On the sharpness of the bound for the Local Converse Theorem of 𝑝-adic GL_{jlcg_}
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Bibliographic record
Abstract
We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>GL</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\operatorname {GL}}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a non-archimedean local field, based on distinguishability by twisted gamma factors. In the case that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is prime and the residual characteristic is greater than or equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left floor StartFraction upper N Over 2 EndFraction right floor"> <mml:semantics> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mfrac> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>⌋</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \lfloor \frac {N}{2}\right \rfloor</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we prove that, for any natural number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i less-than-or-equal-to left floor StartFraction upper N Over 2 EndFraction right floor"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mfrac> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">i\le \left \lfloor \frac {N}{2}\right \rfloor</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there are pairs of cuspidal irreducible representations whose logarithmic distance in this ultrametric is precisely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative i"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi>i</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">-i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This implies that, under the same conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left floor StartFraction upper N Over 2 EndFraction right floor"> <mml:semantics> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mfrac> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>⌋</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \lfloor \frac {N}{2}\right \rfloor</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Local Converse Theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {GL}_N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sharp.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.009 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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