test: substep-scaled tolerance and bitwise populations in window GPU gate

Scale the window-vs-summed-reads tolerance by the actual absolute fine
substep contributions (last_raw after every fine hop), labelled a
conservative engineering tolerance rather than a proof. Compare
populations by finite FP32 bits plus NaN masks and reject infinities.
Test-only.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tests/validation/multigrid/test_exchange_window.py

@@ -1,10 +1,12 @@
 """Optional observation-window exchange ledger: host contract (mocked driver) and GPU gate.
 
 Run CPU and GPU classes in separate processes; GPU callers select the device.
-GPU tolerance: window rows are summed per link over time, the per-step reads over links
+GPU tolerance (conservative engineering tolerance, not a universal proof): the window ledger
-then over time. Reordering a sum of m FP64 terms changes it by at most
+sums h*dt*raw per link over fine substeps, while the reference sums per-base interval ledgers
-2*(m-1)*eps*sum|terms| (each order is within (m-1)*eps*sum|terms| of the exact sum), so the
+over links and then over base steps. Both are FP64 reorderings of the same FP32-exact substep
-bound is scaled by the absolute sum of the per-step row integrals and is robust to cancellation.
+contributions, so the scale is the sum of the ACTUAL absolute substep contributions (read from
+last_raw after every fine hop), not of per-base totals that could hide substep cancellation:
+|window - reference| <= 4 * m * eps * sum|substep terms| + 1e-300, m = links * substeps.
 """
 from contextlib import ExitStack
 import unittest
@@ -26,6 +28,16 @@ def build(m, window=True, raw=True):
                              observe_raw_exchange=raw, observe_window_exchange=window)
 
 
+def same_bits(test, a, b):
+    """Identical NaN masks, no infinities, identical finite FP32 bits (incl. signed zero)."""
+    a, b = np.asarray(a), np.asarray(b)
+    test.assertEqual(a.dtype, b.dtype)
+    test.assertFalse(np.isinf(a).any() or np.isinf(b).any())
+    np.testing.assert_array_equal(np.isnan(a), np.isnan(b))
+    finite = np.isfinite(a)
+    np.testing.assert_array_equal(a[finite].view(np.uint32), b[finite].view(np.uint32))
+
+
 def accumulations(m):
     return [c for c in m["calls"] if c[0] == "BlockAccumulateExchange"]
 
@@ -107,9 +119,22 @@ class ExchangeWindowGPU(unittest.TestCase):
                 windowed.start_exchange_window()
                 b = windowed._level_boundaries[1]
                 count = len(b.tables["cell"])
+                factors = np.array([.25, .25, .125, .25])  # h*dt, h*h*dt, h*h for level-1 h=.5
+                scale = np.zeros(4)
+                substeps = 0
+                kernel = windowed._levels[1].kernel
+
+                def audited(src, dst, *args, **kw):
+                    nonlocal scale, substeps
+                    windowed._stream.synchronize()  # BFL and both accumulations are complete
+                    raw = np.empty((count, 4), np.float32)
+                    cuda.memcpy_dtoh(raw, b.last_raw)
+                    scale = scale + (np.abs(raw.astype(np.float64)) * factors).sum(axis=0)
+                    substeps += 1
+                    kernel(src, dst, *args, **kw)
+                windowed._levels[1].kernel = audited
                 keys = ("impulse", "angular_impulse", "mass_into_fluid")
                 summed = {part: [0., 0., 0., 0.] for part in ("total", "wall", "aperture")}
-                scale = np.zeros(4)
                 steps = 0
                 for speed in (0., .002, .002, -.001, 0., 0.):  # amplitude updates stay in-window
                     windowed.set_jet_amplitude(0, speed)
@@ -123,16 +148,13 @@ class ExchangeWindowGPU(unittest.TestCase):
                         summed[part] = [summed[part][0] + o["impulse"][0], summed[part][1] + o["impulse"][1],
                                         summed[part][2] + o["angular_impulse"],
                                         summed[part][3] + o["mass_into_fluid"]]
-                    rows = np.empty((count, 4), np.float64)
-                    cuda.memcpy_dtoh(rows, b.interval_integral)
-                    scale += np.abs(rows).sum(axis=0)
                     for a, e in zip(windowed._download_raw().values, plain._download_raw().values):
-                        np.testing.assert_array_equal(np.isnan(a), np.isnan(e))
+                        same_bits(self, a, e)
-                        np.testing.assert_array_equal(a[np.isfinite(a)], e[np.isfinite(e)])
                 out = windowed.read_exchange_window(0)
                 self.assertEqual((out["time_end"] - out["time_start"], out["sample_count"]),
                                  (steps, 2 * steps))
-                bound = 2 * (count * steps) * EPS * scale + np.finfo(np.float64).tiny
+                self.assertEqual(substeps, 2 * steps)
+                bound = 4 * (count * substeps) * EPS * scale + 1e-300
                 for part in summed:
                     o = out[part]
                     got = np.array([o["impulse"][0], o["impulse"][1], o["angular_impulse"],