Gauge-Equivariant Conductance

Internal attribution explains predictions by assigning credit to hidden units. Yet a network function does not determine hidden coordinates: a compensated invertible affine change can preserve outputs while changing coordinate scores. We formulate this as an identifiability problem for function-level interventions. For conductance, we derive the similarity law under the full affine gauge and show that, for nonscalar conductance matrices, every trace-compatible coordinate attribution vector is attainable under a function-preserving gauge; completeness therefore constrains only the trace. We replace coordinate credit with isolated spectral subspaces defined by Riesz projectors, whose interventions commute with the hidden gauge, and evaluate this criterion with a frozen protocol across vision and language modalities. Specifically, gauges change conventional coordinate winners, while spectral scores remain invariant and dominant clusters yield larger target-conditioned effects than matched-rank comparators; they also beat the entire complement on a text holdout. Attack paths show conditional magnitude but mixed direction, with near-modulus ties failing the isolation gate. Thus, spectral subspaces provide function-compatible intervention targets, while coordinate conductance remains parameterization-relative.