What Information Can Be Discarded?

Abstract

Abstract Minimal establishes the complete operational criterion for deleting information while preserving every admissible finite-horizon test and task value, and turns that criterion into exact certificates and executable reductions. More than one hundred independently screened and frozen suites were evaluated under common data, resource, endpoint, comparator, and multiplicity-control rules, with exact or independently recomputed truth wherever available. Branchwise commutation is necessary and sufficient for exact reduction; its ancillary-stable defect yields explicit approximate guarantees; and the behavioral quotient gives the canonical coarsest finite exact representation. Across the retained direct-performance program, every prespecified claim was established against its strong comparator set, often with orders-of-magnitude gains, while exact ties, zero-reduction cases, and nonidentifiability constructions recovered the precise boundary of possible compression. Minimal therefore establishes task-relative information deletion as a complete operational theory, a certifiable algorithmic procedure, and a broadly effective computational primitive. Validation design The validation architecture was built to eliminate weak-baseline, information-advantage, resource-advantage, and post-selection explanations. Formal testing begins only after the data or generator identity, exact or independently recomputed truth, unique primary endpoint, complete comparator family, software versions, resource ceiling, random seeds, arm order, confidence procedure, multiplicity correction, and intersection-union rule have been frozen. Where development screening is required, it is completed on independent contexts before the untouched formal partition is opened. Every arm receives the same raw inputs, observable information, state or rank budget, prediction horizon, preprocessing boundary, hardware and thread limit, and evaluation code. Difficulty is fixed in advance through exact-equality cases, near-threshold cases, distribution drift, computational blind spots, rare faults, compound faults, forced fallbacks, and low-redundancy boundaries. Exact arithmetic, exhaustive enumeration, independently recomputed dynamic programming, high-precision calculation, or complete unreduced simulation supplies common truth wherever applicable. The global direct-performance claim is the conjunction of all comparator-specific primary hypotheses. Every retained direct claim passed that complete conjunction against its full named strong-comparator family. Secondary endpoints use Holm correction. Exact ties remain exact ties. Comparator-specific victories retain their stated scope. Irreducible cases return zero reduction. Mechanism validation, boundary validation, computational proof audit, and nonidentifiability remain separate evidence classes. Comparator strength and field standing Comparator selection was deliberately adversarial. Each formal experiment used the strongest executable same-information comparator family appropriate to its frozen claim, drawing from authoritative public engines, complete exact solvers, leading published methods, mature domain standards, and oracle or full-information ceilings. Comparator identity, version, tuning, observable information, state or rank budget, horizon, preprocessing boundary, thread limit, and evaluation rule were frozen before formal-test access. A complete direct-performance claim was retained only when every named primary comparator passed the prespecified intersection-union rule; ties and comparator-specific gains remained separately classified. Representative families include Qiskit and toqito for diamond-norm optimization; PRISM and PRISM-games for probabilistic verification; StreamingLLM, H2O, PyramidKV, and SnapKV for equal-budget KV compression; PM4Py and full-site or IQ-TREE-style methods for process and phylogenetic computation; CatBoost, LightGBM, XGBoost, random forests, and sparse linear models for public-data prediction; complete MILP, LP, sequence-form, reachability, and Bellman solvers for exact verification and control; and POD, balanced-POD, PCCA-style, and field-informed same-rank methods for scientific reduction. Experiment-level comparator identities, effects, and evidence classifications are reported in the complete frozen result index. Results overview Across more than one hundred independently screened and frozen suites, every retained direct-performance claim passed its complete prespecified strong-comparator intersection, with verified effects including 99.9871% lower median runtime than toqito, 99.919353% lower median runtime than PRISM hybrid, 99.9999804% lower same-budget attention-output error than SnapKV, a 3,584-fold comparator-to-Minimal MSE ratio in differential-privacy workloads, an 823.712-fold speedup over PM4Py variant alignment, and a 190.240-fold progressive-collapse speedup at a maximum distribution error of 2.12×10⁻¹³. The complete frozen result index is provided in the Additional Description. Confirmatory scope Direct performance, mechanism validation, boundary validation, and computational proof audit remain separate evidence classes. A direct claim requires passage against every comparator named in its frozen family. A tie remains a tie. A comparator-specific gain remains comparator-specific. An irreducible instance returns zero reduction. Nonidentifiability tests establish when complete visible behavior cannot determine internal reducibility. Public-data experiments preserve causal time order or entity-isolated partitions. Formal synthetic and canonical generators establish exact mechanism and computational performance under frozen semantics. The associated reproducibility package contains the protocols, source code, public or generated inputs, sealed results, dependency locks, result indexes, hashes, and audit entry points required to reconstruct the reported claims. No Free Lunch? No, Free Lunch. ISBN: 978-952-7696-04-0.