Problem size alone does not determine the work performed by a structured computation. This methodological note separates three quantities: the number of possible histories, the number of distinguishable computational states, and the number of samples in a declared feasible region. We give complete proofs for two independent public models. First, exact aggregation by equal partial sums preserves every binary selection count and a verifiable witness while replacing histories by sufficient states. For repeated unit weights, there are 2𝑛 histories, 𝑛 + 1 final states, and exactly 𝑛(𝑛 +1) candidate contributions in the implemented recurrence. Second, an integer grid in a rational interval has an exact, nonincreasing sample count as its scale parameter grows, with a sharp budget threshold. We also prove limits to both interpretations: state aggregation can remain exponential, and thinning a feasible region can change its optimum. A dependency-free Python artifact and independent-oracle tests make all examples reproducible. The contribution is a precise separation of scaling claims and their proof obligations, not a new general-purpose solver or a universal inverse-runtime law.
Exact aggregation replaces redundant histories by sufficient states while preserving counts and witnesses. Rational-grid thinning admits a sharp integer sample-count law and an exact budget certificate. Their limits are equally explicit: state width can remain exponential, and removing samples can remove the optimizer. The public implementation demonstrates precisely these results, with independent oracles and counterexample tests. A credible rethinking of scaling begins by identifying what changes, what is counted, and what must remain true.
References
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📋 How to Cite This Paper
Michael S Yang (2026). Rethinking Scaling through Exact Aggregation and Sampling Thinning. International Journal of Computer Techniques, 13(5), 536–545. ISSN: 2394-2231. DOI: https://doi.org/10.5281/zenodo.23539904
