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 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 | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".