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No algorithm decides whether polynomial equations over the Gaussian integers in 18 variables have solutions.

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Proof reducing the undecidability threshold for Diophantine equations over Z[i] to 18 variables via a new gadget and integer conditions.

T0 review reviewed 2026-06-27 challenge →

load-bearing objection The paper lowers the undecidability threshold over Z[i] to 18 variables via a gadget that folds conditions together, but the AI-generated proof still needs line-by-line checking. the 2 major comments →

arxiv 2606.12776 v1 pith:CKKL6O5N submitted 2026-06-11 math.NT

An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$

Yuchen Ding , Junfeng Li This is my paper
classification math.NT
keywords Diophantine equationsundecidabilityGaussian integersHilbert's tenth problemvariable reductionMatiyasevich theorem
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.

The reading

The paper shows that solvability of polynomial equations over the Gaussian integers cannot be decided by any algorithm once the number of unknowns reaches 18. It improves the prior bound of 20 variables by following an established rationality criterion and integer test while eliminating two variables through targeted changes in the construction. One change uses two integer conditions to replace a clearing-denominator variable. The other folds the remaining nonzero condition into an existing relation-combining lemma via a single auxiliary polynomial. A reader who accepts the construction therefore concludes that the undecidability threshold for this ring lies at or below 18 variables.

Core claim

The central claim is that the Diophantine problem over Z[i] is undecidable for equations in 18 unknowns. The proof reduces an undecidable problem to the existence of solutions in the Gaussian integers by means of a rationality criterion and an integer test, but avoids one variable by imposing two integer conditions in place of a clearing-denominator step and absorbs the remaining nonzero condition into the relation-combining lemma through the one-variable gadget (2R+1)(3R+1).

What carries the argument

The one-variable gadget (2R+1)(3R+1) that absorbs the nonzero condition into the relation-combining lemma, together with the replacement of the clearing-denominator variable by two integer conditions.

Load-bearing premise

The gadget (2R+1)(3R+1) absorbs the nonzero condition without introducing errors and two integer conditions suffice to avoid the clearing-denominator variable.

What would settle it

An explicit algorithm that correctly decides solvability for every polynomial equation in 18 variables over the Gaussian integers, or a concrete case where the gadget (2R+1)(3R+1) fails to preserve the required nonzero property.

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

If this is right

  • Undecidability of Diophantine equations over Z[i] holds already at 18 variables.
  • The variable count required for undecidability results over this ring is at most 18.
  • The same rationality criterion and integer test can be applied with the stated savings in variables.

Where Pith is reading between the lines

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

  • Similar gadget replacements might reduce the variable count below 18 in this or related rings.
  • The technique of replacing a clearing step by two conditions could apply to undecidability proofs over other quadratic integer rings.
  • If the gadget works cleanly, it may also shorten proofs of undecidability results that still rely on separate nonzero checks.
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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

2 major / 0 minor

Summary. The manuscript claims to deliver an AI-assisted proof that there is no algorithm deciding the solvability of polynomial equations over the Gaussian integers in 18 unknowns, improving the 20-variable theorem of Matiyasevich and Sun. It follows their rationality criterion and integer test, saves the clearing-denominator variable via two integer conditions, and absorbs the remaining nonzero condition into the relation-combining lemma via the one-variable gadget (2R+1)(3R+1).

Significance. If the central construction is correct, the result would tighten the known bound on the number of variables needed to establish undecidability of Diophantine equations over Z[i]. The work is a direct, incremental extension of prior techniques rather than a conceptual advance; the AI assistance in generating the proof is methodologically noteworthy but does not alter the mathematical assessment.

major comments (2)
  1. [Abstract] Abstract: the claim that the one-variable gadget (2R+1)(3R+1) absorbs the nonzero condition without introducing new errors or extra variables cannot be verified from the given description; the abstract provides no explicit check that this gadget preserves the rationality criterion and integer test from the referenced prior work.
  2. [Abstract] Abstract: the assertion that two integer conditions suffice to eliminate the clearing-denominator variable is stated without any indication of how the resulting system remains within 18 variables or avoids circularity with the integer test; this step is load-bearing for the reduction from 20 to 18 variables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's review of our manuscript. Below we address the major comments point by point, providing clarifications on the constructions mentioned.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the one-variable gadget (2R+1)(3R+1) absorbs the nonzero condition without introducing new errors or extra variables cannot be verified from the given description; the abstract provides no explicit check that this gadget preserves the rationality criterion and integer test from the referenced prior work.

    Authors: The abstract summarizes the key innovation, but the full paper contains the explicit verification that the gadget (2R+1)(3R+1) preserves the rationality criterion and integer test. Specifically, we substitute the gadget into the relation-combining lemma and confirm that the resulting expressions satisfy the integer conditions without additional variables or errors. If the referee requires, we can include a brief outline of this verification in the abstract in the revised version. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that two integer conditions suffice to eliminate the clearing-denominator variable is stated without any indication of how the resulting system remains within 18 variables or avoids circularity with the integer test; this step is load-bearing for the reduction from 20 to 18 variables.

    Authors: In the manuscript, we describe how the two integer conditions replace the clearing-denominator variable from the prior 20-variable construction, directly resulting in an 18-variable system. The integer test is applied to the modified system after imposing these conditions, which are chosen to be compatible with the test and do not introduce circularity. The variable count is verified by enumerating the variables used in the rationality criterion application and the lemma. We will add a sentence to the abstract clarifying this reduction to address the concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation extends the Matiyasevich-Sun 20-variable result over Z[i] by adding an independent one-variable gadget (2R+1)(3R+1) to absorb the nonzero condition and two integer conditions to eliminate the clearing-denominator variable. It explicitly follows the external rationality criterion and integer test from prior work by different authors rather than re-deriving them. No load-bearing step reduces by definition, by fitted input renamed as prediction, or by a self-citation chain; the central 18-variable claim rests on the new gadget and variable-saving technique applied to the cited external framework. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no information on free parameters, axioms, or invented entities can be extracted or verified.

reviewed 2026-06-27 · how reviews work

0 comments
Cite this review

Pith. "Pith review of An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$." pith.science (2026). https://pith.science/paper/CKKL6O5N

@misc{pith2026260612776,
  author       = {Pith},
  title        = {Pith review of: An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKKL6O5N}},
  note         = {Machine review of arXiv:2606.12776}
}
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abstract

This paper presents an AI proof that there is no algorithm deciding whether a polynomial equation over the Gaussian integers in $18$ unknowns has a solution. The proof improves the $20$-unknown theorem of Matiyasevich and Sun. It follows their rationality criterion and integer test, but saves two variables: the clearing-denominator variable is avoided by imposing two integer conditions, and the remaining nonzero condition is absorbed into the relation-combining lemma by the one-variable gadget $(2R+1)(3R+1)$.

discussion (0)

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Forward citations

Cited by 1 Pith paper

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

  1. On Diophantine equations over the integer rings of quadratic fields

    math.NT 2026-08 conditional novelty 6.0

    For every quadratic field K, deciding solvability of polynomial equations in 16 variables over its ring of integers is undecidable, and 15 variables suffice when K is real quadratic.

Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages · cited by 1 Pith paper

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    M. Davis, H. Putnam and J. Robinson, The decision problem for exponential Diophantine equations, Ann. of Math. 74 (1961), 425–436

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    Y. Matiyasevich, Enumerable sets are Diophantine, Dokl. Akad. Nauk SSSR 191 (1970), 279–282; English translation with addendum, Soviet Math. Dokl. 11 (1970), 354–357

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    Matiyasevich, Hilbert’s Tenth Problem, MIT Press, Cambridge, Massachusetts, 1993

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    Matiyasevich and Z.-W

    Y. Matiyasevich and Z.-W. Sun, Undecidability on Diophantine equations over Z[i] with 20 unknowns, to appear in J. Number Theory, arXiv:2510.18794, 2025

    work page arXiv 2025

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    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. 7

    2021

This paper was first reviewed by grok-4.3 on June 27, 2026.