Datasets:

lylybig8
/

mdm_coverage

#!/usr/bin/env python3
"""E3: numerical fixed-oracle optimum over randomized fair-pair policies.
The LP uses occupation measures, not an average of per-policy TV values.
See docs/policy_lp_reduction.md for the scope and symmetry reduction.
"""
from __future__ import annotations

import argparse
from collections import defaultdict
from dataclasses import dataclass
from functools import lru_cache
from itertools import combinations, combinations_with_replacement, product
import json
import math
from pathlib import Path
import time

import numpy as np
from scipy.optimize import linprog
from scipy.sparse import coo_matrix, csr_matrix, hstack, vstack
from scipy.special import logsumexp
from scipy.stats import binom


@dataclass(frozen=True)
class Oracle:
    probabilities: tuple[float, ...]
    weights: tuple[float, ...]

    def __post_init__(self):
        if len(self.probabilities) != len(self.weights):
            raise ValueError("Component lengths differ")
        if not all(0 < p < 1 for p in self.probabilities):
            raise ValueError("Positive component probabilities are required")
        if not all(w > 0 for w in self.weights):
            raise ValueError("Positive mixture weights are required")
        if abs(sum(self.weights) - 1) > 1e-12:
            raise ValueError("Mixture weights must sum to one")

    @lru_cache(maxsize=None)
    def predictive(self, c: int, m: int) -> float:
        p = np.asarray(self.probabilities)
        logw = np.log(self.weights) + m * np.log(p) + (c - m) * np.log1p(-p)
        posterior = np.exp(logw - logsumexp(logw))
        return float(posterior @ p)

    def predictive_direct(self, c: int, m: int) -> float:
        """Independent small-n evaluation used in recovered-policy checks."""
        w = [a * p**m * (1 - p)**(c - m)
             for a, p in zip(self.weights, self.probabilities)]
        return sum(a * p for a, p in zip(w, self.probabilities)) / sum(w)

    def count_statistics(self, n: int, tau: float) -> dict:
        m = np.arange(n + 1)
        teacher = binom.pmf(m, n, tau)
        log_likelihoods = np.asarray([
            np.log(a) + m * np.log(p / tau)
            + (n - m) * np.log((1 - p) / (1 - tau))
            for a, p in zip(self.weights, self.probabilities)
        ])
        log_ratio = logsumexp(log_likelihoods, axis=0)
        model = sum(a * binom.pmf(m, n, p)
                    for a, p in zip(self.weights, self.probabilities))
        return {
            "actual_joint_kl_P_to_R": float(-teacher @ log_ratio),
            "actual_joint_tv_P_R": float(np.abs(teacher - model).sum() / 2),
            "initial_predictive_drift": self.predictive(0, 0) - tau,
        }


def aggregate_actions(n, state, final):
    c, _, o = state
    f = n - c - o
    if final:
        return [(o, f, 0)]
    return [(u, v, b) for u in range(o + 1)
            for v in range(f + 1) for b in range(f - v + 1)]


def aggregate_transition(oracle, state, action):
    c, m, o = state
    u, v, b = action
    p = oracle.predictive(c, m)
    law = np.convolve(binom.pmf(np.arange(u + 1), u, p),
                      binom.pmf(np.arange(v + 1), v, 0.5))
    return {(c + u + v, m + j, o - u + b): float(prob)
            for j, prob in enumerate(law) if prob > 0}


def direct_aggregate_transition(oracle, state, action):
    """Recompute without the cached PMF/convolution used in LP assembly."""
    c, m, o = state
    u, v, b = action
    p = oracle.predictive_direct(c, m)
    result = defaultdict(float)
    for x in range(u + 1):
        px = math.comb(u, x) * p**x * (1 - p)**(u - x)
        for y in range(v + 1):
            py = math.comb(v, y) * 0.5**v
            result[(c + u + v, m + x + y, o - u + b)] += px * py
    return dict(result)


def occupation_lp(initial, rounds, actions, transition, terminal_index,
                  target, time_limit, action_limit):
    """Generic finite layered MDP LP, also used for literal-coordinate checks."""
    start = time.perf_counter()
    layers = [{initial}]
    records = []
    by_state = defaultdict(list)
    for t in range(rounds):
        following = set()
        for state in sorted(layers[t]):
            for action in actions(state, t == rounds - 1):
                kernel = transition(state, action)
                residual = abs(sum(kernel.values()) - 1)
                if residual > 2e-12 or min(kernel.values()) < 0:
                    raise ArithmeticError(f"Invalid transition: {residual}")
                j = len(records)
                if j >= action_limit:
                    raise RuntimeError(f"State-action guard {action_limit} exceeded")
                records.append((t, state, action, kernel))
                by_state[(t, state)].append(j)
                following.update(kernel)
        layers.append(following)

    na, nt = len(records), len(target)
    rows = {(t, s): i for i, (t, s) in enumerate(
        (t, s) for t in range(rounds) for s in sorted(layers[t]))}
    er, ec, ev, tr, tc, tv = [], [], [], [], [], []
    for j, (t, state, action, kernel) in enumerate(records):
        er.append(rows[(t, state)]); ec.append(j); ev.append(1.0)
        for dest, prob in kernel.items():
            if t + 1 < rounds:
                er.append(rows[(t + 1, dest)]); ec.append(j); ev.append(-prob)
            else:
                tr.append(terminal_index(dest)); tc.append(j); tv.append(prob)
    flow = coo_matrix((ev, (er, ec)), shape=(len(rows), na)).tocsr()
    terminal = coo_matrix((tv, (tr, tc)), shape=(nt, na)).tocsr()
    aeq = hstack([flow, csr_matrix((len(rows), nt))], format="csr")
    beq = np.zeros(len(rows)); beq[rows[(0, initial)]] = 1.0
    ident = csr_matrix(np.eye(nt))
    aub = vstack([hstack([terminal, -ident]),
                  hstack([-terminal, -ident])], format="csr")
    bub = np.concatenate([target, -target])
    cost = np.concatenate([np.zeros(na), np.full(nt, 0.5)])
    assembly_seconds = time.perf_counter() - start
    solution = linprog(cost, A_ub=aub, b_ub=bub, A_eq=aeq, b_eq=beq,
                       bounds=(0, None), method="highs",
                       options={"time_limit": float(time_limit),
                                "threads": 1,
                                "primal_feasibility_tolerance": 1e-9,
                                "dual_feasibility_tolerance": 1e-9})
    elapsed = time.perf_counter() - start
    if not solution.success:
        raise RuntimeError(f"LP did not reach optimum: {solution.message}")
    x = solution.x[:na]
    q = np.asarray(terminal @ x).ravel()
    dual_value = float(beq @ solution.eqlin.marginals
                       + bub @ solution.ineqlin.marginals)
    dual_slack = cost - aeq.T @ solution.eqlin.marginals - aub.T @ solution.ineqlin.marginals
    diagnostics = {
        "solver_status": int(solution.status), "solver_message": solution.message,
        "state_counts_by_layer": [len(s) for s in layers],
        "action_occupancy_variables": na, "lp_variables": na + nt,
        "flow_constraints": len(rows), "assembly_seconds": assembly_seconds,
        "total_seconds": elapsed, "solver_objective": float(solution.fun),
        "objective_recomputed_from_terminal": float(np.abs(q - target).sum() / 2),
        "max_flow_residual": float(np.max(np.abs(aeq @ solution.x - beq))),
        "max_l1_constraint_violation": float(max(0, np.max(aub @ solution.x - bub))),
        "minimum_variable": float(np.min(solution.x)),
        "terminal_mass_residual": float(abs(q.sum() - 1)),
        "dual_objective": dual_value,
        "primal_dual_gap": float(solution.fun - dual_value),
        "max_dual_slack_violation": float(max(0, -np.min(dual_slack))),
        "certificate_type": "floating-point diagnostics, not a rigorous exact certificate",
    }
    # SciPy's <=-constraint marginals are nonpositive.  The upper block is
    # q-d<=P and the lower block is -q-d<=-P, so the terminal test is high
    # where q>P.  Its certificate orientation is min_Q E_Q f - E_P f.
    mu = solution.ineqlin.marginals
    terminal_test = np.clip(0.5 - mu[:nt] + mu[nt:], 0.0, 1.0)
    return {"records": records, "by_state": by_state, "occupancies": x,
            "terminal": q, "terminal_test": terminal_test,
            "diagnostics": diagnostics}


def recover_and_evaluate(result, oracle, n, rounds):
    policies = {}
    serialized = []
    records, occupancies = result["records"], result["occupancies"]
    for (t, state), indices in result["by_state"].items():
        raw = np.asarray([max(0.0, occupancies[j]) for j in indices])
        total = float(raw.sum())
        if total > 0:
            weights = raw / total
        else:
            weights = np.zeros(len(indices)); weights[0] = 1.0
        policy = [(records[j][2], float(a)) for j, a in zip(indices, weights) if a > 0]
        policies[(t, state)] = policy
        if total > 0:
            serialized.append({"round": t + 1, "state_c_m_o": list(state),
                               "state_occupancy": total,
                               "actions": [{"u_v_b": list(action), "probability": a}
                                           for action, a in policy]})
    mass = {(0, 0, 0): 1.0}
    stage_mass_residuals = []
    for t in range(rounds):
        following = defaultdict(float)
        for state, state_mass in mass.items():
            for action, action_prob in policies[(t, state)]:
                for dest, transition_prob in direct_aggregate_transition(oracle, state, action).items():
                    following[dest] += state_mass * action_prob * transition_prob
        mass = dict(following)
        stage_mass_residuals.append(abs(sum(mass.values()) - 1))
    q = np.zeros(n + 1)
    for (c, m, o), prob in mass.items():
        if c != n or o != 0:
            raise AssertionError("Recovered policy leaves unassigned coordinates")
        q[m] += prob
    diagnostics = {
        "recovered_terminal_max_difference": float(np.max(np.abs(q - result["terminal"]))),
        "recovered_stage_mass_max_residual": max(stage_mass_residuals),
        "recovered_terminal": q.tolist(),
    }
    return serialized, diagnostics


def schedule_terminal(oracle, n, cumulative_completions):
    # Round 1 reveals one endpoint of every pair; subsequent rounds complete
    # uniformly selected open pairs. Hidden pair randomization is essential.
    mass = {(0, 0, n): 1.0}
    completed = 0
    for cumulative in cumulative_completions:
        batch = cumulative - completed
        following = defaultdict(float)
        for state, prob in mass.items():
            for dest, conditional in direct_aggregate_transition(oracle, state, (batch, 0, 0)).items():
                following[dest] += prob * conditional
        mass = dict(following)
        completed = cumulative
    q = np.zeros(n + 1)
    for (c, m, o), prob in mass.items():
        if c != n or o != 0:
            raise AssertionError("Incomplete baseline schedule")
        q[m] += prob
    return q


def schedule_baselines(oracle, n, rounds, tau):
    r = rounds - 1
    teacher = binom.pmf(np.arange(n + 1), n, tau)
    equal = tuple(math.ceil(n * j / r) for j in range(1, r + 1))
    geometric = tuple(min(n, math.floor(n ** (j / r) + 1e-12))
                      for j in range(1, r + 1))

    def evaluate(schedule):
        q = schedule_terminal(oracle, n, schedule)
        return float(np.abs(q - teacher).sum() / 2)

    all_grid = [tuple(prefix) + (n,) for prefix in
                combinations_with_replacement(range(n + 1), r - 1)]
    grid_values = [(evaluate(schedule), schedule) for schedule in all_grid]
    best_tv, best_schedule = min(grid_values)
    return {
        "equal": {"tv": evaluate(equal), "cumulative_completions": list(equal)},
        "geometric_heuristic": {"tv": evaluate(geometric),
                                "cumulative_completions": list(geometric)},
        "best_fixed_completion_grid": {
            "tv": best_tv, "cumulative_completions": list(best_schedule),
            "number_of_schedules": len(all_grid),
            "grid": [{"tv": tv, "cumulative_completions": list(schedule)}
                     for tv, schedule in grid_values],
        },
    }


def literal_problem(n, rounds, oracle, tau, time_limit, action_limit):
    """Independent full-coordinate finite MDP; no c,m,o reduction in assembly."""
    complete = list(product((0, 1), repeat=2 * n))
    law = {}
    target = []
    for z in complete:
        mismatch = sum(z[2 * i] != z[2 * i + 1] for i in range(n))
        law[z] = 2.0**(-n) * sum(a * p**mismatch * (1 - p)**(n - mismatch)
                                for a, p in zip(oracle.weights, oracle.probabilities))
        target.append(2.0**(-n) * tau**mismatch * (1 - tau)**(n - mismatch))
    output_index = {z: i for i, z in enumerate(complete)}

    @lru_cache(maxsize=None)
    def literal_heads(state):
        compatible = [(z, p) for z, p in law.items()
                      if all(x == -1 or x == y for x, y in zip(state, z))]
        denominator = sum(p for _, p in compatible)
        return tuple(sum(p * z[j] for z, p in compatible) / denominator
                     for j in range(2 * n))

    def actions(state, final):
        hidden = tuple(j for j, x in enumerate(state) if x == -1)
        if final:
            return [hidden]
        return [subset for size in range(len(hidden) + 1)
                for subset in combinations(hidden, size)]

    @lru_cache(maxsize=None)
    def transition(state, action):
        heads = literal_heads(state)
        kernel = {}
        for outcomes in product((0, 1), repeat=len(action)):
            dest = list(state)
            prob = 1.0
            for j, bit in zip(action, outcomes):
                dest[j] = bit
                prob *= heads[j] if bit else 1 - heads[j]
            kernel[tuple(dest)] = prob
        return kernel

    result = occupation_lp((-1,) * (2 * n), rounds, actions, transition,
                           output_index.__getitem__, np.asarray(target),
                           time_limit, action_limit)
    return result["diagnostics"]


def solve_aggregate(n, rounds, oracle, tau, time_limit, action_limit):
    target = binom.pmf(np.arange(n + 1), n, tau)
    result = occupation_lp((0, 0, 0), rounds,
                           lambda s, final: aggregate_actions(n, s, final),
                           lambda s, a: aggregate_transition(oracle, s, a),
                           lambda s: s[1], target, time_limit, action_limit)
    policy, check = recover_and_evaluate(result, oracle, n, rounds)
    result["diagnostics"].update(check)
    return result, policy


def bellman_terminal_test(result, oracle, n, rounds, tau):
    """Independently rebuild transition probabilities in a MIN Bellman pass.
    This floating-point equality check is not the rigorous certificate; the
    separate rational certifier receives only the test and exact oracle.
    """
    test = result["terminal_test"]
    value = {(n, m, 0): float(test[m]) for m in range(n + 1)}
    for t in reversed(range(rounds)):
        previous = {}
        for (stage, state), indices in result["by_state"].items():
            if stage != t:
                continue
            best = math.inf
            for j in indices:
                action = result["records"][j][2]
                kernel = direct_aggregate_transition(oracle, state, action)
                expectation = math.fsum(p * value[dest] for dest, p in kernel.items())
                best = min(best, expectation)
            previous[state] = best
        value = previous
    target = binom.pmf(np.arange(n + 1), n, tau)
    lower = value[(0, 0, 0)] - float(target @ test)
    gap = lower - result["diagnostics"]["solver_objective"]
    if abs(gap) > 3e-7:
        raise AssertionError(f"Terminal test Bellman/LP mismatch: {gap}")
    return {"floating_bellman_lower": lower,
            "floating_bellman_lp_difference": gap,
            "terminal_test_orientation": "min_Q E_Q[f] - E_P[f]"}


def assert_diagnostics(d):
    for key in ("max_flow_residual", "max_l1_constraint_violation",
                "terminal_mass_residual", "max_dual_slack_violation"):
        if d[key] > 2e-7:
            raise AssertionError(f"{key}={d[key]} exceeds tolerance")
    if abs(d["primal_dual_gap"]) > 2e-7 or d["minimum_variable"] < -2e-8:
        raise AssertionError("Primal/dual or nonnegativity diagnostic failed")
    if abs(d["solver_objective"] - d["objective_recomputed_from_terminal"]) > 2e-7:
        raise AssertionError("Solver objective disagrees with terminal-law TV")
    if d.get("recovered_terminal_max_difference", 0) > 2e-7:
        raise AssertionError("Recovered policy disagrees with occupation solution")


def randomization_regression(time_limit, action_limit):
    oracle = Oracle((0.125,), (1.0,))
    result, policy = solve_aggregate(1, 2, oracle, 0.25, time_limit, action_limit)
    literal = literal_problem(1, 2, oracle, 0.25, time_limit, action_limit)
    assert_diagnostics(result["diagnostics"])
    assert_diagnostics(literal)
    serial = sum(x * action[0] for x, (_, _, action, _) in
                 zip(result["occupancies"], result["records"]))
    cosample = sum(x * action[1] for x, (_, _, action, _) in
                   zip(result["occupancies"], result["records"]))
    if abs(serial - 2 / 3) > 1e-7 or abs(cosample - 1 / 3) > 1e-7:
        raise AssertionError("Expected randomized-policy mixture was not recovered")
    if max(result["diagnostics"]["solver_objective"], literal["solver_objective"]) > 1e-8:
        raise AssertionError("Randomized marginal-TV regression failed")
    return {"status": "PASS", "serial_probability": serial,
            "cosample_probability": cosample, "aggregate": result["diagnostics"],
            "literal": literal, "policy": policy}


def atomic_json(path, obj):
    temporary = path.with_suffix(path.suffix + ".tmp")
    temporary.write_text(json.dumps(obj, indent=2, allow_nan=False) + "\n")
    temporary.replace(path)


def run(config, output):
    cfg = config["e3"]
    if output.suffix == ".json":
        output_file = output
        output_dir = output.parent
        policy_dir = output_dir / (output.stem + "_policies")
    else:
        output_dir = output
        output_file = output_dir / "e3_results.json"
        policy_dir = output_dir / "policies"
    output_dir.mkdir(parents=True, exist_ok=True)
    policy_dir.mkdir(exist_ok=True)
    n_values = cfg.get("n_values", [2, 4, 6, 8])
    rounds_values = cfg.get("rounds", [2, 3, 4])
    tau, w, delta = (float(cfg.get(k, v)) for k, v in
                     [("tau", 0.25), ("w", 0.125), ("delta", 0.25)])
    lam = float(cfg.get("ladder_lambda", 0.2))
    time_limit = min(float(cfg.get("time_limit_seconds", 120)), 120)
    action_limit = min(int(cfg.get("state_action_limit", 250000)), 250000)
    literal_n = set(cfg.get("literal_n_values", [1, 2]))
    if min(n_values) < 1 or min(rounds_values) < 2 or max(rounds_values) > 8:
        raise ValueError("Require n >= 1 and 2 <= K <= 8")
    models = {
        "single": Oracle((tau, tau + delta), (1 - w, w)),
        "two_level": Oracle((tau, tau + delta, tau + delta * (w / 2) * lam),
                            (1 - w, w / 2, w / 2)),
    }
    result = {
        "experiment": "E3 fixed-oracle randomized-policy occupation LP",
        "scope": "inf_policy TV for each fixed R; not a KL-ball supremum or a minimax certificate",
        "arithmetic": "floating-point HiGHS with independent literal-coordinate and forward checks",
        "config": cfg, "status": "running", "cases": [], "errors": [],
        "randomization_regression": randomization_regression(time_limit, action_limit),
    }
    atomic_json(output_file, result)
    print("E3 randomization regression: PASS", flush=True)
    for name, oracle in models.items():
        for n in n_values:
            previous = None
            for rounds in sorted(rounds_values):
                case_id = f"{name}_n{n}_K{rounds}"
                try:
                    solved, policy = solve_aggregate(n, rounds, oracle, tau, time_limit, action_limit)
                    diagnostics = solved["diagnostics"]
                    assert_diagnostics(diagnostics)
                    baselines = schedule_baselines(oracle, n, rounds, tau)
                    optimum = diagnostics["objective_recomputed_from_terminal"]
                    if optimum > min(b["tv"] for b in baselines.values()) + 2e-7:
                        raise AssertionError("LP optimum is worse than a feasible baseline")
                    if previous is not None and optimum > previous + 2e-7:
                        raise AssertionError("Optimal error increased when adding a round")
                    previous = optimum
                    literal = None
                    if n in literal_n:
                        literal = literal_problem(n, rounds, oracle, tau, time_limit, action_limit)
                        assert_diagnostics(literal)
                        difference = abs(literal["objective_recomputed_from_terminal"] - optimum)
                        literal["aggregate_objective_difference"] = difference
                        if difference > 2e-7:
                            raise AssertionError("Literal and aggregate LP objectives disagree")
                    atomic_json(policy_dir / f"{case_id}.json", {
                        "case_id": case_id, "n": n, "rounds": rounds,
                        "oracle_probabilities": list(oracle.probabilities),
                        "oracle_weights": list(oracle.weights), "policy": policy,
                        "unvisited_state_rule": "choose first lexicographic legal action",
                        "lift": "uniform pair selection and fair orientation for fresh singleton births",
                    })
                    case = {"case_id": case_id, "model": name, "n": n,
                            "rounds": rounds, "tau": tau,
                            "oracle_probabilities": list(oracle.probabilities),
                            "oracle_weights": list(oracle.weights),
                            **oracle.count_statistics(n, tau),
                            "optimal_tv": optimum, "diagnostics": diagnostics,
                            "baselines": baselines, "literal_coordinate_check": literal,
                            "policy_file": f"{policy_dir.name}/{case_id}.json"}
                    result["cases"].append(case)
                    print(f"E3 {case_id}: TV={optimum:.8g}, "
                          f"vars={diagnostics['action_occupancy_variables']}, "
                          f"seconds={diagnostics['total_seconds']:.3f}", flush=True)
                except (RuntimeError, ArithmeticError, AssertionError) as exc:
                    result["errors"].append({"case_id": case_id, "error": str(exc)})
                    print(f"E3 {case_id}: FAILED: {exc}", flush=True)
                atomic_json(output_file, result)
    result["status"] = "complete" if not result["errors"] else "completed_with_errors"
    atomic_json(output_file, result)
    return not result["errors"]


if __name__ == "__main__":
    parser = argparse.ArgumentParser()
    parser.add_argument("--config", type=Path, required=True)
    parser.add_argument("--output", type=Path, required=True)
    parser.add_argument("--validate-only", action="store_true",
                        help="Run randomization regression and both fixed oracles at n=2, K=2,3")
    arguments = parser.parse_args()
    config = json.loads(arguments.config.read_text())
    if arguments.validate_only:
        config["e3"].update(n_values=[2], rounds=[2, 3], literal_n_values=[2])
    success = run(config, arguments.output)
    raise SystemExit(0 if success else 1)

	#!/usr/bin/env python3
	"""E3: numerical fixed-oracle optimum over randomized fair-pair policies.

	The LP uses occupation measures, not an average of per-policy TV values.
	See docs/policy_lp_reduction.md for the scope and symmetry reduction.
	"""
	from __future__ import annotations

	import argparse
	from collections import defaultdict
	from dataclasses import dataclass
	from functools import lru_cache
	from itertools import combinations, combinations_with_replacement, product
	import json
	import math
	from pathlib import Path
	import time

	import numpy as np
	from scipy.optimize import linprog
	from scipy.sparse import coo_matrix, csr_matrix, hstack, vstack
	from scipy.special import logsumexp
	from scipy.stats import binom


	@dataclass(frozen=True)
	class Oracle:
	probabilities: tuple[float, ...]
	weights: tuple[float, ...]

	def __post_init__(self):
	if len(self.probabilities) != len(self.weights):
	raise ValueError("Component lengths differ")
	if not all(0 < p < 1 for p in self.probabilities):
	raise ValueError("Positive component probabilities are required")
	if not all(w > 0 for w in self.weights):
	raise ValueError("Positive mixture weights are required")
	if abs(sum(self.weights) - 1) > 1e-12:
	raise ValueError("Mixture weights must sum to one")

	@lru_cache(maxsize=None)
	def predictive(self, c: int, m: int) -> float:
	p = np.asarray(self.probabilities)
	logw = np.log(self.weights) + m * np.log(p) + (c - m) * np.log1p(-p)
	posterior = np.exp(logw - logsumexp(logw))
	return float(posterior @ p)

	def predictive_direct(self, c: int, m: int) -> float:
	"""Independent small-n evaluation used in recovered-policy checks."""
	w = [a * p*m (1 - p)**(c - m)
	for a, p in zip(self.weights, self.probabilities)]
	return sum(a * p for a, p in zip(w, self.probabilities)) / sum(w)

	def count_statistics(self, n: int, tau: float) -> dict:
	m = np.arange(n + 1)
	teacher = binom.pmf(m, n, tau)
	log_likelihoods = np.asarray([
	np.log(a) + m * np.log(p / tau)
	+ (n - m) * np.log((1 - p) / (1 - tau))
	for a, p in zip(self.weights, self.probabilities)
	])
	log_ratio = logsumexp(log_likelihoods, axis=0)
	model = sum(a * binom.pmf(m, n, p)
	for a, p in zip(self.weights, self.probabilities))
	return {
	"actual_joint_kl_P_to_R": float(-teacher @ log_ratio),
	"actual_joint_tv_P_R": float(np.abs(teacher - model).sum() / 2),
	"initial_predictive_drift": self.predictive(0, 0) - tau,
	}


	def aggregate_actions(n, state, final):
	c, _, o = state
	f = n - c - o
	if final:
	return [(o, f, 0)]
	return [(u, v, b) for u in range(o + 1)
	for v in range(f + 1) for b in range(f - v + 1)]


	def aggregate_transition(oracle, state, action):
	c, m, o = state
	u, v, b = action
	p = oracle.predictive(c, m)
	law = np.convolve(binom.pmf(np.arange(u + 1), u, p),
	binom.pmf(np.arange(v + 1), v, 0.5))
	return {(c + u + v, m + j, o - u + b): float(prob)
	for j, prob in enumerate(law) if prob > 0}


	def direct_aggregate_transition(oracle, state, action):
	"""Recompute without the cached PMF/convolution used in LP assembly."""
	c, m, o = state
	u, v, b = action
	p = oracle.predictive_direct(c, m)
	result = defaultdict(float)
	for x in range(u + 1):
	px = math.comb(u, x) * p*x (1 - p)**(u - x)
	for y in range(v + 1):
	py = math.comb(v, y) * 0.5**v
	result[(c + u + v, m + x + y, o - u + b)] += px * py
	return dict(result)


	def occupation_lp(initial, rounds, actions, transition, terminal_index,
	target, time_limit, action_limit):
	"""Generic finite layered MDP LP, also used for literal-coordinate checks."""
	start = time.perf_counter()
	layers = [{initial}]
	records = []
	by_state = defaultdict(list)
	for t in range(rounds):
	following = set()
	for state in sorted(layers[t]):
	for action in actions(state, t == rounds - 1):
	kernel = transition(state, action)
	residual = abs(sum(kernel.values()) - 1)
	if residual > 2e-12 or min(kernel.values()) < 0:
	raise ArithmeticError(f"Invalid transition: {residual}")
	j = len(records)
	if j >= action_limit:
	raise RuntimeError(f"State-action guard {action_limit} exceeded")
	records.append((t, state, action, kernel))
	by_state[(t, state)].append(j)
	following.update(kernel)
	layers.append(following)

	na, nt = len(records), len(target)
	rows = {(t, s): i for i, (t, s) in enumerate(
	(t, s) for t in range(rounds) for s in sorted(layers[t]))}
	er, ec, ev, tr, tc, tv = [], [], [], [], [], []
	for j, (t, state, action, kernel) in enumerate(records):
	er.append(rows[(t, state)]); ec.append(j); ev.append(1.0)
	for dest, prob in kernel.items():
	if t + 1 < rounds:
	er.append(rows[(t + 1, dest)]); ec.append(j); ev.append(-prob)
	else:
	tr.append(terminal_index(dest)); tc.append(j); tv.append(prob)
	flow = coo_matrix((ev, (er, ec)), shape=(len(rows), na)).tocsr()
	terminal = coo_matrix((tv, (tr, tc)), shape=(nt, na)).tocsr()
	aeq = hstack([flow, csr_matrix((len(rows), nt))], format="csr")
	beq = np.zeros(len(rows)); beq[rows[(0, initial)]] = 1.0
	ident = csr_matrix(np.eye(nt))
	aub = vstack([hstack([terminal, -ident]),
	hstack([-terminal, -ident])], format="csr")
	bub = np.concatenate([target, -target])
	cost = np.concatenate([np.zeros(na), np.full(nt, 0.5)])
	assembly_seconds = time.perf_counter() - start
	solution = linprog(cost, A_ub=aub, b_ub=bub, A_eq=aeq, b_eq=beq,
	bounds=(0, None), method="highs",
	options={"time_limit": float(time_limit),
	"threads": 1,
	"primal_feasibility_tolerance": 1e-9,
	"dual_feasibility_tolerance": 1e-9})
	elapsed = time.perf_counter() - start
	if not solution.success:
	raise RuntimeError(f"LP did not reach optimum: {solution.message}")
	x = solution.x[:na]
	q = np.asarray(terminal @ x).ravel()
	dual_value = float(beq @ solution.eqlin.marginals
	+ bub @ solution.ineqlin.marginals)
	dual_slack = cost - aeq.T @ solution.eqlin.marginals - aub.T @ solution.ineqlin.marginals
	diagnostics = {
	"solver_status": int(solution.status), "solver_message": solution.message,
	"state_counts_by_layer": [len(s) for s in layers],
	"action_occupancy_variables": na, "lp_variables": na + nt,
	"flow_constraints": len(rows), "assembly_seconds": assembly_seconds,
	"total_seconds": elapsed, "solver_objective": float(solution.fun),
	"objective_recomputed_from_terminal": float(np.abs(q - target).sum() / 2),
	"max_flow_residual": float(np.max(np.abs(aeq @ solution.x - beq))),
	"max_l1_constraint_violation": float(max(0, np.max(aub @ solution.x - bub))),
	"minimum_variable": float(np.min(solution.x)),
	"terminal_mass_residual": float(abs(q.sum() - 1)),
	"dual_objective": dual_value,
	"primal_dual_gap": float(solution.fun - dual_value),
	"max_dual_slack_violation": float(max(0, -np.min(dual_slack))),
	"certificate_type": "floating-point diagnostics, not a rigorous exact certificate",
	}
	# SciPy's <=-constraint marginals are nonpositive. The upper block is
	# q-d<=P and the lower block is -q-d<=-P, so the terminal test is high
	# where q>P. Its certificate orientation is min_Q E_Q f - E_P f.
	mu = solution.ineqlin.marginals
	terminal_test = np.clip(0.5 - mu[:nt] + mu[nt:], 0.0, 1.0)
	return {"records": records, "by_state": by_state, "occupancies": x,
	"terminal": q, "terminal_test": terminal_test,
	"diagnostics": diagnostics}


	def recover_and_evaluate(result, oracle, n, rounds):
	policies = {}
	serialized = []
	records, occupancies = result["records"], result["occupancies"]
	for (t, state), indices in result["by_state"].items():
	raw = np.asarray([max(0.0, occupancies[j]) for j in indices])
	total = float(raw.sum())
	if total > 0:
	weights = raw / total
	else:
	weights = np.zeros(len(indices)); weights[0] = 1.0
	policy = [(records[j][2], float(a)) for j, a in zip(indices, weights) if a > 0]
	policies[(t, state)] = policy
	if total > 0:
	serialized.append({"round": t + 1, "state_c_m_o": list(state),
	"state_occupancy": total,
	"actions": [{"u_v_b": list(action), "probability": a}
	for action, a in policy]})
	mass = {(0, 0, 0): 1.0}
	stage_mass_residuals = []
	for t in range(rounds):
	following = defaultdict(float)
	for state, state_mass in mass.items():
	for action, action_prob in policies[(t, state)]:
	for dest, transition_prob in direct_aggregate_transition(oracle, state, action).items():
	following[dest] += state_mass * action_prob * transition_prob
	mass = dict(following)
	stage_mass_residuals.append(abs(sum(mass.values()) - 1))
	q = np.zeros(n + 1)
	for (c, m, o), prob in mass.items():
	if c != n or o != 0:
	raise AssertionError("Recovered policy leaves unassigned coordinates")
	q[m] += prob
	diagnostics = {
	"recovered_terminal_max_difference": float(np.max(np.abs(q - result["terminal"]))),
	"recovered_stage_mass_max_residual": max(stage_mass_residuals),
	"recovered_terminal": q.tolist(),
	}
	return serialized, diagnostics


	def schedule_terminal(oracle, n, cumulative_completions):
	# Round 1 reveals one endpoint of every pair; subsequent rounds complete
	# uniformly selected open pairs. Hidden pair randomization is essential.
	mass = {(0, 0, n): 1.0}
	completed = 0
	for cumulative in cumulative_completions:
	batch = cumulative - completed
	following = defaultdict(float)
	for state, prob in mass.items():
	for dest, conditional in direct_aggregate_transition(oracle, state, (batch, 0, 0)).items():
	following[dest] += prob * conditional
	mass = dict(following)
	completed = cumulative
	q = np.zeros(n + 1)
	for (c, m, o), prob in mass.items():
	if c != n or o != 0:
	raise AssertionError("Incomplete baseline schedule")
	q[m] += prob
	return q


	def schedule_baselines(oracle, n, rounds, tau):
	r = rounds - 1
	teacher = binom.pmf(np.arange(n + 1), n, tau)
	equal = tuple(math.ceil(n * j / r) for j in range(1, r + 1))
	geometric = tuple(min(n, math.floor(n ** (j / r) + 1e-12))
	for j in range(1, r + 1))

	def evaluate(schedule):
	q = schedule_terminal(oracle, n, schedule)
	return float(np.abs(q - teacher).sum() / 2)

	all_grid = [tuple(prefix) + (n,) for prefix in
	combinations_with_replacement(range(n + 1), r - 1)]
	grid_values = [(evaluate(schedule), schedule) for schedule in all_grid]
	best_tv, best_schedule = min(grid_values)
	return {
	"equal": {"tv": evaluate(equal), "cumulative_completions": list(equal)},
	"geometric_heuristic": {"tv": evaluate(geometric),
	"cumulative_completions": list(geometric)},
	"best_fixed_completion_grid": {
	"tv": best_tv, "cumulative_completions": list(best_schedule),
	"number_of_schedules": len(all_grid),
	"grid": [{"tv": tv, "cumulative_completions": list(schedule)}
	for tv, schedule in grid_values],
	},
	}


	def literal_problem(n, rounds, oracle, tau, time_limit, action_limit):
	"""Independent full-coordinate finite MDP; no c,m,o reduction in assembly."""
	complete = list(product((0, 1), repeat=2 * n))
	law = {}
	target = []
	for z in complete:
	mismatch = sum(z[2 * i] != z[2 * i + 1] for i in range(n))
	law[z] = 2.0*(-n) sum(a * p*mismatch (1 - p)**(n - mismatch)
	for a, p in zip(oracle.weights, oracle.probabilities))
	target.append(2.0*(-n) tau*mismatch (1 - tau)**(n - mismatch))
	output_index = {z: i for i, z in enumerate(complete)}

	@lru_cache(maxsize=None)
	def literal_heads(state):
	compatible = [(z, p) for z, p in law.items()
	if all(x == -1 or x == y for x, y in zip(state, z))]
	denominator = sum(p for _, p in compatible)
	return tuple(sum(p * z[j] for z, p in compatible) / denominator
	for j in range(2 * n))

	def actions(state, final):
	hidden = tuple(j for j, x in enumerate(state) if x == -1)
	if final:
	return [hidden]
	return [subset for size in range(len(hidden) + 1)
	for subset in combinations(hidden, size)]

	@lru_cache(maxsize=None)
	def transition(state, action):
	heads = literal_heads(state)
	kernel = {}
	for outcomes in product((0, 1), repeat=len(action)):
	dest = list(state)
	prob = 1.0
	for j, bit in zip(action, outcomes):
	dest[j] = bit
	prob *= heads[j] if bit else 1 - heads[j]
	kernel[tuple(dest)] = prob
	return kernel

	result = occupation_lp((-1,) * (2 * n), rounds, actions, transition,
	output_index.__getitem__, np.asarray(target),
	time_limit, action_limit)
	return result["diagnostics"]


	def solve_aggregate(n, rounds, oracle, tau, time_limit, action_limit):
	target = binom.pmf(np.arange(n + 1), n, tau)
	result = occupation_lp((0, 0, 0), rounds,
	lambda s, final: aggregate_actions(n, s, final),
	lambda s, a: aggregate_transition(oracle, s, a),
	lambda s: s[1], target, time_limit, action_limit)
	policy, check = recover_and_evaluate(result, oracle, n, rounds)
	result["diagnostics"].update(check)
	return result, policy


	def bellman_terminal_test(result, oracle, n, rounds, tau):
	"""Independently rebuild transition probabilities in a MIN Bellman pass.

	This floating-point equality check is not the rigorous certificate; the
	separate rational certifier receives only the test and exact oracle.
	"""
	test = result["terminal_test"]
	value = {(n, m, 0): float(test[m]) for m in range(n + 1)}
	for t in reversed(range(rounds)):
	previous = {}
	for (stage, state), indices in result["by_state"].items():
	if stage != t:
	continue
	best = math.inf
	for j in indices:
	action = result["records"][j][2]
	kernel = direct_aggregate_transition(oracle, state, action)
	expectation = math.fsum(p * value[dest] for dest, p in kernel.items())
	best = min(best, expectation)
	previous[state] = best
	value = previous
	target = binom.pmf(np.arange(n + 1), n, tau)
	lower = value[(0, 0, 0)] - float(target @ test)
	gap = lower - result["diagnostics"]["solver_objective"]
	if abs(gap) > 3e-7:
	raise AssertionError(f"Terminal test Bellman/LP mismatch: {gap}")
	return {"floating_bellman_lower": lower,
	"floating_bellman_lp_difference": gap,
	"terminal_test_orientation": "min_Q E_Q[f] - E_P[f]"}


	def assert_diagnostics(d):
	for key in ("max_flow_residual", "max_l1_constraint_violation",
	"terminal_mass_residual", "max_dual_slack_violation"):
	if d[key] > 2e-7:
	raise AssertionError(f"{key}={d[key]} exceeds tolerance")
	if abs(d["primal_dual_gap"]) > 2e-7 or d["minimum_variable"] < -2e-8:
	raise AssertionError("Primal/dual or nonnegativity diagnostic failed")
	if abs(d["solver_objective"] - d["objective_recomputed_from_terminal"]) > 2e-7:
	raise AssertionError("Solver objective disagrees with terminal-law TV")
	if d.get("recovered_terminal_max_difference", 0) > 2e-7:
	raise AssertionError("Recovered policy disagrees with occupation solution")


	def randomization_regression(time_limit, action_limit):
	oracle = Oracle((0.125,), (1.0,))
	result, policy = solve_aggregate(1, 2, oracle, 0.25, time_limit, action_limit)
	literal = literal_problem(1, 2, oracle, 0.25, time_limit, action_limit)
	assert_diagnostics(result["diagnostics"])
	assert_diagnostics(literal)
	serial = sum(x * action[0] for x, (_, _, action, _) in
	zip(result["occupancies"], result["records"]))
	cosample = sum(x * action[1] for x, (_, _, action, _) in
	zip(result["occupancies"], result["records"]))
	if abs(serial - 2 / 3) > 1e-7 or abs(cosample - 1 / 3) > 1e-7:
	raise AssertionError("Expected randomized-policy mixture was not recovered")
	if max(result["diagnostics"]["solver_objective"], literal["solver_objective"]) > 1e-8:
	raise AssertionError("Randomized marginal-TV regression failed")
	return {"status": "PASS", "serial_probability": serial,
	"cosample_probability": cosample, "aggregate": result["diagnostics"],
	"literal": literal, "policy": policy}


	def atomic_json(path, obj):
	temporary = path.with_suffix(path.suffix + ".tmp")
	temporary.write_text(json.dumps(obj, indent=2, allow_nan=False) + "\n")
	temporary.replace(path)


	def run(config, output):
	cfg = config["e3"]
	if output.suffix == ".json":
	output_file = output
	output_dir = output.parent
	policy_dir = output_dir / (output.stem + "_policies")
	else:
	output_dir = output
	output_file = output_dir / "e3_results.json"
	policy_dir = output_dir / "policies"
	output_dir.mkdir(parents=True, exist_ok=True)
	policy_dir.mkdir(exist_ok=True)
	n_values = cfg.get("n_values", [2, 4, 6, 8])
	rounds_values = cfg.get("rounds", [2, 3, 4])
	tau, w, delta = (float(cfg.get(k, v)) for k, v in
	[("tau", 0.25), ("w", 0.125), ("delta", 0.25)])
	lam = float(cfg.get("ladder_lambda", 0.2))
	time_limit = min(float(cfg.get("time_limit_seconds", 120)), 120)
	action_limit = min(int(cfg.get("state_action_limit", 250000)), 250000)
	literal_n = set(cfg.get("literal_n_values", [1, 2]))
	if min(n_values) < 1 or min(rounds_values) < 2 or max(rounds_values) > 8:
	raise ValueError("Require n >= 1 and 2 <= K <= 8")
	models = {
	"single": Oracle((tau, tau + delta), (1 - w, w)),
	"two_level": Oracle((tau, tau + delta, tau + delta * (w / 2) * lam),
	(1 - w, w / 2, w / 2)),
	}
	result = {
	"experiment": "E3 fixed-oracle randomized-policy occupation LP",
	"scope": "inf_policy TV for each fixed R; not a KL-ball supremum or a minimax certificate",
	"arithmetic": "floating-point HiGHS with independent literal-coordinate and forward checks",
	"config": cfg, "status": "running", "cases": [], "errors": [],
	"randomization_regression": randomization_regression(time_limit, action_limit),
	}
	atomic_json(output_file, result)
	print("E3 randomization regression: PASS", flush=True)
	for name, oracle in models.items():
	for n in n_values:
	previous = None
	for rounds in sorted(rounds_values):
	case_id = f"{name}_n{n}_K{rounds}"
	try:
	solved, policy = solve_aggregate(n, rounds, oracle, tau, time_limit, action_limit)
	diagnostics = solved["diagnostics"]
	assert_diagnostics(diagnostics)
	baselines = schedule_baselines(oracle, n, rounds, tau)
	optimum = diagnostics["objective_recomputed_from_terminal"]
	if optimum > min(b["tv"] for b in baselines.values()) + 2e-7:
	raise AssertionError("LP optimum is worse than a feasible baseline")
	if previous is not None and optimum > previous + 2e-7:
	raise AssertionError("Optimal error increased when adding a round")
	previous = optimum
	literal = None
	if n in literal_n:
	literal = literal_problem(n, rounds, oracle, tau, time_limit, action_limit)
	assert_diagnostics(literal)
	difference = abs(literal["objective_recomputed_from_terminal"] - optimum)
	literal["aggregate_objective_difference"] = difference
	if difference > 2e-7:
	raise AssertionError("Literal and aggregate LP objectives disagree")
	atomic_json(policy_dir / f"{case_id}.json", {
	"case_id": case_id, "n": n, "rounds": rounds,
	"oracle_probabilities": list(oracle.probabilities),
	"oracle_weights": list(oracle.weights), "policy": policy,
	"unvisited_state_rule": "choose first lexicographic legal action",
	"lift": "uniform pair selection and fair orientation for fresh singleton births",
	})
	case = {"case_id": case_id, "model": name, "n": n,
	"rounds": rounds, "tau": tau,
	"oracle_probabilities": list(oracle.probabilities),
	"oracle_weights": list(oracle.weights),
	**oracle.count_statistics(n, tau),
	"optimal_tv": optimum, "diagnostics": diagnostics,
	"baselines": baselines, "literal_coordinate_check": literal,
	"policy_file": f"{policy_dir.name}/{case_id}.json"}
	result["cases"].append(case)
	print(f"E3 {case_id}: TV={optimum:.8g}, "
	f"vars={diagnostics['action_occupancy_variables']}, "
	f"seconds={diagnostics['total_seconds']:.3f}", flush=True)
	except (RuntimeError, ArithmeticError, AssertionError) as exc:
	result["errors"].append({"case_id": case_id, "error": str(exc)})
	print(f"E3 {case_id}: FAILED: {exc}", flush=True)
	atomic_json(output_file, result)
	result["status"] = "complete" if not result["errors"] else "completed_with_errors"
	atomic_json(output_file, result)
	return not result["errors"]


	if __name__ == "__main__":
	parser = argparse.ArgumentParser()
	parser.add_argument("--config", type=Path, required=True)
	parser.add_argument("--output", type=Path, required=True)
	parser.add_argument("--validate-only", action="store_true",
	help="Run randomization regression and both fixed oracles at n=2, K=2,3")
	arguments = parser.parse_args()
	config = json.loads(arguments.config.read_text())
	if arguments.validate_only:
	config["e3"].update(n_values=[2], rounds=[2, 3], literal_n_values=[2])
	success = run(config, arguments.output)
	raise SystemExit(0 if success else 1)