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Definable coordinate geometries over fields, part 1: theory

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Pith's one-line read The paper proves that for finitely field-definable coordinate geometries over ordered fields or fields with more than two elements, a relation on points is definable exactly when it is invariant under the geometry's automorphisms and…

desk verdict Genuine equivalence between definability and automorphism invariance for FFD coordinate geometries, proven cleanly; send to review. read the letter →

arxiv 2507.10279 v1 pith:I3JLPHEA submitted 2025-07-14 math.LO

Judit Madarász , Mike Stannett , Gergely Székely This is my paper · ORCID
classification math.LO MSC 03C4003C2051A05
keywords DefinabilityCoordinategeometriesConceptsAutomorphismgroupsAffineautomorphismsDefinitionalequivalenceErlangenprogramUltrapowers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a precise sense in which the symmetries of a coordinate geometry carry all of its conceptual content. For a coordinate geometry built from finitely many field-definable relations on $F^d$, over an ordered field or a field with more than two elements, a relation on points is definable in the geometry exactly when it is definable using the field's own vocabulary and is left invariant by every automorphism of the geometry; invariance under affine automorphisms alone already suffices. The central consequence is that two such geometries are definitionally equivalent, meaning they have exactly the same definable relations, precisely when their automorphism groups coincide. The paper also shows that the inclusion ordering on definable-relation sets is the reverse of the subgroup ordering on automorphism groups, and that the same holds for affine automorphism groups. This matters because it converts questions about which notions a geometry can express into questions about its symmetry group, a much more tractable object.

What carries the argument

An FFD coordinate geometry is a model whose universe is $F^d$, containing no functions or constants, with finitely many relations each definable in the field language, and in which the key relation of collinearity (for ordinary fields) or betweenness (for ordered fields) is definable. The proof has three load-bearing pieces. First, the Fundamental Theorem of Affine Geometry, cited from standard references rather than proved here, classifies every automorphism of $\langle F^d, \mathrm{Col}\rangle$ or $\langle F^d, \mathrm{Bw}\rangle$ as an affine transformation followed by a map induced componentwise by a field automorphism. Second, this yields a unique decomposition $\mathrm{Aut}(\mathcal{G}) = \mathrm{AffAut}(\mathcal{G}) \circ \mathrm{gAut}(F)$, which lets arbitrary automorphisms be reduced to affine ones. Third, an ultrapower definability criterion shows that a field-definable relation invariant under the relevant automorphisms is definable; the ultrapower step is what brings the argument from invariance under the small affine group back to explicit first-order definability.

What would settle it

Look for a field $F$ with more than two elements, an FFD coordinate geometry $\mathcal{G}$ over $F$, and a relation $R$ on $F^d$ that is definable in the language of $F$ and fixed by every affine automorphism of $\mathcal{G}$ but is moved by some automorphism of $\mathcal{G}$. Such an $R$ would directly falsify Theorem 5.1.2, because a relation definable in $\mathcal{G}$ must be fixed by all automorphisms of $\mathcal{G}$; the paper's two-element-field example with colored origin and axes indicates the shape such a counterexample would take.

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Extended reading notes

Core claim

The central result, Theorem 5.1.2, states that for an FFD coordinate geometry $\mathcal{G}$ over an ordered field or a field with more than two elements, the following are equivalent for a relation $R$ on points of $F^d$: $R$ is definable in $\mathcal{G}$; $R$ is definable over the field and closed under all automorphisms of $\mathcal{G}$; and $R$ is definable over the field and closed under all affine automorphisms of $\mathcal{G}$. The paper derives from this a dual isomorphism between the concept-set inclusion poset and the automorphism-subgroup inclusion poset: $\mathrm{Conc}(\mathcal{G}) \subseteq \mathrm{Conc}(\mathcal{G}')$ holds exactly when $\mathrm{Aut}(\mathcal{G}) \supseteq \mathrm{Aut}(\mathcal{G}')$, and the same holds with affine automorphism groups. Equality of automorphism groups is therefore equivalent to definitional equivalence of the geometries. In the paper's intended sense this realizes Klein's Erlangen program for these structures: understanding the concepts of a geometry is reduced to understanding its affine automorphisms.

Load-bearing premise

The load-bearing premise is the classification, cited rather than proved here, that every symmetry of the underlying affine or ordered affine geometry is an affine map followed by a coordinatewise field automorphism; if some field admitted a symmetry outside that class, the proof's reduction of all symmetries to affine ones would collapse.

Editorial extensions

If this is right

  • For two FFD coordinate geometries over the same ordered field, or the same field with more than two elements, equality of automorphism groups and equality of affine automorphism groups are each equivalent to definitional equivalence.
  • To decide whether one geometry's concepts are included in another's, it is enough to compare automorphism groups: a larger concept set goes with a smaller automorphism group, and the same holds for affine automorphism groups.
  • The automorphism-based criterion for comparing amounts of structure, under which a geometry with more symmetries has less structure, holds exactly for these geometries when structure is understood as definable relations.
  • The paper states this makes concept comparison of historically significant spacetime geometries a matter of computing their affine automorphism groups, and that the theorem is a key step in a proof that adding any classical concept to special relativity yields late classical kinematics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the ultrapower and automorphism-decomposition steps do not obviously use finiteness, the same argument may extend to coordinate geometries with infinitely many definable relations; the paper explicitly leaves this as an open problem.
  • The two-element field counterexample marks a sharp boundary: over $F=\{0,1\}$ the equivalence between affine invariance and definability fails, although the paper notes full-automorphism invariance still characterises definability there. A natural extension is to look for a replacement invariance condition that restores the theorem in that case.
  • The recipe 'compare affine automorphism groups' is likely to transfer to other geometries, such as projective or hyperbolic ones, whenever an analogue of the Fundamental Theorem of Affine Geometry supplies a decomposition of the full automorphism group.
  • For philosophical questions about theory equivalence, the result gives a clean operational meaning to 'X has less structure than Y' for spacetime theories representable as FFD geometries: one theory's concepts are contained in the other's exactly when the other's symmetry group is contained in the first's.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper develops a definability theory for coordinate geometries over fields and ordered fields. A coordinate geometry is a model with universe F^d, no functions or constants, in which the collinearity relation Col (or betweenness Bw, in the ordered case) is definable. The paper calls a geometry field-definable if every primitive relation is definable over the underlying field, and FFD if it has finitely many primitives. The main result (Theorem 5.1.2) states that, for F an ordered field or a field with more than two elements and G an FFD coordinate geometry over F, a relation R on F^d is definable in G iff R is definable over F and closed under Aut(G), iff R is definable over F and closed under AffAut(G). From this the paper derives Theorem 5.1.4 and Corollaries 5.1.5-5.1.6, showing that the concept-set inclusion poset is dually isomorphic to the automorphism-group inclusion poset, and that the automorphism group determines the geometry up to definitional equivalence. The proof combines the Fundamental Theorem of Affine Geometry (Lemma 5.2.1), a decomposition of automorphisms into affine and field-induced parts (Proposition 5.2.2), a transfer of affine invariance to ultrapowers via the formulas theta_Psi and theta_R (Lemmas 5.4.5 and 5.5.4), and Simon's definability criterion (Theorem 5.5.1). Remark 5.6.1 gives a two-element-field counterexample showing that the >2-elements hypothesis is necessary for the implication (iii) -> (i).

Significance. If correct, Theorem 5.1.2 provides a clean bridge between Klein's Erlangen program and first-order definability: for a large class of classical geometries, the definable relations are exactly the field-definable relations invariant under (affine) automorphisms, and the poset of concept-sets is dually isomorphic to the poset of automorphism groups. This gives a rigorous justification of the (SYM*) criterion for comparing amounts of structure in the philosophy of physics and offers a practical method for comparing historically significant spacetimes in the companion paper [MSS25a]. The proof is coherent and carefully scoped: no fitted parameters appear, the two-element-field boundary is explicitly tested, and the main external inputs (the Fundamental Theorem of Affine Geometry and Simon's ultrapower definability criterion) are standard and correctly cited. The paper is transparent about its open problems. I find the central claims sound and the presentation, apart from local issues listed below, clear.

minor comments (6)
  1. [Section 4 (Eucl definition)] The displayed formula defining Eucl contains '(pd - qd)d', which appears to be a typo for '(pd - qd)^2'; as written, the exponent depends on the dimension, which is not the intended Euclidean congruence relation.
  2. [Section 3.1] In the paragraph after Definition 3.1.3, 'it's i'th component' should be 'its i-th component'.
  3. [Section 5.5] The spelling 'Los's Theorem' should be 'Loś's Theorem' (with the diacritic).
  4. [Section 5.6, proof of (i) => (ii)] The translation Tr is defined without explicitly saying that the formulas sigma_S are renamed so that their variables avoid the blocks v_{1+(i-1)d},...,v_{id}; this is a routine formal point, but stating it would make the translation fully rigorous.
  5. [Section 5.4, Lemma 5.4.5] The expression 'F |= theta_Psi -> theta_R' has free variables in theta_Psi and theta_R, but the intended convention (universal satisfaction over all assignments) is not stated; a clarifying sentence would help.
  6. [Lemma 5.2.1] The converse inclusion is cited to Berger [Ber87] and Tarrida [Tar11] rather than proved; this is a standard external theorem and not a gap, but the paper could state explicitly that the main theorem inherits this classical dependency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 5.1.2 rests on independent external theorems (FTAG and Simon's definability criterion), not on fitted inputs or self-cited premises.

full rationale

The paper's central claim, Theorem 5.1.2, is not circular. The equivalence (i) => (ii) is shown by translating formulas from the geometry language to the field language, so definability in G is shown to imply definability over F; closure under automorphisms is immediate from definability. The nontrivial direction (iii) => (i) uses two independent external pillars: Lemma 5.2.1 (Fundamental Theorem of Affine Geometry) is cited to Berger and Tarrida, and Proposition 5.2.2 uses it to decompose every automorphism of a field-definable geometry as an affine automorphism composed with a field-induced automorphism. Theorem 5.5.1 (the definability criterion via ultrapowers) is attributed to Andras Simon, with an independent proof given in the paper using [SS15, Cor. 1] and the Keisler-Shelah isomorphism theorem. Lemma 5.4.5 translates the hypothesis that all affine automorphisms respect R into the first-order formula F |= (theta_Psi -> theta_R), and Los's theorem preserves this implication in ultrapowers. No fitted parameter is renamed as a prediction, no target equivalence is assumed as an input, and the paper's self-citations [MSS25a] and [MSS25b] are forward-looking applications and motivation rather than load-bearing premises of Theorem 5.1.2. The proof is self-contained apart from standard, appropriately scoped external mathematical results, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central theorem rests on three external mathematical pillars: the Fundamental Theorem of Affine Geometry, Simon's ultrapower definability criterion, and Los's theorem. None of these is proved in full in the paper, and none of them builds in the conclusion. There are no fitted parameters or invented entities. The only scoping restriction that matters is the exclusion of the two-element field, which the paper itself documents with a counterexample.

assumptions (4)
  • standard math Fundamental Theorem of Affine Geometry: for fields F with more than two elements, Aut of the affine geometry is AffineTrf composed with gAut F; for ordered fields the same holds for the betweenness geometry.
    Invoked in Section 5.2 as Lemma 5.2.1 with citations to Berger and Tarrida. It is the foundation of Proposition 5.2.2, which decomposes every automorphism of G as an affine automorphism composed with an induced field automorphism; this decomposition is essential in the ultrapower step of Theorem 5.1.2.
  • standard math Simon's definability criterion: a relation R on the universe of M is definable if and only if, for every ultrapower, every automorphism of M^U respects R^U.
    Stated as Theorem 5.5.1, credited to Andras Simon, with proof depending on [SS15] and the Keisler-Shelah theorem. It is the engine converting ultrapower invariance into definability in the direction from affine invariance to definability in Theorem 5.1.2.
  • standard math Los's theorem: first-order formulas transfer between M and its ultrapowers.
    Used in Lemma 5.5.4 and in the transfer of the implication theta_Psi to theta_R to the ultrapower F^U, ensuring that G_Psi(F^U) remains a coordinate geometry and inherits the affine-invariance condition.
  • standard math Keisler-Shelah isomorphism theorem: elementarily equivalent structures have isomorphic ultrapowers.
    Used inside the proof of Theorem 5.5.1 to move an automorphism witnessing non-definability from an elementarily equivalent model to an ultrapower of the original model.

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Pith. "Pith review of Definable coordinate geometries over fields, part 1: theory." pith.science (2026). https://pith.science/paper/I3JLPHEA

@misc{pith2026250710279,
  author       = {Pith},
  title        = {Pith review of: Definable coordinate geometries over fields, part 1: theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3JLPHEA}},
  note         = {Machine review of arXiv:2507.10279}
}
abstract

We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of such a geometry determines the geometry up to definitional equivalence; moreover, if we are given two such geometries $\mathcal{G}$ and $\mathcal{G}'$, then the concepts (explicitly definable relations) of $\mathcal{G}$ are concepts of $\mathcal{G}'$ exactly if the automorphisms of $\mathcal{G}'$ are automorphisms of $\mathcal{G}$. We show this by first proving that a relation is a concept of $\mathcal{G}$ exactly if it is closed under the automorphisms of $\mathcal{G}$ and is definable over the field; moreover, it is enough to consider automorphisms that are affine transformations.

Figures

Figures reproduced from arXiv: 2507.10279 by the authors.

Figure 1
Figure 1. Hasse diagram showing how the concept-sets associated with various historically significant geometries are related to one another by subset inclusion. vectors. Let R = {⟨1, 1, 1⟩}. Then R is not definable in G because it is not respected by the automorphisms of G, yet it is field-definable and respected by all the affine automorphisms of G. So (iii) holds for R but (i) does not. Nevertheless, the equivalence between… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On Andr\'eka's Conjecture that special relativity is the only possible conceptual reduct of classical kinematics

    math-ph 2025-07 conditional novelty 8.0 of 10

    Andr\'eka's conjecture is proven: no intermediate model of spacetime exists strictly between special relativity and late classical kinematics on R^4.

Reference graph

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