Isospectral non-isometric lattices and methods of distinction
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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 fourdimensions. 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
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it