kolmogorovComplexity
plain-language theorem explainer
Kolmogorov complexity K(x) is introduced as the length of the shortest program that outputs string x, which Recognition Science equates to the minimum ledger description of x. Information theorists deriving compression bounds from J-cost would cite the definition when linking algorithmic complexity to physical entropy. The implementation is a direct string constant accompanied by comments on incompressibility and the J-cost-to-entropy ratio.
Claim. The Kolmogorov complexity $K(x)$ of a string $x$ is the length of the shortest program that outputs $x$. In Recognition Science this equals the minimum ledger description length of $x$.
background
The module INFO-003 derives fundamental limits on data compression from J-cost. Shannon's source coding theorem states that average code length is at least the entropy $H(X) = -∑ p(x) log₂ p(x)$, so no scheme beats entropy. In Recognition Science, information carries J-cost, compressed forms have lower J-cost, and the entropy limit is the minimum J-cost for faithful representation. Upstream, entropy of a configuration equals its total defect (zero defect yields zero entropy), while cost is the derived cost of a multiplicative recognizer's comparator on positive ratios. The constant K is defined as φ^{1/2}.
proof idea
The definition is a direct string assignment stating 'Shortest program length to output x' together with an inline comment block on incompressibility: at most 2^{n-1} strings of length n compress to n-1 bits, so most strings satisfy K(x) ≈ n and random equals incompressible with maximum J-cost-to-entropy ratio.
why it matters
This definition supplies the RS-native reading of Kolmogorov complexity for the compression module, grounding Shannon limits in J-cost minimization. It supports the claim that entropy equals minimum J-cost for faithful representation and connects to the phi-ladder via upstream defect-based entropy and multiplicative cost functions. No downstream uses are recorded yet.
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