DegreeOfFreedomCounting
plain-language theorem explainer
DegreeOfFreedomCounting packages a radius with explicit naive volume-based and holographic area-based counts of degrees of freedom. Researchers deriving information bounds from ledger projections in Recognition Science would cite the structure when contrasting R³ and R² scalings. The definition is a direct record that equates the naive field to sphere volume over Planck area to the 3/2 power and the holographic field to sphere area over Planck area.
Claim. A structure consisting of radius $r$, naive count $n$, and holographic count $h$ satisfying $n = V(r)/a_P^{3/2}$ and $h = A(r)/a_P$, where $V(r)$ denotes sphere volume, $A(r)$ denotes surface area, and $a_P$ is Planck area.
background
Module Quantum.HolographicBound targets derivation of the holographic bound from Recognition Science ledger projection. The ledger is fundamentally two-dimensional, so volume is reconstructed from boundary data and information is limited to one bit per Planck area. Naive counting therefore scales as volume while holographic counting scales as area, with the supplied structure recording both for a spherical region of given radius.
proof idea
Structure definition that directly introduces the radius field together with naive and holographic count fields and the two equality axioms enforcing the volume and area relations. No lemmas or tactics are applied; the declaration serves as a data type packaging the contrast between the two counting schemes.
why it matters
The structure supplies the concrete type needed to contrast volume and area scalings inside the holographic-bound derivation aimed at a PRD paper on holography from ledger structure. It illustrates redundancy of bulk degrees of freedom with boundary data, consistent with the framework's 2D ledger and the emergence of D=3 from the eight-tick octave. No downstream uses are recorded yet, leaving it as foundational scaffolding for later results such as bekensteinBound and holographicRatio.
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