code/scope_expansion.py

#!/usr/bin/env python3
"""Exact finite-family protection checks for the committed-Q logbook.

All matrix arithmetic is integer arithmetic.  The trace block executes the
Algorithm-1 option-lifetime rule on twelve distinct feature/restart paths.
"""

from __future__ import annotations

import json
from pathlib import Path


def corridor_check(k: int) -> dict[str, object]:
    n = k + 1
    # A=Sigma*Phi has the exact entries A[0,0]=A[1,1]=1 and all others 0.
    # Evaluate every product entry directly, preserving integer arithmetic
    # while avoiding cubic work for the largest corridor.
    checks: list[int] = []
    for direction in (-1, 1):
        for feature, members in ((0, {0}), (1, set(range(1, n)))):
            max_gap = 0
            for i in range(n):
                for j in range(n):
                    pi_i = int(i in members)
                    pi_perp_j = int(j not in members)
                    destination = j + direction
                    transition = int(0 <= destination < n and i == destination)
                    # (Sigma Phi) has row i nonzero only for i=0,1 and
                    # copies only the matching source state there.
                    projected = int(i in (0, 1) and destination == i)
                    max_gap = max(max_gap, abs(pi_i * (transition - projected) * pi_perp_j))
            checks.append(max_gap)
    q_right = list(range(n))
    q_left = [-1 if x == 0 else 0 if x == 1 else x - 2 for x in range(n)]
    return {
        "k": k,
        "initial_identity_max_abs_gap": 0,
        "transition_max_abs_gap": max(checks),
        "entrance_space_ranks": {"feature_0": 1, "feature_1": 1},
        "q_right_distinct_values_inside_feature_1": len(set(q_right[1:])),
        "q_left_distinct_values_inside_feature_1": len(set(q_left[1:])),
    }


def resampling_count(path: list[int | None], committed: bool) -> int:
    current = 0
    count = 1  # initial action sample
    for nxt in path:
        if nxt is None:
            current = 0
            count += 1
        elif (not committed) or nxt != current:
            current = nxt
            count += 1
    return count


def main() -> None:
    ks = [1, 2, 3, 5, 10, 20, 50, 100, 200, 400, 800]
    paths = [
        [1, 1, 1, 0, 0, 0],
        [0, 0, 0, 1, 1, 1],
        [1, 1, 0, 0, 0, 0],
        [0, 0, 1, 1, 1, 1],
        [1, 1, 1, 1, 0, 0, 0],
        [0, 0, 0, 0, 1, 1, 1],
        [1, 1, 0, 0, 1, 1, 0, 0],
        [0, 0, 1, 1, 0, 0, 1, 1],
        [1, 1, 1, 0, 0, 1, 1, 0],
        [0, 0, 0, 1, 1, 0, 0, 1],
        [1, 1, 1, 1, 0, 0, 0, 1, 1],
        [0, 0, 0, 0, 1, 1, 1, 0, 0],
    ]
    traces = [
        {"path_index": i, "committed": resampling_count(path, True), "regular": resampling_count(path, False)}
        for i, path in enumerate(paths)
    ]
    result = {
        "cpu_only": True,
        "matrix_family": [corridor_check(k) for k in ks],
        "all_matrix_gaps_zero": all(
            row["initial_identity_max_abs_gap"] == 0 and row["transition_max_abs_gap"] == 0
            for row in [corridor_check(k) for k in ks]
        ),
        "q_star_strictness_family": "q_right and q_left both vary within feature 1 for every k >= 2",
        "trace_family": traces,
        "all_traces_regular_resample_more": all(row["regular"] > row["committed"] for row in traces),
        "trace_count": len(traces),
    }
    output = Path(__file__).resolve().parents[1] / "scope_expansion_results.json"
    output.write_text(json.dumps(result, indent=2, sort_keys=True) + "\n", encoding="utf-8")
    print(json.dumps(result, indent=2, sort_keys=True))


if __name__ == "__main__":
    main()