The Inductive Aliasing Theorem: Hume's Problem as a Channel Capacity Bound

This paper proves that the problem of induction — Hume's observation that past regularities do not logically necessitate future ones — is aliasing in the precise sense of the adversarial aggregation channel (AAC) framework. The observation channel maps the true law of nature, an infinite-dimensional object, to a finite data sequence, a finite-dimensional signal. The resulting aliasing is Hume's gap: multiple laws produce identical data, and no data-based method can distinguish them. The gap is not a philosophical puzzle but a channel-capacity bound, quantified exactly by the ratio of sample complexity to hypothesis-class complexity. An inductive aliasing theorem proves that for any finite sample and any law space of dimension at least two, the fiber of laws consistent with the observed data has positive dimension. For any dataset, there exist laws that are nearly indistinguishable from the true law in observed data but diverge arbitrarily in their predictions. An inductive Nyquist theorem establishes a sharp phase transition at the Hume boundary n* = d/dim(X), where d is the hypothesis complexity and dim(X) is the information per observation. Below this threshold, induction is fundamentally limited: data cannot distinguish the true law from its aliases regardless of the method used. Above it, consistent identification is possible. For nonparametric hypothesis classes, the Nyquist ratio is zero for all finite samples and the channel is permanently sub-Nyquist. A Popper theorem derives falsificationism as the no-free-lunch theorem for inductive channels: no inductive rule has uniformly bounded error across all law pairs at any finite sample size, but every rule can falsify specific alternatives given sufficient data. Popper's asymmetry between falsification and verification is the asymmetry between pointwise and uniform convergence of total variation. Kuhn's paradigm structure follows as a corollary: a paradigm is a prior concentrated on a region of hypothesis space, and a paradigm shift is a redistribution of discriminability governed by the conservation identity. A Bayesian enrichment theorem proves that priors are additional information subject to the conservation identity: they reduce aliasing on prior-favored law pairs at the cost of increasing it on prior-disfavored ones, with total discriminability conserved across all pairs regardless of the prior. In the limit of infinite data the channel enters the super-Nyquist regime and the prior's influence vanishes by posterior consistency. A Goodhart subsumption theorem for induction proves that overfitting is Goodhart failure: targeting the empirical loss as a proxy for true predictive risk produces divergence proportional to d/n, the reciprocal of the Nyquist ratio. The excess risk decomposes via the conservation identity into meta-adversarial capacity (finite-sample variance of the selected estimator) and residual predictive power. Regularization is identified as Goodhart mitigation: it reduces the adversarial capacity at the cost of increased bias, with the optimal tradeoff governed by the conservation identity. An inductive conservation identity proves that no enrichment — prior, auxiliary covariates, cross-validation, or external knowledge — eliminates aliasing on all law pairs simultaneously when the hypothesis class is infinite-dimensional. Every inductive method has a set of law pairs of positive measure on which its discriminability is zero. A structural identity theorem establishes the formal equivalence of inductive aliasing, Simpson's paradox, and capital reversal: all three are sub-Nyquist projection artifacts governed by the same conservation identity, sharing the same Nyquist threshold structure, the same necessity mechanism, and the same resolution — expand the channel dimensionality to match the intrinsic complexity of the state being identified.