{"id":"8e46b401-6944-4f83-89b3-ffc3fbed9688","arxiv_id":"2603.26949","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"On compact quotients of Euclidean buildings, the Taylor spectrum of the transfer-operator family equals the joint point spectrum away from zero.","lead":"This paper proves that the joint Taylor spectrum of commuting transfer operators on compact quotients of Euclidean buildings coincides with the joint eigenvalue spectrum outside any neighborhood of zero. It extends dynamical resonance theory from rank-one p-adic graphs and smooth higher-rank Anosov flows to higher-rank buildings.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's |χ|≥ϑ is stronger than the proven |χ(k)|>ϑ; the boundary |χ|=ϑ is unhandled and likely in the essential spectrum.","rationale":"The paper's core contribution—discrete joint resonances outside a small radius—is supported by the strict inequality in Theorem 7.27; the reader's conditional acceptance is appropriate. The boundary mismatch between Theorem 1.1 (≥ϑ) and Theorem 7.27 (>ϑ) is the most load-bearing concern: it is an internal inconsistency in the central statement, and the proof gives no argument for the equality case. The standard reduction to a smaller ϑ' fails because the Taylor spectrum on the larger space does not necessarily restrict to the smaller subspace. In rank one, the boundary circle is typically essential spectrum for a transfer operator on a full shift, making the ≥ version likely false. I partially agree with the reader's weakest_assumption: strong regularity is indeed necessary for the framework, but it is explicitly assumed in Section 5 and can be added to Theorem 1.1 as a hypothesis. The boundary issue, by contrast, directly affects the theorem as stated and requires either a proof for the boundary case or a correction to >. The proposed Weyl-sequence test would settle whether the ≥ version fails; if so, the theorem should be restated with |χ|>ϑ.","tokens_in":45751,"tokens_out":24267,"duration_ms":205012,"concrete_test":"Take n=1 (homogeneous tree/full shift) and fix ϑ∈(0,1). For the normalized transfer operator L on the full two-sided shift with Lipschitz constant ϑ, show that the circle {z:|z|=ϑ} is contained in the essential spectrum σ_ess(L) (e.g., by constructing a Weyl sequence φ_m with ||(L-z)φ_m||/||φ_m||→0 and no convergent subsequence), while no z with |z|=ϑ is an eigenvalue. If this holds, the character χ_z(n)=z^n lies in σ_T(L)\\σ_p(L) for the monoid N, disproving Theorem 1.1's ≥ version; the strict > version remains valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.1 asserts inclusion for |χ|≥ϑ, but the proven Theorem 7.27 requires the stronger condition ∃k∈P∨_{++}: |χ(k)|>ϑ. The gap is not a harmless epsilon issue: if α=|χ|=ϑ, then for every k∈P∨_{++}, |χ(k)|≤ϑ, with equality typically only for k0=Σϖ_i. The usual reduction to ϑ'<ϑ fails because one would need χ∈σ_T(L) on F_{ϑ'} to apply Theorem 7.29, and Taylor spectra do not propagate from the larger space Fϑ to the smaller subspace F_{ϑ'} (homology of a subcomplex need not inject). In the rank-one case, a character is χ(n)=a^n with |a|=ϑ; here the circle |z|=ϑ is typically essential spectrum for the transfer operator on a full shift, so such χ can lie in σ_T(L) without being joint eigenvalues. Thus Theorem 1.1 is stronger than the proof supports and is likely false as stated; it should be stated with |χ|>ϑ (or with an ε-slack in the definition of the neighborhood of zero).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a spectral theory for the transfer operators associated with the monoid of dominant coweights acting by shifts on the space of sectors of a compact local building. The authors define ultrametric Lipschitz spaces F_ϑ, prove a Lasota–Yorke-type key inequality, establish quasi-compactness of transfer operators for strongly dominant coweights, and then combine these analytic results with Taylor's homological joint spectrum machinery. The main theorem claims that, outside an arbitrarily small neighborhood of zero, the Taylor spectrum of the commuting family of transfer operators is contained in the joint point spectrum; a stronger technical statement (Theorem 7.28) identifies the relevant Tor/Ext groups with those computed on the finite-dimensional invariant subspace F_1.","tokens_in":46081,"tokens_out":11410,"duration_ms":112551,"significance":"If the main theorem is stated correctly, this is a substantial contribution: it extends the Ruelle–Taylor resonance program from rank-one graphs and trees to higher-rank Euclidean buildings, and it provides a clean mechanism — reduction to the finite-dimensional invariant core F_1 — for converting quasi-compactness of one transfer operator into discreteness of the joint spectrum. The analytic part is standard and essentially complete, the algebraic part is coherent, and the paper works axiomatically without numerical fitting or hidden parameters. The main weakness is a mismatch between the strength of the announced theorem and the strict inequality that the proof actually supports.","major_comments":[{"comment":"Theorem 1.1 asserts inclusion for |χ|≥ϑ, but the proof in Theorem 7.27 requires the strictly stronger condition ∃k∈H° with |χ(k)|>ϑ. Since |χ| is a supremum, |χ|≥ϑ does not imply that any single strongly dominant k satisfies |χ(k)|>ϑ. The gap is not removable by the usual ε-slack: passing to a smaller ϑ′<ϑ would require knowing χ∈σ_T(L) on the smaller space F_{ϑ′}, and Taylor spectra do not restrict from a larger space to a subspace. In the rank-one case, taking χ(n)=a^n with |a|=ϑ, the boundary circle |z|=ϑ is typically essential spectrum for the transfer operator on a full shift, so such characters can lie in σ_T(L) without being joint eigenvalues. The theorem (and the abstract) should be restated with |χ|>ϑ, equivalently ‘for every ε>0, {χ: |χ|>ε}⊂σ_p(L)’.","section":"Theorem 1.1; Definition of |χ|; Theorem 7.27"},{"comment":"The statement writes χ(L_k) where χ is an element of Hom_{C-alg}(C[X],C) and L_k is an operator. This is only meaningful after identifying characters with evaluations on the generated algebra A_H. The identification is made earlier in Remark 7.14, but the notation remains confusing and should be clarified at the point of use.","section":"Section 7.4, Theorem 7.28"}],"minor_comments":[{"comment":"The definition of P∨_++ says ‘We write P∨_++ for the set of all dominant coweights’; this should read ‘strongly dominant coweights’.","section":"Definition 3.20"},{"comment":"Theorem 1.1 states ‘Let C be a compact local building’ without repeating the standing assumptions (locally finite, strongly regular Euclidean building) that are used in Corollary 4.11 and Lemma 5.27 to define M_μ and the semigroup law. The theorem statement should explicitly carry those assumptions.","section":"Theorem 1.1 and Definition 4.3"},{"comment":"The symbol ‘secQuot’ appears undefined; it should presumably be S(C).","section":"Remark 5.6"},{"comment":"The entries [BHW25b] and [GBGHW25b] are duplicates of [BHW25a] and [GBGHW25a]; consolidate to avoid confusion.","section":"References"},{"comment":"In the induction step, the term φ′_m = −z^{-m} \\tilde φ_m is used but not explicitly introduced before the convergence statement; a few words clarifying the definition would improve readability.","section":"Section 6.3, Proposition 6.23"}],"recommendation":"major_revision","confidential_remarks":"The boundary issue in Theorem 1.1 is the only substantive obstruction I see. The proof supports the strict-inequality version, and the overstatement appears fixable within the manuscript's scope by changing ≥ to > in the theorem and abstract. If the authors make that correction and align the statements with the standing assumptions, I would be willing to accept."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This paper builds the first joint resonance theory for the multiparameter Weyl-chamber-type flow on compact quotients of Euclidean buildings—a genuine higher-rank p-adic analogue of the Ruelle-Taylor paradigm—and the core analytic machinery is careful and mostly correct. Second, the headline theorem is overstated: Theorem 1.1 claims inclusion for |χ|≥ϑ, but the proof only establishes the strict version where some strongly dominant k satisfies |χ(k)|>ϑ (Theorem 7.27). The reduction to a smaller ϑ'<ϑ fails because the Taylor spectrum on the smaller space F_ϑ does not propagate to the larger F_{ϑ'}; homology of a subcomplex need not inject. In the rank-one full-shift case, characters with |z|=ϑ sit in the essential spectrum and need not be joint eigenvalues. So the boundary case |χ|=ϑ is genuinely unhandled, and Theorem 1.1 is likely false as stated. The fix is to state it with |χ|>ϑ or add an ε-slack. That is a load-bearing gap in the statement, not in the main construction—the strict version is a real theorem.\n\nWhat is genuinely new: the sector space S(C), the ultrametric, the locally constant approximations F^n, the transfer operators L_μ, and the reduction of Taylor cohomology to the finite-dimensional invariant core F^1 (Theorem 7.28). Section 6 is complete and convincing: Theorem 6.22 (quasi-compactness) is standard but works. The self-citations to [BHW25a/b] and [GBGHW25a/b] are auxiliary and have independent published proofs, so the circularity burden is low.\n\nMinor soft spots: the paper requires strong regularity (the authors note the standard workaround), and the duplicated reference entries for their own papers plus a handful of typos need cleaning. None of that affects the mathematics.\n\nThis paper deserves a serious referee. The referee should focus on the boundary case and whether the strict inequality is the correct statement; I expect the core result for |χ|>ϑ to survive. It is a paper for people working on transfer operators, resonances, and buildings—they will get real value from it. Bring it to reading group; cite it if you work in this area, but quote the corrected statement.","headline":"Solid higher-rank p-adic resonance framework, but Theorem 1.1 overstates the proven result—the boundary case |χ|=ϑ is unhandled and likely false.","tokens_in":46560,"tokens_out":8161,"would_cite":true,"duration_ms":75204,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37C30","37D35","47A13","20E42","51E24"],"pacs":[],"model":"deepseek-v4-flash","headline":"For compact quotients of Euclidean buildings, the Taylor spectrum of the commuting transfer operators equals the joint point spectrum outside any neighborhood of zero.","keywords":["transfer operators","Taylor spectrum","Euclidean buildings","joint point spectrum","Weyl chamber flow","dynamical resonances","quasi-compact operators","Lipschitz spaces"],"falsifier":"Take a compact quotient whose universal covering building is regular but not strongly regular, so that q_s≠q_{σ(s)} for some type-rotating symmetry σ, and compute the preimage counts of a shift σ_μ at two different sectors. If those counts differ, the normalization M_μ in the definition of L_μ is not well-defined and the identity L_μ1=1 fails; checking whether the semigroup law and the joint spectral conclusion survive in such an example would directly test the necessity of strong regularity.","tokens_in":45645,"feed_emoji":"🏗️","tokens_out":5348,"duration_ms":55486,"temperature":0.7,"pith_summary":"This paper establishes that a multiparameter shift flow on the space of sectors of a compact quotient of a Euclidean building has a discrete joint resonance spectrum. The transfer operators are normalized averages over preimages of the shift, acting on ultrametric Lipschitz spaces. The main theorem says that outside an arbitrarily small neighborhood of zero in the space of spectral parameters, the Taylor spectrum of the commuting family coincides with the joint point spectrum. If correct, this gives a well-defined set of dynamical resonances for these discrete higher-rank systems, and the resonances can be studied through a finite-dimensional invariant core of functions constant on unit balls.","feed_headline":"Building flows get discrete joint resonance spectra","feed_subtitle":"On compact quotients of Euclidean buildings, the Taylor spectrum outside zero equals the joint point spectrum.","key_machinery":"The central objects are the sector space S(C) of locally injective type-rotating maps from a fundamental sector S_0 into C, the shift operators σ_μ(s)=s∘t_μ indexed by dominant coweights μ∈P_+^∨, and the transfer operators L_μφ(s)=M_μ^{-1}∑_{σ_μ(s')=s}φ(s'), where M_μ=q_{t_μ} counts preimages. The engine of the proof is a contraction inequality for strongly dominant μ of the form |L_μφ|_ϑ≤ϑ|φ|_ϑ+C‖φ‖_∞, which makes L_μ quasi-compact with essential spectral radius at most ϑ. The Taylor spectrum, the joint spectrum of several commuting operators defined through vanishing of Koszul cohomology, is then controlled by constructing parametrices that reduce the relevant Tor and Ext groups to the fin","core_discovery":"Let C be a compact local building, meaning a compact simplicial complex whose universal cover is a Euclidean building and whose covering map preserves chamber types. For any 0<ϑ<1, the paper proves that every character χ in the Taylor spectrum of the transfer operator family L=(L_μ) with |χ|≥ϑ is a joint eigenvalue: σ_T(L)∩{χ:|χ|≥ϑ}⊂σ_p(L). Since joint eigenvalues always lie in the Taylor spectrum, the two spectra agree outside the ϑ-neighborhood of zero. More strongly, for characters satisfying |χ(L_k)|>ϑ for some strongly dominant coweight k, the Taylor cohomology of the Lipschitz space F_ϑ is isomorphic to the Taylor cohomology of F^1, a finite-dimensional space of functions constant on d","pith_inferences":["Going beyond the paper, the finite-dimensional core F^1 suggests a concrete computational route: the resonances of a given compact local building could be obtained by diagonalizing the commuting finite matrices L_μ restricted to F^1.","If the axiomatic result applies to buildings attached to p-adic Lie groups, it would provide a joint resonance spectrum for Weyl-chamber-type flows on p-adic locally symmetric spaces, a natural counterpart to known rank-one graph and tree results.","The strong-regularity assumption may be relaxable in some cases by choosing a larger root-system type or by inserting weights into the transfer operators; testing whether normalization and the semigroup law survive without strong regularity would clarify the true scope of the theorem.","The discreteness established here invites transfer of tools from smooth Anosov theory, such as counting or exponential-mixing statements, to the combinatorial setting of building quotients."],"forward_implications":["For every compact local building and every ϑ∈(0,1), the resonance spectrum outside the ϑ-disk is discrete and consists entirely of joint eigenvalues of the transfer operators.","All joint eigenspaces for eigenvalues of modulus greater than ϑ are finite-dimensional, because they are contained in the finite-dimensional core F^1.","The Taylor cohomology of the full Lipschitz space can be computed inside F^1, so the joint spectrum is accessible through finite-dimensional linear algebra.","For a strongly dominant coweight, the transfer operator is quasi-compact with explicit essential spectral radius bound ϑ, so eigenvalues outside the ϑ-disk are isolated normal eigenvalues.","The spectrum is independent of the choice of generators of the monoid of dominant coweights, since the Taylor spectrum depends only on the generated operator algebra."],"fun_headline_variants":["Building quotients: transfer operator spectra match point spectra off zero","Joint eigenvalues fill Taylor spectrum on building flows","Away from zero, Taylor and point spectra coincide on buildings","Multiparameter flow spectra on buildings: zero is the only gap","Transfer operators on building quotients have pure point spectra off zero"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes the covering building is locally finite and strongly regular, so that the preimage count M_μ is defined and independent of the sector, which makes the transfer operators normalized (L_μ1=1) and forces the semigroup law L_{μ1}L_{μ2}=L_{μ1+μ2}.","fun_headline_variants_meta":{"raw":{"variants":["Building quotients: transfer operator spectra match point spectra off zero","Joint eigenvalues fill Taylor spectrum on building flows","Away from zero, Taylor and point spectra coincide on buildings","Multiparameter flow spectra on buildings: zero is the only gap","Transfer operators on building quotients have pure point spectra off zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000762,"raw_usage":{"total_tokens":3151,"prompt_tokens":611,"completion_tokens":2540,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":355,"completion_tokens_details":{"reasoning_tokens":2470}},"tokens_in":355,"tokens_out":2540,"duration_ms":15706,"temperature":1.0,"reasoning_tokens":2470,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T17:15:43.833839+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a compact quotient whose universal covering building is regular but not strongly regular, so that q_s≠q_{σ(s)} for some type-rotating symmetry σ, and compute the preimage counts of a shift σ_μ at two different sectors. If those counts differ, the normalization M_μ in the definition of L_μ is not well-defined and the identity L_μ1=1 fails; checking whether the semigroup law and the joint spectral conclusion survive in such an example would directly test the necessity of strong regularity.","supporting_citations":[],"review_version":1}