arXiv:2508.00944v3

Positivity of Nearly Linearly Recurrent Sequences

Amaury Pouly, Mahsa Shirmohammadi, James Worrell

math.DScs.LO11B3711J8111J8711U0593B03

Paper metadata

Original submission: July 31, 2025
Latest revision: July 26, 2026 · version 3
Claimed MSC: 11B37, 11J81, 11J87, 11U05, 93B03
arXiv categories: math.DS, cs.LO
Paper license: CC BY 4.0 ↗

Abstract

Nearly linear recurrences generalise linear recurrences and can be represented as special cases of both linear time-invariant systems in control theory and linear-constraint loops in program analysis. We formulate the Positivity Problem for such recurrences: given a recurrence and initial values, decide whether every sequence satisfying the recurrence is termwise nonnegative. This problem generalises Positivity for linear recurrence sequences and is a special case of halfspace non-reachability for linear time-invariant systems. Our main result is a decision procedure for order-2 recurrences. The termination of the procedure relies on a transcendence theorem of independent interest: we prove that certain convergent series obtained by summing the absolute values of terms of algebraic linear recurrence sequences are transcendental.

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Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The order-two positivity decision procedure and the two transcendence theorems are supported under their stated algebraicity, spectral, and independence hypotheses.

Theorem 1Correct

Positivity is decidable for order-two nearly linear recurrences

Pages 6–7 · Theorem 1 and Equations (7)–(10) · arXiv:2508.00944v3

Minimizing each control contribution independently gives the exact pointwise minimum sequence in Equation (9). If a power of the companion matrix has positive real spectrum, eventual signs and all required partial sums are effective linear recurrences. In the remaining complex-conjugate cases, density of the irrational rotation settles spectral radius at least one. For spectral radius below one, Theorem 2 makes the computable limiting value nonzero, so approximation with an effective tail bound eventually determines its sign and reduces positivity to finitely many algebraic comparisons. The case split and all root-of-unity and modulus tests are effective.

Theorem 2Correct

Transcendence of the absolute-value series for a conjugate pair

Pages 7–14 · Theorem 2, Propositions 4–6, and Claims 8–9 · arXiv:2508.00944v3

The convergent denominators of the irrational argument produce recurrences for $u_m=a\lambda^m+\overline a\,\overline\lambda^m$ whose absolute-value failures occur only at visits to two shrinking intervals. Proposition 6 gives both the required sparseness and the upper return-time bound. Truncating the resulting identity yields a linear form that is exponentially smaller than its height, while Claims 8 and 9 verify both hypotheses of Lemma 7. Multiplicative independence of $\lambda/\overline\lambda$ and the divergence of the relevant exponent differences then exclude every finite-ratio alternative supplied by the Subspace Theorem.

Theorem 12Correct

Transcendence for simple recurrences with two dominant roots

Pages 16–23 · Theorem 12 and Propositions 13–17 · arXiv:2508.00944v3

The cited lower bound for nondegenerate algebraic recurrences makes the sign of the sequence eventually equal to the sign of its two dominant conjugate terms. The higher-order recurrence in Equation (28), the shrinking-interval description of its exceptional indices, and the non-cancellation argument in Propositions 14–17 are valid under multiplicative independence of all characteristic roots. The height estimates and visit-gap bounds in Claims 18 and 19 meet Lemma 7's hypotheses, and Proposition 16 excludes each possible constant quotient of two distinct $S$ -unit coordinates.

02Proofs5 reported findingsCorrect

The reductions, recurrence identities, continued-fraction return estimates, and Subspace-Theorem arguments supporting the central results are correct and complete. No substantive or literal defect survived the mandatory finding gate.

Theorem 1 reductionCorrect and complete

The bang-bang minimum and four spectral cases are exhaustive

Pages 4–7 · Sections 2.2–2.3 · arXiv:2508.00944v3

Equation (8) chooses the correct endpoint of $[\varepsilon_0,\varepsilon_1]$ for each independent control coefficient and therefore gives the actual minimum at each time. The real-spectrum and root-of-unity cases reduce to effectively signed interleaved recurrences. In the non-torsion complex cases, the three alternatives $|\lambda|>1$ , $|\lambda|=1$ , and $|\lambda|<1$ are handled with the correct inequality directions. In the last case, exponential convergence supplies an effective index after the nonzero limit's sign has been isolated.

Lemma 7Correct and complete

The required lower bound for an $S$ -integer plus $S$ -units

Pages 10–16 · Lemma 7 and Appendix A · arXiv:2508.00944v3

The normalized absolute values satisfy the product formula used in the height calculations. The first inequality confines the tuple to finitely many proper subspaces by the stated $p$ -adic Subspace Theorem; eliminating the non-unit coordinate in each resulting hyperplane reduces to the standard finite-ratio theorem for sums of $S$ -units. The one-coordinate residual case is correctly closed by Northcott finiteness. These are exactly the alternatives used later.

Proof of Theorem 2Correct and complete

Exceptional visits and height inequalities

Pages 8–14 · Propositions 4–6 and Section 4.2 · arXiv:2508.00944v3

Each component of $J_n$ has length $2|\epsilon_n|>|\epsilon_n|+|\epsilon_{n+1}|$ , so Proposition 4 gives the claimed return bound. Four subintervals of length $|\epsilon_n|$ give the five-visit separation estimate, and best approximation yields the $q_{n+1}$ gap. The choices $t=s+4$ and Equations (22)–(23) give the two strict height inequalities in the required directions. Every quotient of distinct unit coordinates then has either unbounded modulus exponent or an unbounded non-torsion phase exponent, so it cannot take values in one fixed finite set infinitely often.

Proof of Theorem 12Correct and complete

Higher-order non-cancellation and final Subspace-Theorem step

Pages 17–23 · Appendix B · arXiv:2508.00944v3

Multiplicative independence prevents cancellation of the monomials isolated in Proposition 14. Proposition 15 correctly rules out persistent visits whose indices are affine functions of $q_n$ , and Proposition 16 applies that fact to all possible monomial quotients. This proves the equivalence of the exceptional sets in Proposition 17. The degree- $d$ bounds on the remaining monomials give Claims 18–19, and the three quotient cases at the end exhaust all distinct coordinates of the constructed linear form.

Material cited inputsCorrect and complete

External results are invoked in their stated regimes

Pages 4, 7, 15–17, and 24–25 · cited recurrence, height, and Subspace-Theorem inputs · arXiv:2508.00944v3

Low-order LRS positivity is used only in order three; the recurrence-separation bound is applied to a nondegenerate dominant conjugate pair; and Schlickewei's theorem is applied over a number field with a finite place set containing every Archimedean place and every denominator needed to make the characteristic roots units. The paper verifies the additional height inequalities required by its specialized Lemma 7 rather than assuming them.

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No non-novelty findings.

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Paper: arXiv:2508.00944v3
Authors listed: Amaury Pouly, Mahsa Shirmohammadi, James Worrell
Audit date: August 15, 2026

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Positivity of Nearly Linearly Recurrent Sequences

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Positivity is decidable for order-two nearly linear recurrences

Transcendence of the absolute-value series for a conjugate pair

Transcendence for simple recurrences with two dominant roots

The bang-bang minimum and four spectral cases are exhaustive

The required lower bound for an SSS-integer plus SSS-units

Exceptional visits and height inequalities

Higher-order non-cancellation and final Subspace-Theorem step

External results are invoked in their stated regimes

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The required lower bound for an $S$ -integer plus $S$ -units