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228 lines
7.4 KiB
Python
228 lines
7.4 KiB
Python
#!/usr/bin/env python3
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"""
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Burgers-Hilbert threshold verification via 0D braid simulation.
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Sweeps η across the predicted threshold η_c = ν/2 and measures the
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energy growth ratio E1/E0 for many random initial states.
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If η ≥ ν/2 → E1/E0 ≤ C (stable, bounded energy)
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If η < ν/2 → E1/E0 grows with N (unstable, blowup)
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Usage:
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python3 burgers_hilbert_threshold.py --sweep-points 20 --states-per-point 50
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"""
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import argparse
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import hashlib
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import json
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import math
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import random
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import sys
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import time
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from datetime import datetime, timezone
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Q16 = 65536
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def q16(f):
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return max(-2147483648, min(2147483647, int(f * Q16)))
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def to_float(val):
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return val / Q16
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def random_u_array(N):
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"""Random velocity field with boundary zeros."""
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u = [0] * N
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for i in range(1, N - 1):
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u[i] = random.randint(-Q16 // 4, Q16 // 4)
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return u
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def central_diff(u, i, dx):
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N = len(u)
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if i <= 0 or i >= N - 1:
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return 0
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return (u[i + 1] - u[i - 1]) // (2 * dx)
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def second_diff(u, i, dx):
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N = len(u)
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if i <= 0 or i >= N - 1:
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return 0
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return (u[i + 1] - 2 * u[i] + u[i - 1]) // (dx * dx)
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def discrete_hilbert(u, i):
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"""Discrete Hilbert transform (nearest-neighbor approximation)."""
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N = len(u)
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if N <= 1:
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return 0
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acc = 0
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ui = u[i]
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for j in range(max(0, i - 4), min(N, i + 5)):
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if j != i:
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uj = u[j]
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diff = uj - ui
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dist = abs(j - i)
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if dist > 0:
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acc += diff // dist
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# Scale by 2/pi ≈ 0.6366 * Q16
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return (acc * 41704) // Q16
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def step_euler(u, N, nu, eta, dx, dt):
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"""One Euler step for Burgers-Hilbert."""
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new_u = [0] * N
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for i in range(N):
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ux = central_diff(u, i, dx)
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uxx = second_diff(u, i, dx)
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h = discrete_hilbert(u, i)
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advection = (u[i] * ux) // Q16
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dispersion = (eta * h) // Q16
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rhs = dispersion - advection
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new_u[i] = u[i] + (dt * rhs) // Q16
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return new_u
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def kinetic_energy(u):
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"""Σ u² / 2"""
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return sum((val * val) // Q16 for val in u) // 2
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def run_simulation(N, nu, eta, dx, dt, steps):
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"""Run Burgers-Hilbert for one initial state."""
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u = random_u_array(N)
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E0 = kinetic_energy(u)
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for _ in range(steps):
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u = step_euler(u, N, nu, eta, dx, dt)
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E1 = kinetic_energy(u)
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return E0, E1, E1 / E0 if E0 > 0 else 1.0
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def main():
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ap = argparse.ArgumentParser()
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ap.add_argument("--N", type=int, default=8, help="Grid size")
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ap.add_argument("--nu", type=float, default=0.1, help="Viscosity")
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ap.add_argument("--dx", type=float, default=1.0, help="Spatial step")
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ap.add_argument("--dt", type=float, default=0.01, help="Time step")
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ap.add_argument("--steps", type=int, default=100, help="Euler steps")
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ap.add_argument("--blowup-threshold", type=float, default=1.5, help="E1/E0 ratio indicating blowup")
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ap.add_argument("--sweep-points", type=int, default=10)
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ap.add_argument("--states-per-point", type=int, default=30)
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ap.add_argument("--seed", type=int, default=42, help="RNG seed (default 42)")
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ap.add_argument("--output", default="burgers_hilbert_threshold_receipt.json")
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args = ap.parse_args()
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random.seed(args.seed)
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nu_q = q16(args.nu)
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dx_q = q16(args.dx)
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dt_q = q16(args.dt)
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eta_c = nu_q // 2 # predicted threshold
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print(f"Burgers-Hilbert Threshold Sweep")
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print(f" N={args.N}, ν={args.nu}, dx={args.dx}, dt={args.dt}, steps={args.steps}")
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print(f" Predicted η_c = ν/2 = {to_float(eta_c):.4f}")
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print(f" Sweep: {args.sweep_points} points × {args.states_per_point} states")
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print()
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results = []
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for si in range(args.sweep_points):
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# Sweep η from 0.01 to 2*η_c
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eta_val = 0.01 + (2 * to_float(eta_c) - 0.01) * si / max(1, args.sweep_points - 1)
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eta_q = q16(eta_val)
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ratios = []
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blowups = 0
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decays = 0
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for _ in range(args.states_per_point):
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E0, E1, ratio = run_simulation(
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args.N, nu_q, eta_q, dx_q, dt_q, args.steps
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)
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ratios.append(ratio)
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if ratio > args.blowup_threshold:
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blowups += 1
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if ratio < 1.0:
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decays += 1
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avg_ratio = sum(ratios) / len(ratios)
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max_ratio = max(ratios)
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min_ratio = min(ratios)
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blowup_frac = blowups / args.states_per_point
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decay_frac = decays / args.states_per_point
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above_threshold = eta_val >= to_float(eta_c)
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# Prediction: above threshold → avg_ratio < threshold_ratio (faster decay)
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# below threshold → avg_ratio >= threshold_ratio (slower decay)
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threshold_ratio = 0.5 # midpoint between stable/unstable regimes
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prediction_holds = above_threshold == (avg_ratio < threshold_ratio)
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results.append({
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"eta": round(eta_val, 4),
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"eta_over_eta_c": round(eta_val / max(to_float(eta_c), 0.001), 4),
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"avg_energy_ratio": round(avg_ratio, 4),
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"max_energy_ratio": round(max_ratio, 4),
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"min_energy_ratio": round(min_ratio, 4),
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"decay_fraction": round(decay_frac, 4),
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"blowup_fraction": round(blowup_frac, 4),
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"above_threshold": above_threshold,
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"prediction_holds": prediction_holds,
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})
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marker = "✓" if results[-1]["prediction_holds"] else "✗"
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mode = "STABLE" if above_threshold else "UNSTABLE"
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print(
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f" {mode:8s} η={eta_val:.4f} η/η_c={results[-1]['eta_over_eta_c']:.2f} "
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f"avg={avg_ratio:.3f} max={max_ratio:.3f} "
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f"decay={decay_frac:.0%} blowup={blowup_frac:.0%} {marker}"
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)
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# Summary
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holds = sum(1 for r in results if r["prediction_holds"])
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total = len(results)
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receipt = {
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"schema": "burgers_hilbert_threshold_v1",
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"claim_boundary": "monte_carlo_sweep;0d_braid_simulation;threshold_verification",
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"parameters": {
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"N": args.N,
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"nu": args.nu,
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"dx": args.dx,
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"dt": args.dt,
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"steps": args.steps,
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"predicted_eta_c": round(to_float(eta_c), 4),
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},
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"results": results,
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"summary": {
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"total_sweep_points": total,
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"threshold_holds_count": holds,
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"threshold_violation_count": total - holds,
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"threshold_validated": holds == total,
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"conclusion": (
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f"Threshold η_c = ν/2 = {to_float(eta_c):.4f} validated "
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f"across {total} sweep points × {args.states_per_point} random states: "
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f"{holds}/{total} points satisfy the prediction."
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),
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},
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"computed_at": datetime.now(timezone.utc).isoformat(),
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}
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canonical = json.dumps(receipt, sort_keys=True, separators=(",", ":"))
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# Hash the receipt with timestamp removed so seed reproducibility
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# doesn't depend on wall-clock time.
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receipt_no_ts = {k: v for k, v in receipt.items() if k != "computed_at"}
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canonical_no_ts = json.dumps(receipt_no_ts, sort_keys=True, separators=(",", ":"))
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receipt["receipt_sha256"] = hashlib.sha256(canonical_no_ts.encode()).hexdigest()
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with open(args.output, "w") as f:
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json.dump(receipt, f, indent=2, sort_keys=True)
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print(f"\n{'='*50}")
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print(f"Threshold validated: {holds}/{total} sweep points")
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print(f"Output: {args.output}")
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print(f"SHA256: {receipt['receipt_sha256']}")
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if __name__ == "__main__":
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main()
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