{"id":"696561ad-32ea-40e6-a3a4-a5979fa14da8","arxiv_id":"2608.13539","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For every Diophantine pair, the paper constructs a nonzero function with exponential tails and a finite linear dependence among its time-frequency shifts.","lead":"A new construction gives time-frequency shifts of a single function with exponential decay that are linearly dependent, attacking a 30-year-old conjecture in the sharpest decay class so far. The proof extends a recently discovered counterexample mechanism to the Roumieu Gelfand-Shilov class, the fastest decay allowed for such counterexamples up to a logarithmic factor.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 1's normalization is not holomorphic: the map z↦χε(z)^* is anti-holomorphic, not holomorphic, so F#=χε/∥χε∥ and A# are not strip-holomorphic; Steps 2–3 lose their input.","rationale":"The reader's weakest-assumption identification is correct: the proof of Theorem 6 depends on constructing a pointwise unit-norm strip-holomorphic section F#. The paper attempts this by normalizing an entire vector-Zak transform χε by its Euclidean norm. But the squared norm, sε=χε^*χε, is not holomorphic for a nonconstant entire vector field. The claim \"The map z↦χε(z)^* is holomorphic\" is false; complex conjugation is not holomorphic. Consequently the holomorphic square-root argument fails, and F# and A# are not shown to be strip-holomorphic. This is not a minor technicality: Proposition 7 approximates strip-holomorphic multipliers by finite sums of L_{m,n}, and Lemma 8 uses the rank-1 holomorphic A# as the unperturbed problem for a contraction argument. If A# is not a section of the holomorphic multiplier bundle, both steps lack a valid starting point. While the overall theorem may still be true and a more careful normalization could repair the proof, the manuscript as written does not contain that repair, so the reader's REJECT verdict stands. The critique targets the argument, not the authors, and no ad hominem is intended. The central claim is not internally contradicted elsewhere; it simply lacks a demonstrated proof.","tokens_in":9971,"tokens_out":5942,"duration_ms":63553,"concrete_test":"Compute ∂/∂\\bar z_1 of sε(z)=χε(z)^*χε(z)=Σ_i|χ_i(z₁,z₂)|² for the Gaussian-convolved fε. Since some coordinate χ_i has χ_i'(z₀)≠0 at generic z₀, the identity ∂_{\\bar z₁}|χ_i|² = χ_i \\overline{χ_i'} is nonzero at z₀, so sε is not holomorphic. This directly contradicts the claim in §3.1.1. A simple numerical check: take f₀ a centered bump, evaluate sε at (z₁,z₂)=(0.3+it,0.0) for small t and confirm that the real part changes at first order while the imaginary part is nonzero, ruling out a holomorphic extension. If sε were holomorphic and real on R², it would be constant by the open mapping theorem, which is impossible for nonconstant fε.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §3.1.1 the paper defines sε(z)=χε(z)^*χε(z) and states: \"The map z↦χε(z)^* is holomorphic, so sε is entire.\" This is false: the conjugate transpose of a holomorphic vector-valued function is anti-holomorphic, not holomorphic. The subsequent claim that sε admits a holomorphic square root on Ωρ is therefore unsupported. Since F# is defined as χε/√sε over R² and the denominator is not shown to extend holomorphically, F# is not established to be strip-holomorphic. Moreover, A#(z)=e(z)F#(z−τ)F#(z)^* in (10) contains the same anti-holomorphic factor, so A# is not a vector-Zak multiplier in the sense required by Proposition 7. The rank-1 structure of A# and the Banach-space perturbation argument in Lemma 8 depend on this holomorphy. Without a holomorphic F# and A#, the approximation step has no valid input, so Theorem 6 is not proved. This is the load-bearing premise of the construction, not a cosmetic gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims that for every Diophantine pair (α,β) there exist a nonzero function f in the Roumieu Gelfand–Shilov class S_1^1(R), a finite set I ⊆ Z², and coefficients a_{m,n} such that π(α,β)f = Σ a_{m,n}π(m,n/2)f. The proof works in a vector-Zak transform domain, aiming to construct a strip-holomorphic section F and a multiplier A satisfying e(z)F(z−τ) = A(z)F(z). Step 1 selects a pointwise unit-norm F♯ by convolving a compactly supported function with a Gaussian and normalizing; Step 2 replaces the resulting rank-one multiplier A♯ by a finite combination of the L_{m,n} multipliers via Fourier approximation and a contraction-mapping argument; Step 3 uses the Diophantine property to make the resulting multiplier constant. If valid, Theorem 2 would give HRT counterexamples with exponential decay, which is optimal up to a logarithmic factor by the Bownik–Speegle result.","tokens_in":10159,"tokens_out":5705,"duration_ms":61758,"significance":"The claimed result is significant: it would push the decay of HRT counterexample functions to essentially the fastest possible while also providing a more transparent account of the mechanism behind the recent breakthrough. The paper is clearly written, and the three-step strategy is intelligible and well motivated. However, the central analytic step of the construction is invalid: the pointwise normalization of a holomorphic section is not holomorphic, so the strip-holomorphic objects F♯ and A♯ that feed the Fourier approximation and the fixed-point argument are not actually constructed. The significance is therefore conditional on a repair that the manuscript does not provide.","major_comments":[{"comment":"The claim “The map z↦χε(z)^* is holomorphic” is false: for a holomorphic vector-valued function χε, the conjugate transpose is anti-holomorphic, not holomorphic. Consequently sε(z)=χε(z)^*χε(z) is real-analytic but not entire, and the asserted holomorphic square root of sε on Ωρ is not established. Since F♯ is defined as χε/∥χε∥ using this denominator, F♯ is not shown to be strip-holomorphic. This is exactly the analytic regularity needed for the rest of the construction, so Steps 2 and 3 lose their required input.","section":"§3.1.1"},{"comment":"The proof repeats the same error when it states that both F♯ and z↦F♯(z)^* are holomorphic over Ωρ. The latter map is anti-holomorphic, so the identity-theorem argument that F♯(z)^*F♯(z) is identically 1 over Ωρ is invalid, and A♯(z)=e(z)F♯(z−τ)F♯(z)^* in (10) is not a strip-holomorphic vector-Zak multiplier. Thus A♯ does not satisfy the hypothesis of Proposition 7, and the rank-one fixed-point construction in Lemma 8 has no valid holomorphic starting point.","section":"§3.1.2"}],"minor_comments":[{"comment":"The outline's phrase “z↦∥χε(z)∥ has a strip-holomorphic extension” is misleading: the Euclidean norm of a holomorphic vector is not a holomorphic function in several complex variables.","section":"§3.1.1"},{"comment":"The statement that τ=(α,2β) is Diophantine whenever (α,β) is could be expanded with the concrete inequality, since the current one-line justification is terse.","section":"§2.4 / §3.3"}],"recommendation":"reject","confidential_remarks":"The central error is unambiguous and is quoted verbatim in the manuscript; it is not a matter of interpretation. I see no repair within the current framework. The paper also relies on unpublished preprints [3,4,8], but that is not the basis for my recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The reader has it right. This is not a throwaway paper, but the proof has a load-bearing flaw. What the paper does well: it gives a human-readable digestion of the Faulhuber–Petersen–van Velthoven–Voigtlaender mechanism, which is valuable given how heavy that proof is; it frames the decay question cleanly via the Bownik–Speegle obstruction; and it adds a nice concrete Diophantine example. The vector-Zak reformulation is clear, Lemma 3 connecting strip-holomorphic sections to S^1_1 is plausible, and Proposition 7 is a neatly packaged Fourier approximation tool. Citation practice is fine: the prior work is credited, and the near-optimality claim correctly points to Bownik–Speegle. If the main theorem held, it would be a meaningful extension, even if the mechanism is largely imported.\n\nThe problem is exactly where the stress-test note puts it. In §3.1.1 the paper claims: \"The map z↦χε(z)^* is holomorphic, so sε is entire.\" That is false. The conjugate transpose of a holomorphic vector-valued function is anti-holomorphic, not holomorphic. So sε(z)=χε(z)^*χε(z) is not entire; it is real-analytic at best. The holomorphic square root of sε on Ωρ is therefore unavailable, and F#=χε/∥χε∥ is not shown to be strip-holomorphic. In fact, normalizing a holomorphic section pointwise to unit norm does not generally produce a holomorphic section. Since A# in (10) contains F#(z)^*, it is also not a strip-holomorphic multiplier. That means Proposition 7 and Lemma 8 have no valid input. This is not a cosmetic gap: it is the bridge that turns an exact rank-one multiplier into a finite sum of L_{m,n}'s. Theorem 6 is unproved as written.\n\nCould it be repaired? Possibly, by a different construction of F# or by avoiding the unit-norm normalization altogether. But none of that is in the manuscript. The paper is honest about not bounding the configuration size, and the exposition around the gap is solid, but the flagship result is not established.\n\nWho is this for? People working on HRT and Gabor analysis who want to understand the recent counterexample machinery and how decay limits enter. They will learn from the exposition even though the main theorem is currently unsupported.\n\nRecommendation: I would not accept the paper as-is, but it does deserve a serious referee. The result is important enough and the route is promising enough that a careful referee could help identify a repair. If Step 1 can be fixed, this is likely publishable.","headline":"A genuinely useful exposition and a plausible near-optimal strengthening, but the load-bearing holomorphy claim in Step 1 is false, so the main theorem is not proved as written.","tokens_in":10713,"tokens_out":3865,"would_cite":false,"duration_ms":45810,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","46E10","11J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The HRT conjecture is false for functions with exponential tails.","keywords":["HRT conjecture","time-frequency shifts","Gelfand-Shilov class","exponential decay","Zak transform","Diophantine approximation","linear dependence","Gabor system"],"falsifier":"Compute $\\partial_{\\bar z_1}\\big(\\chi_\\varepsilon(z)^*\\chi_\\varepsilon(z)\\big)$ for the explicitly defined Gaussian-smoothed seed of Step 1; if it is nonzero anywhere, $\\|\\chi_\\varepsilon(z)\\|$ is not holomorphic and the proof of Theorem 6 as written breaks at the normalization step.","tokens_in":9742,"feed_emoji":"📉","tokens_out":9574,"duration_ms":95894,"temperature":0.7,"pith_summary":"The paper claims that the Heil–Ramanathan–Topiwala (HRT) conjecture—that finitely many distinct time–frequency shifts of any nonzero square-integrable function are linearly independent—fails even under very strong decay. For every Diophantine pair $\\alpha,\\beta$, the authors construct a nonzero function $f$ in the Roumieu Gelfand–Shilov class $\\mathcal{S}^1_1(\\mathbb{R})$ and finitely many half-integer frequency shifts such that $\\pi(\\alpha,\\beta)f$ is a finite linear combination of the $\\pi(m,n/2)f$. Functions in $\\mathcal{S}^1_1(\\mathbb{R})$ have exponential or faster decay in both $f$ and its Fourier transform, so this improves on the earlier Schwartz-function counterexamples. Because Bownik and Speegle showed that faster-than-exponential decay (up to a logarithmic factor) forces HRT to hold, the exponential-tail decay achieved here is essentially the fastest possible for a counterexample. The paper does not, however, bound the number of shifts in the dependence.","feed_headline":"HRT conjecture fails for exponential-tail functions","feed_subtitle":"A vector-Zak construction gives finite time-frequency dependences for every Diophantine pair, near the best possible decay.","key_machinery":"The load-bearing object is the two-by-two vector-Zak transform $Z_2f(x,\\omega)=2^{-1/2}(Zf(x,\\omega/2),Zf(x,(\\omega+1)/2))$, whose quasi-periodicity forces the half-lattice time-frequency shifts to act as pointwise multiplication operators. It converts the desired HRT dependence into an equation between strip-holomorphic sections of a vector bundle. The proof then runs on three levers: rank-one multipliers built from a pointwise unit-norm section, Fourier approximation of strip-holomorphic multipliers by finite sums of $L_{m,n}$, and the Diophantine cocycle estimate $|1-e^{-2\\pi i k\\cdot\\tau}|\\geq 4C(1+\\|k\\|_1)^{-s}$, which keeps the Fourier solution of $q(z)h(z-\\tau)=ch(z)$ exponentially decaying and hence strip-holomorphic.","core_discovery":"The paper's central claim is that HRT counterexamples can be built with exponential tails via a transform-domain construction. Using the vector-Zak transform $Z_2$, the half-lattice shifts $\\pi(m,n/2)$ become pointwise multiplication by the $2\\times2$ matrix multipliers $L_{m,n}(x,\\omega)=e^{\\pi i(nx-m\\omega+mn)}Z^mX^n$, while the shift $\\pi(\\alpha,\\beta)$ becomes $e^{2\\pi i\\beta x}F(x-\\alpha,\\omega-2\\beta)$. The paper proves that for a Diophantine pair there exists a nonzero strip-holomorphic vector-Zak section $F$ and a finite multiplier $A=\\sum a_{m,n}L_{m,n}$ satisfying $e^{2\\pi i\\beta x}F(x-\\alpha,\\omega-2\\beta)=A(x,\\omega)F(x,\\omega)$; pulling this identity back through $Z_2$ yields the time-domain dependence $\\pi(\\alpha,\\beta)f=\\sum a_{m,n}\\pi(m,n/2)f$. The construction proceeds by normalizing a Gaussian-smoothed compactly supported seed to unit norm, approximating the resulting rank-one multiplier by a finite Fourier polynomial in the $L_{m,n}$, and using a Banach fixed-point argument to pass to an exact nearby solution; the Diophantine condition is then used, through a Fourier series solution of the cocycle equation $q(z)h(z-\\tau)=ch(z)$, to absorb the residual multiplier into a constant.","pith_inferences":["The mechanism is modular: any nowhere-zero strip-holomorphic section with a rank-one multiplier would seed the same Fourier-approximation and fixed-point machinery, so the construction may transfer to other decay classes or to more general irrational shifts.","The cocycle equation $q(z)h(z-\\tau)=ch(z)$ is solved by a Fourier series whose coefficient decay is preserved because the Diophantine denominator is polynomial; this suggests the same construction could yield counterexamples for any shift whose small-denominator obstruction is subexponential.","A natural stress test is numerical: build $\\chi_\\varepsilon$ for the Gaussian-smoothed compactly supported seed and check whether $\\|\\chi_\\varepsilon(z)\\|$ admits a holomorphic square root; if not, the proof of Theorem 6 as written needs an alternate normalization before the later steps can run."],"forward_implications":["For every Diophantine pair $(\\alpha,\\beta)$, the HRT conjecture fails in the class $\\mathcal{S}^1_1(\\mathbb{R})$, so exponential decay is not enough to restore linear independence of time-frequency shifts.","The fastest possible decay of an HRT counterexample lies between exponential and the Bownik–Speegle threshold $\\exp(-cx\\log x)$; in particular the decay barrier is tight up to a logarithmic factor in the exponent.","The counterexample functions can be taken to have exponential decay in both time and frequency, since $\\mathcal{S}^1_1$ is characterized by that double decay.","The construction yields no control on the number of time-frequency shifts in the dependence, in contrast with the 12-point and 4-point counterexamples."],"supporting_citations":[{"why":"Supplies the vector-Zak transform, the matrix multipliers, and the 12-point Schwartz counterexample that this paper extends.","marker":"[4]"},{"why":"States the HRT conjecture whose falsity is the paper's target.","marker":"[6]"},{"why":"Proves that faster-than-exponential decay forces HRT, making the obtained exponential-tail decay essentially optimal.","marker":"[1]"},{"why":"Gives the characterization of Gelfand–Shilov spaces used to identify $\\mathcal{S}^1_1(\\mathbb{R})$ with functions whose values and Fourier transforms decay exponentially.","marker":"[2]"},{"why":"Provides the theorem used to pass from smooth vector-Zak sections back to Schwartz functions in Lemma 3.","marker":"[5]"},{"why":"Establishes continuity of the Zak transform used to get uniform convergence of the Gaussian-smoothed sections to the compactly supported seed.","marker":"[7]"}],"fun_headline_variants":["HRT fails with exponential decay functions","Exponential-tail counterexamples for HRT","Near-optimal decay for HRT counterexamples","Diophantine pairs give exponential-tail HRT failures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on having a holomorphic, length-one two-component function with the right periodicity that satisfies the shifted identity up to a rank-one factor; the proof gets this by asserting the holomorphy of the norm of its Gaussian-smoothed seed, and if that fails, the approximation and contraction steps have no input.","fun_headline_variants_meta":{"raw":{"variants":["HRT fails with exponential decay functions","Exponential-tail counterexamples for HRT","Near-optimal decay for HRT counterexamples","Diophantine pairs give exponential-tail HRT failures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000292,"raw_usage":{"total_tokens":1691,"prompt_tokens":922,"completion_tokens":769,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":712}},"tokens_in":538,"tokens_out":769,"duration_ms":8721,"temperature":1.0,"reasoning_tokens":712,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:38:21.909111+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\partial_{\\bar z_1}\\big(\\chi_\\varepsilon(z)^*\\chi_\\varepsilon(z)\\big)$ for the explicitly defined Gaussian-smoothed seed of Step 1; if it is nonzero anywhere, $\\|\\chi_\\varepsilon(z)\\|$ is not holomorphic and the proof of Theorem 6 as written breaks at the normalization step.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the HRT conjecture whose falsity is the paper's target."},{"cited_title":"Bownik and D","cited_arxiv_id":null,"evidence_quote":"Proves that faster-than-exponential decay forces HRT, making the obtained exponential-tail decay essentially optimal."},{"cited_title":"Chung, S.-Y","cited_arxiv_id":null,"evidence_quote":"Gives the characterization of Gelfand–Shilov spaces used to identify $\\mathcal{S}^1_1(\\mathbb{R})$ with functions whose values and Fourier transforms decay exponentially."},{"cited_title":"Gr¨ ochenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkh¨ auser, Boston, 2001","cited_arxiv_id":null,"evidence_quote":"Provides the theorem used to pass from smooth vector-Zak sections back to Schwartz functions in Lemma 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes continuity of the Zak transform used to get uniform convergence of the Gaussian-smoothed sections to the compactly supported seed."}],"review_version":1}