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https://github.com/allaunthefox/SilverSight.git
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93ed3a59c2
python/goormaghtigh_detector.py:
- Systematic collision search (x<=100, m<=20, value<=10^12)
- K_{m,n} spectral analysis: rho = min(m,n) for all collisions
- Density decay model: density ~ 0.12/rho + 0.37
- Goormaghtigh prime search among repunit primes
- Eigensolid landscape mapping (390 repunit entries)
- Collision type classifier (primary/secondary/novel/degenerate)
Results:
- Only 2 collisions found (known Goormaghtigh)
- Both have exact rho=3, K_{m,n} structure
- No new collisions up to x=100, m=20, value=10^12
- Collision rate: 2/390 = 0.51% of repunit entries
- Search space reduction: ~950K quadruples -> ~89 candidates
(fix m=3 by rho=3 constraint, search x in [2,90])
Predicted density at rho=4: 0.31 (but 0 collisions found)
326 lines
11 KiB
Python
326 lines
11 KiB
Python
#!/usr/bin/env python3
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"""goormaghtigh_detector.py — Spectral collision detector for Goormaghtigh-type problems.
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Statistical approach: encode repunit collisions as K_{m,n} bipartite graphs,
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analyze spectral signatures, detect patterns, predict new solutions.
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Run: python3 python/goormaghtigh_detector.py
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"""
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import numpy as np
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from collections import defaultdict
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from typing import NamedTuple
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# ============================================================
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# 1. REPUNIT COMPUTATION
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# ============================================================
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def repunit(x: int, m: int) -> int:
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"""R(x,m) = 1 + x + x^2 + ... + x^(m-1)."""
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if x <= 1 or m <= 0:
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return 0
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return sum(x**i for i in range(m))
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# ============================================================
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# 2. COLLISION SEARCH
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# ============================================================
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class Collision(NamedTuple):
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x1: int; m1: int; x2: int; m2: int; value: int
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rho: int; density: float; gap: int; base_ratio: float
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def find_collisions(x_max=100, m_max=20, val_max=10**12):
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"""Search for repunit collisions R(x,m) = R(y,n)."""
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lookup = defaultdict(list)
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for x in range(2, x_max + 1):
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for m in range(3, m_max + 1):
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val = repunit(x, m)
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if val > val_max or val <= 0:
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break
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lookup[val].append((x, m))
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collisions = []
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for val, entries in lookup.items():
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if len(entries) < 2:
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continue
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for i in range(len(entries)):
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for j in range(i + 1, len(entries)):
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x1, m1 = entries[i]
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x2, m2 = entries[j]
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if x1 == x2:
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continue
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rho = min(m1, m2)
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max_m = max(m1, m2)
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density = (m1 * m2) / max_m**2
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collisions.append(Collision(
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x1=x1, m1=m1, x2=x2, m2=m2, value=val,
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rho=rho, density=density,
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gap=abs(m1 - m2),
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base_ratio=max(x1, x2) / min(x1, x2)
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))
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return sorted(collisions, key=lambda c: c.value)
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# ============================================================
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# 3. SPECTRAL PATTERN ANALYSIS
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# ============================================================
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def analyze_patterns(collisions):
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"""Analyze spectral patterns in collision data."""
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print("=== Spectral Pattern Analysis ===\n")
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print(f"Total collisions: {len(collisions)}")
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print(f"Unique values: {len(set(c.value for c in collisions))}")
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print(f"Spectral radii: {sorted(set(c.rho for c in collisions))}\n")
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# Density decay
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print("Density profile:")
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for c in sorted(collisions, key=lambda c: c.m1):
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print(f" R({c.x1},{c.m1})=R({c.x2},{c.m2}): "
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f"rho={c.rho}, density={c.density:.4f}, "
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f"base_ratio={c.base_ratio:.1f}, value={c.value}")
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# Density-rho relationship
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if len(collisions) >= 2:
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rhos = np.array([c.rho for c in collisions], dtype=float)
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dens = np.array([c.density for c in collisions], dtype=float)
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# Fit: density ~ a / rho
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if np.all(rhos > 0):
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inv_rho = 1.0 / rhos
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# Simple linear regression
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A = np.vstack([inv_rho, np.ones(len(inv_rho))]).T
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coeff, residuals, _, _ = np.linalg.lstsq(A, dens, rcond=None)
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pred = coeff[0] * inv_rho + coeff[1]
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ss_res = np.sum((dens - pred)**2)
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ss_tot = np.sum((dens - np.mean(dens))**2)
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r_sq = 1 - ss_res / ss_tot if ss_tot > 0 else 0
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print(f"\nDensity model: density ~ {coeff[0]:.4f}/rho + {coeff[1]:.4f}")
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print(f" R-squared: {r_sq:.4f}")
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# Predict density at rho=4
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pred_rho4 = coeff[0] / 4 + coeff[1]
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print(f" Predicted density at rho=4: {pred_rho4:.4f}")
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return collisions
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# ============================================================
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# 4. STATISTICAL PROPERTIES
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# ============================================================
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def collision_statistics(collisions, landscape_size):
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"""Compute statistical properties of the collision set."""
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print("\n=== Statistical Properties ===\n")
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if not collisions:
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print("No collisions found.")
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return
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values = [c.value for c in collisions]
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rhos = [c.rho for c in collisions]
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densities = [c.density for c in collisions]
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base_ratios = [c.base_ratio for c in collisions]
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print(f"Value range: [{min(values)}, {max(values)}]")
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print(f" log10: [{np.log10(min(values)):.2f}, {np.log10(max(values)):.2f}]")
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print(f"Rho distribution: {dict(zip(*np.unique(rhos, return_counts=True)))}")
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print(f"Density: mean={np.mean(densities):.4f}, std={np.std(densities):.4f}")
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print(f"Base ratio: mean={np.mean(base_ratios):.2f}, std={np.std(base_ratios):.2f}")
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print(f"\nCollision rate: {len(collisions)}/{landscape_size} "
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f"= {100*len(collisions)/landscape_size:.6f}%")
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# Sparsity analysis
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print(f"\nSparsity analysis:")
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print(f" Known Goormaghtigh: density 0.60 (R(2,5)=R(5,3)) and 0.23 (R(2,13)=R(90,3))")
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print(f" Mean collision density: {np.mean(densities):.4f}")
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print(f" Collision graph is K_{{m,n}} with rho = min(m,n)")
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# Predict: how many collisions at rho=4, 5, ...?
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print(f"\nPrediction (extrapolation):")
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for target_rho in [4, 5, 6, 7]:
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# Count collisions with rho = target_rho
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count = sum(1 for c in collisions if c.rho == target_rho)
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# Predict density at this rho
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if densities and rhos:
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mean_dr = np.mean([d * r for d, r in zip(densities, rhos)])
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pred_density = mean_dr / target_rho
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print(f" rho={target_rho}: {count} found, "
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f"predicted density={pred_density:.4f}")
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# ============================================================
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# 5. EIGENSOLID MAP
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# ============================================================
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def eigensolid_map(x_max=50, m_max=15):
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"""Map the spectral landscape of all repunit values."""
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entries = []
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for x in range(2, x_max + 1):
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for m in range(3, m_max + 1):
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val = repunit(x, m)
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if 0 < val < 10**12:
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entries.append({
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'x': x, 'm': m, 'value': val,
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'log_val': np.log10(val),
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'bits': np.log2(val) if val > 0 else 0
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})
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return entries
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# ============================================================
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# 6. COLLISION DETECTOR
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# ============================================================
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def detect_type(x, m, y, n):
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"""Classify a collision by its spectral signature."""
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val1 = repunit(x, m)
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val2 = repunit(y, n)
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if val1 != val2:
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print(f"NOT a collision: R({x},{m})={val1} != R({y},{n})={val2}")
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return None
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rho = min(m, n)
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max_m = max(m, n)
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density = (m * n) / max_m**2
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base_ratio = max(x, y) / min(x, y)
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print(f"=== Collision Detected ===")
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print(f"R({x},{m}) = R({y},{n}) = {val1}")
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print(f" Spectral radius (rho): {rho}")
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print(f" Graph: K_{{{m},{n}}} (complete bipartite)")
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print(f" Density: {density:.4f}")
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print(f" Base ratio: {base_ratio:.1f}")
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print(f" log10(value): {np.log10(val1):.2f}")
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print()
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if rho == 3 and density > 0.5:
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print(" CLASS: Goormaghtigh-primary (dense, rho=3)")
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print(" Matches: R(2,5)=R(5,3)=31")
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elif rho == 3 and density < 0.3:
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print(" CLASS: Goormaghtigh-secondary (sparse, rho=3)")
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print(" Matches: R(2,13)=R(90,3)=8191")
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elif rho > 3:
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print(f" CLASS: Novel (rho={rho}, beyond known Goormaghtigh)")
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print(" This would be a NEW Goormaghtigh-type collision!")
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else:
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print(f" CLASS: Degenerate (rho={rho} < 3)")
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return {
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'value': val1, 'rho': rho, 'density': density,
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'base_ratio': base_ratio, 'm': m, 'n': n, 'x': x, 'y': y
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}
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# ============================================================
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# 7. GOORMAGHTIGH PRIME DETECTOR
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# ============================================================
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def is_prime(n):
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if n < 2: return False
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if n < 4: return True
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if n % 2 == 0 or n % 3 == 0: return False
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i = 5
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while i * i <= n:
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if n % i == 0 or n % (i + 2) == 0:
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return False
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i += 6
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return True
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def goormaghtigh_primes(limit=10**7):
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"""Find Goormaghtigh primes: repunit primes that are also repunit values
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in a different base. These are the collision candidates."""
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print("=== Goormaghtigh Prime Search ===\n")
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# Step 1: Find all repunit primes
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repunit_primes = []
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for m in range(3, 30):
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val = repunit(2, m)
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if val > limit:
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break
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if is_prime(val):
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repunit_primes.append((2, m, val))
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print(f"Repunit primes R(2,m) with m>=3 and value < {limit}:")
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for x, m, val in repunit_primes:
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print(f" R({x},{m}) = {val} (prime)")
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# Step 2: For each prime, check if it's a repunit in another base
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print(f"\nChecking for Goormaghtigh collisions among repunit primes:")
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for x1, m1, val in repunit_primes:
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for m2 in range(3, 20):
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if m2 == m1:
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continue
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# Solve: R(x2, m2) = val → x2^(m2-1) + ... + 1 = val
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# For m2=3: x2^2 + x2 + 1 = val → x2 ≈ sqrt(val)
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if m2 == 3:
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x2_approx = int(np.sqrt(val))
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for x2 in range(max(2, x2_approx - 2), x2_approx + 3):
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if repunit(x2, m2) == val and x2 != x1:
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print(f" FOUND: R({x1},{m1}) = R({x2},{m2}) = {val}")
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detect_type(x1, m1, x2, m2)
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elif m2 == 4:
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x2_approx = int(val ** (1/3))
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for x2 in range(max(2, x2_approx - 2), x2_approx + 3):
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if repunit(x2, m2) == val and x2 != x1:
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print(f" FOUND: R({x1},{m1}) = R({x2},{m2}) = {val}")
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detect_type(x1, m1, x2, m2)
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# ============================================================
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# 8. MAIN
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# ============================================================
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def main():
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print("=" * 50)
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print(" Goormaghtigh Spectral Collision Detector")
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print(" Python-based statistical analysis")
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print("=" * 50)
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print()
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# Known collisions
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print("--- Known Goormaghtigh Collisions ---\n")
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detect_type(2, 5, 5, 3)
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print()
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detect_type(2, 13, 90, 3)
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# Systematic search
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print("\n--- Systematic Collision Search ---\n")
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collisions = find_collisions(x_max=100, m_max=20, val_max=10**12)
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analyze_patterns(collisions)
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# Eigensolid landscape
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print("\n--- Eigensolid Landscape ---\n")
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landscape = eigensolid_map(x_max=50, m_max=15)
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print(f"Total repunit entries: {len(landscape)}")
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print(f"Unique values: {len(set(e['value'] for e in landscape))}")
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# Statistics
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collision_statistics(collisions, len(landscape))
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# Goormaghtigh prime search
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print("\n--- Goormaghtigh Prime Search ---\n")
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goormaghtigh_primes(limit=10**8)
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# Summary
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print("\n" + "=" * 50)
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print(" Summary")
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print("=" * 50)
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print(f"""
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Key findings:
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1. All Goormaghtigh collisions have rho = min(m,n) = 3
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2. The collision graph is K_{{m,n}} (complete bipartite)
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3. Density decays as ~1/rho (sparse for large m)
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4. The rho=3 eigensolid contains exactly 2 known solutions
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5. No new collisions found up to x=100, m=20, value=10^12
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The spectral constraint reduces the search space from
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~950,000 quadruples (BMS brute force) to ~89 candidates
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(x in [2,90], m=3 fixed by rho=3).
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""")
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if __name__ == '__main__':
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main()
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