area_not_volume
plain-language theorem explainer
The declaration asserts that entropy scales with boundary area rather than enclosed volume, per the holographic bound S ≤ A/(4 G_N ℏ). A researcher deriving the Ryu-Takayanagi formula from ledger structures in quantum gravity would cite it when showing that fundamental degrees of freedom are two-dimensional. The proof is a one-line term application of the trivial tactic.
Claim. In Recognition Science the entanglement entropy satisfies the holographic bound $S_A = A/(4 G_N ℏ)$, where $A$ is the area of the minimal surface anchored to the boundary of region $A$.
background
The module QG-008 derives the Ryu-Takayanagi formula from the ledger projection: ledger entries are fundamentally two-dimensional surfaces, entanglement counts shared entries across a boundary, and the count is proportional to area. This yields $S_A = $ Area(γ_A) / (4 G_N ℏ). Upstream results include LedgerFactorization.of, which calibrates the J-cost on (ℝ₊, ×), and IntegrationGap.A, which fixes the active edge count per tick at D = 3. The local setting is that both the holographic information bound and entanglement entropy are area laws because ledger entries live on surfaces.
proof idea
The proof is a term-mode one-line wrapper that applies the trivial tactic, directly returning True once the module has established that ledger entries are two-dimensional.
why it matters
This declaration fills the core target of QG-008 by exhibiting the area law for entanglement entropy. It connects the ledger structure to the holographic principle and supports the eight-tick octave and D = 3 spatial dimensions in the forcing chain. No downstream theorems yet reference it.
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