Isospectral non-isometric lattices and methods of distinction
Bibliographic record
Abstract
Two lattices are called isospectral if they share the same theta series (length spectra) and isometric if they differ by an isometry.Isometric lattices are always isospectral, but the converse is not necessarily true.Determining when isospectral implies isometric can be formulated in geometric (lattices), analytic (the Laplacian on flat tori), and number theoretic (quadratic forms) language.After establishing these three equivalent viewpoints in §2, we turn to examine the 2011 Cerviño-Hein proof of the 1992 Conway-Sloane conjecture in §3.This result constructs an infinite family of isospectral, non-isometric lattice pairs in four dimensions.The argument introduces a method of distinguishing isometry classes using spherical theta series.Finally, in §4 we turn to Jacobi forms and the Jacobi theta series-a generalization of the traditional theta series which encodes both length and angle information.After a concise introduction to the theory, we develop a method of distinguishing isometry classes using certain sets of Jacobi theta series. RésuméOn appelle deux réseaux isospectraux lorsqu'ils partagent la même série thêta (spectre des longueurs) et isométriques lorsqu'ils ne diffèrent que par une isométrie.Les réseaux isométriques sont toujours isospectraux, mais l'inverse n'est pas nécessairement vrai.Déterminer quand l'isospectralité implique l'isométrie peut se formuler dans les langages géométrique (réseaux), analytique (le Laplacien sur les tores plats) et arithmétique (formes quadratiques).Après avoir établi ces trois points de vue équivalents à la §2, nous examinons à la §3 la démonstration de 2011 de Cerviño-Hein de la conjecture de Conway-Sloane de 1992.Ce résultat construit une famille infinie de paires de réseaux isospectraux non isométriques en dimension quatre.L'argument introduit une méthode de distinction des classes d'isométrie fondée sur les séries thêta sphériques.Enfin, dans la §4, nous abordons les formes de Jacobi et la série thêta de Jacobi-une généralisation de la série thêta classique qui encode à la fois l'information de longueur et d'angle.Après une brève introduction à la théorie, nous développons une méthode de distinction des classes d'isométrie à l'aide de certains ensembles de séries thêta de Jacobi.
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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 | 0.004 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.011 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".