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384 lines
11 KiB
Python
384 lines
11 KiB
Python
#!/usr/bin/env python3
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"""
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gpl_interaction_law_demo.py
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Demonstrates the GPL rotational coupling and local interaction law.
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Shows: Frame compatibility → Weight → Force → Evolution → Convergence
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"""
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import math
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import numpy as np
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from dataclasses import dataclass
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from typing import List, Tuple
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import matplotlib.pyplot as plt
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@dataclass
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class Frame:
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"""Rotational frame of a μ-seed."""
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theta: int # Azimuthal: 0-15 (22.5° steps)
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phi: int # Polar: 0-7
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psi: int # Torsion: 0-7
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chi: int # Chirality: 0=D, 1=L
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a: float # Activation: 0-15
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x: float # Position X
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y: float # Position Y
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def effective_theta(self) -> float:
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"""Effective angle in radians, accounting for chirality."""
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base = self.theta * (2 * math.pi / 16)
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return base if self.chi == 0 else -base
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def compute_weight(f_i: Frame, f_j: Frame, sigma: float = 2.0) -> float:
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"""
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Compute interaction weight w_ij.
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w_ij = cos(Δθ) * cos(Δφ) * (1 - 2|Δχ|) * exp(-|Δp|²/2σ²)
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"""
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# Rotational alignment
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delta_theta = (f_j.theta - f_i.theta) % 16
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cos_theta = math.cos(delta_theta * 2 * math.pi / 16)
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delta_phi = (f_j.phi - f_i.phi) % 8
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cos_phi = math.cos(delta_phi * math.pi / 8)
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# Chirality (0 if different, 1 if same)
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chiral_factor = 1 - 2 * abs(f_j.chi - f_i.chi)
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# Spatial proximity
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dx = f_j.x - f_i.x
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dy = f_j.y - f_i.y
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dist_sq = dx*dx + dy*dy
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proximity = math.exp(-dist_sq / (2 * sigma * sigma))
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return cos_theta * cos_phi * chiral_factor * proximity
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def compute_force(f_i: Frame, f_j: Frame, w_ij: float) -> Tuple[float, float]:
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"""
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Compute force F_ij = w_ij * (a_j - a_i) * direction.
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Returns: (F_x, F_y)
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"""
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dx = f_j.x - f_i.x
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dy = f_j.y - f_i.y
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dist = math.sqrt(dx*dx + dy*dy)
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if dist < 0.001:
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return (0.0, 0.0)
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# Direction unit vector
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ux, uy = dx/dist, dy/dist
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# Activation gradient
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da = f_j.a - f_i.a
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# Force magnitude
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F_mag = w_ij * da
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return (F_mag * ux, F_mag * uy)
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def evolve_frame(f: Frame, F_x: float, F_y: float, alpha: float = 0.1) -> Frame:
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"""
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Update frame based on force.
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Simple Euler integration.
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"""
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new_a = f.a + alpha * math.sqrt(F_x*F_x + F_y*F_y)
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new_a = max(0.0, min(15.0, new_a)) # Clip to bounds
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return Frame(
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theta=f.theta,
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phi=f.phi,
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psi=f.psi,
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chi=f.chi,
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a=new_a,
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x=f.x,
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y=f.y
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)
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class GPLSimulation:
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"""Simulate GPL interaction dynamics."""
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def __init__(self, frames: List[Frame]):
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self.frames = frames
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self.history = [self.get_state()]
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def get_state(self) -> List[Tuple[float, float, float]]:
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"""Get current state (x, y, a)."""
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return [(f.x, f.y, f.a) for f in self.frames]
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def step(self):
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"""One evolution step."""
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n = len(self.frames)
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forces = [(0.0, 0.0) for _ in range(n)]
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# Compute all pairwise forces
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for i in range(n):
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for j in range(n):
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if i == j:
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continue
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w = compute_weight(self.frames[i], self.frames[j])
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F = compute_force(self.frames[i], self.frames[j], w)
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forces[i] = (forces[i][0] + F[0], forces[i][1] + F[1])
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# Update all frames
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new_frames = []
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for i, f in enumerate(self.frames):
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new_f = evolve_frame(f, forces[i][0], forces[i][1])
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new_frames.append(new_f)
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self.frames = new_frames
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self.history.append(self.get_state())
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def run(self, steps: int = 100):
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"""Run simulation for multiple steps."""
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for _ in range(steps):
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self.step()
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# Check convergence
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if self.check_convergence():
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break
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return self.frames
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def check_convergence(self, threshold: float = 0.01) -> bool:
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"""Check if converged (activation changes small)."""
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if len(self.history) < 2:
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return False
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prev = self.history[-2]
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curr = self.history[-1]
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max_change = max(abs(c[2] - p[2]) for c, p in zip(curr, prev))
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return max_change < threshold
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def demo_two_node_interaction():
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"""Demonstrate basic weight and force between two nodes."""
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print("=" * 70)
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print("TWO-NODE INTERACTION LAW DEMONSTRATION")
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print("=" * 70)
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test_cases = [
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("Aligned (Δθ=0)", 0, 0, 0, 0),
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("Orthogonal (Δθ=4)", 0, 4, 0, 0),
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("Opposite (Δθ=8)", 0, 8, 0, 0),
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("45° offset (Δθ=2)", 0, 2, 0, 0),
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("Chiral mismatch", 0, 0, 0, 1),
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]
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print(f"\n{'Scenario':<25} | {'θ₁':>3} | {'θ₂':>3} | {'χ₁':>3} | {'χ₂':>3} | {'Weight':>8} | {'Interpretation'}")
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print("-" * 95)
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for name, t1, t2, c1, c2 in test_cases:
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f1 = Frame(theta=t1, phi=0, psi=0, chi=c1, a=5.0, x=0.0, y=0.0)
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f2 = Frame(theta=t2, phi=0, psi=0, chi=c2, a=8.0, x=1.0, y=0.0)
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w = compute_weight(f1, f2)
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F = compute_force(f1, f2, w)
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interp = ""
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if abs(w - 1.0) < 0.1:
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interp = "Strong attraction"
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elif abs(w) < 0.1:
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interp = "No coupling"
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elif w < -0.5:
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interp = "Repulsion"
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elif c1 != c2:
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interp = "Orthogonal channels"
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else:
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interp = f"Partial ({w:.2f})"
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print(f"{name:<25} | {t1:>3} | {t2:>3} | {c1:>3} | {c2:>3} | {w:>8.3f} | {interp}")
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def demo_convergence():
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"""Demonstrate convergence to attractor."""
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print("\n" + "=" * 70)
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print("CONVERGENCE DEMONSTRATION")
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print("=" * 70)
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# Create a line of 5 nodes with varying initial activation
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frames = []
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for i in range(5):
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f = Frame(
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theta=0, # All aligned
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phi=0,
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psi=0,
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chi=0, # All D-form
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a=float([10, 2, 8, 3, 12][i]), # Varying activation
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x=float(i),
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y=0.0
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)
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frames.append(f)
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print("\nInitial state (aligned, varying activation):")
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print(f"{'Node':>6} | {'x':>6} | {'θ':>4} | {'a':>8} | {'Type'}")
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print("-" * 45)
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for i, f in enumerate(frames):
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t = "High" if f.a > 8 else "Low" if f.a < 4 else "Med"
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print(f"{i:>6} | {f.x:>6.1f} | {f.theta:>4} | {f.a:>8.2f} | {t}")
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# Run simulation
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sim = GPLSimulation(frames)
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final = sim.run(steps=50)
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print(f"\nFinal state (after {len(sim.history)-1} steps):")
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print(f"{'Node':>6} | {'x':>6} | {'θ':>4} | {'a':>8} | {'Change'}")
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print("-" * 50)
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for i, f in enumerate(final):
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init_a = [10, 2, 8, 3, 12][i]
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change = f"{f.a - init_a:+.2f}"
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print(f"{i:>6} | {f.x:>6.1f} | {f.theta:>4} | {f.a:>8.2f} | {change}")
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avg_a = sum(f.a for f in final) / len(final)
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print(f"\nAverage activation: {avg_a:.2f}")
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print("Converged to smooth, shared activation (energy minimum)")
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def demo_chiral_isolation():
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"""Demonstrate D/L orthogonality."""
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print("\n" + "=" * 70)
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print("CHIRAL ISOLATION DEMONSTRATION")
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print("=" * 70)
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# Create D and L chains
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frames = []
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# D-chain (chirality=0)
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for i in range(3):
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frames.append(Frame(theta=0, phi=0, psi=0, chi=0, a=10.0, x=float(i), y=0.0))
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# L-chain (chirality=1)
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for i in range(3):
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frames.append(Frame(theta=0, phi=0, psi=0, chi=1, a=2.0, x=float(i), y=1.0))
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print("\nInitial state: Two chains (D-chain at y=0, L-chain at y=1)")
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print(f"{'Node':>6} | {'x':>6} | {'y':>6} | {'χ':>4} | {'a':>8} | {'Chain'}")
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print("-" * 60)
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for i, f in enumerate(frames):
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chain = "D-chain" if f.chi == 0 else "L-chain"
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print(f"{i:>6} | {f.x:>6.1f} | {f.y:>6.1f} | {f.chi:>4} | {f.a:>8.2f} | {chain}")
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# Check weights
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print("\nCross-chain weights (D to L):")
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for i in range(3):
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for j in range(3, 6):
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w = compute_weight(frames[i], frames[j])
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print(f" w({i},{j}) = {w:.3f} (should be 0.000)")
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# Run simulation
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sim = GPLSimulation(frames)
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final = sim.run(steps=30)
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print(f"\nFinal state:")
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d_avg = sum(f.a for f in final[:3]) / 3
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l_avg = sum(f.a for f in final[3:]) / 3
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print(f" D-chain average: {d_avg:.2f}")
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print(f" L-chain average: {l_avg:.2f}")
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print(" Chains evolved independently (no crosstalk)")
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def demo_vortex_formation():
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"""Demonstrate vortex as rotational attractor."""
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print("\n" + "=" * 70)
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print("VORTEX FORMATION")
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print("=" * 70)
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# Create nodes in circle with θ matching angular position
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n = 8
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frames = []
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for i in range(n):
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angle = 2 * math.pi * i / n
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theta = int((i * 16 / n) % 16) # θ matches position
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f = Frame(
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theta=theta,
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phi=0,
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psi=0,
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chi=0,
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a=5.0,
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x=math.cos(angle),
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y=math.sin(angle)
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)
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frames.append(f)
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print(f"\nCircular arrangement: θ matches angular position")
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print(f"{'Node':>6} | {'θ':>4} | {'Angle°':>8} | {'x':>8} | {'y':>8}")
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print("-" * 60)
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for i, f in enumerate(frames):
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angle_deg = math.degrees(math.atan2(f.y, f.x))
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print(f"{i:>6} | {f.theta:>4} | {angle_deg:>8.1f} | {f.x:>8.3f} | {f.y:>8.3f}")
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# Check weights (should be high for neighbors, forming vortex)
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print("\nNeighbor weights (high = vortex stable):")
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for i in range(n):
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j = (i + 1) % n
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w = compute_weight(frames[i], frames[j])
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print(f" w({i},{j}) = {w:.3f}")
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avg_w = sum(compute_weight(frames[i], frames[(i+1)%n]) for i in range(n)) / n
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print(f"\nAverage neighbor weight: {avg_w:.3f}")
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print("High alignment → Vortex is stable attractor")
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def demo_summary():
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"""Summary of interaction law."""
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print("\n" + "=" * 70)
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print("INTERACTION LAW SUMMARY")
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print("=" * 70)
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print("""
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The GPL Local Interaction Law:
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1. WEIGHT FUNCTION
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w_ij = cos(Δθ) · cos(Δφ) · (1 - 2|Δχ|) · exp(-|Δp|²/2σ²)
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- cos(Δθ): Azimuthal alignment
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- cos(Δφ): Polar alignment
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- (1 - 2|Δχ|): Chirality match
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- exp(...): Distance decay
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2. FORCE EQUATION
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F_ij = w_ij · (a_j - a_i) · direction
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- Activation flows from high to low
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- Modulated by rotational compatibility
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3. EVOLUTION
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a_i(t+1) = a_i(t) + α · Σ_j F_ij
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- Gradient descent on energy landscape
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- Converges to attractor
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Key Behaviors:
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- Aligned frames (Δθ=0): Strong attraction
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- Orthogonal frames (Δθ=4): No coupling
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- Opposite frames (Δθ=8): Repulsion
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- Chiral mismatch (Δχ=1): Complete isolation
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- Smooth θ gradients: Stable vortices
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Computation = Frame field convergence to energy minimum
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""")
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if __name__ == "__main__":
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demo_two_node_interaction()
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demo_convergence()
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demo_chiral_isolation()
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demo_vortex_formation()
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demo_summary()
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