A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -analog of the Markoff injectivity conjecture holds
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Bibliographic record
Abstract
The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a longstanding open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -analog. The proof is based on the evaluation at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mo form="prefix">exp</mml:mo> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>π</mml:mi> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Other roots of unity provide some information on the original problem, which corresponds to the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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