IndisputableMonolith.Information.Kolmogorov_Complexity_RS
Defines Kolmogorov-style complexity certificates in Recognition Science units, built from a domain cost functional and a positive canonical threshold. Information theorists working in the RS stack would cite it when bounding description length against J-cost. The module is mostly definitions and elementary positivity or evaluation lemmas, not a deep existence proof.
claimThe module introduces a domain cost $C(\cdot)$, a canonical threshold $\theta>0$, and a Kolmogorov certificate type asserting that a description meets the cost bound relative to $\theta$. Supporting facts include nonnegativity of domain cost and inhabitance of the certificate type at the canonical data.
background
Recognition Science measures recognition effort by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS time quantum $\tau_0=1$ tick and related native units.
This module sits in the Information domain. It packages a domain-level cost (evaluation and nonnegativity lemmas) and a positive canonical threshold against which short descriptions are certified. The certificate type is the RS stand-in for a Kolmogorov bound: a witness that a object's description cost lies under the threshold, rather than a classical prefix-free machine complexity $K(x)$.
proof idea
Definition-first module. Domain cost is introduced with an evaluation identity and a nonnegativity lemma. The canonical threshold is defined and shown positive. KolmogorovCert is a structure or Prop packaging the cost bound; cert builds an instance, and cert_inhabited records that the type is nonempty at the canonical data. No deep tactic proof chain; elementary algebraic or positivity facts over the Cost layer.
why it matters in Recognition Science
Gives the Information layer a Kolmogorov-shaped certificate tied to RS cost rather than Turing-machine bit length. Downstream consumers (none linked yet in the graph) would use KolmogorovCert when comparing description length to J-cost, or when tying information bounds to the phi-ladder and eight-tick octave. It does not yet close a forcing-chain step (T0–T8); it is infrastructure for later information-theoretic claims inside the monolith.
scope and limits
- Does not prove classical Kolmogorov complexity equals RS domain cost.
- Does not construct a universal prefix-free machine or Kraft inequality.
- Does not derive mass, alpha, or forcing-chain (T0–T8) consequences.
- Does not claim used_by edges; the graph currently shows no downstream consumers.
- Does not replace the Cost module's J-uniqueness or RCL identities.