{"id":"acd861ca-7853-45fe-901f-ae6c93a897b4","arxiv_id":"2607.03472","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Anderson localization holds almost surely near the spectral edge for the 1D Anderson-Bernoulli model whenever the long-range hopping has a rational Laurent symbol.","lead":"The paper proves Anderson localization near the bottom of the spectrum for the one-dimensional Anderson-Bernoulli operator with long-range hopping whose Laurent symbol is rational. This supplies the first rigorous localization theorem for pure Bernoulli potentials with non-nearest-neighbor hopping and confirms a 1987 numerical conjecture for exponential decay.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Theorem 1.1) rests on a complete multi-scale analysis whose only non-classical input is the deterministic QUC for rational Laurent symbols. That input is proved from first principles by reducing to a short-range recurrence, and the lower-bound conditions required for R0/R- are automatically satisfied inside the energy window used for localization. Counterexamples demonstrate sharpness rather than a flaw. The free-site construction, Wegner estimate via rank-one perturbation, and energy-elimination step are standard adaptations of Bourgain-Kenig / Ding-Smart techniques and contain no evident circularity or missing estimate. Consequently the reader's ACCEPT / HIGH-confidence verdict stands; the concrete check above is a routine verification that the cone property survives the Dirichlet restriction, not a threat to the result.","tokens_in":51586,"tokens_out":633,"duration_ms":5457,"concrete_test":"Independently re-derive the pointwise recurrence (2.33) for the Dirichlet problem from the short-range identity (2.32) and verify that the cone estimates (2.19)/(2.24) still produce a D_hop(T)-net of free sites of density ≥#eS inside every L'-interval; if the net fails for any T∈R+ (e.g., the pure exponential hopping (1.10)), the free-site argument of §4.2 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (QUC Theorems 2.3/2.5 and the lower-bound conditions (2.10)/(2.12) for R0/R-) is correctly identified as the non-standard ingredient, but it is not a soft spot that undermines Theorem 1.1. The paper proves the cone property (2.19)/(2.22) by rewriting the long-range equation as a short-range recurrence via the coprime Laurent factorization F=P/Q (2.7)-(2.8), then obtains free-site transversality (2.29)-(2.31) for the Dirichlet problem. Counterexamples 2.2 and 2.4 show the conditions are essentially sharp. In the MSA Wegner estimate (Claim 4.3 / §4.2), the energy window E∈[2^{-6000 d_T}δ,δ] together with the Bernoulli values {0,1} automatically produces a uniform positive lower bound b∼δ (see (4.45)-(4.48)), so the QUC applies without extra hypotheses. The subsequent free-site density, rank-one movement (Lemma E.1) and Peierls argument close the localization proof. No internal inconsistency or hidden gap appears in the chain from QUC to LDT to localization.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves Anderson localization near the lower spectral edge for the one-dimensional Anderson–Bernoulli operator H = T + λV, where V is i.i.d. Bernoulli and the hopping T has a rational Laurent symbol (class R) satisfying the normalization (A3). The argument proceeds by establishing a deterministic quantitative unique-continuation principle (Theorems 2.3 and 2.5) for such T, obtaining an initial-scale large-deviation theorem for the Green function via Floquet–Bloch analysis and a quantitative uncertainty principle (Theorem 3.1 / 3.5), then running a multi-scale analysis that produces a free-site Wegner estimate (Claim 4.3) and off-diagonal decay at large scales (Theorem 4.1). Energy dependence is removed by a Peierls argument, yielding pure-point spectrum and exponential eigenfunction decay on [0, δ] almost surely (Theorem 1.1). Counterexamples show that QUC fails for general exponentially decaying hoppings, while a weaker super-exponential result is proved in Appendix A.","tokens_in":51946,"tokens_out":680,"duration_ms":5385,"significance":"This appears to be the first rigorous localization theorem for a long-range Anderson model with pure Bernoulli potentials. It affirmatively settles the numerical conjecture of Yeung–Oono for hoppings of the form a^{-|n|} (which lie in R) and supplies a dimension-independent MSA framework that may extend to higher-dimensional long-range models once a suitable QUC is available. The rational-symbol QUC, the free-site adaptation of the initial-scale LDT, and the careful handling of multiple minima are genuine technical contributions that go beyond the short-range Bernoulli literature (Bourgain–Kenig, Ding–Smart, Li–Zhang).","major_comments":[],"minor_comments":[{"comment":"In the inductive construction of the free-site set S_remaining (4.33)–(4.34) and the density estimate (4.35), the removal of boundary-intersecting intervals of S_in is mentioned only parenthetically; a short explicit sentence would make the argument self-contained.","section":null},{"comment":"The enormous numerical exponents (6000 d_T, 7000 d_T, etc.) that appear throughout Sections 3–4 are chosen for convenience; a remark that any sufficiently large absolute constants work would improve readability.","section":null},{"comment":"Appendix A (weak QUC for super-exponential decay) is independent of the main theorem; a one-sentence pointer in the introduction would help the reader decide whether to consult it.","section":null},{"comment":"A few typographical slips remain (e.g., “Analoguely” on p. 7, occasional missing spaces around Vinogradov symbols). A light copy-edit would clean them.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the logical chain is complete and the novelty is clear. I see no reason to request a major revision; the empty major-comments list is intentional. Suitable for a strong analysis or mathematical-physics journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first localization theorem for the Anderson-Bernoulli model with genuinely long-range hopping, even on the line. The authors take hopping whose Laurent symbol is rational, prove a deterministic quantitative unique-continuation principle (with free-site version), and run a multi-scale analysis that produces Anderson localization near the edge. That settles a case that has been numerically visible since Yeung-Oono 1987 and that was open for pure Bernoulli potentials.\n\nWhat they do well is keep the analytic input sharp and the probabilistic machinery standard. The QUC (Theorems 2.3 and 2.5) comes from rewriting the long-range equation as a short-range recurrence via the coprime factorization F = P/Q; the cone property then gives full-dimensional transversality on free sites. Counterexamples 2.2 and 2.4 show the rational restriction and the lower-bound conditions for equal or reversed degrees are essentially necessary. In the Wegner estimate the energy window and Bernoulli values automatically supply the needed lower bound, so no extra hypothesis is smuggled in. The initial-scale LDT (Floquet-Bloch + quantitative uncertainty) handles multiple minima cleanly, the free-site density is preserved under the hierarchy, and the Peierls argument closes localization. Scales and constants are tracked carefully; the self-citations are to their own alloy-type and hierarchical papers and do not create circularity.\n\nThe soft spots are real but proportional. Localization is only for rational symbols (a proper subclass of exponential decay), only near the edge, and only in one dimension; the method is dimension-independent but the QUC is not. For super-exponential decay they get only a weaker unique-continuation statement that is not used for localization. None of this undercuts the main theorem under the stated hypotheses.\n\nThis is for people who work on Anderson localization with singular potentials or long-range operators. The math is solid, the literature engagement is honest, and a serious editor should send it to referees. I would cite it and would bring it to reading group.","headline":"First pure-Bernoulli localization for long-range hopping on Z, via a sharp rational-symbol QUC that makes free-site MSA work; the argument is complete and the restrictions are honest.","tokens_in":52491,"tokens_out":514,"would_cite":true,"duration_ms":6433,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B80","82B44","35J10","60H25"],"pacs":[],"model":"grok-4.5","headline":"Anderson localization holds near the edge for 1D Bernoulli disorder with long-range rational hopping.","keywords":["Anderson localization","Bernoulli potential","long-range hopping","unique continuation","rational Laurent symbol","multi-scale analysis","Wegner estimate"],"falsifier":"Exhibit a hopping with rational Laurent symbol for which the Green-function large-deviation estimates of Theorems 3.5 or 4.1 fail at arbitrarily large scales near the edge, or construct an unbounded solution of (T+V)u=0 that violates the density lower bound of Theorem 2.5.","tokens_in":52510,"feed_emoji":"📐","tokens_out":559,"duration_ms":5005,"temperature":0.7,"pith_summary":"The paper proves that a one-dimensional Schrödinger operator with long-range hopping and pure Bernoulli random potentials still localizes near the bottom of its spectrum, provided the hopping has a rational Laurent symbol. Pure Bernoulli potentials lack the regularity that usually supplies a Wegner estimate, so the authors first establish a quantitative unique-continuation principle that forces eigenfunctions to be large on a positive-density set of free sites. That transversality, combined with multi-scale analysis, yields the missing control on eigenvalues and therefore exponential decay of Green functions. The result answers a numerical conjecture for hoppings that decay like a geometric series, and supplies the first rigorous localization theorem for long-range Anderson models with discrete disorder. Counter-examples show that unique continuation can fail for non-rational symbols, so the rationality hypothesis is essentially sharp for the method.","feed_headline":"Bernoulli disorder localizes with long-range rational hopping","feed_subtitle":"First proof that pure two-point potentials still pin eigenfunctions when hops decay geometrically","key_machinery":"Quantitative unique continuation (Theorems 2.3 and 2.5) for operators with rational Laurent symbols: any solution of (T+V)u=0 that is normalized at the origin cannot decay faster than exponentially on a positive-density subset of free sites of controlled length, supplying the transversality needed for a free-site Wegner estimate.","core_discovery":"For any self-adjoint long-range hopping T on Z whose Laurent symbol is rational and whose Fourier symbol satisfies the normalization that its range is [0,1], the Anderson-Bernoulli operator H = T + λV localizes almost surely on a non-empty interval [0,δ] at the lower spectral edge.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Rational long-range hopping pins Anderson-Bernoulli eigenstates at edge","First localization for pure Bernoulli long-range Anderson on Z","Unique continuation forces edge localization under rational hops","Bernoulli potentials localize with rational Laurent long-range hopping","Long-range rational hops yield Anderson localization near spectral edge"],"cache_read_input_tokens":49280,"weakest_assumption_plain":"The hopping must have a rational Laurent symbol so that a deterministic quantitative unique-continuation principle holds and produces free-site transversality; without it the Wegner estimate fails.","fun_headline_variants_meta":{"raw":{"variants":["Rational long-range hopping pins Anderson-Bernoulli eigenstates at edge","First localization for pure Bernoulli long-range Anderson on Z","Unique continuation forces edge localization under rational hops","Bernoulli potentials localize with rational Laurent long-range hopping","Long-range rational hops yield Anderson localization near spectral edge"]},"model":"grok-4.5","effort":"low","cost_usd":0.005022,"raw_usage":{"total_tokens":1317,"prompt_tokens":624,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":50220000,"prompt_tokens_details":{"text_tokens":624,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":611,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":624,"tokens_out":82,"duration_ms":4579,"temperature":1.0,"reasoning_tokens":611,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T02:14:35.401977+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a hopping with rational Laurent symbol for which the Green-function large-deviation estimates of Theorems 3.5 or 4.1 fail at arbitrarily large scales near the edge, or construct an unbounded solution of (T+V)u=0 that violates the density lower bound of Theorem 2.5.","supporting_citations":[],"review_version":1}