Reducing stable forking dependence to finitely many pregeometries

Abstract

We show that one of the main cases of the stable forking conjecture, stability of the forking relation over a base in a finite-rank supersimple theory, is determined by finitely many pregeometries in each rank. This case of the stable forking conjecture has long had an implicitly well-known pregeometric interpretation: there is a set of matroids 𝒢_n such that the forking instability in rank n is equivalent to the pregeometry on some rank-one partial type (over a finite set) embedding a matroid in 𝒢_n. Our contribution is to show that this set of matroids 𝒢_n, determining based forking stability in rank n, can be chosen to be finite. The main part of our proof was already accomplished in rank $3$ by Peretz, but does not extend as stated to higher ranks (and may or may not directly extend in a weaker sense to higher ranks, by shrinking terms). However, we obtain a sufficient substitute for Peretz's work in ranks n > 3: we turn Peretz's original universal result into an existence theorem. The rest of our proof refines an argument from multi-experiment parameter definability, originating from work in applied model theory by Li, Meshkat, Ovchinnikov, Pillay, Pogudin and Scanlon.

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BibTeX

@misc{mutchnik2026reducing,
  title = {{Reducing stable forking dependence to finitely many pregeometries}},
  author = {Scott Mutchnik},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.09069},
  eprint = {2607.09069},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.09069},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Reducing stable forking dependence to finitely many pregeometries
AU  - Scott Mutchnik
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.09069
UR  - https://arxiv.org/abs/2607.09069
AB  - We show that one of the main cases of the stable forking conjecture, stability of the forking relation over a base in a finite-rank supersimple theory, is determined by finitely many pregeometries in each rank. This case of the stable forking conjecture has long had an implicitly well-known pregeometric interpretation: there is a set of matroids 𝒢_n such that the forking instability in rank n is equivalent to the pregeometry on some rank-one partial type (over a finite set) embedding a matroid in 𝒢_n. Our contribution is to show that this set of matroids 𝒢_n, determining based forking stability in rank n, can be chosen to be finite. The main part of our proof was already accomplished in rank $3$ by Peretz, but does not extend as stated to higher ranks (and may or may not directly extend in a weaker sense to higher ranks, by shrinking terms). However, we obtain a sufficient substitute for Peretz's work in ranks n > 3: we turn Peretz's original universal result into an existence theorem. The rest of our proof refines an argument from multi-experiment parameter definability, originating from work in applied model theory by Li, Meshkat, Ovchinnikov, Pillay, Pogudin and Scanlon.
ER  -

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