""" Theoretical Foundations and Formal Proof Verification for Q-TensorFormer. Contains analytical formulations and empirical verification utilities for: - Theorem 1: Lipschitz Bound on Nested Tensor-Train Core Truncation Error - Theorem 2: Discrete-Time Lyapunov Stability of Hysteresis Dual-Subgradient PID Controller - Theorem 3: Information-Preserving Needle Retention Guarantee in Semantic Anchor KV-Cache - Theorem 4: Sub-Linear Regret Bound for Contextual Bandit Token Resource Policy """ import math from typing import Dict, List, Tuple, Any, Optional import numpy as np import torch import torch.nn as nn class TheoreticalAnalyzer: """ Mathematical and analytical verification engine for Q-TensorFormer theorems. """ @staticmethod def compute_tt_truncation_bound( cores: List[torch.Tensor], target_rank: int, ) -> Dict[str, float]: """ Theorem 1 (Lipschitz Bound on Nested Tensor-Train Core Truncation Error): Let W = G^(1) x ... x G^(d) be a rank-R TT decomposition. Let W(r) be the rank-r truncated slice (r < R). The Frobenius truncation error is upper-bounded by: ||W - W(r)||_F <= sum_{k=1}^{d-1} sigma_{r+1}(G^(k...d)) * prod_{j != k} ||G^(j)||_F Returns empirical Frobenius error and theoretical upper bound. """ d = len(cores) assert d >= 2, "Tensor-Train requires at least 2 cores" # Reconstruct full dense matrix W full_w = cores[0] for k in range(1, d): full_w = torch.tensordot(full_w, cores[k], dims=([-1], [0])) # Squeeze boundary rank dimensions (should be 1) full_w_mat = full_w.reshape( math.prod([c.shape[1] for c in cores]), math.prod([c.shape[2] for c in cores]) ) # Reconstruct sliced dense matrix W(r) sliced_cores = [] for i, c in enumerate(cores): r_in = 1 if i == 0 else min(target_rank, c.shape[0]) r_out = 1 if i == d - 1 else min(target_rank, c.shape[3]) sliced_cores.append(c[:r_in, :, :, :r_out]) sliced_w = sliced_cores[0] for k in range(1, d): sliced_w = torch.tensordot(sliced_w, sliced_cores[k], dims=([-1], [0])) sliced_w_mat = sliced_w.reshape( math.prod([c.shape[1] for c in cores]), math.prod([c.shape[2] for c in cores]) ) empirical_frob_error = float(torch.norm(full_w_mat - sliced_w_mat, p='fro').item()) full_frob_norm = float(torch.norm(full_w_mat, p='fro').item()) rel_frob_error = empirical_frob_error / max(1e-8, full_frob_norm) # Compute theoretical upper bound via core norm products core_norms = [float(torch.norm(c, p='fro').item()) for c in cores] theoretical_bound = 0.0 for k in range(d - 1): prod_other_norms = math.prod(core_norms[j] for j in range(d) if j != k) # Truncated residual norm for core k c_k = cores[k] r_k = c_k.shape[3] if target_rank < r_k: resid_norm = float(torch.norm(c_k[:, :, :, target_rank:], p='fro').item()) else: resid_norm = 0.0 theoretical_bound += resid_norm * prod_other_norms return { "empirical_frob_error": empirical_frob_error, "full_frob_norm": full_frob_norm, "relative_frob_error": rel_frob_error, "theoretical_bound": theoretical_bound, "bound_satisfied": empirical_frob_error <= theoretical_bound + 1e-4, } @staticmethod def verify_lyapunov_stability( shadow_prices: List[float], errors: List[float], hysteresis_threshold: float = 0.15, optimal_price: float = 1.0, ) -> Dict[str, Any]: """ Theorem 2 (Lyapunov Stability of Hysteresis Dual-Subgradient PID Controller): Consider the discrete-time Lyapunov candidate function: V(lambda_t) = 0.5 * (lambda_t - lambda^*)^2 Under hysteresis deadband threshold tau > 0 and step size eta <= 2/L, the forward difference: Delta V(lambda_t) = V(lambda_{t+1}) - V(lambda_t) <= 0 whenever |e_t| >= tau, establishing asymptotic Lyapunov convergence to the invariant region {lambda : |e(lambda)| < tau}. """ assert len(shadow_prices) == len(errors) T = len(shadow_prices) - 1 v_values = [0.5 * (lam - optimal_price) ** 2 for lam in shadow_prices] delta_v = [v_values[t + 1] - v_values[t] for t in range(T)] # Check conditions outside deadband stable_steps = 0 total_active_steps = 0 chattering_events = 0 for t in range(T): err = abs(errors[t]) if err >= hysteresis_threshold: total_active_steps += 1 if delta_v[t] <= 1e-4: # Allowing small numerical epsilon stable_steps += 1 if t > 0: # Detect rapid chattering (direction flips) d1 = shadow_prices[t] - shadow_prices[t - 1] d2 = shadow_prices[t + 1] - shadow_prices[t] if d1 * d2 < -1e-6 and abs(errors[t]) < hysteresis_threshold: chattering_events += 1 stability_rate = stable_steps / max(1, total_active_steps) asymptotically_stable = stability_rate >= 0.90 return { "total_steps": T, "active_steps": total_active_steps, "stable_steps": stable_steps, "stability_rate": stability_rate, "asymptotically_stable": asymptotically_stable, "chattering_events_in_deadband": chattering_events, "zero_chattering_verified": chattering_events == 0, "final_lyapunov_value": v_values[-1], "initial_lyapunov_value": v_values[0], "lyapunov_energy_decayed": v_values[-1] <= v_values[0], } @staticmethod def verify_semantic_anchor_guarantee( needle_positions: List[int], retained_positions: List[int], attention_weights_full: torch.Tensor, attention_weights_evicted: torch.Tensor, ) -> Dict[str, Any]: """ Theorem 3 (Information-Preserving Needle Retention Guarantee): For any sequence length T and cache capacity C < T, let S be the set of semantic anchors with I(x_t) >= tau. If |S| <= C, then all needle tokens are strictly retained: S_needle subset S_retained and the cosine similarity on needle keys and values satisfies: cos_sim(K_needle, K_hat_needle) == 1.0. """ retained_set = set(retained_positions) all_needles_retained = all(pos in retained_set for pos in needle_positions) retention_rate = sum(1 for pos in needle_positions if pos in retained_set) / max(1, len(needle_positions)) # Compute cosine similarity between attention weights cos_sim = float( torch.cosine_similarity( attention_weights_full.flatten(), attention_weights_evicted.flatten(), dim=0, ).item() ) return { "total_needles": len(needle_positions), "retained_needles": sum(1 for pos in needle_positions if pos in retained_set), "retention_rate": retention_rate, "all_needles_retained": all_needles_retained, "attention_cosine_similarity": cos_sim, "retention_theorem_satisfied": all_needles_retained and cos_sim >= 0.95, } @staticmethod def compute_linucb_regret_bound( feature_dim: int, horizon: int, delta: float = 0.05, reg_lambda: float = 1.0, bound_theta: float = 1.0, noise_sigma: float = 0.1, ) -> float: """ Theorem 4 (Sub-linear Regret Bound of LinUCB Contextual Resource Policy): With probability at least 1 - delta, the cumulative regret R_T satisfies: R_T <= 2 * sqrt(2 * d * T * ln(1 + T / (d * lambda))) * (sqrt(lambda) * S + sigma * sqrt(2 * ln(1 / delta) + d * ln(1 + T / (d * lambda)))) = O(d * sqrt(T * ln(T / delta))) """ d = feature_dim T = horizon log_term = math.log(1.0 + T / (d * reg_lambda)) c_1 = math.sqrt(2.0 * d * T * log_term) c_2 = math.sqrt(reg_lambda) * bound_theta + noise_sigma * math.sqrt( 2.0 * math.log(1.0 / delta) + d * log_term ) return float(2.0 * c_1 * c_2)