To Cast a Stone with Six Birds: A Closure-Deficit Account of Randomness under Packaging and Budget

Ioannis Tsiokos

Abstract:

Randomness is often treated as either ontic indeterminacy or epistemic ignorance. We develop a third, operational account within Six Birds: layer-relative randomness is the predictive residue of non-closure under a chosen packaging and budget.

For a micro process X_t and a packaging Π: X → Y, we define the micro-closure deficit at scale τ by CD_τ(Π) = I(X_t; Y_{t+τ} | Y_t), the information about the next packaged state that remains hidden inside the current macro-object. This yields the exact decomposition H(Y_{t+τ} | Y_t) = H(Y_{t+τ} | X_t) + CD_τ(Π), separating intrinsic substrate uncertainty from randomness introduced by discarded distinctions.

We connect this exact quantity to computable diagnostics native to earlier Six Birds work, including route mismatch for coarse-grained Markov dynamics and predictive log-loss gaps under limited-memory models. In controlled Markov benchmarks, closure deficit vanishes on exactly lumpable partitions and rises with within-fiber heterogeneity, while route mismatch tracks it closely. In budgeted prediction, increasing memory order buys back hidden distinctions and lowers held-out log loss. In toy hashing, one-wayness and random-lookingness emerge as the same packaging-and-budget phenomenon: high-entropy inputs obey q/2^n inversion scaling, while low-entropy inputs collapse the effect.

The result is a measurable account of randomness that unifies coarse-graining, limited prediction, and feasible one-wayness without treating randomness as primitive.