# PH-BC-002

**Name:** Boundary flux imbalance (divergence theorem)

**Severity:** warning

**Input modes:** adapter+dump

**Scope separation (read first):** PH-BC-002's external validation
separates **(i) a harness-level Gauss-Green correctness fixture** from
**(ii) the production rule's Laplace and Poisson divergence-theorem
verdict behavior**. The production rule `src/physics_lint/rules/ph_bc_002.py`
covers the scalar Laplace/Poisson divergence-theorem imbalance —
narrower than the F1 mathematical anchor (the general Gauss-Green /
divergence theorem for arbitrary C¹ vector fields). CITATION.md,
README, and test docstrings do not imply broader production coverage
than the rule provides.

Boundary flux imbalance check: for a Laplace-harmonic field `u`, the
rule verifies `∫_Ω Δu dV ≈ 0` (equivalently `∫_{∂Ω} ∂u/∂n dS ≈ 0` by
Gauss-Green); for a Poisson field (`-Δu = f`) it verifies
`∫_Ω Δu dV + ∫_Ω f dV ≈ 0`. Both compare against relative thresholds
of 0.01 / 0.1 against `||u||_{L²}`.