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Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families

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Pith's one-line read The paper establishes that the L∞ approximate point spectrum of a Koopman operator on Cantor space cannot be computed by any finite tower of algorithms with point-evaluation access, even for continuous measure-preserving maps; the L1 case,

desk verdict The L∞ SCI=∞ result is the real contribution, and the reader's flagged Borelness gap closes under the type-G consistency axiom; this deserves a careful referee. read the letter →

arxiv 2601.12044 v2 pith:2IB6VG7L submitted 2026-01-17 math.LO cs.NAmath.DSmath.NAmath.SP

Christopher Sorg This is my paper · ORCID
classification math.LOcs.NAmath.DSmath.NAmath.SP MSC 03E1503D7847A1047B3337A30
keywords solvabilitycomplexityindexKoopmanoperatorapproximatepointspectrumL∞nonseparabilitytowersofalgorithmsdescriptivesettheoryCantorspaceSilvertree
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that computing the L∞ approximate point spectrum of a Koopman operator is information-theoretically impossible in a strong sense. On the Cantor space with Bernoulli measure, for the input class of continuous measure-preserving maps, the problem F ↦ σ_ap(K_F : L∞ → L∞) has infinite solvability complexity index under point-evaluation queries. This means no finite tower of general algorithms, each reading only finitely many values of F, can converge to the spectrum in the Hausdorff metric, no matter how much internal computation is allowed. The proof works by encoding a known non-Borel decision problem—whether a binary tree contains a Silver tree—into the spectral membership of a fixed non-torsion point, and then showing that any finite tower restricted to these encoded maps would be Borel, a contradiction. At the L1 endpoint, by contrast, the paper shows the same upper bounds as the reflexive Lp theory hold, by proving a uniform finite-dimensional quadrature compatibility lemma.

What carries the argument

The Silver-tree coding. For each binary tree S, the paper constructs a continuous, measure-preserving homeomorphism F_S on Cantor space that acts separately on the clopen blocks Y_m = {first zero at position m}. On block m, it permutes length-m coordinate blocks according to a star-priority pattern u_m(S): certain coordinates are forced to match a witness, while free (star) coordinates carry a cyclic permutation. The number of free coordinates in u_m(S) is unbounded exactly when S contains a Silver tree. Because each free coordinate produces a 2-adic root of unity as an eigenvalue, a fixed non-torsion point z0 lies in σ_ap(K_{F_S}) precisely in that unbounded case. This coding connects a des

What would settle it

Exhibit a finite-height tower of point-evaluation algorithms that computes σ_ap(K_F) in the Hausdorff metric for all continuous measure-preserving maps on 2^N; this directly contradicts Theorem 4.14. More narrowly, since restricting any such tower to the tree-coded maps {F_S} would yield a Borel tower on Tree2 computing S ↦ σ_ap(K_{F_S}), the claim would also collapse if one could construct such a Borel tower, because it would make the Silver-tree predicate V Borel.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.14: on X = 2^N with Bernoulli measure and the input class of continuous measure-preserving maps, the problem Ξσap(F) = σ_ap(K_F : L∞(X,ω) → L∞(X,ω)) satisfies SCIG = ∞ with respect to the point-evaluation oracle. Restated: there is no finite tower of general algorithms whose iterated limits compute the approximate point spectrum in the Hausdorff metric. The proof restricts any hypothetical finite tower to a family {F_S} of tree-coded homeomorphisms; on this restricted family, finite queries are locally constant, so the tower becomes Borel. But membership of a fixed non-torsion point z0 in the spectrum of K_{F_S} is equivalent to S containing a Silver tree, a se

Load-bearing premise

The proof assumes that a general algorithm's query choices are guided only by the finite point values it has already seen; if algorithms were allowed to choose query points based on information about F other than those finite values, the local-constancy step that makes the restricted tower Borel would fail.

Editorial extensions

If this is right

  • Any practical data-driven Koopman scheme on L∞ with finite point samples per stage would need unboundedly many nested limiting stages to recover the approximate point spectrum, even on symbolic measure-preserving systems.
  • The L1 endpoint inherits the reflexive upper bounds: with point evaluations plus fixed quadrature, SCI_G(σ_ap) ≤ 3 and SCI_G(σ_ap,ε) ≤ 2 on the general classes, improving to 2 and 1 on classes with a known modulus.
  • The SCI_G = ∞ lower bound also holds for the 1-Lipschitz measure-preserving classes, so imposing uniform continuity does not restore computability.
  • In the arithmetic tower model, the same L∞ decision problem has SCI_A = ∞ because the acceptance set is analytic-complete and hence non-Borel; thus the obstruction is not merely an effectivity artefact.
  • The appendix constructs a reusable family of decision problems with exact finite type-G tower heights m, providing a reduction source for future SCI lower bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Cantor-subsystem coding can be embedded into broader compact metric spaces, the same L∞ obstruction would likely reappear in many geometric or smooth systems as soon as they contain a coded clopen subsystem; the paper leaves this transfer open.
  • The sharp contrast between finite-height Lp results and infinite-height L∞ suggests that practitioners working with essential-supremum observables should expect strictly worse algorithmic behavior than the familiar L2/DMD picture, even in very simple dynamics.
  • Because the type-G lower bound allows arbitrary post-processing of finitely many point values, the obstruction is not about computability but about finite information: any algorithm whose access to F at each stage is a finite set of point evaluations will fail, regardless of the power of its internal processing.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies SCI classifications for Koopman spectral problems at the endpoint spaces L^1 and L^infty, in the point-evaluation plus fixed-quadrature oracle model. On L^1 it proves a meta upper bound (Theorem 3.2) under axioms (F1)--(F4), and supplies a Riemann-sum lemma (Lemma 3.4) so that L^1 falls under the reflexive upper-bound framework. On L^infty it proves (a) an analytic-completeness result for membership in the approximate point spectrum (Theorems 4.8 and 4.9), giving SCI_A = infinity for a suitable countable oracle model; (b) a Type-G lower bound SCI_G = infinity on the measure-preserving continuous class (Theorem 4.14), via a reduction from the non-Borel Silver-tree set V; and (c) a discontinuity phenomenon for the Hausdorff-distance map F -> sigma_ap(K_F) (Lemma D.12). Appendix E constructs Cantor-matrix predicates with exact finite Type-G heights. The purported gap in Theorem 4.14 concerning input-dependent query sets is not a gap: the type-G consistency axiom, applied with the fixed input F_S as the anchor, makes the restricted tower locally constant and hence Borel.

Significance. If the results are correct, this is a valuable extension of the SCI program to the non-reflexive endpoints: L^1 is brought into the reflexive framework, while L^infty is shown to have information-theoretic hardness that persists even in the unrestricted Type-G model. The explicit Cantor/tree constructions in Appendices A--D are concrete and mostly self-contained, and the paper correctly emphasizes that descriptive-set/Type-2 non-Borelness does not by itself imply Type-G lower bounds. The reusable calibration family in Appendix E is attractive, and Lemma D.12 gives a clean, quantitative discontinuity example. The main theorem, Theorem 4.14, is structurally sound once the cited non-Borelness of V is accepted.

major comments (4)
  1. [Lemma 4.5 / Theorem 4.8, Step 1] The proof of Lemma 4.5 relies on the assertion that, for a standard Borel code c of a Borel set, the map c -> omega(E(c,r)) is Borel in c. In the usual theory of Borel codes this is not available: the set of Borel codes is Pi^1_1 rather than a Borel subset of a Polish space, and measure of a code is not Borel in a Polish parameterization. Since the lemma only needs A_z0 to be analytic, the proof can likely be repaired by using the weak-* compact metrizable unit ball of L^infty and expressing approximate eigenvectors as an existential over sequences in that Polish space. As written, however, Theorem 4.8 Step 1 and consequently Theorem 4.9 rest on an unjustified coding claim.
  2. [Abstract and Sections 3--5, Appendix E] Several claims advertised in the abstract do not appear as theorems in the body: (i) the L^1 'target-split' residual bounds involving R_ap,epsilon and C_ap,epsilon are not defined or proved; (ii) the 'Borel unbounded-period collapse' and 'fixed L^infty point-eigenvalue membership is Borel' are not stated as results; (iii) the 'Koopman point-spectrum calibration families' of the abstract are not constructed: Appendix E gives Cantor-matrix predicates, not Koopman spectra, and Section 4.2 introduces B_m but never reduces B_m to a Koopman spectral problem. These items should be either proved with precise theorem statements or removed from the abstract and motivational sections.
  3. [Section 3, Step 3(d), Eq. (3.5)] The proof that the finite sets Gamma_n2(F) converge to sigma_ap,epsilon(K_F) in Hausdorff metric is not fully justified. The set Gamma_n2(F) uses the strict threshold h_n2(z,F) < epsilon - 1/n2, and the inclusions proved only relate Gamma_n2 to {r_F <= epsilon - 2/n2} and {r_F <= epsilon}. For a point z in sigma_ap,epsilon with r_F(z) = epsilon, a nearby grid point with r_F approximately epsilon need not satisfy the strict threshold, so the argument does not control the Hausdorff distance from boundary points of the pseudospectrum. The standard proof uses two-sided approximations (e.g. Gamma^+ = {h < epsilon + 1/n} and Gamma^- = {h < epsilon - 1/n}) or a finer grid with a margin. The claim is plausible and fixable, but the proof as written is incomplete.
  4. [Section 4.2.1 / Definition 4.12] The subsection is titled 'Reduction To Koopman Approximate Point Spectrum In The m-class,' but no reduction involving B_m is proved or used. Theorem 4.14 uses only the Silver-tree set V of Lemma 4.6 and Lemma 4.13; the predicates B_m play no role. If the B_m family is meant to be a calibration family for Koopman spectra, the missing reduction is a load-bearing gap. If not, the section heading and Definition 4.12 should be removed or clearly separated from the proof of Theorem 4.14.
minor comments (4)
  1. [Throughout] The text contains many conversion artifacts such as '/leftr⫯g⊸tl⫯ne', '/Leftr⫯g⊸tl⫯ne', and 'F AU' in the affiliation. These should be cleaned before publication.
  2. [Lemma 4.6] The proof of the non-Borelness of V is imported from [MZ24, Theorem 3.7] without stating the exact theorem. Since this is the decisive external input for Theorem 4.14, please quote the precise statement and verify that it indeed gives a Borel reduction from IF_omega to V.
  3. [Theorem 4.14] The local-constancy step would be clearer if the authors explicitly invoked the consistency axiom item (3) with anchor input F_S, as the reader's concern suggests. Adding one sentence would prevent confusion.
  4. [Appendix B, Lemma B.4] The statement 'U_{2^{k_m(S)}} subseteq sigma_p(K_S)' should clarify the closure convention: Lemma D.10 shows sigma_ap is the closure of the union of block spectra, not necessarily the union itself. The proof currently reads as if the union were closed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the L∞ SCI lower bound is self-contained; cited prior work is not load-bearing.

full rationale

The paper's central claim (Theorem 4.14: SCIG = ∞ for L∞ approximate point spectra with point evaluations) rests on an explicit tree-to-dynamics construction Ψ (Definition A.4), the spectral equivalence in Corollary A.10, the finite-query localization Lemma 4.13, and the Borel-tower theorem C.3. These are constructive and do not presuppose the target spectrum. The non-Borelness of the Silver-tree set V is imported from the external reduction [MZ24, Thm 3.7], which is independent of the present paper and not a self-citation. The author's [Sor25] is cited for the oracle model and for reflexive Lp sharpness, but the L∞ lower bound does not depend on those results for its validity; the evaluation model is also restated in Assumption 4.1. The local-constancy step in Theorem 4.14 follows from the consistency axiom of general algorithms together with Lemma 4.13: for the finite query set E of a base algorithm on F_S, any S′ agreeing with S through the corresponding level has F_S′ = F_S on E, so the base output is constant on a cylinder around S. No fitted parameters, no data subsets, and no renamed empirical quantities occur. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical free parameters appear; the results are pure existence/classification theorems. The load-bearing inputs are standard descriptive set theory facts and one external Borel-reduction theorem. The star-priority witness u_m(S) and the Cantor/tree maps FS are internal constructions, not extra axioms, and no new physical or formal entity is postulated.

assumptions (5)
  • standard math C(X) is dense in L1(ω) and Lp(ω) for finite Borel measures, and Lipschitz functions are uniformly dense in C(X) on compact metric spaces.
    Used to satisfy axiom (F1) in Theorem 3.2 and Remark 3.3(a)-(b).
  • standard math Density of Borel functions and Borel codes: the Borel σ-algebra of C(X,X) supports standard codings of Borel functions, and measure evaluation on coded Borel sets is Borel.
    Imported from Kechris/Srivastava and used in Lemma 4.5 to prove analyticity of the decision set.
  • domain assumption [MZ24, Theorem 3.7]: There is a Borel reduction from ill-founded trees IFω to the Silver-tree property V.
    This external theorem is the front end of the reduction route in Theorem 4.8 and Theorem 4.14; if it is false, the Σ1^1-completeness lower bounds collapse.
  • standard math SCI framework: general algorithms satisfy consistency, and pointwise limits of Borel maps into a metric target are Borel.
    Used in Appendix C and Theorem 4.14 to convert a finite tower of Borel base maps into a Borel limit map.
  • domain assumption Nonsingularity conditions: for L1 boundedness one needs ρF ∈ L∞; for L∞ well-definedness one needs F#ω ≪ ω.
    These are the standing assumptions on the input class in Section 2 and are needed for the Koopman operator to act on the specified spaces.

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Cite this review

Pith. "Pith review of Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families." pith.science (2026). https://pith.science/paper/2IB6VG7L

@misc{pith2026260112044,
  author       = {Pith},
  title        = {Pith review of: Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2IB6VG7L}},
  note         = {Machine review of arXiv:2601.12044}
}
abstract

We study endpoint Koopman spectral computation from the viewpoint of the Solvability Complexity Index (SCI). Let \((\mathcal X,d)\) be a compact metric space with finite Borel measure \(\omega\), and let \(\mathcal K_F\) be the Koopman operator associated with a continuous nonsingular map \(F:\mathcal X\to\mathcal X\). First, on \(L^1(\mathcal X,\omega)\), we record the endpoint residual upper-bound in the target-split form. The regularized compact fixed-\(\varepsilon\) target $R_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ is separated from the closed fixed-\(\varepsilon\) target $C_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ and from the exact approximate point spectrum $\sigma_{\mathrm{ap}}(\mathcal K_F).$ This endpoint statement uses the same point-evaluation plus fixed-quadrature information model as the \(1<p<\infty\) residual theory. Second, we isolate two obstructions at the nonseparable endpoint \(L^\infty\). Fixed quadrature schemes do not discretize the full \(L^\infty\) unit sphere, and even inside measure-preserving Cantor homeomorphisms the map $F\mapsto \sigma_{\mathrm{ap}}(\mathcal K_F:L^\infty\to L^\infty)$ is maximally discontinuous in Hausdorff distance under arbitrarily small uniform perturbations of \(F\). We also show that finite-period Silver-tree block constructions cannot yield analytic hardness for the \(L^\infty\) approximate point spectrum: for a fixed non-torsion \(z_0\in\mathbb T\), the condition $z_0\in\sigma_{\mathrm{ap}}(\mathcal K_{F}:L^\infty\to L^\infty)$ collapses to a Borel unbounded-period condition. In addition, fixed \(L^\infty\) point-eigenvalue membership is Borel in the measure-preserving continuous class, so one fixed eigenvalue cannot encode a non-Borel tree predicate. Third, we construct Koopman point-spectrum calibration families on the Cantor space.

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