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Learning Graded Sparse Mechanisms: A Representation Theorem and Certified Generalization
Mark Kozdoba, Shie Mannor
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 10/14/2026, 9:09:50 AM
Summary
This paper introduces a theoretical framework for 'graded sparse mechanisms,' where an element's update depends on a weighted subset of other elements. The authors define this class of functions via axioms (value cap, weight regularity, anonymity) and prove a representation theorem stating the class is the Lipschitz ball of a specific transport metric on weighted multisets. This geometric characterization allows for data-dependent generalization bounds based on the sparsity profile of the data and empirical discrepancy computed via optimal matching. The theory is validated using a simulated Vicsek flock model.
Entities (8)
Relation Signals (6)
Graded Sparse Mechanisms → isdefinedby → Axioms
confidence 95% · First, we define the class by axioms, where each element's influence is capped by its weight...
Vicsek Flock → isexampleof → Graded Sparse Mechanisms
confidence 95% · The running example is the Vicsek flock... graded variants of the rule weight the neighbors by a kernel
Graded Sparse Mechanisms → isrepresentedas → Lipschitz Ball
confidence 95% · prove a representation theorem: the class is exactly the Lipschitz ball of a transport metric on weighted multisets
Graded Sparse Mechanisms → exhibits → Generalization Bounds
confidence 90% · the geometry of the representation gives generalization bounds that depend on the data in two ways
Generalization Bounds → dependson → Sparsity Profile
confidence 88% · One is via the sparsity profile of the data... which bounds the rate over all designs with that profile
Graded Sparse Mechanisms → issimilarto → Graph Networks
confidence 80% · neighborhood aggregation of graph networks... received little systematic study as a class of functions with the grading given
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Abstract
Abstract:In a variety of multi-agent and physical environments, an element's update depends strongly on a few other elements and weakly on many, and both the identity and the number of the elements that matter change with the state of the system. Such graded sparse mechanisms are common, for instance, in flocks, crowds, learned particle simulators and the neighborhood aggregation of graph networks, yet they have received little systematic study as a class of functions with the grading given. In this paper we study the supervised learning of the mechanism when the grading is provided by the environment, as a weight on every element. First, we define the class by axioms, where each element's influence is capped by its weight, the dependence on the weights is regular, and no element is distinguished, and prove a representation theorem: the class is exactly the Lipschitz ball of a transport metric on weighted multisets, and the metric is forced by the axioms. Next, the geometry of the representation gives generalization bounds that depend on the data in two ways. One is via the sparsity profile of the data -- the law of how many elements matter and by how much -- which bounds the rate over all designs with that profile, with an exponent from the profile and no dependence on the number of elements. In a number of natural cases we also show matching lower bounds, up to constants in the exponent. The other is empirical: due to the representation result, classical empirical discrepancy bounds apply to the class as they stand, and the resulting bound is the transport distance between the two halves of the sample in the class's own metric, computed exactly by a matching, with an additional concentration term. Finally, we study a simulated Vicsek flock under four interaction rules at two noise levels, and show that the certificate is non-vacuous, a small fraction of the trivial value on every rule.
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- Source: https://arxiv.org/abs/2610.16537v1
- Canonical: https://arxiv.org/abs/2610.16537v1
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Learning Graded Sparse Mechanisms: A Representation Theorem and Certified Generalization Mark KozdobaTechnion, IIT Shie MannorTechnion, IIT and NVIDIA Abstract In a variety of multi-agent and physical environments, an element’s update depends strongly on a few other elements and weakly on many, and both the identity and the number of the elements that matter change with the state of the system. Such graded sparse mechanisms are common, for instance, in flocks, crowds, learned particle simulators and the neighborhood aggregation of graph networks, yet they have received little systematic study as a class of functions with the grading given. In this paper we study the supervised learning of the mechanism when the grading is provided by the environment, as a weight on every element. First, we define the class by axioms, where each element’s influence is capped by its weight, the dependence on the weights is regular, and no element is distinguished, and prove a representation theorem: the class is exactly the Lipschitz ball of a transport metric on weighted multisets, and the metric is forced by the axioms. Next, the geometry of the representation gives generalization bounds that depend on the data in two ways. One is via the sparsity profile of the data—the law of how many elements matter and by how much—which bounds the rate over all designs with that profile, with an exponent from the profile and no dependence on the number of elements. In a number of natural cases we also show matching lower bounds, up to constants in the exponent. The other is empirical: due to the representation result, classical empirical discrepancy bounds apply to the class as they stand, and the resulting bound is the transport distance between the two halves of the sample in the class’s own metric, computed exactly by a matching, with an additional concentration term. Finally, we study a simulated Vicsek flock under four interaction rules at two noise levels, and show that the certificate is non-vacuous, a small fraction of the trivial value on every rule. 11footnotetext: markk@technion.ac.il 1 Introduction In models of complex systems, a property of an element depends on its own state and on the states of some other elements. The dependence is often sparse and graded: a few elements have a strong influence, and the rest a weaker and decreasing influence, or none at all. It is often changing: as the system evolves the grading changes, so the sparse set varies, and varies in size. And it is often anonymous: the elements form a set, with no enumeration that the mechanism reads. One mechanism is applied at every state of the system to the elements the state selects, and we call the environment’s rule that assigns the weights from the state, which elements count and by how much, the selector. The examples of this paper come from multi-agent and physical systems, and in each of them the selector is the environment’s, a rule with at most a few tuned parameters that reads distance, speed, visibility or rank. The running example is the Vicsek flock (Vicsek et al., 1995). Its particles move at a common speed on a periodic square, and at every step each particle turns toward the mean heading of the particles within a fixed distance of itself, up to noise. The selector is the ball: the particles inside it count with equal weight and the rest not at all, and how many are inside varies with the local crowding. Graded variants of the rule weight the neighbors by a kernel of the distance instead, so that the influence fades rather than stops. The selectors of the other examples are of the same kind: distance and speed thresholds in drone flocks (Vásárhelyi et al., 2018), the visible neighbors in crowds (Wirth et al., 2023), the six or seven nearest in starling flocks (Ballerini et al., 2008) and fish schools (Gautrais et al., 2012), a view cone or a sensing radius in driving simulators (Vinitsky et al., 2022; Kazemkhani et al., 2025) and navigation policies (Hüttenrauch et al., 2019; Nayak et al., 2023; Zhang et al., 2025), and a cutoff in learned particle simulators (Sanchez-Gonzalez et al., 2020; Musaelian et al., 2023). The learning problem is this. A configuration is a pair (x,μ)(x,μ), the values x of the d elements and the weights μ the selector gives them, a probability vector: the selector is the environment’s and is given, not learned, so the weights are part of the input, and the unknown is the mechanism f(x,μ)f(x,μ), a member of the class LG_L of Section 3. A sample is n configurations drawn from the design, the law of (x,μ)(x,μ), with the labels f(x,μ)f(x,μ), under the absolute loss, and what we bound is the uniform deviation of the class on the design, the generalization gap of every empirical risk minimizer. While such systems are classical, the graded sparse mechanism as a class of functions, and the complexity of learning one with the selector given, have received little study beyond hard sparsity at a fixed count (Section 2), and this paper develops first results toward both. Taking the selector as given is the decision the graph-network bounds make in taking the graph as given. It is also given in simulators, and in the animal groups some aspects of it are inferred in some cases (Section 2). Axioms and representation. We propose an axiomatic definition of the class. The alternative, a function of a fixed number k0k_0 of arguments fed a different selection of inputs at each configuration, is the form of neighborhood aggregation in the generalization bounds for graph networks, stated under a maximum degree k0k_0 (Garg et al., 2020; Liao et al., 2021), a worst-case parameter that sees neither the average sparsity across configurations nor a grading inside the k0k_0. We define the class instead by three axioms on a mechanism, a function of the elements’ values and of a weight on each element, so that the number of arguments changes with the configuration, and changes gradually. An element’s influence is capped by its weight, none at weight zero. The dependence on the weights is regular: shifting weight between elements changes the output by no more than the same shift can change a weighted mean of the values, the simplest mechanism, which the class must contain. And no element is distinguished, so elements of equal weight are interchangeable. The axioms concern the mechanism alone: the selector enters only through the data distribution, and it may be discontinuous, as a hard radius is. The examples come in two semantics of influence, a mean, fixed in total and diluted by more participants, and a sum, growing with their number, and the framework has a class for each, conservative and accumulative. The main text works with the conservative class. The accumulative class has the same representation and rates of the same exponents, at the price of a factor of the count (Section 3.1, Section B.7). The axioms are a list of inequalities, and the structure of the class is not visible in them. We show that the class is exactly the set of Lipschitz functions on a metric space of weighted configurations (Theorem 3.2): all of the locality structure is in the metric, above it the class is a generic Lipschitz class, and this is what makes the rest of the analysis possible. The metric is the class’s own, each of its terms from one axiom, a transport distance between weighted configurations read as multisets of points on a cone over the value space. We note that the permutation-invariant representation results, Deep Sets (Weyl, 1939; Macdonald, 1995; Zaheer et al., 2017; Wagstaff et al., 2022), have a similar structure: there the sum embedding encodes the invariance, with a generic class above it. There is also an exact connection: permutation-invariant sum networks as above, with a Lipschitz, bounded per-element map and our weights on the sum, ∑iμiφ(xi) _i _i (x_i), are readouts of finitely many linear functionals of our embedding, thus constituting a strict subset of our class. Complexity. Sparsity is a property of the data, and we prove generalization bounds for the class that are data dependent in two ways, both resting on the representation. The first is through the sparsity profile of the design, the law of the weight that a configuration leaves beyond its k heaviest elements: Figure 1(a) is the profile of the flock, the mean weight beyond the k heaviest neighbors under the ball and under three graded kernels at its scale. The bounds have an exponent set by the profile, the worst case over the designs with that profile, and no dependence on the number of elements, and lower bounds in the profile show that the dependence is the right one (Section 4). The second is through the sample itself. It gives no rate, but a bound at the actual sample size, and often a better one. The bound is the classical maximum discrepancy of Bartlett et al. (2002), the largest difference, over the class, between the averages of a function on the two halves of the sample. For a Lipschitz class this is a transport distance (von Luxburg and Bousquet, 2004), and here it is computed exactly by a matching. The result is an empirical generalization certificate, the value of the bound on the data with one concentration term, which on the flock is 0.034κ0.034κ on the hard ball, with κ the value at which the bound says nothing (Section 5.1). To summarize, the contributions in this paper are as follows. (i) Axioms and representation. We define the class of graded sparse mechanisms by axioms and prove that it is the Lipschitz ball of a specific metric space, whose metric encodes the locality (Theorem 3.2). (i) Rates from the profile. We define the sparsity profile of the design and prove generalization bounds with exponent determined by the profile, with no dependence on the number of elements (Theorems 4.3, 4.4 and 4.5), and lower bounds matching up to constants in the exponent in several cases (Theorems 4.4, 4.5 and D.5). (i) The certificate. We show that the generalization gap of the class can be certified from the sample alone, by a matching between its two halves (Theorem 4.6). (iv) The flock. We study the Vicsek flock under four interaction rules at two noise levels, measure their sparsity profiles, compute the certificate for every rule, a thirtieth to a sixth of the trivial value, and read it against a covering certificate and against the gap of a member with zero training risk (Section 5). 2 Literature Graph networks. A message-passing layer is our object with a hard selector, the neighborhood of each node, and it is of the Deep Sets form, a per-neighbor map summed and read out, one readout of finitely many linear sums of our embedding (Section 1), at a fixed width, a bounded degree and no weighting by the environment. Graph networks gain their expressivity from depth, where their sparsity lives: the receptive field is set by the graph and the number of layers, and the bounds of Garg et al. (2020) and Liao et al. (2021) pay for it with the maximum degree k0k_0 per layer, compounded over the depth, Ju et al. (2023) with its spectral relaxation. Here the sparsity is explicit, the weights of the selector, and the bound reads it as the profile: the number of neighbors enters through its law, with rates where there is no nominal sparsity at all and no dependence on the maximum count where there is one (Section 4.1, Remark B.19). Whether any finer statistic of the data than the maximum degree can be used was posed as a question in that literature (Liao et al., 2021; Morris et al., 2023; Morris et al., 2024; Vasileiou et al., 2025b). The two kinds of bound also bound different things: theirs hold over a parametric family at a norm budget and, like our rates, are statements about scaling (Appendix E), and the class adds a bound of a different kind, the certificate, which holds for the whole class at once and is computed from the sample by a matching (Section 4.2). A second line puts a metric on the graphs under which message-passing networks are Lipschitz (Chuang and Jegelka, 2022; Levie, 2023; Böker et al., 2023; Rauchwerger et al., 2025; Amran et al., 2026), and covers the graph space at one scale, or with the sample’s cell proportions at one radius (Carrasco et al., 2025), Covered Forest (Vasileiou et al., 2025a) being its latest form. We differ in the object, a general weighted class whose metric the class itself implies (Theorem 3.2) rather than one metric among several under which a model is Lipschitz, and in the data dependence: rates through the profile (Section 4.1) and, on the data, the two-sample transport in place of a cover (Section 4.2). The matching distance. The metric of Theorem 3.2 is produced by the axioms, which determine it in closed form, a matching on a cone: the identification is new, and the closed form is what makes the rest computable. The metric space itself, a cone with a transport that may leave a point unmatched at a price, is known: unbalanced transport (Chizat et al., 2018; Liero et al., 2018), the matching distances of persistence diagrams (Cohen-Steiner et al., 2010; Turner et al., 2014), of multisets (Chuang and Jegelka, 2022; Davidson and Dym, 2025; Nikolentzos and Skianis, 2026) and of point patterns (Schuhmacher and Xia, 2008; Schuhmacher et al., 2008; Huang et al., 2026), none of them with the correspondence the theorem establishes, between an element’s weight and the mechanism’s continuity in it. Rates from a dimension. The dimensions of a law that bound the transport distance between an empirical measure and its law, the upper Wasserstein dimension of Weed and Bach (2019) and the entropic dimension of Chakraborty and Bartlett (2025), rest on one bulk covered at fixed dimension, the rest of the mass charged at the diameter. Our bounds stratify the design by its weight tail, a device with earlier variants (Fournier and Guillin, 2015; Lei, 2020; Huang et al., 2026), made to work on configurations that are never exactly sparse. Quasi-polynomial rates of the shape of Theorem 4.5 appear before in two other settings, empirical measures in a Hilbert space under an exponential-ellipsoid moment condition (Lei, 2020) and point processes with a geometric count law (Huang et al., 2026). Ours is for the mean weight tail of a configuration, with free weights and values, where the decay is inside each sample and there is no count to stratify by, and on fixed, geometrically decaying weights the class’s metric resembles Lei’s ellipsoid (Section D.4). Interaction kernels. The closest setting to the flock is the learning of interaction laws from trajectories (Lu et al., 2019; Lu et al., 2021), where the mechanism is a mean-field sum with an unknown scalar kernel, a linear sum in the terms of Section 3.1, learned at the one-dimensional rate with no profile in play (Appendix E). On real and simulated flocks, attention networks learn the update with the weights learned and read off the neighbors that matter (Heras et al., 2019; Ha and Jeong, 2021), and relational inference learns the interaction graph (Kipf et al., 2018): the selector is the object learned there, and it is given here. The nearest results in learning theory are on transformers: a target on a few tokens at given positions, a hard selection at a fixed count in a parametric class (Mousavi-Hosseini et al., 2025; Edelman et al., 2022; Sanford et al., 2023; Trauger and Tewari, 2024), an input-dependent importance of the coordinates (Takakura and Suzuki, 2023) or a per-sample sensitivity cap (Nagler, 2023), and further remarks on the literature are in Appendix E. 3 Locality Axioms and Representation This section defines the class by its axioms and proves its representation. 3.1 Axioms Mechanism and selector. Throughout, (,ρ)(V,ρ) is a compact metric space with at least two points, the value space, and R:=12diamR\;:=\; 12\,diamV is the value scale, so that ρ≤2Rρ≤ 2R with equality attained. A configuration is a pair (x,μ)∈:=d×Δd(x,μ) :=V^d× _d: a value xi∈x_i on each of d coordinates and a weight vector μ in the simplex. A mechanism is a function f:→ℝf:C . A selector produces the weights from the state of the system: each element carries a state ωi∈Ω _i∈ , of which its value xi=π(ωi)x_i=π( _i) is the part the mechanism reads, for a fixed map π:Ω→π: . The selector is a map c:Ωd→Δdc: ^d→ _d. The object of the examples is the composite g(ω)=f(π(ω),c(ω)),π(ω):=(π(ω1),…,π(ωd)),g(ω)\;=\;f (π(ω),c(ω) ), π(ω):=(π( _1),…,π( _d)), (1) one mechanism applied at every state to the coordinates the selector weights. In the flock the state of a particle is its position and heading, the values are the headings, the selector is uniform on the particles within the interaction radius, and the mechanism is the mean heading. The axioms concern f alone. The selector enters only through the design, the law Q of (π(ω),c(ω))(π(ω),c(ω)) under the law of the state ω, and nothing is assumed about it: it may be discontinuous, as a hard radius is, and in the examples it usually is. Throughout, the selector is given: the learner observes the configuration (x,μ)(x,μ), weights included, and learns f. Which coordinates of the state are the values, and in which semantics the mechanism is read, are part of the specification (Section C.1, Corollary A.15). Permutations σ of [d][d] act on both arguments, (σx)i=xσ−1(i)(σ x)_i=x_σ^-1(i) and likewise for μ. Write suppμ=i:μi>0suppμ=\i: _i>0\ and μmax=maxiμi _ = _i _i. Cap weights. The dependence allowed on a coordinate is a function of its weight, and the axioms are stated for a generic such function. A cap weight is a continuous map w:Δd→[0,1]dw: _d→[0,1]^d that commutes with permutations, is injective, and has wi(μ)=0w_i(μ)=0 exactly when μi=0 _i=0. The two instances of this paper are wc(μ)=μandwa(μ)=μ/μmax,w^c(μ)\;=\;μ w^a(μ)\;=\;μ/ _ , the weights themselves, the conservative cap of the mean semantics, and the weights relative to the largest one, the shares, the accumulative cap of the sum semantics. On weights uniform over k coordinates the first allots influence L/kL/k to each selected coordinate and the second L, a fixed budget that more participants dilute against a total that accumulates with the count. The conservative class L:=LwcG_L:=G^w^c_L is the class of the main text. The accumulative class ~L:=Lwa G_L:=G^w^a_L has the same representation, Theorem 3.2 below holding for every cap weight, and rates of the same exponents at the price of a factor of the count (Section A.4, Sections B.7 and B.8). Table 1 in Section A.4 sorts the common aggregations between the two classes, with their constants. Definition 3.1 (The class LwG^w_L). For a cap weight w and L≥0L≥ 0, LwG^w_L is the set of mechanisms f:→ℝf:C such that (V) (value cap) if x,y∈dx,y ^d differ in coordinate i only, then for every μ, |f(x,μ)−f(y,μ)|≤Lwi(μ)ρ(xi,yi) f(x,μ)-f(y,μ) ≤ L\,w_i(μ)\,ρ(x_i,y_i). (W) (weight regularity) for every x∈dx ^d and all μ,ν∈Δdμ,ν∈ _d, |f(x,μ)−f(x,ν)|≤LR‖w(μ)−w(ν)‖1 f(x,μ)-f(x,ν) ≤ L\,R\, \|w(μ)-w(ν) \|_1. (A) (anonymity) f(σx,σμ)=f(x,μ)f(σ x,σμ)=f(x,μ) for every permutation σ. (V) is the locality and (A) the anonymity of the introduction, and together they make f a function of the multiset of pairs (xi,μi):μi>0\(x_i, _i): _i>0\ (Lemma A.3). On an interval and for differentiable f the axioms are bounds on partial derivatives (Remark A.1). The weight axiom (W) is the one that needs a justification. Where the weight axiom comes from. The class is meant to contain the weighted sums of a score, ∑iwi(μ)φ(xi) _iw_i(μ)\, (x_i) with φ 11-Lipschitz and bounded by R, the linear sums of Section A.1. (W) is exactly their sensitivity to the weights: a move of weight from μ to ν changes a linear sum by at most R‖w(μ)−w(ν)‖1R\, \|w(μ)-w(ν) \|_1, and some score attains this (Proposition A.4), so (W) is the tightest weight axiom under which every linear sum is still a member at constant 11. The axioms are chosen so that the simplest mechanisms have constant 11, and what Theorem 3.2 adds is that the metric they induce is the matching distance at the same constant (the coordinatewise form of the axiom and the readouts it excludes are in Section A.1). The modulus in (W) is stated in the weights alone, and this is a choice: pricing a weight move by the values it moves between gives the flat metric on the weighted value measure ∑iμiδxi _i _i _x_i, a smaller class in which the weights are mass rather than individuals, that of distribution regression and the measure-input networks (Appendix E), which excludes the selected maximum and the top weight, members here (Remark A.6). A mechanism may also read a distinguished input of its own, the state of the agent evaluating it, as in Section 5, and everything below then holds fiberwise in it (Remark A.2). 3.2 Representation The cone and the lift. The cone over V is :=(×[0,1])/(×0)K:=(V×[0,1])/(V×\0\), the bottom edge collapsed to one point, the apex o. A point other than the apex is a pair (x,u)(x,u) of a value and a height u∈(0,1]u∈(0,1], and the cone cost is ϱ((x,u),(y,v)):=min(u,v)ρ(x,y)+R|u−v|,ϱ((x,u),o):=Ru, ((x,u),(y,v) )\;:=\; (u,v)\,ρ(x,y)+R\, u-v , ((x,u),o )\;:=\;R\,u, (2) a metric on K (Lemma A.7). The lift of a configuration is the counting measure on the cone Φ(x,μ):=∑i∈suppμδ(xi,wi(μ)), (x,μ)\;:=\; _i μ _(x_i,\,w_i(μ)), (3) one atom per coordinate of positive weight, at its value and at the height of its cap weight. Write ℳd:=Φ()M_d:= (C), and nPn_P for the number of atoms of P∈ℳdP _d. Two configurations have the same lift exactly when their multisets of (value, weight) pairs coincide, so Φ identifies C modulo permutations and modulo the values at zero-weight coordinates. Every member of LwG^w_L factors through it (Lemma A.8). The matching distance. For P,Q∈ℳdP,Q _d, a partial matching is a bijection σ between a sub-multiset of the atoms of P and a sub-multiset of the atoms of Q. Its cost is the cone cost summed over the matched pairs plus R times the height of every unmatched atom on either side, and the matching distance is the least cost, Mw(P,Q):=minσ[∑p matchedϱ(p,σ(p))+∑p∈P unmatchedϱ(p,o)+∑q∈Q unmatchedϱ(q,o)].M_w(P,Q)\;:=\; _σ [\, _p matched (p,σ(p) )+ _p∈ P unmatched (p,o)+ _q∈ Q unmatched (q,o) ]. (4) It is a metric on ℳdM_d, and it is a transport distance: padding each side with the other’s number of atoms at the apex, Mw(P,Q)=W1ϱ(P+nQδo,Q+nPδo),M_w(P,Q)\;=\;W_1 (P+n_Q\, _o,\;Q+n_P\, _o ), (5) the Kantorovich distance with ground cost ϱ between two measures of equal mass, whose optimal plans are matchings because the masses are integers (Lemma A.9). We write D:=MwcD:=M_w^c for the conservative instance, heights equal to the weights, and D~:=Mwa D:=M_w^a for the accumulative one, heights equal to the shares. Theorem 3.2 (Representation). Let w be a cap weight and L≥0L≥ 0. (i) A mechanism f belongs to LwG^w_L if and only if f=F∘Φf=F for a function F:ℳd→ℝF:M_d with |F(P)−F(Q)|≤LMw(P,Q) F(P)-F(Q) ≤ L\,M_w(P,Q) for all P,Q∈ℳdP,Q _d. The constant is 11 in both directions, for all configurations. (i) For all P,Q∈ℳdP,Q _d, Mw(P,Q)=supf(z)−f(z′):f∈1w,Φ(z)=P,Φ(z′)=Q.M_w(P,Q)\;=\; \\,f(z)-f(z )\;:\;f ^w_1,\ (z)=P,\ (z )=Q\, \. Consequently, as for any metric and its Lipschitz ball, every metric on ℳdM_d for which the unit ball 1wG^w_1 is 11-Lipschitz dominates MwM_w, and MwM_w is the unique metric whose Lipschitz balls, composed with Φ , are the classes LwG^w_L. Part (i) is the representation: the class is the Lipschitz ball of (ℳd,Mw)(M_d,M_w) in the axioms’ own constant, with nothing lost. Its proof is a routing: the axioms bound f along three moves, one value at fixed weights, one weight at fixed values, a permutation, so f is Lipschitz for the path metric the moves generate, and the theorem computes that metric in closed form as (4) by moving each matched value at the smaller of its two heights, before the weight move where the height rises and after it where it falls, at exactly the matching’s cost (Section A.3; compare Balan et al., 2022; Cahill et al., 2024). Part (i) is a corollary of (i), a metric being the supremum of the differences of its own 11-Lipschitz functions, and it is the sense in which the metric is the class’s own, each of its ingredients one of the axioms (the price R of a unit of height is the modulus of Proposition A.4). The class is convex, closed under Lipschitz readouts of finitely many members, and contains the vector-valued linear sums, so attention pooling and the flocking resultant are members (Section A.3, Lemma A.5). On a fixed hard selection of k coordinates the class is the symmetric Lipschitz functions of the selected values with the constant L/kL/k, the factor k attained by the selected maximum (Corollary A.15). 4 Complexity A design is a Borel probability measure Q on C, the law of the configurations, in the composite (1) the law of (π(ω),c(ω))(π(ω),c(ω)), and its lifted design is Γ:=Φ#Q := _\#Q, a probability measure on (ℳd,D)(M_d,D). The data are Z1,…,ZnZ_1,…,Z_n i.i.d. ∼Q Q, with lifts Pi=Φ(Zi)P_i= (Z_i) and empirical lifted design Γ^n:=n−1∑i≤nδPi _n:=n^-1 _i≤ n _P_i, and κ:=2Rκ:=2R bounds the diameter of (ℳd,D)(M_d,D) (Proposition A.20). The unit ball 1G_1 is the set of F∘ΦF with F 11-Lipschitz on (ℳd,D)(M_d,D) (Theorem 3.2), so the generalization gap of the class on a sample is a transport, ΔS:=supf∈1(Qf−1n∑i=1nf(Zi))=W1(Γ^n,Γ), _S\;:=\; _f _1 (E_Qf- 1n _i=1^nf(Z_i) )\;=\;W_1( _n, ), by Kantorovich–Rubinstein duality (Sriperumbudur et al., 2012). Every bound of this section bounds the mean Δn(Q):=ΔS _n(Q):=E\, _S or its value at the sample at hand, a statement about the design’s geometry: the class itself is never covered. 4.1 Rates from the sparsity profile Definition 4.1 (Tail mass and profile). For (x,μ)∈(x,μ) let μ(1)≥μ(2)≥⋯ _(1)≥ _(2)≥·s be the weights in decreasing order and, for 0≤k≤d0≤ k≤ d, Tk(x,μ):=∑i>kμ(i)T_k(x,μ):= _i>k _(i) the weight beyond the k heaviest coordinates, a continuous function of the lift alone (Lemma B.9). The profile of the design is Gk(t):=Q(Tk>t)G_k(t):=Q(T_k>t) for 0≤k≤d0≤ k≤ d and t∈[0,1)t∈[0,1), nonincreasing in k and in t. Definition 4.2 (Sparsity profiles). A design is exactly k0k_0-sparse if Q(Tk0=0)=1Q(T_k_0=0)=1, a selector that never reads more than k0k_0 coordinates. It has an exponential profile at rate β>0β>0 if Q[Tk]≤Ae−βkE_Q[T_k]≤ Ae^-β k for some A≥1A≥ 1 and all k≥0k≥ 0, a statement about the mean tail (Figure 1(a)). The profile decays in t at fixed k, the weights of one configuration, and in k at fixed t, the law of how many coordinates matter. The rates are stated through the stratum exponent e(k)e(k), the covering exponent of the lifts carried by at most k atoms, for a value space of metric dimension dim , N(ε,,ρ)≤(CR/ε)dimN( ,V,ρ)≤(C_VR/ ) . On interval values e(k)=2k−1e(k)=2k-1, one value and one free weight per atom. On a selector design, one whose cropped weights are, at every crop level, an LcL_c-Lipschitz function of the selected values or are uniform on the support, the ball of Section 5.1 being one, e(k)=kdime(k)=k (Section A.5). No upper bound below depends on d. The stratum of k atoms has covering count Nk(ε)≤(C1R/ε)e(k)N_k( )≤(C_1R/ )^e(k) at resolution ε≤κ ≤κ, by balls of radius ε/2 /2, with C1C_1 depending on the covering constant of V, and on LcL_c on a selector design (Section A.5). Theorem 4.3 (Generalization over the class). Fix K≤dK≤ d and resolutions 0<ε1≤⋯≤εK0< _1≤·s≤ _K with thresholds θk:=εk/(2κ)<1 _k:= _k/(2κ)<1, and let wk:=Q(Tk≤θk,Tk−1>θk−1)w_k:=Q(T_k≤ _k,\ T_k-1> _k-1), with θ0:=θ1 _0:= _1, be the mass of the configurations that k atoms resolve at εk _k and k−1k-1 do not at εk−1 _k-1. Then for every design Q, Δn(Q)≤ 2∑k=1Kwkεk+κ2∑k=1KwkNk(εk)n+κGK(θK). _n(Q)\;≤\;2 _k=1^Kw_k\, _k\;+\; κ2 _k=1^K w_k\,N_k( _k)n\;+\;κ\,G_K( _K). (6) The three terms are each shell’s resolution at its mass, its cell-frequency fluctuation at its count, and the trim beyond the last shell at the diameter, or twice its own tail if cropped (Theorem B.14). The two theorems below follow from it and a multiscale refinement of its coupling (Proposition B.18), each with its converse. Theorem 4.4 (A single sparsity level). For every design and every K≤dK≤ d with e(K)≥3e(K)≥ 3, Δn(Q)≤ 2κQTK+CRn−1e(K), _n(Q)\;≤\;2κ\,E_QT_K\;+\;CR\;n^- 1e(K), with C depending on the covering constant of V only, and on LcL_c on a selector design, uniformly in K and in d (the case e(K)≤2e(K)≤ 2 is in Section B.3): an exactly k0k_0-sparse design learns at n−1/e(k0)n^-1/e(k_0) once e(k0)≥3e(k_0)≥ 3, which is n−1/(2k0−1)n^-1/(2k_0-1) on interval values and n−1/(k0dim)n^-1/(k_0 ) on a selector design. Conversely, on interval values there is an exactly k0k_0-sparse design Qk0Q_k_0, with weights bounded below, such that every n-atom measure νn _n has W1(νn,Γk0)≥ck0n−1/(2k0−1)W_1( _n, _k_0)≥ c_k_0\,n^-1/(2k_0-1), for a constant ck0c_k_0: on interval values the worst-case exponent of a sparsity level k0≥2k_0≥ 2 is 1/(2k0−1)1/(2k_0-1) from both sides. The first term is the mean tail paid as a bias, the cost of cropping every configuration to K atoms, and the second is the rate of the exactly K-sparse design so obtained, by a multiscale coupling on the cropped stratum (Proposition B.18), whose exponent on interval values is the converse’s (Section D.4), and on the converse’s design no learner, from any number of noiseless labels, has risk below cn−1/(2k0−1)c\,n^-1/(2k_0-1) uniformly over the class (Theorem D.5). The simplest members escape the rate: the conservative linear sums (7), the shape of a sum network, learn at n−1/(dim+2)n^-1/( +2) whatever the count and the profile (Proposition B.20), and the class pays the profile for everything beyond Lipschitz readouts of finitely many of them. Theorem 4.5 (Exponential profiles). Let =[−R,R]V=[-R,R] or a circle of diameter 2R2R. Under an exponential profile at rate β, for every n≥1n≥ 1, Δn(Q)≤Ce−βlnn/2, _n(Q)\;≤\;C\,e^- β n/2, and Δn(Q)≤Ce−βlnn _n(Q)≤ C\,e^- β n on a selector design, with C depending on (β,A,R)(β,A,R), and on LcL_c on a selector design. Conversely there is a design QβQ^β with Tk≤e−βkT_k≤ e^-β k at every configuration such that, for every n with d≥2ln(2n)/β+1d≥ 2 (2n)/β+1, every n-atom measure νn _n has W1(νn,Γβ)≥ce−2βln(2n)W_1( _n, ^β)≥ c\,e^- 2β (2n), with c depending on (β,R)(β,R). The rate is slower than every polynomial in n and faster than every power of lnn n, and it is the problem’s own, the constant in the exponent lying between 1/21/ 2 and 2 2. The optimizing cutoff in Theorem 4.4 is Kn≍lnn/βK_n n/β, so the dimension the rate sees grows without bound, until it reaches d, and the profile costs nothing for being a statement about the mean tail, the crop paying each configuration its own tail (Remark B.25). The circle is the value space of the flock (Section C.1). 4.2 The certificate Theorem 4.6 (The two-sample certificate). Let n be even and let S be split, by a rule fixed in advance, into halves A and B of n/2n/2 points each, with empirical lifted designs Γ^A _A and Γ^B _B. With probability at least 1−δ1-δ, W1(Γ^A,Γ)≤W1(Γ^A,Γ^B)+κln(1/δ)n,W_1( _A, )\;≤\;W_1( _A, _B)+κ (1/δ)n, and W1(Γ^A,Γ^B)=minπ2n∑i≤n/2D(Pi,Pπ(i)′)W_1( _A, _B)= _π\, 2n _i≤ n/2D(P_i,P _π(i)), the minimum over the bijections π between the lifted points of the two halves: an assignment problem on n/2n/2 points, each of whose costs is itself an assignment between the atoms of two lifts (Lemma B.3). What is ours is the object: on the class the maximum discrepancy (Bartlett et al., 2002) is a transport by duality (von Luxburg and Bousquet, 2004, eq. (11)), a matching computes it, and the symmetrization of Vapnik and Chervonenkis (1971) carries it to the population with one term (Theorem B.1, Remarks B.2, B.7 and B.8). The matching is the least of the certificates on the halves, every covering-type bound on them being one feasible transport (Proposition B.5): on the flock the cell certificate is twenty times the matching on the ball and trivial on every graded kernel (Section 5.1). 5 Experiments 5.1 Flocking The Vicsek flock of the introduction (Vicsek et al., 1995) is the reference model of collective motion (Vicsek and Zafeiris, 2012; Ginelli, 2016), and its update is a mechanism of the class on the ordered region (Section C.1): the configuration is the set of headings, the selector the ball, uniform on the m+1m+1 particles within distance r of the focal one, itself included, and the update the direction of the mean heading vector, up to an angular noise of amplitude η. We simulate the standard setting, 32003200 particles at density 22 with r=1r=1, at noise η=1.2η=1.2 in the ordered phase (Chaté et al., 2008). Alongside the ball rule we simulate three flocks whose rules are graded, a bump, a Gaussian and a Cauchy kernel, each at the range at which its mean tail matches the ball’s, at a finite cutoff, and each read through the selector of its own family with its lift cropped at a priced tail (Section C.1). A sample is one focal particle of one replica at one instant, one sample per replica, so the samples are independent: 64,00064,000 replicas of each flock, each started at random and read once at step 30003000, after the flock has ordered, the view’s headings rotated by an independent uniform angle: the design is the law of that view. The regime η=2.4η=2.4 and the kernels’ parameters are in Section C.1. The profile. Figure 1(a) is the mean tail of each selector, and Figure 2 in Section C.1 the count law of the ball flock. On the replicas the ball holds 1111 particles on average and 6767 at most, other than the self, from 22 to 2323 between the 5%5\% and 95%95\% samples. Every sample is exactly sparse with uniform weights on its m+1m+1 atoms, so the mean tail is a functional of the count law. It is geometric beyond the knee at the mean support because the count law is, at a decay rate of 0.230.23 per atom. The graded kernels are the other case, a decay inside each sample at a rate set by the local crowding, the Gaussian’s geometric, the average-exponential profile of Theorem 4.5 on a design whose weights fade rather than stop, and the Cauchy’s polynomial. The profile bounds give the rate and the certificate gives the number: the ball design’s support reaches 6868 atoms and a kernel’s is every particle within its cutoff, so a bound in the dimension of the support says nothing at these n, where the certificate does. The rate the sample shows is the design’s own: over the last three doublings the ball’s transport decays with the exponent of a three-dimensional sample and the graded kernels’ with that of thirteen to fifteen, rising by a dimension per doubling as the sample resolves more atoms of the weight vector, both far below the profile’s worst-case exponent, eleven at the ball’s mean count (Section C.1): the ordered flock’s headings are nearly collinear, and the profile bound is over every design with that profile. That number bounds the gap of every Lipschitz function of the local configuration, on a design of strata of dimension one to sixty-eight, twelve on average (Figure 2), the 32003200 headings of the flock’s state nowhere in it. Figure 1: The flock at η=1.2η=1.2, 64,00064,000 independent replicas per flock, one sample each. (a) The profile: the mean tail TkT_k, the weight beyond the k heaviest atoms, under the four selectors, the dashed line the tail of one sample at the mean count m¯+1 m+1. (b) The certificate of Theorem 4.6 for half A at δ=0.05δ=0.05, in units of κ, on samples after the burn-in of 30003000 steps, by which the flock has ordered. The certificate. Figure 1(b) is Theorem 4.6 computed on the split by replicas, half A the first half of the replicas and half B the second, fixed in advance. The transport is one assignment on the halves (Lemma B.3), 32,00032,000 samples per half at the largest n. At η=1.2η=1.2 the certificate is 0.034κ0.034κ on the ball and 0.110.11 to 0.13κ0.13κ on the graded kernels, where the mean distance between two random samples is 0.610.61 to 0.68κ0.68κ and κ=πκ=π: a component of the local mean velocity, of range 22, deviates by at most 0.110.11. Every step here is a theorem on the data: the samples are independent replicas, so the i.i.d. hypothesis of Theorem 4.6 holds as stated, and the concentration term is its 0.007κ0.007κ at n=64,000n=64,000. What the certificate bounds. The certificate is the exact uniform deviation of the class on the two halves, W1(Γ^A,Γ^B)=maxf∈1(Af−Bf)W_1( _A, _B)= _f _1(E_Af-E_Bf), so no bound uniform over the class on these halves is smaller, and by Remark B.22 it bounds the generalization gap of every empirical risk minimizer of LG_L on every target in LG_L by 2L2L times its value. On a member it is nearly attained: for the target LD(⋅,A)L\,D(·,A), zero on half A, the lower Lipschitz extension −LD(⋅,A)-L\,D(·,A) is a member of LG_L with zero training risk whose gap on half B is 2L2L times the mean nearest-neighbor distance from B to A, 0.038κ0.038κ against the certified 0.068κ0.068κ at L=1L=1 on the ball, and within a factor 1.131.13 of the certificate on every graded kernel (Section C.1). For one chosen learner on one target a holdout estimate on half B is tighter by definition, and the certificate does not compete with it: what it gives is uniformity at no labels, over every target on the design and every predictor in the class at once, in hand before a learner is chosen or a label seen, so that a choice made after seeing the data costs nothing (Theorem B.1(i)). Against the covering certificates on the same halves it is the smaller by far: the cell certificate of Proposition B.5, optimized on the data, is 0.68κ0.68κ on the ball, twenty times the matching, and at or above κ on every graded kernel at every n (Section C.1). 6 Conclusions and Future Work We defined graded sparse mechanisms by axioms, showed that they are the Lipschitz ball of a transport metric on weighted configurations, and used that geometry for rates from the design’s sparsity profile and for a certificate computed from the sample by a matching, with non-vacuous bounds on the flock. So although the class, the full Lipschitz ball of the lift, is prohibitively large for any data-independent bound, it is learnable, and the design alone can bound its rate. We pose two questions. The first is the learning of the selector. Although in some situations it is dictated fully by the environment, and this is the setting we assume here, in other situations some aspects of it can be learned (Heras et al., 2019; Ha and Jeong, 2021; Kipf et al., 2018). What makes the problem interesting is its separation: the selector is the environment’s, one rule common to all agents, while the mechanisms of the agents may differ. Second, empirically in this paper we concentrated on various rules on the flock, with the conservative cap. We believe the next natural step would be to evaluate additional scenarios, in terms of the caps, the observed sparsity profiles, and the implied empirical complexity. References Ajtai et al. (1984) Miklós Ajtai, János Komlós, and Gábor Tusnády. On optimal matchings. Combinatorica, 4(4):259–264, 1984. Amir and Dym (2025) Tal Amir and Nadav Dym. Fourier sliced-Wasserstein embedding for multisets and measures. In International Conference on Learning Representations (ICLR), 2025. Amir et al. (2023) Tal Amir, Steven J. Gortler, Ilai Avni, Ravina Ravina, and Nadav Dym. Neural injective functions for multisets, measures and graphs via a finite witness theorem. In Advances in Neural Information Processing Systems (NeurIPS), 2023. Amran et al. (2026) Ofek Amran, Tom Gilat, and Ron Levie. A graphop analysis of graph neural networks on sparse graphs: Generalization and universal approximation. arXiv preprint arXiv:2602.08785, 2026. Ashlagi et al. (2021) Yair Ashlagi, Lee-Ad Gottlieb, and Aryeh Kontorovich. Functions with average smoothness: structure, algorithms, and learning. In Conference on Learning Theory (COLT), 2021. Balan et al. (2022) Radu Balan, Naveed Haghani, and Maneesh Singh. Permutation invariant representations with applications to graph deep learning, 2022. arXiv:2203.07546. Ballerini et al. (2008) M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proceedings of the National Academy of Sciences, 105(4):1232–1237, 2008. Bartlett and Mendelson (2002) Peter L. Bartlett and Shahar Mendelson. Rademacher and Gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3:463–482, 2002. Bartlett et al. (2002) Peter L. Bartlett, Stéphane Boucheron, and Gábor Lugosi. Model selection and error estimation. Machine Learning, 48:85–113, 2002. doi 10.1023/A:1013999503812. Beltagy et al. (2020) Iz Beltagy, Matthew E. Peters, and Arman Cohan. Longformer: The long-document transformer. arXiv preprint arXiv:2004.05150, 2020. Bialek et al. (2012) William Bialek, Andrea Cavagna, Irene Giardina, Thierry Mora, Edmondo Silvestri, Massimiliano Viale, and Aleksandra M. Walczak. Statistical mechanics for natural flocks of birds. Proceedings of the National Academy of Sciences, 109(13):4786–4791, 2012. Bickel and Li (2007) Peter J. Bickel and Bo Li. Local polynomial regression on unknown manifolds. In Complex Datasets and Inverse Problems: Tomography, Networks and Beyond, volume 54 of IMS Lecture Notes–Monograph Series, pages 177–186. 2007. Bietti et al. (2021) Alberto Bietti, Luca Venturi, and Joan Bruna. On the sample complexity of learning under invariance and geometric stability. In Advances in Neural Information Processing Systems (NeurIPS), 2021. arXiv:2106.07148. Block et al. (2022) Adam Block, Zeyu Jia, Yury Polyanskiy, and Alexander Rakhlin. Intrinsic dimension estimation using Wasserstein distance. Journal of Machine Learning Research, 23(313):1–37, 2022. arXiv:2106.04018. Boissard and Le Gouic (2014) Emmanuel Boissard and Thibaut Le Gouic. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 50(2):539–563, 2014. Böker et al. (2023) Jan Böker, Ron Levie, Ningyuan Huang, Soledad Villar, and Christopher Morris. Fine-grained expressivity of graph neural networks. In Advances in Neural Information Processing Systems (NeurIPS), 2023. arXiv:2306.03698. Boucheron et al. (2013) Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013. Bubenik and Elchesen (2022a) Peter Bubenik and Alex Elchesen. Universality of persistence diagrams and the bottleneck and Wasserstein distances. Computational Geometry, 105–106:101882, 2022a. arXiv:1912.02563. Bubenik and Elchesen (2022b) Peter Bubenik and Alex Elchesen. Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces. Journal of Applied and Computational Topology, 6:429–474, 2022b. Bubenik and Wagner (2020) Peter Bubenik and Alexander Wagner. Embeddings of persistence diagrams into Hilbert spaces. Journal of Applied and Computational Topology, 4:339–351, 2020. Cahill et al. (2024) Jameson Cahill, Joseph W. Iverson, Dustin G. Mixon, and Daniel Packer. Group-invariant max filtering. Foundations of Computational Mathematics, 2024. arXiv:2205.14039. Carrasco et al. (2025) Martin Carrasco, Caio F. Deberaldini Netto, Vahan A. Martirosyan, Ehimare Okoyomon, and Caterina Graziani. On the Rademacher complexity of graph neural networks: Unifying expressivity and geometry, 2025. arXiv:2510.10101, preprint. Castin et al. (2024) Valérie Castin, Pierre Ablin, and Gabriel Peyré. How smooth is attention? In International Conference on Machine Learning (ICML), 2024. arXiv:2312.14820. Chakraborty and Bartlett (2024) Saptarshi Chakraborty and Peter L. Bartlett. A statistical analysis for supervised deep learning with exponential families for intrinsically low-dimensional data, 2024. arXiv:2412.09779. Chakraborty and Bartlett (2025) Saptarshi Chakraborty and Peter L. Bartlett. On the statistical properties of generative adversarial models for low intrinsic data dimension. Journal of Machine Learning Research, 26:1–57, 2025. Chaté et al. (2008) Hugues Chaté, Francesco Ginelli, Guillaume Grégoire, and Franck Raynaud. Collective motion of self-propelled particles interacting without cohesion. Physical Review E, 77:046113, 2008. Chazal and Michel (2021) Frédéric Chazal and Bertrand Michel. An introduction to topological data analysis: fundamental and practical aspects for data scientists. Frontiers in Artificial Intelligence, 4:667963, 2021. Child et al. (2019) Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509, 2019. Chizat et al. (2018) Lénaïc Chizat, Gabriel Peyré, Bernhard Schmitzer, and François-Xavier Vialard. Unbalanced optimal transport: dynamic and Kantorovich formulations. Journal of Functional Analysis, 274(11):3090–3123, 2018. Chuang and Jegelka (2022) Ching-Yao Chuang and Stefanie Jegelka. Tree mover’s distance: Bridging graph metrics and stability of graph neural networks. In Advances in Neural Information Processing Systems (NeurIPS), 2022. arXiv:2210.01906. Chuang et al. (2021) Ching-Yao Chuang, Youssef Mroueh, Kristjan Greenewald, Antonio Torralba, and Stefanie Jegelka. Measuring generalization with optimal transport. In Advances in Neural Information Processing Systems 34 (NeurIPS), 2021. arXiv:2106.03314. Cohen et al. (2009) Albert Cohen, Wolfgang Dahmen, and Ronald DeVore. Compressed sensing and best k-term approximation. Journal of the American Mathematical Society, 22(1):211–231, 2009. Cohen-Steiner et al. (2010) David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko. Lipschitz functions have LpL_p-stable persistence. Foundations of Computational Mathematics, 10(2):127–139, 2010. Corso et al. (2020) Gabriele Corso, Luca Cavalleri, Dominique Beaini, Pietro Liò, and Petar Veličković. Principal neighbourhood aggregation for graph nets. In Advances in Neural Information Processing Systems, volume 33, 2020. Cucker and Smale (2007) Felipe Cucker and Steve Smale. Emergent behavior in flocks. IEEE Transactions on Automatic Control, 52(5):852–862, 2007. Davidson and Dym (2025) Yair Davidson and Nadav Dym. On the Hölder stability of multiset and graph neural networks. In International Conference on Learning Representations (ICLR), 2025. De Bie et al. (2019) Gwendoline De Bie, Gabriel Peyré, and Marco Cuturi. Stochastic deep networks. In International Conference on Machine Learning (ICML), 2019. Dereich et al. (2013) Steffen Dereich, Michael Scheutzow, and Reik Schottstedt. Constructive quantization: approximation by empirical measures. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 49(4):1183–1203, 2013. Divol and Lacombe (2021) Vincent Divol and Théo Lacombe. Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport. Journal of Applied and Computational Topology, 5:1–53, 2021. Dudley (1969) Richard M. Dudley. The speed of mean Glivenko–Cantelli convergence. Annals of Mathematical Statistics, 40(1):40–50, 1969. Dziugaite and Roy (2017) Gintare Karolina Dziugaite and Daniel M. Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. In Conference on Uncertainty in Artificial Intelligence (UAI), 2017. arXiv:1703.11008. Edelman et al. (2022) Benjamin L. Edelman, Surbhi Goel, Sham Kakade, and Cyril Zhang. Inductive biases and variable creation in self-attention mechanisms. In International Conference on Machine Learning (ICML), 2022. Elesedy and Zaidi (2021) Bryn Elesedy and Sheheryar Zaidi. Provably strict generalisation benefit for equivariant models. In Proceedings of the 38th International Conference on Machine Learning (ICML), PMLR 139, 2021. arXiv:2102.10333. Esser et al. (2021) Pascal Mattia Esser, Leena Chennuru Vankadara, and Debarghya Ghoshdastidar. Learning theory can (sometimes) explain generalisation in graph neural networks. In Advances in Neural Information Processing Systems (NeurIPS), 2021. Figalli and Gigli (2010) Alessio Figalli and Nicola Gigli. A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. Journal de Mathématiques Pures et Appliquées, 94(2):107–130, 2010. Fornasier et al. (2023) Massimo Fornasier, Pascal Heid, and Giacomo Enrico Sodini. Approximation theory, computing, and deep learning on the Wasserstein space, 2023. arXiv:2310.19548. Fournier and Guillin (2015) Nicolas Fournier and Arnaud Guillin. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, 162(3–4):707–738, 2015. arXiv:1312.2128. Garg et al. (2020) Vikas K. Garg, Stefanie Jegelka, and Tommi Jaakkola. Generalization and representational limits of graph neural networks. In International Conference on Machine Learning (ICML), 2020. Gautrais et al. (2012) Jacques Gautrais, Francesco Ginelli, Richard Fournier, Stéphane Blanco, Marc Soria, Hugues Chaté, and Guy Theraulaz. Deciphering interactions in moving animal groups. PLoS Computational Biology, 8(9):e1002678, 2012. Ginelli (2016) Francesco Ginelli. The physics of the Vicsek model. The European Physical Journal Special Topics, 225:2099–2117, 2016. Gottlieb et al. (2013) Lee-Ad Gottlieb, Aryeh Kontorovich, and Robert Krauthgamer. Adaptive metric dimensionality reduction. In Algorithmic Learning Theory (ALT), 2013. arXiv:1302.2752; journal version in Theoretical Computer Science, 2016. Gottlieb et al. (2017) Lee-Ad Gottlieb, Aryeh Kontorovich, and Robert Krauthgamer. Efficient regression in metric spaces via approximate Lipschitz extension. IEEE Transactions on Information Theory, 63(8):4838–4849, 2017. Ha and Jeong (2021) Seungwoong Ha and Hawoong Jeong. Unraveling hidden interactions in complex systems with deep learning. Scientific Reports, 11:12804, 2021. Hanin (1992) Leonid G. Hanin. Kantorovich–Rubinstein norm and its application in the theory of Lipschitz spaces. Proceedings of the American Mathematical Society, 115(2):345–352, 1992. Heras et al. (2019) Francisco J. H. Heras, Francisco Romero-Ferrero, Robert C. Hinz, and Gonzalo G. de Polavieja. Deep attention networks reveal the rules of collective motion in zebrafish. PLoS Computational Biology, 15(9):e1007354, 2019. Hou et al. (2023) Songyan Hou, Parnian Kassraie, Anastasis Kratsios, Andreas Krause, and Jonas Rothfuss. Instance-dependent generalization bounds via optimal transport. Journal of Machine Learning Research, 24:1–51, 2023. arXiv:2211.01258. Huang et al. (2026) Dongzhou Huang, Tianyi Jiang, and Haonan Wang. Wasserstein convergence rates for empirical measures of point processes, 2026. arXiv:2604.23928. Hüttenrauch et al. (2019) Maximilian Hüttenrauch, Adrian Šošić, and Gerhard Neumann. Deep reinforcement learning for swarm systems. Journal of Machine Learning Research, 20(54):1–31, 2019. Ju et al. (2023) Haotian Ju, Dongyue Li, Aneesh Sharma, and Hongyang R. Zhang. Generalization in graph neural networks: Improved PAC-Bayesian bounds on graph diffusion. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2023. Kantorovich and Rubinstein (1958) Leonid V. Kantorovich and Gennadii Sh. Rubinstein. On a space of completely additive functions. Vestnik Leningrad University, 13(7):52–59, 1958. Kawaguchi et al. (2022) Kenji Kawaguchi, Zhun Deng, Kyle Luh, and Jiaoyang Huang. Robustness implies generalization via data-dependent generalization bounds. In Proceedings of the 39th International Conference on Machine Learning (ICML), PMLR 162, 2022. arXiv:2206.13497. Kazemkhani et al. (2025) Saman Kazemkhani, Aarav Pandya, Daphne Cornelisse, Brennan Shacklett, and Eugene Vinitsky. GPUDrive: Data-driven, multi-agent driving simulation at 1 million FPS. In International Conference on Learning Representations (ICLR), 2025. Kim et al. (2021) Hyunjik Kim, George Papamakarios, and Andriy Mnih. The Lipschitz constant of self-attention. In International Conference on Machine Learning (ICML), 2021. Kipf et al. (2018) Thomas Kipf, Ethan Fetaya, Kuan-Chieh Wang, Max Welling, and Richard Zemel. Neural relational inference for interacting systems. In International Conference on Machine Learning (ICML), 2018. Kolmogorov and Tikhomirov (1959) Andrey N. Kolmogorov and Vladimir M. Tikhomirov. ε -entropy and ε -capacity of sets in function spaces. Uspekhi Matematicheskikh Nauk, 14(2):3–86, 1959. English translation: American Mathematical Society Translations (2) 17 (1961), 277–364. Koltchinskii and Panchenko (2000) Vladimir Koltchinskii and Dmitry Panchenko. Rademacher processes and bounding the risk of function learning. In Evarist Giné, David Mason, and Jon Wellner, editors, High Dimensional Probability I, pages 443–459. Birkhäuser, 2000. Kpotufe (2011) Samory Kpotufe. k-N regression adapts to local intrinsic dimension. In Advances in Neural Information Processing Systems 24 (NIPS), 2011. arXiv:1110.4300. Lacombe et al. (2018) Théo Lacombe, Marco Cuturi, and Steve Oudot. Large scale computation of means and clusters for persistence diagrams using optimal transport. In Advances in Neural Information Processing Systems, volume 31, 2018. Lei (2020) Jing Lei. Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces. Bernoulli, 26(1):767–798, 2020. arXiv:1804.10556. Levie (2023) Ron Levie. A graphon-signal analysis of graph neural networks. In Advances in Neural Information Processing Systems 36 (NeurIPS), 2023. arXiv:2305.15987. Li et al. (2023) Hongkang Li, Meng Wang, Sijia Liu, and Pin-Yu Chen. A theoretical understanding of shallow vision transformers: learning, generalization, and sample complexity. In International Conference on Learning Representations (ICLR), 2023. arXiv:2302.06015. Liao et al. (2021) Renjie Liao, Raquel Urtasun, and Richard Zemel. A PAC-Bayesian approach to generalization bounds for graph neural networks. In International Conference on Learning Representations (ICLR), 2021. Liero et al. (2018) Matthias Liero, Alexander Mielke, and Giuseppe Savaré. Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones Mathematicae, 211(3):969–1117, 2018. Lu et al. (2019) Fei Lu, Mauro Maggioni, Sui Tang, and Ming Zhong. Nonparametric inference of interaction laws in systems of agents from trajectory data. Proceedings of the National Academy of Sciences, 116(29):14424–14433, 2019. arXiv:1812.06003. Lu et al. (2021) Fei Lu, Mauro Maggioni, and Sui Tang. Learning interaction kernels in heterogeneous systems of agents from multiple trajectories. Journal of Machine Learning Research, 22(32):1–67, 2021. arXiv:1910.04832. Lv (2021) Shaogao Lv. Generalization bounds for graph convolutional neural networks via Rademacher complexity. arXiv preprint arXiv:2102.10234, 2021. Macdonald (1995) Ian G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford University Press, 2nd edition, 1995. Mansour and McAllester (2000) Yishay Mansour and David McAllester. Generalization bounds for decision trees. In Proceedings of the 13th Annual Conference on Computational Learning Theory (COLT), 2000. Maskey et al. (2022) Sohir Maskey, Ron Levie, Yunseok Lee, and Gitta Kutyniok. Generalization analysis of message passing neural networks on large random graphs. In Advances in Neural Information Processing Systems (NeurIPS), 2022. Maskey et al. (2024) Sohir Maskey, Gitta Kutyniok, and Ron Levie. Generalization bounds for message passing networks on mixture of graphons, 2024. arXiv:2404.03473. Mileyko et al. (2011) Yuriy Mileyko, Sayan Mukherjee, and John Harer. Probability measures on the space of persistence diagrams. Inverse Problems, 27(12):124007, 2011. Mohri et al. (2018) Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of Machine Learning. MIT Press, second edition, 2018. Morris et al. (2023) Christopher Morris, Floris Geerts, Jan Tönshoff, and Martin Grohe. WL meet VC. In International Conference on Machine Learning (ICML), 2023. Morris et al. (2024) Christopher Morris, Fabrizio Frasca, Nadav Dym, Haggai Maron, İsmail İlkan Ceylan, Ron Levie, Derek Lim, Michael Bronstein, Martin Grohe, and Stefanie Jegelka. Position: Future directions in the theory of graph machine learning. In International Conference on Machine Learning (ICML), 2024. Mousavi-Hosseini et al. (2025) Alireza Mousavi-Hosseini, Clayton Sanford, Denny Wu, and Murat A. Erdogdu. When do transformers outperform feedforward and recurrent networks? A statistical perspective, 2025. arXiv:2503.11272. Musaelian et al. (2023) Albert Musaelian, Simon Batzner, Anders Johansson, Lixin Sun, Cameron J. Owen, Mordechai Kornbluth, and Boris Kozinsky. Learning local equivariant representations for large-scale atomistic dynamics. Nature Communications, 14:579, 2023. Nagarajan and Kolter (2019) Vaishnavh Nagarajan and J. Zico Kolter. Uniform convergence may be unable to explain generalization in deep learning. In Advances in Neural Information Processing Systems (NeurIPS), 2019. arXiv:1902.04742. Nagler (2023) Thomas Nagler. Statistical foundations of prior-data fitted networks. In International Conference on Machine Learning (ICML), 2023. Nayak et al. (2023) Siddharth Nayak, Kenneth Choi, Wenqi Ding, Sydney Dolan, Karthik Gopalakrishnan, and Hamsa Balakrishnan. Scalable multi-agent reinforcement learning through intelligent information aggregation. In Proceedings of the 40th International Conference on Machine Learning (ICML), volume 202 of PMLR, 2023. Nikolentzos and Skianis (2026) Giannis Nikolentzos and Konstantinos Skianis. On the Lipschitz continuity of set aggregation functions and neural networks for sets. In International Conference on Learning Representations (ICLR), 2026. arXiv:2505.24403. Pérez-Ortiz et al. (2021) María Pérez-Ortiz, Omar Rivasplata, John Shawe-Taylor, and Csaba Szepesvári. Tighter risk certificates for neural networks. Journal of Machine Learning Research, 22(227):1–40, 2021. Pevný and Kovařík (2019) Tomáš Pevný and Vojtěch Kovařík. Approximation capability of neural networks on spaces of probability measures and tree-structured domains. In International Conference on Learning Representations (ICLR), 2019. arXiv:1906.00764. Peyré and Cuturi (2019) Gabriel Peyré and Marco Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning, 11(5–6):355–607, 2019. Pham and Warin (2023) Huyên Pham and Xavier Warin. Mean-field neural networks: learning mappings on Wasserstein space. Neural Networks, 2023. arXiv:2210.15179. Piccoli and Rossi (2014) Benedetto Piccoli and Francesco Rossi. Generalized Wasserstein distance and its application to transport equations with source. Archive for Rational Mechanics and Analysis, 211(1):335–358, 2014. Piccoli and Rossi (2016) Benedetto Piccoli and Francesco Rossi. On properties of the generalized Wasserstein distance. Archive for Rational Mechanics and Analysis, 222(3):1339–1365, 2016. Póczos et al. (2013) Barnabás Póczos, Alessandro Rinaldo, Aarti Singh, and Larry Wasserman. Distribution-free distribution regression. In Proceedings of the 16th International Conference on Artificial Intelligence and Statistics (AISTATS), 2013. arXiv:1302.0082. Ramdas et al. (2017) Aaditya Ramdas, Nicolás García Trillos, and Marco Cuturi. On Wasserstein two-sample testing and related families of nonparametric tests. Entropy, 19(2):47, 2017. Rauchwerger et al. (2025) Levi Rauchwerger, Stefanie Jegelka, and Ron Levie. Generalization, expressivity, and universality of graph neural networks on attributed graphs. In International Conference on Learning Representations (ICLR), 2025. arXiv:2411.05464. Redko et al. (2017) Ievgen Redko, Amaury Habrard, and Marc Sebban. Theoretical analysis of domain adaptation with optimal transport. In Machine Learning and Knowledge Discovery in Databases (ECML PKDD), volume 10535 of Lecture Notes in Computer Science, pages 737–753, 2017. Sanchez-Gonzalez et al. (2020) Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter W. Battaglia. Learning to simulate complex physics with graph networks. In Proceedings of the 37th International Conference on Machine Learning (ICML), volume 119 of PMLR, pages 8459–8468, 2020. Sanford et al. (2023) Clayton Sanford, Daniel Hsu, and Matus Telgarsky. Representational strengths and limitations of transformers. In Advances in Neural Information Processing Systems (NeurIPS), 2023. Sannai et al. (2019) Akiyoshi Sannai, Yuuki Takai, and Matthieu Cordonnier. Universal approximations of permutation invariant/equivariant functions by deep neural networks. arXiv:1903.01939, 2019. Sannai et al. (2021) Akiyoshi Sannai, Masaaki Imaizumi, and Makoto Kawano. Improved generalization bounds of group invariant/equivariant deep networks via quotient feature spaces. In Proceedings of the 37th Conference on Uncertainty in Artificial Intelligence (UAI), volume 161 of PMLR, pages 771–780, 2021. Schuhmacher and Xia (2008) Dominic Schuhmacher and Aihua Xia. A new metric between distributions of point processes. Advances in Applied Probability, 40(3):651–672, 2008. arXiv:0708.2777. Schuhmacher et al. (2008) Dominic Schuhmacher, Ba-Tuong Vo, and Ba-Ngu Vo. A consistent metric for performance evaluation of multi-object filters. IEEE Transactions on Signal Processing, 56(8):3447–3457, 2008. Segol and Lipman (2020) Nimrod Segol and Yaron Lipman. On universal equivariant set networks. In International Conference on Learning Representations (ICLR), 2020. Shen et al. (2018) Jian Shen, Yanru Qu, Weinan Zhang, and Yong Yu. Wasserstein distance guided representation learning for domain adaptation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018. Solomon et al. (2022) Justin Solomon, Kristjan Greenewald, and Haikady N. Nagaraja. k-variance: A clustered notion of variance. SIAM Journal on Mathematics of Data Science, 4(3):957–978, 2022. arXiv:2012.06958. Sriperumbudur et al. (2012) Bharath K. Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Schölkopf, and Gert R. G. Lanckriet. On the empirical estimation of integral probability metrics. Electronic Journal of Statistics, 6:1550–1599, 2012. DOI 10.1214/12-EJS722. Struleva et al. (2025) Marina Struleva, Shayan Hundrieser, Dominic Schuhmacher, and Axel Munk. Sharp convergence rates of empirical unbalanced optimal transport for spatio-temporal point processes. arXiv preprint arXiv:2509.04225, 2025. Szabó et al. (2016) Zoltán Szabó, Bharath K. Sriperumbudur, Barnabás Póczos, and Arthur Gretton. Learning theory for distribution regression. Journal of Machine Learning Research, 17:1–40, 2016. arXiv:1411.2066. Tahmasebi and Jegelka (2023) Behrooz Tahmasebi and Stefanie Jegelka. The exact sample complexity gain from invariances for kernel regression. In Advances in Neural Information Processing Systems (NeurIPS), 2023. arXiv:2303.14269. Takakura and Suzuki (2023) Shokichi Takakura and Taiji Suzuki. Approximation and estimation ability of transformers for sequence-to-sequence functions with infinite dimensional input. In International Conference on Machine Learning (ICML), 2023. Tolstaya et al. (2020) Ekaterina Tolstaya, Fernando Gama, James Paulos, George Pappas, Vijay Kumar, and Alejandro Ribeiro. Learning decentralized controllers for robot swarms with graph neural networks. In Conference on Robot Learning (CoRL), volume 100 of Proceedings of Machine Learning Research, 2020. arXiv:1903.10527. Trauger and Tewari (2024) Jacob Trauger and Ambuj Tewari. Sequence length independent norm-based generalization bounds for transformers. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2024. Tsybakov (2009) Alexandre B. Tsybakov. Introduction to Nonparametric Estimation. Springer, New York, 2009. Turner et al. (2014) Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. Fréchet means for distributions of persistence diagrams. Discrete & Computational Geometry, 52(1):44–70, 2014. Vapnik and Chervonenkis (1971) Vladimir N. Vapnik and Alexey Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264–280, 1971. Vásárhelyi et al. (2018) Gábor Vásárhelyi, Csaba Virágh, Gergő Somorjai, Tamás Nepusz, Agoston E. Eiben, and Tamás Vicsek. Optimized flocking of autonomous drones in confined environments. Science Robotics, 3(20):eaat3536, 2018. Vasileiou et al. (2025a) Antonis Vasileiou, Ben Finkelshtein, Floris Geerts, Ron Levie, and Christopher Morris. Covered forest: Fine-grained generalization analysis of graph neural networks. In Proceedings of the 42nd International Conference on Machine Learning (ICML), PMLR 267, 2025a. arXiv:2412.07106. Vasileiou et al. (2025b) Antonis Vasileiou, Stefanie Jegelka, Ron Levie, and Christopher Morris. Survey on generalization theory for graph neural networks. arXiv preprint arXiv:2503.15650, 2025b. Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NeurIPS), 2017. Vicsek and Zafeiris (2012) Tamás Vicsek and Anna Zafeiris. Collective motion. Physics Reports, 517(3–4):71–140, 2012. Vicsek et al. (1995) Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet. Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75(6):1226–1229, 1995. Vinitsky et al. (2022) Eugene Vinitsky, Nathan Lichtlé, Xiaomeng Yang, Brandon Amos, and Jakob Foerster. Nocturne: A scalable driving benchmark for bringing multi-agent learning one step closer to the real world. In Advances in Neural Information Processing Systems 35 (NeurIPS 2022), Datasets and Benchmarks Track, 2022. von Luxburg and Bousquet (2004) Ulrike von Luxburg and Olivier Bousquet. Distance-based classification with Lipschitz functions. Journal of Machine Learning Research, 5:669–695, 2004. Vuckovic et al. (2020) James Vuckovic, Aristide Baratin, and Remi Tachet des Combes. A mathematical theory of attention, 2020. arXiv:2007.02876. Wagstaff et al. (2022) Edward Wagstaff, Fabian B. Fuchs, Martin Engelcke, Michael A. Osborne, and Ingmar Posner. Universal approximation of functions on sets. Journal of Machine Learning Research, 23(151):1–56, 2022. Wang et al. (2024) Zixuan Wang, Stanley Wei, Daniel Hsu, and Jason D. Lee. Transformers provably learn sparse token selection while fully-connected nets cannot. In International Conference on Machine Learning (ICML), 2024. arXiv:2406.06893. Weed and Bach (2019) Jonathan Weed and Francis Bach. Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. Bernoulli, 25(4A):2620–2648, 2019. arXiv:1707.00087. Weyl (1939) Hermann Weyl. The Classical Groups: Their Invariants and Representations. Princeton University Press, 1939. Wirth et al. (2023) Trenton D. Wirth, Gregory C. Dachner, Kevin W. Rio, and William H. Warren. Is the neighborhood of interaction in human crowds metric, topological, or visual? PNAS Nexus, 2(5):pgad118, 2023. Xu and Mannor (2012) Huan Xu and Shie Mannor. Robustness and generalization. Machine Learning, 86(3):391–423, 2012. Xu et al. (2019) Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019. Yao et al. (2022) Rentian Yao, Xiaohui Chen, and Yun Yang. Mean-field nonparametric estimation of interacting particle systems. In Conference on Learning Theory (COLT), volume 178 of Proceedings of Machine Learning Research, 2022. arXiv:2205.07937. Zaheer et al. (2017) Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabás Póczos, Ruslan Salakhutdinov, and Alexander J. Smola. Deep sets. In Advances in Neural Information Processing Systems (NeurIPS), 2017. Zhang et al. (2025) Songyuan Zhang, Oswin So, Kunal Garg, and Chuchu Fan. GCBF+: A neural graph control barrier function framework for distributed safe multiagent control. IEEE Transactions on Robotics, 41:1533–1552, 2025. Zhu et al. (2021) Sicheng Zhu, Bang An, and Furong Huang. Understanding the generalization benefit of model invariance from a data perspective. In Advances in Neural Information Processing Systems 34 (NeurIPS), 2021. arXiv:2111.05529. Zweig and Bruna (2022) Aaron Zweig and Joan Bruna. Exponential separations in symmetric neural networks. In Advances in Neural Information Processing Systems (NeurIPS), 2022. Appendix A Proofs for Section 3 A.1 Basic lemmas, and the weight axiom Two remarks on Definition 3.1 come first. Remark A.1 (Differential form). On an interval =[−R,R]V=[-R,R] and for f differentiable, (V) is the bound |∂f/∂xi|≤Lwi(μ) ∂ f/∂ x_i ≤ L\,w_i(μ) at every configuration, and for the weight w=μw=μ, (W) is the bound max∂if/∂μi−min∂if/∂μi≤2LR _i∂ f/∂ _i- _i∂ f/∂ _i≤ 2LR along the simplex, since a tangent direction h of the simplex has ∑ihi=0 _ih_i=0 and sup‖h‖1≤1⟨∇μf,h⟩ _ \|h \|_1≤ 1 _μf,h is half the spread of the partial derivatives. Remark A.2 (Distinguished inputs). A mechanism may read, besides the anonymous coordinates, inputs of its own that no permutation moves: the state of the agent that evaluates it, or of its environment, as in Section 5. Such an input is one distinguished variable q in a metric space, and the class becomes the set of f(q,x,μ)f(q;\,x,μ) that lie in LwG^w_L for every q and are Lipschitz in q. Everything in the paper holds fiberwise in q, the metric of Section 3.2 becomes a product metric, and the covering numbers of Section A.5 multiply by that of the space of q: with the metric dq(q,q′)+Mw(P,P′)d_q(q,q )+M_w(P,P ) the class is its L-Lipschitz ball, the covering number at ε is at most the product of the two covering numbers at ε/2 /2, and the diameter of Theorem 4.6 is the sum of the two. We keep the notation without the slot. Lemma A.3 (Blindness, factorization, telescope). Let f∈Lwf ^w_L. (i) If μi=0 _i=0 then f(x,μ)f(x,μ) does not depend on xix_i. (i) If the multisets (xi,μi):μi>0\(x_i, _i): _i>0\ and (yj,νj):νj>0\(y_j, _j): _j>0\ coincide, then f(x,μ)=f(y,ν)f(x,μ)=f(y,ν). (i) For all x,y∈dx,y ^d and μ: |f(x,μ)−f(y,μ)|≤L∑iwi(μ)ρ(xi,yi) f(x,μ)-f(y,μ) ≤ L _iw_i(μ)\,ρ(x_i,y_i). Proof. (i) is (V) with wi(μ)=0w_i(μ)=0. (i) Choose a permutation σ matching the positive-weight coordinates of (y,ν)(y,ν) to those of (x,μ)(x,μ) pairwise. Then (σy,σν)(σ y,σν) and (x,μ)(x,μ) agree on every positive-weight coordinate and on the weight vector, and differ at most in values at zero-weight coordinates, which are free by (i). Conclude by (A). (i) Change the coordinates one at a time at fixed μ. Each step is a (V) move and every intermediate point is a configuration. ∎ Where the weight axiom comes from. The principle, one mechanism applied to every weighted coordinate, has a formula: a score φ:→ℝ :V applied to every value and summed with the cap weights. Let :=φ:→ℝ:Lip(φ)≤1,|φ|≤RS:=\ :V \;:\;Lip( )≤ 1,\ ≤ R\ be the score ball, which contains a translate of every 11-Lipschitz score because R=12diamR= 12diamV, and let ℒw:=fφ(x,μ)=∑i=1dwi(μ)φ(xi):φ∈L^w\;:=\; \\,f_ (x,μ)= _i=1^dw_i(μ)\, (x_i)\;:\; \, \ (7) be the linear sums of the cap weight. They contain the weighted means and value aggregations ∑iμiφ(xi) _i _i (x_i) at w=μw=μ, the unnormalized sums ∑i(μi/μmax)φ(xi) _i( _i/ _ ) (x_i) at w=μ/μmaxw=μ/ _ and, with φ≡R ≡ R, the count R/μmaxR/ _ . Every linear sum satisfies (V) with L=1L=1 and (A) whatever the weight axiom (Lemma A.5), and the class is meant to contain them at constant 11. The weight axiom is the tightest one that does. Proposition A.4 (The modulus of the linear sums). For all μ,ν∈Δdμ,ν∈ _d, supx∈d,φ∈|fφ(x,μ)−fφ(x,ν)|=R‖w(μ)−w(ν)‖1. _x ^d,\ \, \,f_ (x,μ)-f_ (x,ν)\, \;=\;R\, \|w(μ)-w(ν) \|_1. So (W) says exactly that f is no more sensitive to a change of weights than the linear sums are, uniformly over the values. Among weight axioms |f(x,μ)−f(x,ν)|≤Lc(μ,ν) f(x,μ)-f(x,ν) ≤ L\,c(μ,ν) with a modulus c that does not look at x, it is the tightest one under which every linear sum keeps constant 11, because any such c dominates the supremum above. In this sense LwG^w_L is the smallest class cut out by (V), (A) and a value-free weight axiom that contains the linear sums in its unit ball. A modulus that does look at x is a transport distance between the weighted value measures, and the class it cuts out is smaller and excludes the selection readouts of the next subsection: two coordinates at the same value R with equal shares, and a share δ shifted from one to the other, leave the weighted value measure unchanged while the selected maximum moves by δRδ R, so no constant serves, where (W) prices the move at 2Rδ2Rδ (Remark A.6). Weights are individuals and not mass, and this is why the metric of Section 3.2 is a matching distance on multisets and not a distance between measures. What (W) excludes is the rank readouts: the weighted median and quantiles, trimmed means and top-k selections jump when a weight crosses a threshold, two atoms at ±R± R with weights 12±δ 12±δ moving the weighted median by 2R2R as δ changes sign, so no constant serves; the selected maximum, which is weighted and not ranked, stays in. Lemma A.5 (Linear sums are members). Every fφ∈ℒwf_ ^w satisfies (V) with L=1L=1 and (A). More generally, let φ:→ℝm :V ^m satisfy |φ(s)−φ(t)|2≤ρ(s,t) (s)- (t) _2≤ρ(s,t) and |φ|2≤R _2≤ R, and put fφ(x,μ):=∑iwi(μ)φ(xi)f_ (x,μ):= _iw_i(μ)\, (x_i). Then for all configurations |fφ(x,μ)−fφ(y,ν)|2≤Mw(Φ(x,μ),Φ(y,ν)), \,f_ (x,μ)-f_ (y,ν)\, _2\;≤\;M_w ( (x,μ), (y,ν) ), so every coordinate of fφf_ , and every 11-Lipschitz function of fφf_ , lies in 1wG^w_1. Proof. A change of xix_i alone changes fφf_ by wi(μ)|φ(xi)−φ(yi)|≤wi(μ)ρ(xi,yi)w_i(μ) (x_i)- (y_i) ≤ w_i(μ)ρ(x_i,y_i), and anonymity is the equivariance of w. For the joint bound write P=Φ(x,μ)P= (x,μ), Q=Φ(y,ν)Q= (y,ν), so that fφ(x,μ)=∑(s,u)∈Puφ(s)f_ (x,μ)= _(s,u)∈ Pu\, (s), and let σ be any partial matching. A matched pair (s,u)↦(t,v)(s,u) (t,v) contributes |uφ(s)−vφ(t)|2≤min(u,v)|φ(s)−φ(t)|2+|u−v|sup|φ|2≤ϱ((s,u),(t,v)) u (s)-v (t) _2≤ (u,v) (s)- (t) _2+ u-v _2≤ ((s,u),(t,v)), and an unmatched atom contributes |uφ(s)|2≤Ru=ϱ((s,u),o) u (s) _2≤ Ru= ((s,u),o). Summing gives the cost of σ, and minimizing over σ gives Mw(P,Q)M_w(P,Q). The last claim is Theorem 3.2(i), or directly: a coordinate of fφf_ satisfies (V) and (A) as above and (W) because |∑i(wi(μ)−wi(ν))φ(xi)|2≤R‖w(μ)−w(ν)‖1 _i(w_i(μ)-w_i(ν)) (x_i) _2≤ R \|w(μ)-w(ν) \|_1. ∎ Proof of Proposition A.4. Upper bound: |∑i(wi(μ)−wi(ν))φ(xi)|≤‖w(μ)−w(ν)‖1sup|φ|≤R‖w(μ)−w(ν)‖1 _i(w_i(μ)-w_i(ν)) (x_i) ≤ \|w(μ)-w(ν) \|_1 ≤ R \|w(μ)-w(ν) \|_1. Lower bound: by compactness pick a,b∈a,b with ρ(a,b)=2Rρ(a,b)=2R, and let φ(z):=R−ρ(z,a) (z):=R-ρ(z,a), which is 11-Lipschitz with φ(a)=R (a)=R, φ(b)=−R (b)=-R and |φ|≤R ≤ R, so φ∈ . Put xi=ax_i=a where wi(μ)>wi(ν)w_i(μ)>w_i(ν) and xi=bx_i=b where wi(μ)<wi(ν)w_i(μ)<w_i(ν), anywhere else otherwise. Then ∑i(wi(μ)−wi(ν))φ(xi)=R∑i|wi(μ)−wi(ν)| _i(w_i(μ)-w_i(ν)) (x_i)=R _i w_i(μ)-w_i(ν) . ∎ Remark A.6 (The pointwise alternative). One could instead take the modulus at the given x, cx(μ,ν):=supφ∈|fφ(x,μ)−fφ(x,ν)|c_x(μ,ν):= _ f_ (x,μ)-f_ (x,ν) , which by Kantorovich–Rubinstein duality on V is a transport distance between the weighted value measures ∑iwi(μ)δxi _iw_i(μ) _x_i and ∑iwi(ν)δxi _iw_i(ν) _x_i: the balanced Kantorovich distance for the conservative weight and the flat distance for the accumulative one. The resulting axiom is stronger and its class smaller, and under it a reweighting among coordinates at equal values is free. That is the right axiom when the members are functionals of the weighted value measure, averages and attention pooling, and the wrong one for the selection readouts. Take =[−R,R]V=[-R,R] and f=maxisixif= _is_ix_i, with a leader at value 00 and share 11 and two coordinates at value R with equal shares s, and shift a share δ from one of the two to the other. The weighted value measure does not change at all, while f changes by δRδ R, so no constant serves. Under (W) the move is priced at 2Rδ2Rδ and the maximum has constant 11 (Proposition A.14). A.2 The cone and the matching distance Lemma A.7 (The cone is a compact metric space). ϱ is a metric on K, (,ϱ)(K, ) is compact, and the distance to the apex is R times the height. Proof. ϱ is well defined on the quotient, since ϱ((x,u),(y,0))=Ru ((x,u),(y,0))=Ru does not depend on y, it is symmetric, and it vanishes only on the diagonal: for u,v>0u,v>0, ϱ=0 =0 forces u=vu=v and then ρ(x,y)=0ρ(x,y)=0. Triangle inequality: let p=(x,u)p=(x,u), q=(z,t)q=(z,t), r=(y,v)r=(y,v) with positive heights. We show ϱ(p,r)≤ϱ(p,q)+ϱ(q,r) (p,r)≤ (p,q)+ (q,r) by the position of t. If t≥u∨vt≥ u v, the right side is uρ(x,z)+vρ(z,y)+R(2t−u−v)≥(u∧v)ρ(x,y)+R|u−v|uρ(x,z)+vρ(z,y)+R(2t-u-v)≥(u v)ρ(x,y)+R u-v , since 2t−u−v≥|u−v|2t-u-v≥ u-v . If t≤u∧vt≤ u v, the right side is t(ρ(x,z)+ρ(z,y))+R(u+v−2t)≥tρ(x,y)+R(u+v−2t)t(ρ(x,z)+ρ(z,y))+R(u+v-2t)≥ tρ(x,y)+R(u+v-2t), and subtracting ϱ(p,r)=(u∧v)ρ(x,y)+R(u∨v−u∧v) (p,r)=(u v)ρ(x,y)+R(u v-u v) leaves (u∧v−t)(2R−ρ(x,y))≥0(u v-t)(2R-ρ(x,y))≥ 0 because ρ≤2Rρ≤ 2R. If u≤t≤vu≤ t≤ v, the other order being symmetric, the right side is uρ(x,z)+tρ(z,y)+R(v−u)≥u(ρ(x,z)+ρ(z,y))+R(v−u)≥ϱ(p,r)uρ(x,z)+tρ(z,y)+R(v-u)≥ u(ρ(x,z)+ρ(z,y))+R(v-u)≥ (p,r). With the apex in the middle, ϱ(p,o)+ϱ(o,r)=R(u+v)≥(u∧v)ρ(x,y)+R|u−v| (p,o)+ (o,r)=R(u+v)≥(u v)ρ(x,y)+R u-v because (u∧v)ρ(x,y)≤2R(u∧v)(u v)ρ(x,y)≤ 2R(u v), and with the apex as an endpoint ϱ(o,r)=Rv≤Rt+R|t−v|≤ϱ(o,q)+ϱ(q,r) (o,r)=Rv≤ Rt+R t-v ≤ (o,q)+ (q,r). Compactness: ϱ((x,u),(y,v))≤ρ(x,y)+R|u−v| ((x,u),(y,v))≤ρ(x,y)+R u-v and ϱ((x,u),o)=Ru ((x,u),o)=Ru, so the quotient map from the compact ×[0,1]V×[0,1] is continuous into (,ϱ)(K, ) and onto. ∎ Lemma A.8 (Factorization). Φ(x,μ)=Φ(y,ν) (x,μ)= (y,ν) if and only if the multisets (xi,μi):μi>0\(x_i, _i): _i>0\ and (yj,νj):νj>0\(y_j, _j): _j>0\ coincide. Consequently every f∈Lwf ^w_L factors as f=F∘Φf=F for a unique F:ℳd→ℝF:M_d , and Φ is a bijection from the quotient of C by permutations and by values at zero-weight coordinates onto ℳdM_d. Proof. If the multisets of (xi,μi)(x_i, _i) coincide, so do those of (xi,wi(μ))(x_i,w_i(μ)), by equivariance. Conversely, if the multisets of pairs (value, cap weight) coincide, there is a permutation π of [d][d] carrying suppνsuppν onto suppμsuppμ with (yπ−1(i),wπ−1(i)(ν))=(xi,wi(μ))(y_π^-1(i),w_π^-1(i)(ν))=(x_i,w_i(μ)) for i∈suppμi μ, so πyπ y agrees with x on suppμsuppμ and w(πν)=w(μ)w(πν)=w(μ) by equivariance. Then πν=μπν=μ by injectivity. The factorization is Lemma A.3(i). ∎ Lemma A.9 (Metric, measure form, duality). (i) MwM_w is a metric on ℳdM_d. (i) Mw(P,Q)=W1ϱ(P+nQδo,Q+nPδo)M_w(P,Q)=W_1 (P+n_Q\, _o,\;Q+n_P\, _o ), the Kantorovich distance with ground cost ϱ between the two apex-padded measures, both of total mass nP+nQn_P+n_Q. (i) With Lip0():=ψ:→ℝ:Lipϱ(ψ)≤1,ψ(o)=0Lip_0(K):=\ψ:K \;:\;Lip_ (ψ)≤ 1,\ ψ(o)=0\, Mw(P,Q)=supψ∈Lip0()[∑(x,u)∈Pψ(x,u)−∑(y,v)∈Qψ(y,v)],M_w(P,Q)\;=\; _ψ _0(K)\; [\, _(x,u)∈ Pψ(x,u)- _(y,v)∈ Qψ(y,v)\, ], and every ψ∈Lip0()ψ _0(K) satisfies |ψ(x,u)|≤Ru ψ(x,u) ≤ Ru. Proof. (i) Symmetry is clear. If Mw(P,Q)=0M_w(P,Q)=0 the optimal σ has zero cost, so no atom is unmatched, an unmatched atom costing Ru>0Ru>0, and every matched pair coincides, hence P=QP=Q. Triangle inequality: given partial matchings σ1 _1 from P to Q and σ2 _2 from Q to S, define σ from P to S by σ(p)=σ2(σ1(p))σ(p)= _2( _1(p)) whenever both are defined, and leave every other atom unmatched. By the triangle inequality of ϱ with the apex included (Lemma A.7): a p matched through q to s costs ϱ(p,s)≤ϱ(p,q)+ϱ(q,s) (p,s)≤ (p,q)+ (q,s), a p matched to a q that σ2 _2 leaves unmatched costs ϱ(p,o)≤ϱ(p,q)+ϱ(q,o) (p,o)≤ (p,q)+ (q,o), an s matched by σ2 _2 to a q that σ1 _1 leaves unmatched costs ϱ(s,o)≤ϱ(q,s)+ϱ(q,o) (s,o)≤ (q,s)+ (q,o), and atoms unmatched in σ1 _1 or σ2 _2 keep their apex cost. Each term of cost(σ1)+cost(σ2)cost( _1)+cost( _2) is used at most once, so cost(σ)≤cost(σ1)+cost(σ2)cost(σ) ( _1)+cost( _2). (i) The Kantorovich problem between the padded measures is a transportation linear program between two counting measures of the same mass, nP+nQn_P+n_Q unit atoms on each side, so by Birkhoff’s theorem its polytope is the convex hull of the permutation matrices (Peyré and Cuturi, 2019, Prop. 2.1) and an optimal integer plan exists, a bijection between the atoms of the two sides. A bijection matches some atoms of P to atoms of Q, the remaining atoms of each side to apex copies, and apex copies to each other at cost 00, and its cost is the cost of the partial matching it induces. Conversely every partial matching extends to such a bijection, the counts of unmatched atoms and spare apex copies agreeing on both sides. Hence the two minima coincide. (i) Kantorovich–Rubinstein duality on the compact metric space (,ϱ)(K, ), for two finite measures of equal mass: W1(α,β)=sup∫ψd(α−β):Lipϱψ≤1W_1(α,β)= \ ψ\,d(α-β):Lip_ ψ≤ 1\. Here α−β=P−Q+(nQ−nP)δoα-β=P-Q+(n_Q-n_P) _o has total mass 00, so ψ↦ψ−ψ(o)ψ ψ-ψ(o) does not change the integral and one may restrict to ψ(o)=0ψ(o)=0, which kills the apex term. The bound is |ψ(x,u)|=|ψ(x,u)−ψ(o)|≤ϱ((x,u),o)=Ru ψ(x,u) = ψ(x,u)-ψ(o) ≤ ((x,u),o)=Ru. ∎ Remark A.10 (The dual ball). For φ∈ the function ψφ(x,u):=uφ(x) _ (x,u):=u\, (x) lies in Lip0()Lip_0(K) and ∫ψφΦ(x,μ)=fφ(x,μ) _ \,d (x,μ)=f_ (x,μ), so the linear sums are the sub-ball of scores that read the height linearly. The dual ball is larger: it contains scores that read the height nonlinearly, such as ψ(x,u)=R2u2ψ(x,u)= R2u^2, whose functional is R2∑iwi(μ)2 R2 _iw_i(μ)^2. This is where the members beyond the linear sums live: the supremum over x in Proposition A.4 enlarged the score ball from uφ(x)\u (x)\ to all Lipschitz functions on the cone vanishing at the apex, and MwM_w is the dual of the enlarged ball. Lemma A.11 (Elementary moves). Let (x,μ)∈(x,μ) . (i) (value move, exact) If y differs from x in coordinate i only, then Mw(Φ(x,μ),Φ(y,μ))=wi(μ)ρ(xi,yi)M_w ( (x,μ), (y,μ) )=w_i(μ)\,ρ(x_i,y_i). (i) (weight move) For every ν, Mw(Φ(x,μ),Φ(x,ν))≤R‖w(μ)−w(ν)‖1M_w ( (x,μ), (x,ν) )≤ R\, \|w(μ)-w(ν) \|_1, with strict inequality possible: two atoms at the same value exchanging heights are at distance 00 while ‖w(μ)−w(ν)‖1>0 \|w(μ)-w(ν) \|_1>0. (i) (re-representation) Φ(σx,σμ)=Φ(x,μ) (σ x,σμ)= (x,μ), and changing xix_i with μi=0 _i=0 does not change Φ . Proof. (i) The identity matching gives ≤ϱ((xi,wi),(yi,wi))=wiρ(xi,yi)≤ ((x_i,w_i),(y_i,w_i))=w_iρ(x_i,y_i). For ≥, take ψ(z,u):=u(R−ρ(z,xi))ψ(z,u):=u\,(R-ρ(z,x_i)), which is ψφ _ for φ=R−ρ(⋅,xi)∈ =R-ρ(·,x_i) , hence in Lip0()Lip_0(K) by Remark A.10. The two lifts differ in one atom and ψ(xi,wi)−ψ(yi,wi)=wiρ(xi,yi)ψ(x_i,w_i)-ψ(y_i,w_i)=w_iρ(x_i,y_i), so Lemma A.9(i) gives the lower bound. (i) Match coordinate i to itself where both weights are positive, at cost R|wi(μ)−wi(ν)|R w_i(μ)-w_i(ν) with no value motion, and send an atom with a zero partner to the apex, at cost RwiRw_i, again R|wi(μ)−wi(ν)|R w_i(μ)-w_i(ν) . The example: (t,u),(t,v)\(t,u),(t,v)\ and (t,v),(t,u)\(t,v),(t,u)\ are the same multiset. (i) is the definition of the lift. ∎ Lemma A.12 (Continuity, compactness, diameter). (i) For all (x,μ),(y,ν)∈(x,μ),(y,ν) , Mw(Φ(x,μ),Φ(y,ν))≤|w(μ)|maxiρ(xi,yi)+R‖w(μ)−w(ν)‖1,M_w ( (x,μ), (y,ν) )\;≤\; w(μ) \, _iρ(x_i,y_i)+R\, \|w(μ)-w(ν) \|_1, where |w(μ)|:=∑iwi(μ) w(μ) := _iw_i(μ). In particular Φ:→(ℳd,Mw) :C→(M_d,M_w) is continuous. (i) (ℳd,Mw)(M_d,M_w) is compact and Φ is a quotient map: a function on ℳdM_d is continuous if and only if its composition with Φ is. In particular the tail masses Tk(μ)=∑i>kμ(i)T_k(μ)= _i>k _(i), with the weights sorted decreasingly, and μmax _ are continuous on (ℳd,Mw)(M_d,M_w). (i) Mw(P,Q)≤R(|P|+|Q|)M_w(P,Q)≤ R\,( P + Q ) with |P|:=∫uP(x,u) P := u\,dP(x,u) the total height, hence diam(ℳd,Mw)≤2Rsupμ|w(μ)|diam(M_d,M_w)≤ 2R _μ w(μ) . Proof. (i) Go through Φ(y,μ) (y,μ): the first leg costs at most ∑iwi(μ)ρ(xi,yi) _iw_i(μ)ρ(x_i,y_i) by the identity matching, the second at most R‖w(μ)−w(ν)‖1R \|w(μ)-w(ν) \|_1 by Lemma A.11(i). Continuity follows with the continuity of w. (i) C is compact and Φ continuous and onto, so (ℳd,Mw)(M_d,M_w) is compact. A continuous map from a compact space onto a Hausdorff space is closed, hence a quotient map. TkT_k and μmax _ are continuous on C and constant on the fibers of Φ . (i) The empty matching sends every atom to the apex. ∎ A.3 Proof of Theorem 3.2 Proof of Theorem 3.2(i), the easy direction. If |F(P)−F(Q)|≤LMw(P,Q) F(P)-F(Q) ≤ L\,M_w(P,Q) for all P,QP,Q and f=F∘Φf=F , then (V), (W) and (A) are Lemma A.11(i), (i), (i) respectively, read through the Lipschitz bound. ∎ Proof of Theorem 3.2(i), the routing direction. Let f∈Lwf ^w_L. By Lemma A.8, f=F∘Φf=F for a unique F, and it remains to show that for all (x,μ),(y,ν)∈(x,μ),(y,ν) , |f(x,μ)−f(y,ν)|≤L⋅Mw(Φ(x,μ),Φ(y,ν)). \,f(x,μ)-f(y,ν)\, \;≤\;L· M_w ( (x,μ),\, (y,ν) ). Write P=Φ(x,μ)P= (x,μ), Q=Φ(y,ν)Q= (y,ν), S=suppμS=suppμ, T=suppνT=suppν, and let σ be an optimal partial matching, viewed at the level of coordinates as a bijection from S′⊆S S onto T′⊆T T. We may take σ maximal: if i∈Si∈ S and j∈Tj∈ T were both unmatched, adding the pair (i,j)(i,j) would change the cost by ϱ((xi,wi(μ)),(yj,wj(ν)))−ϱ((xi,wi(μ)),o)−ϱ((yj,wj(ν)),o)≤0 ((x_i,w_i(μ)),(y_j,w_j(ν)))- ((x_i,w_i(μ)),o)- ((y_j,w_j(ν)),o)≤ 0, by the triangle inequality through the apex. So either S′=S =S or T′=T =T. Relabeling. Define an injection t of T into [d][d] by t(σ(i)):=it(σ(i)):=i for i∈S′i∈ S , and, if T∖T′≠∅T T ≠ (so S′=S =S), by sending T∖T′T T injectively into [d]∖S[d] S, which has d−|S|≥|T|−|T′|d- S ≥ T - T elements. Let π be a permutation of [d][d] with π(j)=t(j)π(j)=t(j) for j∈Tj∈ T, and put ν′:=πν :=πν, y′:=πy :=π y. Then νt(j)′=νjν _t(j)= _j, yt(j)′=yjy _t(j)=y_j, ν′ν vanishes off t(T)t(T), and by equivariance wt(j)(ν′)=wj(ν)w_t(j)(ν )=w_j(ν). By (A), f(y′,ν′)=f(y,ν)f(y ,ν )=f(y,ν), and Φ(y′,ν′)=Q (y ,ν )=Q. Every coordinate i∈[d]i∈[d] is now of one of four kinds: matched (i∈S′i∈ S , hosting (xi,wi(μ))(x_i,w_i(μ)) and (yi′,wi(ν′))=(yσ(i),wσ(i)(ν))(y _i,w_i(ν ))=(y_σ(i),w_σ(i)(ν))), leaving (i∈S∖S′i∈ S S , weight wi(μ)>0=wi(ν′)w_i(μ)>0=w_i(ν )), arriving (i∈t(T∖T′)i∈ t(T T ), weight 0=wi(μ)<wi(ν′)0=w_i(μ)<w_i(ν )), or empty. By maximality, leaving and arriving coordinates do not both occur. The chain. Split the matched coordinates into I↑:=i∈S′:wi(μ)≤wi(ν′)I :=\i∈ S :w_i(μ)≤ w_i(ν )\ and I↓:=S′∖I↑I :=S I . Define z∈dz ^d by zi:=yi′z_i:=y _i for i∈I↑i∈ I and for arriving i, and zi:=xiz_i:=x_i otherwise. Then: Step 1. (x,μ)→(z,μ)(x,μ)→(z,μ): value moves at weights μ on I↑I and on the arriving coordinates. By Lemma A.3(i), |f(x,μ)−f(z,μ)|≤L∑i∈I↑wi(μ)ρ(xi,yi′) f(x,μ)-f(z,μ) ≤ L _i∈ I w_i(μ)\,ρ(x_i,y _i), the arriving coordinates having wi(μ)=0w_i(μ)=0 and contributing nothing. Step 2. (z,μ)→(z,ν′)(z,μ)→(z,ν ): one weight move. By (W), |f(z,μ)−f(z,ν′)|≤LR‖w(μ)−w(ν′)‖1 f(z,μ)-f(z,ν ) ≤ LR\, \|w(μ)-w(ν ) \|_1. Step 3. (z,ν′)→(y′,ν′)(z,ν )→(y ,ν ): value moves at weights ν′ν on I↓I (on the leaving coordinates y′y may be taken equal to x, since their ν′ν -weight is zero and the lift does not see them). By Lemma A.3(i), |f(z,ν′)−f(y′,ν′)|≤L∑i∈I↓wi(ν′)ρ(xi,yi′) f(z,ν )-f(y ,ν ) ≤ L _i∈ I w_i(ν )\,ρ(x_i,y _i). On I↑I the weight in Step 1 is wi(μ)=min(wi(μ),wi(ν′))w_i(μ)= (w_i(μ),w_i(ν )), and on I↓I the weight in Step 3 is wi(ν′)=min(wi(μ),wi(ν′))w_i(ν )= (w_i(μ),w_i(ν )), so the value costs of Steps 1 and 3 add up to ∑i∈S′min(wi(μ),wσ(i)(ν))ρ(xi,yσ(i)) _i∈ S (w_i(μ),w_σ(i)(ν) )\,ρ(x_i,y_σ(i)), while the ℓ1 ^1 sum in Step 2 is ∑i∈S′|wi(μ)−wσ(i)(ν)| _i∈ S w_i(μ)-w_σ(i)(ν) plus the weights wi(μ)w_i(μ) of the leaving coordinates plus the weights wt−1(i)(ν)w_t^-1(i)(ν) of the arriving ones. Adding the three steps, |f(x,μ)−f(y,ν)|≤L[∑i∈S′ϱ((xi,wi(μ)),(yσ(i),wσ(i)(ν)))+∑unmatchedRw]=Lcost(σ), f(x,μ)-f(y,ν) \;≤\;L [ _i∈ S ((x_i,w_i(μ)),(y_σ(i),w_σ(i)(ν)) )+ _unmatchedR\,w ]\;=\;L\,cost(σ), which is LMw(P,Q)L\,M_w(P,Q) by the optimality of σ. ∎ Proof of Theorem 3.2(i). The inequality ≤ is the routing direction. For ≥ take F:=Mw(⋅,Q)F:=M_w(·,Q), which is 11-Lipschitz for MwM_w, so F∘Φ∈1wF ^w_1 by the easy direction, and F(P)−F(Q)=Mw(P,Q)F(P)-F(Q)=M_w(P,Q). If a metric d on ℳdM_d makes every F with F∘Φ∈1wF ^w_1 11-Lipschitz, then Mw(P,Q)=supf[f(z)−f(z′)]≤d(P,Q)M_w(P,Q)= _f[f(z)-f(z )]≤ d(P,Q). And if LipL(d)∘Φ=LwLip_L(d) =G^w_L for all L, the same argument with d in place of MwM_w gives d(P,Q)=supf[f(z)−f(z′)]=Mw(P,Q)d(P,Q)= _f[f(z)-f(z )]=M_w(P,Q). ∎ In the language of persistence diagrams, part (i) is the universality of the Wasserstein distance on the metric pair (,ϱ,o)(K, ,\o\): Bubenik and Elchesen (2022a, Cor. 1.4 and §5.3) prove that on any metric space with a distinguished subset the p-Wasserstein distance on finite multisets is the largest p-subadditive metric compatible with the ground metric, and that at p=1p=1 it satisfies Kantorovich–Rubinstein duality, which is Lemma A.9(i), and Bubenik and Elchesen (2022b) carry the construction to signed measures and their Banach spaces. What is specific here is the pair, the cone with its apex, and that the class comes first. The closure properties named after Theorem 3.2 are those of a Lipschitz ball: the class is convex, and if f1,…,fmf_1,…,f_m are members with constants L1,…,LmL_1,…,L_m and h:ℝm→ℝh:R^m is Lipschitz then h(f1,…,fm)h(f_1,…,f_m) is a member with constant supy∑a|∂ah(y)|La _y _a _ah(y) \,L_a, since h(F1,…,Fm)h(F_1,…,F_m) is Lipschitz for MwM_w with that constant. The linear sums extend to the vector-valued case, ∑iwi(μ)φ(xi) _iw_i(μ) (x_i) with φ:→ℝm :V ^m 11-Lipschitz and bounded by R in norm, which is 11-Lipschitz from (ℳd,Mw)(M_d,M_w) into ℝmR^m jointly (Lemma A.5), so that attention pooling and the flocking resultant are members by post-composition. A.4 The two instances Three levels, and the sum embeddings. Three nested sets order the class: the linear sums (7), of which a weighted Deep Sets network is a Lipschitz readout of finitely many (Remark B.19), the functionals of the lift, the dual ball of the matching distance (Lemma A.9, Remark A.10), and the class of Theorem 3.2, every Lipschitz function of the lift. The conservative linear sums learn at the rate of the value space alone, with no count and no profile (Proposition B.20), while the class sees the law of the lift and pays the profile (Section 4). The two classes of Section 3.1 are the following. Definition A.13 (The two classes). The conservative class L:=LwcG_L:=G^w^c_L is the class of Definition 3.1 for the cap weight wc(μ)=μw^c(μ)=μ, the weights themselves, and the accumulative class ~L:=Lwa G_L:=G^w^a_L is the class for the cap weight wa(μ)=μ/μmaxw^a(μ)=μ/ _ , the vector of shares s(μ)s(μ), the leader at share 11. Table 1 sorts the common aggregations by what they do to influence. The same aggregations are debated in graph networks on expressivity grounds, the sum separating multisets that the mean and the maximum identify (Xu et al., 2019) and a degree-dependent scaler interpolating between the two (Corso et al., 2020). In both classes the weight bound sums over all d coordinates, and ‖s(μ)−s(ν)‖1 \|s(μ)-s(ν) \|_1 can reach d−1d-1: that is the count of Table 1, a member at constant 11, and the size is intended. mechanism form class L weighted mean h(∑iμixi)h ( _i _ix_i ) conservative ‖h′‖∞ \|h \|_∞ value aggregation ∑iμiφ(xi) _i _i\, (x_i), φ∈ conservative 11 attention pooling h(∑iμiv(xi))h ( _i _i\,v(x_i) ), Lipv≤1Lipv≤ 1, |v|≤R v ≤ R conservative Lip(h)Lip(h) top weight RμmaxR\, _ conservative 11 flocking update direction of ∑iμieiθi _i _i\,e^i _i conservative 1/ϖ01/ _0 at polarization ϖ≥ϖ0 ≥ _0 count R/μmax=R∑isiR/ _ =R _is_i accumulative 11 unnormalized sum ∑isiφ(xi) _is_i\, (x_i), φ∈ accumulative 11 force sum ∑isiF(xi) _is_i\,F(x_i), LipF≤1LipF≤ 1, |F|≤R F ≤ R accumulative 11 selected maximum maxisixi _is_i\,x_i accumulative 11 weighted mean∗ ∑iμixi _i _ix_i accumulative 22 Table 1: Common aggregations, the class they belong to and the constant. Values in [−R,R][-R,R] where a value bound is used. The starred row is the mechanism of the first row read against the share cap. The flocking row is Section C.1, the rest is Proposition A.14. No constant depends on d or on the number of selected coordinates. Proposition A.14 (Membership). Each mechanism of Table 1 belongs to the class shown with the constant shown. Moreover L⊆~2LG_L G_2L. Proof of Proposition A.14. Each (V) line is the differential form of Remark A.1 or a direct two-point estimate, and each (W) line is the displayed one-line estimate. Conservative rows, on =[−R,R]V=[-R,R]. Weighted means h(⟨μ,x⟩)h( μ,x ): (V) |h′|μi|xi−yi| h _i x_i-y_i , (W) |h′||⟨μ−ν,x⟩|≤‖h′‖∞R‖μ−ν‖1 h \, μ-ν,x ≤ \|h \|_∞R \|μ-ν \|_1. Value aggregations: Lemma A.5. Attention pooling: the vector-valued linear sum of Lemma A.5 followed by the Lipschitz map h. The top weight RμmaxR\, _ : (V) 00, (W) R|μmax−νmax|≤R‖μ−ν‖∞R _ - _ ≤ R \|μ-ν \|_∞. The flocking update is Section C.1. Accumulative rows, on =[−R,R]V=[-R,R], with s=s(μ)s=s(μ), s′=s(ν)s =s(ν) and |s|:=‖s‖1 s := \|s \|_1. The count R∑isiR _is_i: (V) 00, (W) R||s|−|s′||≤R‖s−s′‖1R\, s - s ≤ R \|s-s \|_1. The sums and the force sums: Lemma A.5 with the share weight. The selected maximum maxisixi _is_ix_i, on values in [0,R][0,R] so that the coordinates off the support, at sixi=0s_ix_i=0, never attain it: (V) |sixi−siyi|≤si|xi−yi| s_ix_i-s_iy_i ≤ s_i x_i-y_i , (W) |maxisixi−maxisi′xi|≤maxi|si−si′||xi|≤R‖s−s′‖∞ _is_ix_i- _is _ix_i ≤ _i s_i-s _i \, x_i ≤ R \|s-s \|_∞. The weighted mean ⟨μ,x⟩=⟨s,x⟩/|s| μ,x = s,x / s : (V) μi≤si _i≤ s_i, and for (W), ⟨μ,x⟩−⟨ν,x⟩=⟨s−s′,x⟩/|s|+⟨s′,x⟩(1/|s|−1/|s′|) μ,x - ν,x = s-s ,x / s + s ,x (1/ s -1/ s ) with |⟨s′,x⟩|≤R|s′| s ,x ≤ R s and |1/|s|−1/|s′||≤‖s−s′‖1/(|s||s′|) 1/ s -1/ s ≤ \|s-s \|_1/( s s ), so the difference is at most 2R‖s−s′‖1/|s|≤2R‖s−s′‖12R \|s-s \|_1/ s ≤ 2R \|s-s \|_1 because |s|≥1 s ≥ 1. Containment. μi≤si _i≤ s_i gives the accumulative (V) from the conservative one, and for the weight axioms ‖μ−ν‖1≤2μmax‖s−s′‖1≤2‖s−s′‖1 \|μ-ν \|_1≤ 2\, _ \, \|s-s \|_1≤ 2 \|s-s \|_1: indeed μ=s/|s|μ=s/ s , so μ−ν=(s−s′)/|s|+s′(1/|s|−1/|s′|)μ-ν=(s-s )/ s +s (1/ s -1/ s ), of ℓ1 ^1 norm at most (‖s−s′‖1+||s′|−|s||)/|s|≤2‖s−s′‖1/|s|( \|s-s \|_1+ s - s )/ s ≤ 2 \|s-s \|_1/ s , and 1/|s|=μmax1/ s = _ . Hence L⊆~2LG_L G_2L, and by Theorem 3.2(i) also D≤2D~D≤ 2\, D on ℳdM_d. ∎ The corollary read in Section 3.2 is the following. Reading a sum as conservative inflates its constant by the count, the price of the semantics. Corollary A.15 (Fibers: the two semantics made exact). Let μS _S be uniform on S⊆[d]S [d] with |S|=k S =k. On the fiber μ=μS\μ= _S\ the accumulative distance is the matching distance between the two k-point multisets of values, D~(Φ(x,μS),Φ(y,μS))=min∑i∈Sτρ(xi,yτ(i)) D ( (x, _S), (y, _S) )\;=\; _τ\, _i∈ Sρ (x_i,y_τ(i) ) over the bijections τ of S, and the restriction of ~L G_L to the fiber is exactly the set of symmetric functions g of xSx_S with per-coordinate constant L, that is |g(xS)−g(xSi→yi)|≤Lρ(xi,yi) g(x_S)-g(x_S^i→ y_i) ≤ L\,ρ(x_i,y_i) with xi→yix^i→ y_i the vector x with coordinate i replaced by yiy_i, every such g extending to a member of ~L G_L. The restriction of LG_L has per-coordinate constant L/kL/k, and the factor k is attained by the selected maximum. Proof of Corollary A.15. On the fiber all shares equal 11, so a matched pair costs ϱ((xi,1),(yj,1))=ρ(xi,yj)≤2R ((x_i,1),(y_j,1))=ρ(x_i,y_j)≤ 2R while an unmatched atom costs R. Leaving a pair unmatched on both sides costs 2R≥ρ2R≥ρ, so a full matching is optimal and the formula follows. A symmetric g with per-coordinate constant L satisfies |g(xS)−g(yS)|≤L∑iρ(xi,yτ(i)) g(x_S)-g(y_S) ≤ L _iρ(x_i,y_τ(i)) for every τ, by symmetry and then the telescope of Lemma A.3(i) with all weights 11, hence is L-Lipschitz for D~ D on the fiber. Extend it to ℳdM_d with the same constant by McShane’s extension and apply the easy direction of Theorem 3.2(i). Conversely the accumulative (V) at si=1s_i=1 gives the per-coordinate constant L, and the conservative (V) at μi=1/k _i=1/k gives L/kL/k. The selected maximum has accumulative constant 11 (Proposition A.14) and, on the fiber, changes by |xi−yi| x_i-y_i under a move of the leading coordinate, so its conservative constant is at least k. ∎ A.5 Strata, coverings, and the crop cost These are the inputs of Section 4: for each number of atoms, a covering of the lifts carried by that many atoms, the cost of cropping a configuration to its heaviest atoms, and the diameter of the space. Definition A.16 (Strata and cells). For k≤dk≤ d, k:=P∈ℳd:nP≤kA_k:=\P _d:n_P≤ k\, the lifts of configurations with at most k participants. For the accumulative instance and m≥0m≥ 0, k[m]:=P∈k:|P|≤2m=k∩μmax≥2−mA_k^[m]:=\P _k: P ≤ 2^m\=A_k∩\ _ ≥ 2^-m\. Since |P|≤nP≤k P ≤ n_P≤ k on kA_k, only 2m<k2^m<k gives a proper subset. Covering numbers N(ε,S,Mw)N( ,S,M_w) use the radius convention. Theorem A.17 (Covering of the strata, conservative). For every k≤dk≤ d and ε>0 >0, N(ε,k,D)≤N(ε2,,ρ)k⋅N(ε2R,Δk,ℓ1)≤N(ε2,,ρ)k(1+16Rε)k−1.N ( ,A_k,D )\;≤\;N ( 2,V,ρ )^k· N ( 2R, _k, ^1 )\;≤\;N ( 2,V,ρ )^k (1+ 16R )^k-1. On =[−R,R]V=[-R,R] and ε≤R ≤ R: logN(ε,k,D)≤(2k−1)log(17R/ε) N( ,A_k,D)≤(2k-1) (17R/ ). Proof. Parametrize: for v∈kv ^k and a∈Δka∈ _k let Ψ(v,a):=Φ(z) (v,a):= (z) where z carries values v and weights a on the first k coordinates and weight 00 elsewhere. By (A) every P∈kP _k is some Ψ(v,a) (v,a). Matching coordinate j to coordinate j, with the apex where one side has aj=0a_j=0, D(Ψ(v,a),Ψ(v′,a′)) D ( (v,a), (v ,a ) )\; ≤∑j=1k[min(aj,aj′)ρ(vj,vj′)+R|aj−aj′|] ≤\; _j=1^k [ (a_j,a _j)\,ρ(v_j,v _j)+R a_j-a _j ] ≤maxjρ(vj,vj′)+R‖a−a′‖1, ≤\; _jρ(v_j,v _j)+R \|a-a \|_1, using ∑jmin(aj,aj′)≤1 _j (a_j,a _j)≤ 1. Take an ε/2 /2-net of V in each value slot and an ε/(2R) /(2R)-net of (Δk,ℓ1)( _k, ^1). Each term is then at most ε/2 /2. For the simplex count: dropping the last coordinate maps Δk _k into the unit ℓ1 ^1-ball of ℝk−1R^k-1 with ‖a−a′‖1≤2‖b−b′‖1 \|a-a \|_1≤ 2 \|b-b \|_1. A δ/4δ/4-net of that ball has at most (1+8/δ)k−1(1+8/δ)^k-1 points, a ball in its own norm, and re-centering the net at points of the image gives a δ/2δ/2-net of the image and hence a δ-net of Δk _k. With δ=ε/(2R)δ= /(2R) this is (1+16R/ε)k−1(1+16R/ )^k-1. On the interval, N(ε/2,[−R,R])≤4R/ε+1≤5R/εN( /2,[-R,R])≤ 4R/ +1≤ 5R/ for ε≤R ≤ R, and klog(5R/ε)+(k−1)log(17R/ε)≤(2k−1)log(17R/ε)k (5R/ )+(k-1) (17R/ )≤(2k-1) (17R/ ). ∎ Theorem A.18 (Covering of the cells, accumulative). For every k≤dk≤ d, m≥0m≥ 0 and ε>0 >0, N(ε,k[m],D~)≤N(2−m−1ε,,ρ)k⋅k(1+2m+3Rε)k−1,N ( ,A_k^[m], D )\;≤\;N (2^-m-1 ,V,ρ )^k· k (1+ 2^m+3R )^k-1, with net points in kA_k. Equivalently, at the cell’s own scale, N(2m−1ε,k[m],D~)≤N(ε/4,,ρ)k⋅k(1+16R/ε)k−1N (2^m-1 ,A_k^[m], D )≤ N( /4,V,ρ)^k· k(1+16R/ )^k-1: the count of a cell resolved at 2m−1ε2^m-1 does not depend on m. On =[−R,R]V=[-R,R] and ε≤R ≤ R: logN(ε,k[m],D~)≤(2k−1)log(2m+5R/ε) N( ,A_k^[m], D)≤(2k-1) (2^m+5R/ ), and 2m≤2k2^m≤ 2k suffices on kA_k. Proof. Parametrize by the share vector: let Σk:=s∈[0,1]k:maxjsj=1 _k:=\s∈[0,1]^k: _js_j=1\ and, for v∈kv ^k, s∈Σks∈ _k, let Ψ~(v,s):=Φ(z) (v,s):= (z) with z carrying values v and weights s/‖s‖1s/ \|s \|_1 on the first k coordinates, so that z has share vector s there. Every P∈k[m]P _k^[m] is Ψ~(v,s) (v,s) with ‖s‖1=|P|≤2m \|s \|_1= P ≤ 2^m. Matching coordinatewise, D~(Ψ~(v,s),Ψ~(v′,s′)) D ( (v,s), (v ,s ) )\; ≤∑jmin(sj,sj′)ρ(vj,vj′)+R‖s−s′‖1 ≤\; _j (s_j,s _j)\,ρ(v_j,v _j)+R \|s-s \|_1 ≤ 2mmaxjρ(vj,vj′)+R‖s−s′‖1. ≤\;2^m _jρ(v_j,v _j)+R \|s-s \|_1. Take a 2−m−1ε2^-m-1 -net of V in each value slot and an ε/(2R) /(2R)-net of the share vectors of the cell in ℓ1 ^1. Each term is then at most ε/2 /2. For the share count: the share vectors of the cell lie in the union over j≤kj≤ k of the faces sj=1\s_j=1\, and on the face sj=1\s_j=1\ the remaining coordinates form a nonnegative vector of ℓ1 ^1-norm at most 2m−1<2m2^m-1<2^m, a point of the ℓ1 ^1-ball of radius 2m2^m in ℝk−1R^k-1. A δ/2δ/2-net of that ball has at most (1+2m+2/δ)k−1(1+2^m+2/δ)^k-1 points, and re-centering at points of the face gives a δ-net of the face inside the face. With δ=ε/(2R)δ= /(2R) this is k(1+2m+3R/ε)k−1k(1+2^m+3R/ )^k-1. The net points Ψ~(v′,s′) (v ,s ) are lifts in kA_k. The second display is the first with 2m−1ε2^m-1 in place of ε . On the interval, N(2−m−1ε,[−R,R])≤2m+2R/ε+1≤2m+3R/εN(2^-m-1 ,[-R,R])≤ 2^m+2R/ +1≤ 2^m+3R/ , and klog(2m+3R/ε)+logk+(k−1)log(2m+4R/ε)≤(2k−1)log(2m+5R/ε)k (2^m+3R/ )+ k+(k-1) (2^m+4R/ )≤(2k-1) (2^m+5R/ ), using logk≤(2k−1)log2 k≤(2k-1) 2. Finally |P|≤k P ≤ k on kA_k, so k[m]=kA_k^[m]=A_k once 2m≥k2^m≥ k. ∎ Both bounds have exponent kdimε()+(k−1)k _ (V)+(k-1), which is 2k−12k-1 on intervals, with the ambient d absent and the cell factor 2m2^m inside the logarithms only, and not at all once the cell is resolved at its own scale. On a selector design, the law of (π(ω),c(ω))(π(ω),c(ω)) for a selector c, the weights are functions of the values and the simplex directions are not there to cover. If the cropped weights depend on the k selected values xSx_S through a map that is LcL_c-Lipschitz from (S,maxiρ)(V^S, _iρ) to (Δk,ℓ1)( _k, ^1), or are uniform on the support, as for a hard selector, then by Lemma A.12(i) the map xS↦Φ(xS,μ^(xS))x_S (x_S, μ(x_S)) is (1+RLc)(1+RL_c)-Lipschitz into (ℳd,D)(M_d,D), and the cropped support of the stratum is covered by N(ε/(2(1+RLc)),,ρ)kN( /(2(1+RL_c)),V,ρ)^k points: exponent kdimε()k _ (V). Both counts are used in Section 4. Proposition A.19 (Crop cost). Let μ μ be a k-crop of μ for k≥1k≥ 1, k of its largest weights kept and renormalized, by a fixed measurable rule where weights tie (the cost below is the same for every choice), and Tk=Tk(μ)T_k=T_k(μ) the weight beyond them. Then D(Φ(x,μ),Φ(x,μ^))≤ 2RTk,D~(Φ(x,μ),Φ(x,μ^))≤RTkμmax(μ).D ( (x,μ), (x, μ) )\;≤\;2R\,T_k, D ( (x,μ), (x, μ) )\;≤\; R\,T_k _ (μ). Proof. Match each kept atom to itself and send the tail to the apex. Conservative: a kept atom moves in height from μi _i to μi/(1−Tk) _i/(1-T_k), in total R∑keptμiTk/(1−Tk)=RTkR _kept _iT_k/(1-T_k)=R\,T_k, and the tail costs RTkR\,T_k. Accumulative: μ^max=μmax/(1−Tk) μ_ = _ /(1-T_k), so the shares of kept atoms are unchanged and they cost nothing, and the tail costs R∑tailsi=RTk/μmaxR _tails_i=R\,T_k/ _ . ∎ Proposition A.20 (Qualitative inputs). For both instances (ℳd,Mw)(M_d,M_w) is compact, TkT_k and μmax _ are continuous on it, the shells Tk≤θ<Tk−1\T_k≤θ<T_k-1\ and the cells μmax≥2−m\ _ ≥ 2^-m\ are Borel, and Kantorovich–Rubinstein duality holds for W1(Mw)W_1(M_w) on probability measures on ℳdM_d. Diameters: diam(ℳd,D)≤2Rdiam(M_d,D)≤ 2R, on |P|≤2m\ P ≤ 2^m\ one has D~(P,Q)≤2m+1R D(P,Q)≤ 2^m+1R, and globally diam(ℳd,D~)≤2Rddiam(M_d, D)≤ 2Rd. Proof. Lemma A.12, and duality on a compact metric space is standard (Peyré and Cuturi, 2019). ∎ Appendix B Proofs for Section 4 Inputs: strata, crop and counts. The stratum k⊆ℳdA_k _d of lifts carried by at most k atoms, the crop of a configuration to its k heaviest coordinates and the covering numbers of the strata are the inputs, all from Section A.5. A configuration with tail TkT_k lies within κTkκ T_k of its crop, a point of kA_k (Proposition A.19), and, when the value space has metric dimension dim , N(ε,,ρ)≤(CR/ε)dimN( ,V,ρ)≤(C_VR/ ) , the strata are covered at Nk(ε):=N(ε2,k,D)≤(C1Rε)e(k)(ε≤κ),N_k( ):=N ( 2,A_k,D )\;≤\; ( C_1R )^e(k) ( ≤κ), with C1C_1 depending on C_V only and e(k)e(k) the stratum exponent, one copy of the value space per atom and one simplex direction per free weight: e(k)=kdim+k−1e(k)=k +k-1 in general, which is 2k−12k-1 on interval values with C1=34C_1=34, and e(k)=kdime(k)=k on a selector design whose cropped weights are Lipschitz in the selected values or constant given the support, since its weights are not there to cover (Theorem A.17, and the paragraph after Theorem A.18 for selector designs). The ball of Section 5.1 is a selector design, its weights being uniform on the support, and the graded kernels read there are not, since their weights are set by positions that the mechanism does not read (Definition C.4). Nothing in these counts depends on d. B.1 The two-sample certificate Theorem B.1 (The two-sample certificate, in full). Let n be even and let S be split, by a rule fixed in advance, into halves A and B of n/2n/2 points each, with empirical lifted designs Γ^A _A and Γ^B _B. (i) W1(Γ^n,Γ)≤W1(Γ^A,Γ)≤W1(Γ^A,Γ^B)≤2W1(Γ^A,Γ)E\,W_1( _n, ) \,W_1( _A, ) \,W_1( _A, _B)≤ 2\,E\,W_1( _A, ). (i) With probability at least 1−δ1-δ, W1(Γ^A,Γ)≤W1(Γ^A,Γ^B)+κln(1/δ)n,W_1( _A, )\;≤\;W_1( _A, _B)+κ (1/δ)n, and with probability at least 1−δ1-δ, W1(Γ^n,Γ)≤W1(Γ^A,Γ^B)+κln(2/δ)n.W_1( _n, )\;≤\;W_1( _A, _B)+κ (2/δ)n. (i) W1(Γ^A,Γ^B)=minπ2n∑i≤n/2D(Pi,Pπ(i)′)W_1( _A, _B)= _π\, 2n _i≤ n/2D(P_i,P _π(i)), the minimum over the bijections π between the lifted points PiP_i of A and Pj′P _j of B: an assignment problem on n/2n/2 points, each of whose costs is itself an assignment between the atoms of two lifts (Lemma B.3). Proof of Theorem B.1, which contains Theorem 4.6. (i). Γ^n=12(Γ^A+Γ^B) _n= 12( _A+ _B) and W1(⋅,Γ)W_1(·, ) is convex, so W1(Γ^n,Γ)≤12W1(Γ^A,Γ)+12W1(Γ^B,Γ)W_1( _n, )≤ 12W_1( _A, )+ 12W_1( _B, ), whose expectation is W1(Γ^A,Γ)E\,W_1( _A, ), the halves being exchangeable. Next, by duality W1(μ,ν)=supF∈Lip1(∫Fμ−∫Fν)W_1(μ,ν)= _F _1( F\,dμ- F\,dν) is a supremum of functionals affine in ν, and ∫FΓ=B∫FdΓ^B F\,d =E_B F\,d _B for every F, so W1(Γ^A,Γ)=supFB[∫FdΓ^A−∫FdΓ^B]≤B[W1(Γ^A,Γ^B)∣A]W_1( _A, )= _FE_B[ F\,d _A- F\,d _B] _B[W_1( _A, _B) A]; take expectations over A. The last inequality is the triangle inequality through Γ and exchangeability. (i). If ν and ν′ν are empirical measures on m points differing in one point then W1(ν,ν′)≤κ/mW_1(ν,ν )≤κ/m, one atom of mass 1/m1/m moved by at most the diameter κ (Proposition A.20). So B↦W1(Γ^A,Γ^B)B W_1( _A, _B) has bounded differences 2κ/n2κ/n over the n/2n/2 points of B, and McDiarmid’s inequality (Boucheron et al., 2013, Thm. 6.2) bounds its conditional mean, which dominates W1(Γ^A,Γ)W_1( _A, ) by the argument of (i), by its value plus κln(1/δ)/nκ (1/δ)/n with probability 1−δ1-δ: the first display. For the second, apply the first display at δ/2δ/2 to A and, with the roles of the halves exchanged, to B: on the intersection, of probability at least 1−δ1-δ, both W1(Γ^A,Γ)W_1( _A, ) and W1(Γ^B,Γ)W_1( _B, ) are at most W1(Γ^A,Γ^B)+κln(2/δ)/nW_1( _A, _B)+κ (2/δ)/n, and W1(Γ^n,Γ)≤12W1(Γ^A,Γ)+12W1(Γ^B,Γ)W_1( _n, )≤ 12W_1( _A, )+ 12W_1( _B, ) by the convexity used in (i). (i). Transport between two uniform measures on n/2n/2 atoms each is a linear program over doubly stochastic matrices, whose extreme points are the permutation matrices (Birkhoff), so the minimum is attained at a bijection. ∎ Remark B.2 (One split suffices, and averaging over splits). (a) The split must be a rule fixed in advance or drawn independently of the data, and the second display of Theorem B.1(i) certifies the learner on the whole sample from the one observed split at the price of δ/2δ/2 in place of δ, a factor ln(2/δ)/ln(1/δ) (2/δ)/ (1/δ), 1.111.11 at δ=0.05δ=0.05. (b) For a fixed sample, the average of W1(Γ^A,Γ^B)W_1( _A, _B) over splits drawn independently of the data has, by exchangeability, the same expectation, and McDiarmid over all n points with differences 2κ/n2κ/n puts the expected deviation of a random half, W1(Γ^n/2,Γ)E\,W_1( _n/2, ), within κ2ln(1/δ)/nκ 2 (1/δ)/n above it. It does not certify the realized half A: conditionally on A the expectation of a mixed split need not dominate W1(Γ^A,Γ)W_1( _A, ), so the argument of (i) does not apply to it, and Section 5.1 uses the one observed split. (c) Section 5.1 reports the certificate of the learner on half A and the supplementary material that of the learner on the whole sample. Lemma B.3 (D between two lifts is a rectangular assignment). Let P carry atoms (xi,ui)(x_i,u_i), i≤si≤ s, and P′P atoms (yj,vj)(y_j,v_j), j≤s′j≤ s , with s≤s′s≤ s . Then D(P,P′)=R∑j≤s′vj+min∑i≤sπciπ(i),cij:=min(ui,vj)ρ(xi,yj)+R|ui−vj|−Rvj,D(P,P )\;=\;R _j≤ s v_j\;+\; _π _i≤ sc_iπ(i), c_ij:= (u_i,v_j)\,ρ(x_i,y_j)+R u_i-v_j -Rv_j, the minimum over injections π:[s]→[s′]π:[s]→[s ]. With uniform weights ui=1/su_i=1/s, vj=1/s′v_j=1/s , D(P,P′)=M/s′+κ(1−s/s′)D(P,P )=M/s +κ(1-s/s ) with M the cost of a minimum-cost injection of the s values into the s′s under ρ. Proof. By (5), D(P,P′)D(P,P ) is the Kantorovich distance with the cone cost between the two apex-padded counting measures, and a transport between counting measures of unit atoms is a bijection between the padded atom sets (Birkhoff). An atom matched to an apex point costs R times its height, and a matched real pair costs ϱ=min(u,v)ρ+R|u−v|≤R(u+v) = (u,v)ρ+R u-v ≤ R(u+v) since ρ≤2Rρ≤ 2R. So if an atom of P and an atom of P′P are both matched to apex points, matching them to each other and the two apex points to each other, at cost 00, does not increase the cost, and iterating, some optimal bijection matches every atom of the smaller lift to a real atom. Its cost is ∑matchedϱiπ(i)+R∑unmatched jvj=∑i(ϱiπ(i)−Rvπ(i))+R∑jvj _matched _iπ(i)+R _unmatched jv_j= _i( _iπ(i)-Rv_π(i))+R _jv_j. With uniform weights cij=ρij/s′+R/s−2R/s′c_ij= _ij/s +R/s-2R/s , and summing over the s matched atoms gives M/s′+R−2Rs/s′M/s +R-2Rs/s , which with R∑jvj=R _jv_j=R is M/s′+2R(1−s/s′)M/s +2R(1-s/s ). ∎ The bound is stated for a resolution profile, a nondecreasing sequence 0<ε1≤⋯≤εK0< _1≤·s≤ _K, K≤dK≤ d, one resolution per number of resolved atoms, with thresholds θk:=εk/(2κ)<1 _k:= _k/(2κ)<1. It is nondecreasing so that the sets of Definition B.4 nest: a configuration that needs more atoms is resolved more coarsely, never more finely. Definition B.4 (Shells). For θ∈[0,1)θ∈[0,1) and 0≤k≤d0≤ k≤ d let Bk(θ):=P∈ℳd:Tk(P)≤θB_k(θ):=\P _d:T_k(P)≤θ\, the lifts within κθκθ of the stratum kA_k, closed sets that increase with k, with B0(θ)=∅B_0(θ)= and Bd(θ)=ℳdB_d(θ)=M_d. For a resolution profile the sets Bk(θk)B_k( _k) are nested, since Tk≤Tk−1T_k≤ T_k-1 and θk−1≤θk _k-1≤ _k, and the shells k:=Bk(θk)∖Bk−1(θk−1),1≤k≤K,θ0:=θ1, Sh_k:=B_k( _k) B_k-1( _k-1), 1≤ k≤ K, _0:= _1, partition BK(θK)B_K( _K). Their masses are increments of the profile, wk:=Γ(k)=Gk−1(θk−1)−Gk(θk)≥ 0(G0≡1),w_k:= ( Sh_k)=G_k-1( _k-1)-G_k( _k)\;≥\;0 (G_0≡ 1), and the mass beyond the last shell, Γ(BK(θK)c)=GK(θK) (B_K( _K)^c)=G_K( _K), is the trim. Shell k holds the configurations that k atoms resolve to within εk/2 _k/2 and k−1k-1 atoms do not resolve to within εk−1/2 _k-1/2: the sparse configurations sit in the low shells and are resolved finely, the crowded ones in the high shells and coarsely, each shell at its own mass. A celling of the shells partitions each k Sh_k into cells, each contained in a closed D-ball of radius εk _k (Definition B.10). One celling is the theorem’s own: the crop of a configuration of shell k to its k heaviest coordinates is a point of the stratum kA_k within εk/2 _k/2 of it (Proposition A.19), so the pullback of an εk/2 _k/2-net of kA_k cells the shell into at most Nk(εk)N_k( _k) cells (Lemma B.12). Another, used in the computations, quantizes the crop on a grid and reads the cell off the point. Both are determined by nothing random, and the shell and the cell of a sample point are computable since TkT_k is. Proposition B.5 (The shells bound the transport). For every resolution profile and every celling of its shells, both of which may be chosen after seeing the sample, write w^kA w^A_k, w^kB w^B_k for the fractions of the two halves in shell k and μ^jA μ^A_j, μ^jB μ^B_j for their fractions in cell j, each point beyond the last shell a cell of its own. Then W1(Γ^A,Γ^B)≤∑k=1K(w^kA+w^kB)εk+κ2∑j|μ^jA−μ^jB|.W_1( _A, _B)\;≤\; _k=1^K ( w^A_k+ w^B_k ) _k\;+\; κ2 _j | μ^A_j- μ^B_j |. The inequality is deterministic, one feasible transport between the two halves: move each point to its cell’s center, couple the common mass within each cell for free, and the residuals at the diameter. The second term is the observed discrepancy of the two halves’ cell frequencies, the points beyond the last shell included as cells of their own at 2/n2/n each, a number of the sample, and its expectation under i.i.d. halves is what the fluctuation term of the next theorem bounds. With Theorem 4.6 the right-hand side is a certificate, with the profile and the celling free to optimize on the data. Proof of Proposition B.5. Move each point of A to the center of its cell, at cost at most ∑kw^kAεk _k w^A_k _k in W1W_1, and likewise B; a point beyond the last shell is its own center and moves at no cost. Between the two measures on the centers, couple mass min(μ^jA,μ^jB) ( μ^A_j, μ^B_j) within each center at cost 00 and the residuals arbitrarily at cost at most κ, whose total mass is 12∑j|μ^jA−μ^jB| 12 _j μ^A_j- μ^B_j . The triangle inequality adds the three. ∎ Proposition B.6 (The expectation of the sample bound). For every resolution profile, with the theorem’s own celling of the shells, at most Nk(εk)N_k( _k) cells of radius εk _k on shell k, fixed before sampling, W1(Γ^A,Γ^B)≤ 2∑kwkεk+κ∑kwkNk(εk)n+κGK(θK),E\,W_1( _A, _B)\;≤\;2 _kw_k _k+κ _k w_kN_k( _k)n+κ\,G_K( _K), which is (8) with κ in place of κ/2κ/2. Hence, by Theorem B.1(i), Δn(Q) _n(Q) obeys the same bound, and every corollary of Theorem B.14 holds along the two-sample route with its constants at most doubled. Proof. Take expectations in Proposition B.5. Floor: (w^kA+w^kB)=2wkE( w^A_k+ w^B_k)=2w_k. Discrepancy: for a cell of mass μj _j, μ^jA−μ^jB μ^A_j- μ^B_j has mean 00 and variance 2μj(1−μj)/(n/2)≤4μj/n2 _j(1- _j)/(n/2)≤ 4 _j/n, so |μ^jA−μ^jB|≤2μj/nE μ^A_j- μ^B_j ≤ 2 _j/n, and over the at most NkN_k cells of shell k, whose masses sum to wkw_k, Cauchy–Schwarz gives ∑j|μ^jA−μ^jB|≤2wkNk/n _jE μ^A_j- μ^B_j ≤ 2 w_kN_k/n; times κ/2κ/2 this is κwkNk/nκ w_kN_k/n. Trim: a trimmed point is a singleton cell of discrepancy 2/n2/n, so the trimmed points contribute (κ/2)(w^trimA+w^trimB)=κGK(θK)(κ/2)\,E( w^A_trim+ w^B_trim)=κ\,G_K( _K). The last claim: a+2b+c≤2(a+b+c)a+2b+c≤ 2(a+b+c) for the three nonnegative terms, so the bound is at most twice (8) at every profile, hence at most twice its infimum, which is what the corollaries optimize. ∎ Remark B.7 (The Rademacher form). The empirical Rademacher complexity of the class on the sample, ℛ^S:=σsupF1n∑iσiF(Pi) R_S:=E_σ _F 1n _i _iF(P_i) over the 11-Lipschitz F bounded by b=κ/2b=κ/2 and independent uniform signs σi _i, gives the classical certificate ΔS≤2ℛ^S+3κln(2/δ)/(2n) _S≤ 2 R_S+3κ (2/δ)/(2n) with probability 1−δ1-δ (Koltchinskii and Panchenko, 2000; Bartlett and Mendelson, 2002): McDiarmid on ΔS _S, the symmetrization of Vapnik and Chervonenkis (1971), McDiarmid on ℛ^S R_S. For the Lipschitz ball, ℛ^S R_S is at most the expected cost of an optimal partial matching of the plus points to the minus points in D with every unmatched point sent to a reservoir at cost b, by pairing the terms of ∑iσiF(Pi) _i _iF(P_i) along the matching, and conditionally on the sizes of the sign classes the plus and minus points are a random split of the sample, so the balanced part of ℛ^S R_S is half the two-sample distance of a random unbalanced split and the rest is the reservoir term, at most b/nb/ n. Against Theorem 4.6 the form pays the factor 22 and three concentration terms, and gains the whole sample in one statement without a split. Its use for a Lipschitz class over the sample’s own geometry is the Rademacher theorem of Gottlieb et al. (2013), in the entropy form through the doubling dimension of the sample, and the cell bound of Proposition B.5 in the Rademacher form is Carrasco et al. (2025, Prop. 4). Remark B.8 (The certificate, read in full). Of Theorem B.1, part (i) is the certificate. The observed distance between the halves, plus one concentration term, bounds the generalization gap of the class for the learner trained on half A and for the learner trained on the whole sample, and by part (i) it is computable: on the flock it is evaluated at up to 32,00032,000 samples per half in Section 5.1. Three remarks on the form. The split must be fixed in advance or drawn independently of the data, and one split suffices: the observed distance is within κ2ln(1/δ)/nκ 2 (1/δ)/n of its mean by McDiarmid on the n points, and averaging over independent splits certifies the expected deviation of a half, not the realized one (Remark B.2). Any quantity computed from the sample that dominates W1(Γ^A,Γ^B)W_1( _A, _B) is a certificate too, whatever was looked at to compute it, a partition chosen on the sample or a profile optimized on it, since the event of (i) does not depend on that choice: this is what lets the shells of the profile enter, in Section B.2, as a bound on the transport rather than as a hypothesis on the design. And the object is classical in its parts: the two-sample distance is the maximum discrepancy of Bartlett et al. (2002) on a Lipschitz ball, where it is a transport by duality (von Luxburg and Bousquet, 2004, eq. (11)) and a two-sample test statistic in its own right (Ramdas et al., 2017), and the certificate is the symmetrization of Vapnik and Chervonenkis (1971) in that form, while the empirical Rademacher complexity of the class (Koltchinskii and Panchenko, 2000; Bartlett and Mendelson, 2002) is the same object with a random sign pattern in place of a balanced one, and gives a certificate with the constant 22 and three concentration terms (Remark B.7). Across two domains the same transport, between the empirical source and target samples, bounds the target risk of a Lipschitz class in domain adaptation (Redko et al., 2017, Thm. 2), (Shen et al., 2018, Thm. 1), with the joint error of the two domains and a concentration term per sample; here the halves share a law, and one term suffices by the convexity step of (i). What is not classical is that the class is exactly the Lipschitz ball of a metric the axioms determine, so that the transport is the class’s own complexity and not an envelope of a model (Section 2). B.2 The shells and the coupling Lemma B.9 (The tails descend to the lift). For every 0≤k≤d0≤ k≤ d there is a unique function Tk:ℳd→[0,1]T_k:M_d→[0,1] with Tk∘Φ=Tk(⋅,⋅)T_k =T_k(·,·), and it is continuous on (ℳd,D)(M_d,D) and on (ℳd,D~)(M_d, D). In particular TkT_k is computable from the lifted point alone. Proof. Well defined: if Φ(x,μ)=Φ(y,ν) (x,μ)= (y,ν) then by Lemma A.8 the multisets of positive (value, weight) pairs coincide, hence so do the multisets of positive weights; padding by zeros does not change TkT_k for k≥1k≥ 1, and T0≡1T_0≡ 1. Continuous: TkT_k is continuous on C (Definition 4.1), C is compact, and Φ is a continuous surjection onto the metric, hence Hausdorff, space ℳdM_d (Lemma A.12), so Φ is a closed map and therefore a quotient map; a continuous function constant on fibers descends to a continuous function through a quotient map. ∎ The sets Bk(θ)=Tk≤θB_k(θ)=\T_k≤θ\ of Definition B.4 are therefore closed, the shells are Borel, and their masses are the profile increments stated there. Definition B.10 (Celling). Let (,ϱ)(X, ) be a metric space and S⊆S Borel. An (r,N)(r,N)-celling of S is a partition of S into at most N Borel sets, each contained in a closed ϱ -ball of radius r about a point of X, its center. Proposition B.11 (Single-partition coupling, grouped). Let (,ϱ)(X, ) be a metric space with diam≤κdiamX≤ _X, Γ a Borel probability measure on X, and Γ^n _n the empirical measure of n i.i.d. draws from Γ . Let B⊆B be Borel, let B=⨆gSgB= _gS_g be a finite Borel partition, and suppose each group SgS_g admits an (rg,Ng)(r_g,N_g)-celling. Then, with wg:=Γ(Sg)w_g:= (S_g) and W1W_1 the Kantorovich distance with ground metric ϱ , W1(Γ^n,Γ)≤ 2∑grgwg+κ2∑gNgwgn+κΓ(Bc).E\,W_1( _n, )\;≤\;2 _gr_g\,w_g\;+\; _X2 _g N_g\,w_gn\;+\; _X\, (B^c). Proof. Let Vcc∈g\V_c\_c∈ g be the celling of SgS_g; together the cells VcV_c, c∈⋃ggc∈ _gg, form a Borel partition of B, the cells of group g having ϱ -diameter at most 2rg2r_g. Write ac=Γ(Vc)a_c= (V_c), bc=Γ^n(Vc)b_c= _n(V_c), mc=ac∧bcm_c=a_c b_c, and M=∑cmcM= _cm_c. One coupling. Inside each cell pair mass mcm_c of Γ with mass mcm_c of Γ^n _n, for instance the product of the two normalized restrictions, at cost at most 2rg2r_g per unit; pair the two residual measures, each of total mass 1−M1-M, arbitrarily, at cost at most κ _X per unit. From a∧b=a+b2−|a−b|2a b= a+b2- a-b 2 and ∑cac=Γ(B) _ca_c= (B), ∑cbc=Γ^n(B) _cb_c= _n(B), 1−M=12∑c|ac−bc|+12(Γ(Bc)+Γ^n(Bc)),1-M\;=\; 12 _c a_c-b_c + 12 ( (B^c)+ _n(B^c) ), hence, using mc≤acm_c≤ a_c and ∑c∈gac=wg _c∈ ga_c=w_g, W1(Γ^n,Γ)≤ 2∑grgwg+κ[12∑c|ac−bc|+12(Γ(Bc)+Γ^n(Bc))].W_1( _n, )\;≤\;2 _gr_g\,w_g+ _X [ 12 _c a_c-b_c + 12 ( (B^c)+ _n(B^c) ) ]. Binomial fluctuations, Cauchy–Schwarz per group. The cells are fixed before sampling, so nΓ^n(Vc)∼Bin(n,ac)n _n(V_c) (n,a_c) and |ac−bc|≤(ac−bc)2≤ac/nE a_c-b_c ≤ E(a_c-b_c)^2≤ a_c/n. Summing over the at most NgN_g cells of group g, ∑c∈g|ac−bc|≤Ng∑c∈gac/n=Ngwg/n _c∈ gE a_c-b_c ≤ N_g _c∈ ga_c/n= N_gw_g/n, and Γ^n(Bc)=Γ(Bc)E _n(B^c)= (B^c). Take expectations in the display. ∎ Lemma B.12 (Celling from a tube). Let S⊆S be Borel, A⊆A , and 0≤s<r0≤ s<r. If every point of S lies within ϱ -distance s of A, and A has a net of N points at radius r−sr-s, then S admits an (r,N)(r,N)-celling centered at the net points. Proof. A point p of S is within s+ηs+η of some point of A for every η>0η>0, which is within r−sr-s of a net point, so p is within r+ηr+η of one of the N net points, and one net point zjz_j serves for a sequence η↓0η 0: ϱ(p,zj)≤r (p,z_j)≤ r, so the closed balls (zj,r) B(z_j,r), j≤Nj≤ N, cover S. Disjointify: Vj:=S∩(zj,r)∖⋃j′<j(zj′,r)V_j:=S∩ B(z_j,r) _j <j B(z_j ,r), Borel, a partition of S, and Vj⊆(zj,r)V_j B(z_j,r). ∎ Proposition B.13 (Single-partition coupling, reference-point form). Let (,ϱ)(X, ) be a metric space, Γ a Borel probability measure on X and Γ^n _n its empirical measure, and fix a reference point z0∈z_0 with ϱ0(⋅):=ϱ(⋅,z0) _0(·):= (·,z_0). Let B=⨆gSgB= _gS_g be a finite Borel partition of a Borel B⊆B , each SgS_g with an (rg,Ng)(r_g,N_g)-celling and with ϱ0≤ϱg _0≤ _g on SgS_g. Then, with wg:=Γ(Sg)w_g:= (S_g), W1(Γ^n,Γ)≤ 2∑grgwg+∑gϱgNgwgn+ 2∫Bcϱ0Γ.E\,W_1( _n, )\;≤\;2 _gr_g\,w_g\;+\; _g _g N_g\,w_gn\;+\;2 _B^c _0\,d . The point of the form is that it needs no bound on the diameter: residual mass is priced at its own distance to the reference point. Proof. Couple within cells as in the proof of Proposition B.11, at cost at most 2rg2r_g per unit. The two residual measures, Γ ’s, consisting of the mass ac−mca_c-m_c in each cell c and of Γ|Bc |_B^c, and Γ^n _n’s, consisting of bc−mcb_c-m_c in each cell and of Γ^n|Bc _n|_B^c, have equal total mass, and any coupling π of them costs ∫ϱπ≤∫(ϱ0(P)+ϱ0(Q))π(P,Q) \,dπ≤ ( _0(P)+ _0(Q))\,dπ(P,Q) by the triangle inequality through z0z_0, that is at most ∫ϱ0 _0 against each residual. A residual unit in a cell of group g costs at most ϱg _g, and the two residuals of a cell together have mass (ac−mc)+(bc−mc)=|ac−bc|(a_c-m_c)+(b_c-m_c)= a_c-b_c ; the residual mass on BcB^c costs ∫Bcϱ0Γ+∫Bcϱ0dΓ^n _B^c _0\,d + _B^c _0\,d _n. Hence W1(Γ^n,Γ)≤ 2∑grgwg+∑gϱg∑c∈g|ac−bc|+∫Bcϱ0d(Γ+Γ^n),W_1( _n, )\;≤\;2 _gr_gw_g+ _g _g _c∈ g a_c-b_c + _B^c _0\,d( + _n), and taking expectations with ∑c∈g|ac−bc|≤Ngwg/n _c∈ gE a_c-b_c ≤ N_gw_g/n as in that proof and ∫Bcϱ0dΓ^n=∫Bcϱ0ΓE _B^c _0\,d _n= _B^c _0\,d gives the display. ∎ Theorem B.14 (Generalization over the class). For every design Q and every resolution profile, Δn(Q)≤ 2∑k=1Kwkεk+κ2∑k=1KwkNk(εk)n+κGK(θK). _n(Q)\;≤\;2 _k=1^Kw_k\, _k\;+\; κ2 _k=1^K w_k\,N_k( _k)n\;+\;κ\,G_K( _K). (8) The count Nk(εk)N_k( _k) may be replaced by the covering number at radius εk/2 _k/2 of any set within εk/2 _k/2 of k Sh_k, in particular of the crop of the shell’s support, whose exponent on a selector design is kdimk . The same bound with κ in place of κ/2κ/2 is the expectation of Proposition B.5 with the theorem’s own celling (Proposition B.6). The trim may be traded for a crop: cropping every configuration beyond the last shell to its K heaviest atoms, which lands it in some shell, (8) holds with κGK(θK)κ\,G_K( _K) replaced by 2κQ[TK;TK>θK]2κ\,E_Q[T_K;\ T_K> _K] and each wkw_k by wk+gkw_k+g_k, where gkg_k is the mass of the cropped configurations whose crop lies in k Sh_k, ∑kgk=GK(θK) _kg_k=G_K( _K). Remark B.15 (Shells by the first threshold met). The monotonicity of the profile serves only to make the shells a partition. Defining instead k′:=Tk≤θk,Tj>θj for j<k Sh _k:=\T_k≤ _k,\ T_j> _j for j<k\, the configurations whose first threshold met is the kkth, gives a partition of the configurations that meet some threshold, for every sequence ε1,…,εK _1,…, _K, with k′⊆Bk(θk) Sh _k B_k( _k) still, and Theorem B.14 and Proposition B.5 hold verbatim with wk:=Γ(k′)w_k:= ( Sh _k) and the trim the mass that meets no threshold (Section B.2). For a nondecreasing profile the two definitions agree. The nested form is the one computed in Section 5.1. Proof of Theorem B.14. By Kantorovich–Rubinstein duality on the compact space (ℳd,D)(M_d,D) (Proposition A.20), W1(Γ^n,Γ)=sup∫Fd(Γ^n−Γ):LipD(F)≤1W_1( _n, )= \ F\,d( _n- ):Lip_D(F)≤ 1\, and by Theorem 3.2 the functions F∘ΦF with LipD(F)≤1Lip_D(F)≤ 1 are exactly the members of 1G_1, so that ∫FdΓ^n=n−1∑if(Zi) F\,d _n=n^-1 _if(Z_i) and ∫FΓ=Qf F\,d =E_Qf for f=F∘Φf=F . The class is symmetric under f↦−f -f, so the supremum of the signed deviation is the supremum of its absolute value, and Δn(Q)=W1(Γ^n,Γ) _n(Q)=E\,W_1( _n, ). Apply Proposition B.11 on (,ϱ)=(ℳd,D)(X, )=(M_d,D) with κ=κ _X=κ (Proposition A.20), B=BK(θK)B=B_K( _K), and the groups Sk=kS_k= Sh_k, k≤Kk≤ K, Borel by Lemma B.9, of masses wkw_k, with Γ(Bc)=GK(θK) (B^c)=G_K( _K). Each group admits an (εk,Nk(εk))( _k,N_k( _k))-celling by Lemma B.12: a point of k⊆Bk(θk) Sh_k B_k( _k) has Tk≤θkT_k≤ _k, hence lies within κθk=εk/2κ _k= _k/2 of its crop, a point of kA_k (Proposition A.19), and kA_k has a net of Nk(εk)=N(εk/2,k,D)N_k( _k)=N( _k/2,A_k,D) points at radius εk/2 _k/2, finite since kA_k is compact (Lemma A.12). So rk=εkr_k= _k, and the proposition’s three terms are the display. For the replacement of the count: Lemma B.12 takes any set A within εk/2 _k/2 of the shell, in particular the image of k∩suppΓ Sh_k under the k-crop, which lies in kA_k and within εk/2 _k/2 of the shell by the same crop cost; on a selector design its covering number has exponent kdimk by the paragraph after Theorem A.18. For Remark B.15: the sets k′=Tk≤θk∩⋂j<kTj>θj Sh _k=\T_k≤ _k\∩ _j<k\T_j> _j\ are Borel by Lemma B.9, they partition ⋃k≤KBk(θk) _k≤ KB_k( _k) for every sequence of thresholds, and k′⊆Bk(θk) Sh _k B_k( _k), so the same cellings apply and Proposition B.11 gives the display with wk=Γ(k′)w_k= ( Sh _k) and the trim Γ(⋂k≤KTk>θk) ( _k≤ K\T_k> _k\); Proposition B.5 needs only a partition into cells of the stated radii. The crop clause. Let φ be the identity on BK(θK)B_K( _K) and the K-crop crKcr_K on its complement, a measurable map into BK(θK)B_K( _K) since TK(crKP)=0T_K(cr_KP)=0, with D(P,φ(P))≤κTK(P) 1TK(P)>θKD(P, (P))≤κ\,T_K(P)\,1\T_K(P)> _K\ by Proposition A.19. Then φ#Γ^n _\# _n is the empirical measure of n i.i.d. draws from Γ′:=φ#Γ := _\# , and coupling along φ on both sides, W1(Γ^n,Γ)≤κn∑i≤nTK(Pi) 1TK(Pi)>θK+W1(φ#Γ^n,Γ′)+κQ[TK;TK>θK],W_1( _n, )\;≤\; κn _i≤ nT_K(P_i)\,1\T_K(P_i)> _K\+W_1( _\# _n, )+κ\,E_Q[T_K;\ T_K> _K], whose expectation is 2κQ[TK;TK>θK]+W1(φ#Γ^n,Γ′)2κ\,E_Q[T_K;T_K> _K]+E\,W_1( _\# _n, ). Apply Proposition B.11 to Γ′ with the same B and the same groups: the cellings are properties of the sets, Γ′(Bc)=0 (B^c)=0, and Γ′(k)=wk+gk ( Sh_k)=w_k+g_k with gk=Γ(TK>θK∩crK−1(k))g_k= (\T_K> _K\ _K^-1( Sh_k)). ∎ Remark B.16 (The three terms of Theorem B.14, read in full). The three terms are a resolution floor, a fluctuation and a trim, and the profile enters each. The floor is mass weighted: shell k pays its own resolution at its own mass, so a coarsely resolved high shell does not tax the finely resolved bulk. The fluctuation is the design’s mixture of local coverings: shell k pays the count of the k-atom stratum, of exponent e(k)e(k) and not e(d)e(d), weighted by the mass the design puts there, and the term is the binomial fluctuation of the sample’s cell counts, which is where the measure enters, with no function class in sight. The trim is the mass beyond the last shell, paid at the diameter. The crop distance κθkκ _k is absorbed into the radius of the cells and paid once, in the floor, never as a bias and never in the counts. Under the crop clause a crowded configuration pays its own tail in place of the diameter, and Theorem 4.4 below is the constant-profile case, with the multiscale coupling of Proposition B.18 on the cropped stratum in place of the single scale. The constant profile εk≡ε _k≡ is the single-scale form, one resolution and one trim, the shape of the trimmed empirical-measure bounds of Weed and Bach (2019); the stratified form, one resolution per shell, is what a profile needs. The stratified form is not the single-scale bound with more indices: the scales index disjoint parts of the design, not refinements of one set, and the mass per shell is what lets each part pay its own count, the per-shell step of Proposition B.11. Every profile gives a valid bound and the left side does not depend on the profile, so the bound is the infimum over K and over profiles. The shells are observable: Tk(Pi)T_k(P_i) is computable from the sample, so an estimator can bin by shell (Remark B.23). The profile is an assumption on one part of the design, its sparsity across and within configurations, and on nothing else: within a shell the distribution is arbitrary and the bound charges the stratum at its worst, which is where the certificate, which sees the sample’s own configurations, improves on it (Section 5.1). B.3 Proof of Theorem 4.4 The upper bound is proved here, from a multiscale coupling on the cropped stratum (Proposition B.18), with C=51C1C=51C_1 for the constant C1C_1 of the stratum count, 17341734 on interval values. The lower bound of Theorem 4.4 is Lemma D.4 in Appendix D: its bound is the smaller of ck0n−1/(2k0−1)c_k_0n^-1/(2k_0-1) and εmax(k0)/2 _ (k_0)/2, and since n−1/(2k0−1)≤1n^-1/(2k_0-1)≤ 1 the statement holds with the smaller of the two constants ck0c_k_0 and εmax(k0)/2 _ (k_0)/2 in place of ck0c_k_0. Lemma B.17 (Nested cellings from covers). Let (,ϱ)(X, ) be a metric space and S⊆S Borel with diamS≤ε0diamS≤ _0 and finite covering numbers, and put εj:=3−jε0 _j:=3^-j _0. For every J≥0J≥ 0 there are Borel partitions 0=S,1,…,JQ_0=\S\,Q_1,…,Q_J of S, each cell of jQ_j of diameter at most εj _j and contained in a cell of j−1Q_j-1, with |j|≤N(εj+1/2,S,ϱ) _j ≤ N( _j+1/2,S, ). Proof. This is the construction of Boissard and Le Gouic (2014, Lemma 2.1) and Weed and Bach (2019, Prop. 3). For 1≤j≤J1≤ j≤ J let B1j,…,BNjjB^j_1,…,B^j_N_j be closed balls of radius εj+1/2 _j+1/2 covering S, Nj:=N(εj+1/2,S,ϱ)N_j:=N( _j+1/2,S, ). Disjointify the finest cover: QℓJ:=S∩BℓJ∖⋃ℓ′<ℓBℓ′JQ^J_ :=S∩ B^J_ _ < B^J_ gives a Borel partition JQ_J of S into at most NJN_J cells of diameter at most εJ+1≤εJ _J+1≤ _J. Given j+1Q_j+1, let QℓjQ^j_ be the union of the cells of j+1Q_j+1 that meet BℓjB^j_ and were not assigned to Qℓ′jQ^j_ for some ℓ′<ℓ < . Every cell of j+1Q_j+1 meets some BℓjB^j_ , so the QℓjQ^j_ form a Borel partition of S into at most NjN_j cells, each a union of cells of j+1Q_j+1; and two points of QℓjQ^j_ lie in cells of j+1Q_j+1, of diameter at most εj+1 _j+1, that both meet the ball BℓjB^j_ , of diameter εj+1 _j+1, so their distance is at most 3εj+1=εj3 _j+1= _j. Finally 0:=SQ_0:=\S\. ∎ Proposition B.18 (Multiscale coupling). Let (,ϱ)(X, ), S and εj _j be as in Lemma B.17, Γ a Borel probability measure on S and Γ^n _n the empirical measure of n i.i.d. draws from Γ . If N(εj+1/2,S,ϱ)≤(A/εj)eN( _j+1/2,S, )≤(A/ _j)^e for every j≥0j≥ 0, with A≥ε0A≥ _0 and e>0e>0, then W1(Γ^n,Γ)≤ 17An−1/eif e≥3,W1(Γ^n,Γ)≤ 4A(1+lnn)n−1/2if e≤2.E\,W_1( _n, )\;≤\;17\,A\,n^-1/e e≥ 3, \,W_1( _n, )\;≤\;4A\,(1+ n)\,n^-1/2 e≤ 2. For 2<e<32<e<3 the first bound holds with a constant depending on e in place of 1717, by the same proof. The exponents used below are integers. Proof. The coupling. Fix J≥0J≥ 0 and the partitions of Lemma B.17, and for two probability measures μ,νμ,ν on S write aC:=μ(C)a_C:=μ(C), bC:=ν(C)b_C:=ν(C) for a cell C. We claim W1(μ,ν)≤ 2εJ+∑j=1Jεj−1⋅12∑C∈j|aC−bC|.W_1(μ,ν)\;≤\;2 _J+ _j=1^J _j-1· 12 _C _j a_C-b_C . (9) Pick a point xCx_C in each cell C∈JC _J and let μJ:=∑CaCδxC _J:= _Ca_C\, _x_C, νJ:=∑CbCδxC _J:= _Cb_C\, _x_C: moving every point to its cell’s representative costs at most εJ _J, so W1(μ,ν)≤2εJ+W1(μJ,νJ)W_1(μ,ν)≤ 2 _J+W_1( _J, _J). Couple μJ _J with νJ _J from the finest level up. First pair, at each xCx_C, the mass aC∧bCa_C b_C of the two, at cost 00; what remains in a cell C∈JC _J is (aC−bC)+(a_C-b_C)_+ of μJ _J and (bC−aC)+(b_C-a_C)_+ of νJ _J. Then, for j=J,J−1,…,1j=J,J-1,…,1 in turn, inside each cell P∈j−1P _j-1 pair as much as possible of the remaining μJ _J-mass of the cells C∈jC _j, C⊆PC P, with the remaining νJ _J-mass of those cells, at cost at most diamP≤εj−1diamP≤ _j-1 per unit. By induction on the level, before the step at level j the mass remaining in a cell C∈jC _j is (aC−bC)+(a_C-b_C)_+ of μJ _J and (bC−aC)+(b_C-a_C)_+ of νJ _J: the two sums ∑C⊆P(aC−bC)+ _C P(a_C-b_C)_+ and ∑C⊆P(bC−aC)+ _C P(b_C-a_C)_+ differ by aP−bPa_P-b_P and add to ∑C⊆P|aC−bC| _C P a_C-b_C , so the step at level j inside P pairs 12∑C⊆P|aC−bC|−12|aP−bP| 12 _C P a_C-b_C - 12 a_P-b_P units and leaves (aP−bP)+(a_P-b_P)_+ of μJ _J and (bP−aP)+(b_P-a_P)_+ of νJ _J in P. At the top, P=SP=S has aS=bS=1a_S=b_S=1 and nothing remains, so the pairing is a coupling of μJ _J and νJ _J, of cost at most ∑j=1Jεj−1∑P∈j−112∑C⊆P|aC−bC| _j=1^J _j-1 _P _j-1 12 _C P a_C-b_C , which is (9). The fluctuations. Take μ=Γ^nμ= _n and ν=Γν= . The cells are fixed before sampling, so as in the proof of Proposition B.11 ∑C∈j|aC−bC|≤|j|/n≤(A/εj)e/2n−1/2E _C _j a_C-b_C ≤ _j /n≤(A/ _j)^e/2n^-1/2, and with εj−1=3εj _j-1=3 _j, W1(Γ^n,Γ)≤ 2εJ+32Ae/2n−1/2∑j=1Jεj 1−e/2.E\,W_1( _n, )\;≤\;2 _J+ 32\,A^e/2\,n^-1/2 _j=1^J _j^\,1-e/2. The rate. Let e≥3e≥ 3, so that εj1−e/2=εJ1−e/2 3−(J−j)(e/2−1)≤εJ1−e/2 3−(J−j)/2 _j^1-e/2= _J^1-e/2\,3^-(J-j)(e/2-1)≤ _J^1-e/2\,3^-(J-j)/2 and the sum is at most εJ1−e/2/(1−3−1/2)≤2.37εJ1−e/2 _J^1-e/2/(1-3^-1/2)≤ 2.37\, _J^1-e/2: the bound reads 2εJ+3.56εJ(A/εJ)e/2n−1/22 _J+3.56\, _J\,(A/ _J)^e/2n^-1/2. Put ε∗:=An−1/e :=A\,n^-1/e. If ε∗≥ε0 ≥ _0 the claim holds since W1≤diamS≤ε0W_1 ≤ _0. Otherwise take J:=⌊log3(ε0/ε∗)⌋≥0J:= _3( _0/ ) ≥ 0, so that ε∗≤εJ<3ε∗ ≤ _J<3 ; then (A/εJ)e/2n−1/2≤(A/ε∗)e/2n−1/2=1(A/ _J)^e/2n^-1/2≤(A/ )^e/2n^-1/2=1 and the bound is at most 5.56εJ<16.7ε∗≤17An−1/e5.56\, _J<16.7\, ≤ 17A\,n^-1/e. For e≤2e≤ 2 take J:=⌈12log3n⌉J:= 12 _3n , so that εJ≤ε0n−1/2 _J≤ _0n^-1/2; each summand is at most ε01−e/2≤A1−e/2 _0^1-e/2≤ A^1-e/2, so the bound is at most 2ε0n−1/2+32AJn−1/2≤A(72+34log3n)n−1/2≤4A(1+lnn)n−1/22 _0n^-1/2+ 32A\,J\,n^-1/2≤ A\,( 72+ 34 _3n)\,n^-1/2≤ 4A\,(1+ n)\,n^-1/2. ∎ Proof of Theorem 4.4. A single level. Let crK:ℳd→Kcr_K:M_d _K be the K-crop, which keeps the K heaviest atoms of a lift and renormalizes their weights, a measurable map with D(P,crKP)≤κTK(P)D(P,cr_KP)≤κ T_K(P) (Proposition A.19). The lifted design ΓK:=(crK)#Γ ^K:=(cr_K)_\# is that of an exactly K-sparse design, the empirical lifted design of the cropped sample is (crK)#Γ^n(cr_K)_\# _n, and by the triangle inequality W1(Γ^n,Γ)≤W1(Γ^n,(crK)#Γ^n)+W1((crK)#Γ^n,ΓK)+W1(ΓK,Γ),W_1( _n, )\;≤\;W_1 ( _n,(cr_K)_\# _n )+W_1 ((cr_K)_\# _n, ^K )+W_1( ^K, ), where the first term is at most κn−1∑iTK(Pi)κ n^-1 _iT_K(P_i) and the third at most κQTKκ\,E_QT_K, by the coupling along the crop; take expectations and apply the exactly sparse case, proved next, to the middle term. Exact sparsity. ΓK ^K is a probability measure on KA_K, of diameter at most κ (Proposition A.20), and N(ε/2,K,D)≤(C1R/ε)eN( /2,A_K,D)≤(C_1R/ )^e for ε≤κ ≤κ with e:=e(K)e:=e(K) (Theorem A.17; on a selector design the cropped support of the stratum replaces KA_K, with the count of the paragraph after Theorem A.18). Apply Proposition B.18 on S=KS=A_K with ε0:=κ _0:=κ: at the scales εj=3−jκ _j=3^-jκ, N(εj+1/2,K,D)≤(C1R/εj+1)e=(3C1R/εj)eN( _j+1/2,A_K,D)≤(C_1R/ _j+1)^e=(3C_1R/ _j)^e, so A:=3C1R≥κA:=3C_1R≥κ serves, and the middle term is at most 51C1Rn−1/e51C_1R\,n^-1/e for e≥3e≥ 3 and 12C1R(1+lnn)n−1/212C_1R\,(1+ n)\,n^-1/2 for e≤2e≤ 2. So C:=51C1C:=51C_1 serves in both cases, C=1734C=1734 on interval values. ∎ Remark B.19 (Graph-network bounds in this currency). The generalization bounds for graph networks bound a different quantity: the gap between the training and the test loss of one trained network, of fixed width and norm budget, over m i.i.d. graphs. The rate in m is m−1/2m^-1/2 whatever the graphs, and the graph enters the constant: the maximum degree d as dl−1d^\,l-1 for an l-step message-passing network, through the Lipschitz constant of the sum over neighbors, and the width h and the stable ranks of the weights as, in effect, the square root of the parameter count (Liao et al., 2021, Theorems 3.2 and 3.4), with the spectral norm of the diffusion matrix in place of d in Ju et al. (2023, Theorem 3.1). The class behind these bounds is a per-neighbor map shared across the neighbors, summed and read out, the weighted Deep Sets form of Section 1: parameter sharing makes its entropy independent of how many neighbors there are, and the sum makes the count a scale, so nothing in these bounds is a rate in the count, and at one layer with a mean the degree is absent altogether. To learn a member of G with such a network the width cannot stay fixed: a sum decomposition that represents every symmetric function of k elements needs latent dimension k (Wagstaff et al., 2022, Theorem 20), and the readout is then a function of that many coordinates. What no parametrization changes is the lower bound: on the stratum design of Appendix D at k atoms the generalization gap of the class is at least n−1/(2k−1)n^-1/(2k-1) (Theorem 4.4), and so is the risk of any learner (Theorem D.5), so the count is in the exponent. In shape, asked to learn the whole class, the graph-network route, one architecture serving every configuration with a readout of kmaxk_ coordinates, sees the design through its maximum count, Theorem 4.4 at k0=kmaxk_0=k_ , where for its own parametric class the degree is a constant in front of the parametric rate. On the flock, where kmax=68k_ =68, that bound is trivial (Section B.3). Where the count is a scale on both sides, in the accumulative class, ours is the mean effective count ζ¯ ζ of Corollary B.28, 1212 on the flock, against the maximum degree of the sample, 6868. The conservative linear sums, the members a sum network with a linear readout learns without the count, are count-free at the rate of Proposition B.20, and the profile is the price of the rest. The graph-network bounds, and Theorem 4.4 at the maximum count. The PAC-Bayes bound of Liao et al. (2021, Theorem 3.2) for an l-layer graph convolutional network of width h trained on m graphs has the form L(fw)≤LS,γ(fw)+O(B2dl−1l2hlog(lh)∏i‖Wi‖22∑i‖Wi‖F2/‖Wi‖22+log(ml/δ)γ2m),L_D(f_w)\;≤\;L_S,γ(f_w)+O ( B^2d^\,l-1l^2h (lh)\, _i \|W_i \|_2^2\, _i \|W_i \|_F^2/ \|W_i \|_2^2+ (ml/δ)γ^2m ), with d−1d-1 the maximum degree of the training graphs, and their Theorem 3.4 for the message-passing network carries the geometric series ((dC)l−1−1)/(dC−1)((dC)^l-1-1)/(dC-1) through a different weight term. Ju et al. (2023, Theorem 3.1) replace the degree by the spectral norm of the diffusion matrix, ‖PG‖2(l−1) \|P_G \|^2(l-1), at most d2(l−1)d^2(l-1) for a sum and at most one for a normalized mean, under a root that carries the width did_i of each layer and the stable ranks ri2∏jsj2r_i^2 _js_j^2 explicitly, summed over the layers. Three features are common to the line. The rate is m−1/2m^-1/2. The count enters as a scale, a power of the degree per layer that comes from the norm of a sum of d bounded vectors, and it is absent at one layer with a mean. And the width enters as roughly the square root of the parameter count, h times the stable rank, so that at a fixed architecture the bound is a constant over m m. These are bounds on the gap of one network, not on the risk of learning a class, and the class they are bounds for reads one neighbor at a time through a shared map and sums. Its entropy does not see the count because the parameters do not, which is the mechanism that makes the node count drop out (Section 2), and the same mechanism keeps the count out of the exponent. To learn a member of G with such a network on a design of maximum count k, the width has to grow with the accuracy. By Wagstaff et al. (2022, Theorem 20) a continuous sum decomposition ρ(∑iφ(xi))ρ( _i (x_i)) that approximates every continuous symmetric function of k scalar elements needs latent dimension at least k, and below it there is a Lipschitz symmetric function, affine in the sorted vector, that no decomposition approximates better than a constant does. So the readout ρ is a function of p≥kp≥ k coordinates, kdimk for vector elements. What the count costs does not depend on the parametrization: on the stratum design of Appendix D at k atoms the generalization gap of the class is at least n−1/(2k−1)n^-1/(2k-1) (Theorem 4.4, Lemma D.4), and the risk of learning likewise (Theorem D.5), so for the class the count is in the exponent whatever the architecture, while the conservative linear sums are count-free, at the rate of Proposition B.20. One architecture serves every configuration, so the count in the exponent is the maximum over the design, where Theorem 4.4 charges the count of the shell and Theorem B.14 the profile. On the flock the maximum support is 6868, and the bound at k0=68k_0=68 is the trivial value κ on the flock, under either exponent convention. B.4 Proof of Proposition B.20 Proposition B.20 (The linear sums are count-free). Let w=μw=μ, and let Δnlin(Q):=supφ∈(Qfφ−n−1∑i≤nfφ(Zi)) ^lin_n(Q):=E _ (E_Qf_ -n^-1 _i≤ nf_ (Z_i) ) be the generalization process of the linear sums (7). Write νz:=∑iμiδxi _z:= _i _i\, _x_i for the weighted value measure of a configuration z=(x,μ)z=(x,μ), a probability measure on V. Then Δnlin(Q)=W1ρ(1n∑i≤nνZi,QνZ)≤infε>0[ 2ε+RN(ε,,ρ)n]≤CRn−1dim+2, ^lin_n(Q)\;=\;E\,W_1^ρ ( 1n _i≤ n _Z_i,\ E_Q\, _Z )\;≤\; _ >0 [\,2 +R N( ,V,ρ)n\, ]\;≤\;C_V\,R\,n^- 1 +2, with dim the covering dimension of the value space and C_V a constant of its covering numbers alone. Neither d, nor the count, nor the profile enters. In one dimension, V an interval or the circle, Δnlin(Q)≤2R/n ^lin_n(Q)≤ 2R/ n. Remark B.21 (The two regimes: the count-free members and the rest). The gap of the linear sums is a transport on the value space, between the mean of n weighted value measures and its expectation, one level below the transport on the lifted space of Section 4, and the count is absent because a weighted mean of k atoms is one point of ()P(V) whatever k: crowding lowers the variance of the cell masses and helps. On the flock dim=1 =1 and the rate is n−1/2n^-1/2 at every crowding, the one-dimensional case being exact (Section B.4). This is the mechanism by which the parameter sharing of a sum network keeps the count out of its entropy (Remark B.19); a Lipschitz readout of p linear sums sees the law of a p-dimensional feature of the mean lift and stays count-free at the price of p, the bottleneck. The class pays the profile for everything beyond, the members that are nonlinear across atoms: the selected maximum, the count, and every functional whose score reads the height nonlinearly (Remark A.10). The proof follows. Proof of Proposition B.20. For z=(x,μ)z=(x,μ) and φ∈ , fφ(z)=∫φdνzf_ (z)= \,d _z, so Qfφ−n−1∑ifφ(Zi)=∫φd(ν¯−ν¯n)E_Qf_ -n^-1 _if_ (Z_i)= \,d( ν- ν_n) with ν¯:=QνZ ν:=E_Q\, _Z and ν¯n:=n−1∑iνZi ν_n:=n^-1 _i _Z_i, both probability measures on V. The score ball S contains a translate of every 11-Lipschitz function on V and ν¯−ν¯n ν- ν_n has mass zero, so the supremum over S is the supremum over the 11-Lipschitz functions, which is W1ρ(ν¯n,ν¯)W_1^ρ( ν_n, ν) by Kantorovich–Rubinstein duality on the compact space (,ρ)(V,ρ); this is the identity. For the bound let c1,…,cNc_1,…,c_N be an ε -net of V, N=N(ε,,ρ)N=N( ,V,ρ), and A1,…,ANA_1,…,A_N its Voronoi cells, every point within ε of its center. Moving a measure to the centers of its cells costs at most ε in W1W_1, and between two measures on the centers W1≤diam⋅12∑j|ν¯n(Aj)−ν¯(Aj)|W_1 · 12 _j ν_n(A_j)- ν(A_j) , so by the triangle inequality W1ρ(ν¯n,ν¯)≤ 2ε+R∑j≤N|ν¯n(Aj)−ν¯(Aj)|.W_1^ρ( ν_n, ν)\;≤\;2 +R _j≤ N ν_n(A_j)- ν(A_j) . The cell masses νZi(Aj) _Z_i(A_j), i≤ni≤ n, are i.i.d. in [0,1][0,1] with mean ν¯(Aj) ν(A_j), so |ν¯n(Aj)−ν¯(Aj)|≤VarνZ(Aj)/n≤ν¯(Aj)/nE ν_n(A_j)- ν(A_j) ≤ Var\, _Z(A_j)/n≤ ν(A_j)/n, using νZ(Aj)2≤νZ(Aj) _Z(A_j)^2≤ _Z(A_j), and by Cauchy–Schwarz ∑jν¯(Aj)/n≤N/n _j ν(A_j)/n≤ N/n. Taking expectations gives the first bound. With N(ε,,ρ)≤(CR/ε)dimN( ,V,ρ)≤(CR/ ) , the choice ε=RCdim/(dim+2)n−1/(dim+2) =R\,C /( +2)\,n^-1/( +2) balances the two terms and gives the rate, with a constant depending on C and dim alone. Nothing in the argument looks at how many atoms carry νz _z, which is where the count and the profile would enter. In one dimension the net is not needed. On an interval I of length ℓ with its length metric, W1(ν¯n,ν¯)=∫I|F¯n(t)−F¯(t)|tW_1( ν_n, ν)= _I F_n(t)- F(t) \,dt with F¯n(t)=n−1∑iνZi((−∞,t]) F_n(t)=n^-1 _i _Z_i((-∞,t]) an average of n i.i.d. variables in [0,1][0,1] of mean F¯(t) F(t), so |F¯n(t)−F¯(t)|≤1/(2n)E F_n(t)- F(t) ≤ 1/(2 n) and W1≤ℓ/(2n)=R/nE\,W_1≤ /(2 n)=R/ n. On the circle cut at a point, the geodesic distance is at most the distance on the cut interval, of length 2π=4R2π=4R, so W1S1≤W1IW_1^S^1≤ W_1^I and W1≤2R/nE\,W_1≤ 2R/ n. On the flock this is the rate of the linear sums at every crowding. ∎ B.5 Regression Remark B.22 (Regression with the absolute loss). The two-sample distance is the maximum discrepancy of the unit ball on the halves, W1(Γ^A,Γ^B)=supf∈1|Af−Bf|W_1( _A, _B)= _f _1 _Af-E_Bf (Section 4.2), which Theorem 4.6 carries to the population with one concentration term. For regression with the absolute loss on a target f∗∈1f^* _1, the loss |f−f∗| f-f^* of a predictor f∈1f _1 is itself a member, of 2G_2, so with RAR_A and RBR_B the empirical risks on the halves, supf∈1|RA(f)−RB(f)|≤2W1(Γ^A,Γ^B) _f _1 R_A(f)-R_B(f) ≤ 2\,W_1( _A, _B): the certificate bounds the generalization gap of every predictor at once, at twice its value, and of the empirical risk minimizer in particular. For a target with m components that is 11-Lipschitz into ℓ2 ^2, under the Euclidean loss, a predictor whose components are each in 1G_1 is m m-Lipschitz into ℓ2 ^2, so the loss |f−f∗|2 f-f _2 has constant m+1 m+1 and the factor is 1+m1+ m in place of 22 (and 2m2 m for a target that is only componentwise in 1G_1), which is 1+21+ 2 for the flock’s update direction, jointly Lipschitz by Proposition C.7 (Section C.1). For labels in a range of length b the loss |f(z)−y| f(z)-y is 11-Lipschitz in the pair (z,y)(z,y) under D+|y−y′|D+ y-y , so the same holds with the transport between the labelled halves in that metric, the same matching with the label distance added to each pair’s cost, and the concentration term at the diameter κ+bκ+b. A loss that is ℓ1 _1-Lipschitz in the prediction costs the factor ℓ1 _1 in place of 11, by Talagrand’s contraction lemma (Mohri et al., 2018). Remark B.23 (The square loss). For the square loss the rate is a different one: least squares has excess risk at the square of a rate, and what governs it is the entropy of the class under the design rather than the transport. The same shells give that entropy and the same fibers give the matching lower bounds, so we sketch the route and omit the statements, the regression that the paper’s number concerns being the absolute-loss one (Remark B.22). Entropy. Cover each shell of Definition B.4 by the celling of Theorem B.14 at its own resolution εk _k, assign to every cell a value from a grid of step εk _k on [−b,b][-b,b], and the value 00 beyond the cutoff shell: this is a cover of the members of 1G_1 bounded by b in L2(Γ)L_2( ), of log-cardinality ∑k≤KNk(εk)ln(3b/εk) _k≤ KN_k( _k) (3b/ _k) at a radius η with η2≤94∑k≤Kwkεk2+b2GK(θK)η^2≤ 94 _k≤ Kw_k _k^2+b^2G_K( _K), on interval values the entropy of a Lipschitz class (Kolmogorov and Tikhomirov, 1959) in the dimension 2K−12K-1 of the cutoff shell, with the mass beyond it charged at b and no dependence on d. Upper bound. Least squares over the cover is the shell-binning regressogram, on each cell the grid value nearest the cell’s sample mean of Y, the cells being observable since TkT_k of a sample is computable from the sample. Bernstein’s inequality with a union bound over the cover gives an excess risk of order η2+b2lnN/nη^2+b^2 N/n over the best member, and the resolution that balances the two terms gives (lnn/n)2/(2k0+1)( n/n)^2/(2k_0+1) at exact sparsity, the nonparametric rate of a Lipschitz class in the dimension 2k0−12k_0-1 of the cutoff shell, and e−2βlnn/7e^-2 β n/7 on an exponential profile, the shape of Theorem 4.5’s rate with another constant in the exponent. Lower bound. On a fiber of fixed weights the metric is bi-Lipschitz to a weighted supremum distance with the weights as semi-axes (Lemma D.1), so bumps of height h with disjoint supports in the cells of the stratum design of Definition D.2 or of the fiber design of Theorem D.6 are 11-Lipschitz for D, and Assouad’s hypercube (Tsybakov, 2009, §2.7) on Gaussian regression over them gives a minimax risk of n−2/(2k+1)n^-2/(2k+1) on the stratum design, n−2/(k+2)n^-2/(k+2) on a selector design at count k and e−8βlnne^- 8β n on the fiber design, with constants polynomial in k; on a mixture of separated populations the hypercubes combine, and the bound is the mass-weighted sum of the populations’ risks at their effective sample sizes npjnp_j. By this sketch the square loss learns at the nonparametric rate of the cutoff shell’s dimension, matched from below in the exponent at exact sparsity and in shape on an exponential profile. B.6 Exponential profiles: proof of Theorem 4.5 The upper bound is proved here, with its constant 2e3β/2Λ2e^3β/2 , Λ:=CR∨2κA :=CR 2κ A for the constant C of Theorem 4.4 (17341734 on interval values), and 2e3βΛ2e^3β with the selector count’s constant in place of C1C_1 on a selector design. The lower bound of Theorem 4.5 is Theorem D.6 in Appendix D, with the constant γ2R/16γ^2R/16, γ:=1−e−βγ:=1-e^-β. Proof. Let n≥2n≥ 2 and d≥2d≥ 2: the bound at n=1n=1 exceeds the diameter κ, and at d=1d=1 the lift is the interval, where Δn(Q)≤2R/n≤2Reβ/4e−βlnn/2 _n(Q)≤ 2R/ n≤ 2Re^β/4e^- β n/2 by the one-dimensional bound of Proposition B.20. Put x:=2lnn/βx:= 2 n/β and K:=d∧(⌈x/2⌉∨2)K:=d ( x/2 2), so that e(K)=2K−1≥3e(K)=2K-1≥ 3, that βK≥βlnn/2β K≥ β n/2 unless K=dK=d, and that 2K−1≤x+32K-1≤ x+3 in every case. Theorem 4.4 at this K, with its constant C, gives Δn(Q)≤ 2κQTK+CRn−1/(2K−1)≤ 2κAe−βK+CRe−lnn/(x+3), _n(Q)\;≤\;2κ\,E_QT_K+CR\,n^-1/(2K-1)\;≤\;2κ A\,e^-β K+CR\,e^- n/(x+3), where the first term vanishes at K=dK=d and is at most 2κAe−βlnn/22κ A\,e^- β n/2 otherwise, and lnn/(x+3)≥lnn/x−3lnn/x2=βlnn/2−3β/2 n/(x+3)≥ n/x-3 n/x^2= β n/2-3β/2. So Δn(Q)≤(2κA+CRe3β/2)e−βlnn/2≤2e3β/2Λe−βlnn/2 _n(Q)≤(2κ A+CR\,e^3β/2)\,e^- β n/2≤ 2e^3β/2 \,e^- β n/2. On a selector design, e(K)=Ke(K)=K: take x:=lnn/βx:= n/β and K:=d∧(⌈x⌉∨3)K:=d ( x 3), so that e(K)≥3e(K)≥ 3 once d≥3d≥ 3, K≤x+3K≤ x+3 and lnn/(x+3)≥βlnn−3β n/(x+3)≥ β n-3β; for d≤2d≤ 2 the design is exactly d-sparse and the clause e≤2e≤ 2 of Theorem 4.4 at K=dK=d, CR(1+lnn)n−1/2CR(1+ n)n^-1/2, is below the claimed rate at a constant depending on β. ∎ The shells route. The same rate with the constant 13 13, at n≥n0n≥ n_0, follows from Theorem B.14 at a constant profile with the trim at the diameter, and the computation serves the accumulative class (Corollary B.30), whose theorem has no crop clause. Write L:=lnnL:= n, let n0:=exp(48β∨9ln2Λ′/β)n_0:= (48β 9 ^2 /β), and let C1:=136C_1:=136, the constant of the accumulative count in Section B.7, which dominates the 3434 of Theorem A.17 and lets one Λ′ serve both classes, so that Nk(ε)≤(C1R/ε)2k−1N_k( )≤(C_1R/ )^2k-1 for ε≤κ ≤κ (Theorem A.17) and Λ′:=C1R∨2κA∨e :=C_1R 2κ A e, a different constant from the crop route’s Λ . Apply Theorem B.14 with the constant profile at ε:=Λ′e−u,θ:=ε/(2κ),u:=13βL, := e^-u, θ:= /(2κ), u:= 13 β L, so that ln(Λ′/ε)=u ( / )=u and Nk(ε)≤e(2k−1)uN_k( )≤ e^(2k-1)u. Three consequences of n≥n0n≥ n_0, that is 48β≤L48β≤ L and 9ln2Λ′≤βL9 ^2 ≤β L, are used repeatedly: (a) u≤L/20u≤ L/20, from β≤L/48β≤ L/48; (b) β≤uβ≤ u and 1≤lnΛ′≤u1≤ ≤ u, from u≥β⋅48β/3≥2βu≥ β· 48β/3≥ 2β and u≥9ln2Λ′/3=lnΛ′u≥ 9 ^2 /3= ; (c) κ≤Λ′κ≤ , since C1≥2C_1≥ 2, and L≥48⋅9lnΛ′≥20L≥ 48· 9\, ≥ 20. In particular ε=Λ′e−u≤1≤κ = e^-u≤ 1≤κ, using R≥12R≥ 12 (so κ=2R≥1κ=2R≥ 1), so the count bound applies and θ≤12θ≤ 12. Markov’s inequality at the working threshold converts the moment profile into the quantile the theorem consumes: Gk(θ)≤ 1∧Ae−βkθ= 1∧e−β(k−h),h:=β−1ln(2κA/ε)∈[0,u/β].G_k(θ)\;≤\;1 Ae^-β kθ\;=\;1 e^-β(k-h), h:=β^-1 (2κ A/ )∈[0,\,u/β]. Set m:=⌈h⌉+1≤u/β+2m:= h +1≤ u/β+2 and apply the theorem with K:=(⌈3u/β⌉∨1)∧dK:=( 3u/β 1) d. Trim: either K=dK=d, or βK≥3uβ K≥ 3u and κGK(θ)≤κe−β(K−h)≤κe−2u≤Λ′e−uκ\,G_K(θ)≤κ\,e^-β(K-h)≤κ\,e^-2u≤ e^-u. Shells: wk≤Gk−1(θ)≤e−β(k−m)+w_k≤ G_k-1(θ)≤ e^-β(k-m)_+, as k−1−h≥k−mk-1-h≥ k-m, so the k-th shell term is at most n−1/2eψ(k)n^-1/2e^ψ(k) with ψ(k):=(2k−1)u2−β2(k−m)+ψ(k):=(2k-1) u2- β2(k-m)_+. Split the sum at m: below m it is geometric with ratio eu≥e^u≥ e; above m, geometric with ratio eu−β/2≥eu/2≥e1/2e^u-β/2≥ e^u/2≥ e^1/2 by (b), and we extend it upward to K~:=⌈3u/β⌉≥K K:= 3u/β ≥ K. Hence ∑k=1KwkNkn≤2eψ(m)+3eψ(K~)n, _k=1^K w_kN_kn\;≤\; 2\,e^ψ(m)+3\,e^ψ( K) n, ψ(m)≤u2β+3u2=L9+3u2,ψ(K~)≤3u2β+u2−(u−β)=L3−u2+β, ψ(m)\;≤\; u^2β+ 3u2\;=\; L9+ 3u2, ψ( K)\;≤\; 3u^2β+ u2- (u-β )\;=\; L3- u2+β, using m≤u/β+2m≤ u/β+2, K~≤3u/β+1 K≤ 3u/β+1, and K~−m≥2u/β−2≥0 K-m≥ 2u/β-2≥ 0 by (b). The middle term of the theorem is thus at most Λ′(eψ(m)+32eψ(K~))/n (e^ψ(m)+ 32e^ψ( K) )/ n, and each piece is at most e−ue^-u: ψ(m)−L2≤L9+3u2−L2≤−L4≤−u, ψ(m)- L2\;≤\; L9+ 3u2- L2\;≤\;- L4\;≤\;-u, ln32+ψ(K~)−L2≤L48+L3+L48−u2−L2=−L8−u2≤−u, 32+ψ( K)- L2\;≤\; L48+ L3+ L48- u2- L2\;=\;- L8- u2\;≤\;-u, using 3u2≤3L40 3u2≤ 3L40, β≤L/48β≤ L/48, ln32≤L/48 32≤ L/48, and u≤L/4u≤ L/4, all from (a)–(c). So the middle term is at most 2Λ′e−u2 e^-u, and the total is 2Λ′e−u+2Λ′e−u+Λ′e−u=5Λ′e−u2 e^-u+2 e^-u+ e^-u=5 e^-u. Remark B.24 (The exponential rate, read in full). The rate e−cβlnne^-c β n is slower than every polynomial in n and faster than every power of lnn n. The optimizing cutoff is Kn≍lnn/βK_n n/β, so the dimension the rate sees at sample size n grows without bound, like lnn n, and no fixed dimension explains the answer. In sample-complexity form, accuracy ε costs n(ε)=exp(c′ln2(1/ε)/β)n( )= (c ^2(1/ )/β ), quasi-polynomial in 1/ε1/ , between every finite-dimensional class and generic nonparametrics, and free of d. The profile is a statement about the mean tail and costs nothing for being one: the crop pays each configuration its own tail and the bound takes the mean (Remark B.25). Since every design is exactly d-sparse, Theorem 4.4 at K=dK=d holds unconditionally and the bound is the minimum of the two branches, which cross at lnn=β(2d−1)2/2 n=β(2d-1)^2/2; on the measured flock, with d in the thousands, the crossover is unreachable at any rate the profiles of Section 5.1 show. Two qualifiers. The content of these bounds is the exponent and the absence of d, and the number at hand is the certificate’s. And the rate is a single level’s, Theorem 4.4 at its best K with the mean tail as bias: what the shells add is the across-configuration axis, where each shell pays its own rate at its own mass. Remark B.25 (The constants in the exponent). The constant 1/21/ 2 is the crop’s. A trim at the diameter, whether Theorem B.14 without its crop clause at a constant profile or the finite-sample dimension of Weed and Bach (2019, Prop. 10) on the trimmed coverings of Section A.5, needs the tail to fall below the working threshold on all but the trimmed mass, which a mean tail gives only through Markov’s inequality, at twice the cutoff: that route returns 12 12 at best, and 13 13 as computed (the shells route of Section B.6, at n≥n0n≥ n_0 and R≥12R≥ 12), and the factor is the trim’s price, not the profile’s; a pointwise bound Tk≤Ae−βkT_k≤ Ae^-β k at every configuration would hand the trim its 1/21/ 2 back, and nothing here needs one. The lower bound of Theorem D.6 has 2 2. B.7 The accumulative class The theorems above are instances of Proposition B.11, and everything specific to the metric entered through four inputs: the stratum covering, the crop cost, the diameter of (ℳd,D)(M_d,D), and the qualitative facts behind Lemma B.9. This subsection instantiates the argument for the accumulative class ~L G_L, the Lipschitz ball of D~ D, the matching distance with the shares si=μi/μmaxs_i= _i/ _ as heights. The lifts of the two instances are bijections from the same quotient of C, so we regard D~ D as a second metric on the same set ℳdM_d, with TkT_k, μmax _ and the shells the same functions and sets, and write |P|:=∑isi=1/μmax(P) P := _is_i=1/ _ (P) for the total height of P, the effective count, with the leader at height 11. Three inputs are imported from Section A.5. The crop cost: for μ μ the k-crop, D~(Φ(x,μ),Φ(x,μ^))≤RTk/μmax=κTk/(2μmax) D( (x,μ), (x, μ))≤ R\,T_k/ _ =κ T_k/(2 _ ), and μmax(μ^)=μmax/(1−Tk)≥μmax _ ( μ)= _ /(1-T_k)≥ _ , the crop does not decrease the top weight (Proposition A.19 and its proof). The qualitative facts: (ℳd,D~)(M_d, D) is compact, TkT_k and μmax _ are continuous on it, and duality holds for W~1 W_1, the Kantorovich distance on measures on ℳdM_d with ground metric D~ D (Proposition A.20); moreover D~(P,Q)≤R(|P|+|Q|) D(P,Q)≤ R( P + Q ), by sending every atom of both to the apex, so for a fixed one-atom lift P0P_0, ϱ0(P):=D~(P,P0)≤R(1μmax(P)+1)≤κμmax(P), _0(P):= D(P,P_0)\;≤\;R ( 1 _ (P)+1 )\;≤\; κ _ (P), while diam(ℳd,D~)≤2Rddiam(M_d, D)≤ 2Rd and no better in general, the count being 11-Lipschitz for D~ D and ranging over [R,Rd][R,Rd]. The celled covering: for m≥0m≥ 0 let k[m]:=P∈k:μmax(P)≥2−m=P∈k:|P|≤2mA_k^[m]:=\P _k: _ (P)≥ 2^-m\=\P _k: P ≤ 2^m\; then for every k≤dk≤ d, m≥0m≥ 0 and ε>0 >0, by the second display of Theorem A.18, N(2m−1ε,k[m],D~)≤N~k(ε):=N(ε4,,ρ)k⋅k(1+16Rε)k−1,N (2^m-1 ,\;A_k^[m],\; D )\;≤\; N_k( )\;:=\;N ( 4,V,ρ )^k· k (1+ 16R )^k-1, with net points in kA_k: resolving the cell |P|≤2m\ P ≤ 2^m\ at scale 2m−1ε2^m-1 costs the conservative count up to the factor k and a constant, independently of m. On interval values and ε≤κ ≤κ, N~k(ε)≤(68R/ε)2k−1 N_k( )≤(68R/ )^2k-1. Definition B.26 (Concentration index, cells, concentration weights). For P∈ℳdP _d let m(P):=⌈log2(1/μmax(P))⌉∈0,1,…m(P):= _2(1/ _ (P)) ∈\0,1,…\, so that 1/μmax(P)≤2m(P)<2/μmax(P)1/ _ (P)≤ 2^m(P)<2/ _ (P); m(P)=0m(P)=0 exactly on point masses. For the stratified shells k Sh_k of Definition B.4 set k,m:=k∩m(P)=m,wk,m:=Γ(k,m), Sh_k,m:= Sh_k∩\m(P)=m\, w_k,m:= ( Sh_k,m), the first and second concentration weights of shell k, ζk:=∑m≥02mwk,m=∫k2m(P)Γ,ξk:=∑m≥04mwk,m, _k:= _m≥ 02^m\,w_k,m= _ Sh_k2^m(P)\,d , _k:= _m≥ 04^m\,w_k,m, and the count-weighted trim HK:=∫BK(θK)cdΓμmax∈[GK(θK),dGK(θK)].H_K:= _B_K( _K)^c d _ \;∈\; [\,G_K( _K),\;d\,G_K( _K)\, ]. Since μmax≥(1−θk)/k _ ≥(1- _k)/k on k Sh_k, only indices m<log2(2k/(1−θk))m< _2(2k/(1- _k)) occur there, at most Mk:=⌈log2(2k/(1−θk))⌉M_k:= _2(2k/(1- _k)) cells per shell, and, from 1/μmax≤2m(P)<2/μmax1/ _ ≤ 2^m(P)<2/ _ , wk≤ζk≤ 2∫kdΓμmax≤2k1−θkwk,ξk≤ 4∫kdΓμmax2≤(2k1−θk)2wk.w_k\;≤\; _k\;≤\;2 _ Sh_k d _ \;≤\; 2k1- _k\,w_k, _k\;≤\;4 _ Sh_k d _ ^2\;≤\; ( 2k1- _k )^2w_k. The design’s concentration moments are ζ¯:=Q[μmax−1] ζ:=E_Q[ _ ^-1] and ξ¯:=Q[μmax−2] ξ:=E_Q[ _ ^-2]. Theorem B.27 (Generalization over the accumulative class). For every design Q and every resolution profile, with the shells and masses of Definition B.4 and the cells, weights and trim of Definition B.26, Δ~n(Q)=W~1(Γ^n,Γ)≤ 2∑k=1Kζkεk+κ∑k=1K∑m<Mk2mwk,mN~k(εk)n+ 2κHK. _n(Q)\;=\;E\, W_1( _n, )\;≤\;2 _k=1^K _k\, _k\;+\;κ _k=1^K _m<M_k2^m w_k,m\, N_k( _k)n\;+\;2κ\,H_K. Supplementary bounds: the floor is at most 4∑kεk∫kμmax−1Γ4 _k _k _ Sh_k _ ^-1\,d , hence at most 4εζ¯4 ζ for the constant profile; the inner sum is at most Mkξk M_k\, _k; and HK≤ξ¯GK(θK)H_K≤ ξ\,G_K( _K). Proof. The identity is duality for W~1 W_1 and Theorem 3.2 for the accumulative instance, as in the proof of Theorem B.14. Apply Proposition B.13 on (,ϱ)=(ℳd,D~)(X, )=(M_d, D) with reference point P0P_0 a point mass, B=BK(θK)B=B_K( _K), and the groups Sk,m=k,mS_k,m= Sh_k,m, k≤Kk≤ K, m<Mkm<M_k, Borel, of masses wk,mw_k,m. Reference costs: on Sk,mS_k,m, 1/μmax≤2m1/ _ ≤ 2^m, so ϱ0≤κ/μmax≤2mκ=:ϱk,m _0≤κ/ _ ≤ 2^mκ=: _k,m; on BcB^c, ϱ0≤κ/μmax _0≤κ/ _ , so ∫Bcϱ0Γ≤κHK _B^c _0\,d ≤κ\,H_K. Cellings: a point P∈k,mP∈ Sh_k,m has Tk(P)≤θkT_k(P)≤ _k, so its k-crop P P satisfies D~(P,P^)≤κθk/(2μmax(P))≤2m−1κθk=2m−2εk D(P, P)≤κ _k/(2 _ (P))≤ 2^m-1κ _k=2^m-2 _k, with P^∈k P _k and μmax(P^)≥μmax(P)≥2−m _ ( P)≥ _ (P)≥ 2^-m, that is P^∈k[m] P _k^[m]; and k[m]A_k^[m] has a net of N~k(εk) N_k( _k) points at radius 2m−1εk≤2mεk−2m−2εk2^m-1 _k≤ 2^m _k-2^m-2 _k, so Lemma B.12 gives a (2mεk,N~k(εk))(2^m _k, N_k( _k))-celling of Sk,mS_k,m. With rk,m=2mεkr_k,m=2^m _k, Nk,m=N~k(εk)N_k,m= N_k( _k) and ϱk,m=2mκ _k,m=2^mκ the proposition’s three terms are the display: 2∑k,m2mεkwk,m=2∑kζkεk2 _k,m2^m _k\,w_k,m=2 _k _k _k, then ∑k,m2mκwk,mN~k/n _k,m2^mκ w_k,m N_k/n, then 2κHK2κ H_K. The supplementary bounds: 2m(P)<2/μmax(P)2^m(P)<2/ _ (P) for the floor; Cauchy–Schwarz over m for the inner sum, ∑m2mwk,m≤Mk∑m4mwk,m _m2^m w_k,m≤ M_k _m4^mw_k,m; and Cauchy–Schwarz for HK≤∫μmax−2ΓΓ(BKc)H_K≤ _ ^-2\,d \, (B_K^c). ∎ Against Theorem B.14 on the same design and the same shells, the profile, the shells and the masses are identical: the accumulative class does not induce a different partition, it is the same partition with cell (k,m)(k,m) resolved at scale 2mεk2^m _k instead of εk _k. The counts are identical in shape. The price of the semantics is a count-weighting of all three terms, the floor by the mean inverse top weight on the bulk, the fluctuation by the second concentration moment, the trim by the count of the trimmed configurations. This is forced, not slack: the count R/μmaxR/ _ is a member of ~1 G_1 (Proposition A.14), so two configurations of counts c and c′c are at D~ D-distance at least R|c−c′|R c-c , and mass misplaced across counts must pay for the difference. In the conservative class the total height is 11 everywhere and all three weights are 11. Corollary B.28 (Exact sparsity, accumulative class, the single-partition route). If Q(Tk0=0)=1Q(T_k_0=0)=1 then 1/μmax≤k01/ _ ≤ k_0 Q-a.s., ζ¯≤k0 ζ≤ k_0, and with K=k0K=k_0 and the constant profile, for every ε>0 >0, Δ~n(Q)≤ 4εζ¯+2k0κk0MN~k0(ε)n,M:=⌈log2(2k0)⌉. _n(Q)\;≤\;4 \, ζ+2k_0κ k_0\,M\, N_k_0( )n, M:= _2(2k_0) . On interval values, optimizing ε , for n≥682k0+1k03n≥ 68^2k_0+1k_0^3, so that the optimizing ε is at most κ, Δ~n(Q)≤CRζ¯n−12k0+1≤CRk0n−12k0+1 _n(Q)\;≤\;C\,R\; ζ\;n^- 12k_0+1\;≤\;C\,R\,k_0\;n^- 12k_0+1 with C absolute. Proof. Gk0(θ)≤Gk0(0)=0G_k_0(θ)≤ G_k_0(0)=0 kills the trim, Hk0=0H_k_0=0. On the support of Q at most k0k_0 weights are positive, so 1/μmax≤k01/ _ ≤ k_0 there, 2m<2k02^m<2k_0 on every occupied cell, at most M values of m occur, and the fluctuation term is at most 2k0κ∑k,mwk,mN~k/n≤2k0κk0MN~k0(ε)/n2k_0κ _k,m w_k,m N_k/n≤ 2k_0κ k_0M\, N_k_0( )/n, by Cauchy–Schwarz over the at most k0Mk_0M occupied cells, ∑k,mwk,m=1 _k,mw_k,m=1, and monotonicity of the counts in k. For the rate insert MN~k0(ε)≤(C1R/ε)2k0−1M\, N_k_0( )≤(C_1R/ )^2k_0-1 with C1=136C_1=136, from N~k0(ε)≤(68R/ε)2k0−1 N_k_0( )≤(68R/ )^2k_0-1 and M≤2k0≤22k0−1M≤ 2k_0≤ 2^2k_0-1, and choose ε:=C1R(k03ζ¯2n)12k0+1≤κ; \;:=\;C_1R\, ( k_0^3 ζ^2n ) 12k_0+1\;≤\;κ; both terms are then at most 4(C1+1)R(k03)1/(2k0+1)ζ¯(2k0−1)/(2k0+1)n−1/(2k0+1)4(C_1+1)R\,(k_0^3)^1/(2k_0+1)\, ζ^(2k_0-1)/(2k_0+1)\,n^-1/(2k_0+1), using κ=2Rκ=2R; conclude with ζ¯≥1 ζ≥ 1 and supk≥1(k3)1/(2k+1)≤2 _k≥ 1(k^3)^1/(2k+1)≤ 2. ∎ Lemma B.29 (Concentration moments from the profile). Let A≥1A≥ 1, β>0β>0, and k1:=⌈β−1ln(2A)⌉∨1k_1:= β^-1 (2A) 1. (i) If Q[Tk]≤Ae−βkE_Q[T_k]≤ Ae^-β k for all k, then ζ¯≤2k1+2+2/β ζ≤ 2k_1+2+2/β, and more generally ζ¯≤2k+4Ae−βk/(1−e−β) ζ≤ 2k+4Ae^-β k/(1-e^-β) for every k≥1k≥ 1. (i) Under the same hypothesis, ξ¯≤4k12+4(2k1+1)(1+1/β)+8/β2 ξ≤ 4k_1^2+4(2k_1+1)(1+1/β)+8/β^2, so ξ¯≤4k1+2+4/β ξ≤ 4k_1+2+4/β; and for every K≥1K≥ 1 and θ, ∫BK(θ)cdΓμmax≤ 2KGK(θ)+4Ae−βK1−e−β. _B_K(θ)^c d _ \;≤\;2K\,G_K(θ)+ 4Ae^-β K1-e^-β. The concentration moments are constants of order β−1(1+ln2A)β^-1(1+ 2A), the effective neighbor count, and its square, independent of d and of n. Proof. For any k≥1k≥ 1, μmax≥(1−Tk)/k _ ≥(1-T_k)/k, so 1/μmax>2k⊆Tk>12\1/ _ >2k\ \T_k> 12\, and Markov gives Γ(1/μmax>2k)≤2Ae−βk (1/ _ >2k)≤ 2Ae^-β k. (i) Layer cake: [μmax−1]≤2k+∑j≥k2Pr(μmax−1>2j)≤2k+4A∑j≥ke−βjE[ _ ^-1]≤ 2k+ _j≥ k2 ( _ ^-1>2j)≤ 2k+4A _j≥ ke^-β j. At k=k1k=k_1, 4Ae−βk1≤24Ae^-β k_1≤ 2 and (1−e−β)−1≤1+1/β(1-e^-β)^-1≤ 1+1/β. (i) For t:=1/μmaxt:=1/ _ and any integer j0≥1j_0≥ 1, pointwise t≤2j0+2∑j≥j0t>2jt≤ 2j_0+2 _j≥ j_01\t>2j\ and t2≤4j02+∑j≥j0(8j+4) 1t>2jt^2≤ 4j_0^2+ _j≥ j_0(8j+4)\,1\t>2j\. Integrating the second against Γ at j0=k1j_0=k_1: ξ¯≤4k12+8A∑j≥k1(2j+1)e−βj≤4k12+8Ae−βk1[(2k1+1)(1+1/β)+2/β2] ξ≤ 4k_1^2+8A _j≥ k_1(2j+1)e^-β j≤ 4k_1^2+8Ae^-β k_1 [(2k_1+1)(1+1/β)+2/β^2 ], using ∑i≥0e−βi≤1+1/β _i≥ 0e^-β i≤ 1+1/β and ∑i≥0ie−βi=e−β/(1−e−β)2≤1/β2 _i≥ 0ie^-β i=e^-β/(1-e^-β)^2≤ 1/β^2, and 8Ae−βk1≤48Ae^-β k_1≤ 4; the square root by 2ab≤a+b2 ab≤ a+b. Integrating the first against Γ restricted to BK(θ)cB_K(θ)^c at j0=Kj_0=K gives the last display. ∎ Corollary B.30 (Exponential profiles, accumulative class, the single-partition route). Let =[−R,R]V=[-R,R] with R≥12R≥ 12, let A and β be as in Theorem 4.5, Λ′:=136R∨2κA∨e :=136R 2κ A e as in the shells route of Section B.6, k1k_1 as in Lemma B.29, and let n0:=exp(48β∨9ln2Λ′/β)n_0:= (48β 9 ^2 /β). Then for n≥n0n≥ n_0, under an exponential profile at rate β, Δ~n(Q)≤32(k1+1+1/β)Λ′e−13βlnn _n(Q)≤ 32\,(k_1+1+1/β)\, \,e^- 13 β n. Proof. Run the shells route of Section B.6 on Theorem B.27 with the constant profile: ε:=Λ′e−u := e^-u, θ:=ε/(2κ)≤12θ:= /(2κ)≤ 12 (as ε≤1≤κ ≤ 1≤κ), u=13βLu= 13 β L, L=lnnL= n; the consequences (a)–(c) hold verbatim, and MkN~k(ε)≤(C1R/ε)2k−1≤e(2k−1)uM_k\, N_k( )≤(C_1R/ )^2k-1≤ e^(2k-1)u with C1=136C_1=136, since Mk≤⌈log24k⌉≤2kM_k≤ _24k ≤ 2k for θ≤12θ≤ 12. Take K:=(⌈3u/β⌉∨1)∧dK:=( 3u/β 1) d as there, so that either K=dK=d or βK≥3uβ K≥ 3u and GK(θ)≤e−2uG_K(θ)≤ e^-2u. Floor: at most 4εζ¯=4Λ′ζ¯e−u4 ζ=4 ζe^-u. Fluctuation: by the supplementary bound of the theorem and ξk≤4ξ¯ _k≤ 4 ξ it is at most 2κξ¯∑k≤KMkN~k(ε)/n≤2κξ¯n−1/2∑k≤Ke(k−12)u≤4κξ¯e(K−12)u−L/22κ ξ _k≤ K M_k N_k( )/n≤ 2κ ξ\,n^-1/2 _k≤ Ke^(k- 12)u≤ 4κ ξ\,e^(K- 12)u-L/2, a geometric sum with ratio eu≥e^u≥ e, and (K−12)u≤3u2/β+u/2=L/3+u/2(K- 12)u≤ 3u^2/β+u/2=L/3+u/2, so the exponent is at most −L/6+u/2≤−u-L/6+u/2≤-u as u≤L/20u≤ L/20: the term is at most 4Λ′ξ¯e−u4 ξ\,e^-u. Trim: if K<dK<d, Lemma B.29(i) gives 2κHK≤4κKe−2u+8κA(1+1/β)e−βK≤4Λ′(3u/β+1)e−2u+4Λ′(1+1/β)e−3u2κ H_K≤ 4κ Ke^-2u+8κ A(1+1/β)e^-β K≤ 4 (3u/β+1)e^-2u+4 (1+1/β)e^-3u, using κ≤Λ′κ≤ , 2κA≤Λ′2κ A≤ , K≤3u/β+1K≤ 3u/β+1 and βK≥3uβ K≥ 3u; since ue−u≤e−1ue^-u≤ e^-1 and e−u≤e−1e^-u≤ e^-1, this is at most 4Λ′e−u[(3e−1+e−2)/β+2e−1]≤8Λ′(1+1/β)e−u4 e^-u [(3e^-1+e^-2)/β+2e^-1 ]≤ 8 (1+1/β)e^-u. If K=dK=d the trim is zero. Total: at most 4Λ′e−u[ζ¯+ξ¯+2+2/β]4 e^-u\,[ ζ+ ξ+2+2/β], and Lemma B.29(i)–(i) give ζ¯+ξ¯≤6k1+4+6/β ζ+ ξ≤ 6k_1+4+6/β, so the bracket is at most 8(k1+1+1/β)8(k_1+1+1/β). ∎ Corollary B.31 (Exact sparsity, accumulative class, the multiscale route). Let =[−R,R]V=[-R,R]. If Q(Tk0=0)=1Q(T_k_0=0)=1 with k0≥2k_0≥ 2, then Δ~n(Q)≤CRk0n−12k0−1,C:=17⋅408=6936. _n(Q)\;≤\;C\,R\,k_0\;n^- 12k_0-1, C:=17· 408=6936. Conversely Δ~n(Q)≥12Δn(Q) _n(Q)≥ 12 _n(Q) on every design, since 1⊆~2G_1 G_2 (Proposition A.14), so on the stratum design Qk0Q_k_0 of Lemma D.4 the exponent 1/(2k0−1)1/(2k_0-1) holds from both sides. Proof. On the support of Q at most k0k_0 weights are positive, so 1/μmax≤k01/ _ ≤ k_0 there and every lift lies in S:=k0[M]S:=A_k_0^[M] with M:=⌈log2k0⌉M:= _2k_0 , so that k0≤2M<2k0k_0≤ 2^M<2k_0. Its diameter: D~(P,P′)≤R(|P|+|P′|)≤2Rk0=:ε0 D(P,P )≤ R( P + P )≤ 2Rk_0=: _0. Its covering: at a radius ϱ≤2M−1κ ≤ 2^M-1κ put ε:=ϱ/2M−1≤κ := /2^M-1≤κ, so that N(ϱ,S,D~)≤N~k0(ε)≤(68R/ε)2k0−1=(68R 2M−1/ϱ)2k0−1≤(68Rk0/ϱ)2k0−1N( ,S, D)≤ N_k_0( )≤(68R/ )^2k_0-1=(68R\,2^M-1/ )^2k_0-1≤(68Rk_0/ )^2k_0-1. Apply Proposition B.18 on S with εj:=3−jε0 _j:=3^-j _0: the radii εj+1/2=εj/6≤ε0/6=Rk0/3≤2M−1κ _j+1/2= _j/6≤ _0/6=Rk_0/3≤ 2^M-1κ are admissible, and N(εj+1/2,S,D~)≤(6⋅68Rk0/εj)2k0−1=(A/εj)eN( _j+1/2,S, D)≤(6· 68Rk_0/ _j)^2k_0-1=(A/ _j)^e with A:=408Rk0≥ε0A:=408Rk_0≥ _0 and e:=2k0−1≥3e:=2k_0-1≥ 3. The proposition gives W~1(Γ^n,Γ)≤17An−1/eE\, W_1( _n, )≤ 17A\,n^-1/e. The converse: a member of 1G_1 is a member of ~2 G_2, so sup~1(Qf−nf)≥12sup1(Qf−nf) _ G_1(E_Qf-E_nf)≥ 12 _G_1(E_Qf-E_nf) pointwise in the sample, and Lemma D.4 bounds the expectation of the right side from below on Qk0Q_k_0. ∎ Corollary B.32 (Exponential profiles, accumulative class, the crop route). Let =[−R,R]V=[-R,R] and let Q have an exponential profile at rate β with constant A≥1A≥ 1. Then for every n≥2n≥ 2 and d≥2d≥ 2, Δ~n(Q)≤CR[A(1+1/β)+e3β/2](lnn2β+3)e−βlnn/2,C=6936: _n(Q)\;≤\;C\,R\, [A(1+1/β)+e^3β/2 ] ( n2β+3 )\,e^- β n/2, C=6936: the exponent of Theorem 4.5, with the effective count at the cutoff, Kn≍lnn/βK_n n/β, as a factor. Proof. The crop. For K≤dK≤ d let crKcr_K be the K-crop, with D~(P,crKP)≤RTK(P)/μmax(P) D(P,cr_KP)≤ R\,T_K(P)/ _ (P) (Proposition A.19), and ΓK:=(crK)#Γ ^K:=(cr_K)_\# , the lift of an exactly K-sparse design. As in the proof of Theorem 4.4, coupling along the crop on both sides, Δ~n(Q)≤ 2RQ[TKμmax]+W~1((crK)#Γ^n,ΓK), _n(Q)\;≤\;2R\,E_Q [ T_K _ ]+E\, W_1 ((cr_K)_\# _n, ^K ), and once K≥2K≥ 2 the second term is at most 6936RKn−1/(2K−1)6936\,RK\,n^-1/(2K-1) by Corollary B.31, the cropped lifts having K atoms of share at most 11. The crop cost. On TK≤12\T_K≤ 12\, μmax≥(1−TK)/K≥1/(2K) _ ≥(1-T_K)/K≥ 1/(2K), so pointwise TK/μmax≤2KTK+μmax−1TK>12T_K/ _ ≤ 2K\,T_K+ _ ^-11\T_K> 12\, and Lemma B.29(i) at θ=12θ= 12, with GK(12)≤2QTK≤2Ae−βKG_K( 12)≤ 2\,E_QT_K≤ 2Ae^-β K by Markov, gives Q[TKμmax]≤ 2KAe−βK+4KAe−βK+4Ae−βK1−e−β≤Ae−βK(6K+4+4/β),E_Q [ T_K _ ]\;≤\;2KAe^-β K+4KAe^-β K+ 4Ae^-β K1-e^-β\;≤\;Ae^-β K\,(6K+4+4/β), and 6K+4+4/β≤6(1+1/β)(K+1)6K+4+4/β≤ 6(1+1/β)(K+1). The cutoff. As in the proof of Theorem 4.5, put x:=2lnn/βx:= 2 n/β and K:=d∧(⌈x/2⌉∨2)K:=d ( x/2 2), so that βK≥βlnn/2β K≥ β n/2 unless K=dK=d, where Td=0T_d=0 and the crop costs nothing, that 2K−1≤x+32K-1≤ x+3, whence n−1/(2K−1)≤e−lnn/(x+3)≤e3β/2e−βlnn/2n^-1/(2K-1)≤ e^- n/(x+3)≤ e^3β/2e^- β n/2, and that K+1≤x/2+3K+1≤ x/2+3. The two terms are then at most 12RA(1+1/β)(K+1)e−βlnn/212RA(1+1/β)(K+1)\,e^- β n/2 and 6936Re3β/2(K+1)e−βlnn/26936\,R\,e^3β/2(K+1)\,e^- β n/2, and 12≤693612≤ 6936. ∎ B.8 The two caps compared, and the accumulative certificate Against Theorem 4.4 and Theorem 4.5, the multiscale and crop routes give the accumulative class the same exponents (Corollaries B.31 and B.32), and the accumulative price is a factor of the count, paid once: the maximal count k0k_0 at exact sparsity, and the effective count at the cutoff, lnn/(2β)+3 n/(2β)+3, on exponential profiles. The single-partition route of Theorem B.27 gives the weaker exponents 1/(2k0+1)1/(2k_0+1) and 1/31/3 (Corollaries B.28 and B.30), with the mean count ζ¯ ζ and a constant of order β−1(1+ln2A)β^-1(1+ 2A) in place of the maximal count, the profile bounding the concentration moments (Lemma B.29). This is the two semantics of Section 3.1 as arithmetic. On a hard selection of k coordinates the accumulative class is the conservative class at scale kLkL (Corollary A.15), and the design pays that scale. Remark B.33 (The accumulative certificate at a truncated diameter). Theorem 4.6 needs a diameter for its concentration step, and (ℳd,D~)(M_d, D) has 2Rd2Rd and no better, the count being a member of the class. Let h(P):=∑isi=1/μmax(P)h(P):= _is_i=1/ _ (P) be the total height, so that D~(P,P′)≤R(h(P)+h(P′)) D(P,P )≤ R(h(P)+h(P )) through the reservoir. Fix ζ>0ζ>0, let τ:=Q(h>ζ)τ:=Q(h>ζ) and Γζ:=Γ(⋅∣h≤ζ) _ζ:= (· h≤ζ). On the event that all n sample points have h≤ζh≤ζ, of probability at least 1−nτ1-nτ, the points are i.i.d. from Γζ _ζ, whose support has D~ D-diameter at most 2Rζ2Rζ, so Theorem 4.6 applies with κζ:=2Rζ _ζ:=2Rζ in place of κ and Γζ _ζ in place of Γ ; and W~1(Γζ,Γ)≤Rτ(ζ+[h∣h>ζ]) W_1( _ζ, )≤ Rτ\,(ζ+E[h h>ζ]), by coupling the heavy mass to light points through the reservoir. Hence, with probability at least 1−δ−nτ1-δ-nτ, W~1(Γ^A,Γ)≤W~1(Γ^A,Γ^B)+2Rζln(1/δ)n+Rτ(ζ+[h∣h>ζ]). W_1( _A, )\;≤\; W_1( _A, _B)+2Rζ (1/δ)n+Rτ (ζ+E[h h>ζ] ). The level comes from the profile, by the first line of the proof of Lemma B.29: μmax<1/(2k) _ <1/(2k) forces Tk>12T_k> 12, so Q(h>2k)≤Q(Tk>12)Q(h>2k)≤ Q(T_k> 12), and Markov gives Q(h>2k)≤2Ae−βkQ(h>2k)≤ 2Ae^-β k, and ζn:=2⌈β−1ln(2An/δ)⌉ _n:=2 β^-1 (2An/δ) , at most (2/β)ln(2An/δ)+2(2/β) (2An/δ)+2, makes nτ≤δnτ≤δ, the display then holding with probability at least 1−2δ1-2δ, with a tail term of order δζn/nδ _n/n; the price is the factor ζn _n on the concentration term, the effective count at the sample size, logarithmic in n, the quantity the bounds in expectation carry as ζ¯ ζ. The trim that works is on the heavy configurations, not on the light atoms: cropping a crowded configuration to K atoms costs R times the removed height, cheap in the conservative class where the height is the weight and expensive here where it is the share. The cell bound of Proposition B.5 and the exact matching need no diameter, a trimmed sample point being priced at its own observed height. Appendix C Examples C.1 Flocking The protocol below stores a sample of Section 5.1, one focal particle at one instant, as a view, the term of the legend of Figure 2. The dynamics. Particles i=1,…,di=1,…,d carry positions pi∈ℓ2p_i ^2_ , the torus of side ℓ , and headings θi∈S1 _i∈ S^1. Writing u(θ)=(cosθ,sinθ)u(θ)=( θ, θ), one step of the drift of the dynamics is θi′=arg∑jcijeiθj,pi′=pi+v0u(θi′),θ _i\;=\; _jc_ij\,e^i _j, p _i\;=\;p_i+v_0\,u(θ _i), where ci⋅∈Δdc_i·∈ _d are the neighborhood weights of particle i, the selector, computed from the distances |pj−pi| p_j-p_i . The classical model (Vicsek et al., 1995) takes ci⋅c_i· uniform on j:|pj−pi|≤r\j: p_j-p_i ≤ r\, the metric interaction on a hard ball. Two variants are in the literature, the second in our roster: the topological interaction, uniform on the particle itself and its k nearest neighbors, the form supported by the starling observations of Ballerini et al. (2008) with k≈7k≈ 7, and soft kernels, cij∝K(|pj−pi|)c_ij K( p_j-p_i ). Convention C.1 (Noise). The stochastic term, angular noise added to θi′θ _i, is exogenous: the mechanism is the drift map above, the noise is added afterwards, and all statements are about the drift. The four flocks. Every flock has d=3200d=3200 particles on the torus of side ℓ=40 =40, density 22, at speed v0=0.5v_0=0.5, and the flocks differ in the rule. The ball rule is the one above at r=1r=1. The three graded rules have the same drift with soft weights cij∝K(|pj−pi|)c_ij K( p_j-p_i ): the bump K(t)=(1−t2/rb2)+2K(t)=(1-t^2/r_b^2)_+^2 at rb=1.4r_b=1.4, the Gaussian K(t)=e−t2/2h2K(t)=e^-t^2/2h^2 at h=0.5h=0.5 truncated at radius 3.53.5, and the Cauchy K(t)=(1+t2/h2)−3K(t)=(1+t^2/h^2)^-3 at h=0.9h=0.9 truncated at radius 44, beyond which lies 0.2%0.2\% of the untruncated kernel’s weight. The range of each graded rule is the one at which its mean tail matches the ball’s at ten atoms, T10T_10 between 0.160.16 and 0.190.19 for every flock in both regimes, so the four rules have the same effective number of partners. Each flock is read through the selector of its own rule at the rule’s parameters: the lift of a sample is the weights the flock’s own update puts on it, and the mechanism whose class the certificate bounds is that update. The two regimes are η=1.2η=1.2, the main text’s, and η=2.4η=2.4, both in the ordered phase (Chaté et al., 2008). The protocol. The protocol is the same for every flock and regime: a flock, a list of seeds, one replica per seed, and every replica viewed at the same steps. A replica draws uniformly random positions and headings from its seed and runs the drift with its noise, 64,00064,000 replicas per flock and regime. A view is one focal particle, drawn uniformly at random at a fixed step of its replica, with its neighborhood within radius 3.53.5 (44 on the Cauchy flock) stored as positions relative to the focal particle and headings rotated by an independent uniformly random angle, the self atom included at position 00: the design certified is the law of the rotated view, a design of the same mechanism, which is equivariant under the rotation. Every replica is viewed at step 30003000, after the burn-in, by which the flock has ordered (the polarization below), and the focal particle and the rotation of a view are drawn from the replica’s generator after its last step. One view per replica: the views are independent samples of the flock’s law at that age, the i.i.d. hypothesis of Theorem 4.6 holds as stated, and nothing is assumed about mixing within a run. The split, the lift and the certificate. Half A is the replicas of the first 32,00032,000 seeds and half B those of the last 32,00032,000, a split fixed before any data were seen, and the nested sample sizes are the prefixes of the seed order, n=500n=500 to 64,00064,000 samples in doublings, both halves counted. The lift of a view is the selector’s weights on its stored atoms, normalized to sum to one, and on the Gaussian and the Cauchy flock the lift is cropped to the heaviest atoms carrying all but 10−310^-3 of the weight, at the cost of Proposition A.19 per sample, paid in the certificate by the triangle inequality: 0.0017κ0.0017κ and 0.0019κ0.0019κ at the largest n. Every distance between a lift of A and a lift of B is the assignment of Lemma B.3, and the transport is one exact assignment on the 32,000×32,00032,000× 32,000 matrix of them, by the Jonker–Volgenant algorithm, at every prefix. The certificate is that of Theorem 4.6 for half A at δ=0.05δ=0.05, the transport plus the crop cost plus κln(1/δ)/nκ (1/δ)/n, and the whole-sample certificate of Theorem B.1(i), with its one term at δ/2δ/2, is larger by 0.0008κ0.0008κ at the largest n. The simulator and the seeds. The simulator is a compiled implementation of the drift above, exact against the reference implementation step by step, and a matching takes from an hour for the ball flock to five hours for the Cauchy flock on a many-core machine, the assignment on the full matrix half an hour of it. The seeds are consecutive integers, one range per flock and regime, stored with every replica and listed in the supplementary material, which regenerates every table and figure of this appendix and of Section 5.1 from them. Figure 2: The count law of the ball flock after the burn-in: the number m of particles inside the ball r=1r=1 other than the self, per sample, on the 64,00064,000 replicas at (left) η=1.2η=1.2, those of Figure 1, and (right) η=2.4η=2.4, those of Figure 3. The count law and the profile. Figure 2 is the count law of the ball flock after the burn-in in both regimes, the number m of particles inside the ball other than the self. At η=1.2η=1.2 the ball holds 10.710.7 particles on average and 6767 at most, from 22 to 2323 between the 5%5\% and 95%95\% samples, with a variance 4.34.3 times the mean, and at η=2.4η=2.4 it holds 10.910.9 on average and 8181 at most, from 22 to 2424, with a variance 4.64.6 times the mean. The survival P(m≥k)P(m≥ k) is geometric beyond the knee, at 0.200.20 per atom on 15≤k≤5015≤ k≤ 50 in both regimes. Every sample of the ball flock is exactly sparse with uniform weights on its m+1m+1 atoms and tail Tk=(1−k/(m+1))+T_k=(1-k/(m+1))_+, so the ball’s mean tail in Figure 1(a) is the functional QTk=∑mP(m)(1−k/(m+1))+E_QT_k= _mP(m)\,(1-k/(m+1))_+ of the count law, and it is geometric beyond the knee because the count law is, at a decay rate of 0.230.23 per atom. The graded rules are the other case, a decay inside each sample at a rate set by the local crowding. Their profiles are Figure 1(a) at η=1.2η=1.2 and Figure 3(a) at η=2.4η=2.4: matched to the ball’s at ten atoms, the bump’s tail is the ball’s shape on a wider support with the boundary atoms discounted, the Gaussian’s is geometric and the Cauchy’s polynomial, above the Gaussian’s from the seventh atom on. At the one-percent level the selectors resolve 2424 (ball), 2727 (bump), 3232 (Gaussian) and 5757 (Cauchy) atoms at η=1.2η=1.2, and 2424, 2929, 3535 and 6666 at η=2.4η=2.4. The profiles of the two regimes are close, the high-noise flock’s tails a little heavier. Table 2: The two-sample transport W1(Γ^A,Γ^B)W_1( _A, _B) and the certificate of Theorem 4.6 for half A at δ=0.05δ=0.05, in units of κ, at n=64,000n=64,000 samples (32,00032,000 per half), each flock through the selector of its own rule, after the burn-in of 30003000 steps, in both regimes, with the cell certificate of Proposition B.5 on the same halves at the same δ, its shells, resolution and cutoff optimized on the data (the covering script of the supplementary material). The trivial value is κ, and the mean distance D between two random samples is 0.560.56 to 0.68κ0.68κ. η=1.2η=1.2 η=2.4η=2.4 flock W1W_1 cert. cell cert. W1W_1 cert. cell cert. ball, r=1r=1 0.0272 0.0340 0.681 0.0435 0.0503 0.881 bump, rb=1.4r_b=1.4 0.1057 0.1125 1.005 0.1386 0.1454 1.004 Gaussian, h=0.5h=0.5 0.1152 0.1238 1.006 0.1498 0.1583 1.004 Cauchy, h=0.9h=0.9, a=3a=3 0.1205 0.1292 1.006 0.1555 0.1643 1.005 The certificate. Table 2 is the certificate of Theorem 4.6 at the largest n on every flock, in both regimes, after the burn-in, with the transport it is built on, and Figure 1(b) and Figure 3(b) are the curves against n. Four readings. The bound is non-vacuous by a wide margin on every flock and regime: the largest value in the table is below 0.17κ0.17κ, against the trivial value κ and a mean distance between two random samples of 0.560.56 to 0.68κ0.68κ, and on the ball at η=1.2η=1.2 the certificate is 0.034κ0.034κ, a thirtieth of the trivial value. The graded flocks sit together, at four times the ball’s transport at η=1.2η=1.2 and three times at η=2.4η=2.4: the weight mismatch between two samples is a cost the ball’s uniform weights never pay, and it depends little on how the grading is done, the three graded certificates lying within 0.02κ0.02κ of one another in both regimes. The matching is close to its floor. The mean distance from a sample of A to its nearest sample of B is 0.950.95 to 0.960.96 of the matching cost on the graded flocks and 0.690.69 (0.790.79 at η=2.4η=2.4) on the ball, 0.0188κ0.0188κ at η=1.2η=1.2, so what remains is the sample size and not slack in the transport. The covering certificates sit at the trivial value. The cell certificate of Proposition B.5 on the cells of the quantized k-crops of the two halves (on the ball, where the weights tie, the crop takes the k headings nearest the view’s circular mean), the resolution floor, the observed discrepancy of the halves’ cell frequencies and the trim, with the concentration term of Theorem 4.6, at the shells, the resolution and the cutoff K≤24K≤ 24 that are best on the data, is 0.68κ0.68κ on the ball at η=1.2η=1.2 and 0.88κ0.88κ at η=2.4η=2.4, twenty and eighteen times the matching certificate, and at or above κ on every graded kernel at every n, where the continuous weights make almost every occupied cell a singleton, and the stratified bound of Theorem B.14 at a fixed covering constant, from the sample’s own tails with the shells optimized, is above 0.98κ0.98κ on every flock and regime. The comparison is between forms of the same bound on the same class, the same halves and the same δ: every cell bound is one feasible transport and the matching is below every one of them (Proposition B.5), and a celling optimized on the sample is a certificate on the same terms (Remark B.8). Not computed is a chained form, the entropy integral over the scales, or the doubling-dimension covers of Gottlieb et al. (2017), which could sit between the two. Figure 3: The flock at η=2.4η=2.4, the twin of Figure 1: 64,00064,000 independent replicas per flock, one sample each. (a) The profile, the mean tail TkT_k under the four selectors. (b) The certificate of Theorem 4.6 for half A at δ=0.05δ=0.05, in units of κ, after the burn-in of 30003000 steps. The implied dimension. The implied dimension of a measured transport is −1/(dlnW1/dlnn)-1/(d W_1/d n), the exponent it shows at the sample sizes attained, read here as the reciprocal of the least-squares slope of lnW1(Γ^A,Γ^B) W_1( _A, _B) against lnn n over a window of three consecutive doublings, at the middle n. No constant enters: the transport is computed exactly and the slope is free of units. On interval values an exactly k-sparse design with k≥2k≥ 2 shows at most 2k−12k-1 as n grows and the stratum design of Appendix D at least 2k−12k-1 (Theorem 4.4), as the bipartite matching on the d-cube decays at n−1/dn^-1/d (von Luxburg and Bousquet, 2004, Thm. 15). In general the number is the dimension of the lifted design at the scales the sample resolves. Figure 4 draws it for the four selectors after the burn-in in both regimes. Over the last three doublings the ball’s transport decays with the exponent of a 33-dimensional sample at η=1.2η=1.2 and of a 55-dimensional one at η=2.4η=2.4, the graded flocks’ with 1313 to 1515: about ten dimensions of weights at the sample’s scale, against the k−1≈10k-1≈ 10 that the stratum count charges at the mean support. The dimensions are local. On the graded flocks the exponent rises with n, from 55 to 88 at n=1000n=1000 to 1313 at n=32,000n=32,000 at η=1.2η=1.2, about one dimension per doubling and decelerating: at a finer scale the sample resolves more atoms of the weight vector, the cutoff Kn≍lnn/βK_n n/β of Theorem 4.5 read off the data, and the number is the dimension of the lifted design at the scales the sample resolves, not a limit. On the ball it is between 22 and 3.53.5 at every n at η=1.2η=1.2, where the support has up to 6868 atoms, and the more ordered flock is the lower-dimensional for every selector. Figure 4: The implied dimension, the reciprocal of the local slope of lnW1(Γ^A,Γ^B) W_1( _A, _B) against lnn n over three consecutive doublings, at the middle n, after the burn-in: (a) on the replicas of Figure 1 at η=1.2η=1.2 and (b) on those of Figure 3 at η=2.4η=2.4. The curves are the transports behind panel (b) of the two figures. The two regimes. Figure 3 is Figure 1 at η=2.4η=2.4, on the replicas of that regime. The profile is nearly the same, the transport is larger, 0.044κ0.044κ on the ball against 0.027κ0.027κ: the high-noise flock is less ordered, with a local polarization of 0.780.78 against 0.940.94 (below), and the bound sees that too. The certificate on a member with zero training risk. For a target f∗∈Lf _L observed on half A, the lower Lipschitz extension f−(z):=maxi∈A(f∗(zi)−LD(z,zi))f_-(z):= _i∈ A(f (z_i)-L\,D(z,z_i)) agrees with f∗f on A and is L-Lipschitz (von Luxburg and Bousquet, 2004, Lemma 7), an empirical risk minimizer of LG_L with zero training risk, and on the target f∗=LD(⋅,A)f =L\,D(·,A), which is zero on A, it is −LD(⋅,A)-L\,D(·,A): its error at z is 2LD(z,A)2L\,D(z,A) and its gap on half B is exactly 2LN¯2L\, N, twice the mean nearest-neighbor distance from B to A. By Remark B.22 the certificate bounds the gap of every empirical risk minimizer of LG_L on every target in LG_L by 2L2L times its value, so on this member the ratio of the bound to the gap is the certificate over N¯ N: 1.81.8 on the ball at η=1.2η=1.2 (0.0340.034 over 0.01880.0188) and 1.51.5 at η=2.4η=2.4, and 1.101.10 to 1.131.13 on the graded kernels in both regimes, where the matching is at its nearest-neighbor floor (Table 2 and the floor above). The bound uniform over the class is within that factor of the truth on this member, and a member is what a uniform bound is judged by. For one chosen learner on one target the gap can be far smaller, the target’s own smoothness at the neighbor scale, which only the labels see (Ashlagi et al., 2021), and a holdout estimate on half B reads it: the certificate competes with nothing there. What it gives is the bound before any learner is chosen or label seen, at once for every target and predictor in the class, so that a choice made after seeing the data costs nothing (Theorem B.1(i)). The bandwidth sweep. The graded kernels of Section 5.1 are at the ball’s own scale: the Gaussian’s h=0.5h=0.5 is the second-moment match to the disc of radius 11 (the disc has mean squared distance 12 12, the kernel 2h22h^2), and the bump’s radius and the Cauchy’s bandwidth match the ball’s tail at ten atoms. Their certificates lie within 0.02κ0.02κ of one another, four times the ball’s: the price of a design of dimension thirteen against one of dimension three. What the bound reads is the profile, and the bandwidth sets the profile, so we sweep it rather than choose it: the Gaussian rule at h∈0.125,0.18,0.25,0.35,0.5h∈\0.125,0.18,0.25,0.35,0.5\, each flock simulated under its own rule as in the protocol, on 8,0008,000 replicas per bandwidth read at 4,0004,000 samples per half and on 32,00032,000 replicas read at 16,00016,000 per half, and the flock of Section 5.1 at the same prefixes (Table 3). The certificate follows the profile, slowly: halving the bandwidth sharpens the profile by a factor of nearly three, from 3232 to 1212 atoms carrying all but one percent of the weight, lowers the implied dimension from 1010 to 77, and lowers the certificate by a tenth at this n, and a quarter of the bandwidth, at 99 atoms and dimension 66, lowers it by three tenths, monotone in the bandwidth over the five values. At 16,00016,000 samples per half every certificate is lower by 15 to 26%, 0.133κ0.133κ, 0.128κ0.128κ, 0.113κ0.113κ, 0.097κ0.097κ and 0.081κ0.081κ at h=0.5h=0.5, 0.350.35, 0.250.25, 0.180.18 and 0.1250.125, against the ball’s 0.043κ0.043κ at that n. The weights the nearest atoms carry vary continuously with their positions whatever the bandwidth, and those are the coordinates the transport pays for; the gain is in the rate, n−1/6n^-1/6 against n−1/10n^-1/10, so it widens with n. The narrow kernels also order the flock less, the polarization falling from 0.940.94 to 0.780.78, since each particle then follows fewer neighbors. Table 3: The bandwidth sweep of the Gaussian rule at η=1.2η=1.2, each flock under its own rule, read at 4,0004,000 samples per half (the pilot sets of 8,0008,000 replicas) and at 16,00016,000 per half (the full sets of 32,00032,000 replicas), h=0.5h=0.5 the flock of Section 5.1 at the same prefixes: the atoms carrying all but 1%1\% and 0.1%0.1\% of the mean weight and the decay rate per atom between them, the mean local polarization within r=1r=1, the implied dimension of the transport over the last three doublings, and the certificate of Theorem 4.6 in units of κ. h k(1%)k(1\%) k(0.1%)k(0.1\%) decay ϖ dim cert. 4,0004,000 samples per half 0.1250.125 9 16 0.33 0.78 5.8 0.109 0.180.18 11 19 0.29 0.85 5.9 0.127 0.250.25 12 21 0.26 0.90 7.2 0.142 0.350.35 20 33 0.18 0.93 7.2 0.155 0.50.5 32 53 0.11 0.94 9.7 0.157 16,00016,000 samples per half 0.1250.125 9 16 0.33 0.78 5.8 0.081 0.180.18 11 19 0.29 0.86 6.7 0.097 0.250.25 13 21 0.29 0.90 8.0 0.113 0.350.35 20 33 0.18 0.93 9.9 0.128 0.50.5 32 53 0.11 0.94 12.0 0.133 The noise sweep. Both regimes of the paper are in the ordered phase. To see where the certificate stops being small we simulated the ball rule above the transition, at η=3η=3, 44 and 55, on 8,0008,000 replicas each under the protocol of Section 5.1, read at 4,0004,000 samples per half, with the flocks of the paper at the same prefix (Table 4). The certificate grows slowly, from 0.070κ0.070κ to 0.107κ0.107κ at η=3η=3, and then holds, as the crowding falls with the order, from 10.710.7 to 6.56.5 neighbors, and the transport’s implied dimension rises from 2.52.5 to 4.94.9: a disordered lift is a count and a spread of headings with no common direction, a few dimensions more than an ordered one. The polarization gate ϖ0=0.6 _0=0.6 of the η=2.4η=2.4 regime, below which the update leaves the class (Proposition C.7), is met on 69%69\%, 27%27\% and 13%13\% of the samples at η=3η=3, 44 and 55: above the transition the update is a member on a minority of the samples, while the certificate is the design’s and does not need the order. Table 4: The noise sweep of the ball rule at 4,0004,000 samples per half: the paper’s regimes at that prefix and the pilots above the transition (8,0008,000 replicas each). The columns of Table 3, with the mean count of neighbors other than the self and the share of samples at the polarization gate ϖ0 _0 of Proposition C.7. η k(1%)k(1\%) k(0.1%)k(0.1\%) β count pol. gate dim. certA ϖ0 _0 1.21.2 24 34 0.23 10.7 0.94 1.00 2.5 0.070 0.8 2.42.4 24 34 0.23 10.9 0.78 0.94 3.5 0.088 0.6 33 26 39 0.18 10.8 0.66 0.69 4.0 0.107 0.6 44 15 20 0.46 7.5 0.47 0.27 4.4 0.101 0.6 55 12 15 0.77 6.5 0.37 0.13 4.9 0.106 0.6 The fully symmetric form. The drift is invariant under torus shifts, and depends on particle i only through the configuration relative to it, with no internal memory. Frame by the focal particle with the shift Ti:p↦p−piT_i:p p-p_i: p~j:=pj−pi∈ℓ2,xj:=θj∈:=S1, p_j:=p_j-p_i\;∈\;T^2_ , x_j:= _j\;∈\;V:=S^1, for all j, including j=ij=i: the framed position p~j p_j is what the selector reads, the heading xjx_j is the value the mechanism reads, and the self atom sits at position p~i=0 p_i=0 with its own heading. Framed positions keep the tilde, and tj:=|p~j|t_j:= p_j is the distance from the focal particle. Headings are untouched (shift-only framing). Then the selector becomes an ordinary anonymous selector c:(ℓ2)d→Δdc:(T^2_ )^d→ _d of the framed positions with per-coordinate scores: with i fixed by the framing, c=c(p~)∈Δdc=c( p)∈ _d is the weight vector, its j-th coordinate cj∝K(tj)c_j K(t_j) depends on p~j p_j alone, and j=ij=i contributes K(0)K(0), the maximal weight. The aggregation is fully anonymous, self included, m=∑jcjeiθj,θi′=argm,pi′=pi+v0m/|m|,m\;=\; _jc_j\,e^i _j, θ _i= m, p _i=p_i+v_0\,m/ m , and the focal state appears only in the framing and un-framing shifts, which are isometries, never as an argument of the aggregation. The composite is therefore θi′=g(x,p~):=f(x,c(p~))θ _i=g(x, p):=f(x,c( p)) with f anonymous, the mechanism reading the headings and the selector the positions, and the design the law of (x,c(p~))(x,c( p)): a move of particle j≠ij≠ i moves the framed positions in coordinate j only, by exactly the world distance, and a move of the focal particle moves the whole frame, an ordinary telescope priced with no new axiom. Convention C.2 (The self atom stays in). The self atom is always part of the configuration. Beyond physical fidelity (the Vicsek average includes the particle itself) it pins the normalization from below: the sum Z=∑lK(tl)Z= _lK(t_l) contains the self term K(0)K(0), so Z≥K(0)Z≥ K(0) however small every neighbor term is, which keeps the selector well defined for an isolated particle and carries the additive locality of compact windows. Remark C.3 (What the framing removes, and what stays). Removed: the self as a distinguished argument. f has no query slot, is symmetric in all d atoms, and cannot tell which atom is the self; it sees a multiset of weighted headings, and the selector sees an atom at position 00 carrying the maximal weight, so a particle exactly on top of the focal one is rightly treated identically. Kept: the self data, as an ordinary atom, necessarily so, because shift-only framing pins the self position but not the self heading, and θi _i is a variable the alignment average reads. Deleting the self atom would force f to remember θi _i, a query slot again. Full SE(2)SE(2) framing, rotating by −θi- _i, removes the self entirely but is wanted only for heading-dependent interactions (vision cones, blind spots) or to make a learned mechanism equivariant by construction; rotations are not isometries of the torus, so that variant lives on a planar relative chart cropped to a ball of a few interaction ranges, at a crop cost that is exactly zero for compact windows and exponentially small for Gaussian kernels. Default: shift-only, self atom in. From here on (x,μ)(x,μ) denotes the framed configuration and neighbor moves are moves of coordinates j≠ij≠ i. The value space and its constants. Definition C.4 (Value space). =S1V=S^1, the heading, with ρ=dS1ρ=d_S^1, the angle in radians. The framed positions are not values: they are read by the selector alone, and the design is the law of the headings and the weights. V is compact and geodesic, diam=πdiamV=π, so R=π/2R=π/2 and κ=πκ=π, and N(ε,,ρ)=⌈π/ε⌉N( ,V,ρ)= π/ : metric dimension 11. A learner who took the positions as values would pay their dimension in the exponents of Section 4, which carry dim . Every statement of Section 4 made on an interval [−R,R][-R,R] holds on S1S^1 with the same constants. With R=π/2R=π/2, N(ε/2,S1)=⌈2π/ε⌉≤5R/εN( /2,S^1)= 2π/ ≤ 5R/ for ε≤R ≤ R, the bound the proof of Theorem A.17 uses for the interval, so C1=34C_1=34 and the constant of Theorem 4.5 are unchanged, and the lower bounds of Theorems 4.4 and 4.5 transfer by placing the design on a half-circle, which is isometric to an interval of length π. By Theorem A.17 the stratum of k-sparse lifted configurations has logN(ε,k,D)≲klog(C/ε)+(k−1)log(C′/ε)=(2k−1)log(C′/ε), N( ,A_k,D)\; \;k\, (C/ )+(k-1) (C / )\;=\;(2k-1)\, (C / ), exponent 2k−12k-1 when the weights are free, as they are for the graded kernels, whose weights are functions of the positions, and k for the ball, whose weights are uniform on its support. Remark C.5 (The domain size is in the selector). The torus enters only through the selector’s scores. The class’s constants are those of the circle: the reservoir price R=π/2R=π/2 is the worst change of a heading average per unit of weight removed, the crop constant κ=πκ=π is a half turn, and neither the side ℓ nor the interaction range r appears in them. The framing shift and the position update pi′=pi+v0u′p _i=p_i+v_0u are an isometry and a Lipschitz image of the heading output, and never enter the aggregation. Membership: the Vicsek statistics are in the class. The update factors through two statistics of the configuration, (x,μ)=∑jμjeiθj∈ℝ2,ϖ(x,μ)=|(x,μ)|∈[0,1],m(x,μ)\;=\; _j _j\,e^i _j\;∈\;R^2, (x,μ)\;=\; (x,μ) \;∈\;[0,1], (10) the local mean velocity, with headings read from the values and weights from the selector slot, and its resultant length, the local polarization: ϖ=1 =1 at perfect alignment, ϖ≈0 ≈ 0 at cancellation, and with uniform weights over the whole flock ϖ is exactly Vicsek’s global order parameter. Membership comes in two layers of different regularity, and the ordered-region hypothesis of the second is a lower bound ϖ≥ϖ0 ≥ _0 on the first. Proposition C.6 (Unconditional layer). For m and ϖ of (10): (i) each component of m is in 1G_1, and jointly |(P)−(Q)|2≤D(P,Q) (P)-m(Q) _2≤ D(P,Q) for all lifted P,QP,Q; (i) ϖ is in 1G_1, so the quantity gating the conditional layer is itself a member, unconditionally. Proof. (i) =∑jμjeiθj=∫uφ(s)Φm= _j _je^i _j= u\, (s)\,d with φ(s)=eiθ(s) (s)=e^iθ(s), the linear sum of Lemma A.5 with a value-only score. For values s,ts,t with headings θ,θ′θ,θ , |eiθ−eiθ′|2=2sin(dS1/2)≤dS1(θ,θ′)=ρ(s,t) e^iθ-e^iθ _2=2 (d_S^1/2)≤ d_S^1(θ,θ )=ρ(s,t). Lemma A.5, in its vector-valued form, gives the joint bound. Anonymity is clear. (i) Post-compose with the 11-Lipschitz map |⋅|2 · _2. ∎ m is position-blind: value moves in position cost nothing and change nothing, consistently with positions entering only through the selector. The update θi′=argθ _i= factors as configuration ↦ (Proposition C.6) followed by the fixed normalization ↦u′=/||m u =m/ of ℝ2→S1R^2→ S^1, which discards the magnitude and keeps the direction. Both outputs pass through it: the position update pi′=pi+v0u′p _i=p_i+v_0\,u consumes the same direction at constant speed, and the magnitude ϖ never enters the dynamics. Proposition C.7 (Direction layer). On the ordered region ϖ≥ϖ0\ ≥ _0\ the normalization u′=/||u =m/ satisfies |u′()−u′(′)|2≤(1/ϖ0)|−′|2 u (m)-u (m ) _2≤(1/ _0) -m _2 and ‖Du′()‖=1/||≤1/ϖ0 \|Du (m) \|=1/ ≤ 1/ _0. Hence the heading-update components lie in the class at scale 1/ϖ01/ _0 on the ordered region, and the position displacement v0/||v_0\,m/ at scale v0/ϖ0v_0/ _0. Proof. For unit vectors u,vu,v and r,s≥ϖ0r,s≥ _0, |ru−sv|22=(r−s)2+rs|u−v|22 ru-sv _2^2=(r-s)^2+rs u-v _2^2, so |u−v|2≤|ru−sv|2/rs≤|ru−sv|2/ϖ0 u-v _2≤ ru-sv _2/ rs≤ ru-sv _2/ _0: take u=u′()u=u (m), r=||r= , v=u′(′)v=u (m ), s=|′|s= . The differential form is the Jacobian (I−u′u′⊤)/||(I-u u )/ . Compose with Proposition C.6(i). ∎ Remark C.8 (The ϖ0 _0 caveat is the phase transition). ϖ≥ϖ0 ≥ _0 fails where neighbors cancel, the rule in the disordered phase and the exception in the ordered one, half a percent of the samples at η=1.2η=1.2. There the true update is genuinely unstable, the direction of a near-zero resultant, so the divergence of the class constant as ϖ0↓0 _0 0 is the constant pricing the physics rather than slack: L=1/ϖ0L=1/ _0 diverges at the order–disorder transition, where the order-parameter fluctuations peak (Chaté et al., 2008). The resultant ϖ is a member unconditionally, and only the direction readout pays 1/ϖ01/ _0. The price is attached to the constant-speed rule, which forces the output onto the unit circle, not to alignment as such: a linear velocity-averaging dynamics of Cucker–Smale type (Cucker and Smale, 2007), vi′=∑jcijvjv _i= _jc_ijv_j with no renormalization, stops at the first layer and is a fully unconditional member. On the replicas after the burn-in the local polarization under the ball selector has mean 0.940.94 at η=1.2η=1.2, with ϖ≥0.8 ≥ 0.8 on 99.5%99.5\% of the samples and ϖ≥0.6 ≥ 0.6 on 99.96%99.96\%, and mean 0.780.78 at η=2.4η=2.4, with ϖ≥0.6 ≥ 0.6 on 94%94\% and ϖ≥0.5 ≥ 0.5 on 97%97\% (64,00064,000 samples per regime, the resultant of the atoms within r=1r=1, self included). The gate ϖ≥ϖ0 ≥ _0 is a condition on (x,μ)(x,μ) jointly, hence with μ=c(ω)μ=c(ω) apparently a restriction on the selector as well as on x. The next lemma decouples it: weighted averages of unit vectors respect cones, so a heading-coherence condition on x alone yields one ϖ0 _0 uniformly over all selectors with a given tail bound. Lemma C.9 (Coherence decouples the gate from the selector). Fix a range rc>0r_c>0 and α<π/2α<π/2, and call the state (α,rc)(α,r_c)-coherent at the focal particle if all headings of the particles within distance rcr_c of it, self included, lie in a closed arc of half-width α, a condition on the headings and the positions together. Then for every μ∈Δdμ∈ _d placing mass at least 1−t1-t on the rcr_c-ball, ϖ(x,μ)≥(1−t)cosα−t≥cosα−2t. (x,μ)\;≥\;(1-t) α-t\;≥\; α-2t. Consequently, for any selector whose tail beyond rcr_c is at most t on the design region, the ordered-region hypothesis holds with ϖ0=cosα−2t _0= α-2t, positive once cosα>2t α>2t, uniformly over the selector family: the restriction is on the state, the standard locally ordered hypothesis of the flocking literature, and on the selector only through its standing tail bound. Proof. Let e be the arc axis. Each in-ball heading vector has ⟨eiθj,e⟩≥cosα e^i _j,e ≥ α, each out-of-ball vector at least −1-1, so ⟨,e⟩≥(1−t)cosα−t ,e ≥(1-t) α-t, and ϖ≥⟨,e⟩ ≥ ,e . ∎ Remark C.10 (Region restrictions are benign). ϖ is a functional of the lift, ϖ=|∫ueiθ(s)P| = u\,e^iθ(s)\,dP , so ϖ≥ϖ0\ ≥ _0\ is a lift-defined region of the same kind as a stratum hypothesis: in the learning setup both are conditions on the lifted design. Membership on the region is direct. m is 11-Lipschitz on all lifts (Proposition C.6) and u′u is (1/ϖ0)(1/ _0)-Lipschitz in m on the region (Proposition C.7), so each component of u′∘u , restricted to the lifts of the region, is (1/ϖ0)(1/ _0)-Lipschitz there, and McShane’s extension, clipped to [−1,1][-1,1], is a member of 1/ϖ0G_1/ _0 that agrees with the update on the region. The characterization itself is not the route: the proof of Theorem 3.2 passes through intermediate configurations that can leave the region. The region is stable under the crop up to a margin: a t-crop moves ϖ by at most 2t/(1−t)2t/(1-t), so after the crop the gate holds at ϖ0−2t/(1−t) _0-2t/(1-t). What the number bounds. The certificate bounds the uniform deviation of the unit ball 1G_1 in the class’s own units, against the trivial κ, and it translates directly into the flock’s own statistics. The components of the local mean velocity are members of 1G_1 with range 22 (Proposition C.6), so on the ball at η=1.2η=1.2 their deviation between the population and the sample is at most 0.034κ=0.110.034κ=0.11 against 22. The heading update is in LG_L at L=1/ϖ0L=1/ _0 on the samples whose local polarization is at least ϖ0 _0 (Proposition C.7), and its extension from that region, clipped to [−1,1][-1,1], is a member of 1/ϖ0G_1/ _0 on every sample (Remark C.10) that differs from the update only on the region B:=ϖ<ϖ0B:=\ < _0\, by at most the range 22. So for a component u of the update and its extension u¯ u, |Γu−Γ^Au|≤|Γu¯−Γ^Au¯|+2Γ(B)+2Γ^A(B)≤1ϖ0W1(Γ^A,Γ)+2Γ(B)+2Γ^A(B),| u- _Au|≤| u- _A u|+2\, (B)+2\, _A(B)≤ 1 _0\,W_1( _A, )+2\, (B)+2\, _A(B), where Γ^A(B) _A(B) is observed and Γ(B) (B) is at most the share of B in the 64,00064,000 samples plus ln(1/δ)/(2⋅64,000)=0.005 (1/δ)/(2· 64,000)=0.005 at δ=0.05δ=0.05, by Hoeffding’s inequality on the indicator of B. At η=1.2η=1.2 and ϖ0=0.8 _0=0.8 the region holds on 99.5%99.5\% of the samples of either half, so the deviation of the update between the population and the sample is at most 0.13+0.03=0.160.13+0.03=0.16 against a range of 22. At η=2.4η=2.4 and ϖ0=0.6 _0=0.6 it holds on 94%94\%, and the deviation is at most 0.26+0.25=0.510.26+0.25=0.51 against 22, the samples outside the gate costing as much as the certificate at that noise. For regression with the absolute loss on a target in the class the gap of every predictor is at most twice the number (Remark B.22), and other losses cost the factor of their Lipschitz constant. Appendix D Lower bounds from the profile The upper bounds of Section 4 are functionals of the profile. This appendix shows that the profile is also what bounds the problem from below: at a fixed stratum the uniform law is at the (2k−1)(2k-1)-dimensional rate, and so is the risk of learning from noiseless labels (Theorem D.5), and on a design with an exponential profile it is quasi-polynomial in n. The technique is classical, a bound on the mass that the lifted design gives a ball, so that n atoms cover at most half of the design at the scale where the bound is 1/(2n)1/(2n), and the content is in the object: the metric D restricted to a fiber of fixed weights is bi-Lipschitz to a weighted supremum distance whose weights are the design’s (Lemma D.1), so the lifted design is locally an anisotropic box with semi-axes the weights, and on an exponential profile its mass at scale ε is exp(−ln2(1/ε)/(2β)) (- ^2(1/ )/(2β)) up to constants. The square loss has lower bounds of the same shape from the same fibers, by Assouad’s hypercube, sketched in Remark B.23. D.1 The device: fibers of fixed weights Lemma D.1 (Fibers of fixed weights). Let μ∈Δdμ∈ _d have support S:=i:μi>0S:=\i: _i>0\ with pairwise distinct entries on S, and set, with γ(μ):=1γ(μ):=1 when S is a single atom, γ(μ):=mini≠j∈S|μi−μj|max(μi,μj)∈(0,1].γ(μ)\;:=\; _i≠ j∈ S _i- _j ( _i, _j)\;∈\;(0,1]. (i) Φ is injective on S×μV^S×\μ\, values off S being forgotten, and for all x,y∈dx,y ^d, γ(μ)2maxi∈Sμiρ(xi,yi)≤D(Φ(x,μ),Φ(y,μ))≤∑i∈Sμiρ(xi,yi). γ(μ)2\, _i∈ S _i\,ρ(x_i,y_i)\;≤\;D ( (x,μ), (y,μ) )\;≤\; _i∈ S _i\,ρ(x_i,y_i). (i) Write Fμ:=Φ(d×μ)F_μ:= (V^d×\μ\) for the fiber over μ. If μ,νμ,ν differ as multisets of positive weights then D(Fμ,Fν)≥Rm(μ,ν)>0D(F_μ,F_ν)≥ R\,m(μ,ν)>0, where m(μ,ν)m(μ,ν) is the minimum, over partial matchings of the positive entries of μ with those of ν, of the sum of |μi−νj| _i- _j over matched pairs plus the sum of the unmatched entries. Proof. (i) Injectivity: Φ(x,μ)=Φ(y,μ) (x,μ)= (y,μ) forces equal multisets of positive pairs (Lemma A.8); the weights being distinct, the pairing is the identity on S. Upper bound: match atom i to atom i, at cone cost μiρ(xi,yi) _iρ(x_i,y_i) each. Lower bound: fix i∈Si∈ S attaining the maximum and follow the atom (xi,μi)(x_i, _i) in an optimal partial matching. Matched to (yi,μi)(y_i, _i) it costs μiρ(xi,yi) _iρ(x_i,y_i). Matched to (yj,μj)(y_j, _j), j≠ij≠ i, it costs at least the height term R|μi−μj|≥Rγ(μ)μi≥12γ(μ)μiρ(xi,yi)R _i- _j ≥ Rγ(μ) _i≥ 12γ(μ)\, _iρ(x_i,y_i), as ρ≤2Rρ≤ 2R. Unmatched it costs Rμi≥12μiρ(xi,yi)R _i≥ 12 _iρ(x_i,y_i). All other terms are nonnegative. (i) Value terms are nonnegative, so every partial matching between Φ(x,μ) (x,μ) and Φ(y,ν) (y,ν) costs at least R times its height cost, which is at least Rm(μ,ν)Rm(μ,ν); the minimum is over finitely many matchings and vanishes only if one of them pairs every entry with an equal entry, that is if the multisets coincide. ∎ D.2 Fixed stratum: the calibration from below Definition D.2 (A region of separated weights). Let =[−R,R]V=[-R,R], k≥2k≥ 2, and put μi0:=2ik(k+1),δ0:=2k(k+1),r:=δ04k,m0:=3δ04,μ^0_i:= 2ik(k+1), _0:= 2k(k+1), r:= _04k, m_0:= 3 _04, so δ0 _0 is both the gap between consecutive coordinates of μ0∈Δkμ^0∈ _k and its minimum, and let Uk:=a∈Δk:|ai−μi0|≤r,i≤k−1,U_k:= \a∈ _k: a_i-μ^0_i ≤ r,\ i≤ k-1 \, a cube of side 2r2r in the chart (a1,…,ak−1)(a_1,…,a_k-1). On UkU_k every coordinate is within δ0/4 _0/4 of μ0μ^0, the last one because (k−1)r≤δ0/4(k-1)r≤ _0/4, hence ai≥m0a_i≥ m_0 for all i, the coordinates of a single a∈Uka∈ U_k are pairwise at distance at least δ0/2 _0/2, and |ai−aj′|≥δ0/2 a_i-a _j ≥ _0/2 for all a,a′∈Uka,a ∈ U_k, i≠ji≠ j. The stratum design is Qk:=Unif([−R,R]k)⊗Unif(Uk)Q_k:=Unif([-R,R]^k) (U_k) on the first k coordinates, weight 00 on the others, normalized Lebesgue measure in the chart: an exactly k-sparse design, Tk=0T_k=0 a.s., with lift Γk _k supported in Sk:=Φ([−R,R]k×Uk)⊆kS_k:= ([-R,R]^k× U_k) _k. At k=1k=1 the stratum is the interval at height 11, and W1(νn,Γ1)≥c/nW_1( _n, _1)≥ c/n for every n-atom νn _n is the quantization bound of the uniform law on an interval, so the converse of Theorem 4.4 holds at k0=1k_0=1 as well. Lemma D.3 (Local bi-Lipschitz bound on the separated region). For (x,a),(y,a′)∈[−R,R]k×Uk(x,a),(y,a )∈[-R,R]^k× U_k with D(Φ(x,a),Φ(y,a′))≤ε<Rδ0/2D( (x,a), (y,a ))≤ <R _0/2, m0‖x−y‖1+R‖a−a′‖1≤ε.m_0\, \|x-y \|_1+R\, \|a-a \|_1\;≤\; . Proof. Both lifts have k atoms, one per coordinate. A partial matching that pairs coordinate i of the first with coordinate j≠ij≠ i of the second pays R|ai−aj′|≥Rδ0/2R a_i-a _j ≥ R _0/2 on that pair, and one that leaves an atom unmatched pays Rai≥Rm0≥Rδ0/2Ra_i≥ Rm_0≥ R _0/2; either exceeds ε . So the optimal matching pairs each coordinate with itself, and D=∑i[min(ai,ai′)|xi−yi|+R|ai−ai′|]≥m0‖x−y‖1+R‖a−a′‖1D= _i [ (a_i,a _i)\, x_i-y_i +R a_i-a _i ]≥ m_0 \|x-y \|_1+R \|a-a \|_1. ∎ Lemma D.4 (Mass bound, and the uniform law at a stratum). Let εmax(k):=Rδ0/5 _ (k):=R _0/5, Ck:=22k−1m0kR2k−1rk−1k!(k−1)!,ck:=12(2Ck)−1/(2k−1).C_k:= 2^2k-1m_0^k\,R^2k-1\,r^k-1\,k!\,(k-1)!, c_k:= 12\,(2C_k)^-1/(2k-1). For every P∈ℳdP _d and ε≤εmax(k) ≤ _ (k), Γk((P,ε))≤Ckε2k−1 _k( B(P, ))≤ C_k\, ^2k-1, and for every n and every n-atom probability measure νn _n on ℳdM_d, W1(νn,Γk)≥minckn−1/(2k−1),εmax(k)/2;W_1( _n, _k)\;≥\; \\,c_k\,n^-1/(2k-1),\ _ (k)/2\, \; in particular this bounds W1(Γ^n,Γk)E\,W_1( _n, _k) from below. Proof. Mass bound. If (P,ε) B(P, ) misses SkS_k there is nothing to prove. Otherwise pick P′=Φ(x′,a′)∈Sk∩(P,ε)P = (x ,a )∈ S_k∩ B(P, ), so that (P,ε)⊆(P′,2ε) B(P, ) B(P ,2 ). Every (y,a)∈[−R,R]k×Uk(y,a)∈[-R,R]^k× U_k with Φ(y,a)∈(P′,2ε) (y,a)∈ B(P ,2 ) satisfies, by Lemma D.3 as 2ε<Rδ0/22 <R _0/2, ‖x′−y‖1≤t:=2ε/m0 \|x -y \|_1≤ t:=2 /m_0 and ‖a′−a‖1≤2ε/R \|a -a \|_1≤ 2 /R. Since QkQ_k is a product measure, Γk((P′,2ε))≤(2t)k!(2R)k⋅(4ε/R)k−1(k−1)!(2r)k−1=Ckε2k−1, _k ( B(P ,2 ) )\;≤\; (2t)^kk!\,(2R)^k· (4 /R)^k-1(k-1)!\,(2r)^k-1\;=\;C_k\, ^2k-1, using the volume (2t)m/m!(2t)^m/m! of the ℓ1 ^1-ball of radius t in ℝmR^m, in the chart for the second factor, where the ℓ1 ^1-norm of the first k−1k-1 coordinates is at most the full one. Uniform law. Let νn=∑i≤naiδPi _n= _i≤ na_i\, _P_i and set ε:=min(2nCk)−1/(2k−1),εmax(k) := \(2nC_k)^-1/(2k-1), _ (k)\, so that nCkε2k−1≤12nC_k ^2k-1≤ 12. By the mass bound, Γk(⋃i(Pi,ε))≤12 _k ( _i B(P_i, ) )≤ 12, and any coupling of νn _n with Γk _k transports the Γk _k-mass outside this union, at least 12 12, across distance >ε> . Hence W1(νn,Γk)≥ε/2W_1( _n, _k)≥ /2. ∎ Theorem D.5 (Learning from below at a stratum). Let k≥2k≥ 2 and let QkQ_k be the stratum design of Definition D.2. A learner is a measurable map from a sample (Zi,f∗(Zi))i≤n(Z_i,f (Z_i))_i≤ n, ZiZ_i i.i.d. ∼Qk Q_k, to a function f f on C, and its risk on the target f∗f is Qk|f^−f∗|E_Q_k f-f . There are ck′>0c _k>0 and nkn_k, depending on k and R only, such that for every n≥nkn≥ n_k and every learner supf∗∈1Qk|f^−f∗|≥ck′n−12k−1: _f _1\;E\,E_Q_k f-f \;≥\;c _k\,n^- 12k-1: at a fixed stratum the labels, however many and noiseless, do not lift the exponent of Theorem 4.4. Proof. Write Γ:=Γk := _k, S:=SkS:=S_k, and fix ε≤εmax′(k):=Rδ0min1/20,2/k ≤ _ (k):=R _0 \1/20,2/k\, so that 2ε≤εmax(k)2 ≤ _ (k) and ε/(8R)≤r /(8R)≤ r. A packing. Let P1,…,PNP_1,…,P_N be a maximal 2ε2 -separated subset of S, finite by compactness. By maximality the balls (Pj,2ε) B(P_j,2 ) cover S, so 1≤∑jΓ((Pj,2ε))≤NCk(2ε)2k−11≤ _j ( B(P_j,2 ))≤ N\,C_k(2 )^2k-1 by the mass bound of Lemma D.4, that is N≥1/(Ck(2ε)2k−1)N≥ 1/(C_k(2 )^2k-1). The mass near a center. Write Pj=Φ(xj,aj)P_j= (x^j,a^j) with (xj,aj)∈[−R,R]k×Uk(x^j,a^j)∈[-R,R]^k× U_k. Matching coordinate i with coordinate i (Equation 4), D(Φ(y,a),Pj)≤‖y−xj‖1+R‖a−aj‖1D( (y,a),P_j)≤ \|y-x^j \|_1+R \|a-a^j \|_1, and the last coordinate of a−aja-a^j is bounded by the sum of the others, so ‖y−xj‖1≤ε/4×‖(a−aj)≤k−1‖1≤ε/(8R)\ \|y-x^j \|_1≤ /4\×\ \|(a-a^j)_≤ k-1 \|_1≤ /(8R)\ lifts into (Pj,ε/2) B(P_j, /2). The ℓ1 ^1-ball of radius t in ℝmR^m has volume (2t)m/m!(2t)^m/m!, and its intersection with a cube containing the center contains one of its 2m2^m orthant pieces, wherever the center sits, once t is at most half the cube’s side, which holds for both factors (ε/4≤R /4≤ R and ε/(8R)≤r /(8R)≤ r). As QkQ_k is the product of the uniform laws on the cube [−R,R]k[-R,R]^k and on the cube UkU_k of side 2r2r in the chart, Γ((Pj,ε/2))≥(ε/2)k2kk!(2R)k⋅(ε/(4R))k−12k−1(k−1)!(2r)k−1=:c2ε2k−1. ( B(P_j, /2) )\;≥\; ( /2)^k2^k\,k!\,(2R)^k· ( /(4R))^k-12^k-1\,(k-1)!\,(2r)^k-1\;=:\;c_2\, ^2k-1. The bumps. Let φj(P):=(ε−D(P,Pj))+ _j(P):=( -D(P,P_j))_+, a 11-Lipschitz function of the lift supported in (Pj,ε) B(P_j, ) and at least ε/2 /2 on (Pj,ε/2) B(P_j, /2), so that ∫φjΓ≥12c2ε2k _j\,d ≥ 12c_2 ^2k. For σ∈−1,1Nσ∈\-1,1\^N put fσ:=∑jσjφj∘Φf_σ:= _j _j\, _j . It is in 1G_1: the supports are disjoint, the centers being 2ε2 apart, and for P in the j-th support and P′P in the l-th, l≠jl≠ j, |fσ(P)−fσ(P′)|≤φj(P)+φl(P′)=2ε−D(P,Pj)−D(P′,Pl)≤D(P,P′) f_σ(P)-f_σ(P ) ≤ _j(P)+ _l(P )=2 -D(P,P_j)-D(P ,P_l)≤ D(P,P ) by the triangle inequality through D(Pj,Pl)≥2εD(P_j,P_l)≥ 2 , while for P′P in no support |fσ(P)−fσ(P′)|=φj(P)≤ε−D(P,Pj)≤D(P′,Pj)−D(P,Pj)≤D(P,P′) f_σ(P)-f_σ(P ) = _j(P)≤ -D(P,P_j)≤ D(P ,P_j)-D(P,P_j)≤ D(P,P ). The signs. Let σ be uniform on −1,1N\-1,1\^N, independent of the sample, and J the set of j whose ball (Pj,ε) B(P_j, ) contains no sample point. The labels fσ(Zi)f_σ(Z_i) are functions of the sample and of (σj)j∉J( _j)_j∉ J alone, so conditionally on both, (σj)j∈J( _j)_j∈ J is still uniform, and for Z′∈(Pj,ε)Z ∈ B(P_j, ) with j∈Jj∈ J, where fσ=σjφjf_σ= _j _j, [|f^(Z′)−σjφj(Z′)|∣sample,(σj)j∉J]≥φj(Z′)E[ f(Z )- _j _j(Z ) ,( _j)_j∉ J]≥ _j(Z ), as 12|a−b|+12|a+b|≥|b| 12 a-b + 12 a+b ≥ b . Integrating over Z′Z and summing over j∈Jj∈ J, the risk averaged over σ, which the supremum over f∗f dominates, is at least ∑jPr(j∈J)∫φjΓ _j (j∈ J) _j\,d , and Pr(j∈J)=(1−Γ((Pj,ε)))n≥1−nCkε2k−1 (j∈ J)=(1- ( B(P_j, )))^n≥ 1-nC_k ^2k-1 by the mass bound. The scale. Take ε:=(2nCk)−1/(2k−1)=2ckn−1/(2k−1) :=(2nC_k)^-1/(2k-1)=2c_kn^-1/(2k-1), which is at most εmax′(k) _ (k) for n≥nk:=(2ck/εmax′(k))2k−1n≥ n_k:=(2c_k/ _ (k))^2k-1. Then Pr(j∈J)≥12 (j∈ J)≥ 12 and supf∗∈1Qk|f^−f∗|≥N⋅12⋅12c2ε2k≥c2ε4Ck 22k−1=c2ck4kCkn−12k−1=:ck′n−12k−1.∎ _f _1E\,E_Q_k f-f \;≥\;N· 12· 12c_2 ^2k\;≥\; c_2\, 4\,C_k\,2^2k-1\;=\; c_2\,c_k4^k\,C_k\;n^- 12k-1\;=:\;c _k\,n^- 12k-1. D.3 Exponential profiles: quasi-polynomial from below Theorem D.6 (Exponential profile: the uniform law from below). Let =[−R,R]V=[-R,R], β>0β>0, q:=e−βq:=e^-β, γ:=1−qγ:=1-q, and d≥2d≥ 2. Let μβ∈Δdμ^β∈ _d have entries μjβ:=γqj−1/(1−qd)μ^β_j:=γ q^j-1/(1-q^d), and let Qβ:=Unif([−R,R]d)⊗δμβQ^β:=Unif([-R,R]^d) _μ^β, with lift Γβ ^β supported in the fiber Fβ:=FμβF^β:=F_μ^β. Then Tk≤e−βkT_k≤ e^-β k for all k at every configuration: an exponential profile at rate β with A=1A=1, to which Theorem 4.5 applies. For every n≥1n≥ 1 with d≥2ln(2n)/β+1d≥ 2 (2n)/β+1 and every n-atom probability measure νn _n on ℳdM_d, W1(νn,Γβ)≥γ2R16e−2βln(2n);W_1( _n, ^β)\;≥\; γ^2R16\,e^- 2β (2n); in particular this bounds W1(Γ^n,Γβ)=Δn(Qβ)EW_1( _n, ^β)= _n(Q^β) from below. Proof. Profile and gaps. Tk(μβ)=∑j>kμjβ=(qk−qd)/(1−qd)≤qkT_k(μ^β)= _j>kμ^β_j=(q^k-q^d)/(1-q^d)≤ q^k, as qk+d≤qdq^k+d≤ q^d. For i<ji<j, μiβ−μjβ=μiβ(1−qj−i)≥γμiβμ^β_i-μ^β_j=μ^β_i(1-q^j-i)≥γμ^β_i, so γ(μβ)≥γ(μ^β)≥γ in Lemma D.1(i): D(Φ(x,μβ),Φ(y,μβ))≥γ2maxjμjβ|xj−yj|(x,y∈[−R,R]d),D ( (x,μ^β), (y,μ^β) )\;≥\; γ2 _jμ^β_j x_j-y_j (x,y∈[-R,R]^d), (11) and Φ is injective on [−R,R]d×μβ[-R,R]^d×\μ^β\. Also μjβ≥γqj−1μ^β_j≥γ q^j-1. The anisotropic box. For t∈(0,γ2R/4]t∈(0,γ^2R/4] put a(t):=ln(γ2R/(4t))≥0a(t):= (γ^2R/(4t))≥ 0, K(t):=⌊a(t)/β⌋+1≥1K(t):= a(t)/β +1≥ 1, and yj(t):=γμjβR/(2t)≥γ2qj−1R/(2t)=2ea(t)−β(j−1)y_j(t):=γμ^β_jR/(2t)≥γ^2q^j-1R/(2t)=2e^a(t)-β(j-1), so that yj(t)≥2y_j(t)≥ 2 for j≤K(t)j≤ K(t), as β(K(t)−1)≤a(t)β(K(t)-1)≤ a(t). Consequently, for K=K(t)≤dK=K(t)≤ d and a=a(t)a=a(t), ∏j≤K1yj(t)≤ 2−Ke−aK+βK(K−1)/2≤ 2−Ke−aK/2≤e−a2/(2β), _j≤ K 1y_j(t)\;≤\;2^-Ke^-aK+β K(K-1)/2\;≤\;2^-Ke^-aK/2\;≤\;e^-a^2/(2β), (12) using β(K−1)≤aβ(K-1)≤ a and then K>a/βK>a/β. The bound. Fix P∈ℳdP _d and ε with 2ε≤γ2R/42 ≤γ^2R/4 and K′:=K(2ε)≤dK :=K(2 )≤ d, and suppose (P,ε) B(P, ) meets FβF^β at P′=Φ(x′,μβ)P = (x ,μ^β). Then (P,ε)⊆(P′,2ε) B(P, ) B(P ,2 ), and by (11) every Φ(y,μβ)∈(P′,2ε) (y,μ^β)∈ B(P ,2 ) has |yj−xj′|≤4ε/(γμjβ)=R/yj(2ε) y_j-x _j ≤ 4 /(γμ^β_j)=R/y_j(2 ) for every j. As QβQ^β is uniform on the cube, Γβ((P,ε))≤∏j≤K′min1,1/yj(2ε)≤e−a(2ε)2/(2β) ^β( B(P, ))≤ _j≤ K \1,1/y_j(2 )\≤ e^-a(2 )^2/(2β) by (12). For an n-atom νn=∑iaiδPi _n= _ia_i _P_i take ε:=18γ2Re−2βln(2n) := 18γ^2R\,e^- 2β (2n), so that a(2ε)2/(2β)=ln(2n)a(2 )^2/(2β)= (2n) and K′=⌊2ln(2n)/β⌋+1≤dK = 2 (2n)/β +1≤ d: then Γβ(⋃i(Pi,ε))≤ne−ln(2n)=12 ^β( _i B(P_i, ))≤ ne^- (2n)= 12, every coupling of νn _n with Γβ ^β moves Γβ ^β-mass at least 12 12 across distance ≥ε≥ , and W1(νn,Γβ)≥ε/2W_1( _n, ^β)≥ /2. ∎ D.4 Reading Fixed stratum. The lower exponent of Lemma D.4, 1/(2k−1)1/(2k-1), is the upper exponent of Theorem 4.4 on interval values for k≥2k≥ 2, by the multiscale coupling of Proposition B.18 on the cropped stratum: at a fixed stratum the uniform law is n−1/(2k−1)n^-1/(2k-1) from both sides. The constants are polynomial in k on the lower side and the comparison of constants is not pursued. At k=1k=1 the stratum is an interval, where the empirical measure converges at n−1/2n^-1/2 and the quantization bound c/nc/n of every n-atom measure is the converse’s form. The exponential rate is the rate of the problem. Between Theorem 4.5 and Theorem D.6 the uniform law of the design QβQ^β is e−cβlnne^-c β n with the constant c lying between 1/21/ 2 and 2 2, up to lower-order factors. The fiber resembles the exponential ellipsoid of Lei (2020, Thm. 4.2) at logγ=β γ=β, whose entropy ln2(1/ε)/(2β) ^2(1/ )/(2β) is his Lemma 4.3 and whose rate e−2logγlnne^- 2 γ\, n has the shape of Theorem D.6. Neither the partition nor the crop is the reason for the quasi-polynomial shape: for every fixed k the lower bound exceeds n−1/(2k−1)n^-1/(2k-1) eventually (for d≥2ln(2n)/β+1d≥ 2 (2n)/β+1), so QβQ^β is harder than every stratum, while the upper bound places it below every logarithmic rate. The design is d-free in the sense that matters: d enters only through that condition, the number of coordinates the construction uses, which is the cutoff Kn≍lnn/βK_n n/β of Theorem 4.5 read from below. The whole lower bound is the mass bound Γβ((P,ε))≤exp(−ln2(γ2R/(8ε))/(2β)) ^β( B(P, ))≤ (- ^2(γ^2R/(8 ))/(2β)): the local dimension of the lifted design at scale ε is ln(1/ε)/β (1/ )/β, the profile’s own cutoff, and upper and lower bounds read the same quantity, one as a covering count and the other as a packing mass. The stratum count of Theorem A.17 resolves every value at the scale ε and covers the weights over all of Δk _k, whereas the fiber itself has the metric entropy of the anisotropic box ∏j[±Rμjβ] _j[± Rμ^β_j] and no weights to cover. Whether a count resolving the atom of rank j at its own scale ε/(kμj) /(k _j), which would recover the fiber’s entropy up to a polylogarithmic factor, raises the constant of Theorem 4.5 is left open. Appendix E The literature in full Section 2 keeps the contrasts the results rest on. This appendix gives the lines in full. Symmetric functions and Deep Sets. The representation of Section 3.2 belongs to a tradition in which an invariant embedding absorbs a symmetry and the functions above it are free. For permutation symmetry the embedding is a sum of per-element features: the fundamental theorem of symmetric polynomials (Weyl, 1939; Macdonald, 1995) gives every continuous symmetric function of n scalars as ρ(∑iφ(xi))ρ( _i (x_i)) with φ the power sums, Deep Sets (Zaheer et al., 2017) is this statement with the two maps learned, and the sum decomposition is the form of the set networks that followed (Sannai et al., 2019; Segol and Lipman, 2020; Amir et al., 2023; Zweig and Bruna, 2022). The width is the price: Wagstaff et al. (2022) show that a continuous sum decomposition of all symmetric functions of n elements needs latent dimension n, and that below it there is a Lipschitz symmetric function that no decomposition approximates better than a constant does. The embedding of these results carries no metric, so the class it represents is all continuous symmetric functions and the complexity sits in the width. Here the embedding carries a metric, the class is its Lipschitz ball, the sum networks are its linear functionals with a readout (Section 3.2), and the complexity sits in the geometry of the design. The count-free rate of the linear sums (Proposition B.20) is the phenomenon of distribution regression, the learning of a function of a distribution from the mean embedding of a sample of it, at rates in the number of distributions and the size of each sample (Póczos et al., 2013; Szabó et al., 2016): a linear sum is a function of the mean lift of the configuration and learns at the value space’s rate whatever the count, and the class beyond the linear sums is what the profile prices. The sample-complexity gain of an invariance is quantified in another line: on an invariant target the least-squares gap of a linear model exceeds the invariant model’s by the dimension of the non-invariant directions over n (Elesedy and Zaidi, 2021), an invariant kernel gains a factor of the group’s order for a finite group (Bietti et al., 2021), the covering of a quotient feature space gains d! d! for a permutation-invariant network (Sannai et al., 2021), and on a manifold under a Lie group the minimax exponent carries the dimension of the quotient (Tahmasebi and Jegelka, 2023). For the symmetric group on d coordinates the quotient of the configuration space still has dimension of order d, so anonymity alone buys a constant, and it is the profile that removes d from the exponent (Section 4.1). The stability results on sum embeddings ask how well a fixed embedding preserves a Wasserstein distance chosen in advance, under which sum-based multiset networks are Hölder in expectation and not bi-Lipschitz (Davidson and Dym, 2025), and which aggregators are Lipschitz for a multiset distance fixed in advance, the sum, the mean, the maximum and attention against three such distances, is Nikolentzos and Skianis (2026), and a sliced-Wasserstein embedding of multisets that is bi-Lipschitz for that distance, with no bi-Lipschitz embedding of measures at all, is Amir and Dym (2025), while De Bie et al. (2019) build networks on weighted point clouds that are Lipschitz for a Wasserstein distance chosen in advance. Networks on spaces of probability measures with a universal approximation theorem are Pevný and Kovařík (2019), and the learning of Lipschitz and Sobolev functions on Wasserstein space from samples is Fornasier et al. (2023) and Pham and Warin (2023): functions of the weighted value measure, the pointwise subclass of Remark A.6, in which the weights are mass. Here the question is which distance the class itself determines. On a fixed hard selection the two views agree, since the restriction of our class is every symmetric Lipschitz function of the selected values (Corollary A.15). Graph networks. A graph network aggregates over the neighborhood of each node, a set that differs from node to node, and its neighborhood aggregation is our object with a hard selector. Its generalization bounds, Rademacher and PAC-Bayes bounds for a fixed architecture (Garg et al., 2020; Liao et al., 2021), charge the maximum degree of the training graphs per layer, or its spectral relaxation (Ju et al., 2023) or norms of the propagation matrix that reduce to degree bounds (Esser et al., 2021), at the rate m−1/2m^-1/2 with the degree a constant in front. Whether any finer statistic of the data can be used was posed as a question in that line (Liao et al., 2021; Morris et al., 2024). The graphon line (Maskey et al., 2022), with its extension to sparse graphs and mixtures of graphons (Maskey et al., 2024), points the other way: with mean aggregation on dense graphs its bounds improve with the node count, with constants that grow as the kernel’s minimal degree shrinks, where ours improve as the design gets sparser and never see the node count. These bounds cover the parameter ball of one network, whose parameters are shared across nodes so that the node count drops out, and with it the neighbor count. What they become when asked to learn our class is Remark B.19, a bound at the maximum count. Lower bounds exist in graph learning, in Weisfeiler–Leman classes and in margins (Vasileiou et al., 2025b), and on the Rademacher complexity of a graph convolutional network at a constant degree on one fixed graph (Lv, 2021), none minimax, and none for a neighbor count that varies with the configuration. A second line puts a metric on the graphs under which message-passing networks are Lipschitz. The metrics are the tree mover’s distance (Chuang and Jegelka, 2022), the cut distance of graphon signals (Levie, 2023), unbalanced transport distances with a mass-difference term (Böker et al., 2023; Rauchwerger et al., 2025), and lately one metric on graphs of every size, sparse and dense, through graphops (Amran et al., 2026). It covers the graph space and applies a generic statistical step, a Rademacher bound or the robustness bound of Xu and Mannor (2012) or its occupied-cell form (Kawaguchi et al., 2022). Covered Forest (Vasileiou et al., 2025a) is its latest form, with covering numbers through Weisfeiler–Leman classes. Ours has the same architecture, the lift with D, the strata with their covering numbers and the coupling, and differs in three places. The direction of the geometry: they start from a model and find a metric in which it is Lipschitz, so the Lipschitz ball is an envelope of the model and the metric one choice among several, where Rauchwerger et al. (2025) come closest with a converse and the model class dense in the continuous functions of their space. We start from the axioms, and the representation theorem is an equality. What is covered: they cover the space at one scale, or with the sample’s cell proportions at one radius (Carrasco et al., 2025, Prop. 4) and per-cell radii (Vasileiou et al., 2025a, Thm. 35), and we cover the measure, by the profile across the shells with a multiscale coupling within each, and by the sample’s own configurations. What is proved, and what is non-vacuous, below the trivial value of the risk in the sense of Dziugaite and Roy (2017) and of the risk certificates of Pérez-Ortiz et al. (2021): the covering number of a graph space is astronomical where it is known (Levie, 2023) and unbounded where it is not (Amran et al., 2026), so the theorems’ values are not reported on a benchmark, and the non-vacuous values Covered Forest reports evaluate the Weisfeiler–Leman classes the dataset occupies, with a Lipschitz constant estimated from the data, a counting of the kind of the occupied-cell bound of Kawaguchi et al. (2022), while the non-vacuous value of Ju et al. (2023) is a PAC-Bayes bound evaluated at the trained network through its loss Hessian, a bound on one trained network rather than over a class. The certificate of Theorem 4.6 is a theorem on the data at every step, and on the flock it is a thirtieth of the trivial value (Section 5.1). That a class large enough to hold a trained network makes every uniform bound vacuous (Nagarajan and Kolter, 2019) is an objection to the size of the class: here the class is the Lipschitz ball of the lift, its uniform deviation on the design is measured rather than bounded from above by a covering, and on the flock it is small. Empirical complexity. The bound behind the certificate is the classical one, the maximum discrepancy of Bartlett et al. (2002) through the symmetrization of Vapnik and Chervonenkis (1971), and what is ours is that on the class it is the transport itself, by the representation, and is computed at one observed split. A similar quantity has recently been used in a different context. Chuang et al. (2021, Thm. 2) take the expected transport between two independent samples of a feature distribution, the k-variance of Solomon et al. (2022), as the complexity term of a margin bound for the readouts of a trained encoder, with an empirical version at one concentration term (their Lemma 5 and Cor. 6), and compute it on data as a measure that tracks generalization rather than evaluating the bound. Hou et al. (2023) evaluate a transport bound with the counts of a fixed partition, and Zhu et al. (2021) compute sample covering numbers on image data with the bound itself not evaluated, in the shape of the sample-cover bound of Mansour and McAllester (2000). None of these is the bound of Theorem 4.6: theirs is a complexity term or a partition count entering a bound on a model, in the metric of its features, ours the observed gap of the class itself in the metric the class implies. The closest form is in domain adaptation, where the target risk of a Lipschitz class is bounded by the source risk plus the transport between the empirical source and target samples (Redko et al., 2017, Thm. 2), (Shen et al., 2018, Thm. 1), with the joint error of the two domains and one concentration term per sample: the two halves of Theorem 4.6 share a law, so the joint error is absent and one term serves, and the transport is a matching in the class’s own metric, computed exactly. Attention. Attention (Vaswani et al., 2017) gives each output its own field of view: the softmax weights over the inputs are a weight vector of the kind this paper is about, they vary with the input, and in some heads and on some inputs a few tokens carry most of the weight. Fixed-pattern attention restricts each output to a window or a sparse pattern chosen in advance (Child et al., 2019; Beltagy et al., 2020), a hard selector given, with the softmax inside the pattern a graded one. The Lipschitz constant of a whole attention map, weights included, is Kim et al. (2021), a statement about the composite rather than about the mechanism with its weights held as an input. Attention as a map on measures, a self-interacting particle system that is Lipschitz in a transport metric (Vuckovic et al., 2020), with a constant bounded by n n in the sequence length on a compact set and independent of it in the mean-field regime (Castin et al., 2024), is the same composite with uniform weights and the selector inside it. The learning-theory results nearest to ours are on transformers whose target depends on a few tokens of the context: a sample complexity almost independent of the context length when the positions of the q tokens are given in the prompt (Mousavi-Hosseini et al., 2025), logarithmic in it for sparse variable creation (Edelman et al., 2022), with the q-sparse average as the representational primitive (Sanford et al., 2023), length-independent norm bounds (Trauger and Tewari, 2024), an input-dependent importance of the coordinates (Takakura and Suzuki, 2023), and a per-sample sensitivity cap as the locality condition (Nagler, 2023), a one-layer transformer that learns the sparse token selection by gradient descent where a fully connected network cannot (Wang et al., 2024), and a sample complexity in the inverse fraction of the label-relevant tokens, a statistic of graded sparsity, with an attention map that training sparsifies (Li et al., 2023). Those have a selection of tokens, hard at a fixed count or graded by an importance the network infers, an ordered link and a parametric class, and ask which architecture learns to retrieve the tokens. Ours has graded weights given as input and a varying count, anonymity, the full Lipschitz class, a profile with lower bounds, and a certificate, and asks the complexity of the mechanism once the weights are given. Nikolentzos and Skianis (2026) show that attention pooling has instances that are Lipschitz for none of the three multiset distances they consider, a statement about the composite (1) with the selector inside it; with the weights an input and the cap an axiom, the mechanism f is Lipschitz in the metric the class determines, and the composite is not treated here. The mechanism is nevertheless outside the scope of this paper. Its weights are learned, together with everything else, and an attention block is not trained on its own, so the problem studied here, learning a mechanism whose selector is given, does not arise in that form. Interaction kernels. The learning of interaction laws in particle systems from trajectories is the subject of Lu et al. (2019); Lu et al. (2021): the mechanism is a mean-field sum N−1∑jφ(|xj−xi|)(xj−xi)N^-1 _j ( x_j-x_i )(x_j-x_i) with the scalar kernel φ unknown, and the least-squares estimator converges at the one-dimensional nonparametric rate, in the L2L^2 norm of the law of the pairwise distances along the trajectories, a measure-level norm. In the terms of Section 3.1 their class is the linear sums with the weights uniform over all agents and the score a fixed function of the pairwise displacement, so the rate is that of the one-dimensional kernel and no profile enters; here the class is every Lipschitz function of the lift, whose rate the profile sets, with the selector given. Their treatment of dependence along a trajectory is what the independent replicas of Section 5 sidestep. The same mean-field object with the drift a nonparametric function of the state and of the empirical measure is estimated from one trajectory per particle at the minimax rate of its Hölder class (Yao et al., 2022), uniform weights again and no selector. In the animal groups the object inferred is the selector itself, the interaction range of a starling flock from field data by a maximum-entropy fit, a fixed number of topological neighbors (Bialek et al., 2012), and in robot swarms a graph network learns the flocking controller with the communication graph given (Tolstaya et al., 2020): the second is the setting of this paper, the first its complement. The matching distance. The metric of Section 3.2 is produced by the axioms, and the derivation is new: none of the lines that built its ingredients starts from a class of functions. Each ingredient was met before from the other end, introduced to compare measures of unequal mass. Mass must be allowed to appear or vanish, the cheapest way to price that is a reservoir at a linear price per unit, and the height coordinate of the cone is the mass itself. The Kantorovich–Rubinstein norm extends from measures of mass zero to all finite measures by paying the variation of the untransported part, which gives the flat metric, whose dual is the space of bounded Lipschitz functions (Kantorovich and Rubinstein, 1958; Hanin, 1992), reached again from transport equations with a source term as the generalized Wasserstein distance (Piccoli and Rossi, 2014; Piccoli and Rossi, 2016). Figalli and Gigli (2010) let mass leak through the boundary of the domain at the cost of its squared distance to it. In unbalanced transport, the least cost of carrying mass m0m_0 at x0x_0 to mass m1m_1 at x1x_1, with transport at unit price and creation or destruction at price δ per unit, is min(|x1−x0|,2δ)min(m0,m1)+δ|m1−m0| ( x_1-x_0 ,2δ) (m_0,m_1)+δ m_1-m_0 (Chizat et al., 2018), the cone cost (2) at δ=Rδ=R. The same function is the marginal perspective of the total-variation entropy-transport problem, which Liero et al. (2018) rewrite as transport on the cone over the base space with mass-weighted marginals. A persistence diagram is a multiset of birth–death pairs, and its distance to another diagram lets a feature vanish at the cost of its distance to the diagonal, available with infinite multiplicity (Cohen-Steiner et al., 2010; Turner et al., 2014): the matching form with a reservoir on multisets of unit masses. On multisets of vectors the matching distances of Chuang and Jegelka (2022), Davidson and Dym (2025) and Nikolentzos and Skianis (2026) charge an unmatched element its norm, the origin as the reservoir. Divol and Lacombe (2021) identify it with boundary transport and compute it by padding each side with the other’s mass at the reservoir, the identity (5) (Lacombe et al., 2018, also). Bubenik and Elchesen (2022a) prove that on any metric space with a distinguished subset the Wasserstein distance on such multisets is universal, the largest subadditive metric compatible with the ground metric, with Kantorovich–Rubinstein duality at p=1p=1: Lemma A.9(i) is that theorem on the cone with its apex. What Theorem 3.2 adds is not the space but the identification. The configuration is a multiset because anonymity says so, the value cost at the smaller height is the value cap, the height is the cap weight rather than a mass, and the price R of a unit of height is the modulus of the linear sums (Proposition A.4). The axioms’ per-coordinate moduli integrate, with constant 11 and for every pair of configurations, to exactly the matching cost (Section A.3). Theorem 3.2(i), that the metric is the smallest making the class Lipschitz, is the class-side statement and a corollary of (i). Two consequences separate this from the mass-bookkeeping reading. The marginals are counting marginals, one atom per participant whatever its weight: with mass-weighted marginals the same cost gives the flat metric, whose Lipschitz ball is a strictly smaller class in which the selected maximum has no constant at all (Remark A.6). And the two semantics differ only in the height coordinate. The statistics done on these spaces asks different questions: consistency and minimax rates for the diagram of a sample under a support assumption (Chazal and Michel, 2021), populations of diagrams and their means (Mileyko et al., 2011; Turner et al., 2014), and plug-in rates for the Kantorovich–Rubinstein distance of unbalanced transport, for point-process intensities, under a covering dimension of the space (Struleva et al., 2025). On the diagram space a hypothesis on the space alone is weak, since with the bottleneck distance every separable bounded metric space embeds isometrically into it (Bubenik and Wagner, 2020), and Section 4 supplies what is missing, the design’s profile over the strata of the lift. Partitions, transport, and one-number dimensions. Bounding the transport distance between two measures by the mass they assign to the cells of a partition is the classical device behind the rates of convergence of empirical measures in Wasserstein distance from Dudley (1969) onward (Ajtai et al., 1984; Dereich et al., 2013; Boissard and Le Gouic, 2014; Fournier and Guillin, 2015), stated by Weed and Bach (2019, Prop. 4) in the form we use, one partition at one scale with a trimmed remainder. The stratified argument behind our rates has been used several times: Fournier and Guillin (2015) stratify ℝdR^d into annuli, Lei (2020) an unbounded function space into shells, and Huang et al. (2026), recently, the space of configurations into cardinalities, for the empirical measures of point processes, a setting that is the unweighted case of ours. We add two things. The first is continuity, the grading and the tubes: Theorem B.14 stratifies by the weight tail instead of the count, so that each stratum is a tube around a sparse set (Lemma B.12) with the crop in place of the truncation (Proposition B.11), and this is what gives rates on kernels of full support, where there is no count to stratify by, including the exponential rate from a decay inside the configuration (Theorem 4.5). The tail TkT_k itself is the best k-term ℓ1 ^1 approximation error of the weight vector (Cohen et al., 2009), so an exponential profile is a geometric compressibility of the weights on average, taken here as a learning parameter. The second is that rate itself, for the mean tail of a configuration with free weights and values: quasi-polynomial rates of the same shape are those of Lei (2020, Thm. 4.2), for empirical measures in a Hilbert space under an exponential-ellipsoid moment condition, and of Huang et al. (2026, Thms. 2, 3), for point processes with a geometric count law, the unweighted case of a hard-ball selector, and neither hypothesis is ours (Section D.4 for the fiber, where the class’s metric is Lei’s ellipsoid). The rates of Weed and Bach are in one number of the measure, its upper Wasserstein dimension, a limit in the scale, or at finite n the trimmed covering count optimized over one scale and one trim (Weed and Bach, 2019, Prop. 10). The entropic dimension of Chakraborty and Bartlett (2025), restated in Chakraborty and Bartlett (2024), is another such number, a trimmed covering in the limit of small scale, with a minimax lower bound in the upper Wasserstein dimension (Theorem 29 of the former), and the regression rates in the intrinsic dimension of the support (Bickel and Li, 2007; Kpotufe, 2011) read the support, which for the lifted flock is every stratum up to the crowded one. A design is not a measure with a dimension but a mixture of strata of different dimension with a mass on each, and the bulk of the lifted flock is such a union (Figure 2, counts from 22 to 2323 other than the self between the 5%5\% and 95%95\% views). The certificate resolves the strata at the sample’s own configurations, and the implied dimension of Section C.1 is the estimator of Block et al. (2022), who read the intrinsic dimension of a measure from the decay in n of the transport between two independent samples, applied to the lifted design. We keep from these works the point of contact, that a measure-level dimension controls rates, and the shape of the minimax lower bound.