{"id":"82c22b40-dbff-4bb1-a593-6e981e040149","arxiv_id":"2608.07604","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Four distinct time-frequency shifts of a nonzero Schwartz function are linearly dependent while every symplectic triangle determinant is strictly below one.","lead":"A four-point set of time-frequency shifts of a nonzero Schwartz function is proven dependent, giving the smallest possible counterexample to the Heil-Ramanathan-Topiwala conjecture. The configuration lies in a subcritical geometric regime, and the proof blends exact analysis with computer-verified inequalities.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof hinges on the unverified interval-arithmetic certificates of Theorem 4.2, which are not executable from the preprint and are explicitly outside the Lean formalization; the conditional verdict is appropriate.","rationale":"I read the manuscript in good faith and checked the non-computational parts of the argument. The vector-Zak reduction, the exact sewing of the flat-step and linear sections, the cone contraction lemma, the dominated splitting, the gauge and winding arguments, and the Fourier cohomology step all appear internally consistent; I found no hidden algebraic or topological obstruction. In particular, the apparent sewing mismatch for the reference section is resolved by noting that cos(pi*s0(x)/2) vanishes at x=1, so the formula does satisfy the U1 sewing relation. The genuinely fragile point is exactly the one identified by the reader: the four certificates of Theorem 4.2 are produced by custom Python/Arb programs whose outputs are quoted but whose code is not in the paper, and the accompanying Lean file does not cover Arb or the certificates. Since every later theorem depends on these inequalities, a flaw in the computational core would invalidate the central existence statement. The printed margins are large but not infinite, and the independent cross-check shares low-level routines. Therefore the conditional verdict should stand: if the supplement reproduces the stated bounds and the Lean file is as described, the proof is convincing; otherwise it is unverified.","tokens_in":12850,"tokens_out":38134,"duration_ms":326206,"concrete_test":"Obtain the Dropbox supplement and run arb_fourier_domination_certificate.py, arb_fourier_overlap_certificate.py, and arb_linear_gauge_certificate.py under the pinned CPython 3.14.6, python-flint 0.8.0, and FLINT/Arb 3.3.1 versions, verifying that the printed global lower bounds are reproduced. Then re-implement the C3 energy ratio and the C4 half-plane polynomials a+kappa*F*c and a+F*c with an independent interval library such as MPFI or Julia's IntervalArithmetic at the same covers, and require agreement with the stated rational margins.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4's Theorem 4.2 is the sole computer-assisted input, and the analytic chain in Sections 5-8 is sound only if certificates (C1)-(C4) hold uniformly on the torus. Specifically: (C1) needs the 16-block singular-gap bound with global lower margin 3.5398; (C2) needs the 32-block junction overlap with margin 15,399,575.8; (C3) needs the overlap ratio <749/1000 with global margin 48.876; and (C4) needs the linear-gauge half-plane bounds with global lower bounds near 74.94. A single error in coefficient expansion, cell derivative allowance, or cover logic in the supplied Python/Arb programs would invalidate the dominated splitting and with it the invariant-line construction, the winding transfer, and Theorem 1.1. The preprint does not include the programs, and Section 10 explicitly states that the Lean file does not formalize Arb or the computational certificates. The 'independent' 1024-squared check is admitted not to be an independent software-stack replication and shares low-level Laurent-product and Hermitian-polynomial routines. Thus the correctness of these hidden computations is the load-bearing condition for the main claim, and it cannot be checked from the manuscript alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a computer-assisted construction of four time-frequency shifts of a nonzero Schwartz function that are linearly dependent, contradicting the HRT conjecture in the four-point, intrinsically subcritical regime. The proof uses the two-component vector Zak transform to reduce the problem to a cocycle over the irrational torus translation z ↦ z − (α, β), with α = ∛2 − 1 and β = ∛4 − 1. Under four numerical certificates (C1)–(C4) stated in Theorem 4.2, the author constructs a dominated invariant line for the sixteen-step cocycle, proves zero winding of the associated multiplier by a rotated-half-plane gauge argument, solves a cohomological equation using an explicit Diophantine bound, and reconstructs the Schwartz eigenfunction. The paper also proves that all four symplectic triangle determinants of the configuration lie strictly between zero and one, so the configuration is intrinsically subcritical.","tokens_in":13158,"tokens_out":24823,"duration_ms":204590,"significance":"If the computational certificates are correct, the result is a substantial advance in the HRT problem: it reduces the known counterexample from twelve points to four and places the counterexample in the previously open subcritical regime. The analytic architecture is largely transparent and exact: the geometry in Section 2, the linear-algebra reductions in Lemma 4.1, the cone estimates in Section 5, and the Diophantine bound in Lemma 7.1 are all presented with explicit constants. The paper is also admirably honest about the scope of its formal verification: Section 10 states that the Lean companion does not formalize Arb or the computational certificates, so the certificates themselves remain the load-bearing input. The main weakness is that this input is not auditable from the manuscript as submitted, because the certificate code and exact logs are not included and no URL is provided.","major_comments":[{"comment":"The four certificates (C1)–(C4) are the sole computer-assisted input to the proof, and every later step—the dominated splitting in Section 5, the gauge overlap in Section 6, the winding transfer, and Theorem 1.1—depends on them. The 'Validated verification' paragraph reports only summary numbers (e.g., global lower bounds 3.5398, 15,399,575.8, 48.876, and 74.94) and the actual Python/Arb programs are not included or linked in the manuscript. Section 10 explicitly says the Lean companion does not formalize Arb or the computational certificates. As submitted, the main theorem cannot be checked from the paper alone, because a single error in coefficient expansion, cell derivative allowance, or cover logic would invalidate the invariant-line construction. I request that the complete certificate code, exact output logs, and the precise rational ball endpoints be made available as ancillary files, and that the relationship between the printed decimal bounds and the exact statements in Theorem 4.2 be spelled out.","section":"§5, Lemma 5.2 and Proposition 5.3; §6, Lemma 6.1"}],"minor_comments":[{"comment":"The sentence 'By Theorems 4.1 and 4.2' should read 'By Lemma 4.1 and Theorem 4.2'; there is no Theorem 4.1.","section":"§4, after Theorem 4.2"},{"comment":"The references to 'Theorem 5.1' and 'Theorem 5.2' should be to Lemma 5.1 and Lemma 5.2.","section":"§5, Proposition 5.3"},{"comment":"The description of the 1024² implementation as 'independently formulated' is misleading, since the next sentence discloses that it shares the low-level Laurent-product and Hermitian-polynomial routines; I suggest using 'separately formulated' and making clear that it is not an independent software-stack check.","section":"§4, 'Validated verification'"},{"comment":"The supplement is said to be available on Dropbox, but no URL or manifest content is included in the manuscript; for a computer-assisted proof, the code and logs should be bundled as ancillary files with the submission.","section":"§10"},{"comment":"The regularity proof of the invariant section is highly compressed; since smoothness of the section is used in the Fourier-cohomology step in Section 7, please provide a precise reference to [9] with a chapter or theorem number, or expand the jet-contraction argument.","section":"§5, Lemma 5.2"},{"comment":"The inference '1 + α + β < 2' does not follow from the stated inequalities 0 < α < 1 and 0 < β < 1 alone; the stronger bounds α < 1/3 and β < 2/3 should be stated explicitly.","section":"§2, Proposition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The author's disclosure that the Lean file does not cover the Arb certificates is commendable, but it means the publication decision rests on the availability and correctness of the certificate programs. I recommend requiring the code and logs to be uploaded as ancillary files, and ideally a second independent implementation, before acceptance. The analytic architecture is otherwise credible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kevin — here's my read on Oussa's four-point HRT counterexample.\n\nThe paper is a genuine strengthening of the Faulhuber–Petersen–van Velthoven–Voigtlaender result. It reduces the dependent configuration from twelve points to four, and it does so in the intrinsically subcritical regime — every triangle determinant is strictly below one, and no re-marking leaves that row of the dashboard. That is new and substantial. The analytic architecture is coherent: exact Zak reduction, the three-term Weyl symbol, the 16/32-block domination certificates, the two-gauge winding argument, and the cubic Diophantine estimate all fit together. The paper is also unusually honest about provenance; Section 9 names what is adapted and what is new, and the Lean file is described at exactly the right level of overstatement — algebraic corroboration, not end-to-end formalization.\n\nThe soft spot is the one you flagged. Theorem 4.2 is the sole computer-assisted input, and the four certificate inequalities carry the construction. The manuscript describes the finite-cover method, the cell allowances, and the concrete margins, but the actual Python/Arb programs are only in a Dropbox supplement, not in the preprint, and the 1024^2 cross-check shares low-level routines with the primary code. So the correctness of the main theorem, as it stands in the arXiv file, depends on computations the reader cannot run or inspect from the text. That is a real limitation, but it is the standard limitation of validated-numerics proofs. The description is detailed enough to make the claims plausible, and the margins are large (the junction margin is ~1.5e7), so this is not a case where a tiny numerical error is likely to flip the result.\n\nMy bottom line: the paper deserves a serious referee, not a desk rejection. The referee should be asked to obtain the supplement, run the Arb programs, and check that the printed bounds match. The authors should also be encouraged to package the code in a way that survives peer review — a GitHub repository or a journal-hosted appendix would be better than a Dropbox link. If the supplement reproduces the stated bounds, the proof is convincing.\n\nFor the reading group: worth a session, but with a laptop ready to poke at the code. I would cite it if I worked in this area.","headline":"A genuine four-point subcritical counterexample to HRT that is analytically coherent but rests on unexecutable validated-numerics certificates, deserving a serious referee who can actually run the accompanying Arb code.","tokens_in":13635,"tokens_out":1988,"would_cite":true,"duration_ms":18305,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","37D30","65G30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper presents a computer-assisted proof that the HRT conjecture fails for four time–frequency shifts of a nonzero Schwartz function, with every symplectic triangle determinant strictly between zero and one.","keywords":["HRT conjecture","time-frequency shifts","Weyl operators","vector Zak transform","dominated cocycle","validated numerics","Schwartz functions","subcritical configuration"],"falsifier":"Recompute the four certificate inequalities on a finer grid (at least $1024^2$ per fundamental rectangle) with a different interval-arithmetic implementation and compare against the paper's printed global lower bounds: gap above $3.5398$, junction above $15\\,399\\,575.8$, overlap lower bound above $48.8768$, and half-plane lower bounds above $74.9429$; a single violation would collapse the domination, the invariant line, and the eigenfunction construction.","tokens_in":12675,"feed_emoji":"📐","tokens_out":8822,"duration_ms":71279,"temperature":0.7,"pith_summary":"This paper disproves the HRT conjecture at the smallest possible cardinality: it exhibits an explicit nonzero Schwartz function whose four time-frequency shifts are linearly dependent. The four phase-space points are built from cube roots of two, and every symplectic triangle they determine has absolute value strictly below one, so the configuration is intrinsically subcritical. The proof passes through a two-component vector Zak transform, turning the problem into a matrix cocycle over an irrational torus translation, then uses computer-assisted interval-arithmetic certificates to construct a smooth invariant line and to solve the corresponding cohomological equation. If the proof is correct, the known counterexample size drops from twelve points to four, and the irrational subcritical regime is the home of such dependent configurations.","feed_headline":"HRT conjecture fails with only four time-frequency shifts","feed_subtitle":"Computer-assisted proof finds a Schwartz function with four dependent Weyl shifts; all triangle areas below one.","key_machinery":"The central object is the two-component vector Zak transform, a unitary map that presents functions on the line as sections of a two-dimensional vector bundle over the torus and converts Weyl shifts into a matrix-valued cocycle $B(x,\\omega)=\\eta(x)A(x,\\omega)$ with $A=I+\\tfrac{3}{5}e^{-\\pi i\\omega}D+\\tfrac{3}{5}e^{\\pi i x}S$, where $D$ and $S$ are the diagonal and permutation matrices, over the irrational translation $T(x,\\omega)=(x-\\alpha,\\omega-\\beta)$. The load-bearing mechanism is a smooth dominated splitting of the iterated cocycle $B^{16}$, obtained from four validated interval-arithmetic certificates that control singular-value ratios, junction overlaps, and a rotated half-plane gauge; this yields a smooth invariant line. On that line the multiplier equation reduces to a cohomological equation $q_{\\mathrm{sm}}(z)h(Tz)=\\lambda h(z)$, solved with an explicit Diophantine bound coming from the algebraic norm in $\\mathbb{Q}(\\sqrt[3]{2})$. The resulting smooth vector-Zak section pulls back to the Schwartz eigenfunction $g$.","core_discovery":"The paper's main theorem asserts that there exist a nonzero complex-valued Schwartz function $g$ and a nonzero constant $\\lambda$ such that $\\left[ I + \\tfrac{3}{5}\\rho(1,0) + \\tfrac{3}{5}\\rho(0,1/2) \\right] \\rho(\\zeta) g = \\lambda g$, with $\\zeta = (\\alpha,\\beta/2)$, $\\alpha=\\sqrt[3]{2}-1$, and $\\beta=\\sqrt[3]{4}-1$. Equivalently, the four shifted copies displayed in (1.5) are linearly dependent. The four points are distinct, no three are collinear, and every absolute symplectic triangle determinant lies strictly between zero and one, so every admissible re-marking of the configuration is subcritical. The construction is computer-assisted: validated interval-arithmetic certificates establish a dominated splitting and a half-plane gauge, and exact cohomological arguments complete the eigenfunction.","pith_inferences":["Varying the two coefficients $3/5$ or replacing $\\sqrt[3]{2}$ by another cubic irrationality may yield further four-point counterexamples, provided the corresponding domination and overlap certificates still pass; this is a testable extension, not a paper claim.","The explicit Diophantine bound (7.1) is strong enough that the cohomological step should work for any pair $\\alpha,\\beta$ of the same algebraic shape, so the construction is not tied to the particular numerical values.","Because all four triangle areas are below one while dependence still occurs, area and covolume thresholds alone do not prevent dependence; the arithmetic of the rogue point's coordinates may be the controlling ingredient.","An independent, differently implemented re-run of the four certificates would either confirm the proof or locate the point where the domination, overlap, or gauge argument breaks; the paper itself flags the computational core as the unformalized part."],"forward_implications":["One-, two-, and three-point systems are always linearly independent, so four is the minimal possible size of a counterexample to the HRT conjecture.","This is the first counterexample in the intrinsically subcritical regime: all four absolute symplectic triangle determinants are below one, and the background lattice covolume is $1/2$.","The maximally irrational subcritical cell of the four-point dashboard is nonempty, so the open classification of dependent four-point configurations now has a concrete member in that cell.","The four certificate inequalities are the only computer-assisted premises; the paper's formal companion verifies exact algebraic steps but deliberately does not formalize the interval-arithmetic computation."],"supporting_citations":[{"why":"Supplies the vector-Zak and cohomological architecture of the twelve-point counterexample that this paper compresses to four points.","marker":"[5]"},{"why":"Introduces the two-component vector Zak transform used to convert Weyl shifts into a matrix cocycle.","marker":"[19]"},{"why":"Provides the interval-arithmetic engine for the outward-rounded certificates (C1)-(C4).","marker":"[10]"},{"why":"Supplies the uniqueness theorem for dominated splittings used to single out the invariant line.","marker":"[1]"},{"why":"Gives the smooth-Zak characterization that upgrades the smooth vector-Zak section to a Schwartz function.","marker":"[6]"},{"why":"States the HRT conjecture and the minimal-independence background that makes four points the smallest possible counterexample.","marker":"[8]"}],"fun_headline_variants":["HRT conjecture fails with only four shifts","Computer-assisted proof: four shifts are dependent","Subcritical four-point HRT counterexample","HRT disproved by four time-frequency shifts","Intrinsically subcritical counterexample with four shifts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's four computer-assisted interval-arithmetic checks are the load-bearing premise: if any of their inequalities is violated, the dominated splitting, the invariant line, and the Schwartz eigenfunction can fail, and the paper's formal companion explicitly does not cover this computational core.","fun_headline_variants_meta":{"raw":{"variants":["HRT conjecture fails with only four shifts","Computer-assisted proof: four shifts are dependent","Subcritical four-point HRT counterexample","HRT disproved by four time-frequency shifts","Intrinsically subcritical counterexample with four shifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001129,"raw_usage":{"total_tokens":4612,"prompt_tokens":782,"completion_tokens":3830,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":3761}},"tokens_in":398,"tokens_out":3830,"duration_ms":27945,"temperature":1.0,"reasoning_tokens":3761,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:32:16.631073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the four certificate inequalities on a finer grid (at least $1024^2$ per fundamental rectangle) with a different interval-arithmetic implementation and compare against the paper's printed global lower bounds: gap above $3.5398$, junction above $15\\,399\\,575.8$, overlap lower bound above $48.8768$, and half-plane lower bounds above $74.9429$; a single violation would collapse the domination, the invariant line, and the eigenfunction construction.","supporting_citations":[{"cited_title":"Linear dependence of time-frequency shifts of a Schwartz function","cited_arxiv_id":"2608.05044","evidence_quote":"Supplies the vector-Zak and cohomological architecture of the twelve-point counterexample that this paper compresses to four points."},{"cited_title":"Zeevi,Analysis of multiwindow Gabor-type schemes by frame methods, Appl","cited_arxiv_id":null,"evidence_quote":"Introduces the two-component vector Zak transform used to convert Weyl shifts into a matrix cocycle."},{"cited_title":"Comput.66(2017), no","cited_arxiv_id":null,"evidence_quote":"Provides the interval-arithmetic engine for the outward-rounded certificates (C1)-(C4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the uniqueness theorem for dominated splittings used to single out the invariant line."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the smooth-Zak characterization that upgrades the smooth vector-Zak section to a Schwartz function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the HRT conjecture and the minimal-independence background that makes four points the smallest possible counterexample."}],"review_version":1}