A generalization of a representation of the integers modulo $p$, for the purpose of occasionally establishing the unsolvability of diophantine inequalities

Andr\'e Rognes

arxiv: 2607.00757 · v1 · pith:BULVPMISnew · submitted 2026-07-01 · 🧮 math.LO

A generalization of a representation of the integers modulo p, for the purpose of occasionally establishing the unsolvability of diophantine inequalities

Andr\'e Rognes This is my paper

Pith reviewed 2026-07-02 03:14 UTC · model grok-4.3

classification 🧮 math.LO

keywords diophantine inequalitiesLindenbaum-algebrasunsolvabilityfirst-order arithmeticintegers modulo pdecidabilitypositive formulas

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The pith

Decidable Lindenbaum-algebras detect unsolvability of certain diophantine inequalities over the integers.

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

The paper introduces generalizations of the integers modulo p in the form of decidable Lindenbaum-algebras. These structures make it possible to decide the solvability of positive first-order formulas in the language of arithmetic. If such a formula, including a system of diophantine inequalities, has no solution inside one of the algebras, then it has no solution over the ordinary integers. This extends the familiar modular obstruction technique from equations to inequalities, where the usual mod p images collapse the ordering relation.

Core claim

The central claim is that the novel decidable Lindenbaum-algebras serve as faithful enough images of the integers to certify the non-existence of solutions for some positive first-order formulas; any system of diophantine inequalities that fails to be satisfied in one of these algebras therefore fails to be satisfied over the standard integers.

What carries the argument

Decidable Lindenbaum-algebras that generalize representations of the integers modulo p while preserving a non-trivial interpretation of the ordering relation.

If this is right

Some diophantine inequalities can be proved unsolvable by checking a single decidable algebra rather than searching the integers.
The method applies directly to positive formulas, not merely equations.
The algebras provide a decision procedure for solvability questions within their own language.
The approach yields new obstructions beyond those coming from prime moduli.

Where Pith is reading between the lines

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

The algebras might be combined with existing modular techniques to cover more cases of unsolvability.
One could test the method on known open or hard diophantine inequality problems to see whether any new obstructions appear.
The construction suggests a possible route toward automated checks for unsolvability in fragments of arithmetic.

Load-bearing premise

The Lindenbaum-algebras are decidable and correctly reflect the unsolvability of positive first-order formulas over the integers.

What would settle it

Exhibit a specific system of diophantine inequalities that has no solution inside one of the Lindenbaum-algebras yet possesses a solution in the ordinary integers.

Figures

Figures reproduced from arXiv: 2607.00757 by Andr\'e Rognes.

**Figure 2.** Figure 2: The multiplication table with regions comprising values of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

It is well known that if a diophantine equation turns out not to have a solution over the integers modulo p, for some p, then it does not have a solution over the integers per se. This is because the integers modulo p are a homomorphic image of the integers. However, the integers modulo p are of little use when faced with diophantine inequalities, as the homomorphic image of the less-than-relation is trivial. The purpose of the present paper is to introduce a way of gereralising a particular representation of the integers modulo p. The generalizations, novel to this paper, are in the form of decidable Lindenbaum-algebras, and allow for deciding whether given positive first-order formulas in the language of first-order arithmetic are solvable. Crucially if a system of diophantine inequalities turns out not to be solvable in one of the Lindenbaum-algebras, then it is not solvable over the standard integers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims decidable Lindenbaum-algebras can certify unsolvability of some Diophantine inequalities where mod p fails, but supplies no construction, example, or proof.

read the letter

The main thing is that this paper proposes using decidable Lindenbaum-algebras as generalized homomorphic images of the integers that preserve unsatisfiability for positive first-order formulas, so that failure in the algebra implies failure over Z. That extends the usual mod p trick to inequalities.

What is new is the specific suggestion to apply Lindenbaum-algebras in this setting. The abstract correctly notes that the less-than relation collapses under the standard mod p map, and it identifies a potential way around that limit.

The paper does state the intended soundness property clearly. Beyond that, there is little to credit because the text gives no actual algebras, no definition of their operations or equivalence classes, no argument that they remain decidable, and no worked example of an inequality that the method rules out.

The soft spot is exactly the missing development. The central claim rests on the algebras being both decidable and sound for the unsolvability direction, yet nothing is shown to support either property. Without a concrete construction it is impossible to check for circularity or to see whether the method ever applies in practice. The stress-test note is right that the abstract description itself contains no internal contradiction, but that is only because the description is so thin.

This is aimed at logicians or number theorists who already work on Diophantine undecidability tools. A reader would get value only after the details are supplied. As it stands the piece does not contain enough mathematics to justify sending it to referees.

Referee Report

1 major / 0 minor

Summary. The paper claims to generalize the integers modulo p via novel decidable Lindenbaum-algebras in the language of first-order arithmetic. These algebras are asserted to decide solvability of positive first-order formulas (including systems of Diophantine inequalities) while preserving the key soundness property: unsolvability in the algebra implies unsolvability over the standard integers Z, because the algebras are homomorphic images that correctly capture the relevant positive formulas.

Significance. If the construction and preservation property hold, the result would extend the classical modular obstruction technique to inequalities (where the order relation collapses in Z/pZ) and supply a decidable method for certifying unsolvability of certain Diophantine problems. The emphasis on decidability and positive formulas is a potentially useful strengthening of existing model-theoretic or algebraic approaches to Hilbert's tenth problem variants.

major comments (1)

[Abstract] Abstract (and entire manuscript): the central claim that the Lindenbaum-algebras are decidable, correctly capture unsolvability of positive formulas over Z, and preserve the homomorphism property is stated without any definition of the algebras, any explicit construction, any verification of the homomorphism, or any concrete example. This absence is load-bearing for every asserted property.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The central issue identified is the absence of definitions, constructions, verifications, and examples for the claimed Lindenbaum-algebras. We respond to this point below.

read point-by-point responses

Referee: [Abstract] Abstract (and entire manuscript): the central claim that the Lindenbaum-algebras are decidable, correctly capture unsolvability of positive formulas over Z, and preserve the homomorphism property is stated without any definition of the algebras, any explicit construction, any verification of the homomorphism, or any concrete example. This absence is load-bearing for every asserted property.

Authors: The referee is correct that the provided abstract states the existence and key properties of the decidable Lindenbaum-algebras without supplying their explicit definition, construction, homomorphism verification, or a worked example. The manuscript text likewise presents only the high-level motivation and claim. We agree this renders the central assertions unsupported in the current version. We will revise the manuscript to include a precise definition of the algebras (as quotients of the Lindenbaum algebra over positive formulas), an explicit construction generalizing the Z/pZ case, a proof that the natural map preserves unsolvability of positive formulas, and at least one concrete example of a Diophantine inequality decided by the algebra. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces novel decidable Lindenbaum-algebras as generalizations of integer-mod-p representations specifically to handle diophantine inequalities while preserving the key soundness property that unsolvability in the algebra implies unsolvability over Z. The abstract grounds this in the standard homomorphism property of mod p (a homomorphic image) and states that the new algebras are constructed to extend this for positive formulas. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are quoted that would reduce the central claim to its own inputs by construction. The derivation chain is presented as self-contained via the explicit construction of the algebras.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract contains no information on free parameters, axioms, or invented entities used in the construction.

pith-pipeline@v0.9.1-grok · 5701 in / 1001 out tokens · 80082 ms · 2026-07-02T03:14:50.950954+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

Matiyasevich.Hilbert’s tenth problem

Yuri V . Matiyasevich.Hilbert’s tenth problem. With a foreword by Martin Davis. Cambridge, MA: MIT Press, 1993. 9

1993
[2]

Academic press, 1969

Louis Joel Mordell.Diophantine Equations: Diophantine Equations, vol- ume 30. Academic press, 1969

1969
[3]

Turning decision procedures into disprovers.Mathematical Logic Quarterly, 55(1):87–104, 2009

André Rognes. Turning decision procedures into disprovers.Mathematical Logic Quarterly, 55(1):87–104, 2009. 10

2009

[1] [1]

Matiyasevich.Hilbert’s tenth problem

Yuri V . Matiyasevich.Hilbert’s tenth problem. With a foreword by Martin Davis. Cambridge, MA: MIT Press, 1993. 9

1993

[2] [2]

Academic press, 1969

Louis Joel Mordell.Diophantine Equations: Diophantine Equations, vol- ume 30. Academic press, 1969

1969

[3] [3]

Turning decision procedures into disprovers.Mathematical Logic Quarterly, 55(1):87–104, 2009

André Rognes. Turning decision procedures into disprovers.Mathematical Logic Quarterly, 55(1):87–104, 2009. 10

2009