{"id":"d27ced31-c3f3-43dc-88bf-0a624688ab91","arxiv_id":"2505.20397","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An algorithm computes all rational linear relations among 1-periods, deciding equality and transcendence.","lead":"The authors give a step-by-step procedure that decides when two integrals of algebraic functions, called 1-periods, are equal or transcendental. This settles a special case of a famous open problem about periods posed by Kontsevich and Zagier.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Supersaturation step (2.0.8) lacks an effective construction of Baker-component splittings: Lemma 8.7.54 substitutes a numerical block-form heuristic for the promised explicit correspondence, so the main theorem's termination and correctness are not unconditional.","rationale":"I read the paper as a serious attempt to make Huber-Wüstholz effective; much of §3–§7 is genuine algorithmic content. The main theorem, however, is only as strong as the supersaturation subroutine. The reader's verdict already set CONDITIONAL; my stress-test isolates the exact missing piece in Lemma 8.7.54 rather than objecting to the global strategy. I do not see evidence of a false statement, but the proof as written leaves an essential effective construction to an 'in practice' numerical step. Therefore an unconditional ACCEPT would be premature, and the conditional verdict stands.","tokens_in":62624,"tokens_out":15839,"duration_ms":163385,"concrete_test":"Work out Lemma 8.7.54 in the minimal nontrivial case: C an elliptic curve, E a point, f a function with div(f) in L_E and [div(f)]χ = 0, and compute the claimed H1([Z → 0]) ↪ H1(J^χ_{C,ψ}) via the proposed period-matrix block form. Verify that the output is an exact Z-basis and is compatible with the pullback by ψ; if the block form yields only approximate or incompatible data, the splitting is not effective.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Algorithm 2.0.1's correctness and termination. Step (2.0.8) requires an explicit isogeny direct sum decomposition M^ss ~ M_B ⊕ M_sat with a saturated push-pull factor and an explicit E-action on H1(M_sat). This is where the paper is least secure. Lemma 8.7.54 is the key place where a Baker component [K^χ_ψ → 0] is split off from J^χ_{C,ψ}. Its proof explicitly says that making the split explicit by correspondences 'is possible but cumbersome' and instead proposes to put the period matrix into block form using effective Baker periods (8.8.60). No algorithm or proof is given for this block-form step: it is not shown how to recover an exact integral basis for H1([Z → 0]) from numerical period computations, nor that the resulting split is compatible with the lattice pullback ψ. This is not a matter of disagreement with consensus; it is an internal completeness gap in the only place where the splitting is established. The same gap propagates: §9.3.52's commutativity of Figure 1, used to compute the E-action on H1(J^χ_{C,ψ}), relies on the splitting maps 4 and 8 being explicit at the level of H1; if Lemma 8.7.54 is only heuristic, then Step (2.0.9) cannot compute the expected period relations of the saturated factor. The theorem is therefore conditional on a missing effective construction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents Algorithm 2.0.1, which takes representations of 1-periods as integrals of rational differentials on algebraic curves and computes the space of Q-linear relations among them, thereby deciding transcendence and equality of 1-periods. The algorithm constructs the mixed Hodge structure of punctured marked curves, forms the associated Jacobian motive, supersaturates it, and then applies a description of period relations of saturated motives based on Huber–Wüstholz. The paper also gives an algorithmic classification of autonomous first-order differential equations. The main theorem is stated unconditionally but relies on several subroutines whose effectiveness is only partially justified.","tokens_in":62948,"tokens_out":5969,"duration_ms":61266,"significance":"If the main theorem holds, this is a major advance: it would be the first effective resolution of the Kontsevich–Zagier equality problem for 1-periods, including transcendence decisions. The paper contains a substantial amount of useful effective machinery: bases for algebraic de Rham cohomology, Betti homology via embedded graphs, push-pull Jacobian motives, explicit correspondences, and a reduction to divisor arithmetic on curves. It is also commendable that the paper cites deep external results (Huber–Wüstholz, Baker, Lombardo, CMSV) and is transparent about the places where arguments are only sketched. The additional classification of first-order autonomous differential equations is a nice application of the developed tools.","major_comments":[{"comment":"The splitting of the Baker component [K^χ_ψ → 0] from J^χ_{C,ψ} is not made effective. The proof of Lemma 8.7.54 explicitly says that making the split explicit by correspondences 'is possible but cumbersome' and instead proposes to put the period matrix into block form using effective Baker periods (8.8.60). No algorithm is given to recover an exact integral basis for H1([Z → 0]) from the numerical block form, nor is it shown that the resulting split is compatible with the lattice pullback ψ. This is load-bearing: the commutativity of Figure 1 in §9.3.52, used to compute the E-action on H1(J^χ_{C,ψ}), requires arrows 4 and 8 to be explicit maps on H1. Since these arrows are constructed using the non-effective splitting of Lemma 8.7.54, Step (2.0.9) of Algorithm 2.0.1 cannot be executed as written. The main theorem is therefore conditional on an effective construction of these splittings.","section":"§8.7, Lemma 8.7.54; §9.3.52"},{"comment":"The termination of the Abel–Jacobi kernel computation is not rigorously established. The algorithm relies on LLL to 'guess' the kernel from numerical period approximations and on lower bounds for the rank of the Néron–Tate height matrix via certified nonzero minors. The text asserts that 'LLL will eventually find a lattice of full rank in KD' but gives no quantitative convergence argument (e.g., a lower bound on the gap between the period lattice and the kernel). Since KD is a direct input to the Baker splitting in (8.4.39) and (8.6.46), and hence to the supersaturation process, this is another load-bearing point that needs a proof or a precise reference.","section":"§8.3, Algorithm (8.3.32)"}],"minor_comments":[{"comment":"Typo: 'On could imagine' should be 'One could imagine'.","section":"§9.2.20"},{"comment":"Typo: 'determening' should be 'determining'.","section":"§1.1.17"},{"comment":"The equation 'h0(Ω1C((−d + 1)p) = 0' is missing a closing parenthesis; it should read 'h0(Ω1C((−d + 1)p)) = 0'.","section":"§3.5.37"},{"comment":"The asterisk in the block matrix (5.8.62.1) is not defined; a sentence explaining the entries would help.","section":"§5.8.62"},{"comment":"In (8.3.31.1), the notation '(F^1 H^1_AdR(C,D))^∨ = (F^1 H^1_AdR(C))^∨' seems to identify a quotient with a subspace; the intended identification should be stated.","section":"§8.3.31"},{"comment":"The reference to a 'day-and-night algorithm analogous to the one given in §8.3' would benefit from a precise pointer to the termination argument.","section":"§8.8.61"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and contains many useful effective constructions, but the central theorem is not yet unconditional because of the gap in Lemma 8.7.54, which the authors themselves acknowledge. The paper should either provide a fully effective splitting construction or explicitly state the main theorem as conditional on an effective Baker-splitting procedure. Similarly, the termination of the LLL-based kernel computation should be made rigorous or stated as a hypothesis. These are localized but load-bearing issues, so major revision seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the bottom line: this is a serious, substantial paper. If the main theorem goes through, it's the first effective algorithm to compute all Q-linear relations among 1-periods, which decides transcendence and equality and resolves Kontsevich–Zagier in that case. The ODE classification is a nice bonus. The authors do not just cite Huber–Wüstholz; they build an effective theory of mixed Hodge structures for punctured marked curves, replacing motivic abstraction with divisor arithmetic on curves. Sections 3 and 4, on effective de Rham cohomology and Betti homology via rectilinear embedded graphs, are careful and genuinely useful. The treatment of endomorphism algebras via CMSV18 and Lombardo is appropriate, and the reliance on deep external theorems is stated plainly.\n\nThe soft spots are real, though. The stress-test note is on target: Lemma 8.7.54 is supposed to make the splitting of Baker components explicit, but its proof says the correspondence-based split is 'cumbersome' and instead proposes to put the period matrix into block form using effective Baker periods. No algorithm or proof is given for that block-form step. In particular, it is not shown how to recover an exact integral basis for H1([Z→0]) from numerical period computations, nor that the resulting split is compatible with the lattice pullback ψ. That gap propagates: the commutativity of Figure 1 in §9.3.52, used to compute the E-action on H1(J^χ_{C,ψ}), depends on those splitting maps being explicit at the level of H1. So Step (2.0.9) of the main algorithm cannot be executed as written. The termination argument for the Abel–Jacobi kernel in §8.3 is also informal: it relies on LLL eventually finding a full-rank sublattice and on certified numerical bounds that 'eventually stop containing 0.' That is plausible but not a proof.\n\nThese are load-bearing omissions, not cosmetic ones. The authors flag supersaturation as a bottleneck (2.0.14), but the paper currently asserts effectivity where it has only a heuristic. A serious referee should engage, but should insist on a rigorous treatment of Lemma 8.7.54 and the block-form reduction, or a clear statement that the main theorem is conditional on that step.\n\nWho this is for: researchers in arithmetic geometry, transcendental number theory, and computational algebraic geometry. The framework in Sections 3–6 is worth studying even if the main theorem needs work. I would send it to peer review, not desk reject, with a request for major revision or at least an explicit caveat.","headline":"First effective algorithm for 1-period relations, but the main theorem is conditional on a real gap in the supersaturation step.","tokens_in":63443,"tokens_out":3187,"would_cite":true,"duration_ms":31214,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14Q05","14C30","14F40","14H40"],"pacs":[],"model":"deepseek-v4-flash","headline":"An algorithm now computes every rational linear relation among 1-periods, deciding transcendence and equality.","keywords":["1-periods","transcendence","period relations","mixed Hodge structures","Jacobian motives","computational algebraic geometry","autonomous differential equations","period equality problem"],"falsifier":"Run the algorithm on the 1-periods log 2 = ∫$_1^{2}$ dx/x and log 4 = ∫$_1^{4}$ dx/x; correctness requires it to return exactly the one-dimensional relation space spanned by (2,−1), and a different answer, a false extra relation, or failure to terminate would refute the central claim.","tokens_in":62451,"feed_emoji":"","tokens_out":6728,"duration_ms":66275,"temperature":0.7,"pith_summary":"The paper presents an algorithm that, given finitely many 1-periods—integrals of univariate algebraic functions over algebraic data—computes a Q-basis for the space of Q-linear relations among them. From that basis it decides whether a single 1-period is transcendental and whether two 1-periods are equal, resolving the period-equality decision problem in the univariate case. The approach converts a theorem stating that all relations among 1-periods arise from 1-motives into explicit arithmetic on divisors on curves, using effective mixed Hodge structures and a saturation procedure. The same machinery yields an algorithmic classification of first-order autonomous differential equations.","feed_headline":"Algorithm computes every rational relation among 1-periods","feed_subtitle":"It can test transcendence and equality, resolving the period-equality decision problem for univariate periods.","key_machinery":"The load-bearing object is the Jacobian motive J^D_{C,E} of a punctured marked curve (C\\D,E): a toric extension of the Jacobian of C marked by divisors supported on E, whose mixed Hodge structure is H1(C\\D,E). The paper computes its Betti homology by rectilinear embedded graphs and its de Rham cohomology by second-kind differentials, then uses push-pull constructions to realise arbitrary 1-period motives as push-pulls of composites. The decisive identity is the theorem that for a saturated 1-motive, the space of period relations equals the expected relations generated by endomorphisms and trivial lattice and toric relations. The supersaturation procedure decomposes any Jacobian motive into a Baker motive—one with trivial abelian core, whose periods are combinations of 1, 2πi, and logarithms of algebraic numbers—and a saturated motive, transferring period relations by linear algebra.","core_discovery":"The central claim is the Main Theorem: given representations (γ_i, ω_i) of 1-periods α_i, Algorithm 2.0.1 computes a Q-basis for the space rel_Q(α_1,...,α_k) of Q-linear relations. In particular, it decides whether a given 1-period is transcendental and whether two 1-periods are equal. The proof makes the dimension estimate for saturated 1-motives effective: it constructs the mixed Hodge structure H1(C\\D,E) of punctured marked curves, builds the associated Jacobian motive, supersaturates it into a Baker motive plus a saturated motive, and reads off period relations from endomorphisms and trivial relations. This gives the first effective resolution of the period-equality problem for 1-periods.","pith_inferences":["If the main theorem is correct, transcendence tests for 1-periods no longer require explicit effective separation constants; the algorithm returns a finite exact description of all relations.","The refined-type routine for differentials suggests a natural test bed: implement the genus-zero and genus-one cases first, where known relations among logarithms and elliptic integrals provide immediate sanity checks.","The same effective mixed-Hodge-structure machinery could be pointed at higher-weight periods or at computing period lattices of families of curves, since the paper's subroutines are built for general punctured marked curves."],"forward_implications":["Equality of any two 1-periods is decidable by a single uniform algorithm, rather than by case-by-case transcendence bounds.","Transcendence of a given 1-period can be certified: the algorithm returns a standard algebraic representation when the period is algebraic.","The Q-vector space of 1-periods is effective, meaning addition, scalar multiplication, and equality are all computable.","First-order autonomous differential equations P(u,u′)=0 over Q are algorithmically classified, refining the classical trichotomy by a toric and abelian type (a,b).","The effective mixed-Hodge-structure and correspondence subroutines provide reusable computational tools for studying algebraic curves and their periods."],"supporting_citations":[{"why":"Supplies the dimension estimate for saturated 1-motives that the algorithm turns into an explicit description of period relations.","marker":"[HW22]"},{"why":"Poses the period-equality decision problem that the main theorem resolves for 1-periods.","marker":"[KZ01]"},{"why":"Provides the analytic subgroup theorem underlying the subgroup theorem for 1-motives and the refined classification.","marker":"[Wüs89]"},{"why":"Provides the practical correspondence-based computation of the endomorphism algebra End(JC) used in supersaturation.","marker":"[CMSV18]"},{"why":"Gives the unconditional endomorphism-algebra algorithm that makes termination theoretical.","marker":"[Lom18]"},{"why":"Provides the linear-independence result for logarithms used to compute Baker-motive period relations.","marker":"[Bak66]"},{"why":"Supplies lattice reduction used to guess kernels in the Abel–Jacobi computation.","marker":"[LLL82]"},{"why":"Supplies the Riemann–Roch space computation that powers divisor arithmetic on curves.","marker":"[Hes02]"}],"fun_headline_variants":["Algorithm decides equality of 1-periods","New algorithm resolves period equality for univariate integrals","Compute all linear relations among 1-periods","Transcendence test for 1-periods solved","Kontsevich-Zagier equality problem solved for 1-periods"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithm is only guaranteed to stop if the internal routine that enlarges the motive until all endomorphisms are visible—a step relying on numerical approximation and lattice reduction—always terminates with exact output.","fun_headline_variants_meta":{"raw":{"variants":["Algorithm decides equality of 1-periods","New algorithm resolves period equality for univariate integrals","Compute all linear relations among 1-periods","Transcendence test for 1-periods solved","Kontsevich-Zagier equality problem solved for 1-periods"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00071,"raw_usage":{"total_tokens":3165,"prompt_tokens":880,"completion_tokens":2285,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":2221}},"tokens_in":496,"tokens_out":2285,"duration_ms":18116,"temperature":1.0,"reasoning_tokens":2221,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:54:52.019104+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the algorithm on the 1-periods log 2 = ∫$_1^{2}$ dx/x and log 4 = ∫$_1^{4}$ dx/x; correctness requires it to return exactly the one-dimensional relation space spanned by (2,−1), and a different answer, a false extra relation, or failure to terminate would refute the central claim.","supporting_citations":[],"review_version":1}