diff --git a/docs/contents/shmc.md b/docs/contents/shmc.md
index 793b359f5..71b380b75 100644
--- a/docs/contents/shmc.md
+++ b/docs/contents/shmc.md
@@ -169,6 +169,13 @@ U^{ba}(x,\mu) T^a = U(x,\mu) T^b U^{-1}(x,\mu)
 \ea
 $$
 
+therefore
+$$
+\ba
+U^{ba}(x,\mu) T^b = U^{-1}(x,\mu) T^a U(x,\mu)
+\ea
+$$
+
 ### Gauge momentum evolve
 
 $$
@@ -211,16 +218,16 @@ $$
 \ba
 \frac{d}{dt}\pi_2(x,\mu)
 &=
--\frac{1}{2} T^a \frac{\delta S}{\delta s_2(x,\mu)^a}
+-\frac{1}{2} T^b \frac{\delta S}{\delta s_2(x,\mu)^b}
 \\
 &=
-\frac{1}{2} U^{ba}(x,\mu) T^a \frac{\delta S}{\delta s_1(x,\mu)^b}
+\frac{1}{2} U^{ba}(x,\mu) T^b \frac{\delta S}{\delta s_1(x,\mu)^a}
 \\
 &=
-\frac{1}{2} U(x,\mu) T^a U^{-1}(x,\mu) \frac{\delta S}{\delta s_1(x,\mu)^a}
+\frac{1}{2} U(x,\mu) T^a U(x,\mu) \frac{\delta S}{\delta s_1(x,\mu)^a}
 \\
 &=
-- U(x,\mu) \frac{d}{dt}\pi_1(x,\mu) U^{-1}(x,\mu)
+- U^{-1}(x,\mu) \frac{d}{dt}\pi_1(x,\mu) U(x,\mu)
 \ea
 $$
 where
diff --git a/examples-py/hmc-pure-gauge-symmetric.log b/examples-py/hmc-pure-gauge-symmetric.log
index 8a5203049..4cb678fa0 100644
--- a/examples-py/hmc-pure-gauge-symmetric.log
+++ b/examples-py/hmc-pure-gauge-symmetric.log
@@ -26,4 +26,28 @@ CHECK: ('meson_tensor_tsep', 1)
 CHECK: ('meson_jwjj_threshold', 0.1)
 CHECK: ('meson_tsep_list', [1, 2, 3, 4, 5, 6, 7])
 CHECK: ('hmc', {'max_traj': 8, 'max_traj_always_accept': 4, 'max_traj_reverse_test': 2, 'md_time': 1.0, 'n_step': 6, 'beta': 2.13, 'c1': -0.331, 'save_traj_interval': 4, 'is_saving_topo_info': True})
+CHECK: 0 'run_hmc: 1 plaq' actual: 0.7349941278702551 ; load: 0.518293870285883 .
+CHECK: target eps: 1e-05 ; actual eps: 0.34581079193798353 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 1 'run_hmc: 2 plaq' actual: 0.6690061680093138 ; load: 0.22815696564136062 .
+CHECK: target eps: 1e-05 ; actual eps: 0.9827626344254248 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 2 'run_hmc: 3 plaq' actual: 0.6347955903849722 ; load: 0.158129326711539 .
+CHECK: target eps: 1e-05 ; actual eps: 1.2022986121280272 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 3 'run_hmc: 4 plaq' actual: 0.6347955903849722 ; load: 0.158129326711539 .
+CHECK: target eps: 1e-05 ; actual eps: 1.2022986121280272 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 4 'run_hmc: 5 plaq' actual: 0.6347955903849722 ; load: 0.158129326711539 .
+CHECK: target eps: 1e-05 ; actual eps: 1.2022986121280272 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 5 'run_hmc: 6 plaq' actual: 0.6347955903849722 ; load: 0.158129326711539 .
+CHECK: target eps: 1e-05 ; actual eps: 1.2022986121280272 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 6 'run_hmc: 7 plaq' actual: 0.6347955903849722 ; load: 0.158129326711539 .
+CHECK: target eps: 1e-05 ; actual eps: 1.2022986121280272 .
+CHECK: ERROR: JSON results value does not match.
+CHECK: 7 'run_hmc: 8 plaq' actual: 0.6347955903849722 ; load: 0.158129326711539 .
+CHECK: target eps: 1e-05 ; actual eps: 1.2022986121280272 .
+CHECK: ERROR: JSON results value does not match.
 CHECK: finished successfully.
diff --git a/examples-py/hmc-pure-gauge-symmetric.log.json b/examples-py/hmc-pure-gauge-symmetric.log.json
index 2fbf15283..2d2fe54ea 100644
--- a/examples-py/hmc-pure-gauge-symmetric.log.json
+++ b/examples-py/hmc-pure-gauge-symmetric.log.json
@@ -1,34 +1,34 @@
 [
  [
   "run_hmc: 1 plaq",
-  0.518293870285883
+  0.7349941278702551
  ],
  [
   "run_hmc: 2 plaq",
-  0.22815696564136062
+  0.6690061680093138
  ],
  [
   "run_hmc: 3 plaq",
-  0.158129326711539
+  0.6347955903849722
  ],
  [
   "run_hmc: 4 plaq",
-  0.158129326711539
+  0.6347955903849722
  ],
  [
   "run_hmc: 5 plaq",
-  0.158129326711539
+  0.6347955903849722
  ],
  [
   "run_hmc: 6 plaq",
-  0.158129326711539
+  0.6347955903849722
  ],
  [
   "run_hmc: 7 plaq",
-  0.158129326711539
+  0.6347955903849722
  ],
  [
   "run_hmc: 8 plaq",
-  0.158129326711539
+  0.6347955903849722
  ]
 ]
\ No newline at end of file
diff --git a/qlat/qlat/lib/hmc.cpp b/qlat/qlat/lib/hmc.cpp
index 8b3639503..e6701fbfd 100644
--- a/qlat/qlat/lib/hmc.cpp
+++ b/qlat/qlat/lib/hmc.cpp
@@ -256,14 +256,14 @@ void set_gm_force_dual(GaugeMomentum& gm_force_dual, const GaugeField& gf,
   qacc_for(index, gf.geo().local_volume(), {
     const Geometry& geo = gf.geo();
     const Coordinate xl = geo.coordinate_from_index(index);
-    Vector<ColorMatrix> gm_dual_v = gm_force_dual.get_elems_const(xl);
+    Vector<ColorMatrix> gm_dual_v = gm_force_dual.get_elems(xl);
     const Vector<ColorMatrix> gf_v = gf.get_elems_const(xl);
     const Vector<ColorMatrix> gm_v = gm_force.get_elems_const(xl);
     qassert(gm_dual_v.size() == 4);
     qassert(gf_v.size() == 4);
     qassert(gm_v.size() == 4);
     for (int mu = 0; mu < 4; ++mu) {
-      gm_dual_v[mu] = -gf_v[mu] * gm_v[mu] * matrix_adjoint(gf_v[mu]);
+      gm_dual_v[mu] = -matrix_adjoint(gf_v[mu]) * gm_v[mu] * gf_v[mu];
     }
   });
 }