The three-loop banana crystal: rank one, the window theorem, and depth one

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22816941
Primary Topic
Polynomial and algebraic computation
Type
preprint
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preprint

The three-loop banana crystal: rank one, the window theorem, and depth one

Eric Yaw
Zenodo (CERN European Organization for Nuclear Research)
Polynomial and algebraic computation
preprint

The three-loop banana crystal: rank one, the window theorem, and depth one

Eric Yaw
preprint en

Abstract

The companion paper "Hasse–Witt obstructions for the Aₙ root polytopes" (DOI 10.5281/zenodo.22135651) shows that the Hasse–Witt conditions fail for those polytopes, and the sufficient criterion of Beukers and Vlasenko therefore does not apply to the banana family and their level-two Frobenius lift does not reach the constants. This paper builds the replacement for three loops. I determine the p-adic structure of the full Cartier matrix beyond its sharp layer, prove a rank-one structure theorem modulo p, and develop a single-term window calculus in which each pairing collapses to one term. The window annihilation theorem is then proved unconditionally at every prime p ≥ 5, by reducing a chain of exact statements to one identity over ℚ that is proved in full. Finally I prove that the layered depth-one system T(1, L) is solvable for every layer at p = 5, p = 7, p = 11 and p = 13, by a finite certified fixed-point computation whose exactness rests on the double-annihilator duality of finite abelian p-groups and whose stabilization is certified by module equality. The depth-one half of the main conjecture is therefore a theorem at the target prime. Falsified intermediate hypotheses are reported as negative results throughout because the proofs are shaped by their failure.

Zenodo (CERN European Organization for Nuclear Research)
Polynomial and algebraic computation
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