Planar kinematics: Cyclic fixed points, mirror superpotential, $k$-dimensional Catalan numbers, and root polytopes
Bibliographic record
Abstract
In this paper, we prove that points in the space X(k,n) of configurations of n points in \mathbb{CP}^{k-1} which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k -element subsets of \{1,\dotsc,n\} , and consequently that their number is bounded above by the number of Lyndon words with k ones and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of G(n-k,n) and the generalized CHY potential on X(k,n) . We also check the recent conjecture that generalized biadjoint amplitudes evaluate to k -dimensional Catalan numbers on PK for several examples including k=3 and n\leq 40 and (k,n)=(6,13) . We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over \mathrm{Trop}^{+} G(k,n) , and we use it to evaluate the amplitude on PK with the purpose of exhibiting how generalized Feynman diagrams glue together. We initiate the study of two minimal lattice polytopal neighborhoods of the planar kinematics point. One of these, the rank-graded root polytope \mathcal{R}_{k,n} , in the case k=2 , is a projection of the standard type A root polytope. The other, denoted by \prod_{k,n} , in the case k=2 , is a degeneration of the associahedron. We check up to and including \mathcal{R}_{3,9} and \mathcal{R}_{4,9} that the relative volume of \mathcal{R}_{k,n} is the multi-dimensional Catalan number C^{(k)}_{n-k} , hinting towards the possibility of deeper geometric and combinatorial interpretations of m^{(k)}(\mathbb{I}_{n},\mathbb{I}_{n}) near the PK point.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".