Statistical learning theory assumed that accumulating infinite data guarantees convergence to underlying empirical truths; this mathematical proof reveals rigorous topological boundaries where specific hypotheses remain fundamentally unlearnable.
For a century, empirical science has operated under the philosophical and mathematical assumption that collecting sufficiently large datasets will eventually allow statistical algorithms to distinguish truth from falsehood.
In complex, high-dimensional hypothesis testing, certain structural relationships exhibit mathematical pathologies where classical statistical power collapses, creating a hidden domain of epistemic blindness.
This foundational mathematical paper establishes the 'Theory of Meager Success' using Baire category theory and algorithmic learning theory, proving that there exist broad classes of well-defined statistical hypotheses where no computable estimator can achieve better than meager learning success.
Mapping the exact mathematical boundaries of the unlearnable establishes profound limits on automated scientific discovery, warning data scientists against claiming definitive algorithmic truth on non-learnable hypothesis spaces.
Meager Success: A Theory of the Unlearnable for Hypothesis Testing
When the standard of pointwise consistency for statistical inference -- convergence to the truth in every possible state of the world -- is provably unachievable, the usual responses are to change the inferential target or to strengthen background assumptions. This paper pursues a third: hold the inference problem fixed and identify the highest standard that remains achievable. I define a hierarchy of standards weaker than pointwise consistency, cast in topological terms, requiring convergence to the truth not everywhere but on a ``large'' set of probability measures. The main result is an impossibility theorem: for finite-precision tests, converging to the truth densely within each hypothesis already forces inconsistency on a comeager -- ``topologically almost all'' -- set of measures, whenever the two hypotheses are dense in their union. Distribution-free testing of conditional independence is one such case. Two further theorems characterize, in purely topological terms, exactly when each weaker standard is achievable, complementing Boeken et al.'s (2026) analysis of pointwise consistency.
Network scientists long treated complex real-world graphs as abstract topological webs in flat space; mapping complex networks into negative-curvature hyperbolic space explains their natural scale-free hierarchies and clustering.
Classical Bell tests were strictly confined to simple two-level qubit pairs; the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality extends quantum nonlocality verification to arbitrarily high-dimensional entangled states.
Shor's quantum algorithm threatens to dismantle RSA and elliptic curve public-key cryptography; multivariate quadratic cryptosystems constructed over extremal algebraic graphs provide robust post-quantum security.
Calculating quantum energy levels inside magnetic confinement fields historically produced intricate mathematical spectral gaps; this spectral analysis maps the precise eigenvalue bounds of the Dirichlet Laplacian under constant magnetic fields.
Nonlinear partial differential equations often exhibit chaotic, unpredictable spatial behaviors; rigorous asymptotic analysis reveals that interior solutions naturally concentrate into sharp, localized energy peaks determined by domain geometry.
Probabilists assumed that the higher-order mathematical moments of normal bell curves were always universally positive; mathematicians discovered an elegant low-dimensional counterexample where the math unexpectedly turns negative. By using computer-assisted optimization to shatter the Gaussian Moments Conjecture, probability theorists have corrected a fundamental assumption in random matrix theory and quantum information science.
Empirical Bayes methods traditionally assumed observational noise errors were completely independent and uncorrelated; this statistical breakthrough establishes minimax optimal estimation for high-dimensional Gaussian sequences under general correlation structures.
Isaac Newton and Gottfried Leibniz built calculus using intuitive infinitesimals that nineteenth-century mathematicians banished as logically inconsistent; the 'chunk and permeate' framework constructs a rigorous, contradiction-tolerant field of infinitesimals.
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