diff --git a/docs/re/structures/bgm-two-stems.md b/docs/re/structures/bgm-two-stems.md
index ceb207dd..031384a1 100644
--- a/docs/re/structures/bgm-two-stems.md
+++ b/docs/re/structures/bgm-two-stems.md
@@ -102,6 +102,28 @@ the other. Controlled first: a real filter of wave 0 reads **0.93–0.94** in ev
 band, a different bank reads **0.001**, and wave 0 against itself misaligned by
 1 s reads **0.004–0.057**.
 
+✅ **2026-08-31 — the identity control, which I had not run, and it strengthens
+this.** `sylpheed-port`'s rule after a disqualified instrument of their own:
+*before asking whether an instrument can measure a difference, ask whether it
+returns zero for no difference.* Applied here:
+
+| | coherence, all bands |
+|---|---|
+| **wave 0 against itself** | **1.0000** |
+| a linear filter of it, **no delay** | **1.0000 / 0.9999** |
+| the same filter **+ 12 ms delay** | 0.9288–0.9380 |
+
+So the estimator is **exact** at identity, and the 0.93–0.94 above is entirely the
+**delay's windowing cost**, not the estimator's ceiling. **The true ceiling for a
+filtered copy is 1.0**, which puts the measured **0.027** midrange further from it
+than the original control implied.
+
+📌 It also adds an argument the first pass did not make: a delay depresses
+coherence **uniformly across bands** (0.9288…0.9380 is flat). The measurement is
+**not** flat — 0.027 in the midrange against 0.83 at HF. So the observed shape is
+inconsistent with a *delayed* filtered copy as well as an undelayed one, which was
+the remaining way a rear pair could have produced it.
+
 | band | w0 vs w1 | w0's own L vs R | energy in w0 |
 |---|---|---|---|
 | 0–200 Hz | 0.169 | 0.321 | 71.4 % |