content(app-G): 明确位置速度混合协议与方向符号

区分候选角导数和位置轨迹的时间导数，给出频移近似误差。
明确保留混合协议及统一可微轨迹时各接口的修改条件。
补充独立差分、固定节点 OLS 与方向符号检查，全部通过。
XeLaTeX 多遍与引用检查通过；PDF 转换受外部进程文件占用阻塞。

check_finite_range_bearing.py

@@ -74,6 +74,35 @@ def geometry(x, t, ell):
     return w, W, wd, WD, b, bx, bd, bdx
 
 
+def mixed_motion(x, t, ell):
+    """Synthetic non-spin rotation: deliberately not an astronomical model."""
+    p0, P0 = intersection(x)
+    a, k = 7.292115e-5*t, 3e-5
+    z = np.array([[np.cos(a), -np.sin(a), 0.],
+                  [np.sin(a), np.cos(a), 0.], [0., 0., 1.]])
+    a = k*t
+    q = np.array([[1., 0., 0.], [0., np.cos(a), -np.sin(a)],
+                  [0., np.sin(a), np.cos(a)]])
+    generator = np.array([[0., 0., 0.], [0., 0., -k], [0., k, 0.]])
+    B = z@q
+    Bd = SPIN@B + B@generator
+    C, Cd = SPIN@B, SPIN@Bd
+    p, pd, vu = B@p0, Bd@p0, C@p0
+    s = SATELLITES[ell] + t*VELOCITIES[ell] + .5*t*t*ACCELERATIONS[ell]
+    vs = VELOCITIES[ell] + t*ACCELERATIONS[ell]
+    r = np.linalg.norm(s-p)
+    w = (s-p)/r
+    S = np.eye(3)-np.outer(w, w)
+    wd = S@(vs-pd)/r
+    proxy = S@(vs-vu)/r
+    W = -S@B@P0/r
+    b = -(vs-vu)@w/WAVELENGTH
+    bg = -(vs-pd)@w/WAVELENGTH
+    bd = -((ACCELERATIONS[ell]-Cd@p0)@w + (vs-vu)@wd)/WAVELENGTH
+    bx = -(-w@C@P0+(vs-vu)@W)/WAVELENGTH
+    return p, pd, vu, w, wd, proxy, b, bg, bd, bx
+
+
 def coefficients(x, ell):
     data = [geometry(x, t, ell) for t in TIMES]
     ref = [geometry(x, t, 0) for t in TIMES]
@@ -168,6 +197,33 @@ def main():
     Tref[:2, :2] = finite_difference(lambda x: bearing(intersection(x)[0], SATELLITES[1]), x)
     Tref[2:, :2] = Cx
     check("reference-change full Jacobian", Tref, finite_difference(switch, z), 2e-6)
+    for ell in range(2):
+        p, pd, vu, w, wd, proxy, b, bg, bd, bx = mixed_motion(x, .41, ell)
+        step = 1e-3
+        check("mixed position velocity", pd,
+              (mixed_motion(x, .41+step, ell)[0]-mixed_motion(x, .41-step, ell)[0])/(2*step),
+              2e-6)
+        check("mixed true LOS time derivative", wd,
+              (mixed_motion(x, .41+step, ell)[3]-mixed_motion(x, .41-step, ell)[3])/(2*step))
+        check("mixed node Doppler Jacobian", bx,
+              finite_difference(lambda x: mixed_motion(x, .41, ell)[6], x))
+        check("mixed Doppler time derivative", bd,
+              (mixed_motion(x, .41+step, ell)[6]-mixed_motion(x, .41-step, ell)[6])/(2*step))
+        check("mixed Doppler discrepancy", b-bg, (vu-pd)@w/WAVELENGTH)
+        check("mixed proxy LOS discrepancy", proxy-wd,
+              -(np.eye(3)-np.outer(w, w))@(vu-pd)/np.linalg.norm(
+                  SATELLITES[ell]+.41*VELOCITIES[ell]+.5*.41**2*ACCELERATIONS[ell]-p))
+        assert np.linalg.norm(vu-pd) > 1, "Fixture must expose the mixed protocol"
+        e = -w
+        check("orientation steering identity",
+              np.exp(-2j*np.pi*ARRAY@w/WAVELENGTH),
+              np.exp(2j*np.pi*ARRAY@e/WAVELENGTH))
+        check("orientation Doppler identity", b, (VELOCITIES[ell]+.41*ACCELERATIONS[ell]-vu)@e/WAVELENGTH)
+    mixed_coeff = lambda x: np.linalg.lstsq(DESIGN, [
+        mixed_motion(x, t, 1)[6]-mixed_motion(x, t, 0)[6] for t in TIMES], rcond=None)[0]
+    mixed_jac = np.linalg.lstsq(DESIGN, [
+        mixed_motion(x, t, 1)[9]-mixed_motion(x, t, 0)[9] for t in TIMES], rcond=None)[0]
+    check("mixed fixed-design OLS Jacobian", mixed_jac, finite_difference(mixed_coeff, x))
     print("PASS: synthetic local formula checks; code-side geometry and CP/IP acceptance still required.")