re: a bank's wave 1 is not a filtered copy of wave 0 -- and my own discriminator cannot finish the job

Coherence on BGM_103, the menu's bank, with controls run first: a real linear
filter of wave 0 reads 0.93-0.94 in every band, a different bank reads 0.001, and
wave 0 misaligned by 1 s reads 0.004-0.057. The measurement reads 0.027 at 1-4 kHz,
so the 'wave 1 is wave 0 filtered' model is refuted.

The frequency structure is inverted relative to any mic-pair or reverb model:
coherence rises with frequency (0.169 -> 0.827) while energy falls (71 % -> 0.2 %),
and a rear pair decorrelates fastest at HF. In the midrange the two waves are 13x
further apart than the two channels of one wave.

But the L-R control is what limits the tool and it is recorded as such: within one
wave, genuinely one performance in two channels, coherence is only 0.221-0.497. So
'same performance' does not imply high coherence here, my positive control was the
wrong model of the rear-pair reading, and the 🟡 is NOT settled. The tool tests for
linear filtering and neither surviving reading requires it.

Also corrects MISSION's Q10 row, which still carried the refuted three-sub-wave
premise and had directed work at a dead question for days. Its gate is in fact met.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Wuu56cE8vJGTBtn1ppsk8v

tools/re-capture/bgm_stem_coherence.py

@@ -0,0 +1,71 @@
+#!/usr/bin/env python3
+"""Are a music bank's two waves the SAME instruments filtered, or DIFFERENT parts?
+
+`structures/bgm-two-stems.md` leaves two readings alive for wave 1 -- the rear
+pair of a 4-channel mix, or a second intensity layer -- and notes that runtime
+simultaneity cannot separate them, since both predict it.
+
+This tries a static discriminator.  Magnitude-squared coherence is ~1 wherever
+one signal is a LINEAR FILTER of the other, and ~0 for independent signals, so
+a rear pair modelled as "front pair, filtered" should read high and a different
+arrangement layer should read low.
+
+⚠️ READ THE CONTROLS BEFORE THE MEASUREMENT.  The L-vs-R control below is the
+one that matters and it is the one that limits this tool: see the docs page.
+
+    bgm_stem_coherence.py <w0.wav> <w1.wav> <control.wav>
+
+Waves come from slb_extract_wave.py + ffmpeg; see the docs page for the exact
+offsets and packet counts.
+"""
+import sys, wave, numpy as np
+
+NFFT, HOP, FS = 8192, 4096, 48000
+BANDS = [(0,200),(200,1000),(1000,4000),(4000,12000),(12000,16000),(16000,24000)]
+
+def load(p, nmax, stereo=False):
+    w = wave.open(p); n = min(nmax, w.getnframes())
+    a = np.frombuffer(w.readframes(n), dtype="<i2").astype(np.float64)
+    a = a.reshape(-1, w.getnchannels())
+    return (a[:,0], a[:,1]) if stereo else a.mean(axis=1)
+
+def _acc(x, y):
+    win = np.hanning(NFFT); Sxx = Syy = Sxy = 0.0; k = 0
+    for i in range(0, min(len(x), len(y)) - NFFT, HOP):
+        X = np.fft.rfft(x[i:i+NFFT]*win); Y = np.fft.rfft(y[i:i+NFFT]*win)
+        Sxx = Sxx + abs(X)**2; Syy = Syy + abs(Y)**2; Sxy = Sxy + X*np.conj(Y); k += 1
+    return np.abs(Sxy)**2/(Sxx*Syy+1e-30), Sxx, Syy, k
+
+def coh(x, y): return _acc(x, y)[0]
+def spec(x):   return _acc(x, x)[1]
+
+def row(name, C, f):
+    print(f"  {name:34} " + "  ".join(
+        f"{lo/1000:g}-{hi/1000:g}k {C[(f>=lo)&(f<hi)].mean():.3f}" for lo,hi in BANDS))
+
+if __name__ == "__main__":
+    p0, p1, pc = sys.argv[1], sys.argv[2], sys.argv[3]
+    D = 60*FS; f = np.fft.rfftfreq(NFFT, 1/FS)
+    w0 = load(p0, D); w1 = load(p1, D); ctl = load(pc, D)
+    L0,R0 = load(p0, D, True); L1,R1 = load(p1, D, True)
+
+    # POSITIVE control: an actual linear filter of w0 (one-pole LP + 12 ms delay)
+    a = 0.06; lp = np.empty_like(w0); acc = 0.0
+    for i, v in enumerate(w0):
+        acc += a*(v-acc); lp[i] = acc
+    d = int(0.012*FS); pos = np.concatenate([np.zeros(d), lp[:-d]])
+    sh = np.concatenate([np.zeros(FS), w0[:-FS]])
+
+    print("CONTROLS")
+    row("POS w0 vs linear-filter(w0)", coh(w0,pos), f)
+    row("NEG w0 vs a different bank",  coh(w0,ctl), f)
+    row("NEG w0 vs w0 shifted 1 s",    coh(w0,sh),  f)
+    row("REF w0 L vs R (one perf.)",   coh(L0,R0),  f)
+    row("REF w1 L vs R (one perf.)",   coh(L1,R1),  f)
+    print("\nMEASUREMENT")
+    row("w0 vs w1",                    coh(w0,w1),  f)
+    print("\nENERGY SHARE")
+    for nm,(L,R) in (("w0",(L0,R0)),("w1",(L1,R1))):
+        S = spec(L)+spec(R); t = S.sum()
+        print(f"  {nm:34} " + "  ".join(
+            f"{lo/1000:g}-{hi/1000:g}k {100*S[(f>=lo)&(f<hi)].sum()/t:5.1f}%" for lo,hi in BANDS))