Advances in the theory of box integrals
Bibliographic record
Abstract
Box integrals—expectations ⟨ | r → | s ⟩ \langle |\vec r|^s \rangle or ⟨ | r → − q → | s ⟩ \langle |\vec r - \vec q|^s \rangle over the unit n n -cube—have over three decades been occasionally given closed forms for isolated n , s n, s . By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of n = 1 , 2 , 3 , 4 n = 1,2,3,4 dimensions the box integrals are for any integer s s hypergeometrically closed (“hyperclosed”) in an explicit sense we clarify herein. For n = 5 n = 5 dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call K 5 {\mathcal K}_5 ; although we do prove that all but a finite set of ( n = 5 n = 5 ) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.015 | 0.006 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".