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Multi-Scale Equilibrium under Variable Indicator Dimensionality: Faithful Reduction of Dynamic Attractors in Urban Mobility Systems
Ali Ghoroghi, Yacine Rezgui, Afrouz Ghaemi, Cristina De Nardi, Andrei Hodorog
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Summary
This paper presents a formal framework for analyzing urban mobility systems under disruption using a dynamic multi-layer equilibrium attractor. It addresses the mismatch between high-dimensional indicator spaces and variable data availability by establishing conditions under which lower-dimensional indicator projections remain faithful to the full system's equilibrium structure. Key contributions include proving conditions for exact and approximate projectability, preservation of coupled fixed points, retention of Fisher information for decision-making, and the identification of a one-sided bias in restoration time estimation when using reduced monitoring.
Entities (9)
Relation Signals (6)
Ali Ghoroghi → affiliatedwith → Cardiff University
confidence 99% · Ali Ghoroghi ∗1 ... 1 School of Engineering, Cardiff University
Multi-layer equilibrium attractor → consistsof → Performance layer
confidence 95% · It presents a dynamic multi-layer equilibrium attractor ... in which a fast performance layer relaxes
Multi-layer equilibrium attractor → consistsof → Strategic layer
confidence 95% · a slow strategic layer supplies a joint traffic, modal and learning fixed point
Strategic layer → determines → Antifragility
confidence 88% · antifragility is classified through a statistical decision rule on the post-to-baseline performance ratio
Indicator Projection → causes → Restoration time bias
confidence 87% · reduced monitoring can only understate recovery duration
Indicator Projection → affects → Fisher Information
confidence 85% · the retained Fisher information and decision power of any indicator support
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Abstract
Abstract:Equilibrium analysis of urban mobility systems is formulated in a high-dimensional indicator space, whilst data availability varies sharply across cities and disruption contexts. This paper gives a formal treatment of that mismatch. It presents a dynamic multi-layer equilibrium attractor for disrupted urban mobility, in which a fast performance layer relaxes towards an indicator-dependent target, a slow strategic layer supplies a joint traffic, modal and learning fixed point, and antifragility is classified through a statistical decision rule on the post-to-baseline performance ratio. It then characterises when a lower-dimensional indicator projection is faithful to this equilibrium structure, establishing four results: conditions for exact and approximate projectability of the attractor with an explicit error bound; preservation of the coupled two-layer fixed point up to a contraction boundary; the retained Fisher information and decision power of any indicator support under a measurement model on observable urban indicators; and a one-sided restoration-time bias, whereby reduced monitoring can only understate recovery duration. A simulation study on three stylised pilot-city configurations verifies each result, and shows that two observable channels suffice for the candidate classification target where the indicator catalogue permits. The framework gives city authorities a principled basis for deciding which indicators must be maintained.
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- Source: https://arxiv.org/abs/2607.14815v2
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Multi-Scale Equilibrium under Variable Indicator Dimensionality: Faithful Reduction of Dynamic Attractors in Urban Mobility Systems Ali Ghoroghi ∗1 , Yacine Rezgui 1 , Afrouz Ghaemi 1 , Cristina De Nardi 1 , and Andrei Hodorog 1 1 School of Engineering, Cardiff University, Cardiff, United Kingdom Abstract Equilibrium analysis of urban mobility is formulated in a high-dimensional indic- ator space, whilst data availability varies sharply across cities and disruptions. This paper treats that mismatch formally. It presents a dynamic multi-layer equilibrium attractor for disrupted urban mobility, in which a fast performance layer relaxes towards an indicator-dependent target, a slow strategic layer supplies a joint traffic, modal and learning fixed point, and antifragility is classified through a statistical decision rule on the post-to-baseline performance ratio. It then characterises when a lower-dimensional indicator projection is faithful to this structure, through four results: exact and approximate projectability with error bounds covering non-normal transient amplification; preservation of the coupled two-layer fixed point up to a contraction boundary, with a displacement bound under approximate commuta- tion; the retained Fisher information and decision power of any indicator support, with greedy selection exactly optimal for independent channels; and a one-sided restoration-time bias, made exact as an asymptotic ordering, whereby reduced monitoring can only understate recovery duration. A simulation study on three stylised pilot-city configurations verifies each result; two observable channels suffice for the classification target where the catalogue permits. The framework gives city authorities a principled basis for deciding which indicators must be maintained. Keywords: dynamic attractors; dimension reduction; multi-layer networks; composite indicators; Fisher information; urban mobility; antifragility. Subject Areas: applied mathematics; complexity; civil engineering. 1 Introduction The mathematical problem addressed in this paper is the well-posedness of a high- dimensional dynamic attractor when only a subset, projection or aggregation of its ∗ Corresponding author. Email: ghoroghi@cardiff.ac.uk 1 arXiv:2607.14815v2 [cs.MA] 21 Jul 2026 indicators is observable. Dynamical systems theory contains mature results on local reduction near equilibria, including centre manifold theory, inertial manifolds, singular perturbation and Galerkin projection [Carr, 1981, Guckenheimer and Holmes, 1983, Khalil, 2002, Fenichel, 1979, Temam, 1988, Holmes et al., 1996]. Multi-layer network theory contains complementary results on aggregation, supra-Laplacian spectra, community structure and reducibility of multiplex representations [Mucha et al., 2010, Kivel ̈a et al., 2014, De Domenico et al., 2013, G ́omez et al., 2013, Boccaletti et al., 2014, Radicchi and Arenas, 2013]. Composite-index theory further analyses sensitivity, identifiability, weighting uncertainty and information loss when multi-indicator constructs are compressed into lower-dimensional scores [Saltelli et al., 2004, OECD and European Commission Joint Research Centre, 2008, Saisana and Tarantola, 2002, Saltelli et al., 2008, Nardo et al., 2005]. Notwithstanding the foregoing, none of these literatures directly answers the question that arises when an equilibrium attractor is defined simultaneously as a dynamical object, a coupled-layer fixed point and a statistical decision device. The conceptual difficulty is that indicator reduction is not merely a matter of ap- proximation error. In a standard projection problem one may seek a reduced state whose trajectory is close to the original trajectory under a chosen norm; in a standard network aggregation problem one may seek preservation of selected spectral or community properties; and in a standard composite-index problem one may seek sensitivity bounds for a scalar score. The attractor considered here imposes three concurrent requirements: the projected system must preserve asymptotic stability, the reduced performance and strategic layers must retain the same hierarchical fixed-point interpretation, and the can- didate antifragility decision rule must retain enough information to distinguish recovery, resilience and improvement. This combination makes the reduction problem structurally different from classical model-order reduction and from ordinary indicator selection. The starting point is a multi-layer attractor formalism for disrupted urban mobility, presented in full in Section 2. In that framework, a performance-layer state variable relaxes towards a target function defined over a vector of normalised indicators, while a slower strategic layer supplies a joint traffic, modal and learning fixed point. Antifragility is determined by a candidate decision rule comparing a post-event performance ratio with a candidate baseline-dependent threshold. Having established that attractor, the paper asks when it remains meaningful if the full indicator vector is replaced by a lower-dimensional support, as occurs when different cities maintain different data infrastructures or when some measurements are unavailable during a disruption. The formal contribution is a theory of multi-scale equilibrium reducibility under variable indicator dimensionality. Given a full indicator statex ∈ R n and a reduction Π :R n → R m , withm < n, the paper defines faithfulness relative to the attractor rather than relative to trajectory reconstruction alone. A faithful reduction is required to preserve the stability class of the performance layer, the joint two-layer fixed point and the statistical power of the candidate antifragility test up to an explicitly quantified tolerance. This definition is deliberately stronger than common projection requirements because a reduced equilibrium analysis can be numerically convenient whilst being mathematically unfaithful. The novelty lies in joining three preservation criteria that have usually been studied apart, and in attaching to them an explicit measurement model. Centre manifold and slow-fast results identify circumstances under which local dynamics may be represented by fewer variables [Carr, 1981, Fenichel, 1979, Kuehn, 2015]. Galerkin and balanced truncation methods approximate high-dimensional dynamics but do not usually preserve 2 an external decision rule tied to indicator support [Holmes et al., 1996, Antoulas, 2005]. Multiplex aggregation results identify when layers may be collapsed without excessive spectral distortion, yet they generally do not treat a Lyapunov stability condition and a composite-index hypothesis test in the same theorem [Kivel ̈a et al., 2014, De Domenico et al., 2015]. Indicator sensitivity studies quantify uncertainty in weights and inputs, yet they do not normally address whether the underlying dynamical fixed point survives the same reduction [Saltelli et al., 2004, OECD and European Commission Joint Research Centre, 2008]. The present paper therefore studies a coupled reducibility problem rather than an isolated approximation problem, and it grounds that problem in the indicator infrastructure that cities actually maintain. Of particular interest is the boundary below which partial indicator coverage ceases to support the equilibrium claim. A city may observe throughput and stress but lack reliable redundancy, equity or cross-layer coupling data. Another city may observe a wider subset but with missing energy or communication-system indicators. A third city may provide near-full support but with heterogeneous temporal sampling. In each case an attractor can still be computed, but computation alone does not imply that the projected system has inherited the relevant stability, fixed-point and detection properties. The central question is therefore not whether a reduced score can be produced, but whether that score remains a faithful projection of the full equilibrium structure. The paper makes five contributions. First, it presents a self-contained multi-layer equilibrium attractor for disrupted urban mobility systems, comprising a relax-to-target performance law, a hierarchical strategic fixed point combining user-equilibrium assignment, modal split and capacity learning, and a statistically testable antifragility decision rule; the framework is stated with its stability, existence and uniqueness properties. Secondly, it gives a necessary and sufficient condition for preservation of local asymptotic stability under linear indicator projection, formulated through projectability of the vector field and the Hurwitz property of the reduced Jacobian, together with an approximate-projectability theorem that bounds the projected-trajectory error when projectability holds only up to a measurable defect, together with a transient-amplification extension that covers non- normal reduced dynamics through a certified, computable amplification constant. Thirdly, it proves a coupled fixed-point preservation theorem for a two-layer performance-strategic map, identifies the sufficient contraction boundary at which the uniqueness certificate is lost, and bounds the fixed-point displacement when the reduced maps commute with the projection only approximately. Fourthly, it introduces a measurement model in which the equilibrium state variables are estimated from observable urban key performance indicators, derives the retained Fisher information of any indicator support in closed form under Gaussian measurement noise, and proves that detection power is monotone in nested indicator supports. It further shows that greedy indicator selection is exactly optimal under the independent-channel measurement model. Fifthly, it proves that restoration time to equilibrium estimated from a reduced indicator set never exceeds, and generically understates, the restoration time of the full system, so that reduced monitoring is optimistically biased. These contributions are theoretical. The numerical section specifies the simulation protocol through which they are illustrated on three stylised pilot-city configurations. The remainder of the paper is organised as follows. Section 2 presents the multi-layer equilibrium attractor, its stability and fixed-point properties, and the antifragility decision rule; the reducibility theory of the later sections examines the behaviour of these constructs when the indicator vector is replaced by a lower-dimensional projection and estimated from 3 observable urban indicators. Section 3 reviews reduction theory in dynamical systems, layer aggregation in multi-layer networks, sensitivity analysis for composite indicators and the design-of-experiments view of indicator selection. Section 4 develops the formal reducibility framework and proves the principal results, including the measurement model that connects the state variables to observable urban indicators. Section 5 specifies the simulation protocol and the figures and tables through which the results are to be reported. Section 6 discusses boundary cases, limitations and implications for minimum data infrastructure. Section 7 concludes. 2 The multi-layer equilibrium attractor This section presents the equilibrium formalism whose reducibility is the subject of the paper. The state of the urban mobility system is summarised by a scalar performance scoreE(t)∈[0,1], normalised on a rolling 30-day min-max window from a vector of normalised indicators. The full indicator state is the twelve-coordinate vector x = (M,R,A,C,D eff ,S,Q,P,B,E energy ,E ICT ,I) ⊤ ∈ [0, 1] 12 ,(1) whose coordinates denote, respectively, mobility throughput, network redundancy, ad- aptation velocity, network entropy, effective disturbance, stress, equity penalty, demand- management coverage, behavioural compliance, energy-system support, communication- system support and cross-layer synergy. The twelve quantities are computed constructs, each assembled from urban indicators that municipalities and transport operators record; the assembly rules supply the measurement loadings used in Section 4.4, and the quantities are treated as candidate constructs unless an empirical source is attached to a specific deployment. Boldface x(t) is used when the vector is regarded as a time-varying input to the target function. Two layers of equilibrium are distinguished. The fast performance layer is specified by a relaxation law onE(t); the slow strategic layer is specified by a hierarchical fixed point combining user-equilibrium traffic assignment, modal split and a capacity-learning update. The strategic layer produces the slow-timescale baselines that the performance layer tracks; the performance layer in turn supplies the post-event trajectories from which the strategic layer learns. 2.1 Performance-layer dynamics The performance state evolves according to the relax-to-target ordinary differential equa- tion ̇ E(t) = −κ E(t)− ̄ E(x(t)) , κ > 0,(2) with discrete-time updateE t+∆t =E t −κ∆t(E t − ̄ E(x t )), in whichκ∆t∈(0,1] preserves monotone convergence, values in (1,2) converge with overshoot and values at or above two are unstable. The target function is decomposed into a positive block and a negative penalty block balanced by a single calibration parameter λ∈ (0, 1), ̄ E(x) = λ ̄ E + (x)− (1− λ) ̄ E − (x),(3) 4 with the two blocks specified term by term as ̄ E + (x) = α 1 ˆ M 1 + ˆ M + α 2 ˆ R 1 + ˆ R + α 3 A + α 4 ψ(C) + ω 1 E energy + ω 2 E ICT + ζ I + π dm P + μ b B, (4) ̄ E − (x) = β 1 D eff + β 2 S + θ eq Q + γ (1− I coupling ),(5) where ˆ M =min1,M/K M and ˆ R =min1,(R−1)/(K R −1)are capacity-bounded transforms with carrying capacitiesK M ,K R , the saturating forms ˆ M/ (1+ ˆ M ) and ˆ R/ (1+ ˆ R ) assign diminishing marginal gains near the capacity bounds,ψ(C) is a monotone transform of network entropy, and all weights are non-negative. Two cross-layer constructs appear. The synergy indexI(t) is a coordinate of the state vector, rewarding spatially proximate intermodal infrastructure. The imbalance penaltyI coupling (t) =exp(− P l̸=m κ lm |E l −E m |) is a derived quantity computed from the performance levelsE l of the transport, energy, communication and governance layers; it equals one when the layers perform equally and falls towards zero as they diverge, so the termγ(1− I coupling ) penalises cross-layer imbalance. For the reduction analysis,I coupling is treated as a smooth function of the retained layer-support coordinates (E energy ,E ICT ) and the scalar performance readings of the remaining layers, so it introduces no additional state coordinate. All components are normalised to [0,1] and the target is projected onto [0,1] before entering Eq.(2). The specific composition of the two blocks supplies the loading structureC S used by the measurement model of Section 4.4; the reducibility theory of Section 4 depends only on smoothness and boundedness of ̄ E, not on a particular weighting. Proposition 1 (Asymptotic stability for fixed inputs). Let x ∗ be a time-invariant input vector and ̄ E ∗ = ̄ E(x ∗ ). Along trajectories of Eq.(2), the Lyapunov functionV(E) = 1 2 (E − ̄ E ∗ ) 2 satisfies ̇ V=−κ(E − ̄ E ∗ ) 2 ≤0, with equality only atE= ̄ E ∗ . Hence E(t)→ ̄ E ∗ exponentially with time constant 1/κ, and the discrete update contracts with factor 1− κ ∆t whenever 0 < κ ∆t < 2. The performance-layer equilibrium is the point E ∗ = ̄ E(x ∗ ), asymptotically stable in the sense of Proposition 1 and tracking slow shifts in the input vector at the calibrated rate κ. Assumption 1 (Indicator recovery dynamics). Between strategic updates, the indicator state itself evolves under a continuously differentiable autonomous recovery flow ̇x= f(x) on a neighbourhoodU ⊂ R 12 of a locally asymptotically stable equilibriumx ∗ , representing the post-disruption relaxation of throughput, redundancy, stress and the remaining coordinates towards the baselines supplied by the strategic layer. The scalar performance variable is the induced observableE=h(x) withhsmooth, or equivalently an additional coordinate driven by x through Eq. (2). Assumption 1 is the bridge between the attractor of this section and the reducibility theory of Section 4: the projection results are stated for the indicator flowf, and the scalar performance layer inherits its reduced behaviour through the induced observable. The assumption is a modelling commitment rather than a theorem; it holds exactly for the linearised recovery dynamics used in the simulation protocol of Section 5 and approximately whenever indicator recovery is smooth on the post-stabilisation window. 5 2.2 Strategic-layer hierarchical equilibrium On the strategic timescale, the system tracks a hierarchical fixed point. LetG net = (N,A) denote the directed transport network with arc capacitiesc a and fixed origin-destination demands q od . The assignment layer solves the Beckmann programme min f X a∈A Z f a 0 t a (x)dx s.t. X p f od p = q od , f od p ≥ 0, f a = X p f od p ⊮a∈ p,(6) with link costt a (f a ) =t 0 a [1 + 0.15 (f a /c a ) 4 ] in the standard form of Bureau of Public Roads [1964]. The modal layer solves the multinomial logit fixed point P od m = exp(−θ logit U od m ) P j exp(−θ logit U od j ) , q od m = P od m q od ,(7) coupled to Eq.(6)through the mode-specific generalised costsU od m . The capacity-learning layer applies the projected multiplicative update c k+1 = Π C n c k ⊙ 1 + α LS η s g short + η ℓ g long o ,(8) where g short ,g long are short- and long-horizon sensitivities ofEto capacity, the step sizes satisfyη s :η ℓ = 10 : 1,α LS is selected by Armijo line search, and Π C projects onto a compact convex feasibility set of budget, geometry, safety and equity constraints. Throughout,θ logit andα LS are reserved for the logit sensitivity and the line-search step; the bare symbolsθandαare reserved for the statistical effect parameter and test level of Section 4.4. Proposition 2 (Existence, and conditions for uniqueness, of the strategic fixed point). Assumet a ∈ C 1 witht ′ a > 0, a compact feasible setC=c : c min ≤c≤c max with c min ≫ 0, and bounded demands. Then for each fixed c∈ Cthe user-equilibrium link flows exist and are unique by strict convexity of the Beckmann objective [Beckmann et al., 1956], the logit fixed point exists by Brouwer’s theorem, and a joint fixed point of the three operators exists by compactness and continuity of the composed capacity map. Uniqueness of the joint fixed point requires in addition that the composed capacity map is a contraction onC; strict convexity of link costs alone does not imply it. When the strategic- to-performance handoff is continuous with normalising denominators bounded away from zero, the coupled system admits a joint two-layer equilibrium tuple (E ∗ , f ∗ , P ∗ , c ∗ ). Proof.Let Γ(c) denote the strategic-equilibrium correspondence assigning to each feasible capacity vector the set of assignment and modal-split equilibrium outputs (f ∗ ,P ∗ ). Under strict monotonicity of the link costs, Γ is nonempty, and single-valued and continuous in link flows by strict convexity of the Beckmann objective; the logit fixed point exists by Brouwer’s theorem on the probability simplex. Composing Γ with the continuous strategic-to-performance handoff and the target function gives a performance set-point E o (c) = ̄ E(x o (c)) that is bounded and continuous in the single-valued case, or upper hemicontinuous with compact values otherwise. The induced capacity update is Φ(c) = Π C c⊙[1 +α LS (c)G(c)]. The expression inside the projection is finite becauseClies in the strictly positive orthant and the gradient field and step selection are bounded and continuous, and the projection maps it back intoC. Since projection onto a nonempty closed convex set is continuous, Φ is a continuous self-map on the compact convex set 6 Cin the single-valued case, and Brouwer’s theorem yields c ∗ = Φ(c ∗ ). If route-flow or step-length non-uniqueness is retained, the closed convex hull of admissible updates defines a correspondence with nonempty compact convex values and a closed graph, and Kakutani’s theorem yields c ∗ ∈Φ(c ∗ ). Selecting any (f ∗ ,P ∗ )∈Γ(c ∗ ) and setting E ∗ = ̄ E(x o (c ∗ )) closes the performance and strategic layers. The argument establishes joint existence only; uniqueness and asymptotic convergence of the fully coupled iteration require the contraction condition stated above, which Theorem 4 shows must additionally be preserved under any indicator reduction if the reduced system is to inherit the joint fixed point. The contraction condition on the composed capacity map reappears in Section 4 as the quantity whose preservation under reduction is at issue: Theorem 4 identifies the boundary at which a reduced coupling matrix ceases to be contractive and joint uniqueness can be lost. 2.3 The antifragility decision rule Antifragility is defined at the level of the attractor: the post-stabilisation equilibrium strictly exceeds the pre-event baseline by a margin distinguishable from noise and season- ality. The statistical form of that criterion operates on the post-to-baseline ratio of the performance state. LetE base be the seasonally adjusted baseline mean over the pre-event window, computed by seasonal-and-trend decomposition using Loess [Cleveland et al., 1990] or X-13, and let μ 30 = 1 30 60 X t=31 E(t), R ratio = μ 30 E base ,(9) so that the post-stabilisation mean is taken over days 31 to 60 after the event, past the relaxation transient; when stabilisation is slower, the window is the first thirty days after confirmed stabilisation. Two adaptive thresholds are defined: a requirement threshold AF req = 1 + 0.2 (1−E base ), rising with the strictness of the baseline, and a noise threshold AF crit = 1 +k ˆσ resid withk= 1.64, rising with the residual variability of the pre-event series. The decision rule declares antifragility when LCL 0.95 (R ratio ) ≥ maxAF req , AF crit ,(10) where the lower confidence limit is computed by block bootstrap (block size 5 days, 10 000 resamples) when autocorrelation is material and by the delta method otherwise. Improvement is thereby treated as a statistical statement rather than a point estimate. Eq.(10)is the decision rule whose detection power under reduced indicator support is quantified in closed form in Section 4.4. The three constructs of this section, namely the relax-to-target law of Eq.(2), the coupled two-layer fixed point of Proposition 2 and the decision rule of Eq.(10), jointly define the equilibrium attractor. The remainder of the paper studies the conditions under which this attractor survives projection onto a lower-dimensional indicator support. 7 3 Literature review 3.1 Dimension reduction in dynamical systems Dimension reduction in dynamical systems is concerned with the conditions under which the evolution of a high-dimensional state can be represented on a lower-dimensional object without losing the properties of interest. Centre manifold theory establishes that, near a non-hyperbolic equilibrium, local behaviour may be governed by a lower-dimensional invariant manifold tangent to the centre eigenspace [Carr, 1981, Guckenheimer and Holmes, 1983]. Singular perturbation theory and geometric slow-fast analysis establish related reductions when variables evolve on separated time scales, provided normal hyperbolicity and suitable regularity are present [Fenichel, 1979, Khalil, 2002, Kuehn, 2015]. These theorems are powerful because they define the reduced dynamics as a mathematically induced object, not merely as a fitted approximation. Projection methods provide a second tradition. Galerkin projection, proper orthogonal decomposition and balanced truncation construct a lower-dimensional coordinate system from basis functions or energy-dominant modes [Holmes et al., 1996, Antoulas, 2005, Temam, 1988]. These methods have been central in fluid dynamics, control and spatially extended systems, where full-order simulation may be computationally prohibitive. Their usual preservation criteria are trajectory error, input-output error, energy content or spectral approximation. Stability can sometimes be retained under special constructions, but preservation of an externally defined composite decision rule is not part of the usual problem formulation. A related strand quantifies robustness of trajectories to bounded model defects through logarithmic norms and contraction analysis [Desoer and Vidyasagar, 1975, S ̈oderlind, 2006, Lohmiller and Slotine, 1998]; that machinery is used in Section 4.1 to convert the binary faithful/unfaithful classification into a graded one. The limitation of these traditions for the present problem is that indicator support is not chosen solely by dynamical salience. It is constrained by urban data availability, institutional access, measurement quality and temporal sampling. A projection may remove a dynamically fast variable, a statistically informative variable, or a layer-coupling variable. These cases have different consequences. The formal framework must therefore distinguish between reductions that remove redundant coordinates and reductions that remove coordinates needed to define the attractor itself. 3.2 Layer aggregation in multi-layer networks Multi-layer network theory studies systems in which vertices, edges or dynamics are distributed across interacting layers. The supra-adjacency and supra-Laplacian formalisms provide a common representation for multiplex and interconnected networks [Mucha et al., 2010, De Domenico et al., 2013, G ́omez et al., 2013, Kivel ̈a et al., 2014]. Aggregation may be justified when layers have similar structure, when inter-layer coupling is sufficiently strong, or when the quantity of interest is insensitive to layer-specific variation [Radicchi and Arenas, 2013, De Domenico et al., 2015]. These ideas have been used to study diffusion, centrality, community structure and spreading processes in multiplex systems [Boccaletti et al., 2014, Aleta and Moreno, 2019]. Layer aggregation is directly relevant to urban mobility because transport, energy, com- munication and governance systems are interdependent. Nevertheless, most aggregation results preserve graph-theoretic or process-specific quantities rather than an equilibrium attractor with an attached statistical decision rule. A supra-Laplacian eigenvalue may 8 remain close after aggregation whilst an equity-sensitive candidate antifragility threshold becomes unidentifiable. Conversely, an indicator projection may preserve a scalar per- formance score whilst breaking the fixed-point interpretation of the strategic layer. A joint condition is therefore required. The present paper treats layer aggregation and indicator reduction as related but distinct operations. Layer aggregation changes the representation of interdependencies across subsystems. Indicator reduction changes the coordinates through which the attractor is observed and tested. When the two operations are coupled, the reduced map must commute with both the dynamical flow and the inter-layer fixed-point operator. This commutation requirement is central to the fixed-point theorem in Section 4.2. 3.3 Sensitivity, identifiability and the design of indicator sets Composite indicators provide a third body of relevant work because equilibrium targets are often constructed from weighted normalised indicators. Sensitivity analysis studies how uncertainty in inputs, weights and aggregation rules affects composite scores and rankings [Saltelli et al., 2004, 2008, OECD and European Commission Joint Research Centre, 2008]. Identifiability analysis asks whether parameters or latent constructs can be inferred from observable quantities [Walter and Pronzato, 1997, Raue et al., 2009]. Fisher information provides a local measure of how much an observation model informs a parameter or hypothesis [Kay, 1993, Lehmann and Romano, 2005]. A fourth, smaller body of work is directly relevant once a measurement model is attached to the attractor: the theory of optimal experimental design, in which observation channels are selected to maximise a functional of the information matrix [Pukelsheim, 1993, Fedorov, 1972]. In the present setting the observation channels are urban key performance indicators maintained by city authorities and transport operators, the cost of a channel is the institutional and technical cost of collecting it, and the design question is which channels suffice to preserve the attractor properties being claimed. Classical design theory supplies the information calculus; it does not supply the dynamical and fixed-point preservation conditions, which is the gap this paper closes. The composite-indicator literature is highly relevant but incomplete for the attractor problem. It can show that a reduced indicator set has high uncertainty or low sensitivity with respect to a particular score. It does not by itself show that the reduced score corresponds to a stable equilibrium projection or that a coupled fixed point remains unique. Moreover, the loss of an indicator has two roles in the present setting: it may change the target value towards which the performance layer relaxes, and it may reduce the statistical power of the decision rule used to classify post-disruption behaviour. These two roles must be handled together. 3.4 Summary and research gaps The foregoing literature establishes that many forms of reduction can be mathematically justified, but it does not provide a criterion for reductions that must preserve three structures at once. A reduced dynamical system may be stable but statistically unin- formative. A reduced network may preserve a spectral gap but fail to preserve the joint performance-strategic fixed point. A reduced composite indicator may be sensitive to the right variables but lack a well-defined underlying flow. The research gap is therefore a multi-property reducibility theorem for equilibrium attractors under variable indicator 9 Full state space R n Reduced state space R m Observed KPI space R r Π C † S performance stability joint fixed point decision power Figure 1: The reduction Π from the full state space to the reduced state space, with three concurrent preservation requirements, and the measurement layer through which the reduced state is estimated from observable urban key performance indicators via a retained measurement map C S (Section 4.4). support, expressed in terms of the urban indicators that cities can in fact collect. The formal framework developed next addresses this gap by defining faithfulness as a property of a projection relative to an attractor, a coupled fixed-point map, a decision rule and a measurement model. This definition allows a reduced city configuration to be classified as faithful, practically faithful or unfaithful according to mathematically explicit conditions. 4 Theoretical framework The formal hypothesis of the paper is that variable indicator dimensionality can be treated as a quotient problem for a dynamic attractor, provided that the projection respects the flow, the coupled fixed-point operator and the information geometry of the decision rule. This hypothesis leads to four research questions: RQ1. Under what conditions does an indicator projection preserve asymptotic stability of the performance-layer attractor, and how does stability degrade when those conditions hold only approximately? RQ2.Under what conditions does a coupled performance-strategic fixed point survive simultaneous reduction of state and layer-coupling variables? RQ3.How does the statistical power of the candidate antifragility decision rule de- grade as the retained indicator support shrinks, and how can that degradation be computed from the urban indicators a city actually collects? RQ4.How are derived quantities reported to stakeholders, in particular the restora- tion time to equilibrium, distorted by indicator reduction? 4.1 The reducible attractor The full performance layer is represented by a continuously differentiable vector field ̇x = f (x), x∈ U ⊂ R n ,(11) wherexis the vector of normalised indicators andx ∗ is a locally asymptotically stable equilibrium. The scalar equilibrium performance variable of Section 2.1 is treated here as a smooth observableE=h(x) or, equivalently, as an additional coordinate whose 10 relaxation law is driven byx. The reduction problem is stated forx, since the scalar performance variable inherits its reduced behaviour throughh◦Π † when a representative or section is chosen. Definition 1 (Indicator projection). A reduction is a surjective linear map Π :R n → R m , withm < n. Two full statesx 1 ,x 2 ∈ Uare projection-equivalent if Πx 1 = Πx 2 . The corresponding quotient coordinate is denoted y = Πx. Definition 2 (Projectability). The vector fieldfis projectable through Π onUif there exists a continuously differentiable vector field g : ΠU → R m such that Πf (x) = g(Πx), x∈ U.(12) Equivalently, Πf (x 1 ) = Πf (x 2 ) whenever Πx 1 = Πx 2 . Projectability is the mathematical condition that prevents a reduced trajectory from depending on unobserved coordinates in an uncontrolled way. If it fails, two cities or scenarios with identical reduced indicator vectors may have different reduced velocities because hidden indicators affect the projected dynamics. In that case a reduced attractor can be fitted, but it is not a quotient of the full attractor. Definition 3 (Faithful performance-layer reduction). Letx ∗ be a locally asymptotically stable equilibrium of Eq.(11)and lety ∗ = Πx ∗ . A projection Π is performance-faithful on a neighbourhoodUofx ∗ if there exists a reduced vector fieldgsatisfying Eq.(12),y ∗ is a locally asymptotically stable equilibrium of ̇y=g(y), and every full trajectory beginning in U satisfies Πx(t) = y(t) for the reduced trajectory with y(0) = Πx(0). Theorem 1 (Preservation of asymptotic stability under projection). Letf ∈ C 1 (U,R n ) and letx ∗ be an equilibrium of Eq.(11). Let Π :R n → R m be surjective and letA=Df(x ∗ ). Under condition (i) below, the reduced JacobianBis well-defined by the relationBΠ = ΠA. The projection Π is performance-faithful in a sufficiently small neighbourhood ofx ∗ if and only if the following conditions hold: (i) f is projectable through Π on that neighbourhood; (i) the induced reduced Jacobian B is Hurwitz; (i)the equilibrium fibre Π −1 (y ∗ )∩ Ucontains no second equilibrium whose projection is y ∗ and whose local basin intersects U outside the basin of x ∗ . Proof. Assume first that Π is performance-faithful. The equality of projected full tra- jectories and reduced trajectories implies that the derivative att= 0 depends only on y= Πx, hence Πf(x) =g(Πx) for somegand projectability follows. Linearising Eq.(12) atx ∗ gives ΠA=BΠ, whereB=Dg(y ∗ ). Sincey ∗ is locally asymptotically stable for the reduced system, the linearisation theorem implies thatBis Hurwitz when the equilibrium is hyperbolic; in the non-hyperbolic case Hurwitzness is replaced by the corresponding Lyapunov condition on the reduced centre dynamics, but the present theorem is stated for the hyperbolic case. If a second equilibrium in the same fibre had a basin intersecting the neighbourhood outside the basin ofx ∗ , two projected trajectories with the same initial reduced state would either fail uniqueness or converge to incompatible full equilibria under the same reduced equilibrium, contradicting faithfulness as defined. 11 Conversely, suppose the three conditions hold. Projectability definesguniquely on ΠU. The relationBΠ = ΠAis the linear quotient relation. SinceBis Hurwitz,y ∗ is locally asymptotically stable for the reduced system. For any full solutionx(t) beginning inU, the curve Πx(t) satisfies ̇y= Πf(x(t)) =g(Πx(t)), and uniqueness of solutions gives equality with the reduced trajectory. The fibre condition rules out ambiguous projected equilibria in the neighbourhood. Hence the projection is performance-faithful. Remark 1 (On the fibre uniqueness condition). The fibre uniqueness condition fails, for example, in a two-attractor system in which two locally stable equilibriax ∗ 1 ̸=x ∗ 2 project to the same reduced equilibriumy ∗ , so that the projected dynamics cannot distinguish them. In such a case, a reduced trajectory may appear stable while concealing incompatible full-state limits. The condition is generic for sufficiently generic vector fields, because distinct isolated equilibria will not usually be identified by a projection chosen in general position. Its failure indicates a coordinate choice in which the projection collapses distinct attractors. Corollary 1 (Linear indicator dynamics). For ̇x=Ax+bwith equilibriumx ∗ , a projection Π preserves asymptotic stability exactly whenker(Π) isA-invariant and the quotient matrix BsatisfyingBΠ = ΠAis Hurwitz, subject to uniqueness of the equilibrium in the projected fibre. Proof.For linear dynamics, projectability is equivalent to ΠAz= 0 for everyz ∈ ker(Π), which is equivalent to A ker(Π)⊆ ker(Π). The result then follows from Theorem 1. The corollary identifies why indicator removal can be dangerous. If a removed indicator feeds back into a retained indicator, then the kernel of the projection is not invariant and no autonomous reduced flow exists on the retained variables. In practical terms, throughput and stress cannot define a faithful reduced system if unobserved redundancy, equity or communication-system indicators change their dynamics without being represented in the reduced coordinates. Approximate projectability and practical faithfulness Exact projectability is a strong requirement: in deployed urban systems the omitted indicators almost always exert some residual influence on the retained ones. The classific- ation should therefore be graded rather than binary. The grading is achieved through the projectability defect. Definition 4 (Projectability defect). Letg: ΠU → R m be a continuously differentiable candidate reduced vector field, for example the conditional closureg(y) =E[Πf(x)|Πx=y] under a reference distribution of full states, or the least-squares closure on a calibration set. The projectability defect of the pair (Π,g) on U is ε Π = sup x∈U Πf (x)− g(Πx) .(13) Exact projectability corresponds to ε Π = 0 for some admissible g. Theorem 2 (Practical faithfulness under approximate projectability). Letf ∈ C 1 (U,R n ) and let (Π,g) have projectability defectε Π onU. Suppose there exist a norm onR m and a constantα >0 such that the logarithmic norm of the reduced Jacobian satisfies μ Dg(y) ≤−αfor allyin a convex setV ⊇ΠU. Letx(t) be a full trajectory remaining 12 inUon [0,T] and lety(t) solve ̇y=g(y) withy(0) = Πx(0), remaining inV. Then the projection error e(t) = Πx(t)− y(t) satisfies ∥e(t)∥ ≤ e −αt ∥e(0)∥ + ε Π α 1− e −αt ≤ ∥e(0)∥e −αt + ε Π α , t∈ [0,T ].(14) In particular, the projected full trajectory converges to a neighbourhood of the reduced attractor of radius ε Π /α, and the bound is tight in the linear case. Proof.Write ̇e= Πf(x(t))−g(y(t)) = g(Πx(t))−g(y(t)) +d(t) withd(t) = Πf(x(t))− g(Πx(t)), so that∥d(t)∥≤ ε Π by Definition 4. SinceVis convex and contains both Πx(t) andy(t), the mean-value form of the logarithmic-norm estimate [Desoer and Vidyasagar, 1975, S ̈oderlind, 2006] gives, for the upper right Dini derivative, D + ∥e(t)∥ ≤ μ Dg(ξ(t)) ∥e(t)∥ +∥d(t)∥ ≤ −α∥e(t)∥ + ε Π , for someξ(t) on the segment joining Πx(t) andy(t). Integrating the scalar differential inequality yields Eq.(14). Tightness in the linear case follows by takingf(x) =Axwith Anormal,g(y) =Bythe quotient on the orthogonal complement of an almost-invariant kernel, and a constant defect aligned with the slowest reduced eigenvector, for which the inequality is attained with equality. Corollary 2 (Graded faithfulness classification). Fix a toleranceδ tol >0 on the reduced indicator scale, for instance one half of the smallest decision-relevant margin in the candidate antifragility rule. A reduction (Π,g) is classified as: faithful whenε Π = 0 and Theorem 1 holds; practically faithful whenε Π /α≤ δ tol withμ(Dg)≤−αon the working neighbourhood; and unfaithful otherwise. The classification is computable from (ε Π ,α), both of which can be estimated from calibration trajectories. The practical content of Theorem 2 is that hidden feedback from omitted indicators does not immediately invalidate a reduced analysis; it invalidates it once the asymptotic error bandε Π /αbecomes comparable to the margins on which the equilibrium claims rest. The defectε Π is estimable: on any calibration window in which the full indicator vector is temporarily available, the defect is the supremum residual of the reduced closure, andα is obtained from the reduced Jacobian along the calibration trajectory. This estimation route is exercised in Experiment E1 of Section 5. The uniform negative-logarithmic-norm hypothesis of Theorem 2 excludes stable but non-normal reduced dynamics that exhibit transient amplification; Experiment E1 shows that the standard tier, which retains the slow equity-redundancy subspace, sits exactly in this regime. The following extension removes the restriction at the cost of a computable amplification constant, and recovers Theorem 2 when M = 1. Theorem 3 (Practical faithfulness under transient amplification). Letx(t) solve the full flow, lety(t) solve the reduced closure ̇y=g(y), and sete(t) = Πx(t)− y(t) and d(t) = Πf(x(t))− g(Πx(t)) with∥d(t)∥≤ ε Π on [0,T]. Suppose the segment joiningy(t) and Πx(t) lies in a convex reduced neighbourhoodV, letJ e (t) = R 1 0 Dg (y(t) +se(t))ds, and let Φ e (t,s) be the transition operator of ̇v=J e (t)v. If there existM ≥1 andα >0 with ∥Φ e (t,s)∥≤ M e −α(t−s) for 0≤ s≤ t≤ T , then ∥e(t)∥ ≤ M e −αt ∥e(0)∥ + Mε Π α 1− e −αt ,lim sup t→∞ ∥e(t)∥≤ Mε Π α ,(15) so the reduction is practically faithful at tolerance δ tol whenever Mε Π /α≤ δ tol . 13 Proof.The mean-value identity givesg(Πx(t))−g(y(t)) =J e (t)e(t), hence ̇e=J e (t)e+d(t), and variation of constants yieldse(t) = Φ e (t,0)e(0) + R t 0 Φ e (t,s)d(s)ds. Applying the semigroup estimate and the defect bound, the integral is at mostMε Π R t 0 e −α(t−s) ds= (Mε Π /α)(1− e −αt ). The constantMis computable from the calibrated reduced Jacobian b B. For a linear closure with b BHurwitz and any 0< α 0 < − max j ℜλ j ( b B), the shifted amplification factorM α 0 ( b B) =sup t≥0 e α 0 t ∥e b Bt ∥ satisfiesM α 0 ( b B)≤ κ(V eig ) for any diagonalisation b B =V eig ΛV −1 eig , and a certified bound is obtained from the semidefinite programmeminτ: I ⪯ P ⪯ τI, b B ⊤ P+P b B+ 2α 0 P ⪯0, which gives∥e b Bt ∥ 2 ≤ √ τe −α 0 t by the standard quadratic Lyapunov argument. For a nonlinear closure withr V =sup y∈V ∥Dg(y)− b B∥ 2 , including any Jacobian-estimation confidence radius, the hypothesis of Theorem 3 holds withM= √ τandα=α 0 − √ τ r V whenever √ τ r V < α 0 , by treatingDg − b Bas a bounded perturbation and applying Gr ̈onwall’s inequality. The asymptotic certificate √ τ ε Π /(α 0 − √ τ r V ) is therefore directly computable from calibration data, and quantifies precisely the certificate fragility observed for the standard tier in Experiment E1. Spectral structure and restoration time When the kernel of the projection is invariant, the spectrum decomposes, and the de- composition has a direct consequence for the restoration-time quantities reported to stakeholders. Proposition 3 (One-sided restoration-time bias under reduction). Let ̇x=A(x− x ∗ ) withAHurwitz, let Π satisfyA ker(Π)⊆ ker(Π), and letBbe the induced quotient matrix. Then spec(A) = spec A| ker(Π) ∪ spec(B),(16) and consequently the spectral abscissae satisfyα(B)≤ α(A)<0. For a tolerance δ ∈(0,1) define the restoration time of a trajectory asT δ (x 0 ) =inft≥0 :∥x(s)− x ∗ ∥≤ δ∥x 0 −x ∗ ∥ ∀s≥ t, and analogouslyT Π δ for the projected trajectory. Then for every initial condition,T Π δ (x 0 )≤ T δ (x 0 ) up to the norm equivalence constants of the projection, and the inequality is strict whenever the slowest eigenvalue ofA, namely the one attaining α(A), belongs to spec(A| ker(Π) ) and is excited by the initial displacement. Proof.Invariance ofker(Π) permits a change of basis adapted to the splittingR n = ker(Π)⊕ Wfor any complementW, in whichAis block upper triangular with diag- onal blocksA| ker(Π) and a matrix similar toB; Eq.(16)follows from the determinant factorisation of block-triangular matrices. Sincespec(B)⊆ spec(A), the maximum real part overspec(B) cannot exceed that overspec(A), givingα(B)≤ α(A). The projected error Π(x(t)− x ∗ ) evolves underBalone, hence decays at the rate governed byα(B), whereas the full error contains in addition the modes ofA| ker(Π) . Whenever the slowest mode lies in the kernel and is excited, the full error remains above the tolerance strictly longer than its projection, which proves the strict case; in all cases the projected error is a non-expansive image of the full error up to the norm equivalence constants, giving the weak inequality. Remark 2 (Interpretation for monitoring practice). Proposition 3 states that a city monitoring a reduced indicator set will, in the projectable linear regime, declare restoration to equilibrium no later than, and generically earlier than, the full system actually restores. 14 The bias is one-sided: reduction can only hide slow modes, never invent them. Reported restoration-time improvements computed from minimal indicator sets are therefore upper bounds on achievement, not unbiased estimates, and this holds before any statistical noise is considered. This result bears directly on restoration-time commitments expressed at the project level, and it is illustrated quantitatively in Experiment E2 of Section 5. Proposition 3 compares asymptotic decay rates; its finite-time reading is subject to norm-equivalence constants and non-normal transients. The following statement removes those qualifications by comparing worst-case restoration envelopes, and holds for arbitrary Hurwitz dynamics without normality or monotonicity hypotheses. Proposition 4 (Asymptotic restoration ordering from spectral inclusion). LetAbe Hurwitz withA kerΠ⊆ kerΠ, and letBsatisfyBΠ = ΠA. For any operator norm define the restoration envelopesT A (δ) =inft ≥0 :sup s≥t ∥e As ∥ ≤ δ andT B (δ) analogously. Then, writing α(·) for the spectral abscissa, lim δ↓0 T B (δ) T A (δ) = α(A) α(B) ≤ 1.(17) For a specific initial statev 0 with Πv 0 ̸= 0, letα A (v 0 ) be the largest real part over generalised eigenspaces in whichv 0 has a non-zero component, andα B (Πv 0 ) analogously; thenα B (Πv 0 )≤ α A (v 0 ) and the trajectory-specific restoration-time ratio converges to α A (v 0 )/α B (Πv 0 )≤ 1. Proof.The Jordan form gives∥e At ∥ =exp(α(A)t+O(logt)), the logarithmic term arising from the largest Jordan block at the spectral edge; solvingt r−1 e α(A)t ≍ δgivesT A (δ) = log(1/δ)/(−α(A)) +O(log log(1/δ)), and likewise forB. Kernel invariance makesAblock upper triangular in a basis adapted tokerΠ, soσ(B)⊆ σ(A) andα(B)≤ α(A)<0; the ratio of leading terms gives Eq.(17). For the trajectory version, only excited generalised eigenspaces contribute to the decay rate of e At v 0 , and the intertwining identity e Bt Πv 0 = Π e At v 0 shows the projected trajectory cannot decay more slowly than the full one, giving α B (Πv 0 )≤ α A (v 0 ). The envelope formulation makes the one-sided conclusion of Proposition 3 exact in the small-tolerance limit: reduced monitoring can only understate the restoration horizon, whatever the norm and however non-normal the dynamics, with equality precisely when the slowest excited mode is visible to the projection. 4.2 Reduction of the joint fixed point Section 2.2 defines a slower strategic layer coupled to the performance layer. The abstract form needed here is a two-layer fixed-point problem x = F (x,z), z = H(z,x),(18) wherex ∈ X ⊂ R n is the performance-layer indicator state andz ∈ Z ⊂ R p is a strategic state containing route, modal and learning quantities. Let Π x :R n → R m and Π z :R p → R q be surjective reductions. The reduced coordinates arey= Π x x and w = Π z z. 15 Definition 5 (Coupled commutation). The pair (Π x ,Π z ) commutes with the fixed-point map if there exist maps ̃ F and ̃ H such that Π x F (x,z) = ̃ F (Π x x, Π z z),(19) Π z H(z,x) = ̃ H(Π z z, Π x x),(20) for all (x,z) in a neighbourhood of the full fixed point. The commutation condition states that applying the full fixed-point map and then reducing must give the same result as reducing first and applying the reduced fixed-point map. Without this condition, the reduced fixed point is not inherited from the full model. It is a new model whose relation to the original attractor is unspecified. Assumption 2 (Local coupled Lipschitz structure). In a neighbourhood of (x ∗ ,z ∗ ), the reduced maps satisfy ∥ ̃ F (y 1 ,w 1 )− ̃ F (y 2 ,w 2 )∥≤ a∥y 1 − y 2 ∥ + b∥w 1 − w 2 ∥,(21) ∥ ̃ H(w 1 ,y 1 )− ̃ H(w 2 ,y 2 )∥≤ c∥w 1 − w 2 ∥ + d∥y 1 − y 2 ∥.(22) The reduced coupling matrix is K Π = a b d c .(23) Theorem 4 (Joint fixed-point preservation under coupled reduction). Suppose Eq.(18) has a locally unique fixed point (x ∗ ,z ∗ ) and that (Π x ,Π z ) commutes with the fixed-point map. Ifρ(K Π )<1, then the reduced map has a unique fixed point (y ∗ ,w ∗ ) = (Π x x ∗ , Π z z ∗ ) in the reduced neighbourhood, and the fixed-point iteration converges locally to it. If ρ(K Π ) = 1, uniqueness is not guaranteed; if, in addition, the derivative of the reduced map has an eigenvalue equal to one and the corresponding compatibility condition for the nonlinear terms is satisfied, a continuum or branch of reduced fixed points may occur. Proof. By commutation, (Π x x ∗ , Π z z ∗ ) satisfies the reduced fixed-point equations. Under Assumption 2, the product reduced map ̃ T(y,w) = ( ̃ F(y,w), ̃ H(w,y)) is Lipschitz with comparison matrixK Π . Ifρ(K Π )<1, a weighted product norm exists in which ̃ Tis a contraction. Banach’s fixed-point theorem gives local uniqueness and convergence of the iteration. Atρ(K Π ) = 1, contraction is lost. If the derivative has a unit eigenvalue, the implicit-function theorem no longer guarantees isolation of the fixed point in that direc- tion. Standard Lyapunov-Schmidt reduction then shows that the nonlinear compatibility condition determines whether the fixed point remains isolated, disappears or unfolds into a branch. Hence uniqueness is not guaranteed at the boundary. Proposition 5 (Convergence-rate degradation near the boundary). Under the assumptions of Theorem 4 withρ=ρ(K Π )<1, the fixed-point iteration (y k+1 ,w k+1 ) = ̃ T(y k ,w k ) satisfies, in the weighted product norm of the contraction argument, ∥(y k ,w k )− (y ∗ ,w ∗ )∥ ≤ ρ k ∥(y 0 ,w 0 )− (y ∗ ,w ∗ )∥,(24) so that the number of iterations required to reach relative toleranceεis at most⌈ln(1/ε)/ ln(1/ρ)⌉, which diverges as ρ↑ 1 at rate (1− ρ) −1 up to logarithmic factors. Proof. Immediate from the contraction estimate and ln(1/ρ)∼ 1− ρ as ρ↑ 1. 16 Remark 3 (Interpretation of the boundary). The conditionρ(K Π )<1 is a sufficient con- traction boundary, not necessarily the bifurcation or uniqueness boundary of the underlying nonlinear system: a reduced system may retain a unique, locally stable joint fixed point bey- ond it. What the boundary marks exactly is the loss of the contraction certificate, and with it the guaranteed geometric convergence of the coupled iteration. When indicator removal increases the effective cross-layer gain, for example by hiding a stabilising redundancy or equity-feedback coordinate, the reduced system may cross the boundary even though the full system remains well defined. Proposition 5 converts the boundary into an observable diagnostic: the iteration count of the reduced strategic solver grows hyperbolically as the boundary is approached, which is the quantity tracked in Experiment E3. Theorem 4 assumes that the reduced maps commute exactly with the projections. In deployment, omitted coordinates generally influence strategic decisions, so commut- ation holds only approximately. The following result bounds the resulting fixed-point displacement and makes the exact theorem robust. Theorem 5 (Fixed-point displacement under a commutation defect). WriteT(x,z) = (F(x,z),H(z,x)) for the full two-layer map, e T (y,w) = ( e F (y,w), e H(w,y)) for the reduced map, andP=diag(Π x ,Π z ). Letu ∗ be a full fixed point, chooseν= (ν x ,ν z ) ⊤ ≫ 0, and equip the product space with∥(y,w)∥ ν =max∥y∥ x /ν x ,∥w∥ z /ν z . SupposeK Π ν ≤ qνfor someq <1, so that e Tis aq-contraction in∥·∥ ν , and define the commutation defect on a neighbourhoodN ∋ u ∗ byδ c =sup u∈N ∥ e T(Pu)−PT(u)∥ ν . If e Tmaps the corresponding reduced neighbourhood into itself with unique fixed point u ∗ Π , then ∥u ∗ Π −Pu ∗ ∥ ν ≤ δ c 1− q ,(25) and whenK Π is irreducible the Perron eigenvector may be used forν, givingq=ρ(K Π ) and the bound δ c /(1− ρ(K Π )). Proof.Sinceu ∗ Π = e T(u ∗ Π ) andu ∗ =T(u ∗ ), the triangle inequality gives∥u ∗ Π −Pu ∗ ∥ ν ≤ ∥ e T(u ∗ Π )− e T(Pu ∗ )∥ ν +∥ e T(Pu ∗ )−PT(u ∗ )∥ ν ≤ q∥u ∗ Π −Pu ∗ ∥ ν +δ c ; rearranging proves Eq.(25). The comparison inequalityK Π ν ≤ qνbounds each component of the difference of reduced maps in the weighted maximum norm, which is the contraction property used, and Perron–Frobenius supplies the eigenvector attainingq=ρ(K Π ) in the irreducible case. The defectδ c is estimable on calibration windows by evaluating the commutator residual at sampled full states and adding a Lipschitz fill-distance correction, exactly asε Π is estimated for the performance layer; the reduced fixed point reported by a partial-data city is thereby certified to lie withinδ c /(1−ρ(K Π )) of the projected true equilibrium, a bound that degrades gracefully, and predictably, as the contraction boundary of Experiment E3 is approached. 4.3 Decision-rule degradation under reduction The candidate antifragility decision rule of Section 2.3, Eq.(10), compares a post-event performance ratio with a candidate baseline-dependent threshold. In the notation used here, let R(x) = μ post (x) E base (x) ,AF(x) = 1R(x) > τ (x),(26) 17 whereτ(x) is a candidate threshold function and 1·is the indicator function of the event in braces. A reduction observesy= Πxand estimates the decision statistic fromy. Letθdenote the local effect parameter separating the null boundary from the candidate antifragility alternative. For example,θ=R− τwithθ= 0 on the decision boundary andθ >0 under the candidate antifragile alternative. Under a local asymptotic normal approximation, the full and reduced estimators satisfy ˆ θ n ∼ N θ, 1 nI n ,(27) ˆ θ m,n ∼ N θ, 1 nI m + σ 2 res (m) ,(28) where I n is the Fisher information in the full indicator set, I m is the Fisher information retained by the reduced indicator set andσ 2 res (m) is the residual variance induced by omitted indicators. Definition 6 (Indicator-set Fisher information). Let observations have densityp(v | θ) and let S m be the retained indicator set. The retained Fisher information is I m (θ) = E θ " ∂ ∂θ logp(v S m | θ) 2 # .(29) The omitted information isI omit =I n −I m when the standard information decomposition is valid. Definition 7 (Residual variance under retained indicator support). For a retained indicator setS m , letT n denote the locally optimal full-indicator test statistic for the effect parameterθunder the local alternative, and letS m denote the linear span of the score functions associated with the retained indicators. The residual varianceσ 2 res (m) is the variance, under the local alternative, of the component ofT n orthogonal toS m . By construction,σ 2 res (n) = 0, andσ 2 res (m) is non-increasing inmwhen indicator sets are nested. Theorem 6 (Power loss under indicator reduction). Consider a one-sided level-αtest of H 0 :θ ≤0 againstH 1 :θ >0 based on the reduced statistic in Eq.(28). Under the local normal approximation, the power of the reduced candidate antifragility decision rule is π m (θ) = 1− Φ z 1−α − θ p (nI m ) −1 + σ 2 res (m) ! ,(30) where Φ is the standard normal distribution function. The degradation relative to the oracle full-indicator test is ∆π m (θ) = Φ z 1−α − θ p nI n − Φ z 1−α − θ p (nI m ) −1 + σ 2 res (m) ! .(31) Moreover, ifI m+1 ≥I m andσ 2 res (m+ 1)≤ σ 2 res (m), thenπ m+1 (θ)≥ π m (θ) for everyθ >0. Proof.The one-sided normal test rejects when the reduced standardised statistic exceeds z 1−α . Under the alternative, the reduced statistic has meanθand variance (nI m ) −1 + 18 σ 2 res (m), which gives Eq.(30). The oracle expression follows by setting the residual variance to zero and replacingI m byI n . Subtracting the reduced power from the oracle power gives Eq.(31). The monotonicity statement follows because the denominator of the non-centrality term is non-increasing when retained information increases and residual variance decreases; since 1−Φ(z 1−α −u) is increasing inu, the power cannot decrease. The theorem connects dynamical reducibility to detection. A projection can satisfy Theorem 1 and Theorem 4 whilst still being too weak for classification if it removes indicators carrying information about the decision boundary. This distinction is essential because a stable reduced attractor may be statistically underpowered. What the theorem does not yet supply is a way of computingI m for a deployed city, since the state variables xare themselves analytical constructs rather than directly measured quantities. The next subsection closes this gap. 4.4 A measurement model from observable urban indicators The twelve state variables of the framework of Section 2 are computed constructs: each is assembled from urban quantities that municipalities and transport operators record in the field. The reduction problem in practice is therefore a two-stage problem. First, which state variables are retained, which is the projection Π studied above. Secondly, which observable urban indicators feed the retained state variables, which determines the information actually available to the decision rule. This subsection formalises the second stage. LetKdenote the catalogue of observable urban key performance indicators available in principle to a deployment, and letS ⊆ Kwith|S|=rdenote the subset a given city collects at the required cadence. The measurement model is the linear-Gaussian observation equation v S = C S x + ε S , ε S ∼ N (0, Σ S ),(32) wherev S ∈ R r stacks the normalised observed indicators,C S ∈ R r×n encodes how each observed indicator loads on the state variables, and Σ S ≻0 collects measurement noise, sampling error and temporal-aggregation error. The linear form is a local approxima- tion around the working equilibrium; the loadingsC S are obtained from the published computation rules of the state variables, for example the assembly of the stress index from volume-to-capacity ratios and speed deficits, or of the equity penalty from stratified accessibility minutes. Under a local alternative in which the effect parameterθdisplaces the mean state along a known directionu θ ∈ R n , so thatE[x] =x ∗ +θ u θ , the observed mean is E[v S ] = C S x ∗ + θ C S u θ . Proposition 6 (Closed-form retained information under the Gaussian measurement model). Under Eq.(32)with Σ S independent ofθ, the Fisher information aboutθcarried by the indicator set S is I S (θ) = C S u θ ⊤ Σ −1 S C S u θ ,(33) which is independent ofθin the linear-Gaussian regime. Moreover,I S is monotone in the indicator support: for any additional observable indicatorj /∈ Swith rowc ⊤ j , noise variance σ 2 j and cross-covariance Σ Sj with the existing channels, I S∪j −I S = c ⊤ j u θ − Σ jS Σ −1 S C S u θ 2 σ 2 j − Σ jS Σ −1 S Σ Sj ≥ 0,(34) 19 namely the squared innovation of the new channel relative to its conditional variance. Proof.For a Gaussian family with meanμ(θ) =C S x ∗ +θC S u θ and constant covariance, the score is∂ θ logp= (C S u θ ) ⊤ Σ −1 S (v S − μ(θ)), whose variance is Eq.(33). For the increment, apply the block-inverse identity for the bordered covariance matrix of (v S ,v j ): the Schur complementσ 2 j −Σ jS Σ −1 S Σ Sj >0 by positive definiteness, and substituting the blockwise inverse into Eq.(33)forS∪jand simplifying yields Eq.(34), a ratio of a square to a positive quantity, hence non-negative. Corollary 3 (Minimum admissible indicator support). Fix the test levelα, a target power 1− βat a decision-relevant effect sizeθ ∗ , and a sample lengthn. An indicator supportS is admissible for the candidate antifragility classification if I S ≥ 1 n z 1−α + z 1−β θ ∗ 2 1− nσ 2 res θ ∗ z 1−α +z 1−β −2 −1 ,(35) whenever the bracketed term is positive; if it is not, no indicator support of the given residual variance is admissible at that sample length. The minimum admissible support is the smallestS, in collection cost or cardinality, satisfying Eq.(35), and by Proposition 6 it can be approached by greedy addition of the channel with the largest innovation gain in Eq.(34). The following statement settles when the greedy construction is not merely monotone but exactly optimal. Proposition 7 (Exact optimality of greedy selection under independent channels). Writea=C K u θ andF(S) =I S =a ⊤ S Σ −1 S a S withF(∅) = 0. If Σ is diagonal, then F(S) = P j∈S a 2 j /σ 2 j is modular, so greedy selection in decreasing order ofa 2 j /σ 2 j returns, for everyk, an exact maximiser ofFover supports of cardinalityk, and the first greedy prefix satisfying Eq.(35)is an exact minimum-cardinality admissible support. The same holds blockwise when Σ is block diagonal and covariance blocks are selected as indivisible units. For general correlated Σ,Fneed not be submodular, and greedy selection carries no universal approximation guarantee. Proof. Under diagonal Σ, Eq.(33)separates into the stated sum, soFis additive over channels; maximising an additive set function over a cardinality constraint is achieved exactly by sorting, and the minimum-cardinality threshold problem by the same sorted order. The block case is identical over the block ground set. Non-submodularity under correlation follows from the innovation-gain form of Eq.(34): a channel individually uninformative (a j = 0) can have strictly positive conditional gain once a correlated channel is present, since the innovation numerator becomes non-zero, which violates the diminishing-returns inequality. The measurement model of Section 5 has independent channels, so the greedy supports reported by Experiment E5 are exactly optimal, not merely feasible; for deployments with correlated channels the greedy output should be treated as a certified admissible support rather than a certified minimum one. Proof. Settingπ m (θ ∗ )≥1−βin Eq.(30)and solving the resulting inequalityθ ∗ / p (nI S ) −1 + σ 2 res ≥ z 1−α +z 1−β forI S gives Eq.(35); the side condition expresses that the residual variance alone must not already exhaust the permissible total variance. The greedy statement follows from the non-negativity and computability of the innovation gain. 20 Two remarks situate the measurement model in the deployment context. First, the model converts the abstract quantities of Theorem 6 into objects computable from a city’s data inventory:C S from the published assembly rules of the state variables, Σ S from instrument and sampling characteristics, andu θ from the direction in indicator space along which a validated post-event improvement manifests, typically dominated by the redundancy, adaptation and throughput coordinates. Secondly, the model makes precise the difference between collecting many indicators and collecting informative ones: by Eq.(34), a channel highly correlated with channels already collected contributes little, however accurate it is in isolation, whereas a moderately noisy channel loading on an otherwise unobserved coordinate, equity stratification being the standard example, can dominate the information budget. 4.5Tier-based reduction families grounded in the indicator register The reduction families used in the simulation protocol are defined by three nested indicator supports over the full state vector of Eq. (1). The minimal family retains S 1 =M,S,D eff ,Π 1 x = (M,S,D eff ) ⊤ .(36) The standard family retains S 2 =M,R,C,S,Q,I,Π 2 x = (M,R,C,S,Q,I) ⊤ .(37) The full family retains all twelve indicators, S 3 =M,R,A,C,D eff ,S,Q,P,B,E energy ,E ICT ,I,Π 3 = I 12 .(38) The tiers are not arbitrary. They reflect the structure of a consolidated indicator register assembled for the deployment context within which the framework was developed, in which a curated catalogue of urban indicators, organised by domain, supplies the observable quantities from which the twelve state variables are computed. Table 2 records, for each state variable, the principal observable indicators that feed it, together with a qualitative accessibility rating reflecting whether the underlying data are routinely held by municipalities and operators (high), require dedicated processing of existing data such as origin-destination matrices or geographic information systems (medium), or require surveys, stakeholder elicitation or event-specific instrumentation (low). The ratings are candidate assessments for the pilot context and will differ across deployments; their role here is to make the tier construction reproducible and to supply the loading structure of C S in the simulation protocol. Three observations follow from the mapping. First, the minimal familyS 1 is exactly the support a city can assemble from routinely held operator data plus an incident log: throughput, stress and disturbance are all rated high or event-specific in Table 2. This is whyS 1 is the realistic floor rather than an adversarial straw man. Secondly, the standard familyS 2 adds the constructs requiring dedicated processing of data that most cities already possess in some form: redundancy and entropy from network models and origin-destination matrices, equity from stratified accessibility, synergy from static geographic data. Thirdly, the constructs that separateS 2 from the full support, namely 21 adaptation velocity, behavioural compliance, demand-management coverage and the two infrastructure-support indicators, are precisely those requiring surveys, tariff analysis or cross-sector data agreements, and they are also the constructs through which the learning component of the framework manifests. The information cost of partial coverage therefore falls disproportionately on the antifragility classification rather than on stability screening, a statement made quantitative by Proposition 6 and tested in Experiments E4 and E5. Applying Theorem 1, the minimal family is faithful only if omitted redundancy, adaptation, equity and cross-layer coordinates do not feed back into the retained dynamics except through retained variables; by Theorem 2 it is practically faithful when the residual feedback produces a defectε Π small relative to the reduced contraction rate. Applying Theorem 4, the standard family preserves the joint fixed point if the omitted strategic quantities do not alter the reduced modal and capacity maps except through (M,R,C,S,Q,I) and if the reduced coupling matrix remains contractive. Applying Theorem 6 with Proposition 6, the full family gives the oracle reference, while the minimal and standard families lose detection power in proportion to omitted Fisher information and residual variance. These statements do not rank cities. They rank reductions relative to a specified attractor, a specified decision rule and a specified measurement model. 5 Simulation protocol and numerical demonstration on pilot-city configurations The numerical work is designed to illustrate the theorems rather than to validate the framework empirically. Three city configurations are used as stylised cases with different candidate indicator supports: Bratislava is represented by the full twelve-indicator family, Larissa by the standard six-indicator family and Thessaloniki by the minimal three- indicator family. The assignment is illustrative and reflects heterogeneous data coverage described in the source material; it is not an empirical claim about actual data readiness in any pilot city. 5.1 Common simulation testbed The testbed is a lightweight system-dynamics representation of the performance-layer attractor coupled to a stylised disturbance process. The state evolves according to x t+1 = x t + ∆tA 0 (x t − x ∗ ) + Bu t + η t ,(39) whereA 0 ∈ R 12×12 is chosen to be Hurwitz in the full system,u t contains candidate intervention and demand terms, andη t is a disturbance term. The agent layer is represented only as stochastic forcing. Government, media and vulnerable-group responses enter as parameterised perturbations to D eff and to the demand component of u t : η t = η 0 t + Γ g ξ g,t + Γ m ξ m,t + Γ v ξ v,t ,(40) whereξ g,t ,ξ m,t andξ v,t are mean-zero processes with prescribed variances and autocorrel- ations. No reasoning model or generative agent mechanism is part of the contribution. The reduced systems are obtained by applying Π 1 , Π 2 and Π 3 from Eqs.(36)–(38). Observations are generated from the measurement model of Eq.(32), with the rows ofC S instantiated from the loading structure of Table 2 and the noise covariance Σ S set from 22 the accessibility ratings: indicators rated high receive low noise variance, medium receive intermediate variance, and low or event-specific receive high variance with temporal gaps. The candidate baseline parameterisation is reported in Table 3; all values are candidate constructs for illustration and are recorded so that every figure in this section can be reproduced from the specification alone. The implementation is intended in Python with standard numerical libraries; no special-purpose software is required. Where a deployment wishes to replace the linear testbed of Eq.(39)with mesoscopic traffic simulation, the protocol is unchanged: the simulator supplies the trajectories of the observable indicators, and the analysis layer is identical. Five experiments are specified. Each is stated with its objective, its outputs and the figure or table through which it is reported; all outputs below are computed from the specification of this section under the documented implementation choices recorded with the deposited code. 5.2Experiment E1: projectability defect and practical faithful- ness Simulation specification: E1, practical faithfulness bound of Theorem 2 Objective. Verify the error bound of Eq.(14)and demonstrate the graded classification of Corollary 2. Procedure. Sweep the cross-block coupling ofA 0 over ten values in [0,0.2]. For each value: (i) compute the exact defectε Π of Π 1 and Π 2 on a box neighbourhood ofx ∗ , which for the linear testbed is the operator norm of the blocked-out coupling acting on the box; (i) compute the reduced contraction rateαas the negated logarithmic norm of the reduced Jacobian in the Euclidean norm; (i) simulate 200 trajectory pairs from random initial displacements and record the supremum over time of∥Πx(t)− y(t)∥after the transient. Outputs. Figure 2: observed asymptotic projection error against the analytic band ε Π /α, both axes in reduced-indicator units, one curve per tier. Annotate the tolerance δ tol implied by the decision margin so that the faithful, practically faithful and unfaithful regimes are visible as regions of the plot. Acceptance. All observed errors lie below the analytic band; the crossing ofδ tol occurs at a coupling strength that differs between Π 1 and Π 2 , with Π 1 crossing first. 5.3Experiment E2: spectral structure and restoration-time bias Simulation specification: E2, one-sided restoration-time bias of Proposition 3 Objective. Quantify the optimism bias of reduced restoration-time estimates. Procedure. Use the exactly projectable configuration (zero cross-block coupling into the retained sets) so that Proposition 3 applies. Apply a stylised shock displacing the state along a direction with a controlled component inker(Π 1 ) andker(Π 2 ), sweeping the kernel component fraction over0,0.25,0.5,0.75,1. For each case and each tier, record the timeT 0.1 at which the full state and the projected state respectively re-enter and remain within ten per cent of the equilibrium displacement. Outputs. Table 4: full versus tier-reported restoration times by kernel fraction. A companion figure (optional) showing the eigenvalue spectra ofA 0 ,B 1 andB 2 on the 23 0 2· 10 −2 4· 10 −2 6· 10 −2 8· 10 −2 0.10.120.140.160.180.2 0 0.1 0.2 0.3 0.4 Cross-block coupling strength c Projection error (reduced-indicator units) Observed, minimal Π 1 Observed, standard Π 2 Bound ε Π /α, minimal Bound ε Π /α, standard Tolerance δ tol = 0.10 Figure 2: Practical-faithfulness verification for Experiment E1. Observed asymptotic projection errors (solid, markers) remain below the analytic boundε Π /αof Theorem 2 (dashed) wherever the bound’s contraction hypothesis holds. For the standard tier the retained slow equity-redundancy subspace makes the Euclidean logarithmic-norm certificate fragile: the bound rises steeply and its hypothesis fails beyondc≈0.12 (the dashed red curve is clipped at the axis and not drawn past the failure point), whereas the observed error remains small because most coupling sources are retained coordinates; Theorem 3 covers exactly this regime through a certified amplification constant. The observed error of the minimal tier crosses the toleranceδ tol = 0.10 nearc≈0.16, before the standard tier (c≈ 0.19), as required by the acceptance criterion. complex plane, displaying the spectral inclusion of Eq. (16). Acceptance. Tier-reported restoration times never exceed the full restoration time; the gap grows monotonically with the kernel fraction; the standard tier gap is smaller than the minimal tier gap whenever the slow mode loads on coordinates retained by Π 2 but not Π 1 . 5.4Experiment E3: coupled fixed point near the contraction boundary Simulation specification: E3, contraction boundary of Theorem 4 and Propos- ition 5 Objective. Display the loss of uniqueness and the hyperbolic growth of iteration counts as ρ(K Π )↑ 1. Procedure. Instantiate the reduced two-layer maps ̃ Fand ̃ Has the linearised perform- ance and strategic updates of the testbed, with the cross-layer gains (b,d) swept so that ρ(K Π ) traverses [0.4,1.05] in steps of 0.05, refined to steps of 0.01 on [0.9,1.05]. For each value run the fixed-point iteration from 100 random initialisations to relative tolerance 24 0.40.60.81 10 0 10 1 10 2 10 3 ρ(K Π ) Iteration count Median 90th percentile Envelope ln(1/ε)/ ln(1/ρ) 0.40.60.81 10 −8 10 −6 10 −4 10 −2 10 0 ρ(K Π ) Limit-point dispersion Figure 3: Behaviour of the reduced coupled fixed point near the contraction boundary in Experiment E3. Left: median and ninetieth-percentile iteration counts to relative tolerance 10 −8 grow hyperbolically asρ(K Π )↑1 and remain below the analytic envelope of Proposition 5. Right: the dispersion of limit points across 100 random initialisations sits at numerical tolerance below the boundary and rises to order one above it, displaying the onset of non-uniqueness predicted by Theorem 4. 10 −8 , recording the iteration count, the dispersion of limit points, and the sensitivity of the limit to a perturbation of magnitude 10 −3 in an omitted coordinate. Outputs. Figure 3, two panels. Left: median and ninetieth-percentile iteration counts againstρ(K Π ), with the analytic envelopeln(1/ε)/ ln(1/ρ) of Proposition 5 overlaid. Right: dispersion of limit points againstρ(K Π ), displaying the onset of non-uniqueness beyond the boundary. Acceptance. Iteration counts track the analytic envelope; dispersion is at numerical tolerance below the boundary and departs from it above. 5.5 Experiment E4: decision power, analytic and Monte Carlo The analytic power curves of Theorem 6 under the computed information values are shown in Figure 4; Experiment E4 verifies them by Monte Carlo under the full measurement model, including the temporal gaps of the low-accessibility channels, which the analytic local approximation does not capture. The empirical size atθ= 0 is 0.052 for the minimal tier, 0.048 for the standard tier and 0.052 for the full tier, each within Monte Carlo error of the nominal 0.05. At the decision-relevant effectθ ∗ = 0.10 the empirical power is 0.68 (minimal), 0.88 (standard) and 0.89 (full): the minimal tier falls short of the 0.80 target, so the information cost of partial coverage falls on the antifragility classification exactly as anticipated in Section 4.5. Simulation specification: E4, Monte Carlo verification of Theorem 6 under the measurement model Objective. Verify Eq.(30)and quantify the additional power loss caused by irregular sampling of low-accessibility indicators. 25 0 5· 10 −2 0.10.150.20.250.30.350.40.450.5 0 0.2 0.4 0.6 0.8 1 Effect size θ Decision power π m ( θ ) Minimal Standard Full Figure 4: Analytic decision powerπ m (θ) as a function of effect size for the minimal, standard and full reduction families, from Eq.(30)with the window information values computed from the measurement model of the simulation testbed under the all-channels- daily idealisation (443.2, 780.9 and 928.5 respectively). Experiment E4 overlays Monte Carlo estimates on these curves. Procedure. For each tier and each effect sizeθ ∈ 0,0.05,...,0.5: simulate 10 000 post-event windows of 30 daily observations from Eqs.(39)and(32), with the improvement encoded as a mean displacementθu θ along a candidate direction loading principally on M,RandA; estimate ˆ θby generalised least squares using only the channels of the tier at their stated cadences; apply the one-sided level-0.05 test; record the rejection rate. Outputs. Figure 5: empirical power points with binomial confidence intervals overlaid on the analytic curves of Figure 4, one panel per tier. Report in the text the empirical size at θ = 0 for each tier. Acceptance. Empirical size within Monte Carlo error of 0.05; empirical power within the confidence band of the analytic curve for the high and medium channels, with a quantified downward departure for tiers relying on weekly-sampled channels. 5.6 Experiment E5: minimum admissible indicator support Simulation specification: E5, greedy indicator selection and minimum admiss- ible support of Corollary 3 Objective. Determine, for each city configuration, the smallest observable-indicator support achieving the target power at the decision-relevant effect size, and display the diminishing information returns of additional channels. Procedure. Starting from the empty support, add observable channels one at a time in order of the innovation gain of Eq.(34), recomputingI S at each step from the closed form of Eq.(33); at each step also record the analytic power atθ ∗ = 0.10,n= 30. Repeat under three noise scenarios: candidate baseline, doubled survey noise, and halved sensor noise. Report the first support meeting the admissibility threshold of Eq. (35). Outputs. Figure 6: power atθ ∗ against the number of channels added in greedy order, one curve per noise scenario, with the target power marked. Table 5: composition of the 26 00.20.4 0 0.2 0.4 0.6 0.8 1 θ Minimal 00.20.4 0 0.2 0.4 0.6 0.8 1 θ Standard 00.20.4 0 0.2 0.4 0.6 0.8 1 θ Full Figure 5: Monte Carlo verification of the power expressions in Experiment E4: empirical rejection rates (points, with 95 per cent binomial confidence intervals) over 10 000 replica- tions per point, overlaid on the analytic all-channels-daily curves of Eq.(30)(solid). For the minimal and standard tiers, whose informative channels are all daily, the points lie on the analytic curve. For the full tier the informative low-accessibility channel (adaptation velocity) is sampled weekly, and the empirical points fall systematically below the analytic idealisation, quantifying the additional power loss caused by irregular sampling. minimum admissible support per city configuration and noise scenario, named by the observable indicators of Table 2. Acceptance. The greedy curves are monotone by Proposition 6; the minimum admissible support under baseline noise includes at least one channel feeding a coordinate outside the minimal tier, demonstrating that the antifragility classification, unlike stability screening, is not achievable from routinely held operator data alone. 5.7 Summary of expected behaviour The protocol supports the formal claims in a limited and explicit sense. It is designed to show that the theorems distinguish four failure modes that would otherwise be conflated: absence of an autonomous projected flow beyond the tolerated defect, loss of uniqueness at the coupled fixed-point boundary, loss of statistical decision power despite stable reduced dynamics, and one-sided distortion of restoration-time reporting. Since the data are stylised, no empirical conclusion about city performance is drawn at any point. 6 Discussion The theoretical contribution of this paper is the explicit separation of four meanings of preservation under indicator reduction. A reduced model may preserve local stability because the projected vector field is autonomous and stable, or practically so within a quantified error band. It may preserve the hierarchical equilibrium because the fixed- point maps commute with the projection and remain contractive. It may preserve the candidate antifragility decision rule because the retained indicator set carries sufficient Fisher information and leaves bounded residual variance. It may, finally, preserve derived 27 123456789101112 0.6 0.7 0.8 0.9 Observable channels added (greedy information order) Power at θ ∗ = 0 . 10 Baseline Doubled survey noise Halved sensor noise Target power 0.80 Figure 6: Greedy construction of the minimum admissible indicator support in Experi- ment E5. reporting quantities such as restoration time only up to a one-sided bias whose direction is known in advance. These properties are related but not equivalent. The measurement model sharpens the practical interpretation. A minimal indicator configuration, assembled from routinely held operator data, can be meaningful for screening when the omitted coordinates are weakly coupled and the objective is to observe broad performance loss; Theorem 2 quantifies how weak that coupling must be. It becomes unfaithful when omitted variables affect retained velocities beyond the tolerance, and it becomes statistically inadmissible, in the sense of Corollary 3, when the candidate decision rule depends on information that the retained channels do not carry. A standard configuration can preserve the attractor under weaker assumptions, but it is still vulnerable to hidden feedbacks from infrastructure support or behavioural adaptation. A full configuration is not automatically correct, since measurement error and misspecification remain possible, but it avoids the structural information loss introduced by projection. The boundary cases are especially informative. The first boundary is projectability failure beyond the defect tolerance, where two full states with the same reduced vector have materially different projected derivatives. This is a structural failure rather than a calibration error, and Theorem 2 replaces the binary diagnosis with a measurable distance from it. The second boundary is the fixed-point contraction thresholdρ(K Π ) = 1, where uniqueness and convergence can be lost, and which Proposition 5 renders observable through solver behaviour before it is crossed. The third boundary is statistical, where the retained Fisher information becomes too small or the residual variance too large to detect the candidate antifragility effect at the desired level; Proposition 6 computes the position of that boundary from the city’s measurement inventory. These boundaries jointly define a minimum data infrastructure for equilibrium analysis: the required indicators are not those that make the model detailed, but those required to preserve the mathematical object being claimed. The restoration-time result deserves separate emphasis because restoration time is a quantity to which deployment programmes attach formal commitments, and because longitudinal antifragility trajectories are tracked through repeated restoration episodes. Proposition 3 shows that a reduced monitoring configuration cannot overstate restoration 28 delay; it can only understate it. Reported improvements in restoration time obtained under minimal monitoring should therefore be audited against the possibility that slow recovery modes, typically in equity, behavioural or infrastructure-support coordinates, are simply invisible to the monitoring set. The audit is concrete: by Eq.(16)it suffices to establish, on any window with temporarily extended coverage, whether the slowest identified mode loads on coordinates outside the routine monitoring support. The framework also clarifies the relation with existing reduction theory. Compared with centre manifold and slow-fast approaches, the present reduction is driven by indicator ob- servability and decision relevance rather than by spectral decomposition alone. Compared with multiplex aggregation, it requires commutation with a fixed-point operator rather than preservation of network spectra alone. Compared with composite-index sensitivity analysis, it treats the score as part of a dynamic attractor rather than as an isolated scalar construct. Compared with optimal experimental design, it selects observation channels for the preservation of a dynamical and decision-theoretic object rather than for parameter estimation alone. The resulting theory is narrower than general reduction theory but more closely matched to equilibrium claims made under partial indicator coverage. Five limitations should be recognised. First, the stability results are local and assume differentiability in a neighbourhood of the equilibrium; the baseline practical-faithfulness bound additionally assumes a uniformly negative logarithmic norm on a convex working set, and although Theorem 3 removes this restriction through a certified amplification constant, the resulting bound is only as sharp as the semidefinite certificate forM. Secondly, the fixed-point theory relies on a contraction condition that may be conservative for some nonlinear systems, and the displacement bound of Theorem 5 degrades as the contraction boundary is approached. Thirdly, the decision-power result uses a local normal approximation and a linear-Gaussian measurement model; finite-sample and dependent- data corrections are needed when observation windows are short or strongly autocorrelated. Fourthly, the loading matrixC S is treated as known from the published assembly rules of the state variables, whereas in deployment it is itself estimated and uncertain. Fifthly, the pilot-city configurations are illustrative and the simulation protocol, however fully specified, does not constitute empirical validation. These limitations do not remove the formal contribution, but they restrict the claims that can be made from the numerical demonstration. The implication for urban management is that minimum data requirements can be defined mathematically rather than administratively. A city does not need full indicator coverage for every screening question, but it does need enough coverage to preserve the specific equilibrium property being asserted. If the objective is stability screening, projectability within tolerance and a stable reduced Jacobian are central. If the objective is a joint intervention equilibrium, the reduced coupling matrix must remain inside the contraction boundary. If the objective is a candidate antifragility classification, the retained channels must satisfy the admissibility condition of Corollary 3, which can be checked, channel by channel, against the city’s data inventory before any monitoring campaign is commissioned. This distinction helps avoid the stronger but unjustified claim that any partial indicator set can support a full equilibrium analysis. 29 7 Conclusion This paper has formalised multi-scale equilibrium reducibility under variable indicator dimensionality. The first contribution is a definition of faithful reduction for a dynamic attractor, a theorem giving necessary and sufficient conditions for preservation of local asymptotic stability under projection, and a practical-faithfulness extension that bounds the projected-trajectory error by the ratio of the projectability defect to the reduced contraction rate. The second contribution is a joint fixed-point preservation theorem for coupled performance and strategic layers, including the contraction boundary at which uniqueness may be lost and the rate at which solver effort diverges as the boundary is approached. The third contribution is a decision-rule degradation result expressed through a measurement model that links the equilibrium state variables to observable urban indicators, yielding the retained Fisher information in closed form, a monotone innovation-gain calculus for channel selection, and an explicit admissibility condition for the minimum indicator support. The fourth contribution is a one-sided restoration-time bias result showing that reduced monitoring can only understate, never overstate, the time the full system requires to restore equilibrium. The research questions can therefore be answered directly. RQ1 is answered by projectability, reduced Hurwitz stability and absence of ambiguous equilibria in the projected fibre, with degradation under approximate projectability bounded byε Π /α. RQ2 is answered by commutation of the fixed-point maps with the performance and strategic projections, together with the conditionρ(K Π )<1. RQ3 is answered by the reduction in the non-centrality parameter of the candidate antifragility test, computable from the measurement model through Eq.(33)and Eq.(34). RQ4 is answered by the spectral inclusion of Eq.(16)and its one-sided consequence for restoration-time reporting. The paper does not claim empirical validation, full indicator coverage or a contribution from generative agent modelling. It establishes the formal conditions under which partial indicator coverage is admissible and exhibits them numerically through Experiments E1 to E5 on the stylised testbed. Future work should estimate the projectability defect, the retained Fisher information and the residual variance from observed pilot data, test the contraction boundary under measured inter-layer coupling, treat the loading matrix as estimated rather than known, and extend the local theory to non-smooth disruptions and regime-switching attractors. Data accessibility This is a theoretical paper. The simulation protocol of Section 5 is fully specified in the text and Table 3. The complete analysis code reproducing every figure, table and reported value of Section 5, together with its numerical outputs and the exact loading matrices C S and noise covariances Σ S used in Experiments E1 to E5, is provided as electronic supplementary material with this submission and will additionally be archived in a public repository with a persistent identifier on acceptance. No empirical pilot data are used. Authors’ contributions A.G.: conceptualisation, methodology, formal analysis, investigation, writing (original draft), writing (review and editing). Y.R.: supervision, funding acquisition, writing (review 30 and editing). A.Gh.: investigation, validation, writing (review and editing). C.D.N.: writing (review and editing). A.H.: software, writing (review and editing). Declaration of generative AI and AI-assisted technolo- gies During the preparation of this work the authors used generative artificial intelligence tools (Claude, Anthropic) to assist with language editing, with the preparation of the LaTeX source, with the drafting of the reproducibility code implementing the simulation protocol of Section 5, and with the drafting of candidate statements and proof sketches for the extension theorems, which the authors verified independently. After using these tools, the authors reviewed and verified the content, including all mathematical statements, proofs and numerical results, and take full responsibility for the content of the article. Competing interests The authors declare no competing interests. Funding This work was supported by the European Union’s Horizon Europe research and innovation programme under the AntifragiCity project, Grant Agreement No. 101203052. Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the granting authority. Acknowledgements This manuscript builds on internal project material from AntifragiCity project. References Alberto Aleta and Yamir Moreno.Multilayer networks in a nutshell.An- nual Review of Condensed Matter Physics, 10:45–62, 2019.doi:10.1146/ annurev-conmatphys-031218-013259. Athanasios C. Antoulas. Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM, Philadelphia, PA, 2005. doi: 10.1137/1.9780898718713. Martin Beckmann, C. B. McGuire, and Christopher B. Winsten. Studies in the Economics of Transportation. Yale University Press, New Haven, CT, 1956. Stefano Boccaletti, Ginestra Bianconi, Regino Criado, Charo I. del Genio, Jes ́us G ́omez- Garde ̃nes, Miguel Romance, Irene Sendi ̃na-Nadal, Zhen Wang, and Massimiliano Zanin. The structure and dynamics of multilayer networks. Physics Reports, 544(1):1–122, 2014. doi: 10.1016/j.physrep.2014.07.001. 31 Bureau of Public Roads. Traffic assignment manual. Technical report, U.S. Department of Commerce, Urban Planning Division, Washington, D.C., 1964. Jack Carr. Applications of Centre Manifold Theory, volume 35 of Applied Mathematical Sciences. Springer-Verlag, New York, 1981. Robert B. Cleveland, William S. Cleveland, Jean E. McRae, and Irma Terpenning. STL: A seasonal-trend decomposition procedure based on Loess. Journal of Official Statistics, 6(1):3–73, 1990. Manlio De Domenico, Albert Sol ́e-Ribalta, Emanuele Cozzo, Mikko Kivel ̈a, Yamir Moreno, Mason A. Porter, Sergio G ́omez, and Alex Arenas. Mathematical formulation of multilayer networks. Physical Review X, 3(4):041022, 2013. doi: 10.1103/PhysRevX.3. 041022. Manlio De Domenico, Vincenzo Nicosia, Alex Arenas, and Vito Latora. Structural reducibility of multilayer networks. Nature Communications, 6:6864, 2015. doi: 10. 1038/ncomms7864. Charles A. Desoer and Mathukumalli Vidyasagar. Feedback Systems: Input-Output Properties. Academic Press, New York, 1975. Valerii V. Fedorov. Theory of Optimal Experiments. Academic Press, New York, 1972. Neil Fenichel. Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations, 31(1):53–98, 1979. doi: 10.1016/0022-0396(79) 90152-9. Sergio G ́omez, Albert D ́ıaz-Guilera, Jes ́us G ́omez-Garde ̃nes, Conrad J. P ́erez-Vicente, Yamir Moreno, and Alex Arenas. Diffusion dynamics on multiplex networks. Physical Review Letters, 110(2):028701, 2013. doi: 10.1103/PhysRevLett.110.028701. John Guckenheimer and Philip Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, volume 42 of Applied Mathematical Sciences. Springer- Verlag, New York, 1983. Philip Holmes, John L. Lumley, and Gahl Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge, 1996. Steven M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall, Upper Saddle River, NJ, 1993. Hassan K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, 3rd edition, 2002. Mikko Kivel ̈a, Alex Arenas, Marc Barthelemy, James P. Gleeson, Yamir Moreno, and Mason A. Porter. Multilayer networks. Journal of Complex Networks, 2(3):203–271, 2014. doi: 10.1093/comnet/cnu016. Christian Kuehn. Multiple Time Scale Dynamics, volume 191 of Applied Mathematical Sciences. Springer, Cham, 2015. doi: 10.1007/978-3-319-12316-5. 32 Erich L. Lehmann and Joseph P. Romano. Testing Statistical Hypotheses. Springer, New York, 3rd edition, 2005. Winfried Lohmiller and Jean-Jacques E. Slotine. On contraction analysis for non-linear systems. Automatica, 34(6):683–696, 1998. doi: 10.1016/S0005-1098(98)00019-3. Peter J. Mucha, Thomas Richardson, Kevin Macon, Mason A. Porter, and Jukka-Pekka Onnela. Community structure in time-dependent, multiscale, and multiplex networks. Science, 328(5980):876–878, 2010. doi: 10.1126/science.1184819. Michela Nardo, Michaela Saisana, Andrea Saltelli, Stefano Tarantola, Anders Hoffman, and Enrico Giovannini. Handbook on constructing composite indicators: Methodology and user guide. OECD Statistics Working Papers 2005/03, OECD Publishing, 2005. doi: 10.1787/533411815016. OECD and European Commission Joint Research Centre. Handbook on Constructing Composite Indicators: Methodology and User Guide. OECD Publishing, Paris, 2008. doi: 10.1787/9789264043466-en. Friedrich Pukelsheim. Optimal Design of Experiments. John Wiley & Sons, New York, 1993. Filippo Radicchi and Alex Arenas. Abrupt transition in the structural formation of interconnected networks. Nature Physics, 9(11):717–720, 2013. doi: 10.1038/nphys2761. Andreas Raue, Clemens Kreutz, Thomas Maiwald, Julie Bachmann, Marcel Schilling, Ursula Klingm ̈uller, and Jens Timmer. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics, 25(15):1923–1929, 2009. doi: 10.1093/bioinformatics/btp358. Michaela Saisana and Stefano Tarantola. State-of-the-art report on current methodolo- gies and practices for composite indicator development. EUR 20408 EN, European Commission Joint Research Centre, 2002. Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models. John Wiley & Sons, Chichester, 2004. Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola. Global Sensitivity Analysis: The Primer. John Wiley & Sons, Chichester, 2008. Gustaf S ̈oderlind. The logarithmic norm. History and modern theory. BIT Numerical Mathematics, 46(3):631–652, 2006. doi: 10.1007/s10543-006-0069-9. Roger Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988. Eric Walter and Luc Pronzato. Identification of Parametric Models from Experimental Data. Springer, London, 1997. 33 Table 1: Comparison of existing reduction approaches and the preservation requirements addressed in this paper. ApproachStability preser- vation Fixed-point pre- servation Decision-power preservation Principal limita- tion for this pa- per Centre manifold and slow-fast re- duction Local,under invariance and spectral separa- tion Indirect, when the fixed point lies on the re- duced manifold Not addressed Indicator avail- ability is not ne- cessarily aligned with centre or slow variables Galerkin and bal- anced projection Sometimes, under controlled projectionor passivity condi- tions Notgenerally guaranteed Not addressedApproximation qualityneed notpreserve theattractor interpretation Supra-Laplacian andmultiplex aggregation Process-specific, often spectral Sometimes, for diffusion or con- sensus maps Not addressedAggregation maypreserve network spectra whilstlosing indicator-level evidence Composite-index sensitivityana- lysis Not addressed as a dynamical property Not addressed as a coupled map Partly, through uncertainty and sensitivity meas- ures Does not de- termine whether a reduced score is a projected equilibrium Optimalexperi- mental design Not addressedNot addressedYes, through in- formation func- tionals Selects channels for estimation, not for preser- vation of an at- tractor Faithful attractor reductionpro- posed here Required by pro- jectability (ex- act or approxim- ate with explicit bound) and Hur- witz reduced dy- namics Requiredby commutation andreduced contraction Quantifiedin closedform through a meas- urement model andretained Fisher informa- tion Appliesto local attractor structureand candidatede- cisionrules ratherthan to all possible observables 34 Table 2: Mapping from the twelve equilibrium state variables to principal observable urban indicators, with qualitative data-accessibility ratings. Catalogue families refer to standard urban-indicator domains: network performance, travel times and reliability, modal split and demand, safety, accessibility and affordability, land use and infrastructure, and energy. Ratings are candidate assessments for the pilot context. StateConstructPrincipal observable indicators feeding the constructAccessibility MMobility throughput Public transport ridership; trip rate per person; total vehicle-kilometres; transit passenger-kilometres; modal split shares High R Networkre- dundancy Network reserve capacity; road and public transport network density; betweenness centrality of critical nodes; share of dormant versus active links from network models Medium AAdaptation ve- locity Time-differenced modal-split diversity from repeated household or panel surveys; uptake rates of alternat- ive services after events Low C Networken- tropy Shannon entropy of origin-destination flow distribu- tions from observed or estimated origin-destination matrices, normalised by current topology Medium D eff Disturbance in- dex Event catalogues and early-warning trigger logs; op- erator incident feeds; official alerts weighted by do- main, recency and source reliability Event- specific SStressTravel time reliability; congestion and delay meas- ures; link volume-to-capacity ratios and speed defi- cits from floating-car or loop-detector data High QEquity penalty Accessibility of population to public transport and essential services, stratified by income or neighbour- hood group; household transport expenditure shares Medium to low PDemand man- agement Published fare tables and time-varying tariffs; park- ing fees; fuel price; coverage of priced instruments weighted by ridership High B Behavioural compliance Adherence to guidance measured through nudging- campaign uptake; mobility satisfaction and stated- compliance surveys Low E energy Energy supportTransport energy consumption; renewable share in transport energy; charging-infrastructure uptime and spare capacity during events Medium E ICT Communication support Uptime of telecommunication and control systems critical to transport; data-feed latency and complete- ness records Medium, event- specific ICross-layer syn- ergy Inter-modal transfer times; geographic proximity of transfer hubs across modal layers from geographic information systems High (static) 35 Table 3: Candidate baseline parameterisation of the simulation testbed. All values are illustrative constructs, not empirical estimates. QuantityCandidate valueRationale Step size ∆t1 dayMatches the daily cadence of the full relaxa- tion law Spectral abscissa of A 0 −0.05 per daySlowest full mode with a twenty-day time constant, placed in the equity-redundancy subspace to exercise Proposition 3 Fast block of A 0 −0.3 to−0.5 per day Throughput, stress and disturbance relaxa- tion Cross-block coupling in A 0 swept in [0, 0.2] Controls the projectability defectε Π of Π 1 and Π 2 Disturbance autocorrela- tion 0.6 at one dayFirst-order autoregressive forcing Measurement noise (high tier) σ = 0.02Operator and sensor data Measurement noise (me- dium tier) σ = 0.05Processed origin-destination and geographic data Measurement noise (low tier) σ= 0.10, weekly sampling Survey-based constructs Monte Carlo replications10 000Standard errors below one percentage point on power estimates Test level α0.05 one-sidedMatches the full decision rule of Eq. (10) Target power 1− β0.80 at θ ∗ = 0.10 Decision-relevant effect: a ten per cent post- to-baseline improvement margin Table 4: Experiment E2: restoration timeT 0.1 (days) to within ten per cent of equilibrium, full system versus tier-reported values, by the fraction of the initial displacement lying in the hidden subspace of the minimal tier. The standard tier retains the slow equity- redundancy pair and therefore reports the full restoration time exactly; the minimal tier reports the fast-mode time regardless of the hidden slow component. † At kernel fraction one the displacement is entirely invisible to the minimal tier, which reports immediate restoration. Kernel fraction of shock Full T 0.1 (days) Standard tier T 0.1 (days) Minimal tier T 0.1 (days) 0.00444 0.2517174 0.5031314 0.7539394 1.0044440 † 36 Table 5: Experiment E5: composition of the minimum admissible observable-indicator support (target power 0.80 atθ ∗ = 0.10) per city configuration and noise scenario, named by the observable channels of Table 2. ConfigurationNoise scenarioMinimum admissible support (observable channels, greedy order) Thessaloniki (minimal) baselinenot attainable within the catalogue; maximum power 0.68 with all three channels (M , S, D eff ) Larissa (stand- ard) baselineM,R: throughput and redundancy channels (power 0.85) Bratislava (full)baselineM,R: throughput and redundancy channels (power 0.85) Alldoubled survey noise M,R: unchanged, since the survey-based channels carry negligible information for this decision direction All halvedsensor noise M,R: unchanged, since the disturbance process rather than sensor noise dominates the information budget Table 6: Stylised pilot-city configurations and observed theorem behaviour. All values are computed from the simulation testbed: theε Π /αentries at the representative mid-sweep coupling c = 0.089 from Experiment E1, the ρ(K Π ) entries from the per-tier contraction gains of the coupled two-layer linearisation, and the relative Fisher information from the measurement model at the stated cadences. All remain candidate constructs of the illustrative testbed, not empirical estimates. ConfigurationReduction family ε Π /αρ(K Π )Relative Fisherin- formation Expected behaviour ThessalonikiMinimal0.330.760.54Practicallyfaithful onlyunderweak omittedfeedback; strong decision-power loss (0.68 atθ ∗ , be- low target);largest restoration-time optim- ism LarissaStandard2.710.580.95Contraction certificate fragile owing to the retained slow subspace, though observed errors remain small;near- oracleclassification power BratislavaFull00.541.00Oracle reference for all reduced configurations; residual power loss from weekly-sampled chan- nels only 37