{"id":"6561813c-f605-4235-97a4-438592802431","arxiv_id":"2507.08138","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conservative Matrix Fields generalize Apéry-type ratios of D-finite sequences to several dimensions and conjecturally have direction-continuous convergence and irrationality measures.","lead":"The paper introduces Conservative Matrix Fields, a high-dimensional generalization of the ratios of D-finite sequences that produced Apéry's irrationality proof for ζ(3). It proves that these objects include classical Apéry limits as special cases and presents numerical evidence, plus conjectures, that their convergence properties vary continuously with direction.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract promises a Benzaid–Lutz/Levinson framework that the body never states; Cor. 4.3.1's companion-form reduction—the only bridge to Apéry limits—is unproved for general CMFs and demonstrated in just one example.","rationale":"The reader's CONDITIONAL verdict is appropriate: the algebraic core (Theorem 3.1, coboundary invariance Proposition 4.3, Corollary 4.1.1) is sound and clearly presented, and the conjectures are labeled as such. However, the paper's own abstract promises a Benzaid–Lutz/Levinson framework, and the body never provides it. The only route from a general CMF ratio to a known Apéry limit with a provable convergence rate is Corollary 4.3.1, whose companion-form hypothesis is verified in exactly one example and is not proved to hold for the CMFs of Section 3 or across the angular domains used in Conjectures 1–4. This is not a matter of disagreement with consensus; it is a missing proof in the central mechanism, and it also affects how Figs. 3–5 should be read. I add two refinements to the reader's weakest_assumption: the fundamental gap is the missing matrix Levinson theorem, and a simple linear-algebra argument shows CMF ratios are already D-finite ratios, so the claimed 'extension' is really a direction-parametrized reformulation whose continuity properties remain unproved. The concrete test on a generic direction would distinguish a one-off algebraic miracle from a general phenomenon, and would decide whether the conjectures' theoretical scaffolding exists.","tokens_in":22735,"tokens_out":19959,"duration_ms":202189,"concrete_test":"For the CMF of Example 2.4 with x=(1,1), p=(0,1), q=(1,1), take v=(2,1), a primitive direction in the convergent angular domain of Figure 3. (1) Compute ρ_est = log|L_{p,q}^{x,v}(N) − L_{p,q}^{x,v}(2N)|/N for N=1000 and compare it with log|λ_2/λ_1|/|v|, where λ_i are the eigenvalues of lim_{n→∞} M_v(1+nv). (2) Perform a symbolic search over A ∈ GL_2(Q(x)) of bounded degree satisfying the companion-form condition of Corollary 4.3.1 for all n. If the rate matches but no such A exists, the companion-form bridge is insufficient and the missing Levinson theorem is the real gap; if the rate deviates, the conjectured direction-continuity mechanism fails for generic directions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.3 reviews only the scalar Poincaré–Perron theorem. The CMF trajectory system y(n+1)=T_{x,v}(n)y(n) is a matrix difference system; its asymptotic analysis requires a Levinson-type theorem (Benzaid–Lutz) for such systems. No such theorem is stated or proved, despite the abstract citing it. The only rigorous bridge from a CMF ratio to a classical D-finite ratio with a known convergence rate is Corollary 4.3.1, which is conditional on the existence of A ∈ GL_r(Q(x)) such that C(n):=A(x+(n−1)v)T_{x,v}(n)A^{−1}(x+nv) is companion and encodes an irreducible Poincaré–Perron recurrence. This existence is not proved for any d>1 CMF constructed in Section 3; the sole example (Examples 2.12 and 4.2) treats one trajectory of one CMF. Conjectures 1–4 implicitly assume the reduction persists over an angular neighborhood of directions, and the red curves in Figures 3–5 comparing convergence rates to log|λ_i|/|v| are not consequences of any proved statement for non-companion trajectories. Additionally, a Krylov argument shows every CMF ratio is a ratio of two solutions of a common P-recursive recurrence over Q(n), so the class of sequences is not actually extended; the high-dimensional content is the direction-parametrized family, whose continuity is precisely what lacks proof. If the companion-form reduction fails or Levinson theory does not apply, the claimed asymptotic and arithmetic continuity phenomena are unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Conservative Matrix Fields (CMFs), defined as cocycles M: Z^d -> GL_r(K(x)) satisfying M_{v+w} = M_v sigma_v(M_w), and uses them to encode families of ratios of D-finite sequences along trajectories x+nv. The main rigorous results are Theorem 3.1 (a construction of CMFs from any D-finite function by basis changes in the Ore-algebra module), Proposition 3.2 (coboundary equivalence corresponds to change of basis), Corollary 4.1.1 (every D-finite ratio is a CMF ratio), and Propositions 4.2 and 4.3 (under companion-form and Poincaré--Perron hypotheses, CMF ratios inherit the classical limits and convergence rates; coboundary equivalence changes the vectors defining the ratio in an explicit way). Section 5 presents numerical experiments for three families of CMFs and formulates Conjectures 1--4 asserting continuity of the limit, irrationality measure, normalized convergence rate, and normalized eigenvalues as functions of the direction v. The conjectures are explicitly labeled as experimental observations, and the paper proposes optimization-based searches for irrationality proofs as a potential application.","tokens_in":23107,"tokens_out":9279,"duration_ms":114189,"significance":"If the conjectures are true, the paper points to a genuinely new phenomenon: Apéry-like limits and their arithmetic qualities varying continuously with the direction of a trajectory in a high-dimensional lattice, a property not visible in the classical one-parameter theory. The rigorous core is coherent and appears correct: the CMF construction from a D-finite function is an exact basis-change computation, the inclusion of Apéry limits as CMF limits is definitionally sound, and the ζ(3) example in Example 4.2 correctly recovers Apéry's recurrence via a coboundary transformation. The paper also has a reproducible, computational flavor: it uses the HolonomicFunctions and Asymptotics packages and gives explicit generators for the examples. The main gap is that the advertised 'continuous asymptotics' and 'continuous arithmetic' are not established theorems but conjectures resting on finite-N estimates, and the only rigorous bridge to classical Poincaré--Perron asymptotics, Corollary 4.3.1, is conditional on an unproved companion-form reduction.","major_comments":[{"comment":"The promised Levinson-type framework is not present in the body of the paper. Section 2.3 states only the scalar Poincaré--Perron theorem (Theorem 2.1), and no Benzaid--Lutz or other discrete Levinson theorem is stated for the matrix systems y(n+1)=T_{x,v}(n)y(n) that underlie CMF ratios. As a result, the only proved bridge from a CMF ratio to a classical D-finite ratio with a known convergence rate is Corollary 4.3.1, whose hypothesis requires an A in GL_r(Q(x)) such that C(n)=A(x+(n-1)v)T_{x,v}(n)A^{-1}(x+nv) is in companion form and encodes an irreducible Poincaré--Perron recurrence. That existence is not proved for the general CMFs constructed in Section 3; Example 4.2 supplies it for one trajectory only. Consequently the red lines in Figures 3--5, comparing normalized convergence rates with log|lambda_i|/|v|, are not consequences of any proved statement for non-companion trajectories. The authors should either state and prove a companion-form/coboundary existence theorem (ideally via a genuine Benzaid--Lutz theorem) or explicitly restrict all asymptotic claims to the cases covered by Corollary 4.3.1 and Proposition 4.2.","section":"Section 2.3 and Corollary 4.3.1"},{"comment":"The 'extension' claim needs re-framing, because a standard Krylov argument shows that every CMF ratio is already a D-finite ratio. For the first-order system y(n+1)=T_{x,v}(n)y(n), the vectors y(n),...,y(n+r) are Q(n)-linearly dependent, so a nontrivial relation gives one common P-recursive recurrence of order at most r satisfied by every coordinate of M_{nv}(x)p' and similarly for q'. Hence L_{p,q}^{x,v}(n) is a ratio of two solutions of a common P-recursive recurrence over Q(n), i.e., exactly a D-finite ratio in the sense of Definition 7. The CMF framework therefore does not enlarge the class of sequences; what is new is the direction-parametrized family {L_{p,q}^{x,v}(.) : v in Z^d}. The abstract's phrase 'extend ratios of D-finite sequences to a high-dimensional setting' and the wording in Section 4 should be adjusted to say that the paper organizes a family of D-finite ratios, rather than introducing a larger sequence class.","section":"Section 4, Definition 17"},{"comment":"The experimental support for Conjectures 1--4 is not calibrated. The estimates hat l, hat rho, and hat delta are computed at N=1000 (Figure 3), N=100 (Figure 4), and N=200 (Figure 5), with no error bars and no demonstration that the finite-N values are close to the true asymptotic quantities. The estimator for delta uses log|H(L(N))| and log|L(N)-L(2N)|/N, which is singular when the convergence rate is close to zero; the plotted non-convergence intervals in Figure 4 correspond exactly to rho-hat = 0 and are where the approximation breaks down. Since Conjectures 1--4 carry the paper's advertised 'continuous asymptotics and arithmetic', the conjectures should be presented strictly as numerical observations with explicit caveats, and the abstract should not state that the paper 'establishes' these properties. In addition, Conjecture 4 implicitly assumes the same unproved companion-form reduction identified in the first major comment.","section":"Section 5.1, Conjectures 1--4"}],"minor_comments":[{"comment":"In the statement of Corollary 5.0.1, the conclusion 'L_{p,q}^{x,v0}(n) -> l' inside the for-all-v sentence should presumably be 'L_{p,q}^{x,v}(n) -> l'; the current wording makes the quantifier over v vacuous.","section":"Corollary 5.0.1"},{"comment":"Definition 7 says u1(n), u2(n) in Q^N, but N is not defined; it should say that u1 and u2 are sequences of rational numbers, or introduce notation for the set of sequences explicitly.","section":"Definition 7"},{"comment":"In Definition 8, if s_n = l then the defining equation |s_n - l| = 1/H(s_n)^{1+delta_n} has no solution delta_n; the definition should explicitly exclude the case s_n = l.","section":"Definition 8"},{"comment":"Proposition 4.2 refers to 'irreducible (in the sense of definition 2 in [32])' without stating the definition; since the proof uses this property to rule out smaller annihilators, the definition should be included or paraphrased in the paper.","section":"Proposition 4.2"},{"comment":"The notation L_{p,p',q,q'}^{x,v} and L_{p,q}^{x,v} is hard to parse because the superscripts carry both the trajectory data and the vector data; a notation such as L(x,v;p,p',q,q')(n) would improve readability.","section":"Definition 17"},{"comment":"The figures would be much more informative with error bars or with multiple starting points for the same direction; the text should also state the stopping criterion used to decide that a trajectory 'does not converge' in Figure 4.","section":"Figures 3--5"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest in labeling its main asymptotic and arithmetic statements as conjectures, and the algebraic core is sound. My main hesitation is the gap between the abstract and the body: the abstract promises a Benzaid--Lutz/Levinson framework and 'established' asymptotic properties, while the body provides only scalar Poincaré--Perron theory plus a conditional companion-form reduction. This is fixable: the authors can either add the missing theorem (if provable) or substantially soften the abstract and title. The Krylov observation that every CMF ratio is already a D-finite ratio should be addressed explicitly, because it changes the framing from 'new class of sequences' to 'direction-parametrized family of known sequences.' For a journal that accepts experimental mathematics, this could become a valuable contribution after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The algebraic core is real and mostly correct: the construction of CMFs of arbitrary rank and dimension from multivariate D-finite functions (Theorem 3.1) is clean, and the coboundary/gauge formalism is a useful way to look at basis changes. The second thing is that the advertised 'continuous asymptotics and arithmetic' are conjectural. Conjectures 1-4 are backed only by finite-N numerics, and the abstract cites a Benzaid-Lutz/Levinson framework that never shows up in the body. That mismatch is the paper's main weakness.\n\nWhat is genuinely new: prior CMF work is almost all rank-2; this paper gives a systematic construction of general rank/dimension CMFs from D-finite functions, defines CMF ratios and limits, and shows that classical Apéry limits are included (Cor 4.1.1). That inclusion is nearly trivial—any D-finite ratio is a CMF ratio via the companion matrix—but the new object is the direction-parametrized family of sequences. The proofs of Props 4.2-4.3 look correct and are genuinely useful: for a companion-form trajectory satisfying Poincaré-Perron, any rational linear combination of shifted solutions yields the same limit and rate as the original ratio. That is a nice generalization of Prop 2.4. The paper also does not fit constants to data; the CMF construction is an exact basis-change computation, and the Apéry inclusion follows definitionally. Credit where due.\n\nThe soft spots, in proportion. The bridge from general CMF trajectories to D-finite ratios is Cor 4.3.1, which is conditional on the existence of A ∈ GL_r(Q(x)) that puts the trajectory into companion form and encodes an irreducible Poincaré-Perron recurrence. That is demonstrated for exactly one example (the ζ(3) CMF, Examples 2.12/4.2). For the higher-dimensional CMFs from Section 3 it is not proved, and Conjectures 1-4 implicitly assume it persists across an angular neighborhood of directions. The numerics in Figures 3-5 are finite-N estimates with no error bars; they are honest motivation but not evidence for the conjectures. Also, the abstract's mention of a 'discrete Levinson-type framework due to Benzaid and Lutz' is not realized in the text; Section 2.3 only reviews scalar Poincaré-Perron. If the authors have a matrix Levinson theorem in their back pocket, they need to state it, because that is exactly the machinery that would turn the companion-form case into a general result. One more observation, not a flaw per se: a Krylov-style argument shows every CMF ratio is a ratio of two solutions of a common P-recursive recurrence over Q(n), so the class of limits is not actually enlarged. The novelty really is the direction family and its conjectured continuity.\n\nWho is this for: people working in Diophantine approximation, experimental mathematics, and symbolic computation. It deserves a serious referee. I would send it out, but expect major revision: fix the abstract, state or remove the Levinson claim, and either prove the companion-form reduction for a nontrivial class or clearly frame the main results as conjectures with supporting numerics. The citation pattern is fine. Recommend engaging, not citing as established.","headline":"Solid algebraic core, but the advertised continuous asymptotics are conjectures, and the abstract cites a Levinson framework the paper never states.","tokens_in":23627,"tokens_out":4452,"would_cite":true,"duration_ms":48209,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J72","11J82","39A06","33C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Conservative Matrix Fields put ratios of D-finite sequences—the engine of Apéry's irrationality proof—into a higher-dimensional directional framework, with conjectured continuous arithmetic along trajectories.","keywords":["Conservative Matrix Field","D-finite functions","Apéry limits","Poincaré-Perron asymptotics","irrationality measures","Ore algebras","cocycle equation","multivariate hypergeometric terms"],"falsifier":"Take the CMF of Example 2.4 with $x=(1,1)$, $p=(0,1)$, $q=(1,1)$, and compute the CMF ratio for primitive directions $v=(2,3)$ and $v=(3,5)$ up to $N=1000$ terms. If either direction's limit estimate misses $\\zeta(3)$ by much more than the predicted error $e^{N\\rho}$, or its measured normalized convergence rate misses the closed-form value, then Conjecture 1 or 3 fails; conversely, agreement for all sampled primitive vectors in a neighborhood of $(1,1)$ is the concrete check the paper's framework predicts.","tokens_in":22562,"feed_emoji":"🧮","tokens_out":10839,"duration_ms":105648,"temperature":0.7,"pith_summary":"The paper tries to establish that ratios of D-finite sequences—the objects behind Apéry's proof that $\\zeta(3)$ is irrational—are special cases of a higher-dimensional structure called a Conservative Matrix Field (CMF). It proves that any such classical ratio is a CMF ratio, and that when a CMF trajectory can be conjugated into companion form encoding an irreducible Poincaré–Perron recurrence, the CMF ratio has the same limit and the same convergence rate as the corresponding D-finite ratio. Numerical experiments on CMF ratios show limits that appear constant over angular regions and normalized convergence rates and irrationality measures that appear continuous in the direction of evaluation; these observations are formalized as four conjectures. If the conjectures hold, they would extend Poincaré–Perron asymptotics to higher dimensions and open the way to optimization-based searches for new irrationality proofs.","feed_headline":"Matrix fields generalize Apéry-style irrationality proofs","feed_subtitle":"CMF ratios inherit Apéry-style limits and convergence rates, with direction-continuous arithmetic conjectured.","key_machinery":"The central object is the Conservative Matrix Field: a map $M:\\mathbb{Z}^d\\to GL_r(K(x))$ satisfying the cocycle equation $M_{v+w}=M_v\\,\\sigma_v(M_w)$, whose evaluation at a point gives path-independent multiplicative matrices. Along a trajectory $x+nv$, the trajectory matrices $T_{x,v}(n)=M_v(x+nv)$ multiply to $M_{nv}(x)$, and when $T_{x,v}(n)$ is in companion form—the matrix form whose last column encodes a linear recurrence—its last column carries the solutions of that recurrence. The other load-bearing pieces are the construction of CMFs from D-finite functions as basis-change matrices of the finite-dimensional module $A'.f$, the notion of coboundary equivalence (change of basis), and the Poincaré–Perron theorem, which converts the eigenvalue ratios of the companion matrix into convergence rates and multiplicative constants.","core_discovery":"The central claim is that the Conservative Matrix Field ($M:\\mathbb{Z}^d\\to GL_r(K(x))$ satisfying $M_{v+w}=M_v\\,\\sigma_v(M_w)$) is the natural multi-dimensional home for ratios of D-finite sequences. Given a multivariate D-finite function $f$ and a basis of its image under the Ore algebra, the basis-change matrices $M^f_v$ form a CMF, and a change of basis is exactly a coboundary transformation. A CMF ratio $L^{p,p',q,q'}_{x,v}(n)=p^t M_{nv}(x)p'/(q^t M_{nv}(x)q')$ along a trajectory $x+nv$ generalizes a D-finite ratio; Corollary 4.1.1 shows every D-finite ratio arises this way, and Corollary 4.3.1 states that under a companion-form irreducibility assumption the CMF ratio inherits the exact limit and convergence rate of the corresponding D-finite ratio. The paper's conjectures assert that the limit, the normalized convergence rate, and the irrationality measure of such ratios depend continuously on the direction $v$, with the limit constant on angular regions between a finite set of discontinuities.","pith_inferences":["The paper's conjectures imply a spectral view of CMF ratios: the normalized eigenvalues $\\log|\\lambda_i|/|v|$ form smooth surfaces over the direction sphere, and the limit should jump exactly where the dominant eigenvalue changes; this pattern is visible in the figures but is not stated as a theorem.","The main algorithmic bottleneck is not the CMF construction but the companion-form reduction; a canonical rational normal form over $K(x)$ for trajectory matrices would turn Corollary 4.3.1 from an existence statement into a computation.","If the continuity conjectures survive, optimization over primitive directions $v$ could be used to improve known irrationality measures for constants like $\\zeta(5)$, $\\pi$, or Catalan's constant, since nearby directions would inherit the proven irrationality while offering different heights and rates.","A quantitative test of Conjecture 2 would be to compute height and convergence rate for many directions and check whether the implied irrationality measure varies smoothly; the paper's Remark 14 already notes that height is determined by the other two."],"forward_implications":["Any D-finite ratio, and therefore any Apéry limit, is a CMF ratio (Corollary 4.1.1), so the new framework contains all previously studied ratios of this kind.","Under the companion-form and irreducibility conditions, every CMF ratio shares the limit and convergence rate of its associated D-finite ratio (Corollary 4.3.1), making CMF ratios legitimate candidates for irrationality proofs.","The $\\zeta(3)$ example reproduces Apéry's sequence exactly: the CMF ratio converges to $\\zeta(3)$ with convergence rate $-8\\log(\\sqrt{2}+1)$ and the same height bound.","If Conjectures 1 and 2 hold, one irrationality-proving CMF ratio yields infinitely many essentially different irrationality-proving sequences in an angular neighborhood of the original direction.","If Conjectures 2 and 3 hold, normalized convergence rate, normalized height, and irrationality measure become continuous functions of direction, so scanning or optimizing over directions $v$ becomes a principled search strategy."],"supporting_citations":[{"why":"supplies the Apéry recurrence for $\\zeta(3)$, the initial conditions, and the irrationality proof that is the prototype D-finite ratio.","marker":"[7]"},{"why":"supplies the multivariate-integral proof of irrationality of $\\zeta(2)$ and $\\zeta(3)$ that motivates evaluating D-finite functions along linear sequences.","marker":"[8]"},{"why":"defines Ore algebras and D-finite functions and provides the Gröbner-basis methods used to compute annihilators and basis-change matrices.","marker":"[24]"},{"why":"provides non-commutative elimination in Ore algebras, used to compute the annihilator generators in the multivariate examples.","marker":"[25]"},{"why":"states the Poincaré–Perron asymptotic theorem used to derive convergence rates from eigenvalue ratios.","marker":"[26]"},{"why":"gives Perron's companion results on difference equations that complete the Poincaré–Perron framework.","marker":"[27]"},{"why":"defines irreducible recurrences, a condition required in Proposition 4.2 for the shared-limit result.","marker":"[32]"},{"why":"implements the holonomic-functions computations used for the experimental CMF ratios and their asymptotic data.","marker":"[33]"},{"why":"computes asymptotic expansions of P-finite recurrence solutions, used to obtain the closed-form rates and eigenvalue data.","marker":"[34]"}],"fun_headline_variants":["CMF ratios generalize Apéry limits to matrix fields","Matrix fields predict continuous directional asymptotics","New tool: Conservative Matrix Fields for D-finite asymptotics","Direction-continuous arithmetic conjectured for CMF limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bridge from CMF ratios to classical Apéry limits rests on the assumption that, for each trajectory, the trajectory matrix can be conjugated by a rational-function matrix $A$ into companion form encoding an irreducible Poincaré–Perron recurrence; the paper supplies such an $A$ explicitly only for the $\\zeta(3)$ example.","fun_headline_variants_meta":{"raw":{"variants":["CMF ratios generalize Apéry limits to matrix fields","Matrix fields predict continuous directional asymptotics","New tool: Conservative Matrix Fields for D-finite asymptotics","Direction-continuous arithmetic conjectured for CMF limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000576,"raw_usage":{"total_tokens":2680,"prompt_tokens":872,"completion_tokens":1808,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":1745}},"tokens_in":488,"tokens_out":1808,"duration_ms":13678,"temperature":1.0,"reasoning_tokens":1745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:28:00.343703+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the CMF of Example 2.4 with $x=(1,1)$, $p=(0,1)$, $q=(1,1)$, and compute the CMF ratio for primitive directions $v=(2,3)$ and $v=(3,5)$ up to $N=1000$ terms. If either direction's limit estimate misses $\\zeta(3)$ by much more than the predicted error $e^{N\\rho}$, or its measured normalized convergence rate misses the closed-form value, then Conjecture 1 or 3 fails; conversely, agreement for all sampled primitive vectors in a neighborhood of $(1,1)$ is the concrete check the paper's framework predicts.","supporting_citations":[{"cited_title":"Irrationalit´ e deζ2 et ζ3","cited_arxiv_id":null,"evidence_quote":"supplies the Apéry recurrence for $\\zeta(3)$, the initial conditions, and the irrationality proof that is the prototype D-finite ratio."},{"cited_title":"A Note on the Irrationality of ζ(2) and ζ(3)","cited_arxiv_id":null,"evidence_quote":"supplies the multivariate-integral proof of irrationality of $\\zeta(2)$ and $\\zeta(3)$ that motivates evaluating D-finite functions along linear sequences."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines Ore algebras and D-finite functions and provides the Gröbner-basis methods used to compute annihilators and basis-change matrices."},{"cited_title":"Non-commutative elimination in Ore algebras proves multivariate holonomic identities","cited_arxiv_id":null,"evidence_quote":"provides non-commutative elimination in Ore algebras, used to compute the annihilator generators in the multivariate examples."},{"cited_title":"Sur les ´ equations lin´ eaires aux diff´ erentielles ordinaires et aux diff´ erences finies","cited_arxiv_id":null,"evidence_quote":"states the Poincaré–Perron asymptotic theorem used to derive convergence rates from eigenvalue ratios."},{"cited_title":"¨Uber Summengleichungen und Poincar´ esche Differenzengleichungen","cited_arxiv_id":null,"evidence_quote":"gives Perron's companion results on difference equations that complete the Poincaré–Perron framework."},{"cited_title":"Irreducible recurrences and representation theorems for 3F2(1)","cited_arxiv_id":null,"evidence_quote":"defines irreducible recurrences, a condition required in Proposition 4.2 for the shared-limit result."},{"cited_title":"Advanced Applications of the Holonomic Systems Approach","cited_arxiv_id":null,"evidence_quote":"implements the holonomic-functions computations used for the experimental CMF ratios and their asymptotic data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"computes asymptotic expansions of P-finite recurrence solutions, used to obtain the closed-form rates and eigenvalue data."}],"review_version":1}