{"id":"89d159ff-3901-45a7-b9d1-0cfeadadad00","arxiv_id":"2607.09376","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Recursive formulas compute all Hankel determinants of weighted binary sum-of-digits functions, yielding closed forms such as det = (±)(k+1) at n=⌈2^{k+2}/3⌉ and structured vanishing at doubled roots of unity.","lead":"The paper derives recursive formulas that compute Hankel determinants of weighted binary digit-sum sequences for every size n, and gives closed forms for infinite families of those determinants. This partially solves an open problem of Allouche and Shallit and maps the vanishing loci of the associated polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim—that Theorem 4.2 supplies a complete, effective recursion for every H_w(n) and that the specialization yields the explicit infinite family of Theorem 4.4—is supported by transparent inductive matrix algebra that never leaves the 2-adic structure of the sum-of-digits function. The reader correctly identified the only potential soft spot (the valuation identity under weight shifts) and that spot is handled explicitly in the proofs. Direct low-order verification and recovery of the known period-doubling formulae supply independent consistency checks. Consequently the ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":33193,"tokens_out":463,"duration_ms":5436,"concrete_test":"Independently recompute the first 32 values of H(n,1) by direct determinant evaluation of the Hankel matrices [s_2(i+j)] and compare them with the values obtained by iterating the recursion of Theorem 4.2 (or Prop. 4.3 with t=x=1); agreement to machine precision confirms that the block reductions preserve the claimed determinants.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption (preservation of the 2-adic valuation formula (3) for d_w under the elementary operations of Props. 3.1–3.2 and Thm. 4.2) is carefully controlled: the proofs track exactly when ν_2(i+j+1) changes (only on the antidiagonal of each 2^k-block) and replace the corresponding weight by the shifted sequence w' or the half-weight ½ w_{k+1}. The resulting identities are verified by induction for both G_w and H_w, and the specializations (Thm. 3.4 product formula, Thm. 4.4 closed form for n_k, Cor. 3.10 roots of G(n,t)) are consistent with direct low-order checks and with the known period-doubling determinants of Fokkink–Kraaikamp–Shallit. No internal inconsistency or hidden cancellation that would invalidate the recursion appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies Hankel determinants H_w(n) = det[s_w(i+j)]_{0≤i,j<n} for the weighted binary sum-of-digits function s_w associated to arbitrary complex weights w. It derives a general recursion (Theorem 4.2) that computes every such determinant from smaller ones via auxiliary first-difference determinants. Specializing to constant weights w_j=1 produces closed forms for several infinite families of indices, including the simple evaluation H(⌈2^{k+2}/3⌉)=(-1)^{(k+2)(k+3)/2}(k+1) (Theorem 4.4), partially solving Allouche–Shallit Problem 4. For polynomial weights w_j=t^j the determinants become polynomials H(n,t); the paper determines their degrees, leading coefficients, factorizations and signs, and proves structured vanishing on long intervals about multiples of high powers of 2 when t=2ζ for a root of unity ζ (Theorems 5.4–5.5), while the complementary non-vanishing set is sparse yet infinite (Theorem 5.10). An explicit product formula for the first-difference Hankel determinants G_w(n) is also obtained (Theorem 3.4), recovering and correcting the period-doubling evaluations of Fokkink–Kraaikamp–Shallit.","tokens_in":33443,"tokens_out":893,"duration_ms":19917,"significance":"Exact closed-form evaluations of Hankel determinants for digital sequences remain rare; most prior work addresses only modular behaviour or automaticity. The recursion of Theorem 4.2 together with the infinite family of explicit values in Theorem 4.4 therefore constitutes a genuine advance on the Allouche–Shallit problem and supplies a practical computational tool. The product formula for G_w and the vanishing theorems for doubled roots of unity give precise structural information that is directly relevant to Padé approximation and irrationality-exponent applications. The proofs are self-contained inductive arguments based on block-matrix row/column operations and Laplace expansions; they include an independent sign correction of earlier literature and are consistent with low-order direct checks. These features make the contribution solid and reusable.","major_comments":[],"minor_comments":[{"comment":"Several typographical errors appear in the manuscript text (e.g., “V anishing”, “prodduct formula”, “eGis proved”, missing spaces after periods). A careful proof-reading pass is needed before final production.","section":null},{"comment":"Section 2 introduces a compact block-matrix notation that suppresses indices and writes block heights to the right; while efficient, a short explicit example of the expansion used in the proofs of Propositions 3.2 and Theorem 4.2 would improve readability for non-specialists.","section":null},{"comment":"The base cases of the recursions (n < 5 for Theorem 4.2, small k for Theorems 3.3–3.4) are asserted to hold by direct computation; listing the explicit 2×2, 3×3 and 4×4 matrices (or their determinants) in an appendix or remark would make the induction completely self-contained.","section":null},{"comment":"Conjectures 4.6, 4.7 and 5.7 are clearly labelled as open; it would be helpful to indicate, even briefly, which of them appear most accessible to the same recursive methods already developed.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, technically solid contribution that fits well in a number-theory or combinatorics-on-words journal. No concerns about novelty disclosure or citation pattern. The empty major-comments list reflects that the central inductive arguments are carefully controlled and free of load-bearing gaps."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a clean, self-contained piece of exact determinant calculus for digital sequences. The real advance is Theorem 4.2: a complete recursion that computes every H_w(n) for arbitrary complex weights by reducing to smaller instances via block-matrix row/column operations. Specializing to ordinary sum-of-digits immediately yields the simple closed form H(⌈2^{k+2}/3⌉) = (-1)^{(k+2)(k+3)/2}(k+1), which is an infinite explicit family and a genuine partial solution to the Allouche–Shallit problem they cite. The companion product formula for the first-difference determinants G_w(n) (Theorem 3.4) is equally useful; it recovers and corrects the sign in the Fokkink–Kraaikamp–Shallit period-doubling result and gives a full root description for the polynomial case w_j = t^j.\n\nThe vanishing analysis for t = 2ζ (Theorems 5.4–5.5, 5.10) is carefully done: long structured zero runs around multiples of high powers of 2, density-zero complement, and an infinite complementary non-vanishing set. All central proofs are transparent inductions on binary length or on n, with base cases checked by hand and the 2-adic valuation identity tracked precisely under the weight shifts. No circularity, no free parameters, no data fitting.\n\nSoft spots are minor and clearly labelled. There is still no single closed form for arbitrary n (they never claim one). Several natural conjectures (monotonicity of |H(n,1)|, automaticity mod 2^k, exact zero set for 2ζ) are left open; that is honest rather than a flaw. The matrix reductions are a bit technical, but they are fully written out and consistent with low-order checks.\n\nThis is for people who work on automatic sequences, Hankel determinants, or digital functions. Anyone who has stared at OEIS A056886 or the Allouche–Shallit problem list will get immediate value. The math is solid enough that a serious editor should send it to referees without hesitation. I would cite the recursion and the n_k formula myself.","headline":"Solid exact-evaluation paper: a usable recursion for weighted sum-of-digits Hankel determinants plus an infinite closed-form family that partially answers Allouche–Shallit Problem 4.","tokens_in":34018,"tokens_out":565,"would_cite":true,"duration_ms":7037,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B85","05A15","11B83","15B05"],"pacs":[],"model":"grok-4.5","headline":"A recursion computes every Hankel determinant of weighted binary digit sums, and for ordinary sum-of-digits it yields the closed form H(⌈2^{k+2}/3⌉)=(±)(k+1).","keywords":["Hankel determinants","sum of digits","weighted binary expansions","period-doubling sequence","roots of unity","automatic sequences"],"falsifier":"Compute the Hankel determinants of ordinary binary sum-of-digits at the indices n_k=⌈2^{k+2}/3⌉ for several large k by direct linear algebra and check whether they equal exactly (-1)^{(k+2)(k+3)/2}(k+1).","tokens_in":34152,"feed_emoji":"∑","tokens_out":709,"duration_ms":5933,"temperature":0.7,"pith_summary":"Hankel determinants built from the binary sum-of-digits function (and its weighted variants) are hard to evaluate exactly; most prior work only controls them modulo an integer. This paper produces a complete recursive scheme that reduces every such determinant of size n to determinants of strictly smaller size, using only the 2-adic structure of the first differences of the sum-of-digits. Specializing the weights to 1 recovers an infinite explicit family, most strikingly H(⌈2^{k+2}/3⌉)=(-1)^{(k+2)(k+3)/2}(k+1), which partially answers a problem of Allouche and Shallit. The same recursion yields product formulas for the first-difference determinants (recovering and correcting the period-doubling case) and a detailed vanishing theory for the polynomial family with weights t^j, showing that when t=2ζ for a root of unity the determinants vanish on long structured blocks of indices while the non-vanishing set remains infinite but sparse.","feed_headline":"Digit-sum Hankel determinants now have a full recursion","feed_subtitle":"Ordinary sum-of-digits yields the closed form H(⌈2^{k+2}/3⌉)=(±)(k+1)","key_machinery":"The block-matrix recursion of Theorem 4.2 (and its first-difference companion Proposition 3.2), driven by the 2-adic valuation formula for the first difference d_w(m) and elementary row/column operations that preserve the Hankel structure under weight shifts.","core_discovery":"There is a uniform recursion (Theorem 4.2) that expresses every Hankel determinant H_w(n) of a weighted binary sum-of-digits sequence in terms of two smaller determinants of the same type (or of the first-difference type). For ordinary sum-of-digits the recursion collapses on the indices n_k=⌈2^{k+2}/3⌉ to the elementary closed form (-1)^{(k+2)(k+3)/2}(k+1), giving an infinite explicit family.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Uniform recursion for every weighted digit-sum Hankel determinant","Binary sum-of-digits Hankels equal ±(k+1) at ceil(2^{k+2}/3)","Infinite closed-form family for ordinary digit-sum Hankel determinants","Recursion reduces H_w(n) to two smaller weighted digit-sum determinants","First-difference digit-sum Hankels admit an explicit product formula"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The recursion assumes that the 2-adic formula for the first difference of the weighted sum-of-digits survives the elementary matrix operations and the weight shifts that appear at each reduction step.","fun_headline_variants_meta":{"raw":{"variants":["Uniform recursion for every weighted digit-sum Hankel determinant","Binary sum-of-digits Hankels equal ±(k+1) at ceil(2^{k+2}/3)","Infinite closed-form family for ordinary digit-sum Hankel determinants","Recursion reduces H_w(n) to two smaller weighted digit-sum determinants","First-difference digit-sum Hankels admit an explicit product formula"]},"model":"grok-4.5","effort":"low","cost_usd":0.005858,"raw_usage":{"total_tokens":1661,"prompt_tokens":931,"num_sources_used":0,"completion_tokens":107,"cost_in_usd_ticks":58580000,"prompt_tokens_details":{"text_tokens":931,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":623,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":931,"tokens_out":107,"duration_ms":6100,"temperature":1.0,"reasoning_tokens":623,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T03:27:21.076010+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the Hankel determinants of ordinary binary sum-of-digits at the indices n_k=⌈2^{k+2}/3⌉ for several large k by direct linear algebra and check whether they equal exactly (-1)^{(k+2)(k+3)/2}(k+1).","supporting_citations":[],"review_version":1}