#!/usr/bin/env python3 """ Frozen-In Gravity Model — Asenjo, Comisso & Winkler (PRL 2026) Extracted equations and their mapping to PIST Extended Encoding. Paper: "Frozen-In Gravitational Fields" DOI: 10.1103/6c4q-kx6f URL: https://phys.org/news/2026-04-frozen-gravity-evolution-spacetime-dynamics.html Core idea: Gravitational field structures remain "frozen" into spacetime dynamics under ideal conditions, preserving topological invariants. PIST analogy: Coordinate structures remain invariant under encoding/decoding, eliminating the need for search-based context modeling. """ import numpy as np from typing import Tuple, Callable # ─── Equation 1: Einstein Field Equations (Standard Form) ─── # # G_μν + Λ g_μν = (8πG / c⁴) T_μν # # G_μν = Einstein tensor (geometry) # T_μν = stress-energy tensor (matter) # Λ = cosmological constant # g_μν = metric tensor # # Asenjo et al. rewrite this in analogy to conducting fluid equations. def einstein_tensor(g_inv: np.ndarray, dg: np.ndarray, ddg: np.ndarray) -> np.ndarray: """ Compute Einstein tensor G_μν from metric and its derivatives. G_μν = R_μν - (1/2) R g_μν where R_μν is Ricci tensor, R is Ricci scalar. """ # Simplified: full computation is complex; this is the symbolic structure # In practice, Christoffel symbols → Riemann → Ricci → Einstein # For PIST analogy, we only need the conceptual mapping pass # Placeholder — full GR computation is beyond scope # ─── Equation 2: Fluid-Dynamics Analog (Asenjo Rewriting) ─── # # The paper rewrites Einstein equations as: # # ∂_t u + (u · ∇) u = -∇p/ρ + ν ∇²u + (other terms) # # where u represents the gravitational "velocity" field, # p is effective pressure, ρ is effective density. # # Key insight: gravitational field lines behave like magnetic field lines in MHD. class GravitationalMHD: """ Model gravitational field as magnetohydrodynamic fluid. Frozen-in theorem: If E + v × B = 0 (ideal condition), then field lines move with the fluid and topology is preserved. """ def __init__(self, grid_size: Tuple[int, int, int] = (64, 64, 64)): self.Nx, self.Ny, self.Nz = grid_size # Gravitational "magnetic" field B_g (analog to magnetic field) self.B_g = np.zeros((*grid_size, 3)) # Gravitational "electric" field E_g self.E_g = np.zeros((*grid_size, 3)) # Velocity field v self.v = np.zeros((*grid_size, 3)) # Gravitational vector potential A_g (B_g = ∇ × A_g) self.A_g = np.zeros((*grid_size, 3)) def ideal_ohm_law(self) -> np.ndarray: """ Equation 3: Ideal Ohm-type condition for gravity. E_g + v × B_g = 0 This is the frozen-in condition. When satisfied, field lines move with the fluid and connectivity is preserved. """ v_cross_B = np.cross(self.v, self.B_g) return self.E_g + v_cross_B # Should be ~0 for frozen-in def is_frozen_in(self, tol: float = 1e-6) -> bool: """Check if ideal condition is satisfied.""" residual = self.ideal_ohm_law() return np.linalg.norm(residual) < tol def evolve(self, dt: float) -> None: """ Equation 4: Evolution equations for frozen-in fields. ∂_t B_g = ∇ × (v × B_g) (induction equation analog) Under ideal condition, this preserves: - Field line connectivity - Gravitational helicity - Topological invariants """ # Compute ∇ × (v × B_g) v_cross_B = np.cross(self.v, self.B_g) curl_v_cross_B = self._curl(v_cross_B) self.B_g += dt * curl_v_cross_B def _curl(self, F: np.ndarray) -> np.ndarray: """Compute curl of vector field F on grid.""" # Simplified finite-difference curl dFz_dy = np.roll(F[..., 2], -1, axis=1) - np.roll(F[..., 2], 1, axis=1) dFy_dz = np.roll(F[..., 1], -1, axis=2) - np.roll(F[..., 1], 1, axis=2) curl_x = dFz_dy - dFy_dz dFx_dz = np.roll(F[..., 0], -1, axis=2) - np.roll(F[..., 0], 1, axis=2) dFz_dx = np.roll(F[..., 2], -1, axis=0) - np.roll(F[..., 2], 1, axis=0) curl_y = dFx_dz - dFz_dx dFy_dx = np.roll(F[..., 1], -1, axis=0) - np.roll(F[..., 1], 1, axis=0) dFx_dy = np.roll(F[..., 0], -1, axis=1) - np.roll(F[..., 0], 1, axis=1) curl_z = dFy_dx - dFx_dy return np.stack([curl_x, curl_y, curl_z], axis=-1) def gravitational_helicity(self) -> float: """ Equation 5: Gravitational helicity (topological invariant). H_g = ∫ A_g · B_g dV This measures the linkedness/knottedness of gravitational field lines. Under frozen-in dynamics, H_g is conserved. """ dot = np.sum(self.A_g * self.B_g, axis=-1) return np.sum(dot) # Discrete integral over grid # ─── PIST Analogy: Coordinate Invariance ─── class PISTFrozenIn: """ Map gravitational frozen-in theorem to PIST coordinate invariance. Gravitational field line → PIST composite address Fluid velocity v → Data stream position n Field line connectivity → Coordinate structure (k,t,tree,surface,torus) Frozen-in condition → Deterministic encoding/decoding Gravitational helicity → Information conservation (lossless roundtrip) """ def __init__(self, basis_size: int = 16): self.basis_size = basis_size self.basis = np.zeros(basis_size, dtype=np.uint8) self.helicity_history = [] def pist_encode(self, n: int) -> Tuple[int, int]: """ n = k² + t (Equation 6: PIST shell decomposition) Analogous to resolving field into components along/orthogonal to a preferred direction in the fluid. """ k = int(np.sqrt(n)) t = n - k * k return k, t def pist_mass(self, k: int, t: int) -> int: """ m(k,t) = t(2k+1-t) for t < 2k+1-t, else mirrored. Analogous to field strength/intensity at a given shell. """ if k == 0: return 0 m = 2 * k + 1 - t tf = t if t < m else m return tf * (2 * k + 1 - tf) def composite_address(self, n: int) -> dict: """ Equation 7: Composite address = (tree, surface, torus, shell) Analogous to full field specification at a point: - Tree address = topological genus/branch label - Surface coords = local field direction - Torus angles = phase/rotation state - Shell coords = radial distance/amplitude """ k, t = self.pist_encode(n) mass = self.pist_mass(k, t) # Tree: base-20 path tree = [(n // (20 ** i)) % 20 for i in range(3)] # Surface: y = 1/x, θ = n·Φ x = 1.0 + (n % 255) y = 1.0 / x theta_surf = (n * 1.618033988749895) % (2 * np.pi) # Torus: Φ-irrational angles phi = (n * 1.618033988749895) % (2 * np.pi) psi = (n * 1.618033988749895 ** 2) % (2 * np.pi) return { 'n': n, 'k': k, 't': t, 'mass': mass, 'tree': tree, 'surface': {'x': x, 'y': y, 'theta': theta_surf}, 'torus': {'phi': phi, 'psi': psi} } def frozen_in_decode(self, stream: bytes, position: int) -> int: """ Equation 8: Decoder = prediction XOR residual. Prediction is a "frozen-in" coordinate function: it depends only on position n and basis, not on data. This guarantees: - Determinism: same n → same prediction - Reversibility: residual = data XOR prediction - Invariance: coordinate structure preserved """ n = position addr = self.composite_address(n) # Prediction from coordinate-derived values pred = self.basis[n % self.basis_size] pred ^= int(addr['torus']['phi'] * 40.5) & 0xFF pred ^= (addr['mass'] % 256) pred ^= int(addr['surface']['theta'] * 40.5) & 0xFF # Residual from stream if position < len(stream): residual = stream[position] else: residual = 0 return pred ^ residual def compute_helicity(self, data: bytes) -> float: """ Equation 9: PIST "helicity" = correlation of address components. Analogous to gravitational helicity but for information coordinates. Measures how tightly the coordinate components are "knotted" together. High helicity = strong correlation = better prediction = lower entropy. """ # Compute pairwise correlations between address components k_vals = [] t_vals = [] mass_vals = [] tree_sum = [] for n in range(len(data)): k, t = self.pist_encode(n) mass = self.pist_mass(k, t) tree = [(n // (20 ** i)) % 20 for i in range(3)] k_vals.append(k) t_vals.append(t) mass_vals.append(mass) tree_sum.append(sum(tree)) # Correlation matrix corr_kt = np.corrcoef(k_vals, t_vals)[0, 1] if len(k_vals) > 1 else 0 corr_km = np.corrcoef(k_vals, mass_vals)[0, 1] if len(k_vals) > 1 else 0 corr_tm = np.corrcoef(t_vals, mass_vals)[0, 1] if len(t_vals) > 1 else 0 # "Helicity" = integrated correlation strength helicity = abs(corr_kt) + abs(corr_km) + abs(corr_tm) return helicity # ─── Demonstration ─── def demonstrate_frozen_in(): """Show that PIST coordinates preserve structure under dynamics.""" print("=" * 60) print("Frozen-In Gravity → PIST Analogy") print("Asenjo, Comisso & Winkler (PRL 2026)") print("=" * 60) # 1. Create PIST system pist = PISTFrozenIn(basis_size=16) # 2. Generate synthetic data np.random.seed(42) data = bytes(np.random.randint(0, 256, size=1000)) # 3. Show address invariance print("\n--- Coordinate Invariance Test ---") for n in [0, 10, 100, 500]: addr1 = pist.composite_address(n) addr2 = pist.composite_address(n) assert addr1['k'] == addr2['k'] assert addr1['t'] == addr2['t'] assert addr1['mass'] == addr2['mass'] print(f"n={n}: k={addr1['k']}, t={addr1['t']}, mass={addr1['mass']}") print(" ✓ Coordinates are deterministic (frozen-in)") # 4. Show reversibility print("\n--- Reversibility Test ---") # Encode: residual = data XOR prediction # Decode: data' = prediction XOR residual residuals = bytearray() for i, b in enumerate(data): pred = pist.basis[i % pist.basis_size] pred ^= (pist.pist_mirror(i) % 256) residual = b ^ pred residuals.append(residual) # Decode decoded = bytearray() for i in range(len(data)): pred = pist.basis[i % pist.basis_size] pred ^= (pist.pist_mirror(i) % 256) out = pred ^ residuals[i] decoded.append(out) assert bytes(decoded) == data print(f" ✓ Roundtrip verified for {len(data)} bytes") # 5. Compute helicity print("\n--- Information Helicity ---") helicity = pist.compute_helicity(data) print(f" PIST coordinate helicity: {helicity:.4f}") print(f" (Higher = stronger coordinate correlations = better compression)") print("\n" + "=" * 60) print("Key insight: Deterministic coordinates = frozen-in structure") print("No search needed. Topology is built into the encoding.") print("=" * 60) def pist_mirror(n: int) -> int: """Helper: PIST mirror operation.""" k = int(np.sqrt(n)) t = n - k * k if k == 0: return 0 return k * k + (2 * k + 1 - t) # Add mirror to class PISTFrozenIn.pist_mirror = staticmethod(pist_mirror) # ─── AngrySphinx Gear Law & FAMM-Coupled Gear Ratio ─── class FAMMRouteMemory: """ Frustration-Aligned Memory Management (FAMM). Records route outcomes as scars that bias future search. """ def __init__(self): self.scars = [] # List of (route_signature, outcome, load) self.torsion = 0.0 self.interlock = 0.0 self.phase_delta = 0.0 self.route_helicity = 0.0 def record_scar(self, route_sig: str, outcome: str, effort: float): """Record a route traversal outcome.""" self.scars.append({ 'route': route_sig, 'outcome': outcome, # 'success', 'failure', 'trap', 'partial' 'effort': effort, 'timestamp': len(self.scars) }) # Update FAMM load components self.torsion += effort * 0.1 self.interlock += 1.0 if outcome == 'trap' else 0.0 self.phase_delta += abs(hash(route_sig) % 100) / 100.0 def load(self) -> float: """ Eq: L_FAMM(t) = Sigma^2(t) + I_lock(t) + Delta_phi(t) """ return self.torsion**2 + self.interlock + self.phase_delta def load_frozen(self) -> float: """ Frozen-FAMM / topology-aware version: L_FAMM+(t) = Sigma^2 + I_lock + Delta_phi + H_route where H_route = preserved route-helicity / connectivity penalty. """ return self.load() + self.route_helicity def hostile_route_count(self) -> int: return sum(1 for s in self.scars if s['outcome'] in ('failure', 'trap')) def repeated_hostile_count(self, route_sig: str) -> int: return sum(1 for s in self.scars if s['route'] == route_sig and s['outcome'] in ('failure', 'trap')) class AngrySphinxShell: """ AngrySphinx Gear Law: O(t) = eta(t) * G_AS(t) * a(t) + F(t) + chi(t) where: a(t) = adversarial input effort G_AS = gear reduction / escalation multiplier eta = efficiency of cost transfer F(t) = FAMM route-scar load chi = cringe / semantic friction Gear ratio (FAMM-coupled escalation): G_AS(t) = 1 + alpha*L_FAMM(t) + beta*R(t) + gamma*U(t) where: L_FAMM = route-scar / frustration load R = repeated hostile route count U = uncertainty or unknown-route risk """ def __init__(self, alpha: float = 0.5, beta: float = 1.0, gamma: float = 2.0, delta: float = 0.3, theta_safe: float = 10.0): self.alpha = alpha self.beta = beta self.gamma = gamma self.delta = delta # H_route coupling self.theta_safe = theta_safe self.famm = FAMMRouteMemory() self.cringe_friction = 0.0 self.semantic_cost = 0.0 self.reality_cost = 0.0 self.constructive_cost = 0.0 def gear_ratio(self, adversarial_effort: float, route_sig: str) -> float: """ Compute the current gear ratio for a given adversarial input. """ L_famm = self.famm.load_frozen() R_repeat = self.famm.repeated_hostile_count(route_sig) U_unknown = 1.0 if self.famm.hostile_route_count() == 0 else 0.0 H_route = self.famm.route_helicity G_as = (1.0 + self.alpha * L_famm + self.beta * R_repeat + self.gamma * U_unknown + self.delta * H_route) return G_as def impose_obligation(self, adversarial_effort: float, route_sig: str) -> dict: """ Convert adversarial input into constructive obligation. C_out = G_AS * C_in + C_semantic + C_reality + C_constructive + C_cringe """ G_as = self.gear_ratio(adversarial_effort, route_sig) eta = 0.85 # efficiency of cost transfer O_compute = eta * G_as * adversarial_effort O_semantic = self.semantic_cost O_reality = self.reality_cost O_constructive = self.constructive_cost O_cringe = self.cringe_friction O_total = O_compute + O_semantic + O_reality + O_constructive + O_cringe # Update FAMM with this engagement self.famm.record_scar(route_sig, 'trap', adversarial_effort) return { 'gear_ratio': G_as, 'compute_obligation': O_compute, 'semantic_obligation': O_semantic, 'reality_obligation': O_reality, 'constructive_obligation': O_constructive, 'cringe_obligation': O_cringe, 'total_obligation': O_total } def is_defensive(self, payload_value: float, auth_recovery_cost: float) -> bool: """ Shell is economically defensive when: S_AS(t) = C_out - V_payload - C_auth > 0 or in full form: S_AS+(t) = C_compute + C_semantic + C_reality + C_constructive + C_cringe + lambda * L_FAMM+ - V_payload - C_auth """ # Use latest obligation as proxy for current state latest_effort = self.famm.scars[-1]['effort'] if self.famm.scars else 1.0 latest_route = self.famm.scars[-1]['route'] if self.famm.scars else 'default' obligation = self.impose_obligation(latest_effort, latest_route) S_as = obligation['total_obligation'] - payload_value - auth_recovery_cost return S_as > 0 def decision_rule(self, payload_value: float, auth_recovery_cost: float) -> str: """ Decision rule based on defensive score: S_AS+(t) > theta_safe => allow shell state to persist 0 < S_AS+(t) <= theta_safe => harden shell S_AS+(t) <= 0 => shell failing or underpriced """ if not self.famm.scars: return "neutral" latest_effort = self.famm.scars[-1]['effort'] latest_route = self.famm.scars[-1]['route'] obligation = self.impose_obligation(latest_effort, latest_route) S_as = obligation['total_obligation'] - payload_value - auth_recovery_cost if S_as > self.theta_safe: return "persist" elif S_as > 0: return "harden" else: return "underpriced" def demonstrate_gear_law(): """Demonstrate AngrySphinx gear reduction with FAMM escalation.""" print("=" * 60) print("AngrySphinx Gear Law — FAMM-Coupled Escalation") print("=" * 60) shell = AngrySphinxShell(alpha=0.5, beta=1.0, gamma=2.0) payload_value = 100.0 auth_cost = 5.0 # Simulate repeated hostile probes on the same route route = "brute_force_shell_0" efforts = [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0] print(f"\nPayload value: {payload_value}, Auth recovery cost: {auth_cost}") print(f"Hostile route: '{route}'") print(f"\n{'Probe':>6} {'Effort':>8} {'Gear G':>10} {'C_out':>10} {'S_AS':>10} {'Decision':>10}") print("-" * 60) for i, effort in enumerate(efforts): obligation = shell.impose_obligation(effort, route) S_as = obligation['total_obligation'] - payload_value - auth_cost decision = shell.decision_rule(payload_value, auth_cost) print(f"{i+1:>6} {effort:>8.2f} {obligation['gear_ratio']:>10.2f} " f"{obligation['total_obligation']:>10.2f} {S_as:>10.2f} {decision:>10}") # Show effect of switching to a novel route print(f"\n--- Novel route probe (high uncertainty load) ---") novel_route = "injection_vector_7" obligation = shell.impose_obligation(1.0, novel_route) S_as = obligation['total_obligation'] - payload_value - auth_cost print(f"Novel route '{novel_route}': G={obligation['gear_ratio']:.2f}, " f"C_out={obligation['total_obligation']:.2f}, S_AS={S_as:.2f}") print("\n" + "=" * 60) print("Gear reduction: cheap attacker speed → expensive defensive torque") print("FAMM scars ratchet the gear ratio up with each hostile engagement") print("=" * 60) if __name__ == "__main__": demonstrate_frozen_in() print("\n") demonstrate_gear_law()