# PH-CON-004

**Name:** Per-element conservation hotspot

**Severity:** warning

**Input modes:** adapter+dump

PH-CON-004 is a 2D mesh-based residual indicator for conservation-defect
localization. The rule operates on `MeshField` inputs from
`physics_lint[mesh]` adapters (scikit-fem 12.0.1+). For each interior
element $K$ of a triangulation, it computes
$\int_K (\Delta_{L^2\text{-proj zero-trace}} u)^2\, dx$ using a P1
finite-element basis and the zero-trace $L^2$-projection of the
Laplacian. The raw value is the ratio
$\max_K / \mathrm{mean}_K$ — a unitless localization indicator.

The rule fires (`APPROXIMATE` or `FAIL`) when the conservation-defect
concentrates on a small number of elements relative to the mesh
average; typical fired ratios exceed 10. It passes on smooth fixtures
where the ratio stays in the 2–5 range, reflecting a well-distributed
residual.

**Scope — localization, not error bounds.** PH-CON-004 is **not** a
Verfürth residual estimator. It omits the $\|hf\|^2$ source term and
the $\|h^{1/2}[\nabla u \cdot n_e]\|^2$ facet-jump terms, and it does
not h-weight per element. Consequently the rule provides neither a
reliability estimate (upper bound on $\|u - u_h\|_{H^1}$) nor an
efficiency estimate (lower bound). It does provide
conservation-defect localization: on fixtures where the zero-trace
projection is clean, hotspots co-locate with regions of large
second-derivative concentration.

The rule emits `SKIPPED` on non-MeshField inputs (no mesh means no
per-element integration) and on PDEs outside its scope. Requires
`physics-lint[mesh]`.