Certified boundary-magic witness for state-dependent proto-area in a holographic code

Abstract

We study a four-qubit product-EPR holographic code whose reconstructing region contains matter and one leg of a geometry bond. Seven deformations compare local, bipartite, bond-stabilizing, and bond-moving operators. Exact spectra show that only matter-controlled bond motion produces a leading quadratic logical-state dependence of the boundary entropy. We select one state-independent physical recovery by optimizing coherent information of the channel Choi state. A separate semidefinite program maximizes entanglement fidelity over all channels. At fixed coupling the physical recovery attains that optimum to numerical precision, while recovered-entropy subtraction leaves a nonzero variational proto-area response. Total stabilizer Renyi magic is also nonzero for deformations with no boundary response, so it does not characterize the effect. Projecting the exact stabilizer-Renyi quadratic form away from pairwise Pauli tangents defines a leading-order boundary witness. Its global convex minimum is positive for the bond-moving deformation and zero for the six comparisons. For the reference-matter-geometry resource partition, however, pairwise tangents span the full projective tangent space at the product-EPR base, so the witness vanishes for every Hermitian deformation. A two-bound extension retains this degeneracy but exhibits a split-geometry structural residual and a logical-state-dependent cut switch. These results establish a finite-code witness, not a universal magic law or continuum gravitational dynamics.