{"id":"de224d36-8551-458a-b0d6-f407630b8ea4","arxiv_id":"2608.08190","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Four time-frequency shifts of a nonzero Schwartz function can be linearly dependent, and no smaller number can, so the minimum cardinality of a dependent Gabor system is exactly 4.","lead":"This paper proves that four time-frequency shifts of one Schwartz function can be linearly dependent, and that three never can, so four is the exact minimum. The result sharpens the recent twelve-shift counterexample to the HRT conjecture down to the smallest possible number.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof hinges on an unreproducible interval certificate (Prop. 3.2 / bounds (32)); the central claim needs an independent recomputation of those bounds before it can be accepted.","rationale":"I read the proof in full. The Zak reduction is standard; the rank-two bundle and sewing are consistent; the determinant bound in (14) is correct; the rational return identities (20)–(21) are plausible, and I independently re-derived the determinant factorization; Lemma 3.1's eigenvalue modulus gap and winding argument work, with the stated winding (1,-1) for trR and detR apparently a harmless typo for (2,-2) that cancels in the quotient; the §4 perturbation estimates give positive margins; the Diophantine bound in Lemma 5.1 is sound because ζ is an algebraic integer with all conjugates O(|m|+|n|); the cohomological flattening is standard once the winding vanishes; and the unfolding to Schwartz functions is legitimate. The single load-bearing unsupported premise is the numerical certificate, which is unreproducible as shipped. This is a verification gap rather than a demonstrated mathematical error, so the reader's CONDITIONAL verdict remains appropriate: the construction should not be fully accepted until an independent computation confirms (32).","tokens_in":10189,"tokens_out":24300,"duration_ms":238934,"concrete_test":"Write an independent implementation of the certificate: evaluate formulas (13), (19), (26), (27) over the 128×128 rational grid with the stated dyadic splitting and outward-rounded 160-bit Arb/Acb arithmetic, and require the output to satisfy the three inequalities in (61). If the recomputed global bounds reproduce (32), the load-bearing numerical premise is validated; if they do not, the central claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction depends on Proposition 3.2: at the rational parameter θ=(1/3,1/3), the three-step projective return has a uniformly contracting graph line, certified solely by the finite interval-arithmetic bounds in (32). Every downstream step—quantitative continuation in §4, winding and cohomology in §5, and unfolding in §6—uses the existence and triviality of this rational dominated line. The certificate is described, but no code, log, or machine-checkable output is supplied. The claimed global bounds d≥15.4044994528652104323, b≤0.2499996482177990124, c≤0.7725133354867000155 rest on evaluating formulas (13), (19), (26), (27) in Arb/Acb at 160-bit precision. A bug in the interval evaluation, a mis-specified sewing holonomy, or an error in the unproved Laurent identities (20)–(21) would invalidate Lemma 3.1 and collapse the construction. These are concrete, checkable issues rather than internal contradictions; the rest of the argument is coherent, and I found no mathematical inconsistency in the analytic structure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the minimum cardinality of a dependent finite Gabor system generated by a nonzero L2(R) function is N* = 4. The main theorem, Theorem 1.1, constructs an explicit Schwartz function f and a scalar λ such that (I + (1/2)W(1,0) + (1/2)W(0,1/2))W(α,β/2)f = λf for α = 1/3 + 10^(-12)√2 and β = 1/3 + 10^(-12)√3. Since the Heil–Ramanathan–Topiwala theorem already excludes dependence for three or fewer shifts, this establishes the sharp threshold. The proof uses a rank-two Zak bundle reduction, analyzes a rational model at θ = (1/3,1/3) whose three-step return has a dominated contracting line, proves the existence of this line by a large interval-arithmetic certificate (Proposition 3.2), then carries the invariant line to the explicit irrational parameter by a quantitative perturbation argument. A winding calculation and a Diophantine cohomology equation flatten the corresponding scalar multiplier, and inverse Zak folding yields the desired Schwartz function.","tokens_in":10356,"tokens_out":21611,"duration_ms":194711,"significance":"If the proof is correct, the result is a substantial advance: it reduces the known dependent Gabor system from twelve time–frequency shifts to four, and it identifies the sharp cardinality threshold, resolving a natural question left open by the recent disproof of the HRT conjecture. The architecture of the proof is original and broadly interesting: a rational finite model supplies a dominated line that cannot itself violate HRT because of Linnell's theorem, and an explicit algebraic perturbation plus Diophantine cohomology turns that geometric object into a genuine counterexample. The paper also gives concrete parameters and a directly checkable linear relation, so the central claim is falsifiable. The main strength is the combination of a clean analytic reduction with quantitative estimates; the main weakness is that the load-bearing interval certificate is not independently reproducible from the manuscript as written.","major_comments":[{"comment":"The entire construction rests on the finite interval certificate that establishes Proposition 3.2, but the manuscript does not supply the code, the exact input formulas, or a machine-checkable certificate that produced the bounds in (32). The description of the 128x128 grid, dyadic splitting, and 160-bit Arb/Acb precision is not enough for an independent referee to verify the global bounds d>=15.404..., b<=0.249999..., c<=0.7725... within reasonable effort. These bounds are load-bearing: Proposition 4.1, the continuation argument, and hence the existence of the invariant line for the explicit parameters all depend on them. Please provide the verifying program and its exact output, or a formal certificate (for example, a proof-producing interval log), so that the computation can be independently rerun.","section":"Appendix A, Proposition 3.2 and bounds (32)"},{"comment":"The identities for trP and detP are stated only as the result of exact multiplication, with the note that they 'may be checked exactly over Q(r)[u^{±1},v^{±1}]' deferred to Appendix A. These identities are used directly in Lemma 3.1 to obtain |trP|^2 = 27/4 and |detP| <= 39/32, which is the basis for the uniform domination gap. An algebraic identity of this kind is easy to verify with a computer algebra system, but it is still a load-bearing step in the proof; please include a derivation or a short script that verifies (20)-(21).","section":"Section 3, Eqs. (20)-(21)"}],"minor_comments":[{"comment":"The box counting '1,043,952 evaluated boxes and 787,060 certified leaf boxes' is not fully transparent: it is unclear whether a box that fails at one level and is split is counted once or several times in the evaluated count. Please clarify the counting convention.","section":"Appendix A"},{"comment":"The nonvanishing of ζ is justified by the linear independence of 1, √2, √3 over Q, but the argument would be clearer if it explicitly stated that ζ = 0 forces m = n = 0 and hence (m,n) = 0, contradicting the choice of (m,n).","section":"Proof of Lemma 5.1"},{"comment":"The displayed inequality '>5.3·10^(-6) > 0' contains a redundant nested inequality; a single lower bound, for example '>= 5.3·10^(-6)', would be read more cleanly.","section":"Section 4, Eq. (44)"}],"recommendation":"major_revision","confidential_remarks":"The result is significant and appears to be within the journal's scope. The main obstacle is reproducibility: the interval certificate is not verifiable from the manuscript, and the algebraic identities (20)-(21) are not proved. If the authors provide code/data and a derivation of those identities, I would be willing to accept. The novelty relative to the twelve-shift construction is clear. No issues with attribution or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The result is the real thing: N*=4, sharp, with an explicit Schwartz function and explicit parameters. The proof architecture is genuinely new—rank-two Zak bundle, rational three-step return with a dominated line, quantitative continuation, winding plus Diophantine cohomology to flatten the multiplier, and inverse unfolding. I read through the analytic steps and found no internal inconsistency; the perturbation estimates in §4 are coarse but the margins are large, and the winding/cohomology section is sound. The reliance on [5] for the Zak reduction is legitimate, and the lower bound from HRT is external. This would be a major within-field result if the certificate holds up.\n\nThe soft spot is exactly the one the stress-test names: Proposition 3.2 depends on a 1,043,952-box interval arithmetic computation, and no code or machine-checkable log is shipped. The description is detailed—Arb/Acb, 160-bit precision, outward rounding, refinement scheme—and the stated bounds have a real margin (e.g., b is just under 1/4, d is ~15.4), which makes a systematic bug less likely. But 'less likely' is not the same as verified. A referee can't re-run the computation without reimplementing it from scratch. The paper also asserts the Laurent identities (20)–(21) without derivation; the authors note they are exact over Q(r) and can be checked, but a four-line verification would remove an annoyance. A second issue: the claim that replacing the holonomy fails as a negative control is a nice sanity check, but it doesn't substitute for a certificate that someone else can execute.\n\nI would not block the paper on these points, but I would insist that the authors upload the certificate-generating code or a machine-checked certificate before publication. The rest of the proof is contingent on those bounds, so the reproducibility burden is high. If the bounds check, the result is almost certainly correct; the analytic argument around them is careful and the margins are comfortable.\n\nBottom line: send this to referees, but ask for the computational artifact as a condition of acceptance. I'd bring it to reading group, and I'd cite it after the certificate is independently confirmed.","headline":"A sharp resolution of the minimal Gabor dependence question, with a proof that rests on a large interval certificate that should be independently reproducible before the result is treated as settled.","tokens_in":10943,"tokens_out":2627,"would_cite":false,"duration_ms":25211,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","42A38","47A16"],"pacs":[],"model":"deepseek-v4-flash","headline":"Four time–frequency shifts of a nonzero Schwartz function can be linearly dependent, and no configuration of three shifts can be.","keywords":["HRT conjecture","Gabor systems","time-frequency shifts","Zak transform","dominated cocycles","Diophantine cohomology","interval arithmetic","linear dependence"],"falsifier":"Re-evaluate the bounds (32) with an independent rigorous interval package or a formal proof checker, evaluating the explicit formulas (13), (19), (26), and (27) on a fine dyadic grid; any certified leaf box violating $d \\ge 15.4044994528652104323$, $b \\le 0.2499996482177990124$, or $c \\le 0.7725133354867000155$ would disprove Proposition 3.2. Alternatively, symbolically verify the Laurent identities (20)–(21); if they fail, the winding and domination arguments collapse.","tokens_in":9925,"feed_emoji":"📐","tokens_out":5795,"duration_ms":55583,"temperature":0.7,"pith_summary":"This paper establishes that the smallest possible number of time–frequency shifts of a nonzero square-integrable function that can be linearly dependent is four. The previous state of the art had shown only that twelve shifts suffice and that three never do. The paper constructs an explicit nonzero Schwartz function and an explicit set of four points in the time–frequency plane at which the shifted functions satisfy a nontrivial linear relation. If the proof is correct, this closes the finite-cardinality version of the HRT question exactly. The construction is complex-valued, in a way that is forced by known four-point independence results for real-valued windows.","feed_headline":"Four time–frequency shifts can be dependent; three never are","feed_subtitle":"An explicit Schwartz window with four linearly dependent shifts settles the sharp threshold.","key_machinery":"The central mechanism is the rank-two Zak bundle over the torus associated with the lattice generated by $(1,0)$ and $(0,1/2)$, together with the matrix field $A(x,\\omega)=I+\\tfrac12 e^{-\\pi i\\omega}Z_0+\\tfrac12 e^{\\pi i x}X_0$, whose uniform invertibility makes the three-term lattice operator a bundle automorphism. The argument concentrates on the three-step projective return $P$ at the rational translation $\\theta=(1/3,1/3)$, whose eigenline is shown by a $1{,}043{,}952$-box interval certificate to be a uniformly contracting graph over a trivial line. Quantitative perturbation carries this dominated line to the explicit algebraic translation, and a winding calculation plus a Diophantine cohomological equation flatten the scalar multiplier to a constant $\\lambda$. Inverse Zak folding turns the resulting section into the Schwartz window $f$.","core_discovery":"The paper's central claim is Corollary 1.2: the minimum cardinality $N^*$ of a dependent finite Gabor system generated by a nonzero $L^2(\\mathbb R)$ function is $4$. Theorem 1.1 provides the witness: with $\\alpha=\\frac13+10^{-12}\\sqrt2$ and $\\beta=\\frac13+10^{-12}\\sqrt3$, there are a nonzero $f\\in\\mathcal S(\\mathbb R)$ and a nonzero scalar $\\lambda$ such that $\\bigl(I+\\tfrac12 W(1,0)+\\tfrac12 W(0,1/2)\\bigr)W(\\alpha,\\beta/2)f=\\lambda f$. Expanding by the Weyl commutation relations turns this into a nontrivial linear dependence among four distinct time–frequency shifts. Since a result of the paper's predecessors shows every three-point system is independent, the four-point example is sharp.","pith_inferences":["The chosen $10^{-12}$ perturbation size is a convenience, so the same construction is likely to work for a wide band of nearby parameters, possibly with simpler explicit values.","The method plausibly extends to other subcritical covolumes where the natural Zak bundle has rank higher than two, potentially yielding sharp thresholds for larger minimal cardinalities.","A machine-checkable formalization of the appendix's interval certificate would remove any doubt about the computer-assisted step, since the paper provides the bounds but not the verifying code."],"forward_implications":["The cardinality threshold $N^*=4$ holds for windows in $L^2(\\mathbb R)$ and for windows in the Schwartz class.","Any attempt to build a dependent system with three time–frequency shifts is provably futile; four is the exact boundary.","The explicit example must be complex-valued, because real-valued windows retain four-point independence.","The near-rational, algebraically irrational parameters are essential: the rational model is lattice-contained and therefore independent by known results, while the irrational perturbation destroys the lattice obstruction."],"supporting_citations":[{"why":"Supplies the rank-two Zak transform machinery and the earlier twelve-shift counterexample that the present construction improves.","marker":"[5]"},{"why":"Proves three-point independence, giving the lower bound $N^* \\ge 4$.","marker":"[11]"},{"why":"Provides the arbitrary-precision interval arithmetic model used by the finite certificate in Appendix A.","marker":"[12]"},{"why":"Shows lattice-contained configurations are independent, explaining why the rational model is only a starting point.","marker":"[13]"},{"why":"Proves four-point independence for real-valued windows, showing the new example must be complex-valued.","marker":"[9]"}],"fun_headline_variants":["Four Gabor shifts can be dependent, three never are","Sharp threshold: dependent Gabor systems need 4 shifts","Minimum dependent Gabor system has exactly four shifts","Smallest dependent Gabor system: four time-frequency shifts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the appendix's interval-arithmetic certificate that the rational three-step return has a uniformly contracting line; a bug in that million-box computation would undo the proof.","fun_headline_variants_meta":{"raw":{"variants":["Four Gabor shifts can be dependent, three never are","Sharp threshold: dependent Gabor systems need 4 shifts","Minimum dependent Gabor system has exactly four shifts","Smallest dependent Gabor system: four time-frequency shifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000888,"raw_usage":{"total_tokens":3860,"prompt_tokens":1000,"completion_tokens":2860,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":2796}},"tokens_in":616,"tokens_out":2860,"duration_ms":22126,"temperature":1.0,"reasoning_tokens":2796,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:20:21.219084+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-evaluate the bounds (32) with an independent rigorous interval package or a formal proof checker, evaluating the explicit formulas (13), (19), (26), and (27) on a fine dyadic grid; any certified leaf box violating $d \\ge 15.4044994528652104323$, $b \\le 0.2499996482177990124$, or $c \\le 0.7725133354867000155$ would disprove Proposition 3.2. Alternatively, symbolically verify the Laurent identities (20)–(21); if they fail, the winding and domination arguments collapse.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves three-point independence, giving the lower bound $N^* \\ge 4$."},{"cited_title":"Linnell, Von Neumann algebras and linear independence of translates,Proc","cited_arxiv_id":null,"evidence_quote":"Shows lattice-contained configurations are independent, explaining why the rational model is only a starting point."},{"cited_title":"The HRT Conjecture for Symmetric Configurations and Real-Valued Functions","cited_arxiv_id":"2607.26878","evidence_quote":"Proves four-point independence for real-valued windows, showing the new example must be complex-valued."}],"review_version":1}