Proof of a K-theoretic polynomial conjecture of Monical, Pechenik, and Searles
Bibliographic record
Abstract
As part of a program to develop K -theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas A ‾ a = ∑ b Q b a ( β ) P ‾ b and Q ‾ a = ∑ b M b a ( β ) F ‾ b , where each of A ‾ a , P ‾ a , Q ‾ a and F ‾ a is a family of polynomials that forms a basis for Z [ x 1 , … , x n ] [ β ] indexed by weak compositions a , and Q b a ( β ) and M b a ( β ) are monomials in β for each pair ( a , b ) of weak compositions. The polynomials A ‾ a are the Lascoux atoms , P ‾ a are the kaons , Q ‾ a are the quasiLascoux polynomials , and F ‾ a are the glide polynomials ; these are respectively the K -analogues of the Demazure atoms A a , the fundamental particles P a , the quasikey polynomials Q a , and the fundamental slide polynomials F a . Monical, Pechenik, and Searles conjecture that for any fixed a , ∑ b Q b a ( − 1 ) , ∑ b M b a ( − 1 ) ∈ { 0 , 1 } , where b ranges over all weak compositions. We prove this conjecture using a sign-reversing involution.
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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 | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.002 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".