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Topological groups with tractable minimal dynamics

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Pith's one-line read The paper defines TMD, the class of topological groups with tractable minimal dynamics, shows it matches the generic point property on Polish groups, and uses forcing to prove the revised Newelski conjecture at all cardinalities.

desk verdict A serious, substantial paper introducing the TMD class and an abstract KPT correspondence; the revised Newelski application is the right target, but the decisive absoluteness proof is cut off in the version I saw. read the letter →

arxiv 2412.05659 v2 pith:YGN7HZLU submitted 2024-12-07 math.DS math.LO

Gianluca Basso , Andy Zucker This is my paper · ORCID
classification math.DSmath.LO MSC 37B0554H1122A05
keywords topologicalgroupsuniversalminimalflowsgenericpointpropertytractabledynamicsstructuralRamseytheoryKPTcorrespondencedefinableNIPNewelskiconjecture
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's aim is to find the right generalization of the generic point property (GPP) — the property of Polish groups whose universal minimal flow has a comeager orbit — to all topological groups, and to show that the resulting class supports structure theorems and model-theoretic applications. The class is called TMD, for tractable minimal dynamics, and is defined by demanding that the universal minimal flow $M(G)$ satisfy the Rosendal criterion: at every scale and in every open region of the flow, some sub-region is topologically transitive. Five equivalent formulations are proved (Theorem 5.17), including a thickness condition on $M(G)$, the statement that every ultracopower of $M(G)$ contains a unique minimal subflow (equivalently, that every ultracopower map is highly proximal), and the compactness of a new meets topology on the space of minimal subflows of any $G$-flow. For Polish groups the new class recovers GPP exactly, and a Polish group is in GPP precisely when $M(G)$ contains a point of first countability (Theorem 7.7), a new result even for the older class. The paper then proves an abstract KPT correspondence: $G$ is in TMD iff it admits a $G$-skeleton — a directed system of metric spaces indexed by the continuous semi-norms on $G$ — satisfying Ramsey, minimality and extremal-disconnectedness properties, in which case the skeleton's folded flow is $M(G)$ (Theorem 9.20). This certificate makes membership in the classes EA, CMD and TMD a $\Delta_1$ property in the Lévy hierarchy (Theorem 10.5), so forcing and absoluteness arguments transfer Polish-group theorems to all TMD groups; the applications are a structure theorem for $M(G)$ of the form $M(G)\cong S_G(\mathrm{UCF}(G))$ with Hausdorff Ellis group, and a proof of the revised Newelski conjecture for groups definable in NIP structures at all cardinalities.

What carries the argument

Three objects carry the argument. (1) The universal minimal flow $M(G)$ together with the equivalence chain of Theorem 5.17: the Rosendal criterion says that for every open $A$ and every neighborhood $U$ of the identity, some open sub-region of $A$ is $U$-topologically transitive, and the theorem equates this with the thickness condition, with uniqueness of the minimal subflow in every ultracopower of $M(G)$, with high proximality of the ultracopower map, and with the compactness of the meets topology on the space of minimal subflows of every $G$-flow; TMD is defined by any one of these conditions holding for $M(G)$ (Definition 5.18). (2) The $G$-skeleton: a directed system of metric spaces $X_\sigma$ indexed by the continuous semi-norms $\sigma$ on $G$, with bonding maps and a compatible $G$-action, satisfying a Ramsey property, a minimality property and an ED (extremally disconnected) property; the folded flow is the inverse limit of the Samuel compactifications of the levels, and the abstract KPT correspondence (Theorem 9.20) says $G\in\mathrm{TMD}$ iff such a skeleton exists, in which case the folded flow is $M(G)$. For $G=\mathrm{Aut}(K)$ the automorphism group of a Fraïssé structure, a $G$-skeleton is exactly a reasonable expansion class and the three properties become the usual Ramsey, expansion and amalgamation properties (Proposition 9.16). (3) $G$-preflows and the Lévy hierarchy: by phrasing every notion over precompact uniform $G$-spaces presented by explicit bases, the paper writes membership in EA, CMD and TMD as both a $\Sigma_1$ and a $\Pi_1$ formula (Lemmas 10.1 and 10.4, Theorem 10.5); since $\Delta_1$ formulas are absolute between transitive models of set theory, this is what makes the forcing transfer work.

What would settle it

Concretely, the claims predict that every Polish group outside GPP has an $M(G)$ with no first-countable points and with some ultracopower containing two distinct minimal subflows, and that every group definable in an NIP structure, at every cardinality, has a Hausdorff Ellis group for its universal minimal externally definable flow. The first counterexample to either prediction — a non-GPP Polish group with a first-countable point in $M(G)$, or a definable NIP group with a non-Hausdorff Ellis group — refutes the corresponding theorem; the paper itself points to countable discrete groups and to oligomorphic permutation groups as the natural candidates to probe.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that 'the universal minimal flow has a comeager orbit' is not a Polish-specific phenomenon: what makes the property work is that $M(G)$ satisfies the Rosendal criterion, and this condition is meaningful and productive for every topological group. Theorem 5.17 proves that this single condition is equivalent to four others: the thickness condition on $M(G)$; the statement that every ultracopower of $M(G)$ contains a unique minimal subflow; the statement that every ultracopower map is highly proximal; and the compactness of the meets topology on $\mathrm{Min}_G(Z)$ for every $G$-flow $Z$. The paper defines TMD as the class of groups for which these equivalent conditions hold (Definition 5.18) and then establishes three large claims about it. First, for Polish groups TMD coincides with GPP, and a Polish group is in GPP iff $M(G)$ has a point of first countability (Theorem 7.7). Second, $G$ is in TMD iff it admits a $G$-skeleton with the Ramsey, minimality and ED properties, in which case the folded flow of the skeleton is $M(G)$; this is the abstract KPT correspondence (Theorem 9.20). Third, the classes EA, CMD and TMD are $\Delta_1$ in the Lévy hierarchy (Theorem 10.5), so membership is absolute under forcing, and a group is TMD iff in some forcing extension its Raikov completion is GPP. The final theorems put this machinery to work: a structure theorem for $M(G)$ for TMD groups with $\mathrm{Aut}(M(G))$ a compact Hausdorff group for the tau-topology, and the revised Newelski conjecture — for any group definable in an NIP structure, the Ellis group of the universal minimal externally definable flow is Hausdorff — proved at all cardinalities.

Load-bearing premise

The load-bearing premise is that the forcing transfer is sound: membership in TMD, EA and CMD, the Hausdorffness of the Ellis group, and tameness can all be expressed as $\Delta_1$ properties of preflows, so that forcing a group to become countable does not change which of these properties it has — if even one of these complexity classifications fails, the transfer theorems and the Newelski proof collapse even though the internal equivalences of Theorem 5.17 could survive.

Editorial extensions

If this is right

  • For Polish groups the new class is exactly the old one: $G\in\mathrm{TMD}$ iff $G\in\mathrm{GPP}$, and membership is decided by a single point of first countability in $M(G)$; in particular every non-compact locally compact Polish group has a universal minimal flow with no first-countable points (Theorem 7.7, Proposition 7.13).
  • TMD is a well-behaved class: it is closed under group extensions, surjective inverse limits and arbitrary products, and it contains no locally compact non-compact groups (Theorems 6.5 and 6.10, Proposition 6.8, Theorem 10.6).
  • Membership in TMD has a certificate: $G\in\mathrm{TMD}$ iff $G$ admits a $G$-skeleton with the Ramsey, minimality and ED properties, and then the folded flow is isomorphic to $M(G)$; adding precompactness of the skeleton characterizes CMD (Theorem 9.20).
  • The classes EA, CMD and TMD are $\Delta_1$ in the Lévy hierarchy, so membership is absolute between transitive models of set theory; a group is TMD iff in some forcing extension its Raikov completion is GPP (Theorem 10.5, Section 10).
  • For TMD groups the universal minimal flow has the concrete form $M(G)\cong S_G(\mathrm{UCF}(G))$, the Gleason completion of the universal compactification flow, and the tau-topology on $\mathrm{Aut}(M(G))$ is compact Hausdorff; the revised Newelski conjecture follows for NIP groups at all cardinalities (Theorems 8.24 and 10.15, Corollary 8.17, Section 11).

Reading between the lines

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

  • Beyond the paper: the abstract KPT correspondence draws a boundary. If TMD is the largest class carrying Ramsey-style certificates, then computing the universal minimal flow of a group outside TMD — a locally compact non-compact group, for example — is not simply hard but structurally impossible by expansion-class methods, and the failure of the Ramsey certificate is a negative criterion one can c
  • Beyond the paper: the $\Delta_1$ absoluteness scheme is a general template. Any dynamical property that can be written as a $\Delta_1$ formula over preflows automatically transfers between a structure and its forcing extensions, so the Newelski-style transfer should extend to further questions about tame flows and Ellis groups at uncountable cardinalities, not just to the Hausdorffness statement s
  • Beyond the paper: the paper leaves open whether TMD equals the pointwise-defined class WCAP (Question 5.20) and whether $M(G)$ itself, rather than only its Gleason completion, is always the universal compactification flow. If WCAP catches up with TMD, the meets-topology criterion is provably the sharp boundary of tractability; if the Gleason step is ever necessary, the correction in the structure
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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

2 major / 5 minor

Summary. The paper introduces the class TMD ('tractable minimal dynamics') for arbitrary topological groups G, defined by requiring that the universal minimal flow M(G) satisfies the equivalent conditions of Theorem 5.17: the Rosendal criterion, the thickness condition, uniqueness of the minimal subflow in every ultracopower of M(G), high proximality of the ultracopower map, and compact Hausdorffness of the meets topology on Min_G(Z) for every G-flow Z. Theorem 5.5 gives the analogous class CMD ('concrete minimal dynamics'); for Polish groups, CMD ∩ Polish = PCMD and, by Theorem 7.7, TMD ∩ Polish = GPP, with GPP also characterized by M(G) having a point of first countability. Section 9 states and proves an abstract KPT correspondence (Theorems 9.19 and 9.20): G is in TMD (resp. CMD, EA) iff it admits a G-skeleton with the Ramsey, minimality, and ED properties (resp. a precompact such skeleton, the trivial skeleton with the Ramsey property). Section 8 studies the tau-topology on Aut(M(G)), proves it is Hausdorff for TMD groups, and gives a structure theorem for minimal flows with Hausdorff Ellis group via proximal/equicontinuous extensions. Section 10 shows that EA, CMD, and TMD are Delta-1 in the Levy hierarchy (Theorem 10.5) and derives forcing-transfer theorems (Theorems 10.6 and 10.8) that recover and extend results previously proved only for Polish groups.

Significance. Assuming the missing Section 11 claims, this is a genuinely novel and unifying contribution. The class TMD is characterized by several independent dynamical conditions, each a natural weakening of the corresponding CMD condition, and the Polish case identifies TMD with the previously studied class GPP while adding a clean first-countability criterion (Theorem 7.7). The abstract KPT correspondence of Theorems 9.19-9.20 genuinely extends the Kechris-Pestov-Todorcevic framework to arbitrary topological groups, with G-skeletons serving as explicit certificates, and the Delta-1 classification of EA/CMD/TMD in Theorem 10.5 is a clever two-sided use of the Levy hierarchy that explains why many results for Polish groups transfer verbatim to all TMD groups. The visible proofs of Theorems 5.5, 5.17, 7.7, 9.19, 9.20, and 10.5 are detailed and largely self-contained; the characterizations are given in closed form with no free parameters. The paper does not include machine-checked proofs or code, so correctness rests on the detailed by-hand arguments, which I found coherent for the portions supplied.

major comments (2)
  1. [Section 11 and final part of Section 10] The Abstract and Introduction claim that, suitably phrased for preflows, the assertions 'X has Hausdorff Ellis group' and 'X is tame' are Delta-1, and that a forcing/absoluteness transfer from the countable case [17] proves the revised Newelski conjecture at all cardinalities and gives a partial Glasner structure theorem for all minimal tame flows. The supplied text does not contain these proofs: Section 10 announces the absoluteness result for Hausdorff Ellis groups in its final subsection and then breaks off at the beginning of the recovery of Bartosova's example, and Section 11 is entirely absent. This is load-bearing: the transfer must show that the externally definable flow of a definable group G in an NIP structure is represented by a G-preflow in the ground model, and that the two properties are absolute between V and a Coll(omega,kappa) extension, with the Pi-1 side witnessed among preflows rather than via unbounded quantifiers over elementary extensions, externally definable subsets, C(beta omega, X), or the enveloping semigroup. The visible Section 10 proofs for EA/CMD/TMD (Lemmas 10.1 and 10.4, Theorem 10.5) have exactly the right quantifier shape and appear sound, but the corresponding Section 11 claims cannot be checked from the manuscript as provided. The authors should either include the missing proofs or explicitly restate the Newelski and tame-structure results as conditional on them.
  2. [Proposition 5.11, Lemma 6.3, Theorem 6.5] The characterization of TMD via the meets topology on Min_G(Z), and the statement that uniqueness of minimal subflows in all ultracopowers of M(G) implies compact Hausdorffness of the meets topology, rest on Proposition 5.11, whose proof cites 'Proposition 4.4 of [69]', an unpublished preprint that the text says 'will be revised and expanded using concepts developed in this paper.' Lemma 6.3 and Theorem 6.5 (closure of TMD under group extensions) subsequently rely on the meets-topology formulation of TMD. A referee cannot verify the cited statement, so the paper's stated results depend on an external, unverifiable source. Please either prove the Vietoris-limit-factor statement in the present paper, or reformulate the proofs of Lemma 6.3 and Theorem 6.5 directly in terms of the other conditions of Theorem 5.17, which are developed in full.
minor comments (5)
  1. [Abstract; Theorem 7.7] The abstract states that 'a Polish group is in GPP iff its UMF has no points of first countability,' which is the negation of the theorem proved in the paper: Theorem 7.7(3) and the Introduction say that G is in GPP iff M(G) has a point of first countability. The abstract should state the characterization with 'a point of first countability.'
  2. [Theorem 5.17] Items (6) and (7) of Theorem 5.17 are printed identically, both reading 'For each sigma in SN(G), partial_sigma has a dense set of compatibility points.' Based on the proof of (6) iff (7) via Proposition 3.3, and on the hierarchy leading to item (8), one of the two items should be the weaker condition 'partial_sigma has some compatibility point'; the statement should be corrected.
  3. [Section 5 proofs] Several cross-references in Section 5 point to 'by Theorem 5.4' and 'by Theorem 5.15' where the intended statements are Lemma 5.4 and Lemma 5.15, respectively; see the proofs of (3) implies (2), (8) implies (2), and (1) iff (6) inside Theorem 5.17.
  4. [Section 4.1 remark] The remark in Section 4.1 says 'see the remark following Theorem 3.6,' but there is no Theorem 3.6; the reference should be to Definition 3.6 or to the remark following it.
  5. [Section 3 numbering] The numbering in Section 3 is out of order: Theorem 3.32 is stated and its proof uses 'Fact 3.31,' but Fact 3.31 is printed after the statement of Theorem 3.32; please renumber or move the fact for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: TMD is introduced via a proved equivalence theorem, and the main results are derived from independent arguments rather than assumed conclusions.

full rationale

The paper's central derivation is not circular. TMD is introduced in Definition 5.18 by declaring that M(G) satisfies the equivalent conditions of Theorem 5.17, and Theorem 5.17 is proved in the paper from ultracoproduct and Vietoris/meets-topology arguments, not by assuming the target GPP or PCMD results. The Polish equivalence (Theorem 7.7) derives GPP = TMD from Rosendal's criterion (Theorem 7.6) and earlier independent results, with no fitted parameter being relabeled as a prediction. The abstract KPT correspondence (Theorems 9.19 and 9.20) proves the equivalence between TMD and existence of G-skeletons with Ramsey/minimality/ED properties by constructing the skeleton from M(G) and by showing the folded flow of a skeleton is M(G); this is a forward implication, not a definitional identification. The Delta-1 absoluteness results in Section 10 use the KPT characterization to exhibit Sigma-1 formulas and Lemma 10.1 for Pi-1 formulas; again, the classification is established rather than assumed. The revised Newelski application is announced and its transfer is said to rely on Delta-1 absoluteness claims, but these claims are not circular: they are unverified in the visible text because Section 11 is truncated, which is a completeness or verifiability gap, not a reduction of a conclusion to its input. Self-citations to [11], [72], and [73] are frequent but carry independent proofs in prior published work and are used as benchmarks; no load-bearing step reduces to an unproved assertion of the present authors.

Assumptions & free parameters 0 free parameters · 11 assumptions · 4 invented entities

No numerical parameters are fitted or tuned; the entries below are the unproved or externally supplied facts the central claims rest on. The main external inputs are Ellis's existence theorem, duality theories, CRTS facts, Zucker's GPP structure theorem [73], Glasner's tame flow structure [30, 34], and the countable Newelski case [17]. The self-referential items are the unpublished preprint [69] used in Proposition 5.11, and Fact 3.31 adapted from the authors' [11]. Standard ZFC plus forcing and Levy absoluteness is assumed throughout Section 10.

assumptions (11)
  • standard math ZFC with forcing and Levy absoluteness for Delta-1 formulas between transitive models with the same ordinals
    Assumed throughout Section 10 to transfer properties between V and forcing extensions; Levy hierarchy facts cited to [42].
  • standard math Ellis's theorem: every topological group has a unique universal minimal flow M(G)
    Foundational input for the definitions of GPP, CMD, and TMD; cited to [21] in Section 2.6.
  • standard math Birkhoff-Kakutani: the topology of a topological group is generated by bounded continuous semi-norms
    Provides SN(G), which indexes the G-fattening structures; cited to [15] in Section 2.5.
  • standard math Stone and Gelfand duality for Boolean algebras and commutative C*-algebras
    Underlies Gleason covers, Samuel compactifications, and near ultrafilter spaces; Section 2.3.
  • standard math Compact right-topological semigroup theory: idempotents, minimal left ideals, coalescence of minimal subflows
    Used for Sa(G) and the Ellis group machinery; cited to [39] and [37] in Section 2.6 and Section 8.
  • domain assumption Zucker's GPP structure theorem [73, Theorem 5.5]: Polish G in GPP iff M(G) is Sa(G/H) for a closed presyndetic extremely amenable H
    Used in Theorems 7.7, 7.11, 7.12, and Proposition 7.15; a published self-citation with independent support.
  • ad hoc to paper Proposition 4.4 of the unpublished preprint [69]: Vietoris limits of minimal subflows are factors of ultracopowers
    Used in the proof of Proposition 5.11; the introduction says [69] will be revised using concepts from the present paper, making the dependency self-referential and unreviewed.
  • domain assumption Fact 3.31: sequences separated by the fattening uniformity induce injective maps from beta-omega
    Proof 'directly adapted from [11, Corollary 5.7]'; it drives Theorem 3.32, which powers the first-countability characterization of GPP.
  • standard math Fraisse theory: Fraisse limits, amalgamation property, Ramsey property of expansion classes
    Background for the KPT analogy and Proposition 9.18; cited to [25] and [44].
  • domain assumption Glasner's structure theory of tame flows and the universal compactification flow [30], [34]
    Used in Section 8.3 for the structure theorems (8.13, 8.22) and as the countable-case input for the Newelski application in Section 11.
  • domain assumption Chernikov-Gannon-Krupinski [17]: the revised Newelski conjecture holds for countable structures
    The base case that the absoluteness argument transfers to all cardinalities in Section 11.
invented entities (4)
  • TMD, the class of topological groups with tractable minimal dynamics independent evidence
    purpose: Extends GPP from Polish groups to all topological groups while preserving a KPT-style correspondence, closure properties, and a structure theorem for M(G)
    Membership has eight provably equivalent handles (Theorem 5.17); on Polish groups TMD provably equals GPP (Theorem 7.7); locally compact non-compact groups provably fail TMD (Theorem 10.6).
  • Fattening spaces and their near ultrafilter spaces NU(X) independent evidence
    purpose: Unify the Samuel compactification and the Gleason completion of G-spaces in one pseudometric framework
    The framework reproduces known objects: NU(X) for a uniform space is exactly the Samuel compactification (Fact 3.33), and SG(X) has the known universal properties of the Gleason completion (Facts 4.4 and 4.5).
  • G-skeletons and the abstract Ramsey property independent evidence
    purpose: Certificate for TMD: G is TMD iff it admits a G-skeleton with Ramsey, minimality, and ED properties, with the folded flow isomorphic to M(G)
    For automorphism groups of Fraisse structures, G-skeletons reduce to expansion classes and the abstract Ramsey property reduces to the classical Ramsey property (Propositions 9.14 and 9.18), providing an external benchmark.
  • Meets topology on Min_G(Z) independent evidence
    purpose: A coarsening of the Vietoris topology whose Hausdorffness characterizes TMD and which is always compact
    Proposition 5.10 proves compactness of the meets topology in general, and Definition 5.18 plus Proposition 5.11 prove the equivalence with TMD, so the characterization is proved rather than assumed.

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Pith. "Pith review of Topological groups with tractable minimal dynamics." pith.science (2026). https://pith.science/paper/YGN7HZLU

@misc{pith2026241205659,
  author       = {Pith},
  title        = {Pith review of: Topological groups with tractable minimal dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGN7HZLU}},
  note         = {Machine review of arXiv:2412.05659}
}
abstract

A Polish group $G$ has the generic point property if any minimal $G$-flow admits a comeager orbit, or equivalently if the universal minimal flow (UMF) does. The class $\mathsf{GPP}$ of such Polish groups is a proper extension of the class $\sf{PCMD}$ of Polish groups with metrizable UMF. Motivated by analogous results for $\mathsf{PCMD}$, we define and explore a robust generalization of $\sf{GPP}$ which makes sense for all topological groups, thus defining the class $\mathsf{TMD}$ of topological groups with tractable minimal dynamics. These characterizations yield novel results even for $\mathsf{GPP}$; for instance, a Polish group is in $\mathsf{GPP}$ iff its UMF has no points of first countability. Motivated by work of Kechris, Pestov, and Todor\v{c}evi\'c that connects topological dynamics and structural Ramsey theory, we state and prove an abstract KPT correspondence which characterizes the class $\mathsf{TMD}$ and shows that $\mathsf{TMD}$ is $\Delta_1$ in the L\'evy hierarchy. We then develop set-theoretic methods which allow us to apply forcing and absoluteness arguments to generalize numerous results about $\mathsf{GPP}$ to all of $\mathsf{TMD}$. We also apply these new set-theoretic methods to first generalize parts of Glasner's structure theorem for minimal, metrizable tame flows to the non-metrizable setting, and then to prove the revised Newelski conjecture regarding definable NIP groups. We conclude by discussing some tantalizing connections between definable NIP groups and $\mathsf{TMD}$ groups.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Some results on NIP groups and their Ellis groups

    math.LO 2026-07 accept novelty 8.0 of 10

    In NIP theories, the Ellis group of any definable group has size at most 2^|T|, independent of the model; under bounded VC-codensity it (and the local quotient G/G^00_φ) is an inverse limit of compact Lie groups of di...

  2. Haar decompression and amenability of Ellis flows

    math.DS 2026-07 accept novelty 7.0 of 10 partial

    For tame flows, Haar measure on an Ellis group decompresses to a regular measure on the enveloping semigroup, and the enveloping flow is amenable iff the original flow is hereditarily amenable.

  3. Some aspects of topological dynamics of Polish groups (with an introduction to descriptive set theory)

    math.LO 2026-02 unverdicted novelty 1.0 of 10

    An attributed, exercise-rich graduate text proving the KPT correspondence linking extreme amenability and Ramsey theory, plus B. Miller's proof of the G₀-dichotomy; no new results are claimed.

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