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

Why does a process look random? This paper develops one precise answer within the Six Birds framework: much of what a description calls randomness is information the description has chosen not to keep. We model a description by a packaging map Π that groups microstates X_t into coarse states Y_t = Π(X_t), and measure the loss with the micro-closure deficit CD_τ(Π) = I(X_t; Y_{t+τ} | Y_t): the information the current microstate still carries about the next packaged state once the current package is known. The exact identity H(Y_{t+τ} | Y_t) = H(Y_{t+τ} | X_t) + CD_τ(Π) splits packaged uncertainty into what would remain even with the full microstate and what the packaging adds. The deficit vanishes for closed (lumpable) packagings and, when every microstate has positive stationary probability, only for them. It bounds a stationary route-mismatch diagnostic through Pinsker's inequality, and for Markov substrates it caps what any amount of packaged memory can recover. Finite counterexamples show that the support condition cannot be dropped, that the bound fails for a uniform lift, and that the deficit need not shrink with longer lags or finer packagings. Three controlled experiments illustrate the account. Across seventeen benchmark conditions (ten finite Markov chains, evaluated at one or two lags), the deficit vanishes on lumpable partitions, grows with within-package heterogeneity, and is strongly correlated with route mismatch (Pearson r = 0.96). In a memory-budget benchmark, exact population entropies fall as memory grows, yet a window of up to eight past packages recovers only about 0.008 of the chain's 0.110 nats of deficit; finite-sample predictors chosen on validation data show the same first two steps of that descent. In a toy hashing experiment with truncated SHA-256, inversion success for uniform inputs agrees with an ideal random-function reference, whereas inputs drawn from a small dictionary are easy to invert for an attacker who enumerates it: a deterministic map can hide only the distinctions its inputs supply. A small Lean 4 development machine-checks the finite Kullback–Leibler bridge behind the deficit's expected-divergence form. Code, data, figure scripts, and formal files: https://github.com/ioannist/six-birds-randomness