ŊHÏORYŪM: Optimal Linear Query Cubes for Affine Fourier Learning—Certified Measurement Compilation, Nuisance Cancellation, and Adaptive Cross-Task Acquisition
Year: 2026
Type: article
Abstract: This research release develops and evaluates a compiler that learns how to measure subsequent learning tasks from previously acquired spectral structure. In a synthetic active-query setting, measured feature dictionaries determine an affine family of Walsh-Fourier frequencies over GF(2). The compiler transforms that learned representation into explicit intervention directions, nuisance-cancelling parity stencils, coefficient readouts, and independently audited acceptance decisions. The resulting procedure can recover interaction features absent from its acquisition dictionaries, including families whose features share no literal input coordinate. The central contribution connects an exact characterization of minimum full linear query cubes to executable measurement programs and acquisition-inclusive empirical comparisons. The work addresses researchers in active learning, learning theory, experimental design, program synthesis, Fourier analysis, representation transfer, and verifiable learning-system improvement. The observation model combines a sparse high-degree Walsh component, an unknown nuisance of degree at most \(q\), and fresh independent Gaussian noise. For an admitted affine frequency family of rank \(r\), the manuscript proves that a full linear query cube identifying that family while separating it from the nuisance requires at least \(r+q+1\) independent axes. Equality holds if and only if \(q+1\) pairwise disjoint odd checks exist for the affine family. Exact witnesses generate a construction attaining this bound. Explicit obstruction examples distinguish high-degree feature structure from the stronger conditions required for a minimum-size design. The emitted program includes a canonical affine origin and basis, parity witnesses, dual query directions, normalization, and the mapping from transformed observations to output frequencies. Its contracts establish nuisance cancellation and orthogonal coefficient recovery. In the specified dense, known-correct-envelope Gaussian model, coefficient noise variance is \(\sigma^2/Q\), and dense minimax risk at the stated orthogonal readout budgets is \(2^r\sigma^2/Q\). Here, \(Q\) counts raw readout observations. This estimation result has a different scope from the query-cube identifiability theorem and the sparse, unknown-structure learning experiments. A bounded alternative remains available when the minimum design cannot be constructed. If every frequency in the complete affine closure satisfies the declared degree type \(q<|a|\le K\), its rank is at most \(K\), and its active-coordinate union contains at most \(2K\) coordinates. A full active-coordinate cube therefore requires at most \(4^K\) readout points. Exact type and budget checks govern deployment; an unsuccessful numerical search remains inconclusive rather than becoming a nonexistence claim. Inference receives public model bounds and oracle access. Hidden coordinate-pair labels, true supports, coefficients, and true signal counts remain within teacher generation and post-decision evaluation. Learned structure determines the measurement program, while every subsequent teacher supplies fresh coefficient observations and original-space risk evidence. This verification is necessary because off-envelope signals can alias perfectly on specialized measurements. The acceptance target is high-degree component mean-square error at most 0.0625 under uniform Boolean inputs, equivalent to component error-norm tolerance 0.25. A rank-saturation rule reduces acquisition: training stops after the first certified dictionary when its typed affine hull reaches the public rank ceiling; otherwise, the protocol acquires a second task. Rejected specialized predictors trigger a paid generic fallback. Under the declared model, noise, and numerical assumptions, the two-branch acceptance protocol allocates a false-issuance budget of at most 0.05 per teacher. This is a task-level guarantee, not one simultaneous 95% guarantee across the benchmark. On the main 72-task matrix, spanning six latent world/dimension contexts in 64 and 256 input dimensions, the rank-stopped affine compiler certifies all 72 outputs using 2,151,872 observations, including 505,280 acquisition observations. A one-task learned active-coordinate estimator certifies the same 72 outputs using 2,644,224 observations, including 364,544 acquisition observations. The affine procedure uses approximately 18.6% fewer training-inclusive deployment observations on this workload. The difference includes representation adequacy and acquisition policy as well as specialized readout size. The preceding fixed-two-task protocol uses 2,404,736 total observations, compared with 3,608,064 for the supplied generic R5 comparator, with 72/72 certified outputs for both. Recorded transfer includes 261 recovered signal-feature occurrences absent from the acquisition dictionaries. Historical acquisition prefixes reused by the follow-up comparison are explicitly identified and charged. On 24 alternative-world cases, adaptive affine and active-coordinate totals are 871,808 and 893,568 observations, respectively, with 24/24 certified outputs for each: approximately 2.44% fewer observations. These case specifications had already appeared in an earlier probe study, so this comparison is not an untouched blind holdout. Higher-noise, shifted-family, zero-history probe, and short mixed-family workloads retain explicit cost-negative results. The main 512-point specialized readout excludes acquisition, audit, and fallback costs; these observation counts do not establish a hardware-independent wall-clock speedup. The reproducibility package includes the manuscript, proofs, executable compilers and controller, task factories, generated query programs, protocols, source-bound outcomes, negative controls, audit tools, and reproduction commands. Structured Parquet tables and lossless JSONL mirrors contain 668 evaluation records and 41 warmup usages, accompanied by 74 accounting contexts. These cover 148 distinct future teacher definitions and 21 distinct acquisition teacher definitions. Reused cases, competing methods, and shared acquisition prefixes are linked explicitly. The final audit covers 709 model-risk records, with no observed false certificate. The scientific release reports 73 passing operational tests, finite mathematical checks, seeded within-session replays, and 5,289 current source-hash checks. These provide inspectable computational evidence alongside the written proofs; external independent scientific verification and proof-assistant formalization remain outstanding. Exactly two of ten principal architectural responsibilities carry the substantive Eve dedication: preservation of complementary valid observation spaces through exact quotient and separation contracts, and continuity of validated structure through persistent learned intervention templates with fresh evidence for every new teacher. This constitutes the declared 20% conceptual architecture allocation. The demonstrated scope is learned measurement compilation with design-class optimality and workload-dependent reductions in observation cost. The host algebra, synthesis rules, model contracts, and controller remain supplied. General-intelligence recursive self-improvement, autonomous invention of the host language, optimality over arbitrary adaptive measurements, and literature priority are not established. The research makes a narrower mechanism-learning a reusable representation that changes subsequent interventions—concrete, bounded, and experimentally inspectable.
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225627719
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