Finite-Resource Stochastics Probability Idealizations as Budgeted Interface Reports

We isolate a single methodological point: many “probability laws” used in modelingare ideal objects—they live on infinite sample spaces, invoke limit statements, or assumecost-free precision. We propose an operational reading compatible with a ledger discipline:whenever such an idealization functions as an observable report (a finite artifact), it mustcome with a declared protocol (depth, truncation/quantization, and precision). We give threeconcrete formal anchors. (1) There is no uniform σ-additive prior on N, hence any generatorover N hides a cutoff or a tail-weight. (2) Finite selection sets cannot realize non-atomic(continuous) laws exactly; they can only realize pushforward approximations under an explicitquantizer. (3) The Gaussian appears as a scaling-limit report from finite-depth path counting(Galton/binomial), with quantitative error controlled by the depth. These are mathematicalfacts; the “finite-resource” stance only adds a disclosure rule for when ideal objects are usedas operational claims.