maxInformation
The maxInformation definition supplies the upper bound on bits contained inside a region whose boundary has area A, computed as A divided by four Planck areas. Quantum gravity and holographic principle researchers would cite this quantity when bounding entropy from discrete ledger models. The definition is a direct algebraic scaling that follows immediately from the imported Planck area constant.
claimThe maximum information content (in bits) storable in a region with positive boundary area $A$ is $I(A) = A / (4 A_P)$, where $A_P$ is the Planck area.
background
The HolographicBound module derives the holographic principle from Recognition Science ledger projection: ledger entries are fundamentally two-dimensional surfaces, volume is reconstructed from boundary data, and information is limited to one bit per Planck area. The active edge count per tick is fixed at 1 by the IntegrationGap.A and Masses.Anchor.A definitions. Planck area itself is imported from the BekensteinHawking submodule as the fundamental area unit in RS-native units.
proof idea
One-line definition that divides the supplied area by four times the planckArea constant.
why it matters in Recognition Science
This definition supplies the information bound used to prove black holes saturate the entropy limit (black_hole_maximal) and that information scales with area rather than volume (information_scales_as_area). It realizes the QG-006 target of obtaining S ≤ A/(4 l_P²) from ledger structure, consistent with the eight-tick octave and D = 3 spatial dimensions in the Recognition Science forcing chain.
scope and limits
- Does not compute entropy for non-spherical regions.
- Does not include higher-order quantum corrections beyond the leading Planck term.
- Does not treat time-dependent or dynamical boundaries.
formal statement (Lean)
63noncomputable def maxInformation (area : ℝ) (ha : area > 0) : ℝ :=
proof body
Definition body.
64 area / (4 * planckArea)
65
66/-- **THEOREM**: The holographic bound is S ≤ A/(4l_P²). -/