# jam1_markov_wall.py — instrument for THE FIRST JAM, line 5.
# Question on the sheet (Grok, line 4): could survival statistics at
# extreme depth depend on the historical sequence of branch events,
# re-admitting biography through the back door?
# Test: is the entire child-residue sequence of a doubling ray a pure
# function of a FINITE residue of its base? If yes, the tree unfolds
# from a finite deterministic automaton on residues: conditioned on
# the current state, the past carries zero further information at any
# depth, and the Markov wall stands.
#
# Ray of odd base m (m not divisible by 3): values m*2^k. Branch
# points are exactly the k of one fixed parity where m*2^k == 4 mod 6;
# there the odd child is c = (m*2^k - 1)/3, odd automatically.

NCHILD = 8  # children examined per ray

def children_mod(m, mod, nchild=NCHILD):
    k = 2 if m % 3 == 1 else 1  # parity class making m*2^k == 1 mod 3
    out = []
    while len(out) < nchild:
        v = m << k
        assert v % 6 == 4, (m, k, v)
        c = (v - 1) // 3
        assert c % 2 == 1, (m, k, c)
        out.append(c % mod)
        k += 2
    return tuple(out)

def minimal_base_modulus(limit, child_mod=18):
    """Smallest tested modulus M with: m mod M determines the whole
    child sequence mod child_mod."""
    for M in (6, 18, 54, 162, 486):
        table = {}
        ok = True
        for m in range(1, limit, 2):
            if m % 3 == 0:
                continue
            s = children_mod(m, child_mod)
            key = m % M
            if key in table:
                if table[key] != s:
                    ok = False
                    break
            else:
                table[key] = s
        if ok:
            return M, table
    return None, None

def two_step_memory_test(limit, M):
    """Grok's statistic, directly: does the type (fertile/sterile) of
    child i+1 depend on the types of children i AND i-1 beyond what
    child i's full state already fixes? With deterministic sequences
    this must come out degenerate; we measure rather than assume."""
    from collections import defaultdict
    one = defaultdict(set)   # state_i           -> next type
    two = defaultdict(set)   # (state_i, type_{i-1}) -> next type
    for m in range(1, limit, 2):
        if m % 3 == 0:
            continue
        seq = children_mod(m, M)
        types = ['F' if c % 3 else 'S' for c in seq]
        for i in range(1, len(seq) - 1):
            one[seq[i]].add(types[i + 1])
            two[(seq[i], types[i - 1])].add(types[i + 1])
    branching_one = max(len(v) for v in one.values())
    branching_two = max(len(v) for v in two.values())
    return branching_one, branching_two

if __name__ == "__main__":
    LIMIT = 2_000_000
    M, table = minimal_base_modulus(LIMIT)
    print(f"bases checked: odd, non-multiples of 3, up to {LIMIT}")
    print(f"minimal modulus M with m mod M -> full child sequence "
          f"mod 18: {M}")
    if M:
        n_classes = len(table)
        print(f"distinct base classes: {n_classes}; each maps to "
              f"exactly ONE child-residue sequence (zero exceptions)")
        b1, b2 = two_step_memory_test(LIMIT, 54)
        print(f"next-type outcomes given current state alone: "
              f"max {b1} (1 = deterministic)")
        print(f"next-type outcomes given current state + previous "
              f"type: max {b2} (no gain possible below {b1})")
        print("VERDICT: the ray's future is a pure function of a "
              "finite residue of its base. The tree is generated by "
              "a finite deterministic automaton on residue classes; "
              "conditioned on the present state, history at ANY "
              "depth adds nothing. The Markov wall stands; the back "
              "door is closed by determinism, not by probability.")