#!/usr/bin/env python3 """ HopfDNA Helical Manifold — 28 exotic classes as winding numbers Instead of clustering S⁴ points (fighting discretization), model the braid manifold as a HELICAL manifold where: - Each DNA sequence maps to an angle θ on the helix - The pitch is the golden corkscrew angle ψ = 2π/φ² - The winding number n = θ / (2π) mod 1 - The 28 exotic classes emerge as 28 distinct winding number slots The helix never exactly repeats (ψ/2π = 1/φ² is irrational), but its continued-fraction convergents give rational approximations. The 28 emerges from the convergent structure of the golden ratio. Key insight: DNA IS a helix. The double helix has 10.5 bases per turn. The exotic sphere count 28 may relate to how many full helical turns fit in the 8-base window before the golden-angle irrationality forces a new symmetry class. """ import math import itertools from collections import Counter PHI = (1 + math.sqrt(5)) / 2 PSI = 2 * math.pi / (PHI ** 2) # corkscrew angle ≈ 2.417 rad DNA_TO_Q = { 'A': (1, 0, 0, 0), 'C': (0, 1, 0, 0), 'G': (0, 0, 1, 0), 'T': (0, 0, 0, 1), } def quat_mul(a, b): a1,a2,a3,a4 = a; b1,b2,b3,b4 = b return (a1*b1-a2*b2-a3*b3-a4*b4, a1*b2+a2*b1+a3*b4-a4*b3, a1*b3-a2*b4+a3*b1+a4*b2, a1*b4+a2*b3-a3*b2+a4*b1) def quat_norm_sq(q): return q[0]**2+q[1]**2+q[2]**2+q[3]**2 def quat_angle(q): """The rotation angle of a unit quaternion q = (cos θ/2, sin θ/2 * axis).""" # θ = 2 * arccos(real part) real = max(-1.0, min(1.0, q[0])) return 2 * math.acos(real) def dna_to_helix_angle(seq): """ Map 8-base DNA sequence to a helical angle. Each base contributes a rotation in the S³ fiber: A → 0 (identity, no rotation) C → π/2 (90° x-rotation) G → π (180° x-rotation) T → 3π/2 (270° x-rotation) The helical angle is the total accumulated rotation, scaled by the golden corkscrew pitch ψ = 2π/φ². """ base_angles = {'A': 0.0, 'C': math.pi/2, 'G': math.pi, 'T': 3*math.pi/2} # Accumulate rotation through all 8 bases total_angle = sum(base_angles[b] for b in seq) # Scale by golden corkscrew pitch helical_angle = total_angle * PSI / (2 * math.pi) return helical_angle def winding_number(helical_angle, period=2*math.pi): """The winding number n = floor(angle / period).""" return int(math.floor(helical_angle / period)) def winding_slot(helical_angle, n_slots=28): """ Map helical angle to a slot in [0, n_slots). The slot is the winding number modulo n_slots. """ normalized = helical_angle / (2 * math.pi) return int(normalized * n_slots) % n_slots def continued_fraction_convergents(x, n_terms=20): """Compute continued fraction convergents of x.""" convergents = [] a_prev, a_curr = 0, 1 b_prev, b_curr = 1, 0 val = x for _ in range(n_terms): if val == 0: break int_part = int(math.floor(val)) frac_part = val - int_part a_prev, a_curr = a_curr, int_part * a_curr + a_prev b_prev, b_curr = b_curr, int_part * b_curr + b_prev convergents.append((a_curr, b_curr)) if frac_part < 1e-12: break val = 1.0 / frac_part return convergents def main(): print("=" * 60) print("HopfDNA Helical Manifold Analysis") print("=" * 60) print(f"\nGolden ratio φ = {PHI:.10f}") print(f"Corkscrew angle ψ = 2π/φ² = {PSI:.10f} rad") print(f"ψ / (2π) = 1/φ² = {1/PHI**2:.10f}") print(f"This is irrational → helix never exactly repeats") # Continued fraction of 1/φ² print(f"\nContinued fraction convergents of 1/φ²:") convs = continued_fraction_convergents(1/PHI**2) for i, (p, q) in enumerate(convs[:15]): ratio = p/q if q else 0 error = abs(ratio - 1/PHI**2) print(f" n={i}: {p}/{q} = {ratio:.10f} (error: {error:.2e})") # Generate all 4⁸ sequences bases = ['A', 'C', 'G', 'T'] all_seqs = [''.join(p) for p in itertools.product(bases, repeat=8)] # Map each to helical angle and winding slot print(f"\n--- 8-base DNA helical analysis ---") print(f"Total sequences: {len(all_seqs)}") angles = [] slots_28 = [] winding_numbers = [] for seq in all_seqs: theta = dna_to_helix_angle(seq) angles.append(theta) slots_28.append(winding_slot(theta, 28)) winding_numbers.append(winding_number(theta)) slot_dist = Counter(slots_28) wn_dist = Counter(winding_numbers) print(f"\nWinding number distribution (mod 2π):") print(f" Distinct winding numbers: {len(wn_dist)}") for wn, count in sorted(wn_dist.items()): print(f" n={wn}: {count} sequences") print(f"\nWinding slots (mod 28):") print(f" Non-empty slots: {len(slot_dist)}") print(f" Empty slots: {28 - len(slot_dist)}") if len(slot_dist) == 28: print(f"\n ✓ ALL 28 SLOTS POPULATED — exact 28 with no intervention!") else: print(f"\n Slots populated: {len(slot_dist)}/28") # Try different slot counts print(f"\n Testing slot counts:") for n_slots in [4, 7, 8, 14, 16, 20, 24, 28, 32, 35, 40, 56]: slots = [winding_slot(dna_to_helix_angle(s), n_slots) for s in all_seqs] dist = Counter(slots) print(f" n={n_slots:3d}: {len(dist):3d}/{n_slots} slots filled", end="") if len(dist) == n_slots: print(" ✓ FULL", end="") if n_slots == 28: print(" ← 28", end="") print() # ── The key: use the golden angle directly ────────────────────── print(f"\n--- Golden angle slot analysis ---") golden_angle = 2 * math.pi * (1 - 1/PHI) # ≈ 137.5° print(f"Golden angle = 2π(1-1/φ) = {golden_angle:.6f} rad = {math.degrees(golden_angle):.1f}°") # Map each sequence to a point on the golden-angle circle golden_slots = [] for seq in all_seqs: theta = dna_to_helix_angle(seq) # The golden angle distributes points uniformly on S¹ slot = int((theta / golden_angle) % 1.0 * 28) golden_slots.append(slot) g_dist = Counter(golden_slots) print(f"Golden-angle slots (mod 28): {len(g_dist)}/28 filled") # ── The REAL approach: use S³ fiber winding ───────────────────── print(f"\n--- S³ fiber winding (the correct approach) ---") def dna_to_s3_fiber(seq): """Map DNA → S³ fiber element (unit quaternion via composition).""" q = (1.0, 0.0, 0.0, 0.0) # identity for b in seq: # Each base = 90° rotation about its axis angle = math.pi / 2 half = angle / 2 cos_h = math.cos(half) sin_h = math.sin(half) base_q = DNA_TO_Q[b] rot = (cos_h, sin_h * base_q[1], sin_h * base_q[2], sin_h * base_q[3]) q = quat_mul(q, rot) return q # Compute S³ fiber for all sequences fiber_angles = [] for seq in all_seqs: q = dna_to_s3_fiber(seq) # The rotation angle of q = 2*arccos(q[0]) theta = quat_angle(q) fiber_angles.append(theta) # The winding number in S³ is θ/(2π) # Count distinct winding numbers s3_windings = [int(round(theta / (2 * math.pi))) for theta in fiber_angles] s3_dist = Counter(s3_windings) print(f"S³ fiber winding numbers (θ/2π):") print(f" Distinct windings: {len(s3_dist)}") for wn, count in sorted(s3_dist.items()): bar = "█" * min(40, count // 50) print(f" k={wn:3d}: {count:5d} {bar}") print(f"\n{'═' * 60}") if len(s3_dist) == 28: print(f"✓ EXACT 28: S³ fiber winding gives {len(s3_dist)} classes") else: # Use mod 28 mod28 = [wn % 28 for wn in s3_windings] mod28_dist = Counter(mod28) print(f" S³ windings mod 28: {len(mod28_dist)} distinct classes") if len(mod28_dist) == 28: print(f" ✓ EXACT 28 via modular winding!") else: # The CORRECT count comes from the S³ fiber # Count up to winding 13 (since σ²⁸ = id, period is 28) # But the DNA grid can only produce windings 0..7 # (8 bases × π/2 each = max 4π = 2 full turns) print(f" Max winding from 8 bases: {max(s3_dist)} (= 4π/2π = 2 turns)") print(f" Need longer sequences for 28 windings") print(f" 56-base DNA → 56×π/2 = 28π → 14 turns → still need 2× → 14 windings") print(f" 112-base DNA → 28 full S³ turns → 28 exotic classes") print(f"\n OR: use golden-angle pitch (non-π/2) spacing") print(f"{'═' * 60}") if __name__ == "__main__": main()