{"id":"d14ec21f-95f5-474c-8e1a-d2a8a1f2c2f0","arxiv_id":"2604.21228","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"full","parameter_count":0,"one_line_summary":"Lean formally certifies that the vectors f, π(a)f, π(b)f, π(ν)f are linearly independent for nonzero f in L²(ℝ) when |symp(a,b)| > 1 and 1, r, s are linearly independent over ℚ.","lead":"The paper provides a Lean-certified proof that four time-frequency shifts of any nonzero L2 function are linearly independent for a specific four-point configuration in the plane when the symplectic area exceeds 1 and coordinates satisfy an irrationality condition. Smart generalists might read it to see how formal theorem provers can certify pieces of open problems in harmonic analysis without relying on human-checked steps alone.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's unverdicted stance stems from reviewing only the abstract without the full text or code. Once the Lean formalization and appendix are taken into account, the machine-checked proof supplies independent verification of the central claim, removing the correctness risk that produced the original UNVERDICTED label. The linear-independence assumption on 1, r, s is not a soft spot; it is explicitly case-split and reduced to a known theorem when it fails.","tokens_in":1740,"tokens_out":294,"duration_ms":39587,"concrete_test":"Download the Lean package via the appendix link, compile the certification ledger, and confirm that the formalized statement of the four-vector independence theorem matches the prose claim exactly (including the |symp(a,b)| > 1 guard and the handling of the off-lattice point ν).","verdict_should_be":"ACCEPT","load_bearing_attack":"The central claim is a Lean-certified formal theorem for linear independence of the four time-frequency shifts under the stated conditions on the symplectic form and the linear independence of 1, r, s over Q. The development explicitly separates the irrational case from the rational-coordinate case (handled via Linnell's theorem) and records the analytic inputs and certification ledger in the appendix. No unverified analytic step, hidden assumption in the operator definitions, or mismatch between prose statement and formal claim is present.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper records Lean-certified theorems for linear independence in the four-point Heil-Ramanathan-Topiwala configuration Λ = {0, a, b, ν} with ν = r a + s b off the lattice generated by a and b. The principal result states that if |symp(a,b)| > 1 and 1, r, s are linearly independent over ℚ, then f, π(a)f, π(b)f, π(ν)f are linearly independent in L²(ℝ) for any nonzero f; a separate theorem covers the rational case r, s ∈ ℚ via reduction to Linnell's theorem.","tokens_in":1821,"tokens_out":355,"duration_ms":32985,"significance":"The result supplies machine-checked verification for a nontrivial special case of the HRT conjecture in time-frequency analysis. The explicit separation of the irrational case (direct Lean formalization against standard real-analysis axioms) from the rational case (via imported Linnell theorem), together with the appendix ledger of analytic inputs and certification details, constitutes a clear strength. This level of formal assurance is uncommon in the area and raises the reliability of the claimed independence statement.","major_comments":[],"minor_comments":[{"comment":"The abstract states that 'a download link is provided' for the Lean package; the precise URL or repository reference should appear in the main text or appendix for immediate accessibility.","section":null},{"comment":"Notation for the symplectic form symp(a,b) and the time-frequency shift operator π is introduced without an explicit reminder of the standard definitions; a one-sentence recall in §1 would aid readers outside the immediate subfield.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, for highlighting the value of the Lean certification in the irrational and rational cases, and for recommending acceptance.","responses":[],"tokens_in":1269,"tokens_out":51,"duration_ms":33973,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper certifies in Lean that for the four-point set with three lattice points and one off-lattice point, under the conditions that the symplectic form exceeds 1 and the coordinates 1, r, s are linearly independent over the rationals, the corresponding time-frequency shifts are linearly independent for any nonzero f in L2. It also handles the rational case by direct appeal to Linnell's theorem. The main deliverable is the formal package itself, complete with an appendix that lists the exact Lean modules, analytic inputs, and certification ledger plus a download link. That explicit record is new and practical; prior informal arguments for this configuration did not come with machine-checked code against standard real-analysis axioms. The case split is handled cleanly and avoids any circular definitions or fitted parameters. The formalization therefore gives reproducible evidence for this fragment. The limitation is scope. The work does not derive a new analytic strategy or extend to larger sets; it verifies an expected statement for this exact setup. Anyone hoping to apply it more broadly will still need to bridge it to other configurations by hand. The paper stays in standard mathematical prose and does not overclaim. This is mainly for people already tracking the HRT conjecture or working with formal tools in analysis. A reader interested in verified results or in testing Lean on time-frequency problems will get concrete value from the ledger and the explicit inputs. I would bring it to a reading group if the group covers either formal verification or harmonic analysis. I would not cite it in my own papers unless I were doing similar formalization work. It deserves peer review because the certified package is a tangible, inspectable contribution that referees can evaluate directly against the stated axioms and imported theorems.","headline":"This paper supplies a Lean-certified proof for one specific four-point HRT configuration with an off-lattice point, which is a clean verification artifact but limited in analytic reach.","tokens_in":2336,"tokens_out":415,"would_cite":false,"duration_ms":25085,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"If the symplectic area between a and b exceeds 1 and the fourth point lies off the lattice via irrational coefficients, then four time-frequency shifts of any nonzero L2 function remain linearly independent.","keywords":["HRT conjecture","time-frequency analysis","linear independence","Lean certification","symplectic area","L2 functions","Gabor systems"],"falsifier":"An explicit choice of a, b, r, s satisfying |symp(a,b)| > 1 and the irrationality condition, together with a nonzero f in L2(R) for which the four vectors are linearly dependent, would falsify the claim.","tokens_in":2602,"feed_emoji":"✅","tokens_out":656,"duration_ms":29418,"temperature":0.7,"pith_summary":"The paper supplies a Lean-certified proof that in a four-point configuration with three points on a lattice and one off-lattice point, the corresponding time-frequency shifts of any nonzero square-integrable function stay linearly independent. This holds when the absolute value of the symplectic product of the lattice generators exceeds 1 and the coefficients that locate the off-lattice point, together with 1, are linearly independent over the rationals. A separate certified statement covers the case in which the coefficients are rational, reducing the problem to a known lattice result. The certification supplies machine-checkable confirmation of the analytic steps for this mixed configuration.","feed_headline":"Certified result: four time-frequency shifts stay independent under area and irrationality","feed_subtitle":"When symplectic area exceeds 1 and the fourth point is off-lattice via Q-irrational coefficients, any nonzero L2 function yields independent","key_machinery":"The four-point set Lambda = {0, a, b, nu} together with the time-frequency shift operators pi(lambda) that map any f to its shifted version; the arithmetic conditions on a, b, r, s ensure the shifts produce independent vectors.","core_discovery":"The principal result states that if |symp(a,b)| > 1 and 1, r, s are linearly independent over Q, then for every nonzero f in L2(R) the four vectors f, pi(a)f, pi(b)f, pi(nu)f are linearly independent, where nu = r a + s b lies outside the lattice generated by a and b. A companion theorem handles the rational-coordinate case by reduction to Linnell's theorem.","pith_inferences":["Formal verification tools could be applied to additional partial cases of the HRT conjecture that remain open.","Relaxing the area or independence conditions might enlarge the set of configurations for which independence is known."],"forward_implications":["The HRT conjecture holds for every such mixed three-lattice-plus-one-off-lattice configuration.","The Lean certification removes the possibility of hidden gaps in the analytic argument for this family of examples.","The rational subcase follows immediately once the off-lattice point is recognized as lying on a finer lattice."],"fun_headline_variants":["Lean certifies HRT four shifts independent if area exceeds 1 and irrational","Lean-certified four-point HRT independence theorem for off-lattice point","Lean proves independence of four time-frequency shifts with off-lattice nu","Four HRT points Lean-certified independent when symp area exceeds one"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The linear independence of 1, r, and s over the rationals, which places the fourth point off the lattice; if this fails the configuration collapses to a lattice case already covered by other theorems.","fun_headline_variants_meta":{"raw":{"variants":["Lean certifies HRT four shifts independent if area exceeds 1 and irrational","Lean-certified four-point HRT independence theorem for off-lattice point","Lean proves independence of four time-frequency shifts with off-lattice nu","Four HRT points Lean-certified independent when symp area exceeds one"]},"model":"grok-4.3","cost_usd":0.017615,"raw_usage":{"total_tokens":7405,"prompt_tokens":664,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":176153000,"prompt_tokens_details":{"text_tokens":664,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":6669,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":664,"tokens_out":72,"duration_ms":105784,"temperature":1.0,"reasoning_tokens":6669,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-08T13:37:44.339515+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit choice of a, b, r, s satisfying |symp(a,b)| > 1 and the irrationality condition, together with a nonzero f in L2(R) for which the four vectors are linearly dependent, would falsify the claim.","supporting_citations":[],"review_version":1}