{"id":"db3789de-deed-4b2b-80a8-cf4ecfb15356","arxiv_id":"2412.16517","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Generating series of q-adic valuations are non-holonomic, non-algebraic over characteristic 0 fields, and take transcendental values at infinitely many rational and algebraic irrational points.","lead":"This paper proves that the power series built from the q-adic valuation function, which counts how many times a base q divides a number, is not a solution to any finite-order linear differential equation with polynomial coefficients, and takes transcendental values at many rational inputs. The result closes off a possible route to computing q-adic valuations quickly and connects to automatic sequences such as the period-doubling sequence.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The transcendence proof as written fails its own positive-segments hypothesis and never establishes distinct Roth approximants; the gap is repairable but real.","rationale":"Good-faith reading: the non-holonomicity proof is essentially correct. The alleged existence gap in Proposition 2.1 is immediate: with q^k>d and m1=q^k u, m2=q^{k+1}v, all m1-i and m2-i for 1<=i<=d have q-adic valuation equal to nu_q(i), so the recurrence subtraction gives nu_q(m1)=nu_q(m2), contradicting k and k+1. The real soft spot is the transcendence machinery. The 'positive segments' definition has the wrong inequality, and the proof of Theorem 5 never establishes that the approximants A_n/B_n are infinitely often distinct. Without that, Roth's theorem does not apply. The fix is to use block ends and the proven positivity A_{k,j}>0; then cumulative partial sums increase. This is a significant but localized gap, so a conditional verdict is appropriate. Separately, Theorem 6's proof breaks for q=2 because no positive rational a/b<1 obeys condition (3); the statement remains true via Polya-Carlson/non-holonomicity, but the written proof needs a different argument. I therefore keep the reader's CONDITIONAL verdict, while disagreeing with the specific weakest assumption identified.","tokens_in":12325,"tokens_out":28637,"duration_ms":246545,"concrete_test":"Set q=3, x=1/2 and compute B_j=f_j(1/2)=2^{-3^j}/(1-2^{-3^j}); B_1 is about 0.1176 and B_2 is about 1.5e-5, so B_1>B_2, contradicting the stated B_j<B_{j+1}. For the Roth step, take V_{3,2}(1/2), index n_j=2j+1, and verify S_{n_{j+1}-1}=S_{n_j-1}+A_{2,j}(1/2) is strictly increasing; this supplies the missing distinct subsequence, showing the concern is a fixable exposition gap rather than a false theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.1's existence step is not a real gap: taking m1=q^k u and m2=q^{k+1}v with u,v not divisible by q and q^k>d gives nu_q(m1-i)=nu_q(i)=nu_q(m2-i) for 1<=i<=d, since 1<=i<q^k. The load-bearing problem is in Theorem 5, the engine of Theorem 2. The definition of 'positive segments' requires blocks B_j(x) to satisfy B_j<B_{j+1}, but Proposition 3.3 assigns V_q the blocks B_j=f_j, and f_j(x)>f_{j+1}(x) on (0,1); for V_{q,k} the paper itself proves A_{k,j}>A_{k,j+1}. Thus the constructed series do not satisfy the stated hypothesis, so Theorem 5 cannot be invoked as written. Moreover, the Roth application needs infinitely many distinct rationals A_n/B_n; the proof never identifies a subsequence on which the partial sums are distinct. The intended argument is visible: block sums are positive, so partial sums at block ends are strictly increasing, but this is not stated or used. A second, smaller gap is that Theorem 6's specialization argument is impossible for q=2, since no a<b with a,b positive satisfies q>2 log b/(log b - log a).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the generating series V_q(X)=sum_{n>=1} nu_q(n)X^n and V_{q,k}(X)=sum_{n>=1}(nu_q(n) mod k)X^n for integer q>=2. Theorem 1 claims that both series are non-holonomic over C, Theorem 2 claims that their values at rationals a/b<1 satisfying q>2 log b/(log b-log a), and at roots-of-unity twists of such rationals, are transcendental, and Theorem 3 claims non-algebraicity over any characteristic-zero field of algebraic numbers. The proofs use the Polya-Carlson theorem for the non-holonomicity, an explicit block decomposition of the series, and Roth's theorem for the transcendence statements. The final sections apply the results to q-automatic sequences, including the period-doubling sequence.","tokens_in":12578,"tokens_out":19170,"duration_ms":169863,"significance":"If the main results are correct, they are natural and interesting: the series of q-adic valuations are shown to lie far outside the holonomic world, and their values are transcendental at an explicit infinite family of rational inputs. The block decompositions (4) and (8) are explicit and checkable, and the applications to automatic sequences give a clean connection to known phenomena. The paper contains no fitted parameters or circular derivations; the external theorems used are standard. However, several load-bearing points in the proofs are not established as written, so the significance can only be assessed after those gaps are repaired.","major_comments":[{"comment":"The definition of 'positive segments' requires B_j(x)<B_{j+1}(x) for all x in (0,1), but for V_q the proposed blocks B_j=f_j(x) satisfy f_j(x)>f_{j+1}(x), and for V_{q,k} the paper itself proves A_{k,j}(x)>A_{k,j+1}(x) immediately before Proposition 3.3. Thus neither series satisfies the hypothesis of Theorem 5 as stated, so the invocation of Theorem 5 in Section 4.3 is blocked. Since the proof of Theorem 5 only uses positivity of the blocks, the likely fix is to weaken the hypothesis to B_j(x)>0 and to prove explicitly that the partial sums at segment endpoints are strictly increasing; this must be stated and proved.","section":"Section 3, definition of positive segments; Proposition 3.3"},{"comment":"The rational approximants A_n/B_n defined in (14) are not shown to be in lowest terms. Theorem 4 is only a valid form of Roth's theorem for reduced fractions; as stated it is false for arbitrary non-reduced representations, because scaling a convergent p_n/q_n by a large common factor c_n gives A_n/B_n=c_n p_n/c_n q_n with |alpha-A_n/B_n|<1/B_n^{2+delta} for any irrational alpha once c_n is large enough. Since B_n may share a large factor with A_n, inequality (15) does not imply the existence of reduced denominators satisfying the Roth bound. The proof must either show that the reduced denominator of A_n/B_n grows like B_n up to a fixed constant, or replace the argument by one adapted to the specific structure of these approximants. This gap is load-bearing for Theorem 2.","section":"Section 4.2, Theorem 5 and Section 4.1"},{"comment":"The specialization argument for q=2 is impossible: the proof needs a rational a/b in (0,1) satisfying q>2 log b/(log b-log a), but for q=2 no positive integers a<b satisfy this strict inequality, since 2 log b/(log b-log a) is always at least 2 (with equality only approached as a/b tends to 0). Therefore Theorem 6 is not proved for q=2, and since Theorem 3 covers q=2, this case needs a separate argument, for example deriving non-algebraicity from the non-holonomicity in Theorem 1, or a different specialization argument.","section":"Section 4.4, proof of Theorem 6"},{"comment":"The proof asserts the existence of m1,m2 with nu_q(m1)=k, nu_q(m2)=k+1 (or with different residues modulo k in Proposition 2.2) and nu_q(m1-i)=nu_q(m2-i) for i=1,...,d, but no construction is given. This existence is load-bearing for the rational case. The assertion is true: taking m1=q^k u and m2=q^{k+1}v with gcd(uv,q)=1, q^k>d, and m1,m2>N+d gives nu_q(m1-i)=nu_q(i)=nu_q(m2-i) for 1<=i<=d. The manuscript should include this construction explicitly and also ensure m1-d and m2-d are at least N.","section":"Proposition 2.1, Case 1, and Proposition 2.2"}],"minor_comments":[{"comment":"The heading 'Proof of Theorem 3' actually proves Theorem 2; the generalized non-algebraicity statement is proved later in Section 4.4 as Theorem 6.","section":"Section 4.3"},{"comment":"The error term is written E_{p,n}(a,b) in (11) but E_{q,n}(a,b) in the following lines; the notation should be unified.","section":"Equation (11)"},{"comment":"The sentence 'the corresponding queue is positive' should read 'tail', and the later remark that transcendence holds 'except for at most a finite number of integers q' is confusing because the condition (3) already specifies which q are covered.","section":"Section 4.2"},{"comment":"The statement of Roth's theorem should explicitly require gcd(A_n,B_n)=1 and use the reduced denominator; otherwise the theorem as printed is not a true statement.","section":"Theorem 4"},{"comment":"In the definition of the automaton, 'the state set S (also called the input alphabet)' should be 'the input alphabet is {0,...,w-1}'; the current phrasing conflates the state set and the alphabet.","section":"Section 5, Theorem 7"}],"recommendation":"major_revision","confidential_remarks":"The paper contains several internal gaps that are likely repairable, but at least two are load-bearing for the transcendence theorem and one blocks the generalized algebraic result for q=2. Given the paper's scope and the elegance of the intended arguments, I would encourage a careful revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The non-holonomicity part is worth your time. The Polya-Carlson argument is standard, but the reduction to a rational-function contradiction is neat, and the missing existence step in Proposition 2.1 is easily filled: take m1=q^k, m2=q^{k+1}; then for i<q^k, nu_q(m1-i)=nu_q(i)=nu_q(m2-i). So Theorem 1 is solid.\n\nThe transcendence part is where I'd be cautious. Theorem 5 requires the series to have \"positive segments\" with B_j < B_{j+1}. But for V_q the natural blocks are just f_j, which are strictly decreasing in x, and the paper itself shows A_{k,j} > A_{k,j+1} for V_{q,k}. So the hypothesis as stated is not satisfied. The proof never uses the increasing property of the B_j; positivity alone gives increasing partial sums at block ends, which is enough for distinct Roth approximants. So this is a repairable technical gap, not a fatal one. But it is real, and a referee would need to ask the authors to fix the definition and the proof.\n\nSecond, Theorem 6's specialization argument fails for q=2, because no a<b with a,b positive satisfies condition (3). The theorem states q>=2, so the proof as written does not cover q=2. Probably the algebraic non-ality for q=2 follows from the same ideas or from Mahler, but it's not in the paper.\n\nFinally, the transcendence results may not be as new as the authors imply. V_q satisfies the Mahler equation f(x)=x^q/(1-x^q)+f(x^q), and standard Mahler method gives transcendence at algebraic points under far weaker conditions than (3). The paper does not cite Mahler or the automatic-sequence transcendence literature. So the advanced take-home is the non-holonomicity, not the transcendence.\n\nOverall: a flawed but promising manuscript. The central non-holonomicity theorem appears correct and new. The transcendence proofs have specific gaps that are fixable. I'd send it to review, expecting major revision before acceptance.","headline":"Non-holonomicity is new and likely correct; the transcendence proof has a real but repairable gap in the \"positive segments\" hypothesis.","tokens_in":13121,"tokens_out":6010,"would_cite":true,"duration_ms":47421,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B37","11J81"],"pacs":[],"model":"deepseek-v4-flash","headline":"The power series of q-adic valuations, and their reductions modulo k, are non-holonomic for every q,k ≥ 2, and take transcendental values at infinitely many rational and algebraic inputs.","keywords":["p-adic valuation","non-holonomic series","transcendental numbers","automatic sequences","period-doubling sequence","generating functions","Roth's theorem"],"falsifier":"Run a recurrence-guessing routine on the first million terms of $\\nu_2(n)$: the paper predicts that no linear recurrence with polynomial coefficients of any fixed order and degree is satisfied, so a recurrence that validates on a long tail of terms would refute Theorem 1. Separately, evaluate $V_3(1/2)$ to several hundred digits and run LLL-based algebraic-number recognition: the paper predicts a transcendental number, so any consistent algebraic candidate of degree below 30 with small height would refute Theorem 2.","tokens_in":12101,"feed_emoji":"🔢","tokens_out":15258,"duration_ms":122069,"temperature":0.7,"pith_summary":"The paper studies the power series whose n-th coefficient is the q-adic valuation of n, the exponent of the largest power of q dividing n, together with the companion series obtained by reducing those coefficients modulo k. Its central claim is that for every q,k ≥ 2 both series are non-holonomic: neither can satisfy any linear differential equation with polynomial coefficients, a far stronger statement than being irrational or algebraic as functions. The paper further proves that at infinitely many rational numbers a/b, namely all coprime a < b with q > 2 log b/(log b − log a), the numerical values of both series are transcendental, and that the same holds after multiplying the input by suitable roots of unity. A byproduct is a sharp contrast for automatic sequences: the mod-k valuation sequence is p-automatic and its finite-field generating function is algebraic, yet the characteristic-0 generating function is not even algebraic.","feed_headline":"No differential equation captures the valuation power series","feed_subtitle":"Even modulo-k versions stay non-holonomic, and their values at many rational numbers are transcendental.","key_machinery":"The load-bearing identity writes $V_q(X)$ as a sum of geometric blocks, $V_q(X)=\\sum_{m\\ge 1} X^{q^m}/(1-X^{q^m})$, grouping terms by valuation level so that each coefficient becomes a count of blocks; the reduced series $V_{q,k}$ is the same sum with block coefficients $a_j=1$ for $k\\nmid j$ and $a_j=1-k$ for $k\\mid j$. Non-holonomicity is obtained in two cases: if the series were rational, the valuation sequence would satisfy a constant-coefficient recurrence, which is contradicted by a pair of integers with valuations k and k+1 whose preceding d neighbors have identical valuations; if it is not rational, a classical dichotomy for integer-coefficient series says the unit circle is a natural boundary, and a holonomic series admits analytic continuation around finitely many singularities, so it cannot have such a boundary. For transcendence, the block structure gives explicit rational approximants $A_n/B_n$ with $B_n=b^{q^n}-a^{q^n}$, and Roth's rational-approximation theorem for algebraic numbers applies once the error term is dominated by $1/B_n^{2+\\delta}$; the hypothesis $q>2\\log b/(\\log b-\\log a)$ is precisely the condition that makes this domination possible. The roots-of-unity extension uses the identity that $V_q(\\omega X)-V_q(X)$ is rational for $\\omega$ of order $q^\\ell$, so an algebraic value at $\\omega a/b$ would force one at $a/b$.","core_discovery":"For integers q,k ≥ 2 let ν_q(n) be the exponent of the largest power of q dividing n. The paper proves that the formal power series $V_q(X)=\\sum_{n\\ge 1}\\nu_q(n)X^n$ and $V_{q,k}(X)=\\sum_{n\\ge 1}(\\nu_q(n)\\bmod k)X^n$ are non-holonomic over $\\mathbb{C}$, meaning neither satisfies any linear differential equation with polynomial coefficients, and that neither is algebraic over the rational function field of any characteristic-0 field whose elements are algebraic numbers. It then proves that whenever a and b are coprime with a < b and q > 2 log b/(log b − log a), the real numbers $V_q(a/b)$ and $V_{q,k}(a/b)$ are transcendental, and the same holds for the complex values $V_q(\\omega a/b)$ and $V_{q,k}(\\omega a/b)$ where $\\omega$ is any root of unity of order $q^\\ell$. These theorems are corollaries of a more general statement about series $W(X)=\\sum_j a_j X^{q^j}/(1-X^{q^j})$ with bounded integer coefficients that are not rational and have positive segments. Finally, the mod-k valuation sequence is shown to be w-automatic, so over finite fields its generating series is algebraic, while the characteristic-0 series remains non-holonomic.","pith_inferences":["Beyond the paper: the threshold in (3) comes from the exponent 2 in Roth's theorem, so any future improvement in uniform rational-approximation bounds would widen the transcendence range, possibly to all $a/b<1$ with the same $q$.","Beyond the paper: the same block decomposition with bounded integer coefficients is a template, so alternating or truncated variants such as $\\sum_j (-1)^j X^{q^j}/(1-X^{q^j})$ are natural test cases that should inherit the same non-holonomicity and transcendence.","Beyond the paper: the contrast between algebraicity over finite fields and non-algebraicity over characteristic-0 fields suggests the obstruction is characteristic-specific; one could test whether other automatic sequences, for instance sum-of-digits sequences, display the same split."],"forward_implications":["No finite-order differential equation with polynomial or rational coefficients has $V_q$ or $V_{q,k}$ as a solution, so the valuation series cannot be captured by any classical D-finite framework.","There are infinitely many rational inputs $a/b$ and infinitely many algebraic-irrational inputs $\\omega a/b$ at which both series take transcendental values.","Over every characteristic-0 field A of algebraic numbers, neither series is algebraic over $A(X)$, ruling out polynomial equations of any degree with coefficients in $A(X)$.","The mod-$k$ valuation sequence is $w$-automatic; in positive characteristic its generating series is algebraic, while in characteristic zero it is not even algebraic, so automaticity does not tame the generating series across characteristics.","The period-doubling sequence serves as the explicit example: its generating series is non-holonomic over $\\mathbb{C}$."],"supporting_citations":[{"why":"Classical dichotomy: an integer-coefficient series on the unit disk is rational or has the unit circle as its natural boundary; this is the engine of the non-holonomicity proof.","marker":"[13]"},{"why":"Rational-approximation theorem for algebraic numbers; the transcendence proof exhibits approximants that beat its exponent 2+δ threshold.","marker":"[18]"},{"why":"Characterization of constant-recursive sequences as rational generating functions, used to rule out the rational case in Proposition 2.1.","marker":"[9]"},{"why":"Companion reference for the same rational-function characterization of C-recursive sequences, cited with [9].","marker":"[19]"},{"why":"Finite-field algebraic characterization of q-automatic sequences, used to prove Theorem 8(ii) that the mod-k valuation series is algebraic over F_q.","marker":"[4]"},{"why":"Automatic-sequence monograph supplying the p^s-automatic iff p-automatic fact and the statement of the Christol criterion used in Theorem 8.","marker":"[1]"},{"why":"Theorem relating p-automatic sequences to algebraic power series over the p-adic integers, used in Theorem 8(iii).","marker":"[5]"},{"why":"Analytic-continuation theory for solutions of linear differential equations, showing a holonomic series cannot have the unit circle as a natural boundary.","marker":"[8]"}],"fun_headline_variants":["Valuation series escape all differential equations","Non-holonomic valuation series yield transcendental values","p-adic valuation series: no DE, values transcend","Valuation series defy holonomy, transcend at many inputs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument that the series cannot be rational depends on the unproved existence, for every recurrence order d, of two integers with q-adic valuations k and k+1 whose preceding d neighbors have identical valuations, and if that combinatorial claim failed the non-holonomicity and the transcendence at roots of unity would lose their footing.","fun_headline_variants_meta":{"raw":{"variants":["Valuation series escape all differential equations","Non-holonomic valuation series yield transcendental values","p-adic valuation series: no DE, values transcend","Valuation series defy holonomy, transcend at many inputs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00052,"raw_usage":{"total_tokens":2504,"prompt_tokens":916,"completion_tokens":1588,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":1525}},"tokens_in":532,"tokens_out":1588,"duration_ms":13371,"temperature":1.0,"reasoning_tokens":1525,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:31:32.822821+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a recurrence-guessing routine on the first million terms of $\\nu_2(n)$: the paper predicts that no linear recurrence with polynomial coefficients of any fixed order and degree is satisfied, so a recurrence that validates on a long tail of terms would refute Theorem 1. Separately, evaluate $V_3(1/2)$ to several hundred digits and run LLL-based algebraic-number recognition: the paper predicts a transcendental number, so any consistent algebraic candidate of degree below 30 with small height would refute Theorem 2.","supporting_citations":[{"cited_title":"¨Uber Potenzreihen mit ganzzahligen Koeffizienten","cited_arxiv_id":null,"evidence_quote":"Classical dichotomy: an integer-coefficient series on the unit disk is rational or has the unit circle as its natural boundary; this is the engine of the non-holonomicity proof."},{"cited_title":"Simion Stoilow","cited_arxiv_id":null,"evidence_quote":"Companion reference for the same rational-function characterization of C-recursive sequences, cited with [9]."},{"cited_title":"Christol, T","cited_arxiv_id":null,"evidence_quote":"Finite-field algebraic characterization of q-automatic sequences, used to prove Theorem 8(ii) that the mod-k valuation series is algebraic over F_q."},{"cited_title":"Algebraic power series a nd diagonals","cited_arxiv_id":null,"evidence_quote":"Theorem relating p-automatic sequences to algebraic power series over the p-adic integers, used in Theorem 8(iii)."},{"cited_title":"Applied and computational complex analysis, vol","cited_arxiv_id":null,"evidence_quote":"Analytic-continuation theory for solutions of linear differential equations, showing a holonomic series cannot have the unit circle as a natural boundary."}],"review_version":1}