holographic_ratio_scales
plain-language theorem explainer
The declaration shows that the holographic ratio for a sphere of positive radius R simplifies exactly to 3 over R. Researchers deriving information bounds in Recognition Science from ledger projections would cite this algebraic identity when connecting surface scaling to volume in three dimensions. The proof is a direct unfolding of the ratio definition followed by field simplification after establishing non-zero conditions for R and pi.
Claim. For a positive real number $R$, the holographic ratio defined as the surface area of a sphere of radius $R$ divided by its volume equals $3/R$.
background
The module QG-006 derives the holographic bound from Recognition Science ledger structure. Ledger entries are fundamentally two-dimensional objects living on causal horizons, so the three-dimensional interior volume is reconstructed from boundary data and information scales with area rather than volume. The standard holographic principle states that entropy satisfies $S ≤ A / (4 l_P^2)$, with black holes saturating the bound as maximally dense ledgers. Upstream results supply the necessary structures: LedgerFactorization calibrates the J-cost on the positive reals, PhiForcingDerived encodes the self-similar fixed point, and PrimitiveDistinction reduces seven axioms to four structural conditions plus definitional facts.
proof idea
The proof unfolds the definitions of holographicRatio, sphereArea, and sphereVolume to expose the explicit ratio $(4πR^2) / ((4/3)πR^3)$. It introduces the auxiliary facts that R is nonzero and pi is nonzero, then applies field_simp to cancel common factors and obtain the simplified form 3/R.
why it matters
This identity supplies the elementary scaling step inside the ledger-projection argument for the holographic bound. It sits inside the module's derivation that information limits arise automatically from two-dimensional ledger entries rather than being imposed by hand. The result aligns with the framework's forcing chain at T8 where three spatial dimensions are fixed, and it supports the area-law behavior required for the alpha band and Planck-scale constants. No downstream theorems are recorded yet, leaving the connection to the full Bekenstein or holographic entropy statements open for later closure.
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