surfaceArea
plain-language theorem explainer
The surfaceArea definition counts boundary vertices in a finite graph subset S: those with at least one adjacency edge crossing to the complement. Holography researchers in Recognition Science cite it to establish that accessible information scales with surface rather than volume in D=3. The definition is a direct one-line cardinality of the filtered boundary set.
Claim. For a finite graph with vertex set $V$ and adjacency relation $adj$, the surface area of a subset $S$ is the cardinality of the boundary vertices: $|{v ∈ S | ∃ w ∉ S : adj(v,w)}|$.
background
The BrainHolography module derives holographic properties from GCIC and local cache forcing. It builds on the chain T5 (J-uniqueness) through graph rigidity at zero cost to local-global information theorems, where every connected subgraph encodes the global ledger state. Surface area enters as the measure of boundary vertices that encode bulk information, consistent with D=3 from the eight-tick octave and SpectralEmergence.V giving vertex counts $2^D$. Upstream structures such as LedgerFactorization.of and PhiForcingDerived.of calibrate the J-cost that forces the underlying graph to minimum cost configurations.
proof idea
One-line definition that applies Finset.filter to isolate vertices in S with an adjacency witness in the complement, then takes .card.
why it matters
This definition supplies the concrete surface-area measure required by the module's derivation chain for info_scales_with_boundary and brain_holography_inevitable. It closes the step from holographic_cache_from_gcic to D=3 boundary scaling, matching the Recognition Science landmark that information access follows surface area in three spatial dimensions. No open scaffolding remains for this object.
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