{"id":"c906bbfb-0854-4208-96e1-a54d96f01b37","arxiv_id":"2412.11806","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a family of nonlinear recurrences, the paper evaluates the asymptotic constant C to 25 digits using a reciprocal transformation and the Mavecha-Laohakosol algorithm.","lead":"Steven Finch computes high-precision constants in asymptotic expansions of nonlinear recurrences by inverting the sequence and applying a known asymptotic algorithm. The paper fills a gap left open by Dumitru Popa, who derived the expansions but did not evaluate the constants.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Several headline constants depend on applying Mavecha-Laohakosol to maps outside the paper's stated hypotheses (non-analytic g in §3.7, non-integer τ in §4.3/§4.5); the needed extension is asserted but not proved.","rationale":"The paper's central claim is the computation of 25-digit constants C for a dozen nonlinear recurrences. The methodology has two pillars: (i) the reciprocal transformation y=1/x, and (ii) the Mavecha-Laohakosol algorithm for the asymptotic expansion of y_k. The reader's concern focuses on pillar (ii)'s numerical side: the truncated series is equated to an exactly computed y_K and solved for C, with no error bound on the remainder. That concern is real, but the more fundamental weakness is that the algorithm is invoked for maps that do not satisfy the paper's own stated hypotheses. If the asymptotic form of y_k is wrong for any of those cases, the corresponding C values are wrong regardless of how carefully the numerical inversion is done. This is not a mere precision issue; it is a validity issue. The paper does present a fully worked example for p=1/2, which gives useful evidence that the method works in the analytic integer-τ setting. But the non-analytic and fractional-τ cases are asserted without derivation, and no reference in the bibliography is shown to cover them. The concrete test proposed—direct coefficient matching for §3.7 and the q=3/2 case—would settle whether the extension is valid. Because the paper's other results may well be correct, the appropriate verdict remains conditional pending this check, matching the reader's CONDITIONAL verdict rather than escalating to rejection.","tokens_in":11886,"tokens_out":13487,"duration_ms":120190,"concrete_test":"Re-derive the displayed y_k expansion in §3.7 by direct substitution. Assume y_k = 4/k^2 + a ln(k)/k^3 + b/k^3 + c ln^2(k)/k^4 + d ln(k)/k^4 + e/k^4 + ..., substitute into y_{k+1}=y_k exp(-√y_k), expand exp(-√y_k) as a series in powers of 1/k and ln k, and match coefficients through order 1/k^4. If the matched coefficients do not equal the paper's (-4, -16C, 3, -2+24C, 4/3-8C+48C^2, ...), then the invoked extension of Mavecha-Laohakosol is invalid and C=0.8791712792948618603132189 is unsupported. A parallel coefficient-matching for g(y)=y/(1+y^{3/2}) with τ=3/2 would resolve the fractional-q cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Appendix (§5) defines the method for an analytic g(y)=y+a1 y^{τ+1}+... with τ≥1 an integer and a1<0. Yet §3.7 uses g(y)=y exp(-√y), which is not analytic at 0, and §4.3/§4.5 use g(y)=y/(1+y^{q}) with q=3/2 and 4/3, whose expansions near 0 contain y^{5/2} and y^{7/3}, i.e. τ=3/2 and τ=4/3, not integers. The text states without proof that the algorithm 'does, however, apply' in the non-analytic case. If the Mavecha-Laohakosol formula genuinely requires analyticity or integer τ, then the asymptotic expansions displayed for these cases—and therefore the 25-digit constants C=0.8791712792948618603132189, 0.8010888849039666437110775, and 0.8248745112329031526004762—are not established by the argument given. This is an internal gap relative to the paper's own stated hypotheses, not merely a disagreement with external consensus.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a computational method for evaluating the constant term C in asymptotic expansions of positive sequences satisfying nonlinear recurrences. For x_{k+1}=f(x_k) with x_k ~ α k + β ln k + C (or analogous growth), the substitution y_k=1/x_k transforms the recurrence to y_{k+1}=g(y_k), and the author applies the Mavecha–Laohakosol algorithm to obtain a truncated asymptotic series for y_k containing C. The constant is then found by matching the series to an exactly computed value y_K. The paper presents expansions and 25-digit constants for a collection of recurrences (p-sequences, radicals, exponential/logarithmic cases, and q-sequences), and corrects two typos in Popa's theorems. The p=1/2 case is worked in full detail in the appendix.","tokens_in":12177,"tokens_out":9452,"duration_ms":75403,"significance":"If the expansions and constants are correct, the paper provides a useful collection of high-precision constants for a family of recurrences that have appeared in various contexts. The reciprocal transformation is a clever and potentially reusable device, and the p=1/2 appendix gives a genuinely worked example with explicit coefficient arrays and the resulting series to order k^{-6}. The procedure is a legitimate shooting/matching method rather than a data fit, since C enters as an integration constant of the asymptotic expansion. The claimed constants, however, are not accompanied by rigorous error bounds, and several cases use the Mavecha-Laohakosol formalism outside the hypotheses stated in the appendix; these gaps must be closed before the numerical claims can be regarded as established.","major_comments":[{"comment":"Section 5 states the Mavecha-Laohakosol algorithm applies to an analytic g(y) whose Taylor series at the origin starts as y + a_1 y^{τ+1} + a_2 y^{2τ+1} + ... with a_1<0 and τ≥1 an integer. However, §3.7 uses g(y)=y exp(-√y), which has expansion y - y^{3/2} + y^2/2 - y^{5/2}/6 + ... and is not analytic at 0, and §4.3 and §4.5 use g(y)=y/(1+y^q) with q=3/2 and q=4/3, corresponding to τ=3/2 and τ=4/3, which are not integers. The text asserts without proof that the algorithm 'does, however, apply' in these cases. Since the constants C=0.8791712792948618603132189, C=0.8010888849039666437110775, and C=0.8248745112329031526004762 rest on these expansions, a proof of the extended validity or an independent verification is required.","section":"§5 vs. §3.7, §4.3, §4.5"},{"comment":"The procedure 'We then set the value y_K equal to our series and numerically solve for C' is used to obtain the 25-digit constants, but the paper gives no remainder bound for the truncated series and no analysis of the numerical solve. The assertion that terms to order 1/k^6 with K≈10^6 might suffice is plausible but unproved. Without such an error estimate, the claimed number of correct digits is not justified. The author should supply a rigorous bound or provide an independent high-precision check, e.g., a second method or comparison of results at different K with rigorous intervals.","section":"§5"},{"comment":"Most of the expansions in Sections 1–4 are presented as 'We find' with no derivations or input data (coefficients a_m, a_{0j}, b_j, or their analogues for the nonstandard cases). The reader cannot reproduce or audit the computations except for the p=1/2 example in the appendix. Since the paper's contribution is precisely these expansions and constants, the author should include enough algorithmic detail or a supplement, such as a Mathematica notebook, to make all results checkable.","section":"§1.2–§4.5"}],"minor_comments":[{"comment":"Reference [7] has the obvious typo 'arXiv:24409.15175'; it should presumably be 'arXiv:2409.15175'.","section":"References"},{"comment":"The displayed series for y_k contains the expression '− (− 1/24 + C + 10C^2 + 16C^3) 1/k^4' with a doubled minus sign; this should be simplified for readability.","section":"§1.1"},{"comment":"The paper does not number its equations, which makes it awkward to refer to specific displayed results; adding equation numbers would help readers and referees.","section":"General"},{"comment":"The introductory reciprocal asymptotic formula is stated only for x_k ~ α k + β ln k + C, but in §3.7 and §3.8 the growth of x_k is quadratic or square-root; a short statement of the generalized reciprocal asymptotic relation used in those sections would improve clarity.","section":"§3.7, §3.8"}],"recommendation":"major_revision","confidential_remarks":"This is a computational note whose main value is a collection of high-precision constants. I would be willing to support publication after the author closes the two load-bearing gaps: justifying the use of the Mavecha-Laohakosol algorithm beyond its stated hypotheses, and providing error control for the 25-digit claims. A supplement with derivations or code would also substantially increase confidence in the many 'We find' results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper computes 25-digit constants for a family of nonlinear recurrences by combining a reciprocal transformation with the Mavecha–Laohakosol algorithm. Several of these constants, e.g. for x exp(1/x) and x + exp(1/x), are new as far as I can tell, and the reciprocity observation itself is clean. The one fully worked example (p=1/2) is internally consistent, and spot-checks of the expansions against Popa's theorems match. The corrections to two typographical errors in Popa's paper are a useful service.\n\nThe soft spot is real and load-bearing. The appendix states that the algorithm applies to analytic g(y) = y + a1 y^{τ+1} + ... with integer τ ≥ 1 and a1 < 0. But §3.7 uses g(y) = y exp(−√y), which is not analytic at 0, and §4.3/§4.5 use g(y) = y/(1+y^q) with q = 3/2 and 4/3, giving τ = 3/2 and 4/3. The text says the algorithm \"does, however, apply\" and moves on. No proof, no reference, no cross-check. Since the constants 0.879171..., 0.801088..., 0.824874... are headline results, this gap matters: the argument given does not establish them. I think the stress-tester is right to flag this.\n\nThe other weakness is the general lack of derivations, code, or error bounds. The expansions are mostly stated with \"We find\". For a paper whose main output is numerical constants to 25 digits, the absence of reproducible artifacts makes independent verification impossible. The p=1/2 example matches an earlier brute-force value, which is reassuring, but the non-analytic and non-integer cases have no such check.\n\nThis paper is for specialists in iterational asymptotics and for anyone who needs these specific constants. It is not a desk-reject: the method appears real and the p=1/2 case is solid. But a serious referee should demand either a proof that Mavecha–Laohakosol extends to the cited non-analytic/non-integer cases, or an independent high-precision numerical verification. If neither is supplied, the affected constants should be labeled conjectural.","headline":"Plausible high-precision constants for a dozen recurrences, several genuinely new, but the Mavecha–Laohakosol algorithm is applied beyond the paper's own stated hypotheses, and the extension is asserted without proof or independent verification.","tokens_in":12635,"tokens_out":3021,"would_cite":false,"duration_ms":27332,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A60","39A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Switching to reciprocals computes a dozen asymptotic constants to 25 digits","keywords":["nonlinear recurrences","asymptotic constants","reciprocal substitution","asymptotic expansions of iterates","Mavecha-Laohakosol algorithm","high-precision computation","recurrent sequences"],"falsifier":"For the reciprocal $p$-sequence with $p=1/2$ (where $g(y)=y/\\sqrt{1+y}$), compute $y_K$ exactly for $K=10^6$ and for $K=2\\cdot10^6$, using the six-term series in Section 5 to solve for $C$ both times; if the two estimates disagree beyond the 25th decimal, the paper's precision claim fails. Alternatively, compare the reported $C=0.8615711875687117305317813\\ldots$ for the $q=2$ sequence against an independent high-precision extrapolation of $\\sqrt{2}\\,x_k - 2\\sqrt{k} - \\frac{1}{4}\\ln(k)/\\sqrt{k}$ from direct iteration; any discrepancy in the last digits would reveal underestimated error.","tokens_in":11695,"feed_emoji":"🔁","tokens_out":7083,"duration_ms":58943,"temperature":0.7,"pith_summary":"This paper tackles a dozen nonlinear recurrences whose iterates grow like $x_k \\sim \\alpha k + \\beta \\ln(k) + C$, where the constant $C$ was previously known to exist but not numerically evaluated. The key move is to study the reciprocal sequence $y_k = 1/x_k$, which obeys its own recurrence and whose asymptotic series can be produced by a standard algorithm for iterates of analytic functions. The paper reports 25-digit values of $C$ for all twelve cases, including several boundary cases where the original asymptotic expansion for $x_k$ does not apply. A sympathetic reader cares because the constants are now concrete numbers, and the reciprocal viewpoint is shown to be the essential computational step rather than a cosmetic shortcut.","feed_headline":"Reciprocal trick yields 25-digit recurrence constants","feed_subtitle":"By iterating reciprocals, a dozen previously open asymptotic constants are computed to full precision.","key_machinery":"The central object is the reciprocal substitution $y_k=1/x_k$ paired with the Mavecha–Laohakosol algorithm, a method for expanding iterates of an analytic map $g$ whose Taylor series at the origin begins $y+a_1 y^{\\tau+1}+a_2 y^{2\\tau+1}+\\cdots$ (with $a_1<0$ and integer $\\tau\\ge1$) into an asymptotic series in powers of $k^{-1/\\tau}$ and $\\ln k$. The substitution converts $x_{k+1}=f(x_k)$ into $y_{k+1}=g(y_k)$ with $g(y)=1/f(1/y)$; the algorithm then delivers a series whose first coefficients are polynomials in $C$, so comparing the series to an exactly iterated value $y_K$ determines $C$. The paper's reported constants all come from this recipe.","core_discovery":"The paper's central claim is that, for any recurrence $x_{k+1}=f(x_k)$ with $x_k \\sim \\alpha k + \\beta \\ln(k) + C$, the constant $C$ can be computed to 25 digits by passing to $y_k = 1/x_k$. The reciprocal sequence satisfies $y_{k+1}=g(y_k)$ with $g(y)=1/f(1/y)$, and the Mavecha–Laohakosol algorithm expands $y_k$ in an asymptotic series in powers of $k^{-1/\\tau}$ and $\\ln k$; setting that series equal to an exactly computed $y_K$ and solving for $C$ yields the constants. Twelve cases are worked out in detail—$p$-sequences, radicals, exponentials, logarithms, and $q$-sequences—and in two boundary cases ($f(x)=x\\exp(\\sqrt{1/x})$ and $f(x)=x(1+\\ln(1+1/x^2))$) the algorithm succeeds even where the original expansion for $x_k$ fails. The paper also corrects two coefficient typos in the source theorems and notes a duality between the $q=1/2$ and $p=1/2$ recurrences.","pith_inferences":["Not pursued in the paper: the claimed 25-digit accuracy is asserted without a rigorous remainder bound, so a numerical-analysis follow-up could certify or slightly revise the final digits of each constant.","The method's success suggests that the open problem of finding a nonlinear recurrence whose constant $C$ has a closed-form expression (independent of the sequence) might be approached by looking for reciprocal maps $g$ whose asymptotic series coefficients simplify.","The same reciprocal-plus-algorithm recipe could be applied to recurrences with asymptotic forms like $x_k \\sim \\alpha k^p + \\beta \\ln k + C$ for other values of $p$, provided the corresponding $g$ is analytic at the origin with the required leading behavior.","The connection between $g(y)=y e^{-y}$ and the Lambert $W$ function hints that some of the computed constants may be expressible in terms of special functions; that link is not developed in the paper."],"forward_implications":["The twelve constants are now available to 25 digits, providing numerical anchors for further asymptotic study of these recurrences and for checking future symbolic derivations.","The reciprocal method extends to boundary cases where Popa's original expansions do not apply, enlarging the class of recurrences whose constants can be computed.","The same procedure can be applied to any recurrence whose reciprocal map $g$ satisfies the algorithm's hypotheses, giving a general numerical tool for constants in similar asymptotic forms.","Correcting the two coefficient typos in the source theorems changes the numerical values derived from the affected corollary, so any prior computations based on those formulas need revision.","The observed duality between the $q=1/2$ and $p=1/2$ recurrences, and between $q=2$ and $p=2$, shows that seemingly different recurrences can share constants up to simple factors, suggesting underlying transformation identities."],"supporting_citations":[{"why":"Supplies the source asymptotic expansions for the recurrences, including Proposition 12 that converts an $x_k$ expansion into one for $1/x_k$.","marker":"[1]"},{"why":"Provides the refined expansions and the reciprocal relation used as a starting point, and contains the theorems whose coefficient typos the paper corrects.","marker":"[2]"},{"why":"Supplies the algorithm that generates the asymptotic series for the reciprocal sequence $y_k$.","marker":"[5]"},{"why":"Contains the formulaic background the appendix assumes when applying the algorithm to the $p=1/2$ reciprocal sequence.","marker":"[9]"},{"why":"Originates the iterational-asymptotics technique that the algorithm later refines.","marker":"[3]"},{"why":"Demonstrates the expansion method for iterates, forming an intermediate step in the algorithm's development.","marker":"[4]"},{"why":"Details the transformation that yields the six-term series used for the $q$-sequence expansions in Section 4.","marker":"[10]"}],"fun_headline_variants":["Reciprocal map cracks open recurrence constants","25-digit constants from reciprocal iteration","Reciprocal trick solves dozen constants","Mavecha–Laohakosol algorithm meets reciprocals","Twelve constants pinned via reciprocal series"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical scheme assumes that, at the chosen index $K$, the truncated asymptotic series for $y_K$ differs from the true value by less than the claimed $10^{-25}$ error, and no rigorous bound on the remainder is provided.","fun_headline_variants_meta":{"raw":{"variants":["Reciprocal map cracks open recurrence constants","25-digit constants from reciprocal iteration","Reciprocal trick solves dozen constants","Mavecha–Laohakosol algorithm meets reciprocals","Twelve constants pinned via reciprocal series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"total_tokens":1280,"prompt_tokens":826,"completion_tokens":454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":388}},"tokens_in":442,"tokens_out":454,"duration_ms":4720,"temperature":1.0,"reasoning_tokens":388,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:33:50.179616+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the reciprocal $p$-sequence with $p=1/2$ (where $g(y)=y/\\sqrt{1+y}$), compute $y_K$ exactly for $K=10^6$ and for $K=2\\cdot10^6$, using the six-term series in Section 5 to solve for $C$ both times; if the two estimates disagree beyond the 25th decimal, the paper's precision claim fails. Alternatively, compare the reported $C=0.8615711875687117305317813\\ldots$ for the $q=2$ sequence against an independent high-precision extrapolation of $\\sqrt{2}\\,x_k - 2\\sqrt{k} - \\frac{1}{4}\\ln(k)/\\sqrt{k}$ from direct iteration; any discrepancy in the last digits would reveal underestimated error.","supporting_citations":[{"cited_title":"Popa, Recurrent sequences and the asymptotic expansion of a function,Gazeta Mat","cited_arxiv_id":null,"evidence_quote":"Supplies the source asymptotic expansions for the recurrences, including Proposition 12 that converts an $x_k$ expansion into one for $1/x_k$."},{"cited_title":"Popa, Refined asymptotic expansions for some recurrent sequences,Gazeta Mat","cited_arxiv_id":null,"evidence_quote":"Provides the refined expansions and the reciprocal relation used as a starting point, and contains the theorems whose coefficient typos the paper corrects."},{"cited_title":"Mavecha and V","cited_arxiv_id":null,"evidence_quote":"Supplies the algorithm that generates the asymptotic series for the reciprocal sequence $y_k$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Originates the iterational-asymptotics technique that the algorithm later refines."},{"cited_title":"Bencherif and G","cited_arxiv_id":null,"evidence_quote":"Demonstrates the expansion method for iterates, forming an intermediate step in the algorithm's development."},{"cited_title":"Exercises in Iterational Asymptotics","cited_arxiv_id":"2411.16062","evidence_quote":"Details the transformation that yields the six-term series used for the $q$-sequence expansions in Section 4."}],"review_version":1}