Pith. sign in

REVIEW 4 major objections 6 minor 21 references

The non-integers among the rationals can be defined by a single polynomial equation in seven unknowns.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-31 02:21 UTC pith:BUJZERAO

load-bearing objection Clean 10-to-7 drop for ∀7-definability of O_{S_0} over any global field, via finite freezing of the torus parameter; architecture holds if the local smoothness sketches check out. the 4 major comments →

arxiv 2607.28606 v1 pith:BUJZERAO submitted 2026-07-30 math.NT math.LO

mathbb Qsetminusmathbb Z is diophantine over mathbb Q with 7 unknowns

Zhi-Wei Sun This is my paper
classification math.NT math.LO MSC 03D3511S1511U0503D2511D99
keywords diophantine setsHilbert's tenth problemglobal fieldsdefinabilityquaternion algebrasundecidabilityS-integers
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows that the set of rational numbers that are not integers is diophantine over the rationals with only seven unknowns: there is one polynomial equation in a parameter t and seven variables such that the equation has a rational solution precisely when t is not an integer. The same bound holds for the ring of S-integers in any global field. Earlier work had brought the number of unknowns down to ten; the improvement comes from freezing a torus parameter in a quaternion-algebra formula so that three existential variables suffice for a key integrality condition, then combining definitions with a variable-saving intersection lemma. Together with a prior undecidability result, the seven-unknown definition yields that a mixed quantifier prefix with nine universal and seven existential quantifiers over the rationals is already undecidable. A sympathetic reader cares because every drop in the number of unknowns tightens the boundary between what is algorithmically decidable and what is not in the arithmetic of the rationals.

Core claim

For every global field K and every finite set S0 of its non-archimedean places, the ring of S0-integers is universally definable in K by a single polynomial inequality in seven variables: x lies in that ring if and only if a fixed polynomial F(x,y1,...,y7) never vanishes for rational (or K-rational) y's. In particular Z is forall-7 definable in Q, so Q minus Z is diophantine over Q with seven unknowns.

What carries the argument

Fixed-parameter quaternion formulas: after finitely many frozen torus parameters are chosen by local Hensel covers, a three-variable equation produces two reduced-norm-one elements whose reduced traces sum to a given c, forcing c to be integral at every ramified place of the associated quaternion algebra; the Hasse principle on the resulting quaternary quadratic form then globalizes the local points.

Load-bearing premise

The argument needs every frozen local equation to stay smoothly solvable after approximation, so that the homogenized quaternary form is non-degenerate and has points everywhere, letting the Hasse–Minkowski theorem supply a global solution.

What would settle it

Exhibit a global field and an S0 for which no seven-variable polynomial works, or find a gap where the frozen-parameter equation fails to have a smooth local point at the extra target place after the global norm approximation that realises the prescribed ramification set.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Z is forall-7 definable over Q, improving the previous forall-10 bound.
  • The forall-9 exists-7 theory of the rationals is undecidable.
  • The same seven-unknown bound holds for S-integers in every global field, not only Q.
  • Unions of maximal ideals outside a finite set become existentially definable with seven variables via the quaternion bridge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Further reduction below seven would likely require a still cheaper existential definition of the quaternion parameter set or a two-variable replacement for the frozen trace-sum equation.
  • The same freezing-plus-Hasse pattern may apply to other integrally closed rings whose local conditions are controlled by central simple algebras.
  • An explicit seven-variable polynomial over Z, even with huge coefficients, would make the undecidability statement fully effective for machine search.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proves that for any global field K and finite S_0 ⊆ V_K, the ring of S_0-integers O_{S_0} is ∀7-definable in K (Theorem 1.1). In particular Z is ∀7-definable in Q, so Q\Z is diophantine over Q with 7 unknowns, improving Daans’ record of 10. The argument freezes the torus parameter in a three-variable reduced-trace quaternion formula to a finite global set Λ, obtains smooth local points at the fixed ramified places and at one additional target place, realises Δ(Q_{a,b})=S∪{w} by global norm approximation, and applies Hasse–Minkowski to the resulting nondegenerate quaternary form. Existential ranks are combined via Daans–Dittmann–Fehm to reach the count 2+(3+3−1)=7. Combined with an earlier result of the author this yields undecidability of ∀9∃7 over Q.

Significance. A reduction from 10 to 7 universal quantifiers for the definability of Z in Q (and of O_{S_0} in arbitrary global fields) is a clear quantitative advance in the Koenigsmann–Poonen–Daans line. The undecidability corollary for the mixed prefix ∀9∃7 is a concrete payoff. The method—finite freezing of the Cayley/Artin–Schreier torus parameter together with a fixed-fiber Hasse principle—is a natural and reusable refinement of Daans’ framework. Strengths include characteristic-independent quaternion formulas, explicit use of the Daans–Dittmann–Fehm rank theorem, and a clean final quantifier arithmetic. If the local-to-global and freezing steps hold as claimed, the result is a solid contribution to definability and undecidability over global fields.

major comments (4)
  1. [§9, Proposition 9.1] Proposition 9.1 (proof, p. 16) asserts that “Sections 7–10 construct (a,b)∈Φ_S^u ////// a frozen parameter ////// and local points” and that “Section 11 then produces a global point.” The manuscript contains only nine sections; the local constructions live in §§5–7 and the Hasse step in §8. These broken cross-references make the main existence argument for D_Θ uncheckable as written and must be rewritten with correct pointers and a self-contained summary of which local points are used where.
  2. [§8, equations (8.16)–(8.17)] The load-bearing local-to-global step (§8) requires that every frozen branch yield a nondegenerate (char ≠ 2) or nonsingular (char 2) quaternary form that is isotropic over every completion, including the places in S where the quaternion algebra is division. The text invokes the finite-freezing cover (Prop. 3.1) and the target-place constructions (§§5–6) but does not spell out, for a general frozen τ∈Λ, why the form remains nondegenerate at places of S and why the smooth K_v-points produced by Prop. 3.1 survive the global choice of (a,b) after the norm approximation of §7. A short verification (or an explicit residual nondegeneracy check) should be added.
  3. [§3 Prop. 3.1; §6] In the characteristic-2 half of Prop. 3.1 (c=0 case) and in §6, smooth solubility rests on producing a unit η in the image of the unramified norm and on a partial derivative of exact valuation 1. The argument is plausible but compressed: the appeal to Lemma 3.2, the construction of q via a nonsquare unit differential, and the subsequent Hensel step for the actual c with v_w(c)≥4 should be written so that the valuations of P and P' are displayed explicitly and the choice of witness variable is unambiguous.
  4. [§9, (9.1)–(9.2); §§5–6] Daans’ bridge (9.2) is applied to h(a,b,x^3) rather than to h(a,b,x). The text inherits the cube from Daans but never records why the cube (as opposed to x itself) is required for the valuation identities v_w(c)=6m−2 and for the residue-field selection lemmas. One sentence clarifying the necessity of the cube would remove a small but recurring ambiguity in §§5–6 and 9.
minor comments (6)
  1. [passim] The running title and several displayed formulas use Q\Z / O_{S_0} notation inconsistently with and without thin spaces; normalise.
  2. [§2, Lemma 2.2] Lemma 2.2: the final adjustment “if z_2=−z_1 replace by −z_1^{−1}” is correct but easy to misread; a one-line check that the new pair still meets z_2≠z_1^{−1} would help.
  3. [§4] In §4 the set Λ is defined differently in odd and even characteristic (four parameters vs one); the notation Θ(a,b,c)=∨_{τ∈Λ} Ψ_τ(a,b,c) is fine, but stating the cardinality bound on Λ explicitly would clarify the later “finite exceptional set” removals.
  4. [§1, Theorem 1.2] Reference [4] is cited as the source of the rank theorem; ensure the published or latest arXiv version is used and that the hypothesis “finitely generated over a perfect subfield” is quoted accurately for global function fields.
  5. [Acknowledgments] Acknowledgments mention AI motivation; this is harmless but unusual in the field—consider shortening to a standard thanks if the journal style is conservative.
  6. [passim] Typos: “undecidablity” (p. 2), “K¨ ahler” spacing (p. 10), “nonempty” vs “non-empty” inconsistency.

Circularity Check

0 steps flagged

No significant circularity: the n=7 bound is obtained by a genuine reduction (finite freezing of the torus parameter plus Hasse–Minkowski on fixed fibers), not by re-labelling inputs or self-citation of the target claim.

full rationale

The derivation chain is self-contained against its stated black-box inputs. Daans’ bridge (9.2), the ∃³-definability of Φ_S^u, and the Daans–Dittmann–Fehm intersection theorem are used as external lemmas whose statements do not contain the new bound n=7; the paper’s own contribution is the construction of a finite family Λ of frozen parameters (Prop. 3.1 and §4) so that the three-variable quaternion formula becomes a fixed-parameter quaternary form to which Hasse–Minkowski applies, yielding the quantifier count 2+(3+3-1)=7. Self-citations to the author’s earlier undecidability-prefix results ([19]) appear only after Theorem 1.1 is proved and are not load-bearing for the definability statement itself. No equation forces the conclusion by normalisation, fitted parameter, or uniqueness imported from the same author. Minor textual defects (references to nonexistent Sections 10–11) do not create definitional circularity. Score 0 is therefore the honest finding.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 2 invented entities

The proof is a pure-existence argument in arithmetic geometry/logic. It imports standard global-field toolkit (Hasse–Minkowski, weak approximation, Brauer exact sequence, Hensel, local class field theory for unramified norms) and several black-box lemmas from Daans. No numerical parameters are fitted. The only ‘new entities’ are definitional constructions (frozen parameter sets, fixed-parameter formulas) introduced to cut quantifier count.

axioms (7)
  • standard math Hasse–Minkowski theorem for quaternary quadratic forms over global fields (including char-2 non-singular forms)
    Invoked in Section 8 to pass from local isotropic points of the homogenised fixed-parameter equation to a global affine solution.
  • standard math Weak approximation in global fields and in finite separable extensions
    Used repeatedly to glue local (a,b,τ) data and to realise global norms (Lemma 7.1, Sections 4–7).
  • standard math Global Brauer exact sequence / Albert–Brauer–Hasse–Noether: period equals index for quaternion algebras
    Section 7 realises a unique quaternion algebra ramified exactly at S∪{w}.
  • domain assumption Daans’ bridge lemma: z lies in some m_w (w∉S) iff ∃(a,b)∈Φ_S^u with h(a,b,z) integral at all places of Δ(Q_{a,b})
    Quoted as (9.2) / [3, Lemma 5.5]; the whole reduction from O_{S_0} to the existential set D_Θ rests on it.
  • domain assumption Daans–Dittmann–Fehm existential-rank inequality: intersection of two ∃^{m_i} sets is ∃^{m_1+m_2−1}
    Theorem 1.2, used in the final variable count 3+3−1.
  • standard math Local trace-sum lemma for norm-one elements of the unramified quadratic extension (Lemma 2.2)
    Supplies the two reduced-norm-one quaternions whose traces sum to c; proved in-text from elementary finite-field counting plus Hensel.
  • domain assumption Φ_S^u is ∃3-definable and each maximal ideal is ∃3-definable
    Taken from Daans [3, Lemma 5.2, Prop. 4.2]; feeds the final 7-variable count.
invented entities (2)
  • Finite freezing datum (I_v, t_{v,i}, U_{v,i}, W_{v,i}) and global frozen set Λ no independent evidence
    purpose: Replace a locally varying torus parameter by a finite list of global constants so that the quaternion equation uses only three witness variables uniformly.
    Constructed in Prop. 3.1 and Section 4 via compactness of P_v and weak approximation; existence is proved but the concrete list is not exhibited.
  • Fixed-parameter formulas Ψ_τ^{odd} / Ψ_τ^{(2)} and their finite disjunction Θ no independent evidence
    purpose: Express the condition ‘c is integral at every place of Δ(Q_{a,b})’ with a uniform three-variable existential formula.
    Defined in (2.1), (2.6), (4.10); the whole ∃7 definition is built by composing Θ with Φ_S^u and h.

pith-pipeline@v1.2.0-daily-grok45 · 20016 in / 3939 out tokens · 75590 ms · 2026-07-31T02:21:39.838577+00:00 · methodology

0 comments
read the original abstract

In 2016 J. Koenigsmann proved that $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\not\in\mathbb Z\iff \exists x_1,\ldots,x_{n}\in\mathbb Q\,[P(t,x_1,\ldots,x_{n})=0].$$ In this paper we show that we may take $n=7$ which improves the previous record $n=10$ obtained by Daans in 2024. (Actually we even extend this to any global field.) This, together with a previous result of Z.-W. Sun, implies that there is no algorithm to decide for any $F(x_1,\ldots,x_{16})\in\mathbb Z[x_1,\ldots,x_{16}]$ whether $$\forall x_1,\ldots,x_9\in\mathbb Q\exists y_1,\ldots,y_{7}\in\mathbb Q\,[F(x_1,\ldots,x_9,y_1,\ldots,y_{7})=0].$$

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 2 linked inside Pith

  1. [1]

    Cassady,Hasse principles for quadratic forms over function fields, J

    C. Cassady,Hasse principles for quadratic forms over function fields, J. Algebra628 (2023), 120–151. Preprint version: arXiv:2204.06368

    Pith/arXiv arXiv 2023

  2. [2]

    Daans,Universally defining finitely generated subrings of global fields, Documenta Math.26(2021), 1851–1869

    N. Daans,Universally defining finitely generated subrings of global fields, Documenta Math.26(2021), 1851–1869

    2021

  3. [3]

    Daans,Universally definingZinQwith10quantifiers, J

    N. Daans,Universally definingZinQwith10quantifiers, J. London Math. Soc.109 (2024), no. 2, Article e12864

    2024

  4. [4]

    Daans, P

    N. Daans, P. Dittmann, and A. Fehm,Existential rank and essential dimension of diophantine sets, preprint, arXiv:2102.06941

    arXiv

  5. [5]

    Gille and T

    P. Gille and T. Szamuely,Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics 101, Cambridge University Press, 2006

    2006

  6. [6]

    Koenigsmann,DefiningZinQ, Annals of Math.183(2016), 73–93

    J. Koenigsmann,DefiningZinQ, Annals of Math.183(2016), 73–93

    2016

  7. [7]

    Matiyasevich,Enumerable sets are diophantine, Dokl

    Y. Matiyasevich,Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR191 (1970), 279–282; English translation with addendum, Soviet Math. Doklady11 (1970), 354–357

    1970

  8. [8]

    Lidl and H

    R. Lidl and H. Niederreiter,Finite Fields, second ed., Encyclopedia of Mathematics and its Applications 20, Cambridge University Press, 1997

    1997

  9. [9]

    O. T. O’Meara,Introduction to Quadratic Forms, Classics in Mathematics, Springer, Berlin, 2000 reprint of the 1963 edition

    2000

  10. [10]

    R. S. Pierce,Associative Algebras, Graduate Texts in Mathematics 88, Springer, New York, 1982

    1982

  11. [11]

    Pollak,Orthogonal groups over global fields of characteristic2, J

    B. Pollak,Orthogonal groups over global fields of characteristic2, J. Algebra15 (1970), 589–595

    1970

  12. [12]

    Poonen,Characterizing integers among rational numbers with a universal- existential formula, Amer

    B. Poonen,Characterizing integers among rational numbers with a universal- existential formula, Amer. J. Math.131(2009), 675–682

    2009

  13. [13]

    Reiner,Maximal Orders, London Mathematical Society Monographs, New Series 28, Oxford University Press, 2003 reprint of the 1975 edition

    I. Reiner,Maximal Orders, London Mathematical Society Monographs, New Series 28, Oxford University Press, 2003 reprint of the 1975 edition

    2003

  14. [14]

    Robinson,Definability and decision problems in arithmetic, J

    J. Robinson,Definability and decision problems in arithmetic, J. Symbolic Logic14 (1949), 98–114

    1949

  15. [15]

    Scharlau,Quadratic and Hermitian Forms, Grundlehren der mathematischen Wis- senschaften 270, Springer, Berlin, 1985

    W. Scharlau,Quadratic and Hermitian Forms, Grundlehren der mathematischen Wis- senschaften 270, Springer, Berlin, 1985

    1985

  16. [16]

    Serre,Local Fields, Graduate Texts in Mathematics 67, Springer, New York, 1979

    J.-P. Serre,Local Fields, Graduate Texts in Mathematics 67, Springer, New York, 1979

    1979

  17. [17]

    Stichtenoth,Algebraic Function Fields and Codes, second ed., Graduate Texts in Mathematics 254, Springer, Berlin, 2009

    H. Stichtenoth,Algebraic Function Fields and Codes, second ed., Graduate Texts in Mathematics 254, Springer, Berlin, 2009. Q\ZIS DIOPHANTINE OVERQWITH 7 UNKNOWNS 19

    2009

  18. [18]

    Sun,Further results on Hilbert’s Tenth Problem, Sci

    Z.-W. Sun,Further results on Hilbert’s Tenth Problem, Sci. China Math.64(2021), 281–306

    2021

  19. [19]

    Sun,Mixed quantifier prefixes over Diophantine equations with integer variables, preprint, arXiv:2103.08302, 2021

    Z.-W. Sun,Mixed quantifier prefixes over Diophantine equations with integer variables, preprint, arXiv:2103.08302, 2021

    Pith/arXiv arXiv 2021

  20. [20]

    Wu,Similarity of quadratic forms over global fields in characteristic2, Adv

    Z. Wu,Similarity of quadratic forms over global fields in characteristic2, Adv. Appl. Clifford Algebras29(2019), Article No. 86

    2019

  21. [21]

    Zhang and Z.-W

    G.-R. Zhang and Z.-W. Sun,Q\Zis diophantine overQwith32unknowns, Bull. Pol. Acad. Sci. Math.70(2022), no. 2, 93–106. School of Mathematics, Nanjing University, Nanjing 210093, People’s Re- public of China Email address:zwsun@nju.edu.cn

    2022