{"id":"170ff64f-f8a9-4921-a85e-8d69f8a36fa7","arxiv_id":"2607.26878","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"HRT holds for all symmetric (2n+1,2) configurations (and commensurable collinear spacings) for arbitrary L2 generators, and for all four-point configurations when the generator is real-valued.","lead":"The paper proves the long-open HRT conjecture for an infinite family of symmetric time-frequency point configurations, for every square-integrable generating function. As a corollary, every four-point configuration is linearly independent when the generator is real-valued.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claims are unconditional HRT statements for an infinite geometric family and, via the real-valued reduction, for all four-point configurations. The proof architecture is transparent: Linnell covers the fully rational subcases; the irrational cases reduce to N-independent two-sided control of |P|-product ratios supplied by Lemmas 1–2; those lemmas rest on a finite product of already-published Demeter–Zaharescu estimates and elementary Riemann-sum bounds. The reader correctly isolated Lemma 1/2 as the sole load-bearing analytic step. After reading the case split (|ρj|=1 versus ≠1), the exceptional-set measure arithmetic, and the combining paragraph, I find no concrete failure mode—only the ordinary need for a specialist to verify the ε-δ constants. That is already reflected in the reader's “low” correctness_risk and ACCEPT verdict. No adjustment is warranted.","tokens_in":17779,"tokens_out":579,"duration_ms":11086,"concrete_test":"Line-check the measure transfer |{t:{sj-at}∈A}|≤Ca|A| and the boundary-factor adjustment between product ranges {0..N-1} and {1..N} in the |ρj|≠1 case of Lemma 1; confirm that the resulting C1,C2 remain finite and N-independent when two on-circle roots collide or when a root lies at distance ≪1/N from the unit circle. If both checks hold, the load-bearing estimate stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption call on Lemma 1/2 is the right technical heart, but the manuscript's factorization-plus-product argument appears internally consistent. After writing P as a product of linear factors in ω=e^{2πiat}, each on-circle root is controlled by Demeter–Zaharescu (Props. 3–4) after removing a small exceptional set whose measure is O(δ) per root, while each off-circle root is controlled by a uniform Riemann-sum comparison whose constants depend only on ∥ϕ′_j∥_∞. The combining step multiplies finitely many (≤2n) such bounds and unions finitely many exceptional sets, producing N-independent C1,C2 on a set of measure ≥1-ε. The subsequent contradiction arguments (periodic telescoping when v∈Q; conjugate-orbit pairing when v∉Q; role-reversed periodicity when a∈Q) then go through exactly as in the classical (2,2) setting. No hidden dependence on root locations or vanishing coefficients survives Lemmas 2–3. I do not see an internal gap that would overturn Theorem 1 or Corollary 1.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves the HRT conjecture for the symmetric (2n+1,2) configurations Λ_n = {(0,k) : −n≤k≤n} ∪ {(a,b),(a,−b)} (ab≠0) for every n≥1 and every nonzero g∈L²(ℝ), and more generally when the collinear frequencies are an arbitrary nonempty commensurable subset of {−n,…,n}. The argument factors the collinear trigonometric polynomial into linear factors in ω=e^{2πiat}, obtains N-independent two-sided product bounds along integer orbits by combining Demeter–Zaharescu estimates (on-circle roots) with Riemann-sum comparisons (off-circle roots), and derives a contradiction to the decay g(t±N)→0 via periodic telescoping or a conjugate-orbit pairing according to the arithmetic of a and v=ab. As a stated consequence, the HRT conjecture is claimed for every four-point configuration when g is real-valued; the body proves this for the normal form {(0,0),(0,1),(s,0),(a,b)} by reducing a hypothetical dependence, via conjugation, to a symmetric (3,2) dependence ruled out by the main theorem.","tokens_in":17988,"tokens_out":1330,"duration_ms":96316,"significance":"The HRT conjecture remains open for general four-point configurations even under strong assumptions on g. Establishing an infinite family of (2n+1,2) configurations for arbitrary L² generators is a genuine advance, and the product-estimate technology (Lemmas 1–2) cleanly extends the Demeter–Zaharescu method beyond linear trigonometric polynomials. The real-valued four-point consequence, even if ultimately restricted to a normal form, is of clear interest and sits naturally with prior (2,2) and (1,3) results. The proofs are explicit, case-divided by arithmetic type, and rely on standard tools (Egorov, continued-fraction convergents, metaplectic invariance, conjugate trick) used in a transparent way.","major_comments":[{"comment":"Abstract and Introduction claim the HRT conjecture for every configuration of four distinct points when g is real-valued. Corollary 1 and its proof establish only the specific normal form Λ̃={(0,0),(0,1),(s,0),(a,b)}. The conjugation argument cancels the (s,0) term and produces a symmetric (3,2) dependence, which works for this geometry. A general four-point set Λ need not reduce to this form by time translation and rescaling alone; full symplectic/metaplectic normalization typically destroys real-valuedness of the window, and Λ∪Λ* need not be a configuration covered by Theorem 1. Either supply a reduction that preserves the real-valued hypothesis, or restrict the abstract/intro claims to the normal form actually proved (and to configurations reducible to it without leaving the real category).","section":"Abstract; §1; Corollary 1 and its proof"},{"comment":"In §2, after (1), non-vanishing of all c_i,d_j is justified by citing resolved (2,2) and (1,3) cases. For n=1 this is fine. Lemma 3 later handles vanishing for general n, but the n=1 write-up should explicitly note that a vanishing coefficient would collapse to a known independent configuration (or to g=0), so the reader can see that the quadratic P and two-term Q used in Lemma 1 are without loss. A one-sentence cross-reference to Lemma 3 (or a local argument) would close this.","section":"§2, paragraph after (1); cf. Lemma 3"}],"minor_comments":[{"comment":"Title on p. 1 and running header: “REAL-V ALUED” contains a spurious space (also “V ALUED” in the section title style).","section":"Title page / headers"},{"comment":"§2, line after the definition of S: “holdsuniformly” needs a space; similarly “casen=1” in the opening of §2.","section":"§2"},{"comment":"Proposition 4 is quoted as giving bounds on products from n=−N to −1 and n=0 to N−1; Lemma 1 uses ranges n=−N…−1 and n=1…N. The boundary-factor adjustment is mentioned for off-circle roots but could be flagged once globally so the range mismatch is not re-checked by the reader in every case.","section":"§2, Prop. 4 and Lemma 1"},{"comment":"In §3.3 Step 1, the integer m is defined by an identity involving t0,t1,n′,v,θ′; a brief remark that m is independent of the orbit index j (so the Q-pairing is exact for all j) would help.","section":"§3.3"},{"comment":"Figure 1 is helpful; labelling the two parallel lines explicitly as the supports of the (2n+1)- and 2-point subsets would match the (m,n)-configuration terminology used in the text.","section":"Figure 1"},{"comment":"Reference [17] is dated 2026 and described as a Lean formalization; ensure the citation is stable (arXiv id or DOI) before publication.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The core analytic contribution (factorization + multi-root Demeter–Zaharescu/Riemann product bounds, Theorems 1–2) looks correct and publishable. The only substantive mismatch is advertising “all four-point configurations for real g” while proving a specific normal form; this is an abstract/intro repair, not a rewrite of the proofs. I would not reject or send for major revision over it. Fit for a strong FA/harmonic-analysis journal is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"Punchline: they close the remaining symmetric (3,2) cases for arbitrary L2 generators, push the same method to the whole symmetric (2n+1,2) ladder (and commensurable collinear subsets), and get every four-point configuration when g is real-valued. That last corollary is the piece people will actually use.\n\nWhat is new is not a new orbit estimate. It is the observation that symmetry makes the collinear trigonometric polynomial a polynomial in one exponential, so it factors into linear terms. Demeter–Zaharescu then applies factor-by-factor; off-circle roots are handled by a plain Riemann-sum comparison. Lemma 1/2 is the technical heart, and the combining step (finite union of exceptional sets, product of finitely many N-independent constants) is written carefully. Rational cases drop to Linnell; the irrational cases recycle the usual telescoping / conjugate-trick contradictions. Coefficient non-vanishing is reduced to known (2,2) and (1,3) results plus a short vanishing lemma. Self-citations to Okoudjou’s earlier partial (3,2) and restriction/extension work are load-bearing but transparent.\n\nSoft spots are minor and proportional. I did not line-check every ε-δ constant in the multi-root product; a specialist should. The method stops at configurations whose collinear points are commensurable—exactly as the authors say—so it does not touch general (m,2). The real-valued four-point corollary is a conjugation trick, not a new geometric idea; still, it removes the main four-point bottleneck under a mild side condition. No data, no free parameters, no formal verification, but the case division is complete and the base-case hygiene looks solid.\n\nThis is for people who already care about HRT geometry. It deserves a serious referee. I would accept it for peer review and would cite the four-point real-valued statement.","headline":"Solid unconditional HRT theorems for symmetric (2n+1,2) configs and all four-point geometries when g is real-valued; the factorization-plus-DZ argument looks clean.","tokens_in":18648,"tokens_out":500,"would_cite":true,"duration_ms":9171,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","42C40","37A30"],"pacs":[],"model":"grok-4.5","headline":"Every symmetric (2n+1,2) time-frequency configuration generates a linearly independent Gabor system for any nonzero L2 function, and every four-point configuration does so when the function is real-valued.","keywords":["HRT conjecture","time-frequency analysis","Gabor systems","linear independence","symmetric configurations","irrational rotations","trigonometric polynomials"],"falsifier":"Exhibit a nonzero g in L2(R) and coefficients making g(t)P(t)=g(t-1)Q(t) hold almost everywhere for some symmetric (3,2) or (2n+1,2) configuration, yet |g(t plus or minus N)| tends to zero fast enough that the product ratio of |P| along those orbits is forced outside every N-independent bound on a full-measure set.","tokens_in":18639,"feed_emoji":"📐","tokens_out":991,"duration_ms":20557,"temperature":0.7,"pith_summary":"The HRT conjecture asks whether any finite set of distinct time-frequency shifts of a nonzero square-integrable function must be linearly independent. It is still open in general, even for four points. This paper proves the conjecture for an infinite family of symmetric configurations: 2n+1 equally spaced (or commensurably spaced) points on one vertical line together with a symmetric pair on a parallel line, for every nonzero g in L2(R). The same argument covers any nonempty commensurable subset of those collinear frequencies. As a direct consequence, when g is real-valued the conjecture holds for every configuration of four distinct points. The authors obtain the result by turning a hypothetical linear dependence into a relation between g along integer orbits and products of trigonometric polynomials, then showing those products cannot decay fast enough to match the L2 decay of g.","feed_headline":"Four-point HRT settled for every real-valued function","feed_subtitle":"Symmetry factors dependence into orbit products that cannot match L2 decay for any nonzero window.","key_machinery":"Factorization of the collinear trigonometric polynomial P into linear factors in omega = e to the 2 pi i a t, followed by simultaneous Demeter-Zaharescu product estimates (and Riemann-sum bounds for off-circle roots) that give N-independent two-sided control of the ratio of backward to forward products of |P| on a large-measure set of starting points.","core_discovery":"For every n greater than or equal to 1 and every nonzero g in L2(R), the Gabor system generated by the symmetric configuration Lambda_n = {(0,k) : -n less than or equal to k less than or equal to n} union {(a,b),(a,-b)} with ab not zero is linearly independent. The conclusion remains valid when the collinear set is replaced by any nonempty subset of those frequencies, provided the spacings stay commensurable. When g is real-valued this implies linear independence for every four distinct points in the time-frequency plane.","pith_inferences":["If the same factorization-plus-orbit method can be adapted to unequally spaced collinear points, a large further class of (m,2) configurations would fall.","The real-valued four-point result suggests that phase cancellation, rather than amplitude decay, is the essential remaining difficulty for complex windows.","Extension principles already in the literature may now push independence from these symmetric skeletons to nearby non-symmetric configurations without extra decay assumptions."],"forward_implications":["The HRT conjecture holds for every symmetric (3,2) configuration and every g in L2(R), closing the cases left open by earlier work.","Every four-point time-frequency configuration generates an independent Gabor system whenever the window is real-valued.","The same independence persists after multiplying a real-valued window by a quadratic phase factor.","Any (m,2) configuration whose collinear frequencies are commensurable reduces to the integer-spaced case already covered.","The remaining obstruction for general four-point HRT is isolated to the phase of a complex-valued window."],"fun_headline_variants":["HRT holds for every four-point set when the window is real-valued","Symmetric (2n+1,2) configs give HRT for all nonzero L2 windows","Four distinct TF points are independent for every real g in L2","Commensurable collinear spacings force HRT on symmetric families","Real-valued generators settle HRT for all four-point configurations"],"cache_read_input_tokens":128,"weakest_assumption_plain":"After factoring the collinear trigonometric polynomial, the product of its absolute values along forward and backward integer orbits stays bounded above and below independently of orbit length on a set of positive measure.","fun_headline_variants_meta":{"raw":{"variants":["HRT holds for every four-point set when the window is real-valued","Symmetric (2n+1,2) configs give HRT for all nonzero L2 windows","Four distinct TF points are independent for every real g in L2","Commensurable collinear spacings force HRT on symmetric families","Real-valued generators settle HRT for all four-point configurations"]},"model":"grok-4.5","effort":"low","cost_usd":0.003484,"raw_usage":{"total_tokens":1107,"prompt_tokens":735,"num_sources_used":0,"completion_tokens":99,"cost_in_usd_ticks":34844000,"prompt_tokens_details":{"text_tokens":735,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":273,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":735,"tokens_out":99,"duration_ms":5314,"temperature":1.0,"reasoning_tokens":273,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T18:20:28.752034+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a nonzero g in L2(R) and coefficients making g(t)P(t)=g(t-1)Q(t) hold almost everywhere for some symmetric (3,2) or (2n+1,2) configuration, yet |g(t plus or minus N)| tends to zero fast enough that the product ratio of |P| along those orbits is forced outside every N-independent bound on a full-measure set.","supporting_citations":[],"review_version":1}