diff --git a/CHANGELOG.md b/CHANGELOG.md
index 77eb29a..0a00df6 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -7,6 +7,17 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+### Fixed
+
+- `assert_unbiased` passed a deterministic estimator however far it missed the
+  truth, because a zero Monte Carlo standard error was reported as a bias t
+  statistic of zero. Identical estimates that miss the truth now raise
+  `ValueError`, since the study cannot tell a deterministic bias from a
+  discrete estimator that happened not to vary; `bias_t` is NaN for them and
+  zero only when they hit the truth.
+- `assert_unbiased` raises `ValueError` on a single-replicate study instead of
+  passing it, as `se_ratio_tolerance` and `assert_narrower` already did.
+
 ## [0.1.0] - 2026-08-09
 
 ### Added
diff --git a/src/simcheck/gates.py b/src/simcheck/gates.py
index 23a35ba..70585d3 100644
--- a/src/simcheck/gates.py
+++ b/src/simcheck/gates.py
@@ -458,14 +458,35 @@ def assert_unbiased(
     bias too small for the study to resolve does not fail, and the study can be
     made more demanding simply by running more replicates.
 
+    A study whose estimates are all equal passes only if they equal the truth.
+    If they miss it, the study has no spread to measure the miss against, and
+    it raises rather than passing: the estimator may be deterministic and
+    biased, or discrete and unlucky, and identical estimates cannot say which.
+
     Args:
         result: A completed Monte Carlo study.
         label: Included in the failure message.
         sigmas: How many Monte Carlo standard errors of slack to allow.
 
     Raises:
+        ValueError: If the study has fewer than two replicates, or if its
+            estimates are all equal and differ from the truth. Either way it has
+            no spread against which to tell bias from noise.
         AssertionError: If the bias t statistic exceeds the gate.
     """
+    if result.reps < 2:
+        raise ValueError(
+            f"{label or 'this study'}: a single replicate has no spread, so its "
+            "distance from the truth cannot be told apart from noise"
+        )
+    if not result.mc_se and result.bias:
+        raise ValueError(
+            f"{label or 'this study'}: all {result.reps} estimates equal "
+            f"{result.bias + result.truth:.6g}, missing the truth by "
+            f"{result.bias:+.6g}, with no spread to measure that against. A "
+            "deterministic estimator is biased by exactly that much; a discrete "
+            "one may need more replicates before it varies."
+        )
     if not abs(result.bias_t) < sigmas:
         raise AssertionError(
             f"{label}: bias {result.bias:+.6f} is {result.bias_t:+.2f} Monte "
diff --git a/src/simcheck/results.py b/src/simcheck/results.py
index fb7a369..b2ce97e 100644
--- a/src/simcheck/results.py
+++ b/src/simcheck/results.py
@@ -22,6 +22,7 @@
 
 from __future__ import annotations
 
+import math
 from dataclasses import dataclass
 
 import numpy as np
@@ -168,11 +169,17 @@ def mc_se(self) -> float:
     def bias_t(self) -> float:
         """Bias in units of its own Monte Carlo standard error.
 
-        A t statistic for the null that the estimator is unbiased. Zero when the
-        estimator is deterministic, in which case there is no sampling variation
-        to test against.
+        A t statistic for the null that the estimator is unbiased. When the
+        estimates did not vary it is zero if every one equals the truth and NaN
+        otherwise. NaN, not zero: zero would certify a constant that misses the
+        truth as unbiased. And NaN, not infinity: identical estimates may come
+        from a deterministic estimator, whose miss is a bias, or from a discrete
+        one that happened not to vary in this study, whose miss may be noise the
+        study was too small to see. The study cannot tell which.
         """
-        return self.bias / self.mc_se if self.mc_se else 0.0
+        if self.mc_se:
+            return self.bias / self.mc_se
+        return math.nan if self.bias else 0.0
 
     @property
     def coverage(self) -> float:
diff --git a/tests/test_negative.py b/tests/test_negative.py
index 96d3194..3f7051b 100644
--- a/tests/test_negative.py
+++ b/tests/test_negative.py
@@ -222,6 +222,68 @@ def test_assert_unbiased_fails_on_a_biased_estimator():
         assert_unbiased(_study(bias=0.5, sd=0.1), "biased")
 
 
+def _constant_study(value: float, truth: float, reps: int = 400) -> MonteCarloResult:
+    """An estimator that returns ``value`` on every replicate.
+
+    Parameters
+    ----------
+    value
+        The estimate every replicate reports.
+    truth
+        The true value.
+    reps
+        Replicates.
+
+    Returns
+    -------
+    MonteCarloResult
+        The constructed study.
+    """
+    return MonteCarloResult(
+        estimates=np.full(reps, value),
+        standard_errors=np.full(reps, 0.1),
+        covered=None,
+        rejected=None,
+        truth=truth,
+    )
+
+
+def test_assert_unbiased_does_not_pass_a_constant_that_misses_the_truth():
+    """Identical estimates that miss the truth must not pass.
+
+    The Monte Carlo standard error is exactly zero here, and the gate used to
+    read that as a t statistic of zero -- so returning 5 for a truth of 2 passed.
+    It raises rather than asserting bias, because the same study arises from a
+    discrete unbiased estimator that happened not to vary.
+    """
+    study = _constant_study(5.0, truth=2.0)
+    assert np.isnan(study.bias_t)
+    with pytest.raises(ValueError, match="no spread"):
+        assert_unbiased(study, "constant, wrong")
+
+
+def test_assert_unbiased_does_not_call_an_unvarying_discrete_estimator_biased():
+    """An unbiased Bernoulli estimator that never varied is not reported biased.
+
+    With a success probability of 0.001, all 400 draws are zero about two times
+    in three. The estimator is unbiased; the study is too small to show it.
+    """
+    study = _constant_study(0.0, truth=0.001)
+    with pytest.raises(ValueError, match="more replicates"):
+        assert_unbiased(study, "rare Bernoulli")
+
+
+def test_assert_unbiased_passes_a_deterministic_exact_estimator():
+    """A constant that equals the truth is unbiased, and must not fire."""
+    assert_unbiased(_constant_study(2.0, truth=2.0), "constant, right")
+
+
+def test_assert_unbiased_rejects_a_single_replicate():
+    """One draw cannot separate bias from noise, however far off it lands."""
+    with pytest.raises(ValueError, match="single replicate"):
+        assert_unbiased(_constant_study(1e6, truth=0.0, reps=1), "one draw")
+
+
 def test_assert_unbiased_resolves_smaller_bias_with_more_replicates():
     """The gate must tighten as the study grows, without any edit.