Sharp Calibration Depth and Robust Prediction in Sparse Uniform Hypergraphs

Sharp Calibration Depth and Robust Prediction in Sparse Uniform Hypergraphs studies how much structural information is required for exact and robust prediction under strong vertex deletion in sparse, nonnegative, weighted uniform hypergraphs. The hypergraph is defined on a known labeled vertex set and contains at most a prescribed number of positive-weight hyperedges. Strong deletion removes every hyperedge intersecting a chosen set of vertices, and the prediction task is to determine the remaining response associated with a specified anchor after such an intervention. Sharp Calibration Depth and Rob… The paper establishes a sharp threshold for the order of labeled containment statistics required to determine every admissible future deletion response. This threshold is controlled jointly by four mechanisms: the hyperedge order, the size of the complementary vertex set, the complexity of the future intervention, and the sparsity of the hidden hypergraph. The result is exact for both nonnegative weighted hypergraphs and the simple unweighted subclass. Necessity is proved using explicit parity-trade constructions that produce distinct sparse hypergraphs with identical lower-order containment data but different post-deletion responses. Sharp Calibration Depth and Rob… The paper then treats calibration as an active identification problem. Complete calibration responses obtained from anchors of fixed size and deletions of bounded depth are shown to contain exactly the same information as containment statistics up to a corresponding structural order. This gives the minimum calibration deletion depth required to guarantee exact prediction for a specified future task. Importantly, the result concerns the maximum number of vertices deleted in a single calibration experiment rather than the total number of measurements or experiments. Sharp Calibration Depth and Rob… A central part of the work concerns robustness to calibration error. Exact identifiability alone does not guarantee stable prediction, so the paper introduces a deterministic minimax framework with coordinatewise bounded errors. For the unbounded nonnegative weighted sparse class, the same calibration threshold that separates insufficient from sufficient exact information also separates two robustness regimes. Below the threshold, the worst-case prediction error is unbounded. At and above the threshold, the minimax prediction error becomes finite and proportional to the calibration error level. The paper provides an explicit sparsity-dependent stability bound that is independent of the ambient number of vertices when sparsity is fixed. In the one-edge case, the stability constant is determined exactly and equals one. Sharp Calibration Depth and Rob… The work also separates nonlinear sparse prediction from exact linear decoding. The minimum calibration depth permitting an exact linear decoder does not benefit from sparsity in the same way as arbitrary nonlinear prediction. As a result, there are sparse regimes in which stable nonlinear prediction is already possible from shallow calibration data while every affine decoder still has infinite worst-case error on the unbounded weighted class. This shows that predictive sufficiency, structural reconstruction, and exact linear inversion are distinct notions and may require different amounts of information. Sharp Calibration Depth and Rob… The mathematical analysis combines Boolean-lattice Möbius inversion, inclusion-exclusion, uniform containment identities, sparse null-space arguments, combinatorial trade constructions, optimal-recovery methods, and finite-dimensional linear programming. The manuscript explicitly distinguishes these classical ingredients from its calibration-specific contributions, which include the sharp prediction and calibration thresholds, the robust minimax transition, the explicit sparse stability guarantee, and the separation from linear decoding. Sharp Calibration Depth and Rob… The paper also states the main limitations of the theory. Calibration assumes access to the complete labeled family of anchor/deletion responses and a reproducible baseline system across interventions. The robustness theorem concerns deterministic coordinatewise errors and an unbounded nonnegative weighted model class, so the infinite-error result below the threshold does not automatically extend to bounded-weight or simple hypergraphs. The proposed recovery procedure is constructive but may require combinatorial support enumeration, and the results should be interpreted as information-theoretic rather than as claims about sample efficiency, optimal experimental design, polynomial-time decoding, or statistical performance under other noise models. Sharp Calibration Depth and Rob… The accompanying reproducibility package contains the complete manuscript source, compiled PDF, exact computational audits, machine-readable outputs, explicit lower-bound witnesses, generated figures and tables, rational primal-dual certificates, provenance metadata, and checksums. The computational verification independently checks containment and deletion identities, prediction thresholds, calibration-depth classifications, deeper parity witnesses, and optimization certificates using exact arithmetic. No external dataset or internet access is required to reproduce the mathematical checks. Sharp Calibration Depth and Rob… The central aim of the work is to determine when limited lower-order structural information is sufficient for intervention prediction, when controlled calibration can recover the missing information, and when exact predictive sufficiency also guarantees robustness. The results show that sparsity can reduce the information required for nonlinear prediction, that controlled deletions can act as measurements of higher-order structure, and that robust prediction can become possible before exact linear inversion.