IndisputableMonolith.Papers.GCIC.ApproximateHolography
The ApproximateHolography module supplies algebraic bounds that convert small J-cost into controlled squared deviations from constancy on graphs. Recognition Science researchers cite it when extending exact rigidity results to realistic near-constant fields in holographic derivations. The argument consists of a chain of lemmas that start from the core J-to-deviation inequality and aggregate it over edges using the imported GraphRigidity summation.
claimIf $J(x) ≤ δ$ and $x > 0$, then $(x-1)^2 ≤ 2xδ$. The module lifts this pointwise bound to graph-level controls on deviation from a constant positive field when the total ratio energy $C_G[x]$ is small.
background
Recognition Science defines the J-cost by $J(x) = (x + x^{-1})/2 - 1$ for positive $x$, which vanishes only at $x=1$ and quantifies ratio mismatch. The upstream GraphRigidity module proves that on any finite connected graph the summed ratio energy $C_G[x] = ∑ J(x_v/x_w)$ is zero if and only if the field $x$ is exactly constant. The present module relaxes the zero-cost hypothesis to a small positive δ and supplies explicit deviation bounds that remain valid under this relaxation.
proof idea
The module proceeds by a sequence of lemmas. The opening inequality converts a local J-cost bound into a squared-deviation bound. Subsequent results apply the inequality edge-wise, sum the resulting terms, and extract global controls on maximum deviation, continuity with respect to δ, and a perturbation certificate. All steps are direct algebraic rearrangements that invoke only the definition of J and the exact rigidity theorem.
why it matters in Recognition Science
This module supplies the approximate-case machinery required by the downstream BrainHolography derivation, which concludes that local ledger regions encode global state with surface-area scaling. It fills the gap between the exact zero-cost rigidity statement and the small-cost regime needed for realistic holographic models in the GCIC paper.
scope and limits
- Does not apply to disconnected or infinite graphs.
- Does not recover exact constancy unless δ = 0.
- Does not derive holographic scaling laws itself.
- Assumes the field remains positive everywhere.