From Trees to All Loops: The Combinatorial Geometry of Two-Site Cosmological Wavefunctions
Abstract
We prove, at arbitrary loop order and including arbitrary tadpole decorations, the shifted-tree decomposition of two-site cosmological wavefunction coefficients proposed in our previous work, and uncover the combinatorial and geometric structure underlying it. For conformally coupled scalars with polynomial interactions, this decomposition expresses each loop coefficient as a finite signed sum of tree coefficients evaluated at shifted energies. For bubble-like graphs, the alternating sum over a Boolean lattice can be written in terms of finite differences or maximal chains. We prove the maximal-chain identity algebraically by induction on loop order and geometrically by decomposing a cubical Laplace integral into simplices. The maximal chains correspond to permutohedron vertices, whose incident facets encode the chain-dependent divisors. For tadpole-decorated graphs, the finite-difference Leibniz rule gives an explicit expansion indexed by ordered partitions and local maximal chains, which glue into global maximal chains and organize the coefficient as a combinatorial "path integral". A natural ordering of the divisors yields a term-by-term match with graph tubings and completes the all-loop proof, while other orderings give alternative tubing representations of the same coefficient. These identities hold before spatial loop-momentum integration and extend to power-law FRW backgrounds by linearity of the twisted integral. Two-site loop coefficients can therefore be obtained directly from known tree coefficients without solving new loop-level differential systems.
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BibTeX
@misc{hang2026from,
title = {{From Trees to All Loops: The Combinatorial Geometry of Two-Site Cosmological Wavefunctions}},
author = {Yanfeng Hang and Cong Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.30797},
eprint = {2605.30797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.30797},
}
RIS
TY - UNPB
TI - From Trees to All Loops: The Combinatorial Geometry of Two-Site Cosmological Wavefunctions
AU - Yanfeng Hang
AU - Cong Shen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2605.30797
UR - https://arxiv.org/abs/2605.30797
AB - We prove, at arbitrary loop order and including arbitrary tadpole decorations, the shifted-tree decomposition of two-site cosmological wavefunction coefficients proposed in our previous work, and uncover the combinatorial and geometric structure underlying it. For conformally coupled scalars with polynomial interactions, this decomposition expresses each loop coefficient as a finite signed sum of tree coefficients evaluated at shifted energies. For bubble-like graphs, the alternating sum over a Boolean lattice can be written in terms of finite differences or maximal chains. We prove the maximal-chain identity algebraically by induction on loop order and geometrically by decomposing a cubical Laplace integral into simplices. The maximal chains correspond to permutohedron vertices, whose incident facets encode the chain-dependent divisors. For tadpole-decorated graphs, the finite-difference Leibniz rule gives an explicit expansion indexed by ordered partitions and local maximal chains, which glue into global maximal chains and organize the coefficient as a combinatorial "path integral". A natural ordering of the divisors yields a term-by-term match with graph tubings and completes the all-loop proof, while other orderings give alternative tubing representations of the same coefficient. These identities hold before spatial loop-momentum integration and extend to power-law FRW backgrounds by linearity of the twisted integral. Two-site loop coefficients can therefore be obtained directly from known tree coefficients without solving new loop-level differential systems.
ER -