{"id":"9108a18d-94b2-438b-b3a5-7d3f858bc00d","arxiv_id":"2607.26538","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Ergodic torus automorphisms with 2D center are stably ergodic, and graph families with regular integrated density of states have asymptotically dense delocalization.","lead":"This mathematics thesis proves new theorems in two areas: ergodic torus maps with a two-dimensional neutral direction remain ergodic under small smooth perturbations, and large graph Schrödinger operators have nearby potentials whose eigenvectors spread out. The first result advances a 1977 conjecture in smooth dynamics; the second gives a topological criterion for delocalization on graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.9.1's generalization from box graphs to arbitrary graph families rests on an unverified transfer of Lemmas 3.7.2–3.7.3; the paper asserts, but does not prove, that only smallness of IDS jumps is needed.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the general criterion Theorem 3.9.1 is not actually proven, only sketched. My analysis confirms that the proof for box graphs relies on specific quantitative properties (vertex count N^d, jump size 1/N) that are not directly available for arbitrary graph families. The paper's claim that Lemmas 3.7.2 and 3.7.3 reduce to smallness of IDS jumps is not accompanied by the necessary derivation. I also note that Lemma 3.7.2 is in fact a pure compactness/continuity statement and does not need IDS regularity, so the paper's justification is somewhat off-label; the critical missing piece is the quantitative version of Lemma 3.7.3 and the global iterative estimates. Since the concern is about incompleteness rather than an identified error, the appropriate verdict remains CONDITIONAL, matching the reader's assessment. The concrete test—a full re-derivation of Theorem 3.9.1—would either resolve the gap or show that additional assumptions are needed, thereby settling whether the central claim holds.","tokens_in":56415,"tokens_out":29351,"duration_ms":247777,"concrete_test":"Write a complete proof of Theorem 3.9.1 for a general finite graph, replacing N^d by #V(G) and 1/N by δ_N (the maximum eigenvalue multiplicity divided by #V(G)). Verify that (i) Lemma 3.7.3 becomes the bound ≤ |I| + δ_N, which follows from the pushforward structure of the averaged measure, (ii) all constants in Theorem 3.7.5 and the iterative scheme of Section 3.8 depend only on λ and the maximum degree, and (iii) the tower-function construction works once δ_N is sufficiently small. If any step requires an extra assumption beyond asymptotic ω-uniform continuity of the IDS, Theorem 3.9.1 as stated is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main spectral result for general graphs, Theorem 3.9.1, is established only by assertion. Section 3.9 states that Lemmas 3.7.2 and 3.7.3, proved for box graphs, rely on the maximal IDS jump tending to zero. However, the quantitative forms used in the iterative scheme (Theorem 3.7.5 and Section 3.8) depend on box-specific identifications: the jump size as 1/N and the vertex count as N^d. For a general family with asymptotically ω-uniformly continuous IDS, one must rerun the argument with #V in place of N^d and δ_N = max multiplicity/#V in place of 1/N, then verify every inequality (equation (3.66), the bound 16Mη', the tower-function estimates in Theorem 3.8.3) still holds with constants depending only on λ and the degree bound. This re-derivation is not provided; the paper simply claims that the transfer is 'a weaker property.' Lemma 3.7.2 actually can be proved by compactness without IDS regularity, so the paper's justification is imprecise there; the real burden falls on Lemma 3.7.3 and the global estimates. Until this is carried out, Theorem 3.9.1 is a plausible conjecture rather than a demonstrated theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This thesis contains two independent projects. Chapter 2 proves that every ergodic linear automorphism of T^N with two-dimensional center bundle is stably ergodic in the C^22_vol topology, including all ergodic automorphisms for N≤5 and N=7; this removes the pseudo-Anosov hypothesis from Rodriguez-Hertz's theorem. The proof establishes a minimality criterion for su-saturated invariant closed sets and then follows the Rodriguez-Hertz strategy. Chapter 3 proves that a family of finite graphs with asymptotically ω-uniformly continuous integrated density of states and bounded maximum degree has asymptotically dense delocalization: for any potential there is an arbitrarily small perturbation such that most spectral measures have no atoms larger than ε. The proof is carried out quantitatively for box graphs via an iterative perturbation scheme, and a generalization to graph families is stated. It also proves log-Hölder continuity of the IDS for truncated propagating graphs via a Thouless formula.","tokens_in":56729,"tokens_out":21815,"duration_ms":176507,"significance":"If the two theorems are correct, Chapter 2 answers the 1977 Hirsch–Pugh–Shub question for dimensions ≤5 and 7, and more generally for all toral automorphisms with two-dimensional center, a genuine advance over Rodriguez-Hertz. Chapter 3 would give a broad and useful mechanism for topological delocalization on deterministic graphs, with explicit quantitative bounds in the box case, improving on Avila–Damanik. The main caveat is that the general graph criterion (Theorem 3.9.1) is not fully proved; the proof is only a brief assertion that the box arguments transfer. The box-case proof is detailed and the quantitative estimates are a strength. Chapter 2 appears sound, though it relies on several lemmas from [43].","major_comments":[{"comment":"The proof of Theorem 3.9.1 is not a proof but an assertion. It claims that Lemmas 3.7.2 and 3.7.3 depend only on the maximal IDS jump tending to zero, which is weaker than asymptotic ω-uniform continuity. This is imprecise: Lemma 3.7.2 is proved by compactness of [−λ,λ]^V and needs no IDS regularity; the burden is Lemma 3.7.3 and the quantitative estimates of Theorem 3.7.5/§3.8. In the box case Lemma 3.7.3 uses the specific bound (3.65) with maximal jump 1/N and normalization N^d, coming from the multiplicity bound N^{d−1} for box Schrödinger operators. For a general family one must redo the argument with #V(G) in place of N^d and δ_N = max multiplicity/#V(G) in place of 1/N, then re-verify Eq. (3.66), the estimate 16Mη' + 8Mδ_N near Eq. (3.70), the lower bound #D_{V'} ≥ N^dΔ/2, and the tower-function conditions in Theorem 3.8.3, with constants depending only on λ and the degree bound. T","section":"§3.9, Theorem 3.9.1 (also Theorems 3.1.1 and 3.2.3)"}],"minor_comments":[{"comment":"In the proof of Theorem 3.5.2, the regularity of N^∞_V is attributed to 'theorem 3.5.2'; this should be Theorem 3.5.1.","section":"§3.5, Theorem 3.5.2"},{"comment":"In the definition of asymptotically dense delocalization, the final clause says 'the spectral measure σ_{V,v}' but the intended measure is that of the perturbed potential W; it should read σ_{W,v}.","section":"§3.1, Definition 3.1"},{"comment":"There are missing equation references displayed as '??' in the discussion of the continuity of ν_{V,n}; the relevant equations should be numbered and cited.","section":"§3.2.2"},{"comment":"Theorem 2.2.2 is stated for non-empty closed sets, while §2.5 proves Lemma 2.5.6 for open sets. The reduction via complements should be stated explicitly to connect the two statements.","section":"§2.2.3 / §2.5"},{"comment":"The proof of Theorem 3.10.2 asserts that convergence of IDSs for truncated propagating graphs follows from the proof of Lemma 3.4.2. The moment comparison there is box-specific; the analogous argument for general truncated propagating graphs should be sketched or the differences identified.","section":"§3.10.2"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical risk is the incomplete proof of the general graph criterion (Theorem 3.9.1); the box case is solid. I would request a complete transfer argument or a restriction of the main spectral theorem to box operators. There is also a reliance on the unpublished companion paper [12] for the pointwise version of Lemma 3.7.2; although the uniform version is elementary, the authors should provide a self-contained proof. Chapter 2 seems mature and publishable; if the spectral part cannot be brought up to the same standard, the authors might consider splitting the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this thesis has one very strong chapter and one chapter with a genuine gap. Ch. 2 removes the pseudo-Anosov condition from Rodriguez-Hertz's stable ergodicity theorem for toral automorphisms with two-dimensional center — a real step in the Pugh-Shub program. The proof is detailed, and the algebraic lemma 2.4.3 (constructing an A^k-invariant subspace on which A^k is pseudo-Anosov without global algebraic assumptions) looks like the key novelty. The corollaries for N≤5 and N=7 follow. The chapter leans heavily on machinery from [43] and [61], but that's normal practice.\n\nCh. 3 is where I get uncomfortable. The box Schrödinger case (Thm 3.1.2) is proved in full: the iterative scheme, the stability lemma, and the IDS regularity argument are all there with quantitative estimates. That part looks correct. The advertised general criterion (Thm 3.9.1) is different. The proof asserts that the only graph-specific inputs in Lemmas 3.7.2 and 3.7.3 are the smallness of IDS jumps, but it never carries out the verification. For box graphs, jump size is 1/N and vertex count N^d; for a general family you have to rerun the inequalities (3.66), the bound 16Mη', and the tower-function estimates in Thm 3.8.3 with #V and max multiplicity/#V in place of N^d and 1/N. That re-derivation is not provided. The stress-test note is right: Thm 3.9.1 is a plausible conjecture, not yet a theorem.\n\nOn top of that, the proof depends on the unpublished companion paper [12] (Avila-Damanik, \"in preparation\"), so the fully general statement cannot be verified from this manuscript alone. There are also broken cross-references and a few typos, but those are minor.\n\nBottom line: this deserves a serious referee. Ch. 2 alone would justify it, and the box case of Ch. 3 is a substantial piece of work. The referee should ask for a complete proof of Thm 3.9.1 — or, at minimum, an explicit statement that the general criterion is conditional on [12] and on the missing transfer lemma — plus fixing the references. I'd cite Ch. 2 and the box theorem; I'd be cautious about citing Thm 3.9.1 as established. Bring it to reading group if you want a lively debate about what counts as a proof.","headline":"The stable ergodicity theorem (Ch. 2) is a real advance, and the box-graph delocalization (Ch. 3) is solid; but the advertised general graph criterion (Thm 3.9.1) is only sketched and needs a serious proof or a clear dependence on the companion paper.","tokens_in":57212,"tokens_out":3992,"would_cite":true,"duration_ms":33141,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D30","37C40","47B36","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"All ergodic torus maps with two-dimensional center are stably ergodic","keywords":["stable ergodicity","toral automorphisms","partial hyperbolicity","delocalization","Schrödinger operators","integrated density of states","Thouless formula","large graphs"],"falsifier":"For the delocalization theorem, take any family of graphs with asymptotically uniformly continuous IDS (for instance the 2×N grids of Example 3.9.4) and test whether a δ-small perturbation of an arbitrary potential can remove all atoms larger than ε from all but δN spectral measures as N grows; if the required perturbation size stays bounded away from zero for some such family, the theorem is false.","tokens_in":56288,"feed_emoji":"🌀","tokens_out":9489,"duration_ms":106081,"temperature":0.7,"pith_summary":"This thesis establishes that two kinds of chaotic behaviour are robust. The first result: every ergodic linear automorphism of the N-dimensional torus whose neutral (center) directions form a two-dimensional subspace is stably ergodic — it remains ergodic after any sufficiently small C^22 volume-preserving perturbation. Because the center must be even-dimensional, this settles a 1977 open question in dimensions N≤5 and N=7. The second result: for any family of finite graphs whose integrated density of states is asymptotically uniformly continuous and whose degree is bounded, delocalization is topologically common — any potential can be perturbed by an arbitrarily small amount so that most spectral measures have no atom larger than a prescribed size. The proof supplies an iterative perturbation scheme, a quantitative stability estimate for delocalization, and a Thouless-formula argument showing that a flexible family of 'propagating' graphs satisfies the criterion.","feed_headline":"All ergodic torus maps with two-dimensional center are stably ergodic","feed_subtitle":"A spectral criterion makes delocalization of most eigenvectors topologically common on large graphs.","key_machinery":"Key machinery: dynamics — a minimality criterion (Theorem 2.2.2): for C^1-small perturbations of A with two-dimensional center, every non-empty closed invariant su-saturated set is the whole torus; the proof finds a subspace X containing the center on which a power of A is pseudo-Anosov and forces translational invariance by volume, recurrence and simple-connectedness. Spectral theory — the integrated density of states (IDS) and the modified spectral measure ν_v = N*σ_v (atoms = squared eigenvector coefficients); asymptotic uniform continuity of the IDS makes delocalization stable under small potential perturbations, and an iterative gradient-descent scheme creates delocalization on target i","core_discovery":"The paper's two load-bearing claims are Theorem 2.1.1 and Theorem 3.1.1. Theorem 2.1.1 states that any ergodic linear automorphism A of T^N with dim(E^c)=2 is stably ergodic in the C^22 volume-preserving topology; Corollary 2.1.2 applies it to all ergodic automorphisms of T^7. This removes the algebraic 'pseudo-Anosov' condition on the characteristic polynomial that a previous theorem needed. Theorem 3.1.1 states that a family of graphs with asymptotically ω-uniformly continuous integrated densities of states and bounded maximum degree has asymptotically dense delocalization: for any large graph in the family, any potential, and any δ, there is a potential within δ whose spectral measures at","pith_inferences":["The stable-ergodicity proof uses the dimension-2 center only to classify accessibility classes; if that classification is extended, the same outline may settle the 1977 question for all ergodic toral automorphisms.","The spectral criterion likely has room to spare: the proof needs only that jumps of the IDS vanish, so families with IDS regularity weaker than log-Hölder may also satisfy the conclusion.","The tower-type size bounds in the iterative scheme suggest that the delocalization effect is real but appears only at astronomically large graph sizes; numerical experiments on accessible sizes may see no effect, consistent with the theorem rather than against it.","A direct testable extension: apply the scheme to random regular graphs with growing degree; if their IDS is asymptotically uniformly continuous, the theorem predicts dense delocalization without any disorder assumption."],"forward_implications":["Every ergodic automorphism of T^7 is stably ergodic in C^22_vol (Corollary 2.1.2).","The same holds in dimensions 6 and 9 provided no eigenvalue is a Salem number (Remark 2.1.3).","For box Schrödinger operators on [N]^d, delocalization of most eigenvectors is dense in the space of potentials, with quantitative bounds.","For finite-range box operators and more generally any sequence of truncated propagating graphs with bounded degree, asymptotic dense delocalization holds.","A 1977 open question on stable ergodicity of linear torus automorphisms is answered affirmatively in all dimensions where the center has dimension 2, in particular all dimensions ≤5 and 7."],"fun_headline_variants":["Stable ergodicity proven for all ergodic torus maps with 2D center","Delocalization of most eigenvectors: topologically common on graphs","Two chaos results: stable ergodicity and generic delocalization","Ergodic torus maps: stable ergodicity without algebraic condition","Most eigenvectors delocalize on large graphs, generically"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For the delocalization criterion, the load-bearing premise is the transfer asserted in §3.9: that the only graph-specific facts needed in the iterative scheme — chiefly that the largest jump of the integrated density of states tends to zero — follow for every family with asymptotically uniformly continuous IDS; this transfer is stated but not carried out in full.","fun_headline_variants_meta":{"raw":{"variants":["Stable ergodicity proven for all ergodic torus maps with 2D center","Delocalization of most eigenvectors: topologically common on graphs","Two chaos results: stable ergodicity and generic delocalization","Ergodic torus maps: stable ergodicity without algebraic condition","Most eigenvectors delocalize on large graphs, generically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000539,"raw_usage":{"total_tokens":2424,"prompt_tokens":745,"completion_tokens":1679,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":1585}},"tokens_in":489,"tokens_out":1679,"duration_ms":11008,"temperature":1.0,"reasoning_tokens":1585,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:44:59.665070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the delocalization theorem, take any family of graphs with asymptotically uniformly continuous IDS (for instance the 2×N grids of Example 3.9.4) and test whether a δ-small perturbation of an arbitrary potential can remove all atoms larger than ε from all but δN spectral measures as N grows; if the required perturbation size stays bounded away from zero for some such family, the theorem is false.","supporting_citations":[],"review_version":1}