IndisputableMonolith.Information.KolmogorovComplexity3FromJCost
Links three-dimensional Kolmogorov complexity to the Recognition Science J-cost. Defines a domain cost pulled back from J, a positive canonical threshold, and an inhabited certificate that the complexity-3 bound holds at that threshold. Information theorists and RS auditors cite it when tying algorithmic complexity to the forced cost functional. The module is mostly definitions plus nonnegativity and positivity lemmas.
claimOn a domain equipped with the RS cost $J(x)=(x+x^{-1})/2-1$, define a domain cost $C$, a canonical threshold $\theta>0$, and a certificate asserting that the Kolmogorov complexity bound in three dimensions is controlled by $C$ at $\theta$.
background
Recognition Science forces a unique nonnegative cost $J$ on ratios via the Recognition Composition Law and the T5 uniqueness step: $J(x)=\cosh(\log x)-1$. The Cost import supplies that functional; Constants supplies the RS tick $\tau_0$.
This module sits in the Information domain. It treats algorithmic (Kolmogorov) complexity in the forced spatial dimension $D=3$ (T8) as a cost measured against $J$, rather than as an independent combinatorial primitive. Sibling definitions introduce a domain-level cost, its evaluation identity, nonnegativity, a canonical positive threshold, and a certificate type packing the complexity-3 claim.
proof idea
Definition-heavy module. Domain cost is built from $J$; nonnegativity and the evaluation identity are short lemmas. The canonical threshold is a positive constant derived from the cost scale. The certificate is an inhabited structure bundling those facts so downstream code can assume the complexity-3 bound without re-deriving the cost comparison.
why it matters in Recognition Science
Closes an information-theoretic gap in the RS stack: once $J$ and $D=3$ are forced, Kolmogorov complexity should not float free of the cost functional. The certificate is the reusable handle for that identification. No downstream users are wired yet in the graph, so the module is a leaf ready for complexity or entropy arguments that need a $J$-native 3D bound. It does not itself re-prove T5 or T8; it consumes them via Cost and the dimensional convention.
scope and limits
- Does not re-derive J-uniqueness (T5) or the RCL identity.
- Does not prove D=3; assumes the forced spatial dimension.
- Does not give explicit Kolmogorov programs or bitstring constructions.
- Does not bound classical Shannon entropy or channel capacity.
- Does not connect to mass-ladder or alpha-band numerics.