Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Mallard's law of observation states that known twins by reticular pseudomerohedry have low twin index m and low obliquity δ between a lattice row and a lattice plane. Crystals in the twin are related by exact reflection in the lattice plane, exact rotation by π about the perpendicular to the lattice plane, or by exact rotation about the lattice row by 2 k π/ q , with q = 2, 3, 4 or 6, and 1 ≤ k < q . In known cases, m is up to five or six, and δ is up to five or six degrees. Mallard's converse problem is then about finding all pairs of indices for rows and planes leading to twin indices not larger than m and to obliquities that are at most δ. Mallard's law is recast as the Diophantine pair constituted by the equality h · u = n and the inequality | h × u | < n tan δ. If a primitive reference system is used, the integer n is either m , 2 m or 3 m . A direct general solution of this system for u given n , h , δ and lattice data is detailed. That straightforward solution involves a tiny two-dimensional grid for u , considerably reducing the number of permutations to be considered. If the primitive reference system is Buerger-reduced, then moduli for indices of solutions h cannot exceed 3 m , thus establishing a simple way to produce complete solutions. A program called OBLIQUE was designed on such principles. An implementation is available for free execution at http://ylp.icpet.nrc.ca/oblique/. OBLIQUE is also an interactive tool in Toth Information Systems' Materials Toolkit (http://www.tothcanada.com/), an exploitation framework currently operating on CRYSTMET and ICSD crystal structure data. The example of quartz cell data is described with m max = 3 and δ max = 6°.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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