Beyond Kron Equivalence: Terminal Detection of Invisible Internal Gain

Kron reduction preserves the terminal response of a network while eliminating the internal degrees of freedom that generate it. Systems with the same reduced operator may therefore have very different internal voltage amplification, hidden response geometry, and sensitivity to changes in the eliminated sector. This work develops an exact framework for detecting and reconstructing this hidden structure. For resistive nodal systems, it identifies the grounding balance that controls internal amplification and separates the passive maximum-principle regime from active configurations in which large hidden gains can remain invisible to the unperturbed terminal response. A controlled common load applied to the hidden ports converts the missing geometry into measurable terminal data. The resulting family of reduced responses recovers the Gram matrix of the harmonic lift, its effective hidden dimension, and the terminally visible part of the hidden spectrum. Finite load increments also provide robust sufficient certificates for hidden active grounding. The framework extends to non-Hermitian systems, where right and left hidden geometries carry an additional relative phase. Their mixed Gram component can be recovered from terminal sensitivity once the two diagonal tail geometries are known. Thus Kron equivalence preserves the external response but does not exhaust the internal geometry compatible with it.