feat: vendor midstream and sublinear-time-solver libraries

Add ruvnet/midstream (AIMDS real-time inference) and
ruvnet/sublinear-time-solver (sublinear optimization algorithms)
as vendored dependencies under vendor/.

Co-Authored-By: claude-flow <ruv@ruv.net>

vendor/sublinear-time-solver/plans/02-algorithms-implementation.md

@@ -0,0 +1,890 @@
+# Algorithm Implementation Plan - Sublinear Time Solver
+
+## Overview
+
+This document outlines the implementation strategy for three complementary algorithmic techniques in the sublinear-time solver: Neumann Series Expansion, Forward & Backward Push Methods, and Hybrid Random-Walk Estimation.
+
+## 1. Neumann Series Implementation
+
+### 1.1 Core Algorithm Structure
+
+```pseudocode
+function neumannSeries(M: Matrix, b: Vector, tolerance: f64) -> Vector:
+    // Pre-process: Ensure ||M|| < 1 for convergence
+    scaling_factor = 1.0 / spectralRadius(M)
+    M_scaled = M * scaling_factor
+    b_scaled = b * scaling_factor
+
+    result = b_scaled.clone()
+    power_term = b_scaled.clone()
+    residual_norm = inf
+    iteration = 0
+
+    while residual_norm > tolerance and iteration < MAX_ITERATIONS:
+        power_term = M_scaled * power_term
+        result += power_term
+
+        // Compute residual: ||b - (I - M)x||
+        residual = b_scaled - (power_term - M_scaled * result)
+        residual_norm = residual.l2_norm()
+
+        iteration += 1
+
+        // Early termination check
+        if power_term.l2_norm() < tolerance * 1e-3:
+            break
+
+    return result
+```
+
+### 1.2 Matrix Scaling Strategies
+
+**Spectral Radius Estimation:**
+- Power iteration method for dominant eigenvalue
+- Gershgorin circle theorem for bounds
+- Adaptive scaling based on matrix structure
+
+**Implementation Details:**
+```rust
+pub struct ScalingStrategy {
+    method: ScalingMethod,
+    max_iterations: usize,
+    tolerance: f64,
+}
+
+enum ScalingMethod {
+    SpectralRadius,
+    GershgorinBounds,
+    FrobeniusNorm,
+    Adaptive,
+}
+
+impl ScalingStrategy {
+    pub fn compute_scaling_factor(&self, matrix: &SparseMatrix) -> f64 {
+        match self.method {
+            ScalingMethod::SpectralRadius => self.power_iteration(matrix),
+            ScalingMethod::GershgorinBounds => self.gershgorin_estimate(matrix),
+            ScalingMethod::FrobeniusNorm => 1.0 / matrix.frobenius_norm(),
+            ScalingMethod::Adaptive => self.adaptive_scaling(matrix),
+        }
+    }
+}
+```
+
+### 1.3 Series Truncation Logic
+
+**Convergence Criteria:**
+1. **Residual-based:** `||r_k|| < tolerance`
+2. **Term magnitude:** `||M^k b|| < epsilon * ||b||`
+3. **Relative improvement:** `||x_{k+1} - x_k|| / ||x_k|| < delta`
+
+**Adaptive Truncation:**
+```pseudocode
+function adaptiveTruncation(term_sequence: Iterator<Vector>) -> usize:
+    terms = []
+    for (i, term) in term_sequence.enumerate():
+        terms.push(term)
+
+        if i >= 3:  // Need minimum terms for analysis
+            // Check for geometric decay
+            ratio = terms[i].norm() / terms[i-1].norm()
+            if ratio > 0.95:  // Poor convergence
+                return i
+
+            // Richardson extrapolation for acceleration
+            if i % 3 == 0:
+                extrapolated = richardsonExtrapolation(terms.last_n(3))
+                if extrapolated.converged():
+                    return i
+
+    return MAX_TERMS
+```
+
+### 1.4 Vectorized Operations Design
+
+**SIMD Optimization:**
+```rust
+use std::simd::{f64x4, Simd};
+
+pub fn vectorized_axpy(alpha: f64, x: &[f64], y: &mut [f64]) {
+    let alpha_vec = Simd::splat(alpha);
+
+    for (x_chunk, y_chunk) in x.chunks_exact(4).zip(y.chunks_exact_mut(4)) {
+        let x_vec = f64x4::from_slice(x_chunk);
+        let y_vec = f64x4::from_slice(y_chunk);
+        let result = alpha_vec * x_vec + y_vec;
+        y_chunk.copy_from_slice(result.as_array());
+    }
+}
+```
+
+### 1.5 Error Bound Calculations
+
+**A Posteriori Error Estimates:**
+```pseudocode
+function errorBounds(x_approx: Vector, M: Matrix, b: Vector) -> ErrorBounds:
+    residual = b - (I - M) * x_approx
+    residual_norm = residual.l2_norm()
+
+    // Condition number estimate
+    condition_estimate = estimateConditionNumber(I - M)
+
+    error_bound = ErrorBounds {
+        absolute: residual_norm / (1 - spectralRadius(M)),
+        relative: residual_norm / (condition_estimate * b.l2_norm()),
+        backward: residual_norm / b.l2_norm(),
+    }
+
+    return error_bound
+```
+
+## 2. Push Methods (Forward/Backward)
+
+### 2.1 Graph Representation for Push
+
+**Sparse Matrix Structure:**
+```rust
+pub struct PushGraph {
+    adjacency: CompressedSparseRow<f64>,
+    reverse_adjacency: CompressedSparseRow<f64>,  // For backward push
+    degrees: Vec<f64>,
+    reverse_degrees: Vec<f64>,
+}
+
+impl PushGraph {
+    pub fn from_matrix(matrix: &SparseMatrix) -> Self {
+        let adjacency = matrix.to_csr();
+        let reverse_adjacency = adjacency.transpose();
+
+        Self {
+            degrees: adjacency.row_sums(),
+            reverse_degrees: reverse_adjacency.row_sums(),
+            adjacency,
+            reverse_adjacency,
+        }
+    }
+}
+```
+
+### 2.2 Residual Vector Management
+
+**Forward Push Algorithm:**
+```pseudocode
+function forwardPush(graph: PushGraph, source: NodeId, alpha: f64, epsilon: f64) -> (Vector, Vector):
+    n = graph.num_nodes()
+    estimate = Vector::zeros(n)
+    residual = Vector::zeros(n)
+    residual[source] = 1.0
+
+    work_queue = PriorityQueue::new()
+    work_queue.push(source, residual[source])
+
+    while let Some((node, _)) = work_queue.pop():
+        if residual[node] < epsilon * graph.degrees[node]:
+            continue
+
+        push_amount = alpha * residual[node]
+        estimate[node] += push_amount
+        residual[node] -= push_amount
+
+        remaining = (1.0 - alpha) * residual[node]
+        residual[node] = 0.0
+
+        // Distribute to neighbors
+        for (neighbor, weight) in graph.adjacency.row(node):
+            delta = remaining * weight / graph.degrees[node]
+            residual[neighbor] += delta
+
+            if residual[neighbor] >= epsilon * graph.degrees[neighbor]:
+                work_queue.push(neighbor, residual[neighbor])
+
+    return (estimate, residual)
+```
+
+### 2.3 Work Queue Optimization
+
+**Priority-Based Processing:**
+```rust
+pub struct WorkQueue {
+    heap: BinaryHeap<WorkItem>,
+    in_queue: BitSet,
+    threshold: f64,
+}
+
+#[derive(PartialEq, PartialOrd)]
+struct WorkItem {
+    priority: OrderedFloat<f64>,
+    node_id: usize,
+}
+
+impl WorkQueue {
+    pub fn push_if_threshold(&mut self, node: usize, residual: f64, degree: f64) {
+        let priority = residual / degree;
+        if priority >= self.threshold && !self.in_queue.contains(node) {
+            self.heap.push(WorkItem {
+                priority: OrderedFloat(priority),
+                node_id: node,
+            });
+            self.in_queue.insert(node);
+        }
+    }
+
+    pub fn adaptive_threshold(&mut self, queue_size: usize) {
+        // Increase threshold if queue too large
+        if queue_size > MAX_QUEUE_SIZE {
+            self.threshold *= 1.1;
+        } else if queue_size < MIN_QUEUE_SIZE {
+            self.threshold *= 0.9;
+        }
+    }
+}
+```
+
+### 2.4 Single-Index vs Full-Solution Modes
+
+**Mode Selection Strategy:**
+```rust
+pub enum PushMode {
+    SingleSource { source: usize, target: Option<usize> },
+    MultiSource { sources: Vec<usize> },
+    FullSolution,
+}
+
+impl PushSolver {
+    pub fn solve(&self, mode: PushMode) -> SolutionResult {
+        match mode {
+            PushMode::SingleSource { source, target } => {
+                let (estimate, residual) = self.forward_push(source);
+                if let Some(t) = target {
+                    SolutionResult::SingleValue(estimate[t])
+                } else {
+                    SolutionResult::SparseVector(estimate)
+                }
+            },
+            PushMode::FullSolution => {
+                self.solve_all_sources()
+            },
+            PushMode::MultiSource { sources } => {
+                self.solve_multiple_sources(&sources)
+            }
+        }
+    }
+}
+```
+
+### 2.5 Visited Node Tracking
+
+**Efficient Set Operations:**
+```rust
+pub struct VisitedTracker {
+    visited: BitSet,
+    visit_order: Vec<usize>,
+    timestamps: Vec<u32>,
+    current_time: u32,
+}
+
+impl VisitedTracker {
+    pub fn mark_visited(&mut self, node: usize) -> bool {
+        if !self.visited.contains(node) {
+            self.visited.insert(node);
+            self.visit_order.push(node);
+            self.timestamps[node] = self.current_time;
+            true
+        } else {
+            false
+        }
+    }
+
+    pub fn reset_for_new_query(&mut self) {
+        self.current_time += 1;
+        if self.current_time == u32::MAX {
+            self.full_reset();
+        }
+    }
+}
+```
+
+## 3. Hybrid Random-Walk
+
+### 3.1 Random Walk Simulation Engine
+
+**Core Random Walk Implementation:**
+```pseudocode
+function randomWalk(graph: Graph, start: NodeId, max_steps: usize, restart_prob: f64) -> WalkResult:
+    current = start
+    steps = 0
+    path = [start]
+
+    rng = RandomGenerator::new()
+
+    while steps < max_steps:
+        if rng.uniform() < restart_prob:
+            return WalkResult::Restart(steps, path)
+
+        neighbors = graph.neighbors(current)
+        if neighbors.is_empty():
+            return WalkResult::Sink(steps, path)
+
+        // Weighted random selection
+        weights = graph.edge_weights(current)
+        next_node = weightedRandomChoice(neighbors, weights, rng)
+
+        current = next_node
+        path.push(current)
+        steps += 1
+
+    return WalkResult::MaxSteps(steps, path)
+```
+
+### 3.2 Sampling Strategies
+
+**Adaptive Sampling:**
+```rust
+pub struct AdaptiveSampler {
+    base_samples: usize,
+    variance_threshold: f64,
+    max_samples: usize,
+    confidence_level: f64,
+}
+
+impl AdaptiveSampler {
+    pub fn sample_until_converged<F>(&self, mut sampler: F) -> SamplingResult
+    where
+        F: FnMut() -> f64,
+    {
+        let mut samples = Vec::with_capacity(self.base_samples);
+        let mut sum = 0.0;
+        let mut sum_squares = 0.0;
+
+        // Initial batch
+        for _ in 0..self.base_samples {
+            let sample = sampler();
+            samples.push(sample);
+            sum += sample;
+            sum_squares += sample * sample;
+        }
+
+        loop {
+            let n = samples.len() as f64;
+            let mean = sum / n;
+            let variance = (sum_squares - sum * sum / n) / (n - 1.0);
+            let std_error = (variance / n).sqrt();
+
+            // Check convergence using confidence interval
+            let margin = self.confidence_level * std_error;
+            if margin / mean.abs() < self.variance_threshold || samples.len() >= self.max_samples {
+                break;
+            }
+
+            // Add more samples
+            let batch_size = (samples.len() / 4).max(10);
+            for _ in 0..batch_size {
+                let sample = sampler();
+                samples.push(sample);
+                sum += sample;
+                sum_squares += sample * sample;
+            }
+        }
+
+        SamplingResult {
+            estimate: sum / samples.len() as f64,
+            variance: sum_squares / samples.len() as f64 - (sum / samples.len() as f64).powi(2),
+            num_samples: samples.len(),
+        }
+    }
+}
+```
+
+### 3.3 Push-Walk Coordination
+
+**Hybrid Strategy:**
+```pseudocode
+function hybridSolver(graph: Graph, source: NodeId, target: NodeId, epsilon: f64) -> f64:
+    // Phase 1: Forward push to reduce problem size
+    (push_estimate, residual) = forwardPush(graph, source, alpha=0.2, epsilon)
+
+    // Phase 2: Random walks from high-residual nodes
+    high_residual_nodes = residual.nonzero_indices_above(epsilon)
+
+    walk_contribution = 0.0
+    for node in high_residual_nodes:
+        // Estimate transition probability from node to target
+        prob_estimate = estimateTransitionProbability(graph, node, target, residual[node])
+        walk_contribution += prob_estimate
+
+    // Phase 3: Backward push from target (if beneficial)
+    if shouldUseBackwardPush(graph, target, high_residual_nodes):
+        (backward_estimate, _) = backwardPush(graph, target, alpha=0.2, epsilon)
+
+        // Combine estimates using residual weights
+        combined = combineBidirectionalEstimates(push_estimate, walk_contribution, backward_estimate, residual)
+        return combined
+
+    return push_estimate[target] + walk_contribution
+```
+
+### 3.4 Bidirectional Exploration
+
+**Meet-in-the-Middle Strategy:**
+```rust
+pub struct BidirectionalWalker {
+    forward_frontier: HashMap<usize, f64>,
+    backward_frontier: HashMap<usize, f64>,
+    meeting_probability: f64,
+}
+
+impl BidirectionalWalker {
+    pub fn explore(&mut self, graph: &Graph, source: usize, target: usize) -> f64 {
+        let forward_steps = self.forward_walk(graph, source);
+        let backward_steps = self.backward_walk(graph, target);
+
+        // Find intersection points
+        let mut total_probability = 0.0;
+        for (&node, &forward_prob) in &self.forward_frontier {
+            if let Some(&backward_prob) = self.backward_frontier.get(&node) {
+                total_probability += forward_prob * backward_prob;
+            }
+        }
+
+        total_probability
+    }
+
+    fn should_meet(&self, forward_depth: usize, backward_depth: usize) -> bool {
+        // Use graph diameter estimate to decide when frontiers should meet
+        forward_depth + backward_depth >= self.estimated_diameter()
+    }
+}
+```
+
+### 3.5 Stochastic Error Estimates
+
+**Confidence Intervals:**
+```pseudocode
+function computeConfidenceInterval(samples: Vec<f64>, confidence: f64) -> (f64, f64):
+    n = samples.len()
+    mean = samples.mean()
+    variance = samples.variance()
+    std_error = sqrt(variance / n)
+
+    // Use t-distribution for small samples, normal for large
+    if n < 30:
+        t_value = tDistributionQuantile(confidence, n - 1)
+        margin = t_value * std_error
+    else:
+        z_value = normalQuantile(confidence)
+        margin = z_value * std_error
+
+    return (mean - margin, mean + margin)
+```
+
+## 4. Algorithm Selection Logic
+
+### 4.1 Condition Number Estimation
+
+**Fast Condition Number Bounds:**
+```rust
+pub fn estimate_condition_number(matrix: &SparseMatrix) -> f64 {
+    // Use power iteration for largest eigenvalue
+    let lambda_max = power_iteration(matrix, 50);
+
+    // Use inverse power iteration for smallest eigenvalue
+    let lambda_min = inverse_power_iteration(matrix, 50);
+
+    lambda_max / lambda_min.abs()
+}
+
+pub fn power_iteration(matrix: &SparseMatrix, max_iter: usize) -> f64 {
+    let n = matrix.nrows();
+    let mut v = vec![1.0 / (n as f64).sqrt(); n];
+    let mut lambda = 0.0;
+
+    for _ in 0..max_iter {
+        let w = matrix * &v;
+        lambda = v.dot(&w);
+        let norm = w.l2_norm();
+        v = w / norm;
+    }
+
+    lambda
+}
+```
+
+### 4.2 Method Auto-Selection Heuristics
+
+**Decision Tree:**
+```rust
+pub enum SolverMethod {
+    NeumannSeries,
+    ForwardPush,
+    BackwardPush,
+    HybridRandomWalk,
+    DirectSolver,
+}
+
+pub struct MethodSelector {
+    matrix_analyzer: MatrixAnalyzer,
+    performance_history: PerformanceTracker,
+}
+
+impl MethodSelector {
+    pub fn select_method(&self, problem: &LinearSystemProblem) -> SolverMethod {
+        let analysis = self.matrix_analyzer.analyze(&problem.matrix);
+
+        // Decision criteria
+        if analysis.condition_number < 10.0 {
+            return SolverMethod::DirectSolver;
+        }
+
+        if analysis.sparsity > 0.99 && problem.query_type == QueryType::SingleEntry {
+            return SolverMethod::ForwardPush;
+        }
+
+        if analysis.spectral_radius < 0.5 {
+            return SolverMethod::NeumannSeries;
+        }
+
+        if problem.precision_requirement < 1e-6 {
+            return SolverMethod::HybridRandomWalk;
+        }
+
+        // Default to adaptive hybrid
+        SolverMethod::HybridRandomWalk
+    }
+}
+```
+
+### 4.3 Adaptive Switching During Solve
+
+**Dynamic Method Switching:**
+```pseudocode
+function adaptiveSolve(problem: LinearSystemProblem) -> Solution:
+    current_method = selectInitialMethod(problem)
+    solution_state = SolutionState::new()
+
+    while not solution_state.converged():
+        progress = executeMethod(current_method, problem, solution_state)
+
+        if progress.stagnated():
+            // Switch to different method
+            candidates = alternativeMethods(current_method, problem)
+            current_method = selectBestCandidate(candidates, solution_state)
+
+        if progress.error_increased():
+            // Fallback to more stable method
+            current_method = conservativeFallback(problem)
+
+        updateSolutionState(solution_state, progress)
+
+    return solution_state.extract_solution()
+```
+
+### 4.4 Performance Profiling Hooks
+
+**Profiling Framework:**
+```rust
+pub struct PerformanceProfiler {
+    timers: HashMap<String, Instant>,
+    counters: HashMap<String, u64>,
+    memory_tracker: MemoryTracker,
+}
+
+impl PerformanceProfiler {
+    pub fn profile<F, R>(&mut self, name: &str, f: F) -> R
+    where
+        F: FnOnce() -> R,
+    {
+        let start = Instant::now();
+        let start_memory = self.memory_tracker.current_usage();
+
+        let result = f();
+
+        let duration = start.elapsed();
+        let memory_delta = self.memory_tracker.current_usage() - start_memory;
+
+        self.record_timing(name, duration);
+        self.record_memory(name, memory_delta);
+
+        result
+    }
+}
+```
+
+## 5. Numerical Stability
+
+### 5.1 Precision Management (f32/f64)
+
+**Mixed Precision Strategy:**
+```rust
+pub enum PrecisionMode {
+    Single,    // f32
+    Double,    // f64
+    Mixed,     // f32 for bulk operations, f64 for critical computations
+    Adaptive,  // Switch based on condition number
+}
+
+pub struct PrecisionManager {
+    mode: PrecisionMode,
+    promotion_threshold: f64,
+    demotion_threshold: f64,
+}
+
+impl PrecisionManager {
+    pub fn should_promote(&self, condition_number: f64) -> bool {
+        condition_number > self.promotion_threshold
+    }
+
+    pub fn execute_with_precision<F, R>(&self, cond_num: f64, f: F) -> R
+    where
+        F: FnOnce(PrecisionLevel) -> R,
+    {
+        let precision = if self.should_promote(cond_num) {
+            PrecisionLevel::Double
+        } else {
+            PrecisionLevel::Single
+        };
+
+        f(precision)
+    }
+}
+```
+
+### 5.2 Overflow/Underflow Handling
+
+**Safe Arithmetic Operations:**
+```rust
+pub trait SafeArithmetic {
+    fn safe_add(&self, other: &Self) -> Result<Self, ArithmeticError>
+    where
+        Self: Sized;
+
+    fn safe_multiply(&self, other: &Self) -> Result<Self, ArithmeticError>
+    where
+        Self: Sized;
+}
+
+impl SafeArithmetic for f64 {
+    fn safe_add(&self, other: &f64) -> Result<f64, ArithmeticError> {
+        let result = self + other;
+        if result.is_infinite() {
+            Err(ArithmeticError::Overflow)
+        } else if result == 0.0 && (*self != 0.0 || *other != 0.0) {
+            Err(ArithmeticError::Underflow)
+        } else {
+            Ok(result)
+        }
+    }
+}
+```
+
+### 5.3 Ill-Conditioned System Detection
+
+**Early Warning System:**
+```pseudocode
+function detectIllConditioning(matrix: Matrix) -> ConditioningReport:
+    // Quick checks
+    diagonal_dominance = checkDiagonalDominance(matrix)
+    if diagonal_dominance < 0.1:
+        return ConditioningReport::PoorlyConditioned
+
+    // Eigenvalue spread estimation
+    eigenvalue_ratio = estimateEigenvalueRatio(matrix)
+    if eigenvalue_ratio > 1e12:
+        return ConditioningReport::IllConditioned
+
+    // Numerical rank estimation
+    rank_deficiency = estimateRankDeficiency(matrix)
+    if rank_deficiency > 0:
+        return ConditioningReport::RankDeficient
+
+    return ConditioningReport::WellConditioned
+```
+
+### 5.4 Residual Computation Strategies
+
+**High-Precision Residual:**
+```rust
+pub fn compute_residual_extended_precision(
+    matrix: &SparseMatrix<f64>,
+    solution: &Vector<f64>,
+    rhs: &Vector<f64>
+) -> Vector<f64> {
+    // Use extended precision for critical computation
+    let matrix_ext: SparseMatrix<f128> = matrix.cast();
+    let solution_ext: Vector<f128> = solution.cast();
+    let rhs_ext: Vector<f128> = rhs.cast();
+
+    let residual_ext = &rhs_ext - &matrix_ext * &solution_ext;
+
+    // Cast back to working precision
+    residual_ext.cast()
+}
+```
+
+## 6. Incremental Updates
+
+### 6.1 Delta Cost Propagation
+
+**Incremental Matrix Updates:**
+```pseudocode
+function incrementalUpdate(solver_state: SolverState, delta_matrix: SparseMatrix) -> SolverState:
+    // Sherman-Morrison-Woodbury formula for low-rank updates
+    if delta_matrix.rank() <= MAX_RANK_UPDATE:
+        return shermanMorrisonUpdate(solver_state, delta_matrix)
+
+    // Incremental push updates for localized changes
+    affected_nodes = delta_matrix.nonzero_pattern()
+    if affected_nodes.len() < solver_state.num_nodes() * 0.1:
+        return localizedPushUpdate(solver_state, delta_matrix, affected_nodes)
+
+    // Full recomputation for major changes
+    return fullRecompute(solver_state.matrix + delta_matrix)
+```
+
+### 6.2 Partial Recomputation Logic
+
+**Smart Invalidation:**
+```rust
+pub struct IncrementalSolver {
+    cached_solutions: HashMap<QueryKey, CachedSolution>,
+    dependency_graph: DependencyGraph,
+    invalidation_frontier: BitSet,
+}
+
+impl IncrementalSolver {
+    pub fn update_matrix(&mut self, changes: &MatrixDelta) {
+        let affected_queries = self.dependency_graph.find_dependent_queries(changes);
+
+        for query in affected_queries {
+            self.invalidate_cached_solution(query);
+            self.invalidation_frontier.insert(query.node_id);
+        }
+
+        // Propagate invalidation using push-based approach
+        self.propagate_invalidation();
+    }
+
+    fn propagate_invalidation(&mut self) {
+        while let Some(node) = self.invalidation_frontier.pop_first() {
+            let dependents = self.dependency_graph.dependents(node);
+            for dependent in dependents {
+                if self.should_invalidate(dependent, node) {
+                    self.invalidation_frontier.insert(dependent);
+                }
+            }
+        }
+    }
+}
+```
+
+### 6.3 State Caching Mechanisms
+
+**Multi-Level Cache:**
+```rust
+pub struct SolutionCache {
+    l1_cache: LruCache<QueryKey, Vector<f64>>,  // Recent exact solutions
+    l2_cache: LruCache<QueryKey, ApproximateSolution>,  // Approximate solutions
+    l3_cache: PersistentCache<QueryKey, CompressedSolution>,  // Compressed historical data
+}
+
+impl SolutionCache {
+    pub fn get_cached_solution(&self, query: &QueryKey) -> Option<CachedSolution> {
+        // Check L1 first
+        if let Some(exact) = self.l1_cache.get(query) {
+            return Some(CachedSolution::Exact(exact.clone()));
+        }
+
+        // Check L2 for approximation
+        if let Some(approx) = self.l2_cache.get(query) {
+            if approx.meets_tolerance(query.tolerance) {
+                return Some(CachedSolution::Approximate(approx.clone()));
+            }
+        }
+
+        // Check L3 for warm start
+        if let Some(compressed) = self.l3_cache.get(query) {
+            let warm_start = compressed.decompress();
+            return Some(CachedSolution::WarmStart(warm_start));
+        }
+
+        None
+    }
+}
+```
+
+### 6.4 Update Verification
+
+**Correctness Checking:**
+```pseudocode
+function verifyIncrementalUpdate(
+    original_solution: Vector,
+    updated_solution: Vector,
+    matrix_delta: SparseMatrix,
+    tolerance: f64
+) -> VerificationResult:
+    // Check solution validity
+    original_residual = computeResidual(original_matrix, original_solution)
+    updated_residual = computeResidual(updated_matrix, updated_solution)
+
+    if updated_residual.norm() > original_residual.norm() * 1.1:
+        return VerificationResult::ResidualIncreased
+
+    // Check incremental consistency
+    expected_change = estimateExpectedChange(matrix_delta, original_solution)
+    actual_change = updated_solution - original_solution
+
+    relative_error = (actual_change - expected_change).norm() / expected_change.norm()
+
+    if relative_error < tolerance:
+        return VerificationResult::Verified
+    else:
+        return VerificationResult::SuspiciousChange(relative_error)
+```
+
+## Complexity Analysis
+
+### Time Complexity Summary
+
+| Algorithm | Single Query | Full Solution | Space |
+|-----------|--------------|---------------|-------|
+| Neumann Series | O(k·nnz) | O(k·n²) | O(n) |
+| Forward Push | O(1/ε) | O(n/ε) | O(n) |
+| Backward Push | O(1/ε) | O(n/ε) | O(n) |
+| Hybrid Random-Walk | O(√n/ε) | O(n√n/ε) | O(√n) |
+
+Where:
+- `k` = number of series terms
+- `nnz` = number of non-zeros
+- `ε` = accuracy parameter
+- `n` = matrix dimension
+
+### Space-Time Tradeoffs
+
+**Memory-Efficient Mode:**
+- Streaming computation for large matrices
+- On-demand residual computation
+- Compressed intermediate storage
+
+**Speed-Optimized Mode:**
+- Full matrix precomputation
+- Aggressive caching
+- Parallel execution of multiple queries
+
+## Implementation Priority
+
+1. **Phase 1:** Core Neumann Series (Week 1-2)
+2. **Phase 2:** Forward Push Method (Week 3-4)
+3. **Phase 3:** Random Walk Engine (Week 5-6)
+4. **Phase 4:** Algorithm Selection & Hybrid Methods (Week 7-8)
+5. **Phase 5:** Numerical Stability & Incremental Updates (Week 9-10)
+6. **Phase 6:** Performance Optimization & Benchmarking (Week 11-12)
+
+## Testing Strategy
+
+- Unit tests for each algorithm component
+- Integration tests for method selection
+- Property-based testing for numerical stability
+- Benchmarking against reference implementations
+- Stress testing with ill-conditioned matrices
+- Performance regression testing
+
+This implementation plan provides a comprehensive roadmap for building a robust, efficient, and numerically stable sublinear-time linear system solver.