Blame for research/code/test_positive_design.py

#!/usr/bin/env python3
"""AËQIRION III: positive probe design and expressive-resource bounds.

All algebraic inequalities, tensor contractions, Walsh identities, gain-graph
ranks, and support-cover checks use exact Fraction/integer arithmetic. Optional
NumPy eigensolver experiments illustrate the proven spectral formulas; their
floating-point results are explicitly separated from exact verification.
"""
from __future__ import annotations

from fractions import Fraction as F
from itertools import product
import json
from math import sqrt
from pathlib import Path
import random


def parity(x):
    return 1 if x.bit_count() % 2 == 0 else -1


def cube_edges(n):
    return [(x, x ^ (1 << i)) for x in range(1 << n)
            for i in range(n) if x < (x ^ (1 << i))]


def dot(a, b):
    return sum((x * y for x, y in zip(a, b)), F(0))


def tensor(factors):
    out = [F(1)]
    for factor in factors:
        out = [a * b for a in out for b in factor]
    return out


def apply_h0(n, x):
    out = [n * a for a in x]
    for u, v in cube_edges(n):
        out[u] += x[v]
        out[v] += x[u]
    return out


def exact_walsh_checks():
    count = 0
    for n in range(1, 9):
        nv = 1 << n
        for mask in range(nv):
            w = [F((-1) ** ((x & mask).bit_count())) for x in range(nv)]
            actual = apply_h0(n, w)
            eigenvalue = 2 * (n - mask.bit_count())
            assert actual == [eigenvalue * a for a in w]
            count += 1
        chi = [F(parity(x)) for x in range(nv)]
        ones = [F(1)] * nv
        indicator = [F(parity(x) == 1) for x in range(nv)]
        # Unit-energy parity mask b=indicator/sqrt(N/2).  Its rank-one update
        # bb^T is rational, so its invariant 2D action is checked exactly.
        def augmented(x):
            update = dot(indicator, x) / (F(nv, 2))
            return [a + update * b for a, b in zip(apply_h0(n, x), indicator)]
        assert augmented(chi) == [(a + b) / 2 for a, b in zip(chi, ones)]
        assert augmented(ones) == [(a + b) / 2 + 2 * n * b
                                    for a, b in zip(chi, ones)]
        # Compression matrix [[1/2,1/2],[1/2,2n+1/2]] has trace2n+1, detn.
        assert F(1, 2) * (2 * n + F(1, 2)) - F(1, 4) == n
        assert dot(indicator, chi) ** 2 == F(nv, 2) * dot(indicator, indicator)
        count += 4
    return count


def exact_resource_checks():
    rng = random.Random(1372026)
    general = rank_one = mixtures = blocks = support = 0
    sharp_examples = 0
    for n in range(2, 9):
        nv = 1 << n
        chi = [F(parity(x)) for x in range(nv)]
        for _ in range(20):
            b = [F(rng.randrange(0, 10), rng.randrange(1, 7)) for _ in range(nv)]
            assert dot(b, chi) ** 2 * 2 <= nv * dot(b, b)
            general += 1
            factors = [[F(rng.randrange(0, 10), rng.randrange(1, 6))
                        for _ in range(2)] for _ in range(n)]
            v = tensor(factors)
            assert dot(v, chi) ** 2 <= dot(v, v)
            rank_one += 1
        for rank in range(1, 6):
            for _ in range(8):
                terms = [tensor([[F(rng.randrange(0, 8), rng.randrange(1, 5))
                                  for _ in range(2)] for _ in range(n)])
                         for _ in range(rank)]
                b = [sum((term[x] for term in terms), F(0)) for x in range(nv)]
                assert dot(b, chi) ** 2 <= rank * dot(b, b)
                assert sum((dot(v, v) for v in terms), F(0)) <= dot(b, b)
                mixtures += 1
        # Any partition into k blocks, with a nonnegative factor per block,
        # has normalized parity overlap squared at most 2^-k.
        for k in range(1, n + 1):
            sizes = [1] * k
            for j in range(n - k):
                sizes[j % k] += 1
            for _ in range(8):
                factors = [[F(rng.randrange(0, 10), rng.randrange(1, 7))
                            for _ in range(1 << size)] for size in sizes]
                b = tensor(factors)
                assert dot(b, chi) ** 2 * (1 << k) <= nv * dot(b, b)
                blocks += 1
        # Exact sharpness of the overlap rank bound for R<=N/2: equal point
        # masses at R even vertices form a nonnegative CP-rank-R tensor.
        even = [x for x in range(nv) if parity(x) == 1]
        for rank in (1, min(3, len(even)), len(even)):
            b = [F(x in even[:rank]) for x in range(nv)]
            assert dot(b, chi) ** 2 == rank * dot(b, b)
            sharp_examples += 1
        for size in (1, 2, nv // 2):
            indices = rng.sample(range(nv), size)
            b = [F(rng.randrange(1, 8) if x in indices else 0) for x in range(nv)]
            assert dot(b, chi) ** 2 <= size * dot(b, b)
            support += 1
    return {"nonnegative_overlap": general, "coordinate_product_overlap": rank_one,
            "nonnegative_CP_mixture_overlap": mixtures,
            "block_product_overlap": blocks, "support_size_overlap": support,
            "rank_bound_sharp_overlap_examples": sharp_examples}


def exact_rectangle_checks():
    checks = 0
    for n in range(1, 9):
        # 0 fixes the bit to0, 1 to1, and2 allows both; each is a nonempty
        # support of one nonnegative one-coordinate factor.
        for pattern in product((0, 1, 2), repeat=n):
            free = pattern.count(2)
            if free:
                # Flipping one free coordinate changes parity within support.
                a = [0 if p == 2 else p for p in pattern]
                b = list(a)
                b[pattern.index(2)] = 1
                assert sum(a) % 2 != sum(b) % 2
            else:
                assert 2 ** free == 1
            checks += 1
    return checks


def matrix_rank(a):
    a = [list(map(F, row)) for row in a]
    if not a:
        return 0
    rank = 0
    for col in range(len(a[0])):
        pivot = next((j for j in range(rank, len(a)) if a[j][col]), None)
        if pivot is None:
            continue
        a[rank], a[pivot] = a[pivot], a[rank]
        value = a[rank][col]
        a[rank] = [v / value for v in a[rank]]
        for j in range(len(a)):
            if j != rank and a[j][col]:
                value = a[j][col]
                a[j] = [x - value * y for x, y in zip(a[j], a[rank])]
        rank += 1
        if rank == len(a):
            break
    return rank


def weighted_balanced(n, edges):
    values = [None] * n
    values[0] = F(1)
    changed = True
    balanced = True
    while changed:
        changed = False
        for u, v, alpha in edges:
            if values[u] is not None:
                proposed = -values[u] / alpha
                if values[v] is None:
                    values[v] = proposed
                    changed = True
                elif values[v] != proposed:
                    balanced = False
            elif values[v] is not None:
                values[u] = -alpha * values[v]
                changed = True
    assert all(x is not None for x in values)
    return balanced


def exact_gain_graph_checks():
    rng = random.Random(2262026)
    count = 0
    for n in range(2, 11):
        for rep in range(18):
            edges = [(i, i + 1, F(rng.randrange(1, 5), rng.randrange(1, 5)))
                     for i in range(n - 1)]
            for _ in range(rep % 4):
                u, v = rng.sample(range(n), 2)
                edges.append((u, v, F(rng.randrange(1, 5), rng.randrange(1, 5))))
            rows = []
            for u, v, alpha in edges:
                row = [F(0)] * n
                row[u], row[v] = F(1), alpha
                rows.append(row)
            assert matrix_rank(rows) == n - int(weighted_balanced(n, edges))
            count += 1
    return count


def numerical_spectral_checks():
    try:
        import numpy as np
    except ImportError:
        return {"status": "SKIPPED", "reason": "NumPy is optional for spectral illustrations"}
    rng = np.random.default_rng(12202026)
    cases = mixtures = pair_cases = 0
    curve = []
    worst_excess = 0.0
    for n in range(1, 8):
        nv = 1 << n
        q = n * np.eye(nv)
        for u, v in cube_edges(n):
            q[u, v] = q[v, u] = 1
        signs = np.array([parity(x) for x in range(nv)], dtype=float)
        even = (signs == 1).astype(float)
        odd = (signs == -1).astype(float)
        b_opt = even / np.linalg.norm(even)
        optimum = (2 * n) / (2 * n + 1 + sqrt(4 * n * n + 1))
        got = float(np.linalg.eigvalsh(q + np.outer(b_opt, b_opt))[0])
        assert abs(got - optimum) < 2e-11
        for _ in range(18):
            b = rng.uniform(0, 1, size=nv)
            b /= np.linalg.norm(b)
            val = float(np.linalg.eigvalsh(q + np.outer(b, b))[0])
            assert val <= optimum + 2e-11
            worst_excess = max(worst_excess, val - optimum)
            cases += 1
        # Total energy1, two positive rows: each parity indicator/sqrt(N).
        h = q + (np.outer(even, even) + np.outer(odd, odd)) / nv
        assert abs(float(np.linalg.eigvalsh(h)[0]) - 0.5) < 2e-11
        pair_cases += 1
        if n >= 2:
            for rank in (1, 2, 3):
                vectors = []
                for _ in range(2):
                    terms = []
                    for _ in range(rank):
                        v = np.array([1.0])
                        for _ in range(n):
                            v = np.kron(v, rng.uniform(0, 1, size=2))
                        terms.append(v)
                    b = sum(terms)
                    b /= np.linalg.norm(b)
                    vectors.append(b)
                h = q + sum(np.outer(b, b) for b in vectors)
                val = float(np.linalg.eigvalsh(h)[0])
                assert val <= 2 * rank / nv + 2e-11
                mixtures += 1
        curve.append({"dimension": n, "vertices": nv,
                      "optimal_one_positive_row": optimum,
                      "two_positive_rows_total_energy_one": 0.5,
                      "single_product_row_upper_bound": 1 / nv,
                      "eigensolver_parity_value": got})
    return {"status": "PASS", "arithmetic": "NumPy float64 symmetric eigensolver",
            "random_single_row_cases": cases, "CP_resource_spectral_cases": mixtures,
            "two_row_optimal_frame_cases": pair_cases,
            "maximum_observed_excess_over_optimum": worst_excess, "curve": curve}


def main():
    return {"status": "PASS", "exact_arithmetic": "integer/Fraction",
            "exact_walsh_and_optimal_invariant_space_checks": exact_walsh_checks(),
            "exact_resource_checks": exact_resource_checks(),
            "exact_parity_rectangle_support_checks": exact_rectangle_checks(),
            "exact_weighted_gain_graph_rank_checks": exact_gain_graph_checks(),
            "spectral_illustrations": numerical_spectral_checks(),
            "scope": ["Proofs, not random search, establish the universal optimum.",
                      "The unscaled baseline is all binary coordinate-fiber edge sums.",
                      "Total probe energy is squared Euclidean row norm, not runtime.",
                      "One dense mask may require exponentially many state accesses.",
                      "CP rank means nonnegative coordinatewise tensor rank."]}


if __name__ == '__main__':
    result = main()
    print(json.dumps(result, indent=2))