Fix physics docs rendering: \implies → \Rightarrow, backtick code params

- Replace \implies with \Rightarrow (Doxygen MathJax compatible)
- Add _format_message() to wrap code parameter references in backticks
  in auto-generated check messages (Fortran accessors, snake_case params,
  known short params like nb/dt/m/n/p)

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

Author: sbryngelson

toolchain/mfc/case_validator.py

@@ -72,7 +72,7 @@
     "check_alpha_rho_consistency": {
         "title": "Alpha-Rho Consistency",
         "category": "Mixture Constraints",
-        "math": r"\alpha_j = 0 \implies \alpha_j \rho_j = 0, \quad \alpha_j > 0 \implies \alpha_j \rho_j > 0",
+        "math": r"\alpha_j = 0 \Rightarrow \alpha_j \rho_j = 0, \quad \alpha_j > 0 \Rightarrow \alpha_j \rho_j > 0",
         "explanation": (
             "Warns about physically inconsistent combinations: "
             "density assigned to an absent phase, or a present phase with zero density."
@@ -106,7 +106,7 @@
     "check_velocity_components": {
         "title": "Velocity Components in Inactive Dimensions",
         "category": "Velocity and Dimensional Consistency",
-        "math": r"n = 0 \implies v_2 = 0, \quad p = 0 \implies v_3 = 0",
+        "math": r"n = 0 \Rightarrow v_2 = 0, \quad p = 0 \Rightarrow v_3 = 0",
         "explanation": "Setting velocity components in dimensions that do not exist is almost certainly a mistake.",
         "exceptions": ["MHD simulations (transverse velocity couples to magnetic field in 1D)"],
     },

toolchain/mfc/gen_physics_docs.py

@@ -8,6 +8,7 @@
 
 from __future__ import annotations
 
+import re
 import sys
 from collections import defaultdict
 from pathlib import Path
@@ -47,6 +48,26 @@
     "warning": "- \\f$\\triangle\\f$ ",
 }
 
+# Regex to detect code-like tokens in validation messages.
+# Matches (in order of priority):
+#   1. Fortran-style accessors:  fluid_pp(i)%gamma, bc_y%beg
+#   2. Known short param names with optional array index: alpha(j), vel(2), Re(1), nb, dt
+#   3. Snake_case identifiers with optional array index: model_eqns, weno_order, alpha_rho(j)
+#   4. Single-letter grid dimension params: m, n, p
+_CODE_RE = re.compile(
+    r"(?<!`)\b("
+    r"\w+(?:\([^)]*\))?%\w+(?:\([^)]*\))?"
+    r"|(?:alpha|vel|Re|dt|nb|sigma|mhd|igr|Bx0|viscous|thermal|polytropic|relativity|rhoref|pref|var)(?:\([^)]*\))?"
+    r"|[a-z]\w*_\w+(?:\([^)]*\))?"
+    r"|[mnp]"
+    r")(?!\w)"
+)
+
+
+def _format_message(msg: str) -> str:
+    """Wrap code-like parameter references in backticks."""
+    return _CODE_RE.sub(r"`\1`", msg)
+
 
 def _stages_str(stages: Set[str]) -> str:
     order = ["common", "pre_process", "simulation", "post_process"]
@@ -88,7 +109,7 @@ def _render_method(doc: dict, method_rules: List[Rule], lines: List[str]) -> Non
         if msgs:
             lines.append("**Enforced checks:**\n")
             for m in msgs[:8]:
-                lines.append(f"{_SEVERITY_ICON.get(m.severity, '- ')}{m.message}")
+                lines.append(f"{_SEVERITY_ICON.get(m.severity, '- ')}{_format_message(m.message)}")
             if len(msgs) > 8:
                 lines.append(f"- *(+{len(msgs) - 8} more)*")
             lines.append("")