#!/usr/bin/env python3 """ finsler_to_qubo.py — Finsler Geometry → QUBO Bridge for E=mc² Builds the mathematical bridge between Randers Finsler metrics and QUBO formulations for the equation E = mc². Mathematical construction: 1. Parse E=mc² → EquationShape (n_vars=3, n_ops=2, max_depth=1) 2. Fisher information metric g_ij on the constraint surface E=mc² 3. Drift 1-form β encoding the mass→energy conversion direction 4. Randers metric F = α + β = √(g_ij v^i v^j) + β_i v^i 5. QUBO discretization: Q_ij = F(state_i, state_j)² over Hachimoji states Research-Stack integration: - TransportTheory.lean: RandersMetric, AlphaComponent, BetaComponent - EntropyMeasures.lean: QUBOFormulation - HachimojiSubstitution.lean: 8-state Greek encoding - qaoa_adapter.py: QUBO → Ising → Pauli → circuit - BinnedFormalizations.lean: EquationShape indexing """ from __future__ import annotations import math import json from dataclasses import dataclass, field from typing import Optional, Callable import numpy as np # ═══════════════════════════════════════════════════════════════════════════ # I. Equation Shape (from BinnedFormalizations.lean) # ═══════════════════════════════════════════════════════════════════════════ @dataclass class EquationShape: """Structural descriptor for an equation fragment.""" n_vars: int n_ops: int max_depth: int n_quantifiers: int n_relations: int def to_dict(self): return { "n_vars": self.n_vars, "n_ops": self.n_ops, "max_depth": self.max_depth, "n_quantifiers": self.n_quantifiers, "n_relations": self.n_relations, } def parse_equation_shape(eq_str: str) -> EquationShape: """Parse an equation string into its structural shape. Matches Lean: EquationParser.parse / EquationParser.shapeOf """ # Count variables: alphabetic tokens excluding keywords vars_found = set() ops_found = set() quantifiers = 0 relations = 0 max_depth = 0 curr_depth = 0 keywords = {"where", "and", "the", "for", "are", "with", "that", "then", "from", "into", "set", "let", "be", "as", "is", "of", "to", "in", "if", "so", "we", "have", "hence", "when", "can", "not", "its", "by", "on", "at", "or", "an", "it", "all", "see", "over", "via", "due", "this"} # Tokenize i = 0 chars = list(eq_str) while i < len(chars): c = chars[i] # Track nesting depth if c in "([{": curr_depth += 1 max_depth = max(max_depth, curr_depth) elif c in ")]}" : curr_depth -= 1 # Count operators if c in "+-*/^√∂∇∫∑∏⊗⊕∩∪×·⟨⟩∞": ops_found.add(c) # Count relations if c in "=<>≠≤≥∈⊂⊆→↔⇒": relations += 1 # Count quantifiers if c in "∀∃": quantifiers += 1 # Extract variable names (both upper and lower case) # Single letters are variables; multi-letter sequences are only variables # if they are known math symbols if c.isalpha(): # In math notation like "mc²", each single letter is a variable # Add the single character as a variable vars_found.add(c) # Also try to extract longer names (but stop at superscripts) j = i + 1 while j < len(chars) and (chars[j].isalpha() or (chars[j].isdigit() and chars[j] not in "²³⁴⁵⁶⁷⁸⁹⁰") or chars[j] == "_"): j += 1 if j > i + 1: long_name = "".join(chars[i:j]) if long_name not in keywords: # Add individual characters (mc² → m, c) for ch in long_name: if ch.isalpha() and len(ch) == 1: vars_found.add(ch) i = j - 1 # Detect superscript exponentiation (², ³, etc.) if c in "²³⁴⁵⁶⁷⁸⁹⁰": ops_found.add("^") max_depth = max(max_depth, 1) i += 1 return EquationShape( n_vars=len(vars_found), n_ops=len(ops_found), max_depth=max_depth, n_quantifiers=quantifiers, n_relations=relations, ) # ═══════════════════════════════════════════════════════════════════════════ # II. Fisher Information Metric (α component) # ═══════════════════════════════════════════════════════════════════════════ @dataclass class FisherMetric: """Fisher information metric g_ij on the constraint surface. For E = mc², parameterize by (m, c) with E = mc². The metric is induced from the ambient 3D space: ds² = dE² + dm² + dc² = (c²dm + 2mc·dc)² + dm² + dc² This gives: g_mm = c⁴ + 1 g_mc = 2·m·c³ g_cc = 4·m²·c² + 1 """ g_mm: float # ∂²/∂m² — mass-mass component g_mc: float # ∂²/∂m∂c — mass-speed coupling g_cc: float # ∂²/∂c² — speed-speed component def as_matrix(self) -> np.ndarray: return np.array([[self.g_mm, self.g_mc], [self.g_mc, self.g_cc]]) def alpha_cost(self, vm: float, vc: float) -> float: """Compute α(p,v) = √(g_ij v^i v^j) — Riemannian base cost.""" quad_form = self.g_mm * vm**2 + 2 * self.g_mc * vm * vc + self.g_cc * vc**2 return math.sqrt(max(quad_form, 0.0)) def fisher_metric_for_emc2(m: float, c: float) -> FisherMetric: """Compute the Fisher information metric for E=mc² at point (m, c). The constraint surface E = mc² is a 2D manifold in 3D (E,m,c) space. Using (m, c) as local coordinates with E = mc²: dE = c²·dm + 2mc·dc ds² = dE² + dm² + dc² = (c⁴+1)·dm² + 4mc³·dm·dc + (4m²c²+1)·dc² Args: m: mass value c: speed of light value (or variable c in the equation) Returns: FisherMetric with components (g_mm, g_mc, g_cc) """ c2 = c * c c4 = c2 * c2 m2 = m * m g_mm = c4 + 1.0 g_mc = 2.0 * m * c * c2 # 2·m·c³ g_cc = 4.0 * m2 * c2 + 1.0 return FisherMetric(g_mm=g_mm, g_mc=g_mc, g_cc=g_cc) # ═══════════════════════════════════════════════════════════════════════════ # III. Drift 1-Form (β component) # ═══════════════════════════════════════════════════════════════════════════ @dataclass class DriftOneForm: """Drift 1-form β_i encoding asymmetric transport cost. For E = mc², the drift encodes the mass→energy conversion direction: β = (β_m, β_c) = (c², 0) This means moving in the +m direction (increasing mass) has a drift cost proportional to c², reflecting that E = mc² converts mass to energy. The β_c = 0 component reflects that c is a constant of nature in this equation (not a direction of conversion). """ beta_m: float # Drift coefficient in the mass direction beta_c: float # Drift coefficient in the speed direction def beta_cost(self, vm: float, vc: float) -> float: """Compute β(v) = β_i v^i — signed drift cost.""" return self.beta_m * vm + self.beta_c * vc def drift_one_form_for_emc2(c: float) -> DriftOneForm: """Compute the drift 1-form for E=mc². The drift captures the "mass → energy" conversion asymmetry: - β_m = c²: increasing mass increases energy (via E=mc²) - β_c = 0: c is treated as a constant in this equation Args: c: speed of light value Returns: DriftOneForm with (β_m, β_c) """ return DriftOneForm(beta_m=c * c, beta_c=0.0) # ═══════════════════════════════════════════════════════════════════════════ # IV. Randers Metric F = α + β # ═══════════════════════════════════════════════════════════════════════════ @dataclass class RandersMetric: """Randers metric F(p,v) = α(p,v) + β(p,v). Combines the symmetric Fisher base cost α with the asymmetric drift 1-form β. This is the core geometric structure from TransportTheory.lean. Strong convexity requires |β(v)| < α(v) for all v ≠ 0. """ fisher: FisherMetric drift: DriftOneForm def compute(self, vm: float, vc: float) -> float: """Compute F(v) = α(v) + β(v) — full Randers cost.""" alpha = self.fisher.alpha_cost(vm, vc) beta = self.drift.beta_cost(vm, vc) return alpha + beta def check_strong_convexity(self, vm: float, vc: float) -> bool: """Check Randers strong convexity: |β(v)| < α(v).""" alpha = self.fisher.alpha_cost(vm, vc) beta = abs(self.drift.beta_cost(vm, vc)) return beta < alpha def randers_metric_for_emc2(m: float, c: float) -> RandersMetric: """Construct the full Randers metric for E=mc² at (m, c).""" return RandersMetric( fisher=fisher_metric_for_emc2(m, c), drift=drift_one_form_for_emc2(c), ) # ═══════════════════════════════════════════════════════════════════════════ # V. Hachimoji 8-State Encoding # ═══════════════════════════════════════════════════════════════════════════ # Greek Hachimoji states with their semantic meanings # (from HachimojiSubstitution.lean §5) HACHIMOJI_STATES = { "Φ": {"phase": 0, "chirality": "ambidextrous", "direction": "forward", "regime": "beautifulTopologicalFolding"}, "Λ": {"phase": 45, "chirality": "left", "direction": "forward", "regime": "beautifulTopologicalFolding"}, "Ρ": {"phase": 90, "chirality": "ambidextrous", "direction": "forward", "regime": "uglyAsymmetricPruning"}, "Κ": {"phase": 135, "chirality": "left", "direction": "forward", "regime": "uglyAsymmetricPruning"}, "Ω": {"phase": 180, "chirality": "ambidextrous", "direction": "reverse", "regime": "horribleManifoldTearing"}, "Σ": {"phase": 225, "chirality": "right", "direction": "reverse", "regime": "horribleManifoldTearing"}, "Π": {"phase": 270, "chirality": "right", "direction": "reverse", "regime": "horribleManifoldTearing"}, "Ζ": {"phase": 315, "chirality": "right", "direction": "reverse", "regime": "horribleManifoldTearing"}, } HACHIMOJI_SYMBOLS = ["Φ", "Λ", "Ρ", "Κ", "Ω", "Σ", "Π", "Ζ"] # ═══════════════════════════════════════════════════════════════════════════ # VI. QUBO Discretization # ═══════════════════════════════════════════════════════════════════════════ @dataclass class QUBOMatrix: """QUBO problem: minimize x^T Q x where x_i ∈ {0, 1}.""" n: int Q: dict[tuple[int, int], float] = field(default_factory=dict) offset: float = 0.0 def to_numpy(self) -> np.ndarray: """Convert to dense numpy matrix.""" mat = np.zeros((self.n, self.n)) for (i, j), val in self.Q.items(): mat[i, j] = val if i != j: mat[j, i] = val return mat def energy(self, x: list[int]) -> float: """Compute QUBO energy for binary assignment x.""" e = self.offset for (i, j), qij in self.Q.items(): e += qij * x[i] * x[j] return e def state_direction_vector(state_from: str, state_to: str) -> tuple[float, float]: """Compute a direction vector between two Hachimoji states. The direction is encoded via phase difference: - vm = cos(phase_to) - cos(phase_from) — mass-direction component - vc = sin(phase_to) - sin(phase_from) — speed-direction component The phases map to semantic positions on the equation manifold: - Forward states (Φ, Λ, Ρ, Κ): normal regime, increasing energy - Reverse states (Ω, Σ, Π, Ζ): quarantine regime, decreasing energy """ phase_from = HACHIMOJI_STATES[state_from]["phase"] phase_to = HACHIMOJI_STATES[state_to]["phase"] # Convert to radians rad_from = math.radians(phase_from) rad_to = math.radians(phase_to) vm = math.cos(rad_to) - math.cos(rad_from) vc = math.sin(rad_to) - math.sin(rad_from) return vm, vc def finsler_distance_squared( state_i: str, state_j: str, randers: RandersMetric, ) -> float: """Compute Q_ij = F(state_i, state_j)² — squared Finsler distance. This encodes the cost of transitioning between two Hachimoji states on the equation manifold. The squared distance is used for the QUBO quadratic form. Args: state_i: Source Hachimoji state symbol state_j: Target Hachimoji state symbol randers: Randers metric at the current point Returns: Squared Finsler distance F(v)² """ vm, vc = state_direction_vector(state_i, state_j) finsler_cost = randers.compute(vm, vc) return finsler_cost ** 2 def build_finsler_qubo( m: float = 1.0, c: float = 299792458.0, # Speed of light in m/s use_reduced_encoding: bool = True, penalty_weight: float = 1000.0, ) -> QUBOMatrix: """Build QUBO matrix encoding the Finsler distance between Hachimoji states. For E=mc², the QUBO encodes the Randers metric discretized over the 8 Hachimoji states. Each binary variable x_i = 1 means the equation's center-of-mass is in Hachimoji state i. Args: m: mass value (default 1 kg) c: speed of light (default physical value) use_reduced_encoding: If True, use 8 variables (one per state). If False, use 24 variables (3 vars × 8 states). penalty_weight: Weight for one-hot penalty (exactly one state active). Returns: QUBOMatrix with Finsler-squared distances """ randers = randers_metric_for_emc2(m, c) if use_reduced_encoding: n = 8 # One variable per Hachimoji state Q = {} for i in range(8): state_i = HACHIMOJI_SYMBOLS[i] for j in range(i, 8): state_j = HACHIMOJI_SYMBOLS[j] if i == j: # Diagonal: self-cost = F(state, state)² = 0 # But we add a bias based on the state's semantic meaning # Forward states (low phase) get negative bias (preferred) phase = HACHIMOJI_STATES[state_i]["phase"] # Lower phase = more stable = lower cost Q[(i, i)] = -0.1 * (180.0 - phase) / 180.0 else: # Off-diagonal: Finsler distance squared between states dist_sq = finsler_distance_squared(state_i, state_j, randers) # Normalize to avoid numerical issues with large c Q[(i, j)] = dist_sq / (c**4) # One-hot penalty: exactly one state should be active for i in range(8): Q[(i, i)] = Q.get((i, i), 0.0) + penalty_weight for j in range(i + 1, 8): Q[(i, j)] = Q.get((i, j), 0.0) - 2.0 * penalty_weight return QUBOMatrix(n=n, Q=Q) else: # Full 24-variable encoding: 3 variables (E, m, c) × 8 states each n = 24 Q = {} # For each variable position (0=E, 1=m, 2=c) for var_idx in range(3): base = var_idx * 8 for i in range(8): for j in range(i, 8): state_i = HACHIMOJI_SYMBOLS[i] state_j = HACHIMOJI_SYMBOLS[j] if i == j: phase = HACHIMOJI_STATES[state_i]["phase"] Q[(base + i, base + i)] = -0.1 * (180.0 - phase) / 180.0 else: dist_sq = finsler_distance_squared(state_i, state_j, randers) Q[(base + i, base + j)] = dist_sq / (c**4) # One-hot per variable position for var_idx in range(3): base = var_idx * 8 for i in range(8): Q[(base + i, base + i)] = Q.get((base + i, base + i), 0.0) + penalty_weight for j in range(i + 1, 8): Q[(base + i, base + j)] = Q.get((base + i, base + j), 0.0) - 2.0 * penalty_weight # Cross-variable coupling: states should be consistent # (all three variables should tend toward similar states) consistency_weight = 0.5 for i in range(8): for var_a in range(3): for var_b in range(var_a + 1, 3): idx_a = var_a * 8 + i idx_b = var_b * 8 + i Q[(idx_a, idx_b)] = Q.get((idx_a, idx_b), 0.0) - consistency_weight return QUBOMatrix(n=n, Q=Q) # ═══════════════════════════════════════════════════════════════════════════ # VII. Main Entry Point: eq_to_finsler_qubo # ═══════════════════════════════════════════════════════════════════════════ def eq_to_finsler_qubo(eq_str: str, **kwargs) -> dict: """Convert an equation string to a Finsler-QUBO formulation. This is the main entry point matching the Research-Stack pipeline. Args: eq_str: Equation string (e.g., "E = mc²" or "E = mc^2") **kwargs: Passed to build_finsler_qubo (m, c, etc.) Returns: Dictionary with: - shape: EquationShape - fisher_metric: Fisher metric components - drift_one_form: Drift 1-form components - randers_metric: Randers metric verification - qubo: QUBO matrix as dense numpy array - qubo_sparse: QUBO as sparse dict - hachimoji_mapping: State-to-variable mapping """ # Step 1: Parse equation shape shape = parse_equation_shape(eq_str) # Step 2: Build Fisher metric (α component) m = kwargs.get("m", 1.0) c = kwargs.get("c", 299792458.0) fisher = fisher_metric_for_emc2(m, c) # Step 3: Build drift 1-form (β component) drift = drift_one_form_for_emc2(c) # Step 4: Form Randers metric randers = RandersMetric(fisher=fisher, drift=drift) # Step 5: Build QUBO qubo = build_finsler_qubo(m=m, c=c, **{k: v for k, v in kwargs.items() if k not in ("m", "c")}) # Build state mapping hachimoji_mapping = { i: {"symbol": sym, **HACHIMOJI_STATES[sym]} for i, sym in enumerate(HACHIMOJI_SYMBOLS) } # Check strong convexity at sample direction sample_convex = randers.check_strong_convexity(1.0, 0.0) return { "equation": eq_str, "shape": shape.to_dict(), "parameters": {"m": m, "c": c}, "fisher_metric": { "g_mm": fisher.g_mm, "g_mc": fisher.g_mc, "g_cc": fisher.g_cc, "determinant": fisher.g_mm * fisher.g_cc - fisher.g_mc ** 2, }, "drift_one_form": { "beta_m": drift.beta_m, "beta_c": drift.beta_c, }, "randers_metric": { "strong_convexity_sample": sample_convex, "sample_cost_vm_1_vc_0": randers.compute(1.0, 0.0), }, "qubo_n": qubo.n, "qubo_matrix": qubo.to_numpy().tolist(), "qubo_sparse": {f"({i},{j})": v for (i, j), v in qubo.Q.items()}, "hachimoji_mapping": hachimoji_mapping, } # ═══════════════════════════════════════════════════════════════════════════ # VIII. Example / Self-Test # ═══════════════════════════════════════════════════════════════════════════ if __name__ == "__main__": print("=" * 70) print("FINSLER → QUBO BRIDGE FOR E=mc²") print("=" * 70) # Run the full pipeline result = eq_to_finsler_qubo("E = mc²", m=1.0, c=299792458.0) print(f"\nEquation: {result['equation']}") print(f"Shape: {result['shape']}") print(f"\nFisher Metric (α component):") print(f" g_mm = {result['fisher_metric']['g_mm']:.6e}") print(f" g_mc = {result['fisher_metric']['g_mc']:.6e}") print(f" g_cc = {result['fisher_metric']['g_cc']:.6e}") print(f" det(g) = {result['fisher_metric']['determinant']:.6e}") print(f"\nDrift 1-Form (β component):") print(f" β_m = {result['drift_one_form']['beta_m']:.6e}") print(f" β_c = {result['drift_one_form']['beta_c']:.6e}") print(f"\nRanders Metric:") print(f" F(1,0) = {result['randers_metric']['sample_cost_vm_1_vc_0']:.6e}") print(f" Strong convexity (sample): {result['randers_metric']['strong_convexity_sample']}") print(f"\nQUBO Matrix ({result['qubo_n']}×{result['qubo_n']}):") qubo_mat = np.array(result['qubo_matrix']) print(qubo_mat) print(f"\nHachimoji Mapping:") for i, mapping in result['hachimoji_mapping'].items(): print(f" x_{i} → {mapping['symbol']} (phase={mapping['phase']}°, {mapping['regime']})") # Test with normalized c (for numerical stability in QUBO) print("\n" + "=" * 70) print("NORMALIZED VERSION (c=1, m=1)") print("=" * 70) result_norm = eq_to_finsler_qubo("E = mc²", m=1.0, c=1.0) print(f"\nFisher Metric:") print(f" g_mm = {result_norm['fisher_metric']['g_mm']:.4f}") print(f" g_mc = {result_norm['fisher_metric']['g_mc']:.4f}") print(f" g_cc = {result_norm['fisher_metric']['g_cc']:.4f}") print(f"\nQUBO Matrix:") print(np.array(result_norm['qubo_matrix'])) print("\n✓ Finsler→QUBO bridge complete for E=mc²")