diff --git a/wizard-work/beam/find_duplicate.py b/wizard-work/beam/find_duplicate.py
index ceefac4..25c8643 100644
--- a/wizard-work/beam/find_duplicate.py
+++ b/wizard-work/beam/find_duplicate.py
@@ -1,18 +1,43 @@
-"""Find the duplicate in an array of n+1 integers each in [1, n].
+"""Find a duplicate in an array of n+1 integers each in [1, n].
 
 Classic Floyd's tortoise-and-hare on the implicit linked list defined by
 f(i) = nums[i]. Guarantees O(n) time, O(1) extra space, and does not
-mutate the input. Returns the single duplicated value (input is assumed
-to contain exactly one value appearing twice, all others once).
+mutate the input. Returns a duplicated value; when several values repeat,
+returns one of them (the entry point of the cycle reachable from index 0).
 
 Edge cases handled:
   - len(nums) < 2  -> ValueError
-  - values out of [1, n]  -> ValueError
-  - no duplicate present  -> ValueError (defensive; Floyd would loop)
+  - values out of [1, n], or not a plain int (bool and float included)
+                    -> ValueError
+  - no duplicate present  -> unreachable, and therefore never raised:
+    a no-duplicate array cannot satisfy both preconditions at once. With
+    len(nums) = n + 1 values drawn from 1..n, pigeonhole forces a repeat;
+    the only arrays that avoid one are shorter than n + 1 or carry a value
+    outside 1..n, and both are refused by the gates above before Floyd
+    runs (see `_no_duplicate_guard()`). The count check after phase 2 is
+    kept as an invariant, not as a handler: an entry node has two distinct
+    predecessors by definition, so its value occurs at least twice.
 """
 from typing import List
 
 
+def _no_duplicate_guard(candidate: int, nums: List[int]) -> int:
+    """Invariant: the phase-2 entrance value is duplicated, or raise.
+
+    Unreachable on validated input (see the module docstring). It is here so
+    that an edit which widens the gates above -- or shortens phase 2, the other
+    half of the argument -- fails loudly instead of returning a value as if it
+    were a duplicate. Measured: the raise fires 0 times over every validated
+    input for n = 1..4, and it *does* fire when phase 2's loop is deleted, which
+    is the edit it exists for. `test_find_duplicate.py` pins that this is the
+    module's only unexecuted line, so removing it, or growing new dead code
+    beside it, moves a cell.
+    """
+    if nums.count(candidate) < 2:
+        raise ValueError("no duplicate present in nums")
+    return candidate
+
+
 def find_duplicate(nums: List[int]) -> int:
     n = len(nums) - 1
     if n < 1:
@@ -33,12 +58,10 @@ def find_duplicate(nums: List[int]) -> int:
     while slow2 != slow:
         slow2 = nums[slow2]
         slow = nums[slow]
-    candidate = slow
-    # Verify the candidate actually occurs more than once; a permutation
-    # (no duplicate) also has cycles, so Floyd alone is not sufficient.
-    if nums.count(candidate) < 2:
-        raise ValueError("no duplicate present in nums")
-    return candidate
+    # Floyd alone would return an entrance value even for a permutation; the
+    # guard below states the invariant that the range gates make that
+    # impossible. It is a proof checkpoint, not a scenario handler.
+    return _no_duplicate_guard(slow, nums)
 
 
 def _self_test() -> None:
@@ -81,11 +104,19 @@ def _self_test() -> None:
         else:
             raise AssertionError(f"expected ValueError for {bad!r}")
 
-    # No duplicate (all distinct) -> must not hang; we detect via timeout-ish
-    # guard: a distinct permutation of [1..n] has no cycle reachable from 0? Actually
-    # f on a permutation is a set of cycles; Floyd may find a cycle that is NOT a
-    # duplicate. We therefore do NOT promise detection of the no-duplicate case;
-    # instead we assert it raises OR returns something, but never hangs.
+    # The no-duplicate case never reaches Floyd: a permutation of 1..n has
+    # length n, which is one short of a valid input, so the range gate
+    # refuses it with an out-of-range message. The `no duplicate present`
+    # branch cannot fire on any validated input -- pinned as the module's
+    # only unexecuted line in test_find_duplicate.py.
+    for permutation in ([1, 2, 3, 4], [2, 1], [1, 2, 3]):
+        try:
+            find_duplicate(list(permutation))
+        except ValueError as exc:
+            assert "out of range" in str(exc), (permutation, exc)
+        else:
+            raise AssertionError(f"expected ValueError for {permutation!r}")
+
     print("ALL TESTS PASSED")
 
 
diff --git a/wizard-work/beam/test_find_duplicate.py b/wizard-work/beam/test_find_duplicate.py
new file mode 100644
index 0000000..f334e2b
--- /dev/null
+++ b/wizard-work/beam/test_find_duplicate.py
@@ -0,0 +1,216 @@
+"""Cells for `find_duplicate` (Q95, volt, 2026-10-28).
+
+The module had no cell in any gate, and the record that opened Q95 measured a
+`raise ValueError("no duplicate present in nums")` that no validated input can
+reach. These cells do two things a green gate could not do before: they grade
+the behaviour over the whole validated domain (n = 1..4, every array), and they
+grade the *shape* of the code -- the no-duplicate guard is declared as the
+module's only unexecuted line, so deleting it, or growing new dead code beside
+it, moves a cell.
+
+Run from the lane directory:  python3 -m pytest -q test_find_duplicate.py
+"""
+import ast
+import itertools
+import sys
+import unittest
+from pathlib import Path
+
+import find_duplicate as fd
+
+MODULE_PATH = Path(fd.__file__)
+
+
+def _module_source():
+    return MODULE_PATH.read_text(encoding="utf-8")
+
+
+def _traced_call(func, *args):
+    """Call func(*args), returning (result_or_exception, executed_line_set)."""
+    executed = set()
+
+    def local(frame, event, arg):
+        if event == "line" and frame.f_code.co_filename == str(MODULE_PATH):
+            executed.add(frame.f_lineno)
+        return local
+
+    previous = sys.gettrace()
+    sys.settrace(local)
+    try:
+        try:
+            result = func(*args)
+        except Exception as exc:  # noqa: BLE001 - the caller grades the class
+            result = exc
+    finally:
+        sys.settrace(previous)
+    return result, executed
+
+
+def _valid_arrays(n):
+    return itertools.product(range(1, n + 1), repeat=n + 1)
+
+
+NO_DUPLICATE_INPUTS = ([], [1], [0, 1], [1, 3], [1, 2, 3, 4], [True, 1],
+                       [1.0, 1], ["1", 1])
+NO_DUPLICATE_PERMUTATIONS = ([1, 2, 3, 4], [2, 1], [1, 2, 3])
+
+
+class KnownDuplicates(unittest.TestCase):
+    def test_each_documented_case_returns_its_duplicate(self):
+        cases = [
+            ([1, 3, 4, 2, 2], 2),
+            ([3, 1, 3, 4, 2], 3),
+            ([1, 1], 1),
+            ([2, 2, 2], 2),
+            ([1, 2, 3, 4, 5, 5], 5),
+            ([2, 4, 3, 2, 1], 2),
+            # Phase 1's meeting point is inside the cycle and its value (3)
+            # occurs once; only phase 2's entrance (2) is the duplicate. A
+            # suite whose cases all have meeting == entrance cannot see phase 2
+            # go missing -- this case can.
+            ([1, 2, 3, 4, 2], 2),
+        ]
+        for nums, expected in cases:
+            with self.subTest(nums=nums):
+                self.assertEqual(fd.find_duplicate(list(nums)), expected)
+
+    def test_input_is_not_mutated(self):
+        nums = [3, 1, 3, 4, 2]
+        snapshot = list(nums)
+        fd.find_duplicate(nums)
+        self.assertEqual(nums, snapshot)
+
+    def test_repeated_calls_are_deterministic(self):
+        nums = [4, 3, 4, 2, 1]
+        reads = {fd.find_duplicate(list(nums)) for _ in range(5)}
+        self.assertEqual(reads, {4})
+
+
+class ValidatedDomain(unittest.TestCase):
+    def test_domain_size_is_the_pigeonhole_domain(self):
+        # n**(n+1) for n = 1..4 -> 1 + 8 + 81 + 1024
+        self.assertEqual(sum(n ** (n + 1) for n in range(1, 5)), 1114)
+
+    def test_every_validated_array_returns_a_value_occurring_twice(self):
+        total = 0
+        for n in range(1, 5):
+            for array in _valid_arrays(n):
+                total += 1
+                nums = list(array)
+                with self.subTest(nums=nums):
+                    got = fd.find_duplicate(nums)
+                    self.assertGreaterEqual(nums.count(got), 2)
+
+    def test_no_validated_array_raises_anything(self):
+        total = raises = 0
+        for n in range(1, 5):
+            for array in _valid_arrays(n):
+                total += 1
+                try:
+                    fd.find_duplicate(list(array))
+                except Exception as exc:  # noqa: BLE001 - any raise is the finding
+                    raises += 1
+                    self.fail("validated input raised %r: %r" % (array, exc))
+        self.assertEqual((total, raises), (1114, 0))
+
+
+class RefusedInputs(unittest.TestCase):
+    def test_short_arrays_are_refused_by_the_length_gate(self):
+        for bad in ([], [1]):
+            with self.subTest(bad=bad):
+                with self.assertRaises(ValueError) as ctx:
+                    fd.find_duplicate(list(bad))
+                self.assertIn("at least 2 elements", str(ctx.exception))
+
+    def test_out_of_range_values_are_refused_by_the_range_gate(self):
+        for bad in ([0, 1], [1, 3], [1, 2, 3, 4], [-1, 1]):
+            with self.subTest(bad=bad):
+                with self.assertRaises(ValueError) as ctx:
+                    fd.find_duplicate(list(bad))
+                self.assertIn("out of range", str(ctx.exception))
+
+    def test_bools_floats_and_strings_are_not_ints_for_this_contract(self):
+        for bad in ([True, 1], [1.0, 1], ["1", 1]):
+            with self.subTest(bad=bad):
+                with self.assertRaises(ValueError) as ctx:
+                    fd.find_duplicate(list(bad))
+                self.assertIn("out of range", str(ctx.exception))
+
+    def test_no_duplicate_shape_is_refused_by_the_range_gate_not_by_floyd(self):
+        # Q95's reading: a no-duplicate array of length n+1 over 1..n cannot
+        # exist, so the nearest no-duplicate shape is one short and dies on the
+        # range gate -- never on the "no duplicate present" branch.
+        for permutation in NO_DUPLICATE_PERMUTATIONS:
+            with self.subTest(permutation=permutation):
+                with self.assertRaises(ValueError) as ctx:
+                    fd.find_duplicate(list(permutation))
+                message = str(ctx.exception)
+                self.assertIn("out of range", message)
+                self.assertNotIn("no duplicate present", message)
+
+
+class DeadBranchIsDeclared(unittest.TestCase):
+    """The no-duplicate guard is the module's only unexecuted line."""
+
+    def _guard_raise_lineno(self):
+        tree = ast.parse(_module_source())
+        for node in ast.walk(tree):
+            if isinstance(node, ast.Raise) and isinstance(node.exc, ast.Call):
+                for arg in node.exc.args:
+                    if isinstance(arg, ast.Constant) and isinstance(arg.value, str):
+                        if "no duplicate present" in arg.value:
+                            return node.lineno
+        self.fail("the no-duplicate raise is gone from find_duplicate.py")
+
+    def _statement_lines(self):
+        tree = ast.parse(_module_source())
+
+        def is_docstring(node):
+            return (
+                isinstance(node, ast.Expr)
+                and isinstance(node.value, ast.Constant)
+                and isinstance(node.value.value, str)
+            )
+
+        lines = set()
+        for node in tree.body:
+            if isinstance(node, (ast.FunctionDef, ast.AsyncFunctionDef)):
+                if node.name == "_self_test":
+                    continue  # its own asserts are not this module's behaviour
+                # `def` itself emits no line event, and a docstring is an
+                # Expr the interpreter never steps into.
+                for inner in ast.walk(node):
+                    if isinstance(inner, ast.stmt) and not is_docstring(inner):
+                        lines.add(inner.lineno)
+                lines.discard(node.lineno)
+        return lines
+
+    def test_only_the_no_duplicate_raise_is_unexecuted(self):
+        executed = set()
+        for n in range(1, 5):
+            for array in _valid_arrays(n):
+                _, seen = _traced_call(fd.find_duplicate, list(array))
+                executed |= seen
+        for bad in NO_DUPLICATE_INPUTS + NO_DUPLICATE_PERMUTATIONS:
+            _, seen = _traced_call(fd.find_duplicate, list(bad))
+            executed |= seen
+
+        declared_dead = self._guard_raise_lineno()
+        unexecuted = self._statement_lines() - executed
+        self.assertEqual(
+            unexecuted, {declared_dead},
+            "unexecuted lines (declared dead line %d): %s"
+            % (declared_dead, sorted(unexecuted)),
+        )
+
+    def test_the_guard_cannot_fire_on_validated_input(self):
+        # Independent of line numbers: no validated call may produce the
+        # "no duplicate present" message.
+        for n in range(1, 5):
+            for array in _valid_arrays(n):
+                result, _ = _traced_call(fd.find_duplicate, list(array))
+                self.assertNotIsInstance(result, ValueError, array)
+
+
+if __name__ == "__main__":
+    unittest.main()