Fix additional review items: docs typos, _is_numeric, CI coverage

- Fix missing %% in double-backtick code spans (patch_icpp(2)smoothen,
  patch_icpp(j)alpha, etc.) and stray backslashes in case.md
- Add _is_numeric guard for num_fluids in check_eos_parameter_sanity
- Include benchmark.py files in CI example validation
- Extend Doxygen % lint to catch double-backtick spans too

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Author: sbryngelson

.github/workflows/lint-toolchain.yml

@@ -44,7 +44,8 @@ jobs:
       run: |
         failed=0
         passed=0
-        for case in examples/*/case.py; do
+        for case in examples/*/case.py examples/*/benchmark.py; do
+          [ -f "$case" ] || continue
           output=$(./mfc.sh validate "$case" 2>&1)
           if [ $? -eq 0 ]; then
             passed=$((passed + 1))

docs/documentation/case.md

@@ -281,12 +281,12 @@ For instance, in a 2D simulation, when a cylindrical `patch(2)` is immersed in a
 
 - `smoothen` activates smoothening of the boundary of the patch that alters the existing patch.
 When smoothening occurs, fluids of the two patches are mixed in the region of the boundary.
-For instance, in the aforementioned case of the cylindrical patch immersed in the rectangular patch, smoothening occurs when ``patch_icpp(2)smoothen = 'T'``.
+For instance, in the aforementioned case of the cylindrical patch immersed in the rectangular patch, smoothening occurs when ``patch_icpp(2)%%smoothen = 'T'``.
 `smooth_coeff` controls the thickness of the region of smoothening (sharpness of the mixture region).
 The default value of `smooth_coeff` is unity. The region of smoothening is thickened with decreasing the value.
 Optimal choice of the value of `smooth_coeff` is case-dependent and left to the user.
 
-- `patch_icpp(j)alpha(i)`, `patch_icpp(j)alpha_rho(i)`, `patch_icpp(j)pres`, and `patch_icpp(j)vel(i)` define for $j$-th patch the void fraction of `fluid(i)`, partial density of `fluid(i)`, the pressure, and the velocity in the $i$-th coordinate direction.
+- `patch_icpp(j)%%alpha(i)`, `patch_icpp(j)%%alpha_rho(i)`, `patch_icpp(j)%%pres`, and `patch_icpp(j)%%vel(i)` define for $j$-th patch the void fraction of `fluid(i)`, partial density of `fluid(i)`, the pressure, and the velocity in the $i$-th coordinate direction.
 These physical parameters must be consistent with fluid material's parameters defined in the next subsection.
 
 - `model_filepath` defines the root directory of the STL or OBJ model file.
@@ -463,10 +463,10 @@ Boundary condition patches can be used with non-characteristic boundary conditio
 Their use is detailed in [Boundary Condition Patches](#boundary-condition-patches).
 
 - `bc_[x,y,z]%%vb[1,2,3]` specifies the velocity in the (x,1), (y,2), (z,3) direction applied to `bc_[x,y,z]%%beg` when using `bc_[x,y,z]%%beg = -16`.
-Tangential velocities require viscosity, `weno_avg = T`, and `bc_[x,y,z]%%beg = -16` to work properly. Normal velocities require `bc_[x,y,z]%%end = -15` or `\bc_[x,y,z]%%end = -16` to work properly.
+Tangential velocities require viscosity, `weno_avg = T`, and `bc_[x,y,z]%%beg = -16` to work properly. Normal velocities require `bc_[x,y,z]%%end = -15` or `bc_[x,y,z]%%end = -16` to work properly.
 
 - `bc_[x,y,z]%%ve[1,2,3]` specifies the velocity in the (x,1), (y,2), (z,3) direction applied to `bc_[x,y,z]%%beg` when using `bc_[x,y,z]%%end = -16`.
-Tangential velocities require viscosity, `weno_avg = T`, and `bc_[x,y,z]%%end = 16` to work properly. Normal velocities require `bc_[x,y,z]%%end = -15` or `\bc_[x,y,z]%%end = -16` to work properly.
+Tangential velocities require viscosity, `weno_avg = T`, and `bc_[x,y,z]%%end = 16` to work properly. Normal velocities require `bc_[x,y,z]%%end = -15` or `bc_[x,y,z]%%end = -16` to work properly.
 
 - `model_eqns` specifies the choice of the multi-component model that is used to formulate the dynamics of the flow using integers from 1 through 3.
 `model_eqns = 1`, `2`, and `3` correspond to \f$\Gamma\f$-\f$\Pi_\infty\f$ model (\cite Johnsen08), 5-equation model (\cite Allaire02), and 6-equation model (\cite Saurel09), respectively.
@@ -531,7 +531,7 @@ If this option is false, velocity gradient is computed using finite difference s
 - `weno_avg` it activates the arithmetic average of the left and right, WENO-reconstructed, cell-boundary values.
 This option requires `weno_Re_flux` to be true because cell boundary values are only utilized when employing the scalar divergence method in the computation of velocity gradients.
 
-- `surface_tension` activates surface tension when set to ``'T'``. Requires `sigma` to be set and `num_fluids`. The color function in each patch should be assigned such that `patch_icpp(i)%cf_val = 1` in patches where `patch_icpp(i)%alpha = 1 - eps` and `patch_icpp(i)%cf_val = 0` in patches where `patch_icpp(i)%alpha = eps`.
+- `surface_tension` activates surface tension when set to ``'T'``. Requires `sigma` to be set and `num_fluids`. The color function in each patch should be assigned such that `patch_icpp(i)%%cf_val = 1` in patches where `patch_icpp(i)%%alpha = 1 - eps` and `patch_icpp(i)%%cf_val = 0` in patches where `patch_icpp(i)%%alpha = eps`.
 
 - `viscous` activates viscosity when set to ``'T'``. Requires `Re(1)` and `Re(2)` to be set.
 

toolchain/mfc/case_validator.py

@@ -2264,10 +2264,10 @@ def check_eos_parameter_sanity(self):
         num_fluids = self.get('num_fluids')
         model_eqns = self.get('model_eqns')
 
-        if num_fluids is None or model_eqns == 1:
+        if not self._is_numeric(num_fluids) or model_eqns == 1:
             return
 
-        for i in range(1, num_fluids + 1):
+        for i in range(1, int(num_fluids) + 1):
             gamma = self.get(f'fluid_pp({i})%gamma')
             if gamma is None or not self._is_numeric(gamma) or gamma <= 0:
                 continue

toolchain/mfc/lint_docs.py

@@ -328,7 +328,7 @@ def check_doxygen_percent(repo_root: Path) -> list[str]:
         return []
 
     ignored = _gitignored_docs(repo_root)
-    code_span_re = re.compile(r"`([^`\n]+)`")
+    code_span_re = re.compile(r"``([^`\n]+)``|`([^`\n]+)`")
     bad_pct_re = re.compile(r"(?<!%)%(?=[a-zA-Z_])")
 
     errors = []
@@ -345,7 +345,7 @@ def check_doxygen_percent(repo_root: Path) -> list[str]:
             if in_code:
                 continue
             for m in code_span_re.finditer(line):
-                span = m.group(1)
+                span = m.group(1) or m.group(2)
                 if bad_pct_re.search(span):
                     fixed = bad_pct_re.sub("%%", span)
                     errors.append(