IndisputableMonolith.Papers.GCIC.ApproximateHolography
ApproximateHolography supplies quantitative bounds that convert small J-cost into controlled squared deviations from constancy on finite graphs. Researchers extending exact rigidity to near-holographic regimes in Recognition Science cite these results when the total ratio energy is small but nonzero. The proofs proceed by direct algebraic rearrangement of the local inequality relating J(x) to (x-1)^2, followed by summation over edges.
claimIf $J(x) \leq \delta$ and $x > 0$, then $(x-1)^2 \leq 2x\delta$. Further lemmas bound total edge deviation by the global J-cost and recover exact constancy in the zero-cost limit.
background
The module sits inside the GCIC paper sequence and imports the J-cost definition from IndisputableMonolith.Cost together with the exact rigidity theorem from GraphRigidity. The upstream GraphRigidity result states that on any finite connected graph the ratio energy $C_G[x] = \sum J(x_v/x_w)$ vanishes if and only if the positive field $x$ is constant. The present module supplies the quantitative version needed when $C_G[x]$ is merely small.
proof idea
The module opens with the elementary inequality relating $J(x)$ to quadratic deviation, then derives edge-wise and global bounds by summing the local estimate. Subsequent lemmas establish small-deviation certificates, continuity in the cost parameter, and exact recovery when the cost reaches zero. All steps are algebraic; no external lemmas beyond the definition of J are required.
why it matters in Recognition Science
These bounds feed directly into the BrainHolography module, which derives that local ledger regions must encode global information with surface-area scaling. The module therefore closes the approximate case in the GCIC derivation chain that begins from the exact rigidity statement and the T0-T8 forcing sequence.
scope and limits
- Does not treat infinite or disconnected graphs.
- Does not supply explicit constants for large-diameter graphs.
- Does not derive the functional form of J.
- Does not address fields that change sign.