{"id":"bad61dd4-8191-4f1e-b10a-5413e09fb9e7","arxiv_id":"2608.05044","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":9.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"An explicit 12-term linear dependence among time-frequency shifts of a Schwartz function is constructed, giving the first counterexample to the HRT conjecture.","lead":"A team of mathematicians and a large language model built a set of 12 waves, shifted in time and frequency, that add up to zero, contradicting the 30-year-old HRT conjecture. If the construction and its computer-certified bound hold up, this settles a central open question in time-frequency analysis.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 1b of Theorem 5.1 expands A0 with unconjugated phase factors, so the certified δ may bound a different matrix field than the one used in the fixed-point argument.","rationale":"The reader identified the certified numerical bound δ<1/3 as the most fragile point, and I agree that Theorem 5.1 is load-bearing. However, the more specific and serious problem is that the analytic reduction in Step 1b of Theorem 5.1 appears to expand the wrong matrix field: the coefficients d_{r,s,ℓ} use η and σ where the definition A0=χ(z)w(z)^* with w(z)=η(x)χ(Tτz) requires η^{-1} and σ^{-1}. This is not a matter of interval arithmetic or script execution; it is an internal inconsistency in the proof as written. If the printed formulas are what the script implements, then the certified bound concerns a comparison with a phase-shifted replacement of A0, not with the actual A0 used in Lemma 6.1 and Proposition 6.3. If the script secretly uses the correct conjugates, then the paper's proof text is still inaccurate and needs correction. Either way, the present manuscript does not establish the crucial estimate. I do not see a comparable defect in the other major components: the vector-Zak framework, the Fourier solution of the scalar multiplier equation, and the regularity argument of Lemma 6.5 all seem coherent at the level checked here. The constructive and numerical parts of the paper are extensive and deserve credit, but the central link between the certified computation and the fixed-point argument is broken by this phase error. The verdict should therefore be REJECT for the current version, pending a corrected derivation and a re-run of the certificate with the correct conjugate phases.","tokens_in":38860,"tokens_out":23101,"duration_ms":221483,"concrete_test":"Evaluate both sides of equation (5.4) at a generic point, e.g. x=0.8 (so x≥α) and ω=0.3, using (4.5)–(4.9) for the left side and the printed d_{r,s,ℓ} formulas for the right side; a nonzero difference confirms the phase error. If the difference is zero, the concern is resolved, and the remaining check is to re-run the audit script with the conjugate phases to verify that the corrected δ still satisfies δ<1/3.","verdict_should_be":"REJECT","load_bearing_attack":"In Section 5.2, Step 1b, the paper expands A0=χ(z)w(z)^*, with w(z)=η(x)χ(Tτz) from (4.5)–(4.9). Since w^* contains \\overline{η(x)}=η(x)^{-1} and, for x≥α, \\overline{e^{πiβ}e^{-πiω}}=e^{-πiβ}e^{πiω}=σ^{-1}U^{-1}, every coefficient d_{r,s,ℓ} should carry η^{-1} and σ^{-1}. The printed coefficients instead carry η and σ; for example, at x≥α the paper gives d_{r,s,0}=η/2(a(x)A_- + ϵ_rϵ_s b(x)B_- σ), whereas the correct value is η^{-1}/2(a(x)A_- + ϵ_rϵ_s b(x)B_- σ^{-1}). The same pattern appears for x<α, where the printed d_{r,s,-2}=η/2 a(x)B_- σ² should instead be η^{-1}/2 a(x)B_- σ^{-2}. The displayed product in Step 1b also writes (w)_{s+1} rather than (w^*)_{s+1}. Consequently, the function B(x) certified to be <0.328813 bounds ∥A*(x,ω)-Ã0(x,ω)∥ for a matrix field Ã0 that is not the A0 of (4.9). Since (4.13), Lemma 6.1, and Proposition 6.3 all use A0 (equivalently B0=χχ(Tτ)^*), the numerical certificate does not connect to the fixed-point argument as written. This defect is separate from the size of the interval-arithmetic margin and from the question of whether the script was actually executed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a counterexample to the Heil–Ramanathan–Topiwala (HRT) conjecture in the Schwartz class: twelve time-frequency shifts of a nonzero Schwartz function are linearly dependent. The proof constructs a nonzero Schwartz eigenfunction f* for an eleven-term Weyl polynomial P*, then adds the constant term to obtain a twelve-term relation. The construction uses the vector Zak transform, reducing the eigenvalue equation to a matrix-valued difference equation B*(z)F(Tτz)=cF(z). The key quantitative step is a certified numerical estimate δ≤41629/125000<1/3 for the uniform distance between B* and a rank-one matrix field B0 built from an explicit smooth Zak function χ. With this estimate, a fixed-point argument produces a smooth nonzero vector-valued solution v*, a scalar cocycle argument reduces the multiplier q* to a constant c*, and a Fourier-series correction gives a smooth function h satisfying q*(z)h(Tτz)=c*h(z). The product h·v* is shown to be a smooth vector-Zak function whose inverse vector Zak transform lies in the Schwartz class. The paper also contains appendices giving the Faà di Bruno estimates, the sewing-relation verifications, and a description of the computer-assisted certificate.","tokens_in":39173,"tokens_out":11440,"duration_ms":108824,"significance":"If the main theorem is correct, it disproves a long-standing open conjecture of Heil, Ramanathan, and Topiwala, including its Schwartz-class formulation. The paper provides an explicit finite configuration, exact dyadic coefficients, and a reproducible certified numerical script, which are valuable strengths. The analytic architecture—vector Zak transform, fixed-point contraction, and Fourier cocycle reduction—is coherent and detailed. However, the decisive estimate δ<1/3 currently rests on an algebraic error in the expansion of A0, so the central claim is not yet established by the written proof. With a corrected certificate the result could be a major advance; as submitted, the key bound is not proven.","major_comments":[{"comment":"The Laurent expansion of A0 is computed incorrectly. Since A0(z)=χ(z)w(z)^* with w(z)=η(x)χ(Tτz), the (r,s) entry equals χ_r(z) overline{w_s(z)}, which is overline{η(x)} χ_r(z) overline{χ_s(Tτz)}. For x≥α this yields d_{r,s,0}=overline{η}/2(a A_- + ϵ_r ϵ_s b B_- σ^{-1}), d_{r,s,-1}=overline{η}/2 ϵ_s a B_- σ^{-1}, and d_{r,s,1}=overline{η}/2 ϵ_r b A_-, with the analogous σ^{-2} correction in the x<α case. The printed formulas instead contain η, σ, and σ^2 and multiply by w_s rather than overline{w_s}. Consequently, the certified bound in (5.11) applies to ∥A*(x,ω)-Ã0(x,ω)∥ for a different matrix field Ã0(x,ω)=χ(x,ω)w(x,ω)^T, not to the A0 of (4.9). Since (4.13), Corollary 5.3, Lemma 6.1, and Proposition 6.3 all rely on δ<1/3 for A*-A0, the numerical certificate does not connect to the fixed-point argument. The expansion must be corrected and the certified computation rerun; the margin 0.333032 versus 1/3 is far too small for this to be dismissed as a purely cosmetic issue.","section":"§5.2, Step 1b, Eqs. (5.4)–(5.8)"}],"minor_comments":[{"comment":"The displayed identity w(z)^* = η(x)χ(Tτz)^* is false; since η(x) is a unit-modulus phase, the correct expression is overline{η(x)}χ(Tτz)^*. The subsequent identity B0(z)=A0(z)η(x)I2 remains true because η overline{η}=1, but the typo should be fixed as it is the likely source of the Step 1b error.","section":"§4.2, after Eq. (4.6)"},{"comment":"The text says the outer product A0=χw^* is expanded entry by entry as (χ)_{r+1}(w)_{s+1}; it should be (χ)_{r+1} overline{(w)_{s+1}}. This is the concrete form of the conjugation error that invalidates the current certificate.","section":"§5.2, Step 1b, Case 1"},{"comment":"The proof of Theorem 5.1 delegates the decisive bound to a Python/Arb script. The description of the interval arithmetic is clear, but since the script implements the incorrect d-coefficients, the reproducibility statement alone cannot compensate for the missing corrected certificate.","section":"§5.1 and Remark 5.2"}],"recommendation":"major_revision","confidential_remarks":"The potential result is of very high interest, and the analytic framework is sophisticated. However, the incorrect expansion of A0 in Theorem 5.1 is load-bearing and the numerical margin is only about 3×10^{-4}, so I cannot recommend acceptance without a corrected, independently verifiable certificate. The extensive LLM-provenance disclosure in Section 1.2 makes it especially important that the final proof contain a fully checkable numerical argument; I would ask the authors to provide the corrected script output and, ideally, an independent verification of the new bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper claims a 12-term linear dependence of time-frequency shifts of a Schwartz function, which would refute the HRT conjecture. The construction is genuinely new, and the analytic framework is impressive. But I found a serious flaw in the printed numerical reduction, and the proof as written does not connect the certificate to the fixed-point argument.\n\nWhat is new: the explicit configuration of eleven lattice points plus the origin, the use of the vector-Zak transform to convert the Weyl polynomial into a multiplication operator, the rank-one approximation B0, and the contraction argument for the fixed point. The Fourier step in Section 7, which reduces the scalar multiplier to a constant using a diophantine estimate, is elegant. The paper is also unusually transparent about the role of LLMs in generating the initial proof, which is honest but means every step needs checking.\n\nThe soft spot is load-bearing. In Section 5.2, Step 1b, you expand A0 = χ w^* entrywise, but you write (A0)_{r+1,s+1} = (χ)_{r+1} (w)_{s+1}. That is χ w^T, not χ w^*. The printed d-coefficients carry η and σ where the correct values are η^{-1} and σ^{-1} (check the x≥α case: the constant term should have σ^{-1}, not σ). So the matrix field being certified, call it Ã0, is not the A0 that appears in Lemma 6.1, Proposition 6.3, and ultimately the definition of B0. The inequality δ < 1/3 may hold for the wrong object. This is not a typo that leaves the argument intact; the fixed-point equation and the bound on q_X both depend on the relation B0(z) = χ(z)χ(Tτ z)^*.\n\nI could not run the script hrts_fixed_12_point_arb_1d_audit.py, and you note the certified margin is about 3e-4 below 1/3. The conjugation error makes that margin irrelevant until the script is checked: if the script uses the correct conjugated formulas, the paper's text needs correction; if the script mirrors the text, the certificate proves nothing about the fixed-point setup.\n\nThe rest of the architecture—the diophantine estimate, the smoothness bootstrap, the Fourier series for the multiplier—is coherent and looks solid. The claimed result is important enough that the paper deserves refereeing, but it should not be cited until the Step 1b issue is resolved and the certificate is independently reproduced.\n\nRecommendation: send it to peer review with a referee who can run the script and verify the conjugation. I would not desk-reject it; I also would not trust it yet.","headline":"The paper claims a counterexample to HRT, but the printed numerical certificate appears to bound a different matrix field due to a conjugation error in Step 1b; worth refereeing, not citable yet.","tokens_in":39769,"tokens_out":8504,"would_cite":false,"duration_ms":79457,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","43A65"],"pacs":[],"model":"deepseek-v4-flash","headline":"HRT conjecture falls to a 12-term linear dependence","keywords":["HRT conjecture","time-frequency shifts","Weyl polynomial","vector Zak transform","Schwartz function","linear dependence","computer-assisted proof","contraction fixed point"],"falsifier":"Reproduce the supplementary interval-arithmetic script in an independent certified-arithmetic implementation and check whether the two key enclosures still hold: the bound $B(x_j) < 0.328813$ at all 2048 grid centers and the derivative allowance $D_*/2N < 0.004219$. If either fails by more than the small margin, or if a point $z$ is found with $\\|B_*(z)-B_0(z)\\|_{\\mathrm{op}} \\ge 1/3$, then the contraction argument collapses and the claimed counterexample would no longer be established.","tokens_in":38591,"feed_emoji":"📉","tokens_out":4082,"duration_ms":43284,"temperature":0.7,"pith_summary":"This paper claims to settle a 30-year-old open problem, the HRT conjecture, by constructing a nonzero Schwartz function whose twelve time-frequency shifts are linearly dependent. The conjecture predicted that any finite set of time-frequency shifts of a nonzero square-integrable function must be independent, and the paper produces an explicit counterexample with twelve distinct shift points. Doing so requires a genuinely new mechanism: a vector-valued Zak transform that escapes the boundary obstructions of the scalar transform, followed by a computer-assisted estimate and a contraction argument. If correct, the result would close the HRT question in the negative and show that even the Schwartz class does not restore independence.","feed_headline":"HRT conjecture falls to a 12-term linear dependence","feed_subtitle":"It predicted finite time-frequency shifts are independent; this paper constructs a genuine 12-term counterexample.","key_machinery":"The central objects are the vector Zak transform $\\mathcal{Z}_2$, mapping $L^2(\\mathbb{R})$ to $\\mathbb{C}^2$-valued functions satisfying sewing relations, and the certified bound $\\delta \\le \\tfrac{41629}{125000} < \\tfrac13$ on the operator-norm distance between the matrix field $B_*$ and the rank-one field $B_0$. The vector Zak transform is necessary because the scalar Zak transform forces any continuous quasiperiodic function to vanish somewhere, which would destroy the nowhere-vanishing solution needed for the eigenvalue problem. The smallness bound $\\delta < 1/3$ makes the contraction argument work: it keeps the scalar multiplier $q_*$ bounded away from zero and gives a Lipschitz constant below 1 in the fixed-point iteration for the smooth vector-Zak function.","core_discovery":"The central discovery is Theorem 1.1: there exist nonzero complex coefficients and twelve pairwise distinct points in the plane such that a nonzero Schwartz function satisfies an exact linear relation among its time-frequency shifts. The proof reformulates this as an eigenvalue problem for an explicit eleven-term Weyl polynomial, transforms the problem through the vector Zak transform into a matrix-valued difference equation, and then proves existence of a smooth, nowhere-vanishing solution by a contraction fixed-point argument near an explicit rank-one matrix field. The decisive ingredient is a certified uniform bound, less than 1/3, on the distance between the true matrix field and that rank-one field; the remaining steps reduce the scalar multiplier to a constant by a Fourier argument that uses arithmetic properties of the cube root of two.","pith_inferences":["The same vector-Zak framework may be adaptable to produce counterexamples with fewer than twelve shifts or with other algebraic irrational translation parameters, since the proof only needs a strong diophantine separation property of the translation vector.","The certified gap between $\\delta$ and $1/3$ is only about $3\\times 10^{-4}$, so a purely analytic proof of the needed bound may be within reach, which would replace the computer-assisted component by a rigorous closed-form estimate.","A natural testable extension is to search numerically for eigenvalues of the associated Weyl polynomial for the same or slightly perturbed coefficients; a robust nonzero eigenvalue would support the existence mechanism beyond the single certified configuration.","The failure of the HRT conjecture for $\\mathbb{R}^2$ does not by itself settle the discrete zero-divisor conjecture for the Heisenberg group, but it removes the primary analytic evidence that the two conjectures must share the same fate."],"forward_implications":["The HRT conjecture is false in $L^2(\\mathbb{R})$, since the constructed function is in fact Schwartz.","Even the Schwartz-space formulation of the conjecture, raised by several authors, is disproved.","Because Linnell's theorem guarantees independence for any finite subset of a lattice, the counterexample necessarily uses an irrational translation vector; the construction does exactly that, with the origin added to a translated lattice.","The proof is fully explicit: the eleven coefficients are fixed dyadic numbers, and the validity reduces to a finite computer-assisted certificate that can be independently audited.","The vector Zak transform of dimension two is shown to be both necessary and minimal for this construction, since a scalar version cannot work."],"supporting_citations":[{"why":"States the HRT conjecture that the paper disproves.","marker":"[15]"},{"why":"Proves the lattice case, which forces the counterexample to use an irrational translation and the origin.","marker":"[19]"},{"why":"Supplies the Zak transform machinery, the quasiperiodicity facts, and the theorem that smoothness of the Zak transform implies Schwartz decay.","marker":"[12]"},{"why":"Provides the Fourier coefficient decay and smoothness criteria used for the scalar multiplier reduction.","marker":"[11]"},{"why":"Documents the interval-arithmetic backend that makes the certified bound 1/3 rigorous.","marker":"[17]"},{"why":"Provides the multivariate Faà di Bruno formula underlying the regularity argument for the fixed point.","marker":"[5]"}],"fun_headline_variants":["12-term counterexample disproves HRT conjecture","Explicit 12-term dependence refutes HRT conjecture","Twelve time-frequency shifts yield explicit HRT counterexample","HRT conjecture false: 12-term counterexample","HRT conjecture falls to explicit 12-term linear relation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction breaks if the computer-certified bound that the matrix field stays within one-third of the rank-one field is wrong, since that strict inequality drives the contraction, the nonzero scalar multiplier, and the smoothness of the fixed point.","fun_headline_variants_meta":{"raw":{"variants":["12-term counterexample disproves HRT conjecture","Explicit 12-term dependence refutes HRT conjecture","Twelve time-frequency shifts yield explicit HRT counterexample","HRT conjecture false: 12-term counterexample","HRT conjecture falls to explicit 12-term linear relation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001198,"raw_usage":{"total_tokens":4823,"prompt_tokens":714,"completion_tokens":4109,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":330,"completion_tokens_details":{"reasoning_tokens":4031}},"tokens_in":330,"tokens_out":4109,"duration_ms":29801,"temperature":1.0,"reasoning_tokens":4031,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:40:20.116220+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Reproduce the supplementary interval-arithmetic script in an independent certified-arithmetic implementation and check whether the two key enclosures still hold: the bound $B(x_j) < 0.328813$ at all 2048 grid centers and the derivative allowance $D_*/2N < 0.004219$. If either fails by more than the small margin, or if a point $z$ is found with $\\|B_*(z)-B_0(z)\\|_{\\mathrm{op}} \\ge 1/3$, then the contraction argument collapses and the claimed counterexample would no longer be established.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the HRT conjecture that the paper disproves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the lattice case, which forces the counterexample to use an irrational translation and the origin."},{"cited_title":"Gröchenig.Foundations of time-frequency analysis","cited_arxiv_id":null,"evidence_quote":"Supplies the Zak transform machinery, the quasiperiodicity facts, and the theorem that smoothness of the Zak transform implies Schwartz decay."},{"cited_title":"Comput., 66(8):1281–1292, 2017","cited_arxiv_id":null,"evidence_quote":"Documents the interval-arithmetic backend that makes the certified bound 1/3 rigorous."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the multivariate Faà di Bruno formula underlying the regularity argument for the fixed point."}],"review_version":1}