{"id":"93c81f6e-f3d0-4fe5-8358-1417db66aa30","arxiv_id":"2507.12914","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For square and rectangular tori, the paper gives explicit non-holomorphic minimal tori in R^4 with total curvature -8π and one end, and proves uniqueness on the square torus.","lead":"This paper constructs explicit minimal tori, the mathematical models of soap films, in four-dimensional space for square and rectangular tori, and proves no such surface exists for one exceptional torus. It gives a complete classification on the square torus and explicit examples on all rectangular tori, extending the classical Chen-Gackstatter torus to four dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rectangular-torus existence proof rests on a non-reproducible Sagemath sign check in §7.7; without interval-arithmetic certification the theorem is not fully established.","rationale":"The reader's weakest_assumption identifies the Sagemath numerical check in §7.7 as a main reason for the conditional verdict, and I agree that this is the most load-bearing unverified step: it is the only place where the proof of A < 0 for all R > 1 is not backed by explicit analytic estimates. However, I do not share the reader's emphasis on Assumption 4 as a logical weakness. Restricting the search to real unknowns with the symmetry t = w, u = −y, v = −z does not weaken an existence theorem: if the restricted system has a solution, then a solution to the original system exists. The symmetry ansatz is only problematic if it silently prevents the construction from covering all rectangular tori, but Theorem 5 asserts existence, not completeness, so a special solution suffices. The genuine gap is instead that the restricted system's solvability is established by an unreported numerical computation, and the derived explicit solution also requires positivity of t^2 which is not separately justified. The square-torus classification and equianharmonic nonexistence are algebraic and do not depend on this numerical step; they appear solid. Thus the conditional verdict is appropriate: the paper should either supply the Sagemath script or replace the middle-interval estimates by certified interval arithmetic, and should explicitly verify all positivity conditions needed for the §7.8 formulas to define real minimal tori.","tokens_in":16030,"tokens_out":4667,"duration_ms":56729,"concrete_test":"Recompute the §7.7 estimates with certified interval arithmetic, e.g. using SageMath's RealIntervalField or arb, on the ten subintervals [1 + k/100, 1 + (k+1)/100] for k = 5,...,14. For each subinterval, evaluate the three terms A, B, C in (85)-(87) using the q-series (51)-(52), keeping rigorous upper bounds, and verify that A + B + C < 0 with margin. Also certify that T(R) > 0 and c(R) − u^2(R) > 0 on each subinterval, so that the §7.8 formulas produce real t. If all certified bounds are negative, the numerical-gap objection is resolved; if any upper bound is nonnegative, Theorem 5 is unproved in the current form.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5 claims existence for every rectangular torus, and the decisive step is the proof that the left-hand side of (65), equivalently the quantity in (78), is negative for all R > 1. The cases R ≥ 1.15 and 1 < R ≤ 1.05 are handled analytically in §7.5 and §7.6, but the interval 1.05 ≤ R ≤ 1.15 is covered in §7.7 only by ten numerical estimates obtained with Sagemath. The paper does not ship the code, does not give the intermediate values of A, B, C, and does not use interval arithmetic or certified rounding. Since the theorem's existence conclusion for rectangular tori depends on these ten numbers being strict upper bounds for (78) on each subinterval, a small bug or rounding error there would invalidate the existence proof exactly in the range where the other analytic estimates do not reach. This is not merely a stylistic weakness: the expression in (78) is a rational function of q-series with explicit formulas (51)-(52), so the numerical step is replaceable by rigorous interval bounds, but as written the proof is not independently checkable. A secondary gap is that the formulas in §7.8 define t^2 = (c − u^2)T(R); positivity of t^2 is not explicitly proved for all R > 1, although it is needed for Assumption 4 to yield real solutions. This should be checked together with the main sign condition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies complete proper non-holomorphic minimal immersions of a punctured 2-torus into R^4 with one end and total curvature -8\\pi. It translates the geometric problem into a system of algebraic equations in the Weierstrass data (Propositions 4 and 5), then treats three conformal classes: the square torus, where it claims a complete classification and an explicit one-complex-parameter family generalizing the Chen--Gackstatter torus (Theorem 3); the equianharmonic torus, where it proves nonexistence (Proposition 6); and rectangular tori, where it claims existence for every R>1 with explicit rational formulas (Theorem 5). The paper also introduces tools involving Gauss maps, links, braids, and writhe at infinity in Section 2, and uses them to analyze the ends and conclude non-embeddedness of the square family. The rectangular theorem is proved under an additional symmetry ansatz, Assumption 4, and relies in part on a Sagemath numerical check in Section 7.7.","tokens_in":16312,"tokens_out":8266,"duration_ms":95906,"significance":"If the square-torus uniqueness and the equianharmonic nonexistence hold, they are strong and explicit results in the still-underdeveloped theory of minimal tori in R^4, and the explicit deformation family of the Chen--Gackstatter torus is a useful addition. The algebraic reduction of the period and conformality conditions to the 10-equation system is a clear methodological contribution, and the equianharmonic proof in Section 6 is short and rigorous. The rectangular existence theorem would also be significant, but as written it is not fully established: the decisive sign check on [1.05,1.15] is delegated to unreported numerical evaluations, and the positivity of t^2 required for real solutions is not proved. The paper does not ship code or certified interval arithmetic, so the numerical part is not independently checkable. I regard the main results as plausible and likely correct, but the rectangular theorem needs additional work before it can be accepted as stated.","major_comments":[{"comment":"The proof that the inequality (65), equivalently the negativity of (78), holds for all R>1 is not complete. The cases R≥1.15 and 1<R≤1.05 are handled analytically in §7.5 and §7.6, but the remaining interval 1.05≤R≤1.15 is treated only by ten asserted Sagemath estimates of A+B+C. The manuscript gives neither the code nor the intermediate values of A, B, C, nor any interval-arithmetic or certified-rounding bounds. Since Theorem 5 asserts existence for every rectangular torus, a single incorrect or non-strict estimate on any of the ten subintervals would invalidate the existence conclusion exactly where the analytic estimates do not reach. Because the expression in (78) is an explicit rational function of q-series with known formulas (51)-(52), this numerical step is replaceable by rigorous interval bounds; as written, however, the proof is not independently checkable.","section":"§7.7, Theorem 5"},{"comment":"After the formulas for u^2 and c are obtained, the paper sets t^2 = (c - u^2)T(R) in (55). For Assumption 4 to yield real solutions, t^2 must be nonnegative. Lemma 2 only rules out c = u^2; it does not establish c - u^2 > 0 for all R>1. The sample value R=2 is not a substitute for a general argument. Since t appears in the explicit map (89), the existence of a real t is part of the construction, and the missing positivity check is therefore load-bearing for Theorem 5.","section":"§7.8, Assumption 4"},{"comment":"The uniqueness proof for the square torus is not fully self-contained as written. It depends on Assumption 3, whose 'without loss of generality' status is not demonstrated: the remark after Assumption 3 reduces the case u℘ + \\bar v\\bar{℘} to R^3, but the excluded case u℘' + \\bar v\\bar{℘'} is merely ruled out by assumption, not by an argument. In addition, after deriving that s=vz=uy=0 (or the corresponding vanishing statements), the text concludes v=z=0 using that t and w are not zero, but this nonvanishing has not been proved at that point; it would require, for instance, a justification that a=-4tw is nonzero in the first equation of (40). Without these steps, the claimed completeness of the classification for all non-holomorphic minimal square tori is not established.","section":"§5.3, Theorem 3(2)"}],"minor_comments":[{"comment":"There are several typos and stylistic slips that should be corrected in revision: 'finit union' in §2, 'prelimineries' in §1.3, 'Exemple' in §7.9, '4 D Chen-Gacksatter' in §8, and inconsistent capitalization of 'Sagemath' (should be 'SageMath').","section":"Throughout"},{"comment":"The ten numerical estimates A+B+C are presented as a bare list; a table with the interval endpoints and the individual bounds for A, B, and C would greatly improve verifiability, even before adding certified arithmetic.","section":"§7.7"},{"comment":"The displayed formula for the second coordinate contains the term 't℘+ t℘', which appears to be a typo; it should presumably be t℘ + t\\bar{℘} or an equivalent expression. The notation for antiderivatives involving \\bar{℘} is also used without a clear convention and should be spelled out.","section":"§7.8, Eq. (88)"},{"comment":"The analytic bounds in Lemmas 8 and 9 quote numerical constants such as g2(1)≈ and g3(1)≈ without listing their exact expressions or the elementary estimates used. Since these numbers enter the conclusion (65) ≤ -0.5, they should be stated explicitly.","section":"§7.6"}],"recommendation":"major_revision","confidential_remarks":"The square-torus and equianharmonic results appear to be the strongest and most convincing parts of the paper. The rectangular theorem is the main risk: it depends on an extra symmetry ansatz, a non-reproducible Sagemath sign check, and an unproved positivity condition. I would recommend requesting a certified or at least fully documented numerical supplement, and asking the authors to state Theorem 5 explicitly as conditional on Assumption 4 if the numerical part cannot be made rigorous. There is no apparent novelty or attribution concern; the issue is solely the completeness of the existence proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper earns a careful read for the square-torus result alone. Theorem 3 gives an explicit one-parameter family of non-holomorphic minimal tori in R^4 with total curvature -8π and one end, and proves these are the only ones on the square torus, reducing the problem to a 10-equation system and solving it. That is a genuine advance beyond Xie-Ma's local existence, and the algebra in §5 is coherent. The equianharmonic non-existence (Prop 6) is a short clean argument; no complaints there.\n\nThe rectangular-torus theorem is the conditional piece, and the stress-test note lands. The existence claim for every rectangular torus rests on the negativity of (78) over 1.05 ≤ R ≤ 1.15, and §7.7 covers that with ten Sagemath estimates for which no code or interval arithmetic is shipped. Since (78) is a rational function of explicit q-series, this gap is formally fixable, but as written the proof of Theorem 5 is not independently checkable at its decisive step. The paper does flag Assumption 4 as an extra ansatz rather than a WLOG normalization, and it also flags the unresolved Type (I) case, so the authors are not hiding the limitations. The secondary point about t^2 positivity is real but minor; if the sign of (78) is certified, t^2 likely follows from the same numerics, but it should be stated.\n\nCitation pattern looks reasonable: Chen-Gackstatter, Lopez, Ma-Xie, Eells-Salamon, etc. Self-citation is not an issue here; the new results are not restatements of earlier work.\n\nWho is this for? Anyone working on explicit minimal surfaces in R^4, the Gauss-map/period-system approach, or the Chen-Gackstatter family. It deserves a serious referee. My recommendation: send it out, but instruct the referee to require either the Sagemath worksheet or rigorous interval bounds for §7.7 before the rectangular theorem is accepted as fully proved. The square-torus theorem should remain acceptable regardless.","headline":"The square-torus classification and equianharmonic no-go are solid; the rectangular existence theorem is real but its Sagemath sign check needs certification before the main claim rests on it.","tokens_in":16833,"tokens_out":1622,"would_cite":true,"duration_ms":17982,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53A10","53C42"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs explicit non-holomorphic minimal tori in $\\mathbb{R}^4$ for every rectangular torus and proves a unique family for the square torus.","keywords":["minimal surfaces in R4","Chen-Gackstatter torus","total curvature -8π","Weierstrass elliptic functions","square torus","rectangular torus","Gauss maps","braid at infinity"],"falsifier":"Evaluate the left-hand side of inequality (65) with rigorous interval arithmetic for all R>1, particularly on [1.05,1.15]; if it is nonnegative at any R, the theorem's rectangular claim fails under its own ansatz. Alternatively, find a rectangular torus where the explicit formulas of Section 7.8 fail to satisfy the original conformality and period equations.","tokens_in":15755,"feed_emoji":"🧮","tokens_out":8337,"duration_ms":89610,"temperature":0.7,"pith_summary":"The paper's goal is a classification and existence statement for complete proper minimal tori in $\\mathbb{R}^4$ with one puncture, one end, and total curvature $-8\\pi$. It turns the geometric problem into a finite algebraic system in eleven real unknowns built from the Weierstrass $\\wp$-function, then solves the system for three families of tori: the square torus, the rectangular tori, and the equianharmonic torus. The results give a complete answer on the square torus (a unique family generalizing the Chen-Gackstatter torus), explicit existence on every rectangular torus, and a nonexistence proof on the equianharmonic torus. A sympathetic reader would care because this is the four-dimensional version of a problem that in $\\mathbb{R}^3$ has exactly one solution.","feed_headline":"Explicit minimal tori found for every rectangular torus in R4","feed_subtitle":"Solving the 10-equation system yields one-ended non-holomorphic tori; the square torus has a unique family.","key_machinery":"The machinery is the Weierstrass $\\wp$-function representation of a torus end: four meromorphic functions $e',f',g',h'$ with a common pole at the removed point, conformality enforced by $e'f'+g'h'=0$, and periodicity enforced by two period equations. Because the pole order is fixed by the total curvature, the functions take the form of $\\wp^2,\\wp',\\wp,1$ combinations, and the whole problem becomes a real system of ten quadratic or linear equations in eleven unknowns whose coefficients are $\\tau,g_2,g_3,\\eta_1$. For the square torus the system collapses to a unique family; for rectangular tori an extra symmetry ansatz reduces it to one linear equation in $u^2$, and the sign of a rational expression $A(g_2,g_3,\\eta,R)$ decides existence.","core_discovery":"On its own terms, the paper's central discovery is that the one-ended, total-curvature $-8\\pi$ minimal torus problem in $\\mathbb{R}^4$ is algebraically tractable: it is equivalent to a finite system of real equations, and in the cases studied the system has explicit answers. For the square torus $\\tau=i$, Theorem 3 gives a complete classification: the maps (30), parameterized by $\\lambda\\in\\mathbb{C}^*$, are the only non-holomorphic proper minimal immersions with one end and total curvature $-8\\pi$, and they reduce to the Chen-Gackstatter torus in $\\mathbb{R}^3$ exactly when $|\\lambda|=1$. For every rectangular torus $\\tau=Ri$, Theorem 5 constructs a non-holomorphic minimal torus $T_R$ whose coordinates are rational functions of $R,g_2,g_3,\\eta_1$. The equianharmonic torus is excluded by Proposition 6.","pith_inferences":["Editorial inference: if the rectangular family is continuous in $R$, then the square torus is not isolated: tori with $R$ close to 1 produce one-ended, total-curvature $-8\\pi$ minimal tori near the 3D Chen-Gackstatter torus, suggesting a one-dimensional family of conformal types.","Editorial inference: the symmetry Assumption 4 is a genuine restriction, so the same 10-equation system may have additional rectangular solutions with $t\\neq w$ or complex parameters; a numerical search could test this.","Editorial inference: the writhe-at-infinity invariant computed for square tori gives a blueprint for testing embeddedness of the rectangular tori; the paper's final Question could be settled by computing $w_\\infty$ for the rectangular ends.","Editorial inference: if a torus satisfying the Type (I) condition (33) exists, it would lie outside the Type (II) classification, so the current 'no equianharmonic' and uniqueness results would not cover the full problem."],"forward_implications":["If Theorem 3 is correct, the square torus's solution set is completely described by the one-complex-parameter family (30), and the 3D Chen-Gackstatter torus is exactly the $|\\lambda|=1$ slice.","If Theorem 5 is correct, every rectangular torus $T_{Ri}$ admits an explicit non-holomorphic minimal torus with one end and total curvature $-8\\pi$, so the Main Problem has a positive answer on a one-parameter family of conformal types.","If Proposition 6 is correct, the equianharmonic torus is an obstruction example: total curvature $-8\\pi$ and one end are not sufficient for existence.","The paper's integral formula (Theorem 2) and braid-at-infinity computation imply that the 4D Chen-Gackstatter square tori with $|\\lambda|\\neq1$ are not embedded.","For every solution, the end has order $N=3$ by the Jorge-Meeks formula, so the end behavior is fixed: $z\\mapsto(z^3+o(|z^3|),o(|z^3|))$."],"supporting_citations":[{"why":"Provides the 3D Chen-Gackstatter torus and the constant A used in the square-torus family; it is the object being generalized.","marker":"[2]"},{"why":"Establishes local existence of such surfaces near the square torus by the implicit function theorem, giving the deformation context.","marker":"[10]"},{"why":"Supplies the Jorge-Meeks formula relating total curvature, Euler characteristic, and end orders; fixes N=3 and total curvature -8\\pi.","marker":"[7]"},{"why":"Gives the twistor definition of Gauss maps used to show the constructed tori are not complex.","marker":"[4]"},{"why":"Provides the classical Gauss map definition and integral curvature formulas, including the degrees d+ and d-.","marker":"[6]"},{"why":"Supplies the q-series for eta1, g2, g3 used in the rectangular tori estimates.","marker":"[8]"},{"why":"Together with [13], classifies the 3D Chen-Gackstatter torus; used in Assumption 3.","marker":"[9]"},{"why":"Together with [9], supports Theorem 4 on the uniqueness of the Chen-Gackstatter torus in R3.","marker":"[13]"}],"fun_headline_variants":["Explicit minimal tori in R4 for every rectangular torus","Square torus yields unique minimal torus family in R4","Equianharmonic torus has no minimal immersion in R4","10-equation system solves minimal tori problem in R4"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The existence proof for rectangular tori assumes the unknowns are real and obey the symmetry t=w, u=-y, v=-z; if that ansatz does not cover all solutions, or the numerical sign check on the interval 1.05\\le R\\le 1.15 has an error, the theorem is not established for every R.","fun_headline_variants_meta":{"raw":{"variants":["Explicit minimal tori in R4 for every rectangular torus","Square torus yields unique minimal torus family in R4","Equianharmonic torus has no minimal immersion in R4","10-equation system solves minimal tori problem in R4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1452,"prompt_tokens":894,"completion_tokens":558,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":486}},"tokens_in":510,"tokens_out":558,"duration_ms":5846,"temperature":1.0,"reasoning_tokens":486,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:37:42.253937+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the left-hand side of inequality (65) with rigorous interval arithmetic for all R>1, particularly on [1.05,1.15]; if it is nonnegative at any R, the theorem's rectangular claim fails under its own ansatz. Alternatively, find a rectangular torus where the explicit formulas of Section 7.8 fail to satisfy the original conformality and period equations.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the 3D Chen-Gackstatter torus and the constant A used in the square-torus family; it is the object being generalized."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes local existence of such surfaces near the square torus by the implicit function theorem, giving the deformation context."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Jorge-Meeks formula relating total curvature, Euler characteristic, and end orders; fixes N=3 and total curvature -8\\pi."},{"cited_title":"Eells, S","cited_arxiv_id":null,"evidence_quote":"Gives the twistor definition of Gauss maps used to show the constructed tori are not complex."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical Gauss map definition and integral curvature formulas, including the degrees d+ and d-."},{"cited_title":"Lang Elliptic functions , Graduate Texts in Mathematics, Springer, 1987 29","cited_arxiv_id":null,"evidence_quote":"Supplies the q-series for eta1, g2, g3 used in the rectangular tori estimates."},{"cited_title":"Lopez The classification of complete minimal surfaces with total curvature greater than −2π, Trans","cited_arxiv_id":null,"evidence_quote":"Together with [13], classifies the 3D Chen-Gackstatter torus; used in Assumption 3."},{"cited_title":"Weber Period quotient maps of meromorphic 1-forms and minimal surfaces on tori Marc Soret: Universit´ e F","cited_arxiv_id":null,"evidence_quote":"Together with [9], supports Theorem 4 on the uniqueness of the Chen-Gackstatter torus in R3."}],"review_version":1}