PA-CDM: Position-Aware Character Detection Matching for Evaluating Handwritten Mathematical Expression Recognition

Handwritten mathematical expression recognition (HMER) is conventionally scored by exact-match rates and string-similarity metrics that are blind to where an error occurs: two predictions with identical token-error counts receive identical scores whether they misplace a subscript or swap the operands of a fraction. Render-based character detection matching (CDM) aligns glyphs robustly but remains position-blind---on controlled fraction-operand swaps it scores 0.8595 where position-aware scoring yields 0.6253. Tree-edit metrics exhibit a complementary blind spot: rewrites outside the parser's normalization coverage are penalized as structural errors (0.8552 where render-based metrics score 1.0). We propose PA-CDM, a position-aware metric that couples character detection matching with position-forest encoding and divergence-level weighting; StructPerturb v2.0, a frozen benchmark of 1,340 controlled perturbation pairs across 15 type--intensity cells; and a cross-metric consistency protocol combining a sensitivity matrix, a human study, and LLM-judge calibration. In a six-annotator study, PA-CDM attains the highest correlation with human judgments among seven automatic metrics (Spearman rho=0.9535, n=990). A frontier LLM judge correlates slightly higher (rho=0.9613) but is costly, nondeterministic, and API-dependent; PA-CDM approaches it at zero marginal cost with deterministic, diagnosable behavior.