{"id":"e630d34a-e04a-4c55-87e3-4d5983ec2086","arxiv_id":"2506.21950","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A noncommutative geometry toolbox: a Peller-style multiple operator integral calculus for abstract pseudodifferential operators, plus Dixmier trace formulas for the density of states and for truncated spectral triples, with a quantum ergodicity analogue.","lead":"This mathematics thesis builds a unified theory of multiple operator integrals for abstract pseudodifferential operators, yielding heat trace expansions and new Dixmier trace formulas in noncommutative geometry. It connects those formulas to quantum ergodicity, the density of states in solid state physics, and Roe's index on open manifolds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.4.11 asserts an NCG quantum-ergodicity theorem by citing Zelditch's Lemma 2.1, but that lemma is neither stated nor shown to apply to the noncommutative C*-dynamical system (S*A,R,σ_t); without verification, the flagship Part II theorem is unsupported.","rationale":"The reader's weakest_assumption points to the local Weyl law, and that is genuinely fragile: Example 3.6.5 shows it can fail, and the thesis flags it as an explicit hypothesis. But the unresolved delegation in Theorem 4.4.11 is more load-bearing because it is presented as a headline result, is not conditional on a stated external assumption, and rests entirely on an unstated lemma whose hypotheses may not hold for noncommutative S*A. The reader already identified this as a reason for CONDITIONAL, so this stress-test agrees with the verdict rather than moving it. The Part I MOI construction appears internally consistent: the functional calculus and Taylor expansions are lengthy but the order bookkeeping checks out, and the counterexample in Example 3.6.5 is concrete and checkable. The direct derivation from Proposition 4.1.1 would likely resolve the concern, which is why I recommend keeping the CONDITIONAL verdict rather than rejecting the thesis; if the direct derivation fails, the theorem as written remains unproven.","tokens_in":861,"tokens_out":818,"duration_ms":215686,"concrete_test":"State [Zel96, Lemma 2.1] verbatim and verify its hypotheses for (S*A,R,σ_t). In parallel, attempt to re-derive Theorem 4.4.11 directly from Proposition 4.1.1: under the local Weyl law, Tr(P_λ aP_λ)/Tr(P_λ) converges to the noncommutative integral, so the normalized partial sums of ⟨e_k,ae_k⟩ converge; then use the standard Cesàro-to-density-one subsequence argument (with a countable dense subset of the separable closure of A) to get the conclusion without invoking ergodicity or Zelditch. If this direct derivation succeeds, the theorem is true and the citation is unnecessary; if it fails, the missing verification of Zelditch's lemma is load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Part II's headline result (Theorem 4.4.11) is proven by a one-line reduction: 'classical ergodicity means precisely that (S*A,R,σ_t) has a unique vacuum state ... hence ... consequence of [Zel96, Lemma 2.1].' The thesis never states Lemma 2.1, nor checks that its hypotheses hold when S*A is a noncommutative algebra. In the classical manifold case the Shnirelman–Zelditch–Colin de Verdière theorem needs Egorov estimates and positivity of the local Weyl measure; a unique invariant vector in L^2 alone only gives mean ergodic convergence (Proposition 4.4.15), which is not obviously equivalent to density-one convergence of matrix coefficients for every eigenbasis. Example 4.4.12(2) even asserts a correction to Zelditch, so the transfer of his Lemma 2.1 to this noncommutative setting is exactly the missing step. If the lemma does not apply, the theorem—and the claimed NCG analogue of 'ergodic flow implies quantum ergodicity'—does not follow from the written proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This PhD thesis develops trace formulas in noncommutative geometry in two parts. Part I adapts the theory of multiple operator integrals to the abstract pseudodifferential calculus of Connes–Moscovici: it constructs functional calculi for elliptic and zero-order operators, builds multiple operator integrals whose entries may be unbounded pseudodifferential operators (Theorem 3.2.5), derives a noncommutative Taylor expansion (Theorem 3.5.5), and obtains asymptotic trace expansions for perturbations of regular s-summable spectral triples (Corollary 3.6.6), thereby addressing a question of Eckstein and Iochum. Part II studies Connes' integral formula through spectral truncations: Theorem 4.1.7 identifies the noncommutative integral with suitable averaged limits of normalized traces on spectral projections under a Weyl law, Theorem 4.2.2 gives a Szegő limit theorem, and Theorem 4.4.11 claims an NCG analogue of 'ergodic flow implies quantum ergodicity'. The final chapters give Dixmier trace formulas for the density of states on discrete metric spaces and on manifolds of bounded geometry, with an application to Roe's index on open manifolds.","tokens_in":76306,"tokens_out":8450,"duration_ms":94258,"significance":"If the results are valid, the thesis makes a substantial contribution. The construction of multiple operator integrals for abstract pseudodifferential operators is a genuine generalization of Peller's theory and provides a common language for operator integral arguments in NCG; the noncommutative Taylor expansion and the trace expansions are parameter-free theorems with explicit constants, not fitted quantities. The counterexample in Example 3.6.5 is concrete and checkable, and it precisely shows the limitations of Weyl-law assumptions. Part II offers a fresh bridge between NCG and quantum ergodicity and gives general Dixmier trace formulas for the density of states, including a Roe-index application. The proofs of the core Part I results are detailed, and the Borel lemma and Hadamard three-line argument are written out. The main weakness is the proof of the Part II flagship ergodicity theorem, which rests on an unverified transfer of Zelditch's Lemma 2.1 to noncommutative C*-dynamical systems.","major_comments":[{"comment":"The proof of Theorem 4.4.11 is a one-line reduction: 'Classical ergodicity ... means precisely that (S*A,R,σ_t) has a unique vacuum state ... hence ... a consequence of [Zel96, Lemma 2.1].' The lemma is not stated, and its hypotheses are not checked for the noncommutative C*-dynamical system (S*A,R,σ_t). The thesis itself notes that Zelditch's results are mostly formulated for 'quantised abelian' systems, and Example 4.4.12(2) explicitly corrects a Zelditch corollary in the noncommutative setting. A unique invariant vector in L^2 gives mean ergodic convergence (Proposition 4.4.15), which is not obviously equivalent to density-one convergence of matrix coefficients for every eigenbasis. To make Theorem 4.4.11 load-bearing, the author must either state and prove the needed version of Zelditch's lemma for noncommutative S*A, or prove the density-one convergence directly; as written, the claimed NCG analogue of 'ergodic flow implies quantum ergodicity' is unsupported.","section":"Section 4.4, Theorem 4.4.11"},{"comment":"The Weyl law assumption is the load-bearing premise of Part II: Theorem 4.1.7, Theorem 4.2.2, and Theorem 4.4.11 all depend on the global and local growth of the eigenvalue counting function. The thesis correctly demonstrates in Example 3.6.5 that the local Weyl law is not automatic even when the global Weyl law holds. This is not an internal inconsistency, but the presentation should more prominently state that the local Weyl law is a genuine additional hypothesis whose verification for a given spectral triple is a separate problem, and the reader should be pointed to the precise places where failure of the local law invalidates the conclusions. In particular, the proof of Theorem 4.1.7 uses the local trace only through the constants C(a), so the statement should make explicit that the result is conditional on the local Weyl law for the specific a in question.","section":"Section 4.1, Definition 4.1.3 and Theorem 4.1.7"},{"comment":"The independence of the multiple operator integral from the chosen integral representation of the symbol is asserted by reference to [ACDS09, Lemma 4.3]. Since the present setting allows unbounded self-adjoint operators H_i and the entries are abstract pseudodifferential operators rather than bounded operators, the transfer of that argument is not automatic. The text indicates the rank-one operators θ_{η,ξ} lie in op^{-∞}(Θ), but the proof of independence in [ACDS09] is formulated for bounded Hilbert-space operators. Please expand the argument here, or at least spell out why the unbounded case follows from the same computations on H^s for every s. This point is central because all later MOI identities and trace expansions inherit the well-definedness of the construction.","section":"Section 3.2, Theorem 3.2.5"}],"minor_comments":[{"comment":"The text says 'we can choose ε = 0' in applying Corollary 3.6.6, but Corollary 3.6.6 states the error term for arbitrary ε>0. Please justify why the endpoint ε=0 is admissible in this example or rephrase to avoid the appearance of an inconsistent choice.","section":"Example 3.6.8"},{"comment":"The symbol B is overloaded: it denotes both the algebra generated by A and D and, later in the same corollary, the operator |D+V|-|D|. Please use different notation for one of these objects to avoid confusion.","section":"Corollary 3.6.6"},{"comment":"The statement that τ defines a finite positive trace on S*A is labeled 'well-known' and the proof is only sketched. Since the quotient by K(H) is involved, a short argument showing well-definedness on the quotient (i.e., vanishing on compact perturbations) would improve readability and make the subsequent L^2 construction self-contained.","section":"Section 4.4, Proposition 4.4.6"},{"comment":"The proof of part 1 of Lemma 3.3.1 first establishes the integral representation for compactly supported smooth functions and then invokes density of C_c^∞ in T_α(R). Please add a sentence explaining why the divided differences f^[n] depend continuously on f in the relevant norm, since this is not entirely immediate from the definition of the ⊠_i seminorms.","section":"Chapter 3, Lemma 3.3.1"}],"recommendation":"major_revision","confidential_remarks":"The thesis is largely based on four joint papers, and the boundary between the author's own contribution and the co-authored material is not always clearly marked. This is a presentation issue rather than a mathematical one, but it should be clarified. The main mathematical concern is the quantum ergodicity theorem: the proof of Theorem 4.4.11 relies on Zelditch's Lemma 2.1 without checking its hypotheses in the noncommutative setting, and the manuscript itself acknowledges that Zelditch's framework is 'quantised abelian'. I recommend asking the author to either prove the needed lemma in the noncommutative case or state the theorem under hypotheses that make the transfer explicit. The rest of the thesis, in particular the Part I MOI construction and the density-of-states results, is detailed and convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The part worth taking seriously is Part I. Hekkelman builds a multiple operator integral theory for Connes–Moscovici's abstract pseudodifferential calculus: functional calculus for Θ-elliptic operators, MOIs with divided-difference symbols, a noncommutative Taylor expansion, and trace expansions. The proofs are mostly written out, including the Borel lemma and the Hadamard three-line argument. This genuinely extends Peller's construction to unbounded operators and gives a language that should simplify operator integral arguments in the local index formula and spectral action literature. The counterexample in Examples 3.6.5 and 3.6.8, showing that global Weyl law data does not control local heat trace expansions, is concrete and checkable, and it earns its claim to answer part of the Eckstein–Iochum open question.\n\nPart II is more uneven. The results on truncated spectral triples and the Szegő theorem are reasonable, and the link to Connes' integral formula is sensible. The density of states chapters are competent and connect to earlier work. The weak point is Theorem 4.4.11, the NCG quantum ergodicity claim. The proof is a one-line appeal to [Zel96, Lemma 2.1], but the lemma is never stated and no argument is given that it applies to the noncommutative dynamical system (S*A,R,σ_t). The author even corrects Zelditch in Example 4.4.12(2), so this is not a routine transfer. It may be true, but the written proof does not establish it. That theorem should be treated as a conjecture until the Zelditch argument is checked in this setting.\n\nThe local Weyl law assumption is a real restriction, but the author is transparent about it and shows with the ℓ∞ example that it is non-automatic. Some proofs are delegated to the author's own papers and the Sukochev–Zanin book; for a thesis that is fine, but a referee should verify the imports.\n\nWho is the reader? Anyone working on operator integrals in NCG or on trace asymptotics for spectral triples. Part I alone is worth a serious referee. Part II needs a fix in the quantum ergodicity section.\n\nRecommendation: send to peer review, but require the authors to prove or properly cite a version of Zelditch's lemma that covers noncommutative S*A.","headline":"The MOI framework in Part I is a real contribution; the quantum ergodicity theorem in Part II rests on an unverified Zelditch lemma.","tokens_in":77035,"tokens_out":2380,"would_cite":true,"duration_ms":28705,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L87","47G30","46L51","47B10","35P20","81Q50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The thesis builds a multiple-operator-integral calculus for the abstract pseudodifferential operators of noncommutative geometry, derives a noncommutative Taylor formula and asymptotic heat-trace expansions, and obtains Dixmier trace…","keywords":["noncommutative geometry","multiple operator integrals","abstract pseudodifferential calculus","noncommutative Taylor formula","heat trace expansions","Dixmier traces","density of states","quantum ergodicity"],"falsifier":"Take any regular $s$-summable spectral triple satisfying the local Weyl law for all $P\\in\\mathcal{B}$, with a bounded self-adjoint $V$; a computation that violated the stated remainder $O(t^{N+1-s})$ in Corollary 3.6.6 for some $N$ would refute the heat-trace expansion claim. Similarly, a spectral triple satisfying the Weyl law for which the logarithmic mean of $\\operatorname{Tr}(P_{\\lambda_n}aP_{\\lambda_n})/\\operatorname{Tr}(P_{\\lambda_n})$ differs from $\\operatorname{Tr}_\\omega(a\\langle D\\rangle^{-d})/\\operatorname{Tr}_\\omega(\\langle D\\rangle^{-d})$ would refute Theorem 4.1.7.","tokens_in":75792,"feed_emoji":"📐","tokens_out":21718,"duration_ms":215119,"temperature":0.7,"pith_summary":"This thesis establishes that multiple operator integrals—the operator-analytic tool that makes rigorous sense of differences like $f(A+B)-f(A)$ by integrating divided differences against spectral measures—can be defined inside the abstract pseudodifferential calculus of Connes and Moscovici, the Hilbert-space calculus built from a single operator $\\Theta=(1+D^2)^{1/2}$. From that construction it derives a noncommutative Taylor formula $f(H+V)\\sim\\sum_{n\\geq 0}T^H_{f^{[n]}}(V,\\ldots,V)$ whose remainder terms drop in operator order, and consequently asymptotic expansions for heat traces such as $\\operatorname{Tr}(f(tD+tV))$, $\\operatorname{Tr}(Pe^{-t(D+V)^2})$ and $\\operatorname{Tr}(Pe^{-t|D+V|})$ on regular $s$-summable spectral triples. These expansions answer an open question of Eckstein and Iochum: the global heat trace $\\operatorname{Tr}(e^{-tD^2})$ alone does not control the perturbed traces, but a local Weyl law for the full algebra does. The second part of the thesis proves Dixmier trace formulas in the same spirit: spectrally truncated traces approximate the noncommutative integral, a noncommutative Szegő limit theorem holds, classical ergodicity of a spectral triple implies quantum ergodicity of its eigenbasis, and the density of states on discrete metric spaces and open manifolds is realised as a Dixmier trace.","feed_headline":"Operator calculus yields heat-trace expansions in noncommutative geometry","feed_subtitle":"Adapting multiple-operator integrals to the Connes–Moscovici calculus answers an open question on perturbed heat traces.","key_machinery":"The central object is the multiple operator integral for the abstract pseudodifferential calculus: for symmetric $\\Theta$-elliptic operators $H_i\\in \\mathrm{op}^{h_i}(\\Theta)$ with spectral measures $E_i$, and a symbol $\\phi$ in the integral projective tensor product $\\mathcal{L}^{\\beta_0}_\\infty(E_0)\\ \\hat{\\otimes}_i\\cdots \\hat{\\otimes}_i\\ \\mathcal{L}^{\\beta_n}_\\infty(E_n)$, the integral $T^{H_0,\\ldots,H_n}_\\phi(X_1,\\ldots,X_n)$ integrates $a_0(H_0,\\omega)X_1a_1(H_1,\\omega)\\cdots X_na_n(H_n,\\omega)$ as a Bochner integral in every $H^s(\\Theta)$. Divided differences $f^{[n]}$—the finite-difference quotients that reduce to $f^{(n)}/n!$ on the diagonal—provide the symbols: for $f\\in T^\\alpha(\\mathbb{R})$ they lie in the corresponding tensor-product classes, and the MOI identity that moving an operator $a$ past an argument $X_j$ costs the commutator $[H_j,a]$ is what drives the Taylor expansion and its remainder estimates. For the second part the carrying device is the (local) Weyl law, $\\operatorname{Tr}(e^{-tD^2})\\sim C t^{-d/2}$ and $\\operatorname{Tr}(a e^{-tD^2})\\sim C(a)t^{-d/2}$, which upgrades logarithmic averages of diagonal matrix entries to the Dixmier trace $\\operatorname{Tr}_\\omega(a\\langle D\\rangle^{-d})$, together with Widom's lemma on the vanishing of $\\operatorname{Tr}(P_\\lambda A(1-P_\\lambda)BP_\\lambda)/\\operatorname{Tr}(P_\\lambda)$ that yields the Szegő limit theorem.","core_discovery":"On the author's own terms, the discovery is that the theory of multiple operator integrals, which Peller built for bounded operators using integral projective tensor products of $L^\\infty$ spaces, survives transplant into the Connes–Moscovici calculus of abstract pseudodifferential operators. The thesis constructs a functional calculus for symmetric $\\Theta$-elliptic operators of positive order and for zero-order operators, then defines $T^{H_0,\\ldots,H_n}_\\phi(X_1,\\ldots,X_n)$ by a Bochner integral in every Sobolev space $H^s(\\Theta)$, with symbol $\\phi$ in the integral projective tensor product $\\mathcal{L}^{\\beta_0}_\\infty(E_0)\\ \\hat{\\otimes}_i\\cdots \\hat{\\otimes}_i\\ \\mathcal{L}^{\\beta_n}_\\infty(E_n)$. The payoff is a noncommutative Taylor expansion (Theorem 3.5.5) and asymptotic trace expansions on regular $s$-summable spectral triples (Corollary 3.6.6), together with the counterexample showing the local Weyl law is genuinely necessary. In the second part, spectrally truncated quotients $\\operatorname{Tr}(P_\\lambda a P_\\lambda)/\\operatorname{Tr}(P_\\lambda)$ are shown to compute the noncommutative integral under a Weyl law, a Szegő limit theorem follows, and the density of states on discrete metric spaces and bounded-geometry manifolds is expressed through Dixmier traces, yielding a Dixmier trace formula for Roe's index on open manifolds.","pith_inferences":["Extension: because the MOI construction only needs the abstract Hilbert-scale calculus, the same noncommutative Taylor expansion should hold in any concrete pseudodifferential calculus that fits the $\\Theta$-scale format, including filtered manifolds and Lie groupoids, without repeating the symbol-level estimates.","Extension: the truncated-triple formula links the noncommutative integral to numerically computable finite-dimensional data; one could use it to test ergodicity of noncommutative spaces by computing logarithmic means of eigenbasis matrix entries, which the paper does not do.","Extension: the Dixmier-trace formula for the density of states suggests defining the DOS as a Dixmier trace even when the spatial average defining it is not known to converge; the paper shows Dixmier measurability is strictly weaker than existence on discrete spaces but leaves the question open for Schrödinger-type operators on manifolds.","Extension: the commutator-decay hypothesis $\\delta^k_H(V)\\in\\mathrm{op}^{r+k(h-\\varepsilon)}(\\Theta)$ is stronger than a mere order bound on $V$; if it could be relaxed to allow $\\varepsilon=0$ with log-type losses, the Taylor expansion would extend to perturbations of the same order as $H$, which the current statement excludes."],"forward_implications":["Heat traces on perturbed spectral triples admit computable asymptotic expansions: $\\operatorname{Tr}(P e^{-t(D+V)^2})$ and $\\operatorname{Tr}(P e^{-t|D+V|})$ are expanded to arbitrary order in $t$ with coefficients built from commutators of $D$ and $V$, on every regular $s$-summable spectral triple (Corollary 3.6.6).","The Eckstein–Iochum question is settled: expanding $\\operatorname{Tr}(e^{-tD^2})$ alone is insufficient—the diagonal counterexample has no local Weyl law—but expanding $\\operatorname{Tr}(P e^{-tD^2})$ for all $P\\in\\mathcal{B}$ does transfer to the perturbed operator.","Truncated spectral triples are faithful: under a Weyl law, the noncommutative integral of $a$ is the logarithmic mean of the finite-dimensional quotients $\\operatorname{Tr}(P_\\lambda a P_\\lambda)/\\operatorname{Tr}(P_\\lambda)$, and Widom's lemma upgrades this to a noncommutative Szegő limit theorem for $f(A)$.","Classical ergodicity implies quantum ergodicity in NCG: a classically ergodic spectral triple has a density-one subsequence of eigenbasis vectors along which $\\langle e_j, a e_j\\rangle$ converges to the noncommutative integral of $a$ (Theorem 4.4.11).","The density of states becomes a Dixmier trace: for Hamiltonians on discrete metric spaces and on manifolds of bounded geometry, the DOS is computed by a Dixmier trace, and this yields a Dixmier trace formula for Roe's index on open manifolds."],"supporting_citations":[{"why":"the joint preprint on which Part I is based, supplying the multiple-operator-integral construction for abstract pseudodifferential operators.","marker":"[HMN24]"},{"why":"Peller's multiple-operator-integral construction that the thesis generalises to the abstract pseudodifferential calculus.","marker":"[Pel06]"},{"why":"Connes–Moscovici's abstract pseudodifferential calculus and local index formula, the NCG setting the MOIs are adapted to.","marker":"[CM95]"},{"why":"poses the open question on heat-trace asymptotic expansions under perturbations that Corollary 3.6.6 and the accompanying examples answer.","marker":"[EI18]"},{"why":"Connes' trace theorem and integral formula, the foundation for the noncommutative integral approximated by truncated spectral triples.","marker":"[Con88]"},{"why":"the theory of singular and Dixmier traces and the noncommutative integral, supplying the technical trace results used throughout Part II.","marker":"[LSZ21]"},{"why":"Widom's Szegő limit theorem and the lemma on truncated products that Theorem 4.2.2 translates into NCG.","marker":"[Wid79]"},{"why":"Colin de Verdière's theorem that ergodic geodesic flow implies quantum ergodicity, the classical statement behind Theorem 4.4.11.","marker":"[Col85]"},{"why":"Zelditch's result on unique vacuum states, the C*-dynamical system statement used as the core of the NCG quantum ergodicity theorem.","marker":"[Zel96]"},{"why":"Simon's definition of the density of states, the object whose Dixmier trace formula is proved in Chapters 5 and 6.","marker":"[Sim82]"}],"fun_headline_variants":["Multiple operator integrals yield heat trace expansions in noncommutative geometry","Noncommutative Taylor formula via multiple operator integrals","Dixmier trace formulas for density of states and Roe index","Operator integral calculus proves trace formulas in noncommutative geometry","Heat trace and Dixmier trace formulas unified in noncommutative geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $D^2$ satisfies a (local) Weyl law—eigenvalue growth $\\lambda_k(D^2)\\sim C k^{2/d}$ with the matching behaviour of $\\operatorname{Tr}(a e^{-tD^2})$—so that spectral averaging converges to the noncommutative integral, and, for the Taylor expansions of Part I, that the commutators $\\delta^k_H(V)$ decay in operator order by a fixed $\\varepsilon>0$; the thesis itself constructs a spectral triple where the global Weyl law holds but the local ones fail, so neither premise is automatic.","fun_headline_variants_meta":{"raw":{"variants":["Multiple operator integrals yield heat trace expansions in noncommutative geometry","Noncommutative Taylor formula via multiple operator integrals","Dixmier trace formulas for density of states and Roe index","Operator integral calculus proves trace formulas in noncommutative geometry","Heat trace and Dixmier trace formulas unified in noncommutative geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001436,"raw_usage":{"total_tokens":5892,"prompt_tokens":1149,"completion_tokens":4743,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":4656}},"tokens_in":765,"tokens_out":4743,"duration_ms":30778,"temperature":1.0,"reasoning_tokens":4656,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:17:32.379982+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any regular $s$-summable spectral triple satisfying the local Weyl law for all $P\\in\\mathcal{B}$, with a bounded self-adjoint $V$; a computation that violated the stated remainder $O(t^{N+1-s})$ in Corollary 3.6.6 for some $N$ would refute the heat-trace expansion claim. Similarly, a spectral triple satisfying the Weyl law for which the logarithmic mean of $\\operatorname{Tr}(P_{\\lambda_n}aP_{\\lambda_n})/\\operatorname{Tr}(P_{\\lambda_n})$ differs from $\\operatorname{Tr}_\\omega(a\\langle D\\rangle^{-d})/\\operatorname{Tr}_\\omega(\\langle D\\rangle^{-d})$ would refute Theorem 4.1.7.","supporting_citations":[],"review_version":1}