in a system with a spectral gap, the entanglement entropy of a region scales with the area of its boundary, not the volume it encloses. the interior holds no more information than its surface can carry
proven in one dimension (Hastings 2007) as a consequence of the gap; measured in cold-atom experiments via Rényi entropy (Islam 2015); the working assumption behind every tensor-network method that succeeds
the same shape appears in gravity: black-hole entropy is proportional to horizon area, not volume — the observation that started the holographic principle
for cyber the statement is computational rather than quantum. because the tri-kernel is gapped, a focus update inside a region is determined by flows across its boundary, so a light client verifies by checking boundary commitments instead of recomputing the bulk. compute-verify symmetry is the area law read as a cost model
see Lieb-Robinson bound · spectral gap · physical analogies
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