{"id":"53356daa-325e-4e92-bf46-38c49f94f05a","arxiv_id":"2511.10358","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Observability for Schrödinger and heat equations on lattices is equivalent to a local arithmetic condition on the set, revealing a discrete obstruction absent in Euclidean settings.","lead":"The paper shows that for Schrödinger equations on one-dimensional lattices, a set is observable precisely when it meets a local arithmetic condition, unlike the thickness requirement in continuous space. The same holds for the heat equation, with extensions to higher-dimensional lattices and discrete tori where density alone fails.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Necessity of the local arithmetic condition may fail without a rate on V(n)→c, as slow-varying tails can distort quasi-mode constructions used for the 'only if' direction.","rationale":"The reader's weakest assumption correctly isolates the asymptotic hypothesis on V. The load-bearing risk is not the mere presence of the hypothesis but whether convergence alone suffices for the explicit constructions that establish necessity; a slow tail introduces a technical gap that is not addressed by the abstract statement and would move the verdict from UNVERDICTED to CONDITIONAL pending a rate assumption or a proof that works with plain convergence.","tokens_in":1621,"tokens_out":440,"duration_ms":51451,"concrete_test":"Fix c=0 and the arithmetic-violating set E=2ℤ. Construct V_ε(n)=ε·sign(n)/log(2+|n|) for ε>0 small. Analytically re-derive the necessity estimate of §3 (or the corresponding section) replacing the constant-potential plane waves by WKB or scattering solutions for this V_ε; if the error term cannot be made smaller than the observation deficit uniformly in ε, the necessity proof requires a decay rate on V−c.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim asserts an if-and-only-if between observability and a local arithmetic condition on E, under the sole hypothesis that V(n)→c. For the necessity direction, the argument presumably constructs (approximate) solutions to the time-dependent Schrödinger equation that remain small on E for all times by perturbing explicit oscillatory solutions of the constant-potential operator. When V converges to c with no quantitative rate, the perturbation is not compact in the usual sense and can accumulate phase shifts over long distances. This risks producing a non-trivial solution whose observation on E does not vanish, violating necessity, or conversely allowing an arithmetic-violating E to become observable because the tail of V localizes energy away from the arithmetic gaps. The sufficiency direction is less sensitive to the rate, but the equivalence therefore rests on an unstated quantitative control that is not implied by mere convergence.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves if-and-only-if characterizations of observable sets for Schrödinger equations on combinatorial graphs. For the one-dimensional discrete Laplacian plus potential with V(n)→c as |n|→∞, a set E⊂Z is observable at some time (equivalently, at every time) if and only if it satisfies a local arithmetic condition; the same criterion governs observability for the associated heat equation. In higher-dimensional lattices the complement of any finite set is observable. On discrete tori the authors supply arithmetic criteria showing that positive density alone does not guarantee observability.","tokens_in":1834,"tokens_out":501,"duration_ms":17972,"significance":"If the central equivalences hold, the work is significant because it isolates an arithmetic obstruction to observability that has no counterpart in the Euclidean theory (where thickness is decisive) and supplies explicit, checkable conditions on graphs. The extension to the heat equation and the higher-dimensional and toroidal results further strengthen the contribution.","major_comments":[{"comment":"§1 (main theorem for 1D lattices): the necessity direction asserts that any E violating the local arithmetic condition fails to be observable, yet the argument is stated under the sole hypothesis V(n)→c with no quantitative rate. Without a rate, the perturbation of constant-potential oscillatory solutions may accumulate uncontrolled phase shifts, so that the constructed approximate solution need not remain small on E; this directly threatens the claimed equivalence.","section":"§1 (main theorem for 1D lattices)"},{"comment":"§3 (proof of necessity): the quasi-mode construction for the 'only if' direction is not shown to be robust under mere convergence of V; a concrete counter-example or a lemma establishing that the error remains o(1) uniformly in time would be required to close the argument.","section":"§3 (proof of necessity)"}],"minor_comments":[{"comment":"The precise statement of the 'local arithmetic condition' should be displayed as a numbered definition or displayed equation rather than introduced only in prose.","section":"Introduction"},{"comment":"Notation for the discrete Laplacian and the observation operator should be fixed once at the beginning and used consistently; minor inconsistencies appear in the toroidal section.","section":"Notation and §4"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying the need to strengthen the presentation of the necessity argument in the one-dimensional case. The comments correctly note that the current write-up of the quasi-mode construction under the mere assumption V(n)→c would benefit from more explicit error control. We address both points below and will revise the manuscript accordingly.","responses":[{"response":"We agree that an explicit quantitative control is desirable for clarity. The quasi-mode is constructed by taking a high-frequency oscillatory solution for the constant-potential operator and localizing it far from the origin, where |V(n)−c| can be made arbitrarily small. Because the time horizon T is fixed in advance, the location of the support can be chosen after T so that the Duhamel integral of the perturbation remains o(1) uniformly on [0,T]. We will add a short lemma (new Lemma 3.4) that makes this estimate precise, showing that the L²-norm of the error on E stays below any prescribed δ for sufficiently large frequency. This removes any possibility of uncontrolled phase accumulation and confirms the claimed equivalence.","revision_made":"yes","referee_comment":"§1 (main theorem for 1D lattices): the necessity direction asserts that any E violating the local arithmetic condition fails to be observable, yet the argument is stated under the sole hypothesis V(n)→c with no quantitative rate. Without a rate, the perturbation of constant-potential oscillatory solutions may accumulate uncontrolled phase shifts, so that the constructed approximate solution need not remain small on E; this directly threatens the claimed equivalence."},{"response":"We thank the referee for this precise remark. Rather than a counter-example (which we do not expect to exist), we will insert the lemma mentioned above. The lemma applies Duhamel’s formula to the difference of the two evolution operators, bounds the potential difference by an arbitrary ε outside a large but finite interval, and integrates over the fixed time interval [0,T]. The resulting error is then made smaller than any positive constant by choosing the quasi-mode support sufficiently far out. The revised proof will cite this lemma explicitly, thereby establishing the required uniform o(1) control.","revision_made":"yes","referee_comment":"§3 (proof of necessity): the quasi-mode construction for the 'only if' direction is not shown to be robust under mere convergence of V; a concrete counter-example or a lemma establishing that the error remains o(1) uniformly in time would be required to close the argument."}],"tokens_in":1308,"tokens_out":546,"duration_ms":24100,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that for the discrete Schrödinger operator on Z with V(n) tending to a constant, a set E is observable at some time (and hence any time) precisely when it meets a local arithmetic condition. The same condition works for the heat equation. In higher dimensions the complement of any finite set is observable, and on tori positive density is not enough; arithmetic gaps still matter. These distinctions from the continuous theory are the real novelty here and they are stated cleanly in the abstract and introduction.","headline":"The paper gives a clean arithmetic criterion for observability on 1D lattices that has no direct Euclidean analog, but the necessity direction may need a rate on how V approaches its limit.","tokens_in":2282,"tokens_out":187,"would_cite":false,"duration_ms":17103,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/ArithmeticFromLogic.lean","rs_theorem":"LogicNat recovery / embed_injective","paper_passage":"a set E⊂Z is observable at some time, equivalently at any time, if and only if it satisfies a local arithmetic condition"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"Theorem 2.2 (discrete Logvinenko-Sereda) via Beurling sampling on PW_a"}],"headline":"Arithmetic observability criteria on Z-lattices via Beurling density and Bohr sets; no overlap with RS forcing","alignment":"orthogonal","rationale":"Paper derives discrete Logvinenko-Sereda (Thm 2.2 via Beurling sampling), thickness thresholds γ=1/2, periodic Bohr-set arithmetic criterion (Thm 2.5: subgroup generated by D(R) = Z/pZ), and exterior observability on Z^d. These are harmonic-analysis results on combinatorial graphs. RS framework (reality_from_one_distinction, AbsoluteFloorClosure, AlexanderDuality_circle_linking forcing D=3, ArithmeticFromLogic.LogicNat recovery, J-cost uniqueness) contains no statements about observability inequalities, Schrödinger flows, or sampling sets on Z. Domain mismatch is total.","tokens_in":61268,"confidence":"high","tokens_out":344,"duration_ms":10437,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"On one-dimensional lattices with potentials approaching a constant, a set is observable for the Schrödinger equation exactly when it meets a local arithmetic condition.","keywords":["observable sets","Schrödinger equation","combinatorial graphs","lattice operators","arithmetic condition","observability","heat equation","discrete tori"],"falsifier":"A concrete set E on the integers that satisfies the local arithmetic condition yet fails to be observable for some potential converging to a constant at infinity.","tokens_in":2528,"feed_emoji":"","tokens_out":641,"duration_ms":24763,"temperature":0.7,"pith_summary":"The paper shows that for the discrete Schrödinger operator on the integers whose potential tends to a fixed constant far away, observability of a subset E holds at one time if and only if it holds at every positive time, and this occurs precisely when E obeys a local arithmetic condition. The same arithmetic test decides observability for the associated heat equation. In higher-dimensional lattices any cofinite set works, while on discrete tori the arithmetic conditions remain decisive and positive density alone is not enough. The result isolates an arithmetic obstruction that has no direct counterpart in the continuous Euclidean setting where thickness is the main requirement.","feed_headline":"Arithmetic condition decides observability on 1D lattices","feed_subtitle":"Sets must satisfy a local number-theoretic rule to control Schrödinger or heat flows when the potential settles to a constant at infinity.","key_machinery":"The local arithmetic condition on the set E, which is the necessary and sufficient requirement for observability and encodes the discrete arithmetic obstruction absent from continuous theory.","core_discovery":"For the one-dimensional lattice Schrödinger operator H = −Δ_disc + V with V(n) → c ∈ R as |n| → ∞, a set E ⊂ Z is observable at some time, equivalently at any time, if and only if it satisfies a local arithmetic condition. The same criterion characterizes observability for the heat equation on Z. In higher dimensions observability holds from the complement of any finite set, and on discrete tori arithmetic criteria apply with positive density insufficient.","pith_inferences":["The arithmetic obstruction suggests that discreteness introduces number-theoretic barriers to control that thickness conditions cannot capture in the continuous case.","Similar local arithmetic tests may characterize observability on other regular graphs or lattices with different connectivity.","Explicit periodic or lacunary sets can be checked against the condition to map the precise boundary between observable and non-observable sets."],"forward_implications":["Observability at one positive time is equivalent to observability at every positive time.","The identical local arithmetic condition governs observability for the heat equation on the same lattice.","In higher-dimensional lattices every set whose complement is finite is observable.","On discrete tori arithmetic criteria determine observability and positive density by itself does not suffice."],"fun_headline_variants":["Local arithmetic condition required for Schrödinger observability on Z","Same arithmetic rule controls both Schrödinger and heat equation on lattices","Higher dimensional lattices observable outside any finite set","Positive density insufficient for observability on discrete tori"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The potential must converge to a finite constant at infinity.","fun_headline_variants_meta":{"raw":{"variants":["Local arithmetic condition required for Schrödinger observability on Z","Same arithmetic rule controls both Schrödinger and heat equation on lattices","Higher dimensional lattices observable outside any finite set","Positive density insufficient for observability on discrete tori"]},"model":"grok-4.3","cost_usd":0.009144,"raw_usage":{"total_tokens":4066,"prompt_tokens":602,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":91437000,"prompt_tokens_details":{"text_tokens":602,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3404,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":602,"tokens_out":60,"duration_ms":38350,"temperature":1.0,"reasoning_tokens":3404,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-17T22:27:26.805703+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete set E on the integers that satisfies the local arithmetic condition yet fails to be observable for some potential converging to a constant at infinity.","supporting_citations":[],"review_version":1}