holographicBits
holographicBits computes the total information capacity in bits for a surface of area A under the holographic principle. Workers deriving recognition rates or information bounds in quantum gravity cite it when linking boundary area to ledger throughput. The definition is a direct one-line algebraic division of area by four Planck areas.
claimThe total holographic capacity in bits of a surface with area $A$ is $A/(4 ell_P^2)$, where $ell_P^2$ is the Planck area.
background
Recognition Science connects the holographic bound (maximum information proportional to boundary area over four Planck areas) to the per-bit recognition cost and the eight-tick cadence. The module defines recognition bandwidth as the maximum rate of recognition events inside a holographically bounded region, given by $R_max = S_holo / (k_R · 8 tau_0)$. Upstream, k_R is defined as the natural logarithm of phi, the fundamental cost per ledger bit, while planckArea is the square of the Planck length.
proof idea
This is a one-line definition that directly implements the holographic entropy formula by dividing the input area by four times the Planck area.
why it matters in Recognition Science
This definition supplies the numerator for the bandwidth theorems that equate bandwidth to holographic bits divided by (k_R times eight-tick cadence) and that show bandwidth times cost equals the bit count. It fills the holographic capacity step in the Recognition Bandwidth module, linking the Bekenstein-Hawking entropy formula to the per-bit cost k_R and the eight-tick octave from the forcing chain.
scope and limits
- Does not incorporate the recognition cost k_R or the eight-tick cadence.
- Does not prove positivity or monotonicity of the capacity.
- Does not address quantum corrections or higher-dimensional cases.
- Does not specify physical units or embedding into spacetime.
formal statement (Lean)
115noncomputable def holographicBits (area : ℝ) : ℝ :=
proof body
Definition body.
116 area / (4 * planckArea)
117
118/-- Bandwidth equals holographic bits divided by (k_R · 8τ₀). -/