Interestingness as an Inductive Heuristic for Future Compression Progress

We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) - O(1) <= K_R(u_{0:n}) <= K(Pi_n) + O(1), where K_R(u_{0:n}) is the length of the shortest program answering every query in the class R, and the O(1) overhead is independent of both the sequence length n and the stack depth k. Sufficiency follows from the classical wiping property of the Preisach hysteresis operator. Minimality is established via a finite indicator family whose rate-independence is verified explicitly. Any compression of a hysteresis-driven stream that preserves the full class R must therefore retain at least K(Pi_n) - O(1) bits; the stack-based compression algorithm implied by the result carries a Kolmogorov optimality guarantee that none of the standard time-series compression methods provide.

Comparing intelligent systems under deployment constraints requires more than predictiveaccuracy.This paper develops Compression-Prediction-Resource Intelligence Efficiency (CPR-IE) as a protocol-relative ordering by representational economy, predictive quality, and resourceburden. The analysis separates two questions-how raw resource consumption is represented, andhow the resulting attributes are aggregated. Proportional-increment composition uniquely yieldslogarithmic cumulative burden, and context-independent ratio response yields power responsesto compression, prediction, and burden; with reference normalization the representation is I(C,P,T).We prove Pareto consistency, unit invariance, boundary behavior, trade-off identities, ranking-stability regions, and cross-task aggregation. A translog parent model makes interaction restrictions explicit, and further results establish cardinal and ordinal identification, sub-Gaussianfinite-sample ranking guarantees, robust selection under exponent uncertainty, and deterministicregret bounds. Minimum description length, algorithmic complexity, proper scoring rules, varia-tional inference, and Landauer's principle motivate measurement choices but do not entail theformula. CPR-IE is a constructed efficiency representation, not a universal law or a definition ofintelligence itself.

Compression progress is a long-standing proposal for intrinsic motivation: reward an agent when its world model becomes better at predicting or compressing experience. The folk claim is that this reward is "credible" because it is paid only for learning. We make this precise and prove it. If intrinsic reward is the signed decrease of a fixed sealed-audit loss, r_t = E(theta_{t-1}) - E(theta_t), then cumulative reward telescopes exactly to endpoint audit improvement, so no policy can push reward up indefinitely while true audit performance stagnates or degrades. For finite audit panels the same result holds with a sharp false-positive budget: cumulative empirical reward is at most true audit improvement plus 2 Delta_n(F, delta), the uniform audit deviation of the model class. This is horizon-free: adaptivity over time costs nothing once the sealed panel uniformly controls the class. The theorem also identifies the failure modes: the guarantee disappears if progress is clipped, scored on the agent's own stream, exposed to a high-capacity model on a reusable panel, or applied to a neural class that makes Delta_n vacuous. We give a Lean 4 mechanization of the structural core (telescoping, the finite-audit bound, finite Gibbs, and the entropy floor) and an experiment suite on ARC-TGI grid-transformation generators with adaptive holdout attacks. Experiments confirm the theory: finite-audit deviation scales as n^{-0.527}; signed progress resists clip-farming, stream leakage, and noisy-TV curiosity; naive reusable audits are exploitable by black-box scalar feedback, while standard release defenses keep the attack below the 2 Delta_n threshold. Signed compression progress on a sealed audit is an accounting signal of genuine improvement.