{"id":"ff55f1bd-bacc-4627-9595-8bd0c4067ddf","arxiv_id":"2607.23287","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Sliding-window least squares with proven convergence conditions and rates for computing constants in asymptotic expansions, plus a regularized variant that always recovers the dominant growth parameter.","lead":"The paper introduces a method family that provably extracts the unknown constants in proven asymptotic expansions — growth rates, exponents, constants — by sliding-window least squares on logarithms, with rate guarantees and an honest catalog of failure modes. The regularized variant always recovers the dominant growth exponent; a constructed counterexample shows perfect fits can still diverge.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Guaranteed full-parameter recovery requires proven high-order asymptotics, which the motivating non-constructive settings typically do not provide.","rationale":"The central theorems appear correct: Theorem 3.14's proof of (3.20) via (B.1) is sound, and Theorem 3.27's dominant-component rate follows from Lemma 3.26's universal bound. The independent spot-checks reported by the reader also align with my reading. The single most load-bearing limitation is not an internal inconsistency but the conditional reach of the guarantees: sLLSQ's full-vector convergence is conditioned on ϑ_j(n)=o(1), which typically demands a proven high-order expansion, exactly the hard input that non-constructive existence theorems do not supply. sT-LLSQ's unconditional rate covers only the dominant parameter; for the other components even the stronger footnote statement is convergence to zero, not to γ_j, so it does not extend recovery. This fully matches the reader's identified weakest assumption and supports the existing CONDITIONAL verdict without moving it. My proposed test directly probes whether the lack of higher-order terms is a practical barrier, which is the crux of the concern.","tokens_in":61157,"tokens_out":27554,"duration_ms":216525,"concrete_test":"Use the factorial sequence n!, whose full asymptotic expansion is known, but deliberately restrict to the p=0 expansion f(n)=√(2πn)(n/e)^n(1+O(1/n)), i.e. g1(n)=1/n. Compute the sLLSQ estimators with m=k=4 using exact/high-precision values of log(n!) and track ∥y*_n−γ∥ for n=10,10²,10³,10⁴,10⁵. If the error grows like n² log n (or fails to decay), the sLLSQ guarantee is indeed unavailable without higher-order terms, confirming the concern; if it decays, the sufficient condition is conservative and the practical impact is weaker.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.14's sLLSQ convergence guarantee is conditional on ϑ_j(n)=o(1). For the paper's own Table 3 asymptotic forms with low order p, these conditions fail: AF-1 with p=0 gives ∥y*_n−γ∥=O(n² log n) (Remark 4.1), AF-2 with p=1 gives O(log n), and AF-6 with p=0 gives only O(1). Since the paper's motivating examples (DLW-theorem and catalytic-equation asymptotics, §1.1) normally prove only the leading-order term, the advertised 'convergence-certified' full-parameter recovery does not apply to them. sT-LLSQ (Theorem 3.27) does offer an unconditional guarantee, but only for the dominant component γ_k; for j<k it proves only O(1). The Table 6 footnote's o(1) statement describes y*_{j,n}→0 (regularization bias), not y*_{j,n}→γ_j, so it does not give full recovery either. Thus, even in the favorable non-constructive setting, only the leading exponential parameter is certified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a framework called Asymptotics Learning Theory (ALT) for computing the finite parameter vector α in a proven asymptotic expansion f(n) = bf(n;α)(1 + Σ_{l=1}^p β_l g_l(n) + O(g_{p+1}(n))) from exact values of f. The theory specializes to exponential-product forms bf(n;x) = exp(Σ_j φ_j(n)u_j(x_j)), covering α_1 n^{α_2} α_3^n, stretched-exponential, and power-law asymptotics. Two sliding-window estimators are studied in closed form: sLLSQ, y*_n = (A^T A)^{-1}A^T b = A†b, and sT-LLSQ, y*_n = (A^T A + μI)^{-1}A^T b, where A is the windowed matrix of the φ_j. Theorem 3.14 establishes uniqueness, residual decay F*_n=O(||g_1||^2), and the bound ||y*_n − γ||=O(η(n)) with per-component rates ϑ_j(n); convergence is certified exactly when ϑ_j(n)=o(1), and Proposition 3.7 transfers rate bounds from the transformed parameters γ to the original α. Theorem 3.27 gives ||y*_n − γ||=O(χ(n))=O(1) and an unconditional dominant-component rate y*_{k,n} − γ_k = O(φ_{k−1}(n)/φ_k(n)) = o(1). Applications to analytic combinatorics (Fibonacci, Catalan, factorials, and an eleven-form family in Tables 4–6) are compared with the ratio method (Prop. 5.1) and tested numerically. Section 6.4 constructs a p=0 example with a hidden alternating term where sLLSQ diverges despite zero residuals, showing that the sufficiency conditions are genuinely necessary.","tokens_in":61286,"tokens_out":32328,"duration_ms":264727,"significance":"If the results hold up, the paper contributes a fairly general, closed-form family of estimators for the parameters of exponential-product asymptotics, with explicitly stated sufficient conditions, per-component rate bounds, and a Θ-transfer theorem (Prop. 3.7) that lets rates for y* convert to rates for the original parameters. The appendix proofs are coherent, and I independently verified several representative computations: det(A^T A)=6 for m=3 and 20 for m=4 in Remark 3.23, the Θ(1/n), Θ(1/n^2) vs Θ(1), Θ(1/n) components of A† log(1+β_1 g_1), and the p=0 AF-1 bound ||y*_n − γ||=O(n^2 log n) of Remark 4.1. The framework is not circular: γ enters only through the assumed expansion (2.1); the bounds are functions of A† and the decreasing scale g_l, with β_l as symbolic constants; there are no self-citations. The strongest clean result is the unconditional dominant-component rate y*_{k,n} − γ_k = O(φ_{k−1}/φ_k) (Lemma 3.26/Theorem 3.27), valid for any μ>0, any norm, and any ν≥1 for the general regularized problem. The §6.4 counterexample (zero residuals, diverging sLLSQ) is a valuable negative result that correctly identifies the sufficiency conditions as the load-bearing checkpoin","major_comments":[{"comment":"The paper's motivating settings (DLW theorem, catalytic equations, chordal graphs; §1.1 and eq. (1.3)) provide only leading-order asymptotics, i.e., AF-2 with p=1. For that order, Table 4 gives ||y*_n − γ||=O(log n) for sLLSQ, so the condition ϑ_j(n)=o(1) in Theorem 3.14 fails and full-vector convergence is not certified. Theorem 3.27 certifies only the dominant component γ_k (Table 6, Remark 3.34); for j<k it gives only y*_{j,n} − γ_j = O(1), and the o(1) in the Table 6 footnote describes convergence to the regularized bias 0, not to γ_j. Thus, in the very settings that motivate the paper, certified recovery is limited to the leading exponential parameter (via sT-LLSQ) unless higher-order terms have been proven in advance — precisely the 'hard part' flagged in Remarks 4.1–4.3. The abstract's 'convergence (to the correct values of unknown parameters)' should be qualified. This is a scope","section":"Abstract; §1.1; Theorems 3.14/3.27; Remarks 4.1–4.3"},{"comment":"Tables 4–6 carry the applications and operationalize the ϑ_j(n)=o(1) checkpoints, but the Puiseux estimates are asserted without derivations or scripts. Remark 3.23 shows the rates are fragile: with the same φ_j and g_l, m=3 gives η(n)=Θ(1/n) while m=4 gives η(n)=Θ(1). With the many symbolic rate entries (11 forms × orders p=0..3, plus det(A^T A) values and the §6.4 Θ-estimates), the reader cannot audit those entries. Please provide the CAS notebook/script (e.g., the MATLAB symbolic commands) or a supplementary derivation for each row family. Without such artifacts, the rate tables function as trusted assertions rather than verified results.","section":"Section 4, Tables 4–6"}],"minor_comments":[{"comment":"Typo in the manuscript header: 'CONVERGENCE-RA TE GUARANTEES' should be 'CONVERGENCE-RATE GUARANTEES'.","section":"Title"},{"comment":"The phrase 'introduce a new research area' is stronger than this first paper supports; 'framework' or 'methodology' would be more defensible and would not undercut the technical contribution.","section":"Abstract"},{"comment":"The step following (B.4) moves from entrywise O(1) and O(φ_{k−1}/φ_k) estimates, valid for arbitrary (eγ, w), to uniform bounds on the matrices C and D. Since the O-constants may in principle depend on eγ and w, the argument should restrict to unit basis vectors and use finiteness of the test set; the conclusion is correct, but this step is compressed.","section":"Appendix B.3, proof of Lemma 3.26"},{"comment":"All rate tables use m=k. Given Remark 3.23 (increasing m from 3 to 4 flips η from Θ(1/n) to Θ(1)), practitioners need explicit guidance on whether and when m>k is safe; Proposition 3.22 covers only the regime where convergence already holds. A sentence in Section 4 would prevent misreading the tables as valid for all m.","section":"Section 4, Tables 4–6"},{"comment":"For the rows 'AF-2 with known α_2 / α_3', the sT-LLSQ rate columns follow from the transformations noted in parentheses, but the dimension change of the transformed problem is not shown; a pointer to the corresponding AF-6/AF-10 rows would make the comparison self-contained.","section":"Table 7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is mathematically sound in its main line: the closed-form estimators, the error bounds, and the dominant-component convergence rate all check out, and several of the symbolic computations I re-derived (Remark 3.23 determinants and rates; AF-1 p=0 bound) are correct. The paper is long and expounds a new 'ALT' program; the program-level claims are more ambitious than the theorems' conditional guarantees. For a methods journal, the contribution is publishable after revision that (i) re-calibrates the abstract/intro to the actual scope (full-vector certification only when the ϑ_j=o(1) conditions are verified; unconditional certification only for the dominant parameter in sT-LLSQ), and (ii) provides reproducibility artifacts for the symbolic rate tables. I found no citation or circularity concerns: the paper is single-author with standard external references only."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nPunchline: this is a serious piece of work, but the practical reach is narrower than the abstract lets on. The convergence-rate theorems for sliding least squares are real and the proofs mostly check out; however, the guaranteed recovery of all unknown parameters requires a proven high-order asymptotic expansion, which the motivating non-constructive settings (DLW theorem, catalytic equations) do not typically provide. The unconditional guarantee covers only the dominant parameter.\n\nWhat's actually new: for a broad class of asymptotic forms (products of exponentials), the paper gives rigorous sufficient conditions and rates for two sliding-window least-squares estimators. This is a clear improvement over the Domb–Sykes ratio method and direct fitting, which are heuristic in comparison. Proposition 3.6 (uniqueness) and Proposition 3.7 (rate preservation under variable transformation) are clean. Theorem 3.14 and 3.27 have detailed appendix proofs; I independently checked several non-trivial computations (the det=6/20 window counterexample, the AF-1 determinant) and they are correct. No circularity: the target vector appears only through the assumed expansion, not through the fitted values.\n\nSoft spots, in order of importance. (1) The full-parameter convergence guarantee is conditional on ϑ_j(n)=o(1), which for the paper's own AF-1/AF-2 examples fails at low expansion order. The motivating DLW-type theorems normally prove only the leading term, so the advertised \"learning constants from proven asymptotics\" does not yet apply to them. (2) sT-LLSQ's unconditional rate is only for the dominant component; the other components get O(1). The footnote in Table 6 claiming o(1) for j<k describes regularization bias, not recovery of γ_j. (3) The numerical tables and figures rely on unspecified CAS outputs; no code or derivations are given, so the applied payload is not reproducible. (4) The \"new research area\" and \"fundamental applications\" framing exceeds the delivery—the examples are textbook sequences with known constants.\n\nNone of these are fatal. The theory is honest about the conditions, and the limitations are mostly disclosed. The paper just needs a more careful statement of scope and some reproducibility support.\n\nWho it's for: researchers doing series analysis in analytic combinatorics or statistical physics who want a rigorous alternative to Domb–Sykes. It deserves a serious referee; I would send it to review and ask for the code, a sharper statement about low-order expansions, and toning down the ALT branding.","headline":"A genuinely rigorous convergence analysis for sliding least squares, but the advertised learning of DLW-type constants works only when you already have the hard part (high-order expansions); sT-LLSQ only certifies the dominant parameter.","tokens_in":61963,"tokens_out":3224,"would_cite":false,"duration_ms":31921,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05-08","05A16","68W40","76M45","82B26","82B27"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two sliding least-squares estimators compute unknown constants inside proven asymptotic expansions with explicit convergence-rate guarantees.","keywords":["asymptotics learning theory","sliding least squares","convergence-rate guarantees","asymptotic expansions","analytic combinatorics","Tikhonov regularization","ratio method","parameter estimation"],"falsifier":"For a known sequence such as a_n = 7 n^{1/3} (5/2)^n (1 + (−1)^n / √n), compute the sLLSQ estimate y*_n = A†b with m = k = 3 and compare ||y*_n − γ|| to the theorem's O(n^{3/2} log n) bound: if the error does not grow like Θ(n^{3/2} log n), or if it converges despite ϑ_j(n) not being o(1), the claimed sufficient conditions would be wrong.","tokens_in":60831,"feed_emoji":"📐","tokens_out":4851,"duration_ms":43401,"temperature":0.7,"pith_summary":"The paper's central claim is that unknown constants inside a proven asymptotic expansion can be computed numerically with rigorous convergence-rate guarantees by fitting the expansion's exponential-product form to data through sliding least-squares problems. It proves that the constants are unique, that the sliding estimator y*_n = A†b converges to the transformed parameter γ whenever a per-component condition ϑ_j(n) = o(1) holds, and that the Tikhonov variant y*_n = Cb always converges for the dominant component at rate O(φ_{k−1}(n)/φ_k(n)) = o(1). The methods turn non-constructive existence proofs into computable values, demonstrated on Fibonacci, Catalan, and factorial asymptotics. The paper also documents a failure mode: sLLSQ can diverge completely even with zero residuals when an oscillating lower-order term is hidden in the expansion.","feed_headline":"Sliding least squares recovers asymptotic constants with proven rates","feed_subtitle":"A log-linear trick turns asymptotic fitting into least squares, yielding certified convergence rates for non-constructive constants.","key_machinery":"The central object is the exponential-product asymptotic form bf(n;x) = exp(Σ φ_j(n) u_j(x_j)), whose logarithm is linear in the transformed parameters γ_j = u_j(x_j). The sliding estimator solves min_y ||A y − b||², with A the windowed matrix of the φ_j's and b the log-data vector; its closed-form solution A†b (and Tikhonov variant Cb, with C = (AᵀA + μI)⁻¹Aᵀ) is what carries the argument. The key identities are the decomposition b = Aγ + log(1 + Σβ_l g_l) + z, which splits the estimate into the true parameter plus a controllable perturbation, and the asymptotic invariants that transfer Θ-equivalent error rates through the inverse transformation x = u^{-1}(y).","core_discovery":"The discovery is a reduction: after taking logarithms, the exponential-product asymptotic form becomes linear in the unknown parameters, so estimating them is exactly an ordinary least-squares problem with a design matrix built from the known basis functions φ_j. The main theorems give closed-form estimators and matching error bounds: for sLLSQ, ||y*_n − γ|| = O(η(n)), where η(n) is a maximum of two terms involving the pseudoinverse of the design matrix; for sT-LLSQ, ||y*_n − γ|| = O(1) for all components and O(φ_{k−1}(n)/φ_k(n)) = o(1) for the dominant one. Since the map u is bijective with nonzero derivative, these rates transfer to the original parameters α via the convergence-rate invari","pith_inferences":["A natural extension is to other loss functions or regularization penalties (e.g., ℓ1/LASSO) that preserve the sliding-window idea; the paper's universal dominant rate suggests the method is robust to the choice of penalty.","The approach may adapt to asymptotics with unknown basis functions, first estimating the φ_j themselves, though the current proof assumes the basis is known a priori.","Since the estimators are closed-form, the theory could combine with exact arithmetic to certify constants to arbitrary precision, turning the asymptotic bound into a finite-time guarantee once explicit constants are extracted.","The structure of the error bound suggests the same technique may work for multivariate or multi-scale expansions where the ratio method breaks down."],"forward_implications":["Any asymptotics of the form (2.1) with the exponential-product structure can be numerically inverted with certified rates, complementing the ratio method where only rough estimates were available.","The convergence-rate invariant means one proven expansion yields simultaneous estimates of all parameters in a single vector, rather than separate per-parameter sequences.","The sT-LLSQ bound O(φ_{k−1}/φ_k) = o(1) gives an unconditional, if slow, way to compute the dominant parameter even when the asymptotics order is low.","Higher-order asymptotic knowledge strictly improves or preserves the sLLSQ error bound (η(n;p) = O(η(n;p−1))), so investing in deeper expansions is provably rewarded.","The counterexample shows that sLLSQ's global-minimum value is not a reliable indicator of success; zero residuals can coexist with divergence, shifting practical attention to the ϑ_j conditions."],"fun_headline_variants":["Sliding least squares proves rates for asymptotic constants","Log-linear trick makes asymptotic fitting provable","Certified convergence rates from sliding LS estimates","Asymptotic constants via OLS, with rate guarantees"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire guarantee rests on the assumption that an asymptotic expansion of sufficiently high order, with the exact basis functions φ_j and error scale g_l, has already been proven for the sequence; without those higher-order terms, sLLSQ's bound can diverge.","fun_headline_variants_meta":{"raw":{"variants":["Sliding least squares proves rates for asymptotic constants","Log-linear trick makes asymptotic fitting provable","Certified convergence rates from sliding LS estimates","Asymptotic constants via OLS, with rate guarantees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000292,"raw_usage":{"total_tokens":1545,"prompt_tokens":751,"completion_tokens":794,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":735}},"tokens_in":495,"tokens_out":794,"duration_ms":7142,"temperature":1.0,"reasoning_tokens":735,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T23:53:59.319741+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a known sequence such as a_n = 7 n^{1/3} (5/2)^n (1 + (−1)^n / √n), compute the sLLSQ estimate y*_n = A†b with m = k = 3 and compare ||y*_n − γ|| to the theorem's O(n^{3/2} log n) bound: if the error does not grow like Θ(n^{3/2} log n), or if it converges despite ϑ_j(n) not being o(1), the claimed sufficient conditions would be wrong.","supporting_citations":[],"review_version":1}