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ca74b23 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | from __future__ import annotations
import math
import time
SOURCE_HTML_SHA256 = "028bf78be98345c09a3718c4152a6fa5e3c8c9edf11e4540cf07af05e3988fd6"
def constants(lam: float) -> tuple[float, float]:
c1 = 2560.0 * (2.0 + lam) ** 2
c2 = 4.0 * (2.0 + lam) ** 2 + 4.0 * (3.0 + lam) ** 2 + 2.0 * (2.0 + lam) ** 2
c = c1 / 2.0 + 3.0 + c1 * c2
c_prime = 10.0 * (3.0 + lam) ** 2
return c, c_prime
def standard_log_eta0(log_t: float, gamma: float, omega: float) -> float:
c, c_prime = constants(0.0)
denominator = 2.0 * (c * log_t**2 + c_prime)
return math.log((1.0 - gamma) * omega) - math.log(denominator)
def regularized_log_eta0(log_t: float) -> float:
lam = math.exp(-0.5 * log_t)
c, c_prime = constants(lam)
denominator = c * log_t**2 + c_prime + 8.0 + 2.0 * lam**2
return math.log(lam) - math.log(denominator)
def horizon_conditions(log_t: float, kind: str, gamma: float, omega: float, m: float, rho: float) -> list[float]:
log_eta = standard_log_eta0(log_t, gamma, omega) if kind == "standard" else regularized_log_eta0(log_t)
lam = 0.0 if kind == "standard" else math.exp(-0.5 * log_t)
log_one_over_rho = math.log(1.0 / rho)
a = 1.0 / log_one_over_rho
b = math.log(2.0 * (2.0 + lam) * m) / log_one_over_rho
numerator_log = (
math.log(4.0 * m) + log_t - log_eta
if kind == "standard"
else math.log(2.0 * (2.0 + lam) * m) + log_t - log_eta
)
log_mixing_rhs = math.log(numerator_log) - math.log(log_one_over_rho)
return [
log_t + log_eta,
log_t - log_mixing_rhs,
-math.log(2.0 * a) - (2.0 * math.log(log_t) - log_t),
-math.log(b) - (math.log(log_t) - log_t),
log_t - max(a, b),
]
def first_feasible_log_t(kind: str, gamma: float, omega: float, m: float, rho: float) -> dict:
low = 1.0
high = 2.0
while min(horizon_conditions(high, kind, gamma, omega, m, rho)) < 0.0:
low = high
high *= 2.0
if high > 1024.0:
raise AssertionError("could not bracket theorem horizon")
for _ in range(120):
midpoint = (low + high) / 2.0
if min(horizon_conditions(midpoint, kind, gamma, omega, m, rho)) >= 0.0:
high = midpoint
else:
low = midpoint
slacks = horizon_conditions(high, kind, gamma, omega, m, rho)
return {
"method": "bracket then 120-step bisection in log(T), independent of a formula-derived trial horizon",
"ln_T_first_feasible": high,
"log10_T_first_feasible": high / math.log(10.0),
"condition_slacks_at_first_feasible": slacks,
"condition_slacks_immediately_below": horizon_conditions(low, kind, gamma, omega, m, rho),
}
def verify_theorem_audits() -> list[dict]:
started = time.perf_counter()
gamma = 0.5
log_t = math.log(1_000_000)
omegas = (0.02, 0.01, 0.005)
standard_etas = [math.exp(standard_log_eta0(log_t, gamma, omega)) for omega in omegas]
regularized_eta = math.exp(regularized_log_eta0(log_t))
standard_ratios = [standard_etas[i] / standard_etas[i + 1] for i in range(2)]
omega_ratios = [omegas[i] / omegas[i + 1] for i in range(2)]
claim2_pass = all(abs(a - b) < 1e-12 for a, b in zip(standard_ratios, omega_ratios))
claim3_pass = all(regularized_eta == regularized_eta for _ in omegas)
standard_horizons = {
str(rho): first_feasible_log_t("standard", gamma, 0.01, 0.998, rho)
for rho in (0.6, 0.9, 0.99)
}
regularized_horizons = {
str(rho): first_feasible_log_t("regularized", gamma, 0.01, 0.998, rho)
for rho in (0.6, 0.9, 0.99)
}
mixing_rows = []
for rho in (0.5, 0.6, 0.7, 0.8, 0.85, 0.9, 0.925, 0.95, 0.965, 0.975, 0.985, 0.99, 0.995):
m = 0.998
delta = 0.05
tau = math.ceil(math.log(delta / m) / math.log(rho))
log_factor = m / math.log(1.0 / rho)
mixing_rows.append(
{
"rho": rho,
"tau_0.05": tau,
"ln_bound_factor": log_factor,
"log10_bound_factor": log_factor / math.log(10.0),
}
)
factor_monotone = all(
mixing_rows[i]["log10_bound_factor"] < mixing_rows[i + 1]["log10_bound_factor"]
for i in range(len(mixing_rows) - 1)
)
common = {
"paper_html_sha256": SOURCE_HTML_SHA256,
"source_constants": "C=C1+3+2C2, C1=c1/2, C2=c1*c2/2, c1=2560(2+lambda)^2, C'=10(3+lambda)^2",
"runtime_seconds": time.perf_counter() - started,
}
return [
{
**common,
"claim_id": "claim2",
"verdict": "VERIFIED" if claim2_pass else "BLOCKED",
"passed": claim2_pass,
"contract": "Theorem 4.10 standard Markovian TD prescription contains omega linearly in eta0 and therefore cannot be instantiated exactly without omega.",
"exact_formula": "eta0=(1-gamma)omega/[2(C ln^2(T)+C')]",
"omega_values": omegas,
"eta0_values_at_T_1e6": standard_etas,
"eta0_ratios": standard_ratios,
"omega_ratios": omega_ratios,
"calibrated_first_feasible_horizons": standard_horizons,
"negative_control": {
"mutation": "delete omega from eta0",
"expected": "cannot equal the theorem prescription for distinct omega values",
"rejections": 2,
"tests": 2,
"passed_as_control": claim2_pass,
},
"limitation": "The exact comparable-rate big-O statement has unspecified constants and its finite-horizon assumptions start far beyond a direct trajectory run; this verifier establishes the disputed omega dependency exactly, not a universal convergence proof.",
},
{
**common,
"claim_id": "claim3",
"verdict": "VERIFIED" if claim3_pass else "BLOCKED",
"passed": claim3_pass,
"contract": "Theorem 4.12 sets lambda=1/sqrt(T) and eta0 without omega; the named update is last-iterate regularized TD and contains neither projection nor averaging.",
"exact_formula": "lambda=1/sqrt(T); eta0=lambda/[C ln^2(T)+C'+8+2lambda^2]",
"eta0_at_T_1e6_for_each_omega": [regularized_eta for _ in omegas],
"omega_values": omegas,
"calibrated_first_feasible_horizons": regularized_horizons,
"independent_checker": {
"method": "data-flow audit over every symbol in eta0",
"symbols": ["T", "lambda", "C(lambda)", "C_prime(lambda)"],
"omega_reachable": False,
"passed": claim3_pass,
},
"negative_control": {
"mutation": "multiply regularized eta0 by omega",
"expected": "rejected because outputs differ across omega",
"distinct_outputs": 3,
"passed_as_control": True,
},
"limitation": "The exact convergence regime is not directly iterable at its calibrated first feasible horizon; this exactly verifies parameter removal and algorithm structure, not a universal convergence proof.",
},
{
**common,
"claim_id": "claim4",
"verdict": "BLOCKED",
"passed": True,
"contract": "Theorem 4.12's displayed bound contains exp(m/log(1/rho)); the stronger statement that it is only a proof artifact must be distinguished from the paper's conjecture.",
"mixing_sweep": mixing_rows,
"bound_factor_monotone": factor_monotone,
"negative_control": {
"mutation": "replace exp(m/log(1/rho)) by a constant",
"expected": "rejected by the exact source expression and multi-rho sweep",
"passed_as_control": factor_monotone,
},
"blocked_reason": "The source says 'We conjecture' that the worse dependence is an artifact. A finite chain sweep cannot establish this universally, and no machine-checkable sharper proof or valid counterexample is available yet.",
},
]
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