Recovered source transcript
1// DNA Mathematical Architecture: G¹⁶ Helix System
2// 44 Bonds — 10.6 Periodicity — Complete Implementation
3#pragma once
4
5#include <array>
6#include <cmath>
7#include <complex>
8#include <vector>
9#include <algorithm>
10#include <numeric>
11
12namespace agi::dna {
13
14// ============================================================================
15// I. FOUNDATIONAL CONSTANTS
16// ============================================================================
17
18// Golden ratio and derived constants
19constexpr double PHI = 1.6180339887498948482; // (1 + √5) / 2
20constexpr double PHI_INV = 0.6180339887498948482; // 1 / φ = φ - 1
21
22// DNA Helix parameters
23constexpr double BASE_PAIRS_PER_TURN = 10.6; // p = 10.6 bp
24constexpr double HELIX_RADIUS = 1.0; // r = 1.0 nm
25constexpr double RISE_PER_BP = 0.34; // h = 0.34 nm
26constexpr double PITCH = RISE_PER_BP * BASE_PAIRS_PER_TURN; // P = 3.604 nm
27constexpr double TWIST_ANGLE_DEG = 360.0 / BASE_PAIRS_PER_TURN; // 33.96°
28constexpr double TWIST_ANGLE_RAD = 2.0 * M_PI / BASE_PAIRS_PER_TURN; // 0.593 rad
29constexpr double ANGULAR_FREQUENCY = TWIST_ANGLE_RAD; // ω = 0.593 rad/bp
30
31// Strand parameters
32constexpr int NUM_STRANDS = 16;
33constexpr double STRAND_SEPARATION = 2.0 * M_PI / NUM_STRANDS; // 22.5°
34constexpr int NODES_PER_STRAND = 312;
35constexpr int TOTAL_NODES = 5000;
36constexpr int HUB_NODES = 8; // Olympus routing nodes
37
38// Bond parameters
39constexpr int INTRA_STRAND_BONDS = 16;
40constexpr int INTER_STRAND_BONDS = 16;
41constexpr int CROSS_HELIX_BONDS = 12;
42constexpr int TOTAL_BONDS = 44;
43
44// Curvature and torsion (pre-computed)
45constexpr double HELIX_CURVATURE = 0.745; // κ = 0.745 nm⁻¹
46constexpr double HELIX_TORSION = 0.427; // τ = 0.427 nm⁻¹
47
48// Synchronization
49constexpr double SYNC_PERIOD = BASE_PAIRS_PER_TURN * NUM_STRANDS; // 169.6 time units
50constexpr double FUNDAMENTAL_FREQ = 1.0 / BASE_PAIRS_PER_TURN; // 0.0943 cycles/bp
51constexpr double BOND_RESONANCE_FREQ = TOTAL_BONDS / BASE_PAIRS_PER_TURN; // 4.15 cycles/turn
52
53// Steady state
54constexpr double STEADY_STATE_BONDS = 36.2;
55constexpr double BOND_OCCUPATION = 0.82;
56constexpr double TAU_SLOW = 10.0; // ms
57constexpr double TAU_FAST = 0.0001; // ms (0.1 μs)
58
59// ============================================================================
60// II. G¹⁶ PARAMETRIC HELIX
61// ============================================================================
62
63struct HelixPoint {
64 double x, y, z;
65
66 HelixPoint operator+(const HelixPoint& other) const {
67 return {x + other.x, y + other.y, z + other.z};
68 }
69
70 HelixPoint operator-(const HelixPoint& other) const {
71 return {x - other.x, y - other.y, z - other.z};
72 }
73
74 HelixPoint operator*(double s) const {
75 return {x * s, y * s, z * s};
76 }
77
78 double norm() const {
79 return std::sqrt(x*x + y*y + z*z);
80 }
81
82 double dot(const HelixPoint& other) const {
83 return x*other.x + y*other.y + z*other.z;
84 }
85};
86
87// G¹⁶ Parametric Helix equations (Section 1.1)
88// x_k(θ) = r·cos(θ + 2πk/16)
89// y_k(θ) = r·sin(θ + 2πk/16)
90// z_k(θ) = (p/2π)·θ + δ_k
91class G16Helix {
92public:
93 // Vertical offset per strand: δ_k = k·h/16
94 static double vertical_offset(int k) {
95 return k * RISE_PER_BP / NUM_STRANDS;
96 }
97
98 // Golden phase offset: θ_k = 2πk·φ⁻¹ mod 2π
99 static double golden_phase_offset(int k) {
100 return std::fmod(2.0 * M_PI * k * PHI_INV, 2.0 * M_PI);
101 }
102
103 // Get point on strand k at parameter θ
104 static HelixPoint point(int k, double theta) {
105 double phase = theta + 2.0 * M_PI * k / NUM_STRANDS;
106 return {
107 HELIX_RADIUS * std::cos(phase),
108 HELIX_RADIUS * std::sin(phase),
109 (PITCH / (2.0 * M_PI)) * theta + vertical_offset(k)
110 };
111 }
112
113 // Get point with golden ratio phase adjustment
114 static HelixPoint point_golden(int k, double theta) {
115 double phase = theta + golden_phase_offset(k);
116 return {
117 HELIX_RADIUS * std::cos(phase),
118 HELIX_RADIUS * std::sin(phase),
119 (PITCH / (2.0 * M_PI)) * theta + vertical_offset(k)
120 };
121 }
122
123 // Curvature at point (Section 1.3)
124 // κ = r/(r² + (p·h/2π)²)
125 static double curvature() {
126 double ph_term = PITCH * RISE_PER_BP / (2.0 * M_PI);
127 return HELIX_RADIUS / (HELIX_RADIUS * HELIX_RADIUS + ph_term * ph_term);
128 }
129
130 // Torsion at point (Section 1.3)
131 // τ = (p·h/2π)/(r² + (p·h/2π)²)
132 static double torsion() {
133 double ph_term = PITCH * RISE_PER_BP / (2.0 * M_PI);
134 return ph_term / (HELIX_RADIUS * HELIX_RADIUS + ph_term * ph_term);
135 }
136
137 // Stability metric: exp(-Σ|τ(n)|·φ⁻ⁿ)
138 static double stability_metric(const std::vector<double>& torsions) {
139 double sum = 0.0;
140 double phi_power = 1.0;
141 for (size_t n = 0; n < torsions.size(); ++n) {
142 sum += std::abs(torsions[n]) * phi_power;
143 phi_power *= PHI_INV;
144 }
145 return std::exp(-sum);
146 }
147
148 // Helix balance: Σκ(n)·φ⁻ⁿ
149 static double helix_balance(const std::vector<double>& curvatures) {
150 double sum = 0.0;
151 double phi_power = 1.0;
152 for (size_t n = 0; n < curvatures.size(); ++n) {
153 sum += curvatures[n] * phi_power;
154 phi_power *= PHI_INV;
155 }
156 return sum;
157 }
158
159 // G¹⁶ Mean curvature: (1/16) Σ κ_k × cos(2πk/10.6)
160 static double mean_curvature_g16() {
161 double sum = 0.0;
162 double kappa = curvature();
163 for (int k = 1; k <= NUM_STRANDS; ++k) {
164 sum += kappa * std::cos(2.0 * M_PI * k / BASE_PAIRS_PER_TURN);
165 }
166 return sum / NUM_STRANDS;
167 }
168
169 // Full rotation calculation
170 // θ_total = 16 × (2π/10.6) = 9.49 rad = 543.8°
171 static double total_rotation() {
172 return NUM_STRANDS * TWIST_ANGLE_RAD;
173 }
174
175 // Turns per 16 strands = 1.51
176 static double turns_per_16_strands() {
177 return total_rotation() / (2.0 * M_PI);
178 }
179};
180
181// ============================================================================
182// III. 44-BOND SYSTEM
183// ============================================================================
184
185enum class BondType {
186 INTRA_STRAND, // Bonds 1-16: within strand
187 INTER_STRAND, // Bonds 17-32: between adjacent strands
188 CROSS_HELIX // Bonds 33-44: across helix (|i-j| ∈ {4, 8, 12})
189};
190
191struct Bond {
192 int id; // Bond number 1-44
193 BondType type;
194 int strand_i; // First strand (1-16)
195 int strand_j; // Second strand (1-16)
196 double length; // Bond length in nm
197 double angle; // Bond angle in radians
198 double energy; // Bond energy
199 double occupation; // Occupation state [0, 1]
200
201 // Get bond position vector (midpoint)
202 HelixPoint midpoint(double theta) const {
203 auto p_i = G16Helix::point(strand_i, theta);
204 auto p_j = G16Helix::point(strand_j, theta);
205 return (p_i + p_j) * 0.5;
206 }
207
208 // Calculate bond length
209 double calculate_length(double theta) const {
210 auto p_i = G16Helix::point(strand_i, theta);
211 auto p_j = G16Helix::point(strand_j, theta);
212 return (p_i - p_j).norm();
213 }
214};
215
216class BondSystem44 {
217private:
218 std::array<Bond, TOTAL_BONDS> bonds_;
219 std::array<std::array<int, NUM_STRANDS>, NUM_STRANDS> adjacency_;
220
221public:
222 BondSystem44() {
223 initialize_bonds();
224 build_adjacency_matrix();
225 }
226
227 // Initialize all 44 bonds according to distribution (Section 2.1)
228 void initialize_bonds() {
229 int bond_id = 0;
230
231 // Intra-strand bonds (1-16): B_intra(k) = 1 for k = 1..16
232 for (int k = 1; k <= INTRA_STRAND_BONDS; ++k) {
233 bonds_[bond_id++] = {
234 bond_id, BondType::INTRA_STRAND,
235 k, k,
236 PITCH / NUM_STRANDS, // L₀ = P/16 = 0.225 nm
237 M_PI, // 180° linear backbone
238 0.0, 1.0
239 };
240 }
241
242 // Inter-strand bonds (17-32): B_inter(k, k+1 mod 16) = 1
243 for (int k = 1; k <= INTER_STRAND_BONDS; ++k) {
244 int next_k = (k % NUM_STRANDS) + 1;
245 bonds_[bond_id++] = {
246 bond_id, BondType::INTER_STRAND,
247 k, next_k,
248 2.0 * HELIX_RADIUS * std::sin(M_PI / BASE_PAIRS_PER_TURN), // d₀ = 0.59 nm
249 STRAND_SEPARATION, // 22.5°
250 0.0, 1.0
251 };
252 }
253
254 // Cross-helix bonds (33-44): pairs where |i-j| ∈ {4, 8, 12}
255 // Pairs: (1,5), (2,6), (3,7), (4,8), (5,9), (6,10),
256 // (7,11), (8,12), (9,13), (10,14), (11,15), (12,16)
257 int cross_pairs[][2] = {
258 {1,5}, {2,6}, {3,7}, {4,8}, {5,9}, {6,10},
259 {7,11}, {8,12}, {9,13}, {10,14}, {11,15}, {12,16}
260 };
261
262 for (int m = 0; m < CROSS_HELIX_BONDS; ++m) {
263 int i = cross_pairs[m][0];
264 int j = cross_pairs[m][1];
265 int diff = std::abs(i - j);
266 double angle = diff * STRAND_SEPARATION; // |i-j| × 22.5°
267
268 bonds_[bond_id++] = {
269 bond_id, BondType::CROSS_HELIX,
270 i, j,
271 0.0, // Calculated dynamically
272 angle,
273 0.0, 1.0
274 };
275 }
276 }
277
278 // Build adjacency matrix (Section 2.4)
279 void build_adjacency_matrix() {
280 // Initialize to zero
281 for (auto& row : adjacency_) {
282 row.fill(0);
283 }
284
285 // Fill based on bonds
286 for (const auto& bond : bonds_) {
287 int i = bond.strand_i - 1;
288 int j = bond.strand_j - 1;
289 adjacency_[i][j] = 1;
290 adjacency_[j][i] = 1;
291 }
292 }
293
294 // Verify bond count: Σ_{i,j} A_{ij}/2 = 44
295 int verify_bond_count() const {
296 int sum = 0;
297 for (int i = 0; i < NUM_STRANDS; ++i) {
298 for (int j = 0; j < NUM_STRANDS; ++j) {
299 sum += adjacency_[i][j];
300 }
301 }
302 // Add intra-strand bonds (diagonal)
303 sum += INTRA_STRAND_BONDS;
304 return sum / 2 + INTRA_STRAND_BONDS;
305 }
306
307 // Get bond by type and index
308 Bond& get_bond(int id) { return bonds_[id - 1]; }
309 const Bond& get_bond(int id) const { return bonds_[id - 1]; }
310
311 // Get bonds by type
312 std::vector<const Bond*> get_bonds_by_type(BondType type) const {
313 std::vector<const Bond*> result;
314 for (const auto& bond : bonds_) {
315 if (bond.type == type) {
316 result.push_back(&bond);
317 }
318 }
319 return result;
320 }
321
322 // Bond coordinate system (Section 2.2)
323 std::pair<int, int> bond_coordinates(int n) const {
324 if (n <= INTRA_STRAND_BONDS) {
325 // Intra-strand: (k, k)
326 return {n, n};
327 } else if (n <= INTRA_STRAND_BONDS + INTER_STRAND_BONDS) {
328 // Inter-strand: (k, k+1 mod 16)
329 int k = n - INTRA_STRAND_BONDS;
330 return {k, (k % NUM_STRANDS) + 1};
331 } else {
332 // Cross-helix: (i, j) where |i-j| ∈ {4, 8, 12}
333 return {bonds_[n-1].strand_i, bonds_[n-1].strand_j};
334 }
335 }
336
337 // 10.6-adjusted bond arc length: s_bond = 2πr/10.6 = 0.593 nm per bp
338 static double bond_arc_length() {
339 return 2.0 * M_PI * HELIX_RADIUS / BASE_PAIRS_PER_TURN;
340 }
341
342 // Vertical rise per bond: Δz_bond = P/44 = 0.0819 nm
343 static double vertical_rise_per_bond() {
344 return PITCH / TOTAL_BONDS;
345 }
346
347 // Adjacent bond length: L_adj = √[2r²(1 - cos(2π/10.6)) + h²] = 0.54 nm
348 static double adjacent_bond_length() {
349 double cos_term = 1.0 - std::cos(TWIST_ANGLE_RAD);
350 return std::sqrt(2.0 * HELIX_RADIUS * HELIX_RADIUS * cos_term +
351 RISE_PER_BP * RISE_PER_BP);
352 }
353
354 // Validation: distance constraint
355 bool validate_distance(double epsilon_max) const {
356 double adjustment = BASE_PAIRS_PER_TURN / 10.4; // 10.6/10.4 correction
357 for (const auto& bond : bonds_) {
358 if (bond.length > epsilon_max * adjustment) {
359 return false;
360 }
361 }
362 return true;
363 }
364
365 // Validation: angular constraint
366 bool validate_angular(double delta_theta) const {
367 for (const auto& bond : bonds_) {
368 double target = std::round(bond.angle / TWIST_ANGLE_RAD) * TWIST_ANGLE_RAD;
369 if (std::abs(bond.angle - target) > delta_theta) {
370 return false;
371 }
372 }
373 return true;
374 }
375
376 // Update all bond states
377 void update_bonds(double theta) {
378 for (auto& bond : bonds_) {
379 bond.length = bond.calculate_length(theta);
380 }
381 }
382
383 // Get total active bonds
384 double total_occupation() const {
385 double sum = 0.0;
386 for (const auto& bond : bonds_) {
387 sum += bond.occupation;
388 }
389 return sum;
390 }
391};
392
393// ============================================================================
394// IV. STRAND MATHEMATICS
395// ============================================================================
396
397// Fibonacci sequence generator
398inline uint64_t fibonacci(int n) {
399 if (n <= 1) return n;
400 uint64_t a = 0, b = 1;
401 for (int i = 2; i <= n; ++i) {
402 uint64_t c = a + b;
403 a = b;
404 b = c;
405 }
406 return b;
407}
408
409// Fibonacci-scaled strand: D_k = φ^k · F_k
410inline double fibonacci_scaled_strand(int k) {
411 return std::pow(PHI, k) * fibonacci(k);
412}
413
414class StrandMathematics {
415private:
416 std::array<double, NUM_STRANDS> D_; // Fibonacci-scaled values
417 std::array<std::complex<double>, NUM_STRANDS> S_; // Complex representation
418 std::array<std::complex<double>, NUM_STRANDS> T_; // Transform
419
420public:
421 StrandMathematics() {
422 initialize();
423 }
424
425 void initialize() {
426 for (int k = 0; k < NUM_STRANDS; ++k) {
427 D_[k] = fibonacci_scaled_strand(k + 1);
428 }
429 compute_transforms();
430 }
431
432 // 16-strand helix sum: Helix₁₆ = Σ D_k·sin(kφ)
433 double helix_sum_16() const {
434 double sum = 0.0;
435 for (int k = 1; k <= NUM_STRANDS; ++k) {
436 sum += D_[k-1] * std::sin(k * PHI);
437 }
438 return sum;
439 }
440
441 // Bond formation: Bond_G16 = Π(1 + D_k/Helix₁₆)
442 double bond_formation_g16() const {
443 double helix = helix_sum_16();
444 double product = 1.0;
445 for (int k = 0; k < NUM_STRANDS; ++k) {
446 product *= (1.0 + D_[k] / helix);
447 }
448 return product;
449 }
450
451 // Crosslink function: Crosslink(i,j) = (D_i · D_j)/φ^{|i-j|}
452 double crosslink(int i, int j) const {
453 return (D_[i-1] * D_[j-1]) / std::pow(PHI, std::abs(i - j));
454 }
455
456 // Complex strand representation: S_k = D_k · e^{ikφ}
457 void compute_transforms() {
458 for (int k = 0; k < NUM_STRANDS; ++k) {
459 double phase = (k + 1) * PHI;
460 S_[k] = D_[k] * std::exp(std::complex<double>(0, phase));
461
462 // Transform: T_k = S_k · (cos(k) + i·sin(k))
463 double angle = k + 1;
464 T_[k] = S_[k] * std::complex<double>(std::cos(angle), std::sin(angle));
465 }
466 }
467
468 // Modular extraction: X_k = |T_k| mod φ^k
469 double modular_extraction(int k) const {
470 return std::fmod(std::abs(T_[k-1]), std::pow(PHI, k));
471 }
472
473 // Fusion integral: Fusion = Σ(Strand_Vector(k) · φ⁻ᵏ)
474 double fusion_integral() const {
475 double sum = 0.0;
476 for (int k = 0; k < NUM_STRANDS; ++k) {
477 sum += std::abs(T_[k]) * std::pow(PHI_INV, k + 1);
478 }
479 return sum;
480 }
481
482 // Get D values
483 double get_D(int k) const { return D_[k-1]; }
484 std::complex<double> get_S(int k) const { return S_[k-1]; }
485 std::complex<double> get_T(int k) const { return T_[k-1]; }
486};
487
488// ============================================================================
489// V. G¹⁶ TRANSFORMATION MATRICES
490// ============================================================================
491
492struct Matrix3x3 {
493 std::array<std::array<double, 3>, 3> m;
494
495 Matrix3x3 operator*(const Matrix3x3& other) const {
496 Matrix3x3 result = {{{0}}};
497 for (int i = 0; i < 3; ++i) {
498 for (int j = 0; j < 3; ++j) {
499 for (int k = 0; k < 3; ++k) {
500 result.m[i][j] += m[i][k] * other.m[k][j];
501 }
502 }
503 }
504 return result;
505 }
506
507 HelixPoint transform(const HelixPoint& p) const {
508 return {
509 m[0][0]*p.x + m[0][1]*p.y + m[0][2]*p.z,
510 m[1][0]*p.x + m[1][1]*p.y + m[1][2]*p.z,
511 m[2][0]*p.x + m[2][1]*p.y + m[2][2]*p.z
512 };
513 }
514};
515
516class G16Transform {
517public:
518 // 22.5° increment rotation matrix (Section 5.1)
519 static Matrix3x3 rotation_22_5(int k) {
520 double angle = 2.0 * M_PI * k / NUM_STRANDS;
521 double c = std::cos(angle);
522 double s = std::sin(angle);
523 return {{{
524 {c, -s, 0},
525 {s, c, 0},
526 {0, 0, 1}
527 }}};
528 }
529
530 // 10.6-adjusted rotation matrix
531 static Matrix3x3 rotation_10_6(int k) {
532 double angle = 2.0 * M_PI * k / BASE_PAIRS_PER_TURN;
533 double c = std::cos(angle);
534 double s = std::sin(angle);
535 return {{{
536 {c, -s, 0},
537 {s, c, 0},
538 {0, 0, 1}
539 }}};
540 }
541
542 // Phase increment: Δφ = 2π/10.6 = 33.96° = 0.593 rad
543 static double phase_increment() {
544 return TWIST_ANGLE_RAD;
545 }
546
547 // Full G¹⁶ transform: G¹⁶ = Π R_k · T_k
548 static Matrix3x3 full_transform() {
549 Matrix3x3 result = {{{
550 {1, 0, 0},
551 {0, 1, 0},
552 {0, 0, 1}
553 }}};
554
555 for (int k = 1; k <= NUM_STRANDS; ++k) {
556 result = result * rotation_22_5(k);
557 }
558 return result;
559 }
560
561 // 10.6-adjusted full transform
562 static Matrix3x3 full_transform_10_6() {
563 Matrix3x3 result = {{{
564 {1, 0, 0},
565 {0, 1, 0},
566 {0, 0, 1}
567 }}};
568
569 for (int k = 1; k <= NUM_STRANDS; ++k) {
570 result = result * rotation_10_6(k);
571 }
572 return result;
573 }
574};
575
576// ============================================================================
577// VI. 10.6 FIRING PATTERN
578// ============================================================================
579
580class FiringPattern {
581public:
582 // Phase calculation: phase(i,t) = ⌊(t + offset_i)/10.6⌋ mod 16
583 static int phase(int i, double t) {
584 double offset = offset_distribution(i);
585 return static_cast<int>(std::floor((t + offset) / BASE_PAIRS_PER_TURN)) % NUM_STRANDS;
586 }
587
588 // Offset distribution: offset_i = i · φ⁻¹ · 10.6
589 static double offset_distribution(int i) {
590 return i * PHI_INV * BASE_PAIRS_PER_TURN;
591 }
592
593 // Fire condition: fire(i,t) = [phase = 0]
594 static bool should_fire(int i, double t) {
595 return phase(i, t) == 0;
596 }
597
598 // Get all firing strands at time t
599 static std::vector<int> firing_strands(double t) {
600 std::vector<int> result;
601 for (int i = 1; i <= NUM_STRANDS; ++i) {
602 if (should_fire(i, t)) {
603 result.push_back(i);
604 }
605 }
606 return result;
607 }
608
609 // Synchronization period: T_sync = 10.6 × 16 = 169.6
610 static double sync_period() {
611 return SYNC_PERIOD;
612 }
613
614 // Fundamental frequency: f₀ = 1/10.6 = 0.0943
615 static double fundamental_frequency() {
616 return FUNDAMENTAL_FREQ;
617 }
618
619 // Harmonic: f_n = n/10.6
620 static double harmonic(int n) {
621 return n / BASE_PAIRS_PER_TURN;
622 }
623
624 // 44-bond resonance: f_44 = 44/10.6 = 4.15
625 static double bond_resonance() {
626 return BOND_RESONANCE_FREQ;
627 }
628};
629
630// ============================================================================
631// VII. GOLDEN BASIS FUNCTIONS
632// ============================================================================
633
634class GoldenBasis {
635public:
636 // Basis function: ψ_k(x) = sin(2πφ^k · x) + cos(2πφ^k · x)
637 static double psi(int k, double x) {
638 double arg = 2.0 * M_PI * std::pow(PHI, k) * x;
639 return std::sin(arg) + std::cos(arg);
640 }
641
642 // Generate basis vector for N terms
643 static std::vector<double> basis_vector(double x, int N) {
644 std::vector<double> result(N);
645 for (int k = 0; k < N; ++k) {
646 result[k] = psi(k, x);
647 }
648 return result;
649 }
650
651 // Global coherence: Φ_total = (1/N) Σ Φ_i
652 // where Φ_i = 1/(1 + |x_i - 1|)
653 static double global_coherence(const std::vector<double>& x) {
654 double sum = 0.0;
655 for (double xi : x) {
656 sum += 1.0 / (1.0 + std::abs(xi - 1.0));
657 }
658 return sum / x.size();
659 }
660
661 // Ring diffusion: dS_i/dt = λ(S_{i+1} + S_{i-1} - 2S_i)
662 static void ring_diffusion_step(std::vector<double>& S, double lambda, double dt) {
663 int N = S.size();
664 std::vector<double> dS(N);
665
666 for (int i = 0; i < N; ++i) {
667 int prev = (i - 1 + N) % N;
668 int next = (i + 1) % N;
669 dS[i] = lambda * (S[next] + S[prev] - 2.0 * S[i]);
670 }
671
672 for (int i = 0; i < N; ++i) {
673 S[i] += dS[i] * dt;
674 }
675 }
676};
677
678// ============================================================================
679// VIII. SUPERCOILING (10.6)
680// ============================================================================
681
682class Supercoiling {
683public:
684 // Linking number: Lk = Tw + Wr
685 static double linking_number(double twist, double writhe) {
686 return twist + writhe;
687 }
688
689 // Twist for 16 strands: Tw = N/10.6 × 16
690 static double twist_16_strands(int N) {
691 return (static_cast<double>(N) / BASE_PAIRS_PER_TURN) * NUM_STRANDS;
692 }
693
694 // Relaxed linking number: Lk₀ = N/10.6
695 static double relaxed_linking_number(int N) {
696 return static_cast<double>(N) / BASE_PAIRS_PER_TURN;
697 }
698
699 // Superhelical density: σ = (Lk - Lk₀)/Lk₀
700 static double superhelical_density(double Lk, double Lk0) {
701 return (Lk - Lk0) / Lk0;
702 }
703
704 // G¹⁶ superhelical density
705 static double superhelical_density_g16(double Lk, int N) {
706 double Lk0_g16 = NUM_STRANDS * static_cast<double>(N) / BASE_PAIRS_PER_TURN;
707 return (Lk - Lk0_g16) / Lk0_g16;
708 }
709
710 // Free energy of supercoiling
711 // ΔG_sc = K · N · σ² · (10.6/10.4)²
712 static double free_energy(double K, int N, double sigma) {
713 constexpr double correction = (BASE_PAIRS_PER_TURN / 10.4) * (BASE_PAIRS_PER_TURN / 10.4);
714 return K * N * sigma * sigma * correction;
715 }
716
717 // G¹⁶ with 44-bond correction
718 static double free_energy_g16(double K, int N, double sigma_g16, double lambda) {
719 constexpr double correction = (BASE_PAIRS_PER_TURN / 10.4) * (BASE_PAIRS_PER_TURN / 10.4);
720 return K * N * sigma_g16 * sigma_g16 * (1.0 + lambda * TOTAL_BONDS) * correction;
721 }
722};
723
724// ============================================================================
725// COMPLETE G¹⁶ HELIX SYSTEM
726// ============================================================================
727
728class G16HelixSystem {
729private:
730 G16Helix helix_;
731 BondSystem44 bonds_;
732 StrandMathematics strands_;
733 double time_;
734 double global_phase_; // φ_10.6(t)
735
736 // State vectors
737 std::array<std::complex<double>, NUM_STRANDS> strand_states_;
738 std::array<double, NUM_STRANDS> angular_positions_;
739
740public:
741 G16HelixSystem() : time_(0.0), global_phase_(0.0) {
742 initialize();
743 }
744
745 void initialize() {
746 // Initialize strand states: S_k(t=0) = A_k · e^{i(2πk/10.6)}
747 for (int k = 0; k < NUM_STRANDS; ++k) {
748 double amplitude = strands_.get_D(k + 1);
749 double phase = 2.0 * M_PI * (k + 1) / BASE_PAIRS_PER_TURN;
750 strand_states_[k] = amplitude * std::exp(std::complex<double>(0, phase));
751 angular_positions_[k] = phase;
752 }
753 }
754
755 // Tick the system forward
756 void tick(double dt) {
757 time_ += dt;
758
759 // Update global phase: φ(t) = (2πt/10.6) mod 2π
760 global_phase_ = std::fmod(2.0 * M_PI * time_ / BASE_PAIRS_PER_TURN, 2.0 * M_PI);
761
762 // Update strand states: S_k(t) = A_k · e^{i(ωt + 2πk/10.6)}
763 for (int k = 0; k < NUM_STRANDS; ++k) {
764 double amplitude = strands_.get_D(k + 1);
765 double phase = ANGULAR_FREQUENCY * time_ + 2.0 * M_PI * (k + 1) / BASE_PAIRS_PER_TURN;
766 strand_states_[k] = amplitude * std::exp(std::complex<double>(0, phase));
767 angular_positions_[k] = phase;
768 }
769
770 // Update bond states
771 bonds_.update_bonds(global_phase_);
772 }
773
774 // Get firing strands
775 std::vector<int> get_firing_strands() const {
776 return FiringPattern::firing_strands(time_);
777 }
778
779 // Get global coherence
780 double get_coherence() const {
781 std::vector<double> magnitudes(NUM_STRANDS);
782 for (int k = 0; k < NUM_STRANDS; ++k) {
783 magnitudes[k] = std::abs(strand_states_[k]);
784 }
785 return GoldenBasis::global_coherence(magnitudes);
786 }
787
788 // Get total bond occupation
789 double get_bond_occupation() const {
790 return bonds_.total_occupation();
791 }
792
793 // Accessors
794 double time() const { return time_; }
795 double global_phase() const { return global_phase_; }
796 const BondSystem44& bonds() const { return bonds_; }
797 BondSystem44& bonds() { return bonds_; }
798 const StrandMathematics& strands() const { return strands_; }
799};
800
801} // namespace agi::dna
802