module module high

`IndisputableMonolith.Quantum.HolographicBound`

This module supplies the core definitions for the holographic bound in Recognition Science, expressed in natural units where the Planck length equals 1. It introduces Planck area, bits per Planck area, maximum information capacity, the Bekenstein bound, and related scaling relations. Workers on quantum information or RS unification cite these when linking boundary area to ledger events. The module is purely definitional with no proofs.

claimIn RS-native units with $l_P = 1$, the module defines Planck area $A_P$, bits per Planck area, maximum information $I_0(A) = A/(4 l_P^2)$, the holographic bound, Bekenstein bound, sphere area and volume, information scaling as area, holographic ratio, and holography from ledger.

background

The module sits inside the Quantum domain and imports Constants, whose fundamental object is the RS time quantum τ₀ = 1 tick. In this setting all quantities are expressed with c = 1 and lengths measured in Planck units, so the Planck length is fixed at l_P = 1. The sibling definitions then build the standard holographic quantities: Planck area, bits per Planck area, maxInformation, holographic_bound, bekensteinBound, and the scaling statements information_scales_as_area and holography_from_ledger.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the holographic-bound definitions required by the downstream RecognitionBandwidth unification. That module states: 'Five elements of Recognition Science have never been formally connected: 1. Holographic bound — max information ∝ boundary area / (4 Planck areas)'. The present definitions close the first of those five gaps and enable the later connection to recognition cost per bit k_R = ln(φ) and the 8-tick cadence.

scope and limits

Does not derive the holographic principle from the Recognition Composition Law.
Does not include quantum corrections or higher-curvature terms.
Does not compute numerical values for specific geometries beyond the definitions.
Does not address the relation to the phi-ladder mass formula.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Unification.RecognitionBandwidth module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (23)

def planckLength L51
def planckArea L54
def bitsPerPlanckArea L57
def maxInformation L63
theorem holographic_bound L67
def bekensteinBound L74
def sphereArea L80
def sphereVolume L83
theorem information_scales_as_area L87
def holographicRatio L95
theorem holographic_ratio_scales L98
theorem holography_from_ledger L116
theorem bulk_from_boundary L122
def blackHoleEntropy L131
theorem black_hole_maximal L136
theorem exceed_bound_makes_black_hole L142
structure DegreeOfFreedomCounting L150
theorem no_lost_dof L163
structure AdSCFT L174
theorem ryu_takayanagi L185
def holographicPredictions L197
structure HolographicFalsifier L211
def experimentalStatus L218