ +#eval! burgersStateToSpectralWindow 5 fixtureBurgersSmooth + +/- Spectral window extraction from shock field. -/ +#eval! burgersStateToSpectralWindow 5 fixtureBurgersShock + +/- Regime classification: smooth parabola → bloch. -/ +#eval! burgersStateToRegime 5 fixtureBurgersSmooth + +/- Regime classification: shock step → neel. -/ +#eval! burgersStateToRegime 5 fixtureBurgersShock + +/- 16D φ-NUVMAP projection of smooth Burgers field. -/ +#eval! burgersFieldToPhiNUVMAP 5 fixtureBurgersSmooth + (Q16_16.ofRatio 1 10) Q16_16.zero (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100) + +/- 16D φ-NUVMAP projection of shock Burgers field. -/ +#eval! burgersFieldToPhiNUVMAP 5 fixtureBurgersShock + (Q16_16.ofRatio 1 10) Q16_16.zero (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100) + +/- Golden dissipation step on smooth field. -/ +#eval! burgersPhiDissipationStep 5 fixtureBurgersSmooth + (Q16_16.ofRatio 1 10) (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100) + +/- Golden dissipation step on shock field. -/ +#eval! burgersPhiDissipationStep 5 fixtureBurgersShock + (Q16_16.ofRatio 1 10) (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100) + +/- Compare 16D states: smooth vs shock. -/ +#eval! let smooth16 := burgersFieldToPhiNUVMAP 5 fixtureBurgersSmooth + (Q16_16.ofRatio 1 10) Q16_16.zero (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100); + let shock16 := burgersFieldToPhiNUVMAP 5 fixtureBurgersShock + (Q16_16.ofRatio 1 10) Q16_16.zero (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100); + phiContract smooth16 shock16 + end Semantics.PistSimulation