CIv6-SBD Solution Proposal: Geometric-Topological Structural Break Detection

CIv6-SBD Solution Proposal: Geometric-Topological Structural Break Detection

🚧 Updated Implementation Plan (Aligned with ADIA Challenge + Topological Reasoning Engine)

🧠 Overview

This solution expands the original proposal by integrating concrete code from a working topological reasoning notebook. It aligns directly with the ADIA Lab Structural Break Detection challenge and operationalizes the CIv6 hypothesis through latent geometry and attention topology.

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1. Preprocessing and ECA Encoding

Input

• X_train: Pandas DataFrame, indexed by id, with value and period columns.
• y_train: Boolean labels indicating structural breaks.
• X_test: List of test DataFrames.

Transformation Pipeline

1. Symbolic Encoding

   • Convert each time series segment (pre/post boundary) into symbolic binary strings.
   • Use delta-sign encoding or permutation-based symbolic embedding.

2. ECA Evolution

   • Apply Elementary Cellular Automata (e.g. Rule 110) to each binary sequence.
   • Run for 32–64 time steps.
   • Flatten 2D evolution into token-like input strings.

# Example: run_eca function
RULE_MAPS = {
    110: {f"{i:03b}": b for i, b in enumerate(f"{110:08b}"[::-1])}
}

def run_eca(initial_bits, rule=110, steps=32):
    rule_map = RULE_MAPS[rule]
    current = ''.join(initial_bits)
    history = [current]
    for _ in range(steps):
        padded = '0' + current + '0'
        next_state = ''.join(rule_map[padded[i:i+3]] for i in range(len(current)))
        history.append(next_state)
        current = next_state
    return history

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2. Model: Topological Attention Analyzer

Model Architecture

• Use a Transformer model capable of outputting attentions and hidden embeddings.

  • Pretrained options: GPT-2, Qwen2.5, or TransformerECA (if symbolically trained).

Core Modules

1. Semantic Loop Monitor

def extract_cycles_and_log_wilson(head_idx, attn_matrix, tokens, threshold=0.05):
    import networkx as nx
    import numpy as np
    A = attn_matrix[head_idx].cpu().numpy()
    idx = {t: i for i, t in enumerate(tokens)}
    G = nx.DiGraph()
    for i in range(len(tokens)):
        for j in range(len(tokens)):
            if A[i, j] > threshold:
                G.add_edge(tokens[i], tokens[j], weight=A[i, j])
    cycles = [c for c in nx.simple_cycles(G) if 3 <= len(c) <= 6]

    def log_wilson_loop(cycle):
        return sum(np.log(A[idx[s], idx[t]] + 1e-12) for s, t in zip(cycle, cycle[1:] + cycle[:1]))

    return [(cycle, log_wilson_loop(cycle)) for cycle in cycles]

1. Holonomy Analyzer

def compute_holonomy_spectrum(cycle, Q, K, token_idx):
    import torch, numpy as np
    H = torch.eye(Q.shape[-1])
    for s, t in zip(cycle, cycle[1:] + cycle[:1]):
        i, j = token_idx[s], token_idx[t]
        qi = Q[i].unsqueeze(1)
        kj = K[j].unsqueeze(0)
        transport = qi @ kj
        H = transport @ H
    eigvals = torch.linalg.eigvals(H).cpu().numpy()
    return eigvals

1. Topological Divergence Detector

   • Compare:

     • Loop energy stats (mean, max, count)
     • Eigenvalue spectrum: dispersion, real-imag range
     • Use statistical distance (e.g., cosine/Mahalanobis) to assess change

2. Entropy Divergence Tracker

   • Use attention matrices to compute entropy per token:

def compute_attention_entropy(attn_matrix):
    import scipy.stats
    entropy_per_head = []
    for head_attn in attn_matrix:
        probs = head_attn / head_attn.sum(axis=-1, keepdims=True)
        entropy = scipy.stats.entropy(probs, axis=-1)
        entropy_per_head.append(entropy.mean())
    return entropy_per_head

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3. Detection Logic

Break Score Computation

• Define break score as:

  • Drop in mean loop energy
  • Spectral divergence (eigenvalue distance)
  • Entropy jump or FIM curvature collapse

• Can be a trained classifier or a rule-based scoring function.

Output

• ROC-AUC compatible structural break scores ∈ [0, 1]

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4. Evaluation Protocol (ADIA-Aligned)

1. Apply the full pipeline to each training ID.
2. Extract topological metrics pre/post.
3. Train a shallow classifier or compute thresholded break score.
4. Use y_train for supervised evaluation.
5. Apply trained detector to X_test (submission-ready).

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🔁 CIv6 System View

Time Series
  └▶ Symbolic Encoding (delta/permutation)
       └▶ ECA Evolution (Rule 110)
            └▶ Transformer Attention Probing
                  ├▶ Loop Energy Analyzer
                  ├▶ Holonomy + Curvature Spectrum
                  ├▶ Entropy/FIM Divergence
                  └▶ Structural Break Scoring

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🧪 Implementation Modules to Build

• run_eca() → Evolve bit sequences
• extract_cycles_and_log_wilson() → Detect closed attention paths
• compute_holonomy_spectrum() → Eigenvalues from Q/K loop transport
• compare_pre_post_topology() → Vector of topological metrics
• structural_break_score() → Final ROC-compatible scalar

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✅ Ready for Prototyping

This proposal is now concretely aligned with:

• The ADIA Lab challenge schema
• Transformer latent state observables
• Geometric-topological signal tracking
• CIv6 cybernetic consistency modeling

It is fully modular and ready to be integrated into the ADIA baseline notebook.