Planar kinematics: Cyclic fixed points, mirror superpotential, $k$-dimensional Catalan numbers, and root polytopes
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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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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| 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 it