{"id":"d3d258be-49e5-4af2-8de7-a961d2b4ef00","arxiv_id":"2411.16062","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For three families of nonlinear recurrences, the paper gives asymptotic expansions and high-precision numerical constants, including a new reciprocity relation for the q=3/2 case.","lead":"This paper finds very precise formulas for how fast certain number sequences, defined by repeating the same operation, shrink or grow over many steps. It gives researchers a set of new constants and expansion formulas they can use in later calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved self-referential template from [3] drives all Section 2 coefficients and constants; one hidden error would shift every reported value, and the only stated check is a correction to Popa's δ in [18].","rationale":"The reader's weakest_assumption correctly identifies the imported template as the hinge of the paper. I agree. The central claim consists of explicit asymptotic expansions and high-precision constants; both are generated by a black-box formula from [3] and by Popa's theorems. Since [3] is authored by the same person and not peer-reviewed, the paper does not supply an independent verification. The only hint of a check is the correction to Popa's δ coefficient, which cuts both ways: it shows the author is reading the sources, but it also confirms that the sources contain erratum-level errors. A single sign or factor error in the template would change every polynomial P_m and hence the constant C extracted from the expansion. The proposed test, direct substitution into the recurrence, is the standard way to confirm an asymptotic expansion and requires no knowledge of [3]. If it passes for Section 2, the main remaining risk moves to Section 3's Popa-based formulas, for which the same substitution test is recommended. No change to the reader's CONDITIONAL verdict is needed; the requested revision should include this verification.","tokens_in":6969,"tokens_out":7897,"duration_ms":67609,"concrete_test":"Directly re-derive the Section 2 expansion by substituting the ansatz x_k = 4/k^2 + ∑_{m=3}^{6} ∑_{j=0}^{⌊m/2⌋} d_{m,j} ln(k)^j / k^m into x_k = x_{k-1}(1 − sqrt(x_{k-1})) and matching coefficients in powers of 1/k and ln(k) up to O(k^{-7}), using symbolic computation. Check whether the resulting d_{m,j} match the paper's coefficients with C treated as a free parameter (e.g., A3=−12, B3=−8C). If they match, the template is validated for this case; if not, the exact mismatch identifies the error. Optionally, the same substitution-check can be performed on the q=2 case of Section 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 opens by stating that 'formulaic knowledge of sections 1, 2, 3 of [3] is assumed' and then applies a template with parameters τ=1/2, a1=−1, λ=2 to generate the polynomials P2–P6 and the resulting six-term expansion. No proof, statement of hypotheses, or numerical cross-check is provided, and [3] is an unpublished preprint by the same author, making the chain self-referential. Because the reported constant C=1.98803983644549695008812308629512... is estimated 'by a simple numerical method [2] using the preceding expansion,' any error in the template propagates directly into the headline constant. Section 3 similarly relies on Popa's Theorems 6 of [17] and 5 of [18], and the paper itself notes a correction ('the lead coefficient 1/2 of δ in [18] should be 1/4'), demonstrating that the imported theorems are not being applied blindly yet also signaling that the sources contain errors. If the template or the corrected formula has a subtle factor error, every listed coefficient and constant would shift, so the central claim is not independently verifiable from the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents three worked exercises in iterational asymptotics. Section 1 proves existence of the convergence-rate constant C for the recurrences x_k = p x_{k-1}(1 - x_{k-1}) and x_k = p x_{k-1}(1 + x_{k-1}), and tabulates numerical estimates for several p. Section 2 derives a six-term asymptotic expansion for x_k = x_{k-1}(1 - sqrt(x_{k-1})) by applying a template from the author's preprint [3], and reports the constant C = 1.98803983644549695008812308629512... for x0 = 1/2 and a second value for x0 = 4/9. Section 3 derives expansions for x_k = x_{k-1} + x_{k-1}^{1-q} using theorems of Popa [17,18], reports constants c(2), c(3), and c(3/2), and corrects a coefficient in [18]. The Addendum completes the q = 3/2 case via an asserted reciprocity transformation.","tokens_in":7292,"tokens_out":4039,"duration_ms":34824,"significance":"If the imported templates and theorems are valid, the paper provides new explicit asymptotic expansions, new high-precision constants, and a correction of a published coefficient, which would be useful contributions. The existence arguments in Section 1 are complete and self-contained, and the algebra after the imports is coherent. However, the central Section 2 expansion rests entirely on an unpublished preprint by the same author, and the Section 3 results rest on imported theorems with a stated but unproved correction. No error bounds or independent numerical cross-checks are supplied for the many-digit constants. Thus the significance is real but conditional on the validity of the external machinery, which the manuscript does not make independently verifiable.","major_comments":[{"comment":"The line 'formulaic knowledge of sections 1, 2, 3 of [3] is assumed' is not a substitute for stating the hypotheses of the template. The paper gives no theorem, proof, or precise statement of the class of functions f for which the template applies, and [3] is an unpublished preprint by the same author. Because the displayed polynomials P_2 through P_6, the coefficients {b_j}, {a_0j}, {c_i}, and the resulting six-term expansion all follow from this template, and because C = 1.98803983644549695008812308629512... is estimated using that same expansion, the central numerical claim is not independently verifiable from the manuscript. Please either include the relevant theorem with hypotheses and proof (or a precise published reference), or provide an independent numerical cross-check that validates the first few coefficients.","section":"Section 2, template from [3]"},{"comment":"The reciprocity transformation used to complete the q = 3/2 case is asserted without proof. The Addendum states 'A certain reciprocity has been found' and then gives the limiting relation for ξ_k from which c(3/2) = 0.8010888849039666437110775... follows. Since this is the only route to the reported value, the transformation and the limiting relation need a derivation or a reference containing a proof. As written, the value is an unexplained additional axiom of the paper.","section":"Section 5 (Addendum), reciprocity transformation"},{"comment":"The note 'beware: the lead coefficient 1/2 of δ in [18] should be 1/4' is a correction to an imported theorem, but the corrected formula is used without re-derivation or numerical test. Because the expansions in this section are only as reliable as the imported theorems, this correction should be justified—ideally by a short derivation of δ or by checking the resulting expansion against a numerical example—rather than stated as a parenthetical warning.","section":"Section 3, correction to Popa's δ"},{"comment":"The constants c(2), c(3), and c(3/2) are quoted to 25 or more decimal places, but the paper provides no error analysis or convergence test for the 'brute-force matching-coefficient method' that produced them. The reader cannot tell how many digits are reliable. Please report an error estimate or a comparison of successive truncations of the expansions for each constant.","section":"Section 3, constants precision"}],"minor_comments":[{"comment":"The phrase 'a s far as is known' contains a spacing error; it should read 'as far as is known'.","section":"Abstract"},{"comment":"The text 'More ge nerally' has a stray space; it should read 'More generally'.","section":"Section 1"},{"comment":"The parenthetical remark 'Assu ming x0 = 4/9' contains a spacing error; it should read 'Assuming x0 = 4/9'.","section":"Section 2"},{"comment":"The phrase 'blead coefficient' is a typo for 'lead coefficient'.","section":"Section 3"},{"comment":"The captions state that 'no closed-form expressions are known'; this is an unsupported empirical claim and should be softened, for example to 'none are given here'.","section":"Tables 1 and 2"},{"comment":"The sentence 'The elegant technique from [13, 14], useful in Section 2, does not apply here' is clear, but the subsequent description of the brute-force method would benefit from a short example showing how the series for x_{k+1} is compared with the series for x_k + x_k^{-(q-1)}.","section":"Section 3, brute-force method"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is heavily self-referential: the main Section 2 expansion is imported from an unpublished preprint by the same author, and several of the constants are then estimated using that same expansion. The paper also relies on Popa's theorems, one of which it corrects without proof. This makes the central claims difficult to verify from the manuscript alone. For a journal publication, I would want either the relevant theorems reproduced with hypotheses/proofs or an independent, reproducible numerical verification of the leading expansion terms. The paper is more of a collection of worked exercises than a self-contained research article, which may be relevant to the editor's assessment of fit with the journal's scope. I have no concerns about the integrity of the author; the issues are strictly about support for the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful, honest collection of new explicit expansions and high-precision constants for a few classic nonlinear recurrences. Its main weakness is exactly what the reader flagged: the Section 2 template is imported from your own unpublished preprint [3], with no proof and no independent numerical check, and the headline constant C is estimated using that same expansion.\n\nWhat is actually new: the six-term expansion for x_k = x_{k-1}(1 – sqrt(x_{k-1})) with constants C(1/2) = 1.9880... and C(4/9) = 1.9684... is, as far as I know, new. The analogous treatment of x_{k-1}(1 – x^2) and of x_{k-1} + x_{k-1}^{1-q} gives new constants c(2), c(3), c(3/2). The Addendum's reciprocity relation is genuinely original and ties c(3/2) to a different iteration, which is a clever observation. The paper is also honest: it explicitly corrects a coefficient in Popa's [18] from 1/2 to 1/4, and it is candid about the difficulty of c(3/2). The Section 1 existence arguments are complete and self-contained.\n\nThe soft spots are real but not disqualifying. The biggest is the self-citation chain: Section 2 opens by assuming formulaic knowledge of [3], and the polynomials P_m and the entire expansion come from that template. If the template has a factor error, every coefficient and the constant shift. The author provides no proof or independent cross-check. The numerical method is also described only as 'a simple numerical method [2]' and the brute-force matching is sketched, so the constants are not fully reproducible from the text alone. Section 3 rests on Popa's theorems; the correction to [18] shows the author is reading carefully, but it also signals that the external theorems are not beyond scrutiny. The stress-test concern that one hidden error would shift every reported value is right, though it remains a hypothesis, not a demonstrated flaw.\n\nI read the 'exercises' framing as honest; this is a working paper with the gritty details left in previous preprints. For a specialist in iterational asymptotics or a constant collector, the paper is valuable. It deserves a serious referee who can check the template proof or demand one, and who can ask for a clearer numerical description. I would not desk reject it; I'd send it to a referee with a request to verify the Section 2 algebra and the numerical constants. My own verdict would be conditional acceptance, pending those checks.","headline":"A useful collection of new explicit expansions and constants for nonlinear recurrences, undercut by heavy reliance on an unproved self-cited template; deserve a careful referee, not a desk reject.","tokens_in":7804,"tokens_out":3791,"would_cite":false,"duration_ms":31414,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["39A10","40A05","65Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For three nonlinear recurrences, including $x_k=x_{k-1}(1-\\sqrt{x_{k-1}})$ and $x_k=x_{k-1}+1/x_{k-1}^{q-1}$, explicit asymptotic expansions are derived and the new constants in them are evaluated to high precision.","keywords":["nonlinear recurrences","asymptotic expansion","iterates","logistic map","slow decay","numerical constants","reciprocity","power series matching"],"falsifier":"Take $x_0=1/2$, iterate $x_k=x_{k-1}(1-\\sqrt{x_{k-1}})$ in high-precision arithmetic up to $k=10^6$, and compare $x_k$ to the six-term expansion with $C=1.98803983644549695008812308629512$. If the difference after subtracting all displayed terms through order $1/k^6$ does not decay like $o(k^{-6})$ (or at least like $O(k^{-7})$), then the template or the constant is wrong. For $q=2$, iterate $x_k=x_{k-1}+1/x_{k-1}$ from $x_0=1$ and test that $2^{1/2}x_k - 2k^{1/2} - \\frac{1}{4}\\frac{\\ln(k)}{k^{1/2}} - c(2)\\frac{1}{k^{1/2}}$ converges to $0$ at the predicted rate.","tokens_in":6742,"feed_emoji":"🔁","tokens_out":12543,"duration_ms":99927,"temperature":0.7,"pith_summary":"This paper presents worked asymptotic analyses of three families of one-dimensional nonlinear recurrences. For the map $x_k=x_{k-1}(1-\\sqrt{x_{k-1}})$, it derives a complete expansion in powers of $1/k$ and $\\log k$ through $1/k^6$, including the new numerical constant $C=1.98803983644549695008812308629512\\ldots$ for the midpoint initial value, and analogous constants for a starting value that minimizes $C$. For the slow-growth recurrence $x_k=x_{k-1}+1/x_{k-1}^{q-1}$ with $q>1$, it derives expansions in powers of $k^{-1/q}$ and $\\log k$ and reports constants $c(2)=0.8615711875687117305317813\\ldots$, $c(3)=1.3784186157718345713984647\\ldots$, and $c(3/2)=0.8010888849039666437110775\\ldots$, with the last computed via a reciprocity relation stated in the addendum. A first exercise determines the geometric decay constant for $p$-scaled logistic maps $x_k=p\\,x_{k-1}(1\\pm x_{k-1})$. The paper presents these as new problems and solutions; if correct, they give explicit closed-form asymptotics where only numerical iteration was previously available.","feed_headline":"Three iterated maps yield new asymptotic constants","feed_subtitle":"Expansions through order 1/k^6 and new constants for three nonlinear recurrences.","key_machinery":"The argument rests on an asymptotic template for iterations $f(x)=x(1-a x^\\tau)$: the paper assumes the form $x_k \\sim (\\lambda/k)^{1/\\tau}\\{1+\\sum_{m=1}^6 P_m(-(1/\\tau)(b_1\\ln(k)+C))/k^m\\}$, where $\\tau$, $\\lambda$, and the sequences $a_m$, $b_j$, $c_i$ determine the polynomials $P_m$; for $f(x)=x(1-\\sqrt{x})$ this is $\\tau=1/2$, $\\lambda=2$, and it produces the displayed expansion through $1/k^6$. For the $q$-recurrence, the transformation $y_k=x_k^q$ reduces to $y_k=y_{k-1}(1+1/y_{k-1})^q$, whose asymptotics follow from cited theorems for iterates of functions with $\\phi(0)=1$ and $\\phi'(0)\\neq0$; a brute-force matching-coefficient method, comparing series for $x_{k+1}$ and $x_k+x_k^{1-q}$, extends the expansion to higher order. The addendum's reciprocity relation---that a limit involving iterates of $\\xi_k=\\xi_{k-1}/(1+\\xi_{k-1}^{3/2})$ equals $2\\Lambda/3$ with $\\Lambda=c(3/2)$---brings the hard $q=3/2$ case back into the range of the earlier convergent-iteration method.","core_discovery":"The central discovery is that iterates of $f(x)=x(1-\\sqrt{x})$ have the asymptotic form $x_k \\sim 4/k^2 - 12\\ln(k)/k^3 - 8C/k^3 + \\cdots$, with $C=1.98803983644549695008812308629512\\ldots$ when $x_0=1/2$, and that the full expansion through order $1/k^6$ is given by the displayed polynomial template with $P_2$ through $P_6$. The paper likewise establishes that for $x_k=x_{k-1}+x_{k-1}^{1-q}$, substituting $y_k=x_k^q$ yields $y_k \\sim q k + \\frac{q-1}{2}\\ln(k) + C + \\cdots$, and it reports $c(2)=0.8615711875687117305317813\\ldots$, $c(3)=1.3784186157718345713984647\\ldots$, and $c(3/2)=0.8010888849039666437110775\\ldots$ after invoking a reciprocal limit in the addendum. All of these are stated as new as far as is known, and are meant to quantify the long-term trend of the respective recurrences exactly.","pith_inferences":["The polynomial template used for $\\tau=1/2$ likely applies to other maps tangent to the identity at $0$, such as $u_k=u_{k-1}(1-u_{k-1}^2/2)$, $v_k=v_{k-1}\\cos(v_{k-1})$, and $w_k=w_{k-1}\\exp(-w_{k-1}^2/2)$, which the paper explicitly leaves open; deriving their $\\tau$ and $\\lambda$ would give comparable expansions.","The reciprocity in the addendum suggests a general two-parameter duality between iterates of $x/(1+x^r)$ and $x+x^{1-q}$; if the duality holds more broadly, the hardest regimes (for example irrational $q$) could be handled by the convergent-iteration method rather than the brute-force matching one.","The brute-force matching-coefficient method appears to be a general \"shift and compare\" tool: because it uses only the recurrence and the leading asymptotics, it could extend many slow recurrences to arbitrary order without additional analytic theorems.","The same reciprocity could be pushed further: expanding the companion iterations $\\eta_k$ and $\\zeta_k$ to higher order should reproduce $c(3)$ and $c(2)$ from a convergent-iteration calculation, making the brute-force step unnecessary at those parameters."],"forward_implications":["For $f(x)=x(1-\\sqrt{x})$, the expansion through $1/k^6$ lets one compute $x_k$ accurately for large $k$ from the constant $C$ alone, without iterating the map millions of times.","For the $q$-recurrence, the displayed formulas give explicit closed-form asymptotics for every $q>1$ at three benchmark values, with $c(2)$, $c(3)$, and $c(3/2)$ as numerical anchors.","The addendum shows the constants for divergent iterations where $x_k$ grows like $k^{1/q}$ and convergent iterations where $x_k$ decays are the same constants under a reciprocal transformation, so computing one gives the other.","For $p$-scaled logistic maps with $p<1$, the product formula $C=x_0\\prod_{j=0}^{\\infty}(1-x_j)$ gives a direct way to estimate the geometric decay rate from finitely many iterations."],"supporting_citations":[{"why":"Provides the simple numerical method used to estimate the constants $C$ in Section 2.","marker":"[2]"},{"why":"Supplies the asymptotic expansion template for $f(x)=x(1-a x^\\tau)$ that generates the Section 2 formulas.","marker":"[3]"},{"why":"Gives the convergent-iteration method for iterates of $\\sin(x)$ that underlies the Section 2 expansion and the addendum's reciprocity calculation.","marker":"[13]"},{"why":"Extends the asymptotic-expansion method to iterates of classical functions, used alongside [13] for the convergent iterations.","marker":"[14]"},{"why":"Earlier treatment of the cubic recurrence $x(1-x^2)$ by a different approach, used as comparison and for the brute-force method.","marker":"[15]"},{"why":"Supplies the theorem for asymptotics of $y_k=y_{k-1}(1+1/y_{k-1})^q$ used in Section 3.","marker":"[17]"},{"why":"Supplies the refined asymptotic theorem and Proposition 7 for the transformed variable, including the coefficient the paper corrects.","marker":"[18]"}],"fun_headline_variants":["Three iterated maps yield new asymptotic constants","New constants from iterating x(1-sqrt x)","Sixth-order expansions for three nonlinear recurrences","New constants in long-term iterated recurrences","Asymptotic constants from iterating three maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The expansions all depend on asymptotic formulas imported from earlier work being valid for these particular recurrences; if any recurrence falls outside those formulas' hypotheses, the displayed constants shift.","fun_headline_variants_meta":{"raw":{"variants":["Three iterated maps yield new asymptotic constants","New constants from iterating x(1-sqrt x)","Sixth-order expansions for three nonlinear recurrences","New constants in long-term iterated recurrences","Asymptotic constants from iterating three maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000995,"raw_usage":{"total_tokens":4145,"prompt_tokens":803,"completion_tokens":3342,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":419,"completion_tokens_details":{"reasoning_tokens":3270}},"tokens_in":419,"tokens_out":3342,"duration_ms":25380,"temperature":1.0,"reasoning_tokens":3270,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:35:00.810010+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $x_0=1/2$, iterate $x_k=x_{k-1}(1-\\sqrt{x_{k-1}})$ in high-precision arithmetic up to $k=10^6$, and compare $x_k$ to the six-term expansion with $C=1.98803983644549695008812308629512$. If the difference after subtracting all displayed terms through order $1/k^6$ does not decay like $o(k^{-6})$ (or at least like $O(k^{-7})$), then the template or the constant is wrong. For $q=2$, iterate $x_k=x_{k-1}+1/x_{k-1}$ from $x_0=1$ and test that $2^{1/2}x_k - 2k^{1/2} - \\frac{1}{4}\\frac{\\ln(k)}{k^{1/2}} - c(2)\\frac{1}{k^{1/2}}$ converges to $0$ at the predicted rate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the simple numerical method used to estimate the constants $C$ in Section 2."},{"cited_title":"Mavecha and V","cited_arxiv_id":null,"evidence_quote":"Extends the asymptotic-expansion method to iterates of classical functions, used alongside [13] for the convergent iterations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier treatment of the cubic recurrence $x(1-x^2)$ by a different approach, used as comparison and for the brute-force method."},{"cited_title":"Popa, Recurrent sequences and the asymptotic expansion of a function, Gazeta Mat","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem for asymptotics of $y_k=y_{k-1}(1+1/y_{k-1})^q$ used in Section 3."},{"cited_title":"Popa, Refined asymptotic expansions for some recurrent s e- quences, Gazeta Mat","cited_arxiv_id":null,"evidence_quote":"Supplies the refined asymptotic theorem and Proposition 7 for the transformed variable, including the coefficient the paper corrects."}],"review_version":1}