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Priority Coordination Games: Hodge Decomposition and a Sharp Design Limit
Zhihao Lin, Jianglin Lan, Anh-Tu Nguyen, Yoshinobu Kawahara
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 10/6/2026, 10:00:27 AM
Summary
This paper analyzes decentralized priority coordination games using Hodge decomposition to separate incentives into potential (common objective) and harmonic (non-potential) components. It derives closed-form expressions for these energies on conflict graphs, demonstrating that the harmonic component represents a fundamental design limit that cannot be eliminated by payoff design. The harmonic energy correlates with entropy production in log-linear learning and sets a lower bound on the error of potential game approximations.
Entities (8)
Relation Signals (7)
Hodge Decomposition → decomposes → Priority Coordination Games
confidence 95% · This paper determines what that surrogate misses, using the Hodge decomposition of the incentives...
Hodge Decomposition → produces → Potential Component
confidence 95% · using the Hodge decomposition of the incentives into a potential component... and a harmonic component
Hodge Decomposition → produces → Harmonic Component
confidence 95% · using the Hodge decomposition of the incentives into a potential component... and a harmonic component
Potential Component → represents → Common Objective
confidence 90% · the potential component, which a common objective can represent
Harmonic Component → determines → Entropy Production
confidence 88% · its energy sets the entropy-production rate
Harmonic Component → causes → Stationary Probability Current
confidence 85% · the missed component is... the stationary probability current
Affine Payoffs → minimizes → Harmonic Share
confidence 85% · affine payoffs attain this value [relative error of at least 1/N]
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Abstract
Abstract:In decentralised priority coordination, agents announce priority levels and a shared resource serves them in decreasing order, as at an unsignalised intersection; the levels form the decision layer of a hierarchical controller. Such interactions are routinely replaced by a potential game, i.e.\ by a common objective, for analysis and design. This paper determines what that surrogate misses, using the Hodge decomposition of the incentives into a potential component, which a common objective can represent, and a harmonic component, which it cannot. For the linear payoff, both components are obtained in closed form on every conflict graph and for every deterministic tie-breaking protocol: in common units, the harmonic energy is the number of conflicts and the potential energy adds the number of adjacent pairs of conflicts. Consequently, for every rationality parameter, the best common-objective model of the agents' choice log-odds, weighted uniformly over unilateral moves, has a relative squared error of at least $1/(d_{\max}+1)$, where $d_{\max}$ is the largest number of conflicts of one agent; for an eight-vehicle intersection it is exactly one fifth, for any number of priority levels. Invisible to strict-improvement dynamics, the missed component is, under low-rationality log-linear learning with uniform revision and to leading order, the stationary probability current, and its energy sets the entropy-production rate. Payoff design cannot remove it: on the complete conflict graph of $N$ agents, under a total-order protocol and with at least three priority levels, every nonconstant rank-based payoff leaves a relative error of at least $1/N$, with equality exactly for affine payoffs.
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- Source: https://arxiv.org/abs/2610.05832v1
- Canonical: https://arxiv.org/abs/2610.05832v1
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Priority Coordination Games: Hodge Decomposition and a Sharp Design Limit Zhihao Lin Jianglin Lan Anh-Tu Nguyen Yoshinobu Kawahara †thanks: This work was supported in part by the Japan Society for the Promotion of Science (JSPS) through the JSPS Postdoctoral Fellowship for Research in Japan (Fellow ID P26385), in part by JSPS KAKENHI Grant Number JP26H02498, and in part by the Japan Science and Technology Agency (JST) Moonshot R&D Program under Grant No. JPMJMS263K. (Corresponding author: Yoshinobu Kawahara.)†thanks: Zhihao Lin is with the Graduate School of Information Science and Technology, The University of Osaka, Suita, Osaka 565-0871, Japan (e-mail: zhihao.lin@ist.osaka-u.ac.jp).†thanks: Jianglin Lan is with the James Watt School of Engineering, University of Glasgow, Glasgow G12 8Q, United Kingdom (e-mail: Jianglin.Lan@glasgow.ac.uk).†thanks: Anh-Tu Nguyen is with the LAMIH UMR CNRS 8201 Laboratory, Université Polytechnique Hauts-de-France, 59313 Valenciennes, France, and also with INSA Hauts-de-France, 59313 Valenciennes, France (e-mail: tnguyen@uphf.fr).†thanks: Yoshinobu Kawahara is with the Graduate School of Information Science and Technology, The University of Osaka, Suita, Osaka 565-0871, Japan, and also with the RIKEN Center for Advanced Intelligence Project, Tokyo 103-0027, Japan (e-mail: kawahara@ist.osaka-u.ac.jp). Abstract In decentralised priority coordination, agents announce priority levels and a shared resource serves them in decreasing order, as at an unsignalised intersection; the levels form the decision layer of a hierarchical controller. Such interactions are routinely replaced by a potential game, i.e. by a common objective, for analysis and design. This paper determines what that surrogate misses, using the Hodge decomposition of the incentives into a potential component, which a common objective can represent, and a harmonic component, which it cannot. For the linear payoff, both components are obtained in closed form on every conflict graph and for every deterministic tie-breaking protocol: in common units, the harmonic energy is the number of conflicts and the potential energy adds the number of adjacent pairs of conflicts. Consequently, for every rationality parameter, the best common-objective model of the agents’ choice log-odds, weighted uniformly over unilateral moves, has a relative squared error of at least 1/(dmax+1)1/(d_ +1), where dmaxd_ is the largest number of conflicts of one agent; for an eight-vehicle intersection it is exactly one fifth, for any number of priority levels. Invisible to strict-improvement dynamics, the missed component is, under low-rationality log-linear learning with uniform revision and to leading order, the stationary probability current, and its energy sets the entropy-production rate. Payoff design cannot remove it: on the complete conflict graph of N agents, under a total-order protocol and with at least three priority levels, every nonconstant rank-based payoff leaves a relative error of at least 1/N1/N, with equality exactly for affine payoffs. Index Terms: Multi-agent coordination, potential games, utility design, log-linear learning, Hodge decomposition, priority protocols. I Introduction Many multi-agent systems resolve conflicts by priority. Vehicles at an unsignalised intersection must decide who crosses first [1, 2, 3, 4, 5, 6, 7]; robots meeting at a corridor junction and transmitters contending for a channel face the same problem. In decentralised priority coordination, each of N agents announces one of K priority levels, and the shared resource serves the agents in decreasing order of level, ties being broken by a fixed protocol. Every agent prefers to be served early, so the agents play a game. Priority selection is a control problem, not only a static game. At an intersection, priority levels fix the order in which vehicles enter the conflict zone, and trajectory controllers then realise that order [2, 3]. Read this way, the priority levels form the decision layer of a hierarchical controller, the conflict graph encodes the coupling of the agents through shared resources, and the learning rule governs how decisions are updated over time. We study this decision layer and assume that the execution-layer controllers execute the service order it produces. This is an abstraction, since an ordering of the agents restricts what the execution layer could do; here we take it as given and analyse the incentives inside it. A standard way to analyse, predict and design such decision layers is to model them as potential games [8, 9]. In a potential game, the change in any agent’s payoff caused by its own deviation equals the change of one common function, the potential. This brings two guarantees. First, the potential is a Lyapunov function for asynchronous best-response dynamics with inertia, so these dynamics converge [8]. Second, under log-linear learning the stationary distribution is explicit: it is the Gibbs distribution of the potential [10, 11]. For this reason, utility design for self-interested agents has become a standard problem in control [12, 13, 14], and for interacting vehicles, potential-game models are used for prediction and planning [15], a game that is not potential being projected onto the nearest potential game and the residual discarded [16]. Whether such a surrogate is adequate depends on how much it misses, and that quantity is rarely computed. 0010011100111100(a) game flow u==0010011112 1212 1212 1212 12(b) potential δϕ⋆δφ ++0010011112 1212 1212 1212 12(c) harmonic h Fig. 1: The smallest priority game: two conflicting agents, levels 0,1\0,1\, agent 00 served first on a tie. Vertices are profiles s0s1s_0s_1; arrows are unilateral raises, labelled with the mover’s payoff gain. (a) The game flow; (b) its potential part (blue), the common incentive to raise one’s level; (c) its harmonic part (red), a circulation recording who goes first. Priority coordination is not a potential game as soon as two agents are in conflict (Propositions 1(d) and 2). The reason is elementary: when two conflicting agents choose the same level, only one of them can go first. Fig. 1(a) shows the smallest case. Agent 11 gains by raising its level when both are low, agent 00 then gains by raising its level too, and nobody gains by lowering. In a potential game, the gains collected around any closed cycle of profiles add up to zero [8]; here they add up to 22. This paper asks three questions about the part of priority coordination that a potential surrogate misses: How large is it? What does it do to the agents’ behaviour? Can payoff design remove it? Classical utility design does not address the last question: there, utilities are designed mainly to improve welfare or the price of anarchy [12, 13], whereas in priority coordination the total number of waits is the same in every profile and for every payoff, because on every conflict exactly one agent waits. What a payoff can change is how far the game is from a potential game, and we determine a limit on this distance that no payoff can overcome. Our tool is the Hodge decomposition of finite games [17]. It splits the incentives of every game, viewed as a flow on the graph of profiles, into two orthogonal components (Fig. 1(b)–(c)). The potential component is the part that a common objective can represent; the harmonic component is the remainder, which circulates. The squared norms of the two components, the potential and harmonic energies, measure how much of the incentives each carries, and the harmonic energy is exactly the smallest squared error of any potential surrogate. The decomposition can always be computed numerically; our aim is to obtain it in closed form and to determine what it implies. Our answers to the three questions are as follows. Size. The harmonic component of a pairwise game is the antisymmetric part of its interactions (Proposition 1). For the linear payoff f(r)=−rf(r)=-r, nothing in an interaction is reciprocal, because exactly one of two conflicting agents goes first, so both energies are counts on the conflict graph (Proposition 2). Since the harmonic energy is the smallest squared error of any potential surrogate, the best common-objective model of the agents’ choice log-odds misses at least 1/(dmax+1)1/(d_ +1) of the incentive energy for every rationality β>0β>0 (Corollary 1); the floor is set by the conflicts per agent, not by the number of agents. For the eight-vehicle intersection of Fig. 2, with 1616 conflicts and 4848 adjacent pairs, it is one fifth. Behaviour. Under log-linear learning at low rationality, the missed component is what makes learning irreversible: the stationary law is still Gibbs in the Hodge potential up to second-order terms, but the leading-order stationary current is proportional to the harmonic flow and the leading-order entropy-production rate to the harmonic energy (Proposition 6). It is, in contrast, invisible to ordinal analysis: paths of strictly improving moves always terminate (Proposition 4), and, under the index protocol, all pure-strategy Nash equilibria induce the same service order (Proposition 5); in Fig. 1 both equilibria, 1010 and 1111, serve agent 00 first. Discarding the harmonic component is therefore harmless for best-response reasoning and costly for any model that uses the magnitudes of the incentives. Design. The harmonic component cannot be designed away. On the complete conflict graph, under a total-order protocol and with at least three priority levels, no payoff that depends only on the rank brings the harmonic share, and hence the surrogate error, below 1/N1/N, and affine payoffs attain this value (Theorem 2); for continuous priorities the full spectrum of the problem is explicit (Theorem 1). This is a limit on potential-game approximation, not a recommendation of the affine payoff. The main contributions of this paper are the following. 1. The error of a potential surrogate, in closed form (Proposition 2, Corollary 1): for the linear payoff, on every conflict graph, for every K≥2K≥ 2 and every deterministic protocol. Prior work assumes a bound on this distance [18, 19] or discards it [16]. 2. Learning identities and ordinal invisibility (Propositions 6, 4 and 5): exact leading-order identities for arbitrary payoffs as the rationality tends to zero, where the potential share was so far an empirical predictor [20]; termination and a unique equilibrium service order for every strictly decreasing payoff. 3. A sharp design limit (Theorems 1 and 2): on the complete conflict graph, under a total-order protocol, for every K≥3K≥ 3 and every nonconstant payoff-from-rank function; it can fail for K=2K=2 and, at finite K, off the complete graph (Section VII-D). It is an impossibility result on the distance to a potential game, not a welfare optimisation [12, 13]; the computer-assisted steps of its proof are exact certificates. I-A Related Work Decompositions of games. The potential–harmonic decomposition is due to Candogan et al. [17]; related decompositions and characterisations appear in [21, 22, 23, 24, 25], and, in the control literature, through the semi-tensor product [26, 27, 28, 29, 30]. For graphical games the decomposition acts edge by edge [28, 31]; Proposition 1 specialises these results to centred interaction kernels, and we claim no novelty beyond this form. Symmetrised interactions give approximate potentials of network games with continuous actions [32, 33]. Potential games in control and learning. Potential-game design is a standard tool of cooperative control [9, 11]. The potential share has been proposed as a predictor of learning behaviour [20]; near-potential games inherit approximate guarantees [18, 19]; harmonic structure makes no-regret dynamics recurrent [34, 35]; bounded-rationality log-linear learning has been analysed for symmetric coordination games [36]; and currents and entropy production driven by non-reciprocal interactions appear in statistical physics and evolutionary games [37, 38, 39]. We are not aware of closed-form energies for rank-order games, of identities linking the discarded residual to learning, or of sharp bounds on the potential share over a family of payoffs. Rank-order games and the affine case. Rank-order games are studied computationally in [40]. In weighted congestion games, affine costs are singled out by an exact potential [41]; here affine payoffs minimise, rather than remove, the harmonic share. The spectral analysis uses the Hoeffding decomposition of functions of ranks [42] and discrete Chebyshev polynomials [43]. I-B Organisation and Notation Organisation. Section I states the model and the three problems; Sections I–VI answer them in turn, and Section VII reports the numerical study. Short proofs are given in full in the text. The proofs of Theorems 1 and 2 are developed in the Appendix, with lengthy algebraic reductions and exact computer-assisted certificates deferred to the supplementary material (SM), which also contains further results. For these, the text also explains why the result holds. Notation. [N]=0,…,N−1[N]=\0,…,N-1\; #A\#A is the number of elements of a finite set A; 1 is the all-ones vector; xk¯=x(x−1)⋯(x−k+1)x k=x(x-1)·s(x-k+1) is the falling factorial; Δf(r)=f(r+1)−f(r) f(r)=f(r+1)-f(r) is the forward difference and Δm ^m its m-th iterate; Bin(n,p)Bin(n,p) is the binomial law; M⪰0M 0 means that M is positive semidefinite. For a finite set S, functions on S carry the uniform probability measure, ‖g‖2=[g2]\|g\|^2=E[g^2], and VarVar, CovCov refer to it. Flows (functions on edges) carry the counting inner product ⟨x,y⟩=∑ex(e)y(e) x,y = _ex(e)y(e); the factor KNK^N between the two conventions is always written out. AiA_i averages over the i-th coordinate, gTg_T is the Hoeffding component of g on T⊆[N]T [N] (Section I), and [ui]T[u_i]_T is that of the payoff uiu_i. I Model and Problem Statement This section defines pairwise games, for which the Hodge decomposition has a simple form, and the priority games of interest, which are pairwise when the payoff is affine; it then states the three problems and the lemma from which the closed forms are derived. I-A Pairwise Games on an Interconnection Graph Let C=(,E)C=(V,E) be a finite undirected graph on the agent set =[N]V=[N], the interconnection (or conflict) graph; (i)N(i) is the neighbourhood of i and di=|(i)|d_i=|N(i)| its degree. Each agent chooses a priority level (level for short) si∈0,…,K−1s_i∈\0,…,K-1\, K≥2K≥ 2; the profile is s∈S:=0,…,K−1Ns∈ S:=\0,…,K-1\^N. A pairwise game on C has payoffs ui(s)=γi(si)+∑j∈(i)ψij(si,sj),u_i(s)= _i(s_i)+ _j (i) _ij(s_i,s_j), (1) with arbitrary own terms γi _i and arbitrary kernels ψij _ij, one for each ordered pair of neighbours. Terms that do not depend on sis_i are strategically irrelevant and are omitted. Decomposing each kernel by the two-variable ANOVA under the uniform measure gives ψij(x,y)=ψ¯ij+aij(x)+bij(y)+ωij(x,y), _ij(x,y)= ψ_ij+a_ij(x)+b_ij(y)+ _ij(x,y), (2) where aija_ij, bijb_ij are centred and the interaction ωij _ij is centred in each argument. The own effect of agent i is i(x):=γi(x)−γ¯i+∑j∈(i)aij(x), a_i(x):= _i(x)- γ_i+ _j (i)a_ij(x), (3) where γ¯i γ_i is the mean of γi _i. Each interaction splits into its symmetric and antisymmetric parts under the exchange of the two agents, ωij±(x,y):=12(ωij(x,y)±ωji(y,x)),ω^±_ij(x,y):= 12( _ij(x,y)± _ji(y,x)), (4) so that ‖ωij±‖\|ω^±_ij\| depends only on the edge i,j\i,j\. I-B Priority (Rank-Order) Games Agents are served in decreasing order of level, ties being broken by a fixed protocol. Since only neighbours compete, we model a deterministic protocol as an orientation of the conflict edges: we write j→ij→ i if, on a tie, j is served before i. Throughout the paper, “any deterministic protocol” means any such orientation. Unless stated otherwise we use the index protocol, j→ij→ i iff j<ij<i, i.e. the acyclic orientation induced by a total order of the agents; after relabelling, every total order is of this form. The conflict position of agent i is the number of its neighbours served before it, posC(i,s)=#j∈(i):sj>sior(sj=si,j→i),\!\!pos_C(i,s)=\#\j (i):s_j>s_i\ or\ (s_j=s_i,\ j→ i)\,\!\! (5) and ui(s)=f(posC(i,s))u_i(s)=f(pos_C(i,s)) for a payoff-from-rank function f, which is strictly decreasing unless stated otherwise. The linear game (C,K)G(C,K) has f(r)=−rf(r)=-r; it is the pairwise game with γi≡0 _i≡ 0 and ψij(x,y)=−[y>x]−[y=x] 1[j→i]. _ij(x,y)=-1[y>x]-1[y=x]\,1[j→ i]. (6) For general f we write f(C,K)G_f(C,K), which is pairwise iff f is affine on 0,…,dmax\0,…,d_ \, with dmaxd_ being the maximum degree of C (an agent of degree d only ever sees ranks 0,…,d0,…,d). Since every conflict edge contributes one unit to the position of whichever endpoint is served second, ∑iposC(i,s)=|E|,∀s∈S. _ipos_C(i,s)=|E|, ∀ s∈ S. (7) Thus the total number of waits, i.e. of conflicting agents served before each agent, is the same in every profile. The aggregate payoff ∑if(posC(i,s)) _if(pos_C(i,s)) is also constant when f is affine, but for a nonlinear f it need not be. An orientation with directed cycles still defines posCpos_C and the game, but only as a system of local comparisons (three mutually tied agents on a directed triangle admit no global service order); statements about pure-strategy Nash equilibria (Proposition 5) are made for the index protocol, and, for the linear payoff, the energies are the same for every orientation. Example 1 (Intersection). At a two-lane four-arm intersection with eight vehicles, four turning on the inner lanes and four going straight on the outer ones, the conflict graph has an edge for every pairwise crossing of the nominal paths (Fig. 2): it is 44-regular with 1616 edges, the turning vehicles forming a 44-cycle, the straight ones another, and each turning vehicle crossing two straight lanes. A vehicle’s delay is, to first order, proportional to posCpos_C, the number of conflicting vehicles it lets through. This graph is our running example. Fig. 2: Running example. (a) Eight vehicles at a two-lane intersection: A1–A4 turn left from the inner lanes (solid), A5–A8 go straight on the outer lanes (dashed); circles mark the 1616 pairwise crossings of the nominal paths. (b) The resulting conflict graph; arrows show the index protocol (on a tie, tail first). I-C Flows and Hodge Decomposition We use the flow representation of [17]. The game graph has vertex set S and an edge e=(i,s→s′)e=(i,s→ s ) for every agent i, profile s, and level si′>sis _i>s_i (with s−i′=s−is _-i=s_-i). The game’s flow is u(i,s→s′)=ui(s′)−ui(s)u(i,s→ s )=u_i(s )-u_i(s); the gradient of ϕ:S→ℝφ:S is (δϕ)(i,s→s′)=ϕ(s′)−ϕ(s)(δφ)(i,s→ s )=φ(s )-φ(s); and (divx)(s)=∑eintosx(e)−∑eout ofsx(e)(divx)(s)= _e\,into\,sx(e)- _e\,out of\,sx(e), so that ⟨δϕ,x⟩=∑sϕ(s)(divx)(s) δφ,x = _sφ(s)(divx)(s). Every flow decomposes uniquely and orthogonally as u=δϕ⋆+h,divh=0,u=δφ +h, =0, (8) with δϕ⋆δφ the projection of u onto imδimδ; the game is an exact potential game iff h=0h=0. We call ℰpot:=‖δϕ⋆‖2,ℰharm:=‖h‖2E_pot:=\|δφ \|^2, _harm:=\|h\|^2 (9) the potential and harmonic energies, respectively; “energy” means this squared norm, not physical energy. The potential share ρpot:=ℰpot/(ℰpot+ℰharm) _pot:=E_pot/(E_pot+E_harm) and ρharm:=1−ρpot _harm:=1- _pot are normalised summaries (the discussion after Corollary 2 states when they can be compared across designs). I-D Problem Statement Once a conflict graph C, a number K of levels, a protocol and a payoff-from-rank function f are fixed, the game f(C,K)G_f(C,K) and the pair (ℰpot,ℰharm)(E_pot,E_harm) of (9) are determined. We address three problems. (i) Closed form: determine ℰpotE_pot and ℰharmE_harm as functions of C, K and f (Proposition 2 for the linear game, for every protocol). (i) Meaning: determine what the two energies imply for the behaviour of the agents, both for the convergence of improvement dynamics (Propositions 4–5) and for the stationary behaviour of log-linear learning (Proposition 6). (i) Design: under a total-order protocol on the complete conflict graph, determine the smallest achievable harmonic share, ρharmmin(N,K):=inffnonconstantρharm(f(KN,K)). _harm (N,K):= _f\ nonconstant _harm (G_f(K_N,K) ). (10) The infimum is over all nonconstant f, not only decreasing ones (Theorems 1 and 2). Lemma 1 below reduces (i) and (i), and the learning part of (i), to computations with the Hoeffding components of a single function. I-E The Number-Operator Lemma Let AiA_i denote averaging over the i-th coordinate. Every g:S→ℝg:S has the Hoeffding (ANOVA) decomposition g=∑T⊆[N]gTg= _T [N]g_T, gT=∏j∈T(I−Aj)∏j∉TAjg_T= _j∈ T(I-A_j) _j∉ TA_j\,g; the components are mutually orthogonal, gTg_T depends on sTs_T only and is centred in each of its variables. Put gm=∑|T|=mgTg_m= _|T|=mg_T. The number operator :=∑i(I−Ai) N:= _i(I-A_i) multiplies gTg_T by |T||T|. Lemma 1 yields all closed forms: since the game graph is a product of complete graphs, its Laplacian multiplies an m-agent Hoeffding component by m, so the Hodge potential divides each component by its order. Its content is implicit in [17, Sec. 4.1], and (11) follows from the inverse of N in [30] and is stated in [25] (Section I); we state it in the graded form in which it is used. Lemma 1 (Number-operator lemma). For any payoffs (ui)(u_i) on S, let g:=∑i(ui−Aiui)g:= _i(u_i-A_iu_i). Then, up to an additive constant, ϕ⋆=−1g=∑T≠∅gT|T|,φ = N^-1g= _T≠ g_T|T|, (11) and ℰpot _pot =KN+1∑m≥1‖gm‖2m, =K^N+1 _m≥ 1 \|g_m\|^2m, (12) ℰpot+ℰharm _pot+E_harm =KN+1∑i‖ui−Aiui‖2. =K^N+1 _i\|u_i-A_iu_i\|^2. Proof. The game graph is the Cartesian product of N copies of the complete graph on K vertices, whose Laplacian is K(I−A)K(I-A); hence divδ=K∑i(I−Ai)=Kdivδ=K _i(I-A_i)=K N. At a profile s, (divu)(s) (divu)(s) =∑i∑t[ui(s)−ui(t,s−i)] = _i _t[u_i(s)-u_i(t,s_-i)] =K∑i(ui−Aiui)(s)=Kg(s). =K _i(u_i-A_iu_i)(s)=Kg(s). Thus the normal equation divδϕ⋆=divudivδφ =divu reduces to ϕ⋆=g Nφ =g. Since each AiA_i preserves the uniform mean, g has zero mean. Moreover, for the orthogonal decomposition g=∑TgTg= _Tg_T, one has gT=|T|gT Ng_T=|T|g_T. Therefore ϕ⋆=−1gφ = N^-1g up to an additive constant, which is (11). By adjointness and the normal equation, ℰpot _pot =⟨δϕ⋆,δϕ⋆⟩edge=∑sϕ⋆(s)(divu)(s) = δφ ,δφ _ edge= _sφ (s)(divu)(s) =KN+1⟨−1g,g⟩. =K^N+1 N^-1g,g . Writing gm=∑|T|=mgTg_m= _|T|=mg_T and using orthogonality gives ⟨−1g,g⟩=∑m≥1‖gm‖2/m N^-1g,g = _m≥ 1\|g_m\|^2/m, hence the first identity in (12). For the second, fix i and s−is_-i. Since ∑x<y(f(y)−f(x))2=K2Var(f) _x<y(f(y)-f(x))^2=K^2Var(f), ∑x<y(ui(y,s−i)−ui(x,s−i))2=K2Varsiui(⋅,s−i). _x<y (u_i(y,s_-i)-u_i(x,s_-i) )^2=K^2Var_s_iu_i(·,s_-i). Summing over s−is_-i gives ‖u‖edge2=KN+1∑i‖ui−Aiui‖2\|u\|_ edge^2=K^N+1 _i\|u_i-A_iu_i\|^2. Since the potential and harmonic components are orthogonal, ℰpot+ℰharm=‖u‖edge2E_pot+E_harm=\|u\|_ edge^2, proving the second identity. ∎ I Closed-Form Energies of Priority Coordination This section solves the first problem described in Section I-D: computing the potential and harmonic components of priority coordination in closed form. The main result, Proposition 2, shows that for the linear payoff the harmonic energy counts the conflicts and the potential energy adds the adjacent pairs of conflicts; Section I-C then reads the two components for a single move. I-A Symmetric and Antisymmetric Interactions Consider two neighbours i and j. Their interaction ωij _ij splits, by (4), into a symmetric part ωij+ω^+_ij, which affects both agents in the same way, and an antisymmetric part ωij−ω^-_ij, in which what one agent gains the other loses. Intuitively, only the symmetric part can be represented by a common objective that changes in the same way for both agents, whereas under the antisymmetric part their interests are opposed. Proposition 1 makes this exact and gives the two energies. Proposition 1 (Pairwise games, kernel form). Let G be the pairwise game (1) on a graph C with at least one edge, K≥2K≥ 2. Then: (a) the Hodge potential is, up to a constant, ϕ⋆(s)=∑i(si)+∑i,j∈Eωij+(si,sj).φ (s)= _i a_i(s_i)+ _\i,j\∈ Eω^+_ij(s_i,s_j). (13) (b) the harmonic flow h is the flow of the payoffs hi(s)=∑j∈(i)ωij−(si,sj)h_i(s)= _j (i)ω^-_ij(s_i,s_j). (c) the energies are ℰpot _pot =KN+1(∑i‖i‖2+∑i,j∈E‖ωij+‖2), =K^N+1 ( _i\| a_i\|^2+2\!\! _\i,j\∈ E\!\!\|ω^+_ij\|^2 ), (14) ℰharm _harm =2KN+1∑i,j∈E‖ωij−‖2. =2K^N+1\!\! _\i,j\∈ E\!\!\|ω^-_ij\|^2. (d) G is an exact potential game iff every interaction is symmetric, ωij(x,y)=ωji(y,x) _ij(x,y)= _ji(y,x), for every edge. Proof. Step 1. I−AiI-A_i annihilates every term of (1)–(2) not depending on sis_i and fixes i(si) a_i(s_i) and ωij(si,sj) _ij(s_i,s_j), which are centred in sis_i. Hence vi:=ui−Aiui=i(si)+∑j∈(i)ωij(si,sj)v_i:=u_i-A_iu_i= a_i(s_i)+ _j (i) _ij(s_i,s_j). Step 2. In g=∑ivig= _iv_i, the term i a_i is a Hoeffding component on T=iT=\i\, and ωij _ij is a Hoeffding component on T=i,jT=\i,j\. Collecting by index set gives gi=i,gi,j=ωij(si,sj)+ωji(sj,si)=2ωij+(si,sj)g_\i\= a_i, g_\i,j\= _ij(s_i,s_j)+ _ji(s_j,s_i)=2ω^+_ij(s_i,s_j) for i,j∈E\i,j\∈ E, and gT=0g_T=0 otherwise. Step 3. Lemma 1 gives ϕ⋆=∑TgT/|T|φ = _Tg_T/|T|, which is (13). The payoffs u^i:=i+∑jωij+ u_i:= a_i+ _jω^+_ij differ from ϕ⋆φ by a function of s−is_-i, so their flow is δϕ⋆δφ ; subtracting from viv_i gives (b). Step 4. By (12) and orthogonality, ℰpot/KN+1=∑i‖i‖2+∑E‖2ωij+‖2/2E_pot/K^N+1= _i\| a_i\|^2+ _E\|2ω^+_ij\|^2/2 and (ℰpot+ℰharm)/KN+1=∑i‖i‖2+∑E(‖ωij‖2+‖ωji‖2)(E_pot+E_harm)/K^N+1= _i\| a_i\|^2+ _E(\| _ij\|^2+\| _ji\|^2). Since the uniform measure is exchange-invariant, ‖ωij‖2+‖ωji‖2=2‖ωij+‖2+2‖ωij−‖2\| _ij\|^2+\| _ji\|^2=2\|ω^+_ij\|^2+2\|ω^-_ij\|^2, and (c) follows by subtraction; (d) is immediate from (c). ∎ What an agent gains from its own level, and what two neighbours gain reciprocally, admits a common objective; what one agent gains exactly at the other’s expense does not. The proposition is not new in substance ([17, Sec. 4.3] for two players; [28, 31] on graphs; [25, 21] for (a) and (d)); we include the short proof because the constants in (c) are used throughout. I-B Priority Games: Every Interaction is Antisymmetric In a priority game, the interaction between two conflicting agents is strictly competitive: exactly one of them is served first, so whatever one gains the other loses. For the linear payoff, which makes the game pairwise, Proposition 1(d) therefore shows that the game is never an exact potential game once there is a conflict. The result in Proposition 2 makes this quantitative: both energies are determined by the degrees of the conflict graph. Proposition 2 (Energies of priority coordination). For the linear rank-order game (C,K)G(C,K), every interaction is purely antisymmetric, ωij+≡0ω^+_ij≡ 0, and, up to an additive constant, the Hodge potential is ϕ⋆(s)=K−1∑idisiφ (s)=K^-1 _id_is_i. With cK:=KN−1(K2−1)/12c_K:=K^N-1(K^2-1)/12, ℰpot=cK∑idi2,ℰharm=cK∑idi=2cK|E|.E_pot=c_K _id_i^2, _harm=c_K _id_i=2c_K|E|. (15) Consequently, ρpot=∑idi2/∑idi(di+1) _pot= _id_i^2/ _id_i(d_i+1), independently of K, while ρharm≥1/(dmax+1) _harm≥ 1/(d_ +1) on every non-empty graph of maximum degree dmaxd_ . Thus the energy split is controlled by the degree structure rather than explicitly by K or N. In particular, ρpot=d/(d+1) _pot=d/(d+1) on a d-regular graph and ρpot=(N−1)/N _pot=(N-1)/N on the complete graph. All these conclusions hold for every protocol, i.e. every orientation of the conflict edges, acyclic or not. Proof. For any protocol, ψij(x,y)+ψji(y,x)≡−1 _ij(x,y)+ _ji(y,x)≡-1, since exactly one of two conflicting agents is served first. Double centring removes the constant, so ωij(x,y)+ωji(y,x)=0 _ij(x,y)+ _ji(y,x)=0 and hence ωij+=0ω^+_ij=0. Next, yψij(x,y)=−(K−1−x)/KE_y _ij(x,y)=-(K-1-x)/K if the protocol favours i on the edge and −(K−x)/K-(K-x)/K otherwise. Both equal (x−s¯)/K(x- s)/K up to a constant, where s¯=(K−1)/2 s=(K-1)/2. Hence i(x)=di(x−s¯)/K a_i(x)=d_i(x- s)/K, so Proposition 1(a) gives ϕ⋆(s)=K−1∑idisiφ (s)=K^-1 _id_is_i up to an additive constant, and ‖i‖2=di2(K2−1)/(12K2)\| a_i\|^2=d_i^2(K^2-1)/(12K^2). For either orientation, Varψij=(K2−1)/(4K2)Var _ij=(K^2-1)/(4K^2) and ‖aij‖2=‖bij‖2=(K2−1)/(12K2)\|a_ij\|^2=\|b_ij\|^2=(K^2-1)/(12K^2). By (2) and orthogonality, ‖ωij‖2=(K2−1)/(12K2)\| _ij\|^2=(K^2-1)/(12K^2); since ωij+=0ω^+_ij=0, also ‖ωij−‖2=‖ωij‖2\|ω^-_ij\|^2=\| _ij\|^2. Substituting into (14) gives (15). Finally, ρpot=∑idi2/∑idi(di+1) _pot= _id_i^2/ _id_i(d_i+1) and ρharm=∑idi/∑idi(di+1) _harm= _id_i/ _id_i(d_i+1). Since di+1≤dmax+1d_i+1≤ d_ +1, ρharm≥1/(dmax+1) _harm≥ 1/(d_ +1). The d-regular and complete-graph formulas follow by setting di=d_i=d and di=N−1d_i=N-1, respectively. ∎ Remark 1 (Reading the two energies). Let E2E_2 be the set of pairs of conflicts that share an agent, so that |E2|=∑i(di2)|E_2|= _i d_i2. Since ∑idi2=2|E|+2|E2| _id_i^2=2|E|+2|E_2|, in units of 2cK2c_K, ℰharm=|E|,ℰpot=|E|+|E2|,ρpot=|E|+|E2|2|E|+|E2|.E_harm=|E|,\;\;E_pot=|E|+|E_2|,\;\; _pot= |E|+|E_2|2|E|+|E_2|. (16) Every conflict carries one unit of harmonic energy, whatever the rest of the graph; potential energy is generated, in addition, by adjacent conflicts, because what is potential is the shared incentive to escalate, which is an own effect, and not the interaction. Removing the conflict i,j\i,j\ eliminates one unit of harmonic energy and di+dj−1d_i+d_j-1 units of potential energy. For the intersection of Example 1, |E|=16|E|=16 and |E2|=48|E_2|=48, so ℰharm:ℰpot=16:64E_harm:E_pot=16:64 and one fifth of the incentive energy is harmonic. The error of a potential surrogate. Let Φ:S→ℝ :S be a common objective that replaces the agents [15, 16]. Under log-linear learning with rationality β>0β>0, a revising agent i at profile s chooses between sis_i and si′s _i with log-odds β(ui(s′)−ui(s))=βu(e)β(u_i(s )-u_i(s))=β u(e), where e is the corresponding edge of the game graph; the common-objective model predicts βδΦ(e)β\,δ (e) instead. Corollary 1 (Best common-objective model of choice odds). For every game, every β>0β>0 and every Φ:S→ℝ :S , the squared prediction error of the log-odds, summed over all NKN(K−1)/2NK^N(K-1)/2 edges of the game graph, satisfies ∑eβ2(u(e)−δΦ(e))2≥β2ℰharm, _eβ^2 (u(e)-δ (e) )^2\ ≥\ β^2E_harm, (17) with equality iff Φ−ϕ⋆ -φ is constant. Hence, for every β>0β>0, the relative squared error of the best common-objective model is ρharm _harm. For the linear game (C,K)G(C,K) it is at least 1/(dmax+1)1/(d_ +1), on every conflict graph with at least one conflict and for every deterministic protocol. Proof. By (8), u−δΦ=h+δ(ϕ⋆−Φ)u-δ =h+δ(φ - ) with h⟂imδh δ, so ‖u−δΦ‖2=ℰharm+‖δ(ϕ⋆−Φ)‖2\|u-δ \|^2=E_harm+\|δ(φ - )\|^2. Dividing by ∑eu(e)2=ℰpot+ℰharm _eu(e)^2=E_pot+E_harm gives the relative error; the bound for (C,K)G(C,K) is Proposition 2. ∎ The error weights every unilateral move equally, as low-rationality learning does to first order (Proposition 6(a)); at high rationality play concentrates near the equilibria of Proposition 5, and an error weighted by the visited profiles can be smaller. The corollary measures what a surrogate misses about the incentives, not how often it mispredicts along a trajectory. Pricing and the limits of the two energies. For the linear game, a price on the levels changes the potential energy but not the harmonic one. This follows from a general property of pairwise games described in Corollary 2. Corollary 2 (Rigidity of the harmonic energy). ℰharmE_harm is unchanged by any modification of the own terms γi _i, the marginals aij,bija_ij,b_ij, the constants ψ¯ij ψ_ij, or the symmetric parts ωij+ω^+_ij. In particular, no separable pricing γi(si) _i(s_i) changes ℰharmE_harm. If ℰharm>0E_harm>0, pricing alone can move ρpot _pot to any value in [ρmin,1)[ _ ,1), where ρmin=∑E‖ωij+‖2/∑E(‖ωij+‖2+‖ωij−‖2) _ = _E\|ω^+_ij\|^2/ _E(\|ω^+_ij\|^2+\|ω^-_ij\|^2). Proof. By (14), ℰharmE_harm depends only on ω−ω^-. Hence changes to γi,aij,bij,ψ¯ij _i,a_ij,b_ij, ψ_ij, or ωij+ω^+_ij leave it unchanged. Pricing changes only i a_i: choosing γi=−∑jaij _i=- _ja_ij gives i≡0 a_i≡ 0 and hence ρpot=ρmin _pot= _ , while scaling any nonconstant pricing makes ∑i‖i‖2→∞ _i\| a_i\|^2→∞ and thus ρpot→1 _pot→ 1. Since ρpot _pot varies continuously with the scaling parameter, every value in [ρmin,1)[ _ ,1) is attained when ℰharm>0E_harm>0. ∎ Corollary 2 reveals the reason why we treat the pair (ℰpot,ℰharm)(E_pot,E_harm), rather than the ratio, as the fundamental quantity: ρpot _pot is comparable across designs only at a fixed normalisation of the own effects. It also separates pricing from transfers. A transfer is a payment τi:S→ℝ _i:S added to each payoff, giving the game with payoffs ui+τiu_i+ _i. Transfers that make this game an exact potential game must cancel the harmonic flow, so the energy of their flow is at least ℰharmE_harm; for a pairwise game this minimum is attained by the budget-balanced side payment τi=−∑j∈(i)ωij−(si,sj) _i=- _j (i)ω^-_ij(s_i,s_j) (Proposition 1(b)), which no pricing of the levels can achieve. Pricing, in turn, can change the behaviour drastically while leaving ℰharmE_harm fixed: with payoffs −posC(i,s)−csi-pos_C(i,s)-c\,s_i under the index protocol, an arbitrarily small price c∈(0,1)c∈(0,1) destroys every pure-strategy Nash equilibrium on any graph with a conflict (Proposition S44 in Appendix S-D of the SM). Neither energy predicts the existence of pure-strategy Nash equilibria, and we make no such claim. Nor do they measure efficiency: by (7), the total number of waits is constant across profiles. I-C Reading a Single Move: Mean and Fluctuation Proposition 2 gives the energies of the whole game. The results in Proposition 3 explain the decomposition for a single move: when an agent raises its level, it identifies which part of the gain is potential and which part is harmonic. Proposition 3 (Mean and fluctuation). For (C,K)G(C,K), along an edge (i,s→s′)(i,s→ s ) with jump ℓ=si′−si≥1 =s _i-s_i≥ 1, a neighbour j is overtaken iff si<sj<si′,or(sj=si,j→i),or(sj=si′,i→j),s_i<s_j<s _i,\ or\ (s_j=s_i,\ j→ i),\ or\ (s_j=s _i,\ i→ j),\!\! (18) and the game flow is the number W of overtaken neighbours. Then h(i,s→s′)=W−diℓ/Kh(i,s→ s )=W-d_i /K: the potential part of a move is the expected number of neighbours overtaken under uniformly random levels, and the harmonic part is the deviation from it. Proof. By definition of the rank-order payoff, the flow along (i,s→s′)(i,s→ s ) equals the number W of neighbours satisfying (18), so u=Wu=W. By Proposition 2, δϕ⋆=diℓ/Kδφ =d_i /K, hence h=u−δϕ⋆=W−diℓ/Kh=u-δφ =W-d_i /K. For each neighbour j, exactly ℓ of the K possible values of sjs_j satisfy (18): the ℓ−1 -1 interior levels si<sj<si′s_i<s_j<s_i , together with exactly one endpoint selected by the orientation of i,j\i,j\. Thus, under a uniform level for sjs_j, the overtaking probability is ℓ/K /K. By linearity of expectation, W=diℓ/KEW=d_i /K, proving the claim. ∎ Proposition 3 identifies the potential part of a move with the mean of W. In fact, the whole distribution of W is explicit. Given the move, the did_i overtaking events are independent under the uniform reference measure, each with probability ℓ/K /K, so W∼Bin(di,ℓ/K)W (d_i, /K). The potential part of the move is the mean diℓ/Kd_i /K, and the harmonic part is the fluctuation around it, with variance diℓK(1−ℓK)d_i K (1- K ). The squared mean grows like di2d_i^2 and the variance like did_i, which is the scaling ℰpot∝∑idi2E_pot _id_i^2 and ℰharm∝∑idiE_harm _id_i of Proposition 2. The constants also match. Summing over all moves weights each jump length ℓ by the number K−ℓK- of level pairs at distance ℓ , and ∑ℓ=1K−1(K−ℓ)(ℓK)2=∑ℓ=1K−1(K−ℓ)ℓK(1−ℓK)=K2−112. _ =1^K-1(K- ) ( K )^2= _ =1^K-1(K- )\, K (1- K )= K^2-112. The two sums are not equal term by term; rather, the substitution ℓ↦K−ℓ K- maps one onto the other. This is why the potential share of the linear game does not depend on K. Thus the Hodge potential collects what an agent can expect from its own level against uniformly random opponents, and the harmonic flow is what depends on who the opponents actually are (cf. the tournament reading in [44]); its dependence on ties is computed in Theorem S36 of the SM. For the linear payoff, if ties are broken uniformly at random and payoffs are taken in expectation, ℰpotE_pot is unchanged and ℰharmE_harm is multiplied by (K−2)/(K+1)(K-2)/(K+1) (Proposition S38). For K≥3K≥ 3 a deterministic protocol therefore amplifies the harmonic component without creating it, whereas for K=2K=2 randomised tie-breaking yields an exact potential game. IV Dynamics and Learning This section addresses the second problem posed in Section I-D. It first shows that a nonzero harmonic component need not induce cycles in improvement dynamics, and then identifies what the harmonic component does affect: the stationary behaviour of log-linear learning. In Propositions 4 and 5, C is an arbitrary conflict graph and f an arbitrary strictly decreasing payoff-from-rank function. Proposition 4 (Finite improvement). For every conflict graph C, every protocol and every strictly decreasing f, both V(s)=∑isiV(s)= _is_i and the Hodge potential of the linear game, ϕ⋆(s)=1K∑idisiφ (s)= 1K _id_is_i, are generalised ordinal potentials of f(C,K)G_f(C,K). Every improvement path has length at most N(K−1)N(K-1). Proof. Fix s−is_-i. If si′>sis _i>s_i and j is served before i at s′s , then sj≥si′>sis_j≥ s _i>s_i, so j is also served before i at s. Hence posC(i,s)pos_C(i,s) is non-increasing in sis_i, and since f is strictly decreasing, every strict improvement must have si′>sis _i>s_i. The mover cannot be isolated, so di>0d_i>0; therefore both V(s)=∑isiV(s)= _is_i and ϕ⋆(s)=K−1∑idisiφ (s)=K^-1 _id_is_i strictly increase. Since each improvement raises the integer-valued V by at least one and 0≤V≤N(K−1)0≤ V≤ N(K-1), every improvement path has length at most N(K−1)N(K-1). ∎ Proposition 4 concerns strictly improving moves. If a revising agent may switch between equally good levels, play can cycle: for N=K=2N=K=2 with agent 00 favoured, (0,0)→(0,1)→(1,1)→(1,0)→(0,0)(0,0)→(0,1)→(1,1)→(1,0)→(0,0) consists of best responses, the last two being indifferent. Hence best-response dynamics converge with inertia (an agent moves only when it strictly gains) and not without it. Thus ϕ⋆φ is a Lyapunov function for asynchronous strict better-response dynamics of every f(C,K)G_f(C,K), although (C,K)G(C,K) is not an exact potential game. Since every improving move (i,s→s′)(i,s→ s ) raises the level of i by some ℓ≥1 ≥ 1, δϕ⋆=diℓ/K>0δφ =d_i /K>0 along it: the harmonic flow changes the size of the gain but cannot create an improving move against the potential. Proposition 5 (Pure-strategy Nash equilibria). Under the index protocol, let ℬ(s)=i:si<K−1B(s)=\i:s_i<K-1\ be the set of agents below the top level. A profile s is a pure-strategy Nash equilibrium of f(C,K)G_f(C,K) iff every i∈ℬ(s)i (s) has a larger index than each of its neighbours; in particular ℬ(s)B(s) is then an independent set of C. Consequently, in every pure-strategy Nash equilibrium and on every conflict edge, the agent favoured by the protocol is served first: all pure-strategy Nash equilibria induce the same service order. Mixed equilibria are not considered. Proof. By monotonicity of posC(i,⋅)pos_C(i,·) in sis_i, i’s best deviation is to K−1K-1, after which its predecessors are exactly j∈(i):sj=K−1,j<i\j (i):s_j=K-1,\ j<i\. Hence s is a Nash equilibrium iff every i∈ℬ(s)i (s) has no neighbour in ℬ(s)B(s) and no neighbour j>ij>i, i.e. iff every such i has larger index than all its neighbours. Thus ℬ(s)B(s) is independent. Consequently every conflict edge has at least one endpoint at level K−1K-1; if exactly one does, it is the protocol-favoured endpoint, while if both do, the index protocol itself determines the order. Hence all pure-strategy Nash equilibria induce the same service order. ∎ IV-A Log-linear Learning: Stationary Law and Current Best response ignores magnitudes. Log-linear learning is sensitive to them, and at low rationality it gives both energies a dynamical meaning to leading order. At each step an agent i is selected with probability αi>0 _i>0, where ∑iαi=1 _i _i=1, and resamples its level from Pr(si=x)∝exp(βui(x,s−i)) (s_i=x) (β u_i(x,s_-i)). This is an irreducible Markov chain PβP_β on S with stationary law πβ _β; for a potential game with potential ϕφ it is reversible and πβ∝eβϕ _β e^βφ [10]. For a general game, let ϕα _α and hα:=u−δϕαh_α:=u-δ _α be the Hodge decomposition in the weighted norm ‖x‖α2=∑eαi(e)x(e)2\|x\|_α^2= _e _i(e)x(e)^2, where i(e)i(e) is the agent moving along e; for uniform selection, αi=1/N _i=1/N, these are ϕ⋆φ and h. Write β(e)=πβ(s)Pβ(s,s′)−πβ(s′)Pβ(s′,s)J_β(e)= _β(s)P_β(s,s )- _β(s )P_β(s ,s) for the stationary probability current along e=(i,s→s′)e=(i,s→ s ), and σβ=∑eβ(e)logπβ(s)Pβ(s,s′)πβ(s′)Pβ(s′,s)≥0 _β= _eJ_β(e) _β(s)P_β(s,s ) _β(s )P_β(s ,s)≥ 0 for the entropy production rate, which vanishes iff the chain is reversible. Proposition 6 (Low-rationality learning). Let N≥2N≥ 2 agents each have the K≥2K≥ 2 actions 0,…,K−1\0,…,K-1\, let (ui)i∈[N](u_i)_i∈[N] be arbitrary payoffs on S=0,…,K−1NS=\0,…,K-1\^N, and let log-linear learning be run with fixed selection probabilities αi>0 _i>0, ∑iαi=1 _i _i=1. Then, with N, K, (ui)(u_i) and (αi)( _i) fixed and β→0β→ 0, (a) πβ∝eβϕα(1+O(β2)) _β e^β _α\, (1+O(β^2) ): the occupation measure is Gibbs in the Hodge potential; (b) β(e)=βαihα(e)/KN+1+O(β2)J_β(e)=β\, _i\,h_α(e)/K^N+1+O(β^2) on every edge e=(i,s→s′)e=(i,s→ s ): the probability current is the harmonic flow; (c) σβ=β2‖hα‖α2/KN+1+O(β3) _β=β^2\,\|h_α\|_α^2/K^N+1+O(β^3). Under uniform selection, πβ∝eβϕ⋆(1+O(β2)) _β e^βφ (1+O(β^2)), β=βh/(NKN+1)+O(β2)J_β\!=\!β h/(NK^N+1)\!+O(β^2) and σβ=β2ℰharm/(NKN+1)+O(β3) _β\!\!=\!\!β^2E_harm/(NK^N+1)\!+O(β^3). Proof. Since αi>0 _i>0 and every logit choice has positive probability, the finite chain is irreducible for all β, so its stationary distribution is unique and smooth near β=0β=0. Expanding Pβ=P0+βP1+O(β2)P_β=P_0+β P_1+O(β^2) gives P0=∑iαiAiP_0= _i _iA_i and, for a transition changing coordinate i, P1(s,s′)=αiK(ui(s′)−(Aiui)(s))P_1(s,s )= _iK(u_i(s )-(A_iu_i)(s)). Since P0P_0 is symmetric and doubly stochastic, π0=K−N _0=K^-N1 and πβ=π0+βπ1+O(β2) _β= _0+β _1+O(β^2) with π1⟂ _1 1. (a) The O(β)O(β) stationarity equation is π1(I−P0)=π0P1 _1(I-P_0)= _0P_1. Here I−P0=α:=∑iαi(I−Ai)I-P_0= N_α:= _i _i(I-A_i) and π0P1=K−Ngα _0P_1=K^-Ng_α, where gα:=∑iαi(ui−Aiui)g_α:= _i _i(u_i-A_iu_i). The weighted Hodge normal equation is αϕα=gα N_α _α=g_α; since α N_α is invertible on ⟂1 , π1=K−N(ϕα−ϕ¯α) _1=K^-N( _α- φ_α). This is exactly the first-order expansion of the normalized Gibbs measure proportional to eβϕαe^β _α, proving (a). (b) Let β=0+β1+O(β2)J_β=J_0+ _1+O(β^2). Since P0P_0 is symmetric, 0=0J_0=0. For e=(i,s→s′)e=(i,s→ s ), using P0(s,s′)=αi/KP_0(s,s )= _i/K and (Aiui)(s)=(Aiui)(s′)(A_iu_i)(s)=(A_iu_i)(s ), 1(e)=αiKN+1[(ui(s′)−ui(s))−(ϕα(s′)−ϕα(s))]=αihα(e)KN+1.J_1(e)\!\!=\!\! _iK^N+1 [(u_i(s )-u_i(s))-( _α(s )- _α(s)) ]\!\!=\!\! _ih_α(e)K^N+1. (c) Let qe=πβ(s)Pβ(s,s′)q_e= _β(s)P_β(s,s ) and qe′=πβ(s′)Pβ(s′,s)q _e= _β(s )P_β(s ,s). Then qe−qe′=β(e)=O(β)q_e-q _e=J_β(e)=O(β) and qe,qe′=αi/KN+1+O(β)q_e,q _e= _i/K^N+1+O(β), so log(qe/qe′)=KN+1β(e)/αi+O(β2) (q_e/q _e)=K^N+1J_β(e)/ _i+O(β^2). Hence β(e)logqeqe′=β2αihα(e)2KN+1+O(β3).J_β(e) q_eq _e= β^2 _ih_α(e)^2K^N+1+O(β^3). Summing over one orientation of each edge yields σβ=β2‖hα‖α2/KN+1+O(β3) _β=β^2\|h_α\|_α^2/K^N+1+O(β^3). For uniform selection, αi=1/N _i=1/N, so α=/N N_α= N/N and gα=g/Ng_α=g/N, hence ϕα=ϕ⋆ _α=φ and hα=h_α=h. Also ‖hα‖α2=ℰharm/N\|h_α\|_α^2=E_harm/N, giving the stated formulas. ∎ The harmonic component is thus invisible in the occupation measure to first order and appears as the current it sustains; its energy determines, to leading order, the entropy production rate. For the linear priority game under uniform revision, Proposition 2 gives σβ=β2(K2−1)|E|/(6NK2)+O(β3) _β=β^2(K^2-1)|E|/(6NK^2)+O(β^3): every conflict contributes the same amount to the leading-order entropy production, whatever the rest of the graph. With the L2L^2 projection the first-order discrepancy from a Gibbs law vanishes identically, whereas near-potential games are only guaranteed a stationary law within O(βδ)O(βδ) of that of a nearby potential game [19]. The selection probabilities αi _i enter the stationary law at first order, through ϕα _α, only when ℰharm>0E_harm>0. Remark 2 (Range of validity). Proposition 6 describes the noise-dominated regime and asserts only the orders of the remainders. For the intersection, the law (c) is accurate to about 10%10\% for βmaxidi≲1β _id_i 1 (Fig. 4(d)), an observation for one example, not a uniform bound. As β grows, logit responses approach best responses, which see only the ordinal structure of Propositions 4–5; we do not attempt a large-β theorem. V General Payoffs: The Hodge Spectrum Sections I–IV concern the linear payoff. This section and the next address the third problem described in Section I-D: whether the harmonic share can be reduced by choosing a different payoff-from-rank function. We work on the complete conflict graph C=KNC=K_N, where agents compete with every other, and, for finitely many levels, with the index protocol (every total order is equivalent after relabelling). This section treats continuous priorities, where the answer can be computed completely; Section VI returns to finitely many levels. The first step is to turn the design question into a matrix problem. A payoff-from-rank function is determined, up to an irrelevant constant, by its increments Fj=f(j−1)−f(j)F_j=f(j-1)-f(j), j=1,…,N−1j=1,…,N-1, i.e. by how much an agent loses when it drops by one rank. The game flow is linear in F: along an edge e, uf(e)=∑jFjIj(e)u_f(e)= _jF_jI_j^(e) with Ij(e)=pos′<j≤posI^(e)_j=1\pos <j \, where pospos and pos′pos are the positions of the moving agent at the two ends of e (a move upwards can only lower the position). Both energies are therefore quadratic forms in F. With the Gram matrices [Mtot]jk=⟨Ij,Ik⟩[M_tot]_jk= I_j,I_k and [Mpot]jk=⟨PIj,PIk⟩[M_pot]_jk= PI_j,PI_k , where P is the orthogonal projection onto imδimδ, the potential share is a Rayleigh quotient, ρpot(f)=F⊤MpotF⊤MtotF,QK:=N−1NMtot−Mpot. _pot(G_f)= F M_potFF M_totF, Q_K:= N-1NM_tot-M_pot. (19) The bound ρpot≤(N−1)/N _pot≤(N-1)/N for all F is thus equivalent to QK⪰0Q_K 0, and the equality cases are the kernel of QKQ_K. The linear payoff corresponds to F=F=1. Lemma 2 (Exact eigenvector). For every N≥2N≥ 2 and K≥2K≥ 2, QK=0Q_K1=0: the Rayleigh quotient (19) is stationary at the linear payoff, with value (N−1)/N(N-1)/N. The proof, given in Appendix B-A, uses only (7) and the conditional independence of opponents given sis_i, and the dependence on K cancels. To compute the whole spectrum, we pass to continuous priorities, where ties have probability zero. Each agent chooses si∈[0,1]s_i∈[0,1], the reference measure is the uniform law on [0,1]N[0,1]^N, and ui(s)=f(posi(s))u_i(s)=f(pos_i(s)) with posi(s)=#j≠i:sj>sipos_i(s)=\#\j≠ i:s_j>s_i\. A flow assigns to every agent i, every pair of levels t,t′t,t and every s−is_-i a value xi(t→t′;s−i)x_i(t→ t ;s_-i) that is antisymmetric in (t,t′)(t,t ). The game flow is ui(t′,s−i)−ui(t,s−i)u_i(t ,s_-i)-u_i(t,s_-i), and the gradient of ϕ:[0,1]N→ℝφ:[0,1]^N is ϕ(t′,s−i)−ϕ(t,s−i)φ(t ,s_-i)-φ(t,s_-i). Flows carry the norm ∥x∥2=∑is−i[12∬[0,1]2xi(t→t′;s−i)2dtdt′],\|x\|^2= _iE_s_-i [ 12 _[0,1]^2x_i(t→ t ;s_-i)^2\,dt\,dt ], (20) the continuous analogue, up to normalisation, of the counting norm of Section I, and P denotes the orthogonal projection onto the closure of the space of gradients. The potential and the total energy are ‖Pu‖2\|Pu\|^2 and ‖u‖2\|u\|^2, and ρpot=‖Pu‖2/‖u‖2 _pot=\|Pu\|^2/\|u\|^2. With the continuum form of Lemma 1 (Appendix A, Step 1), the two energies become ‖Pu‖2 \|Pu\|^2 =∑m=1N−1(N−m)2mJm(f), = _m=1^N-1 (N-m)^2mJ_m(f), (21) ‖u‖2 \|u\|^2 =N∑m=1N−1(N−m)Jm(f), =N _m=1^N-1(N-m)J_m(f), where Jm(f)=Jm(f,f)J_m(f)=J_m(f,f) is the order-m form Jm(f,g)=(Nm)∬κ(q,q′)mdm[f](q)dm[g](q′)qdq′J_m(f,g)= Nm κ(q,q )^md_m[f](q)d_m[g](q )\,dq\,dq , with κ(q,q′)=min(q,q′)−qq′κ(q,q )= (q,q )-q the covariance of the Brownian bridge and dm[f](q)=[Δmf(Bin(N−1−m,q))]d_m[f](q)=E[ ^mf(Bin(N-1-m,q))]; each JmJ_m is positive semidefinite. Intuitively, Jm(f)J_m(f) measures how much of the incentive structure of the payoff is carried by interactions that involve m agents at once. Comparing the equations in (21), the order m contributes to the potential energy with a relative weight that decreases with m, so interactions among many agents are relatively more harmonic. Theorem 1 (Hodge spectrum). Let the discrete Chebyshev polynomials be tp(r)=∑k=0p(−1)k(pk)(p+k)rk¯(N−1)k¯t_p(r)= _k=0^p(-1)^k pk p+kk r k(N-1) k, p=0,…,N−1p=0,…,N-1, i.e. the orthogonal polynomials of the uniform distribution on 0,…,N−1\0,…,N-1\. (a) Every order form is diagonal. For all m and all p≠p′p≠ p , Jm(tp,tp′)=0J_m(t_p,t_p )=0, and Jm(tp,tp)=(Nm)m(p+m)!p(p+1)(2p+1)(p−m)![(N−1)m¯]2J_m(t_p,t_p)= Nm\,m\,(p+m)!p(p+1)(2p+1)(p-m)!\,[(N-1) m]^2 (22) for m≤pm≤ p, while Jm(tp,tp)=0J_m(t_p,t_p)=0 for m>pm>p. Consequently t1,…,tN−1t_1,…,t_N-1 diagonalise ∑mwmJm _mw_mJ_m for every choice of weights, in particular both forms in (21). (b) Eigenvalues. The generalised eigenvalues of (21) are λp=(N−p)(N+p+1)Np(p+1)[1−(Np)(N+p+1p)], _p= (N-p)(N+p+1)N\,p(p+1) [1- Np N+p+1p ], (23) for p=1,…,N−1p=1,…,N-1. In particular λ1=(N−1)/N _1=(N-1)/N and λp+1<λp _p+1< _p. The proof is given in Appendix A; we explain here why the theorem holds. Each order form JmJ_m pairs the smoothed differences dm[f]d_m[f] with the kernel κmκ^m, and two classical facts about orthogonal polynomials make the two compatible. First, averaging tpt_p over a binomial law turns it into the Legendre polynomial of degree p, and taking m differences then turns it into a Jacobi polynomial of degree p−mp-m [43]. Second, the integral operator with kernel κmκ^m maps each of these Jacobi polynomials to a multiple of itself times [q(1−q)]m[q(1-q)]^m, and [q(1−q)]m[q(1-q)]^m is exactly the weight function for which these Jacobi polynomials are orthogonal. Hence, tpt_p and tp′t_p are orthogonal in every order form, which is part (a), and the constant in (22) is the norm of a Jacobi polynomial. Part (b) follows by inserting (22) into (21), where the resulting finite sums can be evaluated in closed form. A payoff of degree p has Hoeffding components of order up to p, which is why λp _p decreases. For fixed p and N→∞N→∞, 1−λp=(p+12)/N+O(N−2)1- _p= p+12/N+O(N^-2). Corollary 3 (Continuum optimality). For continuous priorities, ρpot(f)≤(N−1)/N _pot(f)≤(N-1)/N for every nonconstant f, with equality iff f is affine. Proof. For nonconstant f, ‖u‖2>0\|u\|^2>0. By (21), N−1N‖u‖2−‖Pu‖2=∑m≥1(N−m)Jm(f)N(m−1)m≥0, N-1N\|u\|^2-\|Pu\|^2= _m≥ 1(N-m)J_m(f) N(m-1)m≥ 0, since Jm(f)≥0J_m(f)≥ 0. Hence ρpot(f)=‖Pu‖2/‖u‖2≤(N−1)/N _pot(f)=\|Pu\|^2/\|u\|^2≤(N-1)/N. Equality holds iff Jm(f)=0J_m(f)=0 for every m≥2m≥ 2. By the representation of JmJ_m in terms of dm[f]d_m[f], this is equivalent to dm[f]≡0d_m[f]≡ 0 for all m≥2m≥ 2, hence to f being affine. Conversely, affine f has dm[f]≡0d_m[f]≡ 0 for m≥2m≥ 2, so equality holds. ∎ Corollary 4 (Loss of potential share for non-affine payoffs). For continuous priorities and nonconstant f, let Rf2R_f^2 be the squared correlation between f(r)f(r) and r under the uniform law on ranks, i.e. the fraction of VarrfVar_rf explained by linear regression on r. Then 2N+2(1−Rf2)≤λ1−ρpot(f)≤(λ1−λN−1)(1−Rf2), 2N+2(1-R_f^2)≤ _1- _pot(f)≤( _1- _N-1)(1-R_f^2), (24) with equality on the left iff f is a polynomial of degree at most two. Proof. By Theorem 1, ρpot(f)=∑p=1N−1λpwp _pot(f)= _p=1^N-1 _pw_p, where wp≥0w_p≥ 0 is the variance share of the tpt_p component and ∑pwp=1 _pw_p=1. Since t1t_1 spans the centred linear functions of rank, w1=Rf2w_1=R_f^2. Hence λ1−ρpot(f)=∑p≥2(λ1−λp)wp,∑p≥2wp=1−Rf2. _1- _pot(f)= _p≥ 2( _1- _p)w_p, _p≥ 2w_p=1-R_f^2. As λp _p is strictly decreasing, the bounds follow from λ1−λ2=2/(N+2) _1- _2=2/(N+2) and λ1−λp≤λ1−λN−1 _1- _p≤ _1- _N-1. Equality in the lower bound holds iff wp=0w_p=0 for all p≥3p≥ 3, i.e. iff f∈spant0,t1,t2f \t_0,t_1,t_2\, equivalently f has degree at most two. For N=2N=2, every f on the two ranks is affine and the claim is immediate. ∎ Thus, the loss of potential share relative to the linear payoff is bounded on both sides by the payoff’s deviation from linearity. For example, when N=8N=8, the potential share decreases from 0.8750.875 for the linear payoff to 0.8600.860 for f(r)=−r2f(r)=-r^2, and further to 0.6470.647 for winner-take-all. VI Finitely Many Priority Levels Corollary 3 shows that, for continuous priorities, no payoff-from-rank function has a smaller harmonic share than the affine one. In applications the number of priority levels is finite and often small, e.g. a few levels of urgency. Ties then occur with positive probability, and a deterministic protocol resolves them in favour of some agents, which breaks the exchangeability that the continuum argument rests on. The bound nevertheless remains valid for every K≥3K≥ 3, and it can fail for K=2K=2. Theorem 2 (Finite quantisation). For every K≥3K≥ 3 and every N≥2N≥ 2, QK⪰0Q_K 0 with kerQK=span() Q_K=span(1). Equivalently, among all nonconstant payoff-from-rank functions on KNK_N under a total-order protocol with K≥3K≥ 3 priority levels, ρpot(f)≤(N−1)/N _pot(f)≤(N-1)/N with equality iff f is affine. The hypothesis K≥3K≥ 3 cannot be dropped: for K=2K=2 and N=7N=7 the strictly decreasing payoff with increments F=(9,8,6,4,2,1)F=(9,8,6,4,2,1) has F⊤Q2F=−7237/480<0F Q_2F=-7237/480<0, i.e. ρpot=53454277/62354880>6/7 _pot=53454277/62354880>6/7. The proof is provided in Appendix B, with the lengthy computations in the SM. We explain here its structure and where the hypothesis K≥3K≥ 3 enters. Lemma 2 gives the kernel; positivity is established in four stages, each carried out in the corresponding part of Appendix B. 1) Splitting by interaction order (Appendix B-B). Decomposing the payoffs into their Hoeffding components splits QKQ_K into a sum of forms, one for each interaction order m, i.e. for each number of agents whose levels jointly determine a component. Every form of order m≥3m≥ 3 is positive semidefinite for every payoff (Proposition 8). The reason is a sign structure: of two conflicting agents exactly one is served first, so the corresponding indicators are complementary and their cross terms are nonpositive. It remains to treat the one- and two-agent terms. 2) Reduction to bond variables (Appendix B-C). The one- and two-agent terms depend on the payoff only through N−1N-1 vectors in ℝKR^K, one for each pair of adjacent ranks, which we call bonds. In these variables the form is a sum of squares minus a single negative term (Proposition 9). 3) A sharp inequality on the levels (Appendix B-D). The negative term is dominated, bond by bond, by an inequality on functions of the level whose constant is K−2K-2 and cannot be improved (Lemma 3 below). This is the only place where K≥3K≥ 3 is used. 4) Closing the argument (Appendices B-E and B-F). What remains is a matrix inequality on the path of bonds, which a Perron-type argument reduces to one scalar condition per bond. This condition is proved analytically for K≥5K≥ 5, and with exact symbolic computation for K=3,4K=3,4, in each case for all but finitely many N; the finitely many remaining pairs (K,N)(K,N) are certified by exact rational matrix factorisations. No floating-point computation enters the proof; the scripts are listed in Appendix S-E of the SM. Since stage 3) is where the hypothesis K≥3K≥ 3 enters, we state its inequality. For D∈ℝKD ^K, viewed as a function of the level, let G(a)=∑x≤aD(x)G(a)= _x≤ aD(x) and H(a)=∑x<aD(x)H(a)= _x<aD(x) be its inclusive and exclusive cumulative sums, let Var(g)=1K∑x=0K−1(g(x)−g¯)2Var(g)= 1K _x=0^K-1 (g(x)- g )^2, with g¯=1K∑xg(x) g= 1K _xg(x), be the variance of g∈ℝKg ^K under the uniform law on the levels (so Var(D)Var(D), Var(G)Var(G), Var(H)Var(H) are defined), and let (g)=∑y=1K−1y(K−y)K(g(y)−g(y−1))2D(g)= _y=1^K-1 y(K-y)K (g(y)-g(y-1) )^2, a weighted Dirichlet form for which ≥2VarD≥ 2Var. Lemma 3 (Sharp level inequality). For K≥3K≥ 3, μ∈[0,1]μ∈[0,1], ν=1−μν=1-μ and every D∈ℝKD ^K, μ(−2Var)(H)+ν(−2Var)(G)≥(K−2)μνVar(D),μ\,(D-2Var)(H)+ν\,(D-2Var)(G)\ ≥\ (K-2)\,μν\,Var(D), (25) and the constant K−2K-2 cannot be improved. The proof (Appendix B-D; details in Lemma S16 of the SM) writes the difference of the two sides as a Schur product of the Green’s function of a path with a positive semidefinite matrix. Remark 3 (why K≥3K≥ 3). For K=2K=2 the right-hand side of (25) vanishes, the negative term of stage 2) is no longer compensated, and the statement is indeed false: exact computation gives Q2⪰0Q_2 0 for N≤6N≤ 6, while at N=7N=7 both the payoff of Theorem 2 and its reversal beat the linear one. Numerically the violation persists for 7≤N≤127≤ N≤ 12 (Table S3 in the SM). Proposition 7 describes how finite quantisation perturbs the continuum picture. Let a protocol be given by a total order ≺ of the agents, let ≺rev ^rev be the reversed order, and let ≺(K) m (K) denote either of the two energy forms of f(C,K)G_f(C,K) divided by KN+1K^N+1. Proposition 7 (Reciprocity and expansion). (a) ≺(K) m (K) is a polynomial in 1/K1/K of degree at most 2N−12N-1, whose constant term is the corresponding form for continuous priorities. (b) As polynomials in x=1/Kx=1/K, ≺(−x)=≺rev(x) m (-x)= m ^rev(x): changing the sign of K reverses the protocol. (c) If an automorphism of C reverses the protocol on every edge, as i↦N−1−i N-1-i does on KNK_N, then ≺(K) m (K) is a polynomial in K−2K^-2 of degree at most N−1N-1 vanishing at K=1K=1. In particular K−(N+1)QK=Q∞+∑j=1N−1K−2jj,Q∞+∑j=0,K^-(N+1)Q_K=Q_∞+ _j=1^N-1K^-2j\, C_j,~Q_∞+ _j C_j=0, (26) where Q∞:=limK→∞K−(N+1)QKQ_∞:= _K→∞K^-(N+1)Q_K is the continuum form of Section V, i.e. the form N−1N‖u‖2−‖Pu‖2 N-1N\|u\|^2-\|Pu\|^2 of (21) written in the increments F, diagonal in the Chebyshev basis, and j C_j do not depend on K. The proof (Section S-A.15 of the SM) counts the lattice points that induce a given order of the coordinates; part (b) is the reciprocity theorem for P-partitions [45], since changing the sign of the dilation exchanges weak and strict inequalities, i.e. who wins a tie. Hence finite quantisation perturbs the two forms of Theorem 1 by a terminating expansion in K−2K^-2, and the generalised eigenvalues equal λp+O(K−2) _p+O(K^-2). Without an automorphism that reverses the protocol, odd powers of 1/K1/K do occur. The expansion also locates the failure at K=2K=2: extended to real K via (26), the critical value at which the smallest nontrivial eigenvalue vanishes is 1.7501.750, 1.8751.875, 1.9531.953, 2.0062.006 for N=4,…,7N=4,…,7. Theorem 2 turns Corollary 1 into a limit over all payoffs. Corollary 5 (Design limit for potential surrogates). Under the settings of Theorem 2, let K≥3K≥ 3 and β>0β>0. For every nonconstant f and every Φ:S→ℝ :S , ∑eβ2(u(e)−δΦ(e))2≥1N∑eβ2u(e)2, _eβ^2 (u(e)-δ (e) )^2\ ≥\ 1N _eβ^2u(e)^2, (27) with equality iff f is affine and Φ−ϕ⋆ -φ is constant. Proof. Combine (17) with ℰharm≥1N(ℰpot+ℰharm)E_harm≥ 1N(E_pot+E_harm), which is Theorem 2. ∎ Fig. 3: Closed forms against independent computation. (a) Energies of Proposition 2 against a least-squares Hodge projection. (b) Finite-K spectrum against Theorem 1. Remark 4 (Interpretation of Theorem 2). Within the family f\G_f\ there are no own terms to rescale, so the normalisation issue of Corollary 2 does not arise and the shares of different payoffs are comparable. Theorem 2 is therefore a limit on reward design. By Proposition 6, under uniform revision, σβ≥β2(ℰpot+ℰharm)/(N2KN+1)+O(β3) _β≥β^2(E_pot+E_harm)/(N^2K^N+1)+O(β^3): relative to the total incentive energy, no payoff from rank makes low-rationality learning less irreversible, to leading order, than the linear payoff. Corollary 5 gives the same limit at every rationality, without an expansion in β. The limit concerns the magnitudes of the incentives, not their direction, which an ordinal potential captures (Proposition 4). Fig. 4: Log-linear learning (Proposition 6), computed from the transition matrix. (a) Occupation, on C5C_5 (K=3K=3): distance between the stationary law and the Gibbs law of a Hodge potential. (b) Circulation: stationary current against the harmonic flow on every edge of the game graph of the intersection. (c) Irreversibility: entropy production against harmonic energy for four families of games; the line is the prediction, without fitted parameters; the random games are not pairwise. (d) Range of validity on the intersection: the low-β law gives way to the ordinal regime, in which mass shifts towards the set of pure-strategy Nash equilibria of Proposition 5 without being absorbed by it. VII Numerical Study The experiments check the closed forms against computations that do not use them and show what the results mean on concrete systems: the eight-vehicle intersection of Example 1 with K=3K=3 (65616561 profiles, 52 48852\,488 edges), families of small conflict graphs, games that are not priority games, and alternative layouts. All quantities are computed by enumerating the profile space in double precision. The Hodge components are obtained by a sparse least-squares projection of the flow onto imδimδ, which uses neither Lemma 1 nor any closed form; stationary laws, currents and entropy production are computed from the transition matrix of log-linear learning, not by simulation. None of these computations enters any proof. VII-A Verification of the Closed Forms Figure 3 checks Proposition 2 and Theorem 1. Panel (a) shows that the least-squares energies of (C,K)G(C,K) on 2424 conflict graphs with four to six vertices and K∈2,3,4K∈\2,3,4\, divided by cKc_K, equal ∑idi2 _id_i^2 and ∑idi _id_i to within 4×10−154× 10^-15, and the markers of one graph coincide across K, as the independence of the potential share from K requires. Panel (b) shows the generalised eigenvalues of (Mpot,Mtot)(M_pot,M_tot) for N=5N=5 and K≤12K≤ 12: they tend to the values (23) at rate O(K−2)O(K^-2), as Proposition 7 predicts, and (N−1)/N(N-1)/N is an eigenvalue for every K (Lemma 2), the largest one in this example. TABLE I: Four layouts for eight agents, K=3K=3. Energies in units of 2cK2c_K; σβ/β2 _β/β^2 measured at βmaxidi=0.02β _id_i=0.02. Layout |E||E| |E2||E_2| ℰpotE_pot ρpot _pot σβ/β2 _β/β^2 Fully channelised (matching) 44 00 44 1/21/2 0.07410.0741 Single merge cycle C8C_8 88 88 1616 2/32/3 0.14810.1481 One arbiter (star) 77 2121 2828 4/54/5 0.12960.1296 Unprotected crossing (Example 1) 1616 4848 6464 4/54/5 0.29630.2963 VII-B Log-linear Learning Figure 4 tests the three parts of Proposition 6, one per panel, and delimits their range of validity. Panel (a) shows, for the rank-order game on C5C_5 (K=3K=3), that the stationary law is Gibbs in the Hodge potential to second order. It also shows that under non-uniform revision rates (αi∝i+1 _i i+1), this is true of the weighted potential ϕα _α and false of ϕ⋆φ : the revision rates enter at first order. Panel (b) compares, on all edges of the game graph of the intersection (β=0.02β=0.02), the stationary current with βh/(NKN+1)β h/(NK^N+1), where h is evaluated from Proposition 3; the harmonic flow takes finitely many values, hence the clusters. Panel (c) is the main check. It plots σβ/β2 _β/β^2 against ℰharm/(NKN+1)E_harm/(NK^N+1), with no fitted parameter, for four families of different origin: rank-order games on all 112112 non-isomorphic connected graphs on six vertices; the same games with randomised tie-breaking, whose harmonic energy is smaller by the factor (K−2)/(K+1)(K-2)/(K+1) (Section I-C); the interpolation uϑ=δϕ⋆+ϑhu_ =δφ + h on C5C_5, ϑ∈[0.15,1.6] ∈[0.15,1.6]; and thirty games on four agents with i.i.d. standard normal payoffs. The last family is not pairwise, yet obeys the same law, as Proposition 6 asserts for arbitrary payoffs. For each game βmaxe|u(e)|=0.02β _e|u(e)|=0.02; the largest relative deviation from the line over all 284284 points is 1.2×10−31.2× 10^-3. Panel (d) shows the two regimes of Remark 2 on the intersection: the ratio of σβ _β to its prediction is 0.990.99 at β=0.1β=0.1, 0.880.88 at β=0.35β=0.35, 0.420.42 at β=1β=1, and 0.030.03 at β=8β=8, while the stationary mass of the three pure-strategy Nash equilibria of Proposition 5 rises from 10−310^-3 and remains near 0.650.65 up to β=8β=8. VII-C Reading a Layout This experiment shows how the closed forms can be used for design, before any simulation. Table I compares four ways of routing eight agents: separate channels, so that the conflicts form a perfect matching; a single merge cycle; one agent that conflicts with all others; and the unprotected crossing of Example 1. By (16) and Proposition 6, the leading-order entropy production counts the conflicts, and the measured σβ/β2 _β/β^2 agrees with the prediction (K2−1)|E|/(6NK2)(K^2-1)|E|/(6NK^2) to five digits. This has two consequences. First, each removed conflict lowers the leading-order entropy production by the same amount, whatever happens to the ratio: the fully channelised layout has the smallest potential share, 1/21/2, and is the closest to a reversible system. Second, the ratio does not rank layouts: the star and the unprotected crossing share ρpot=4/5 _pot=4/5, yet the crossing has 16/716/7 times the entropy production. A hub concentrates conflicts on one agent, which raises |E2||E_2| and hence ℰpotE_pot at almost no harmonic cost; whether that is desirable is for the designer to decide. VII-D Beyond the Complete Graph This experiment probes the hypotheses of Theorem 2. The theorem is stated for KNK_N, which has the protocol-reversal symmetry of Proposition 7(c). Off KNK_N the affine payoff can be slightly beaten at finite K. Under the index protocol on K3,3K_3,3 the best payoff, found by maximising the Rayleigh quotient (19) over F, reaches ρpot=0.7635 _pot=0.7635 for K=3K=3 and 0.75040.7504 for K=7K=7, against 3/43/4 for the linear payoff; the prism and the octahedron (linear values 3/43/4 and 4/54/5) are likewise beaten; on the cycles we tested the linear payoff was optimal. The excess decreases with K; we do not claim a single cause. Randomised tie-breaking on KNK_N behaves likewise (N=5N=5, K=3K=3: 0.9550.955 at the optimum against 0.9410.941 for the linear payoff). VIII Conclusion This paper has measured what a potential-game surrogate misses in priority coordination. For the linear payoff the missed energy is the number of conflicts, on every conflict graph and for every deterministic protocol, so the best common-objective model of the agents’ choice odds errs by at least 1/(dmax+1)1/(d_ +1) for every rationality β>0β>0, a floor set by the conflicts per agent rather than by the size of the system. The missed part is invisible to strict better-response paths and to the equilibrium service order, but it is, to leading order, the probability current of low-rationality learning, and on the complete conflict graph no rank-based payoff brings it below 1/N1/N. Since the counts are available from the layout before any simulation, layouts should be compared by the pair of energies rather than by their ratio: the star and the unprotected crossing of Table I have the same potential share, yet the crossing has 16/716/7 times the entropy production. At finite K the extremality of affine payoffs can fail off the complete graph and under randomised tie-breaking (Section VII-D); whether it holds on every regular graph for continuous priorities, for which N it fails at K=2K=2, and whether Theorem 2 extends to orientations with directed cycles remain open. The analysis has taken the ordering abstraction and an ideal execution layer as given; coupling the decision layer with the continuous dynamics of the agents is the subject of ongoing work. Appendix A Proof of Theorem 1 Write n=N−1n=N-1, m(q)=[q(1−q)]m w_m(q)=[q(1-q)]^m, and pj(m)(q)=Pj(m,m)(1−2q)p^(m)_j(q)=P^(m,m)_j(1-2q), which are orthogonal on [0,1][0,1] with weight m w_m. The required integral identities are collected in Appendix S-C of the SM. Step 1: the two energies. By the continuum analogue of Lemma 1, ‖Pu‖2=∑m‖gm‖2/m\|Pu\|^2= _m\|g_m\|^2/m and ‖u‖2=∑i‖ui−Aiui‖2\|u\|^2= _i\|u_i-A_iu_i\|^2. Conditional on s−is_-i, Aiui=∫01f(Bt)tA_iu_i= _0^1f(B_t)\,dt, where Bt=#j≠i:sj>tB_t=\#\j≠ i:s_j>t\ is a sum of independent Bernoulli(1−t)(1-t) variables. Its Hoeffding components are products of centred indicators with covariance kernel κ. Lemmas S40 and S41 of the SM, together with the conservation law, therefore give ‖gm‖2=(N−m)2Jm(f)\|g_m\|^2=(N-m)^2J_m(f) and the corresponding total form, yielding (21). Step 2: from discrete Chebyshev to Jacobi polynomials. Let f(q)=f(Bin(n,q)) b_f(q)=Ef(Bin(n,q)) be the Bernstein polynomial of f. Since [Bin(n,q)r¯]=nr¯qrE[Bin(n,q) r]=n rq^r, tp(q)=Pp(1−2q) b_t_p(q)=P_p(1-2q), where PpP_p is the Legendre polynomial of degree p. The Bernstein derivative identity gives dm[f]=f(m)/nm¯d_m[f]= b_f^(m)/n m, and the m-th derivative of PpP_p is a multiple of Pp−m(m,m)P^(m,m)_p-m; hence dm[tp]=cp,mp−m(m),cp,m=(−1)m(p+m)!p!nm¯,d_m[t_p]=c_p,mp^(m)_p-m, c_p,m= (-1)^m(p+m)!p!\,n m, (28) for m≤pm≤ p, and dm[tp]=0d_m[t_p]=0 for m>pm>p. Step 3: the kernel κmκ^m. Let (mg)(q)=∫01κ(q,q′)mg(q′)dq′( K_mg)(q)= _0^1κ(q,q )^mg(q )\,dq . Splitting at q′=q =q shows that m K_m maps polynomials of degree at most j to m w_m times polynomials of degree at most j. Thus mpj(m)=mqj K_mp^(m)_j= w_mq_j with degqj≤j q_j≤ j. By self-adjointness, qjq_j is m w_m-orthogonal to all polynomials of degree <j<j, so qjq_j is proportional to pj(m)p^(m)_j. The proportionality constant, obtained from the Rodrigues formula at q→0q→ 0 (Lemma S42 of the SM), is mpj(m)=m(j+m)(j+m+1)mpj(m). K_mp^(m)_j= m(j+m)(j+m+1)\, w_mp^(m)_j. (29) Step 4: part (a). For m≤min(p,p′)m≤ (p,p ), (28)–(29) give Jm(tp,tp′) J_m(t_p,t_p ) = = (Nm)cp,mcp′,mmp′(p′+1)∫01mp−m(m)p′−m(m)q. Nmc_p,mc_p ,m mp (p +1) _0^1 w_m\,p^(m)_p-mp^(m)_p -m\,dq. This vanishes for p≠p′p≠ p by Jacobi orthogonality. For p=p′p=p , ∫01m(p−m(m))2q=(p!)2/[(2p+1)(p+m)!(p−m)!] _0^1 w_m(p^(m)_p-m)^2dq=(p!)^2/[(2p+1)(p+m)!(p-m)!], giving (22); for m>pm>p, it vanishes by Step 2. Step 5: part (b). Substituting (22) into (21), the potential form of tpt_p is a positive common factor times ∑m=1p(N−m)!(p+m)!/[m!(p−m)!] _m=1^p(N-m)!(p+m)!/[m!(p-m)!]. The Chu–Vandermonde identity gives ∑m=0p(N−m)!(p+m)!m!(p−m)!=(N+p+1)!(N−p)!(N+1)!, _m=0^p (N-m)!(p+m)!m!(p-m)!= (N+p+1)!(N-p)!(N+1)!, so subtracting the m=0m=0 term and dividing by the total form of tpt_p, which is N+1∑ktp(k)2 NN+1 _kt_p(k)^2 by Step 1, with ∑k=0N−1tp(k)2=(N+p)!(N−1−p)!/[(2p+1)(N−1)!2] _k=0^N-1t_p(k)^2=(N+p)!(N-1-p)!/[(2p+1)(N-1)!^2] [43], yields (23). Direct substitution gives λ1=(N−1)/N _1=(N-1)/N. Finally, write Nλp=Dp(1−εp)N _p=D_p(1- _p), where Dp=N(N+1)/[p(p+1)]−1D_p=N(N+1)/[p(p+1)]-1 and εp=(Np)/(N+p+1p) _p= Np/ N+p+1p. The inequality λp+1<λp _p+1< _p is equivalent to 1−εpεp>p(p+1)(N−p−1)N(N+1). 1- _p _p> p(p+1)(N-p-1)N(N+1). Since εp−1=∏i=0p−1(1+p+1N−i)≥1+p(p+1)N _p^-1= _i=0^p-1(1+ p+1N-i)≥ 1+ p(p+1)N, the left-hand side is at least p(p+1)/Np(p+1)/N, which is strictly larger than the right-hand side. This proves the claim. □ Appendix B Proof of Theorem 2 This appendix contains the proof arguments for Theorem 2, following the four stages of Section VI. Lengthy algebraic expansions, the exact symbolic computations for K=3,4K=3,4 and the rational certificates are in Appendix S-A of the SM, whose steps are cited below. Throughout, C=KNC=K_N, the protocol is the index protocol, and ui=f(posi)u_i=f(pos_i). B-A The linear payoff is a null vector: proof of Lemma 2 The identity QK=0Q_K1=0 is equivalent to ⟨Pulin−N−1Nulin,uf⟩=0 Pu lin- N-1Nu lin,u^f =0 for every f, where ulinu lin and ufu^f are the flows of the linear game and of f(KN,K)G_f(K_N,K). By Proposition 2, with di=N−1d_i=N-1, PulinPu lin is the flow generated by (N−1)(si−s¯)/K(N-1)(s_i- s)/K, while ulin−Pulinu lin-Pu lin is the flow of the interaction terms in (6). Writing the resulting flow inner products as conditional covariances and using (7) gives the equivalent identity ∑i[posi][f(posi)]−∑r=0N−1rf(r) _iE[pos_i]\,E[f(pos_i)]- _r=0^N-1rf(r) = = N+1K∑iCov(si,f(posi)). N+1K _iCov (s_i,f(pos_i) ). (30) Both sides are linear in f, so it suffices to verify (30) for f(r)=xrf(r)=x^r for N distinct values of x. Conditional on sis_i, the events that the other agents precede i are independent, with probabilities (K−si)/K(K-s_i)/K or (K−1−si)/K(K-1-s_i)/K according to the protocol orientation. The resulting generating functions are products aiaN−1−i A_a^\,i B_a^\,N-1-i with a=a+1 A_a= B_a+1; the telescoping calculation in Step S-A.2 of the SM proves (30). Hence QK=0Q_K1=0. Finally, Proposition 2 on KNK_N, where di=N−1d_i=N-1, gives the Rayleigh quotient ρpot=(N−1)/N _pot=(N-1)/N. □ B-B Stage 1: splitting by interaction order By Lemma 1 and (7), gT=∑i∈T[ui]Tg_T= _i∈ T[u_i]_T, and QK=KN+1∑m≥1QK(m)Q_K=K^N+1 _m≥ 1Q^(m)_K with QK(m)=∑|T|=m[N−1N∑i∈T‖[ui]T‖2−1m‖∑i∈T[ui]T‖2].Q^(m)_K= _|T|=m [ N-1N _i∈ T\|[u_i]_T\|^2- 1m \| _i∈ T[u_i]_T \|^2 ]. (31) Proposition 8 (Higher orders). For N≥3N≥ 3, K≥2K≥ 2 and every T with |T|=m|T|=m, ‖∑i∈T[ui]T‖2≤2∑i∈T‖[ui]T‖2\| _i∈ T[u_i]_T\|^2≤ 2 _i∈ T\|[u_i]_T\|^2; hence QK(m)⪰0Q^(m)_K 0 for all m≥3m≥ 3. Proof. Write Xi=[ui]T=(I−Ai)[Ψ(i)D(i)(si)]X_i=[u_i]_T=(I-A_i)[ ^(i)D^(i)(s_i)]. By Lemma S6 of the SM, for i≠ki≠ k the bilinear kernel of ⟨Xi,Xk⟩ X_i,X_k in (D(i),D(k))(D^(i),D^(k)) is entrywise nonpositive, while the kernel subtracted in ‖Xi‖2\|X_i\|^2 is entrywise nonnegative. Let X~i X_i be obtained by replacing D(i)D^(i) with |D(i)||D^(i)|. These sign properties imply ⟨Xi,Xk⟩≤−⟨X~i,X~k⟩ X_i,X_k ≤- X_i, X_k for i≠ki≠ k and ‖X~i‖≤‖Xi‖\| X_i\|≤\|X_i\|. Hence ∑i≠k⟨Xi,Xk⟩ _i≠ k X_i,X_k ≤−∑i≠k⟨X~i,X~k⟩=∑i∥X~i∥2−∥∑iX~i∥2 ≤- _i≠ k X_i, X_k = _i\| X_i\|^2- \| _i X_i \|^2 ≤∑i‖Xi‖2. ≤ _i\|X_i\|^2. Therefore ‖∑i∈TXi‖2≤2∑i∈T‖Xi‖2\| _i∈ TX_i\|^2≤ 2 _i∈ T\|X_i\|^2. Consequently, QK(m)≥(N−1N−2m)∑|T|=m∑i∈T‖[ui]T‖2≥0,Q_K^(m)≥ ( N-1N- 2m ) _|T|=m _i∈ T\|[u_i]_T\|^2≥ 0, since N,m≥3N,m≥ 3. Thus QK(m)⪰0Q_K^(m) 0 for every m≥3m≥ 3. ∎ The argument uses no property of f. It remains to show QK(1)+QK(2)⪰0Q^(1)_K+Q^(2)_K 0, where QK(1)⪯0Q^(1)_K 0. B-C Stage 2: reduction to bond variables For k=0,…,N−2k=0,…,N-2 and a level a, define the bond vector D^k(a)=[Δf(Bin(k,K−aK)+Bin(N−2−k,K−1−aK))] D_k(a)=E [ f (Bin(k, K-aK)+Bin(N-2-k, K-1-aK) ) ], and let D^(f)=(D^k)k=0N−2 D(f)=( D_k)_k=0^N-2. Let μk=N−1−kN _k= N-1-kN and νk=k+1N _k= k+1N. For a general array D=(Dk)k=0N−2D=(D_k)_k=0^N-2 with Dk∈ℝKD_k ^K, write Gk,HkG_k,H_k for the inclusive and exclusive cumulative sums of DkD_k, as in Section VI. Proposition 9 (Two-body reduction). For every payoff f, QK(1)(f)+QK(2)(f)=N,K(D^(f)),Q^(1)_K(f)+Q^(2)_K(f)= B_N,K( D(f)), where N,K B_N,K is the quadratic form in arrays D defined in Step S-A.4 of the SM. With Zk=N−22[μk(−2Var)(Hk)+νk(−2Var)(Gk)],Z_k= N-22[ _k(D-2Var)(H_k)+ _k(D-2Var)(G_k)], it satisfies K2N,K(D)=∑kZk+Sq(D)−∑k(2Nμkνk−12)Var(Dk),K^2 B_N,K(D)\!=\! _kZ_k+Sq(D)- _k(2N _k _k- 12)Var(D_k), (32) where Sq(D)≥0Sq(D)≥ 0 is the sum of variances Sq(D)=12N∑k<l[ (D)= 12N _k<l [ (N−1−l)Var(Hk−Hl+Dl) (N-1-l)Var(H_k-H_l+D_l) (33) +(k+1)Var(Gk−Gl+Dk) +(k+1)Var(G_k-G_l+D_k) +(l−k)Var(Hk−Gl)]. +(l-k)Var(H_k-G_l) ]. Moreover, Sq(D)=0Sq(D)=0 iff D∈span(⊗)D (1 1), and D^(f)∈span(⊗) D(f) (1 1) iff f is affine. Proof. Given si=as_i=a, the events that opponents precede i are independent, with probability (K−a)/K(K-a)/K for opponents favoured by the protocol and (K−1−a)/K(K-1-a)/K otherwise. First order. Lowering a by one increases each such probability by 1/K1/K. Since f(R+Bern(p+1/K))−f(R+Bern(p))=K−1Δf(R)Ef(R+Bern(p+1/K))-Ef(R+Bern(p))=K^-1E f(R), one obtains, up to an additive constant, φj(a):=[uj∣sj=a]=−K−1Ωj _j(a):=E[u_j s_j=a]=-K^-1 _j, where Ωj=∑k<jHk+∑k≥jGk _j= _k<jH_k+ _k≥ jG_k. Hence QK(1)=−1NK2∑jVar(Ωj)Q_K^(1)=- 1NK^2 _jVar( _j). Second order. Conditioning also on sj=bs_j=b isolates the indicator of whether j precedes i: [ui∣a,b]=χ(a)+[j≺i]D^k(a)E[u_i a,b]=χ(a)+1[j i]\, D_k(a), with k∈i−1,ik∈\i-1,i\. Thus [ui]i,j[u_i]_\i,j\ is a centred Bernoulli indicator times the corresponding bond vector. Substitution into (31) gives the diagonal Bernoulli-covariance terms and, by complementarity of the two orientations, cross terms −K−2Cov(Hk,Gl)-K^-2Cov(H_k,G_l). Summation by parts and n∑kVar(zk)−Var(∑kzk)=∑k<lVar(zk−zl)n _kVar(z_k)-Var( _kz_k)= _k<lVar(z_k-z_l) then yield (32)–(33); the detailed rearrangement is given in Steps S-A.4–S-A.6 of the SM. Since every coefficient in (33) is nonnegative, Sq=0Sq=0 iff all displayed variance terms vanish. Step S-A.6 of the SM shows that this happens iff D∈span(⊗)D (1 1). Finally, by the definition of D^(f) D(f), this occurs iff all second differences of f vanish, equivalently iff f is affine. ∎ B-D Stage 3: the level inequality By Lemma 3, Zk≥N−22(K−2)μkνkVar(Dk)Z_k≥ N-22(K-2) _k _kVar(D_k), so the only negative term of (32) is compensated bond by bond. The proof of Lemma 3 works in the step coordinates cm=D(m)−D(m−1)c_m=D(m)-D(m-1). In these coordinates the difference of the two sides of (25) is the Hadamard product of the Green’s function min(m,m′)(K−max(m,m′))/K (m,m )(K- (m,m ))/K of the path with a one-pair matrix minus a rank-one term. The inverse of the one-pair matrix is a path Laplacian with Robin ends, whose row sums show that the second factor is positive semidefinite (and singular), and the Schur product theorem concludes (Lemma S16 of the SM). The constant K−2K-2 cannot be improved, and for K=2K=2 the inequality gives no margin. B-E Stage 4: the Perron argument and three branches Let ϱ(μ):=μν+12c⋆(μ) (μ):=μν+ 12c (μ), where c⋆(μ)c (μ) is the best constant in (25) for the given μ. Retaining part of SqSq, positivity of N,K B_N,K reduces to NL−2diag(1/θk)⪰0NL-2diag(1/ _k) 0, where L is the Dirichlet Laplacian of the path of bonds and θk=(N−2)ϱ(μk)−Nμkνk+12 _k=(N-2) ( _k)-N _k _k+ 12. The vector vk=(k+1)(N−1−k)v_k=(k+1)(N-1-k) satisfies Lv=2⋅Lv=2·1 exactly, so by the M-matrix criterion it suffices that, for every bond, (N−2)ϱ(μk)−2Nμkνk+12≥ 0.(N-2)\, ( _k)-2N _k _k+ 12\ ≥\ 0. (34) Lemma 3 gives ϱ(μ)≥Kμν/2 (μ)≥ Kμν/2, and with μν≤14μν≤ 14 condition (34) holds whenever (N−2)(K−4)≥4(N-2)(K-4)≥ 4. This covers K≥5K≥ 5 up to finitely many N. At K=4K=4 the bound from Lemma 3 never suffices, and one uses the true ϱ(μ) (μ), the smallest root of an explicit cubic; root isolation by Sturm sequences certifies ϱ≥2μν+5(μν)3 ≥ 2μν+5(μν)^3 and hence (34) for N≥10N≥ 10. At K=3K=3 the level space modulo constants is two-dimensional; an explicit congruence removes the same-bond coupling and leaves a sum of squares plus a Z-matrix with a positive Perron vector for N≥17N≥ 17, the required polynomial identities in N being established by exact computer algebra. The finitely many remaining pairs (K,N)(K,N) are settled by LDL⊤LDL factorisations in exact rational arithmetic (Table I). Steps S-A.8–S-A.13 of the SM give the details. TABLE I: Coverage of Theorem 2 for K≥3K≥ 3, N≥3N≥ 3: by the large-N branch of Appendix B-E or by an exact rational certificate. K all large N bond-space cert. image cert. 33 N≥17N≥ 17 – 4≤N≤16∗4≤ N≤ 16 44 N≥10N≥ 10 5≤N≤95≤ N≤ 9 N=3,4N=3,4 55 N≥6N≥ 6 N=4,5N=4,5 N=3N=3 6,76,7 N≥4N≥ 4 N=3N=3 – ≥8≥ 8 N≥3N≥ 3 – – ∗At (K,N)=(3,3)(K,N)=(3,3) the two-body form vanishes identically on the image, and Q3=83(1−1−11)Q_3= 83 ( smallmatrix1&-1\\ -1&1 smallmatrix ) is computed exactly from the definition. 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Thomas, “The network HHD: Quantifying cyclic competition in trait-performance models of tournaments,” SIAM Review, vol. 64, no. 2, p. 360–391, 2022. [45] R. P. Stanley, Enumerative Combinatorics, Volume 1, 2nd ed. Cambridge University Press, 2012. [46] F. R. Gantmacher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems. Providence, RI: AMS Chelsea Publishing, 2002, revised ed. Zhihao Lin received the Ph.D. degree in autonomous systems and connectivity from the University of Glasgow, Glasgow, U.K., in 2026. He is currently a JSPS Postdoctoral Fellow at the Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan. His research interests include multi-agent coordination, game theory and learning in games, operator-theoretic methods for data-driven control, and safe autonomous driving. Jianglin Lan received the Ph.D. degree in Engineering from the University of Hull in 2017. He is currently a Lecturer at the University of Glasgow and an Honorary Research Fellow with Imperial College London. From 2022 to 2025, he held a Leverhulme Early Career Fellowship at University of Glasgow. He is an Associate Editor for IEEE Transactions on Intelligent Transportation Systems, an Editor for International Journal of Adaptive Control and Signal Processing, and a member of the IFAC SAFEPROCESS Technical Committee. His research interests include safe AI, and control theory. Anh-Tu Nguyen (Senior Member, IEEE) received the Ph.D. degree in automatic control from the Université Polytechnique Hauts-de-France, Valenciennes, France, in 2013. He is currently an Associate Professor with INSA Hauts-de-France, Université Polytechnique Hauts-de-France, Valenciennes, France. His research interests include robust control and estimation, cybernetic control systems, and mechatronic applications. Dr. Nguyen serves as an Associate Editor for several international journals, including the IEEE/ASME Transactions on Mechatronics, the IEEE Transactions on Automation Science and Engineering, the IEEE Transactions on Intelligent Transportation Systems, Control Engineering Practice, Engineering Applications of Artificial Intelligence, and ISA Transactions. Yoshinobu Kawahara received the B.S., M.S., and Ph.D. degrees in engineering from the University of Tokyo, Tokyo, Japan, in 2003, 2005, and 2008, respectively. He is currently a Professor with the Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan, and the Team Director of the Dynamical Systems Learning Team at the RIKEN Center for Advanced Intelligence Project, Japan. He is also a Project Manager of the JST Moonshot R&D Program (Goal 3). He was previously a Professor with the Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan, and has held visiting positions at the Max Planck Institute for Intelligent Systems, Tübingen, Germany, and the University of Washington, Seattle, WA, USA. His research interests lie at the intersection of machine learning and dynamical systems, including operator-theoretic methods and their applications to scientific and engineering problems. Dr. Kawahara received the Young Scientists’ Prize, the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology of Japan, in 2020, and Best Paper Awards from the Japanese Society for Artificial Intelligence (2011 and 2020) and the Japanese Neural Network Society (2020). He currently serves as an Action Editor for Transactions on Machine Learning Research and Neural Networks, and has served as an Area Chair or Senior Area Chair for conferences including NeurIPS, ICML, and ICLR. Supplementary Material for Priority Coordination Games: Hodge Decomposition and a Sharp Design Limit This document gives the details behind the proofs in the appendices of the main text, the proofs of the further results stated there, and documents the computer-assisted steps. References of the form “Theorem 2” or “(19)” point to the main text; auxiliary results proved here are numbered S1, S2, …, and equations (S1), (S2), …. Appendix S-A comes first because it proves the main technical result, Theorem 2; Appendix S-C gives a self-contained version of the proof of Theorem 1 in Appendix A of the main text, and uses one lemma of Appendix S-A. • Appendix S-A proves Theorem 2 (Finite quantisation), in fourteen steps following the four-stage outline of Section VI; its last subsection (Section S-A.15) proves Proposition 7 (reciprocity and expansion in K−2K^-2), independent of the rest. • Appendix S-B proves the two statements about the harmonic component of the linear game made after Proposition 3 of the main text: the tie-pattern law (Theorem S36) and the effect of randomised tie-breaking (Proposition S38). • Appendix S-C proves Theorem 1 (Hodge spectrum) and the identities (21). • Appendix S-D proves the statement about priced priorities made after Corollary 2 of the main text (Proposition S44) and records the corresponding exact computation on the intersection. • Appendix S-E lists the scripts behind every computer-assisted step, describes an end-to-end check that certifies Theorem 2 for small (K,N)(K,N) without using any step of its proof, and says which computations are not part of any proof. Proofs given in full in the main text (Lemma 1, Proposition 1 and its corollary, Propositions 2, 3, 4, 5 and 6) are not repeated. Fig. S1: Logical structure of the supplementary material. Boxes are the fourteen steps of the proof of Theorem 2 (Appendix S-A); arrows point from a step to the steps that use it, and the horizontal collector above Step S-A.14 gathers everything the assembly needs: the higher orders (Step S-A.3), the kernel of SqSq (Step S-A.6), the three branches (Steps S-A.10–S-A.12) and the certificates (Step S-A.13), which together cover every pair (K,N)(K,N) with K≥3K≥ 3; Step S-A.2 feeds both the kernel statement and the image certificates. Dashed boxes are the three computer-assisted steps; all computations are exact. Section S-A.15 and Appendices S-B–S-D are independent of the proof and of one another, except that Appendix S-C uses the conservation law of Step S-A.1. What depends on what. Figure S1 shows the logical structure of this document. The four appendices are independent of one another and depend on the main text only through Lemma 1 and the definitions of Section I, except that Appendix S-C uses Lemma S2 of Step S-A.1; Section S-A.15 is not used in the proof of Theorem 2. Conventions. Notation is that of the main text. Throughout, [N]=0,…,N−1[N]=\0,…,N-1\; functions on the profile space S carry the uniform probability measure and ‖g‖2=[g2]\|g\|^2=E[g^2]; flows carry the counting inner product ⟨x,y⟩=∑ex(e)y(e) x,y = _ex(e)y(e); AiA_i is the average over the i-th coordinate, gTg_T the Hoeffding component of g on T⊆[N]T [N], and [ui]T[u_i]_T that of the payoff uiu_i. The bond weights of Appendix S-A are μk=N−1−kN _k= N-1-kN and νk=k+1N _k= k+1N, as in the main text; β is reserved for the rationality parameter and appears only in Appendix S-D. In Appendix S-A “Step k” refers to one of the fourteen steps of the proof of Theorem 2, which are numbered in reading order; parts of a single proof are labelled (1), (2), … and are local to that proof. Three kinds of statement occur and are marked as such: results proved by hand; polynomial identities in N established by computer algebra, whose scripts are named at the point of use; and finitely many exact rational certificates. Nothing in any proof depends on a floating-point computation. Appendix S-A Proof of Theorem 2 (finite quantisation) Throughout this appendix the conflict graph is complete, C=KNC=K_N, the protocol is the index protocol, and ui=f(posi)u_i=f(pos_i) for an arbitrary payoff-from-rank function f with increments Fj=f(j−1)−f(j)F_j=f(j-1)-f(j); the factor KNK^N between the expectation norm of functions and the counting norm of flows appears explicitly wherever one is converted into the other. The proof is organised in fourteen steps, numbered in the order in which they are read; Appendix B of the main text follows the four stages of Section VI and cites the steps each consists of. Roadmap. Table S1 lists the steps, what each proves, where its result is used, and how it is verified. A reader interested only in the logical structure can read Steps S-A.1–S-A.6 and S-A.14 and take Table S2 on trust; a reader auditing the computer-assisted parts needs Steps S-A.11–S-A.13 and Appendix S-E. TABLE S1: Roadmap of the proof of Theorem 2. “exact” means rational or symbolic arithmetic. Step proves used in verified by S-A.1 Laplacian divδ=Kdivδ=K N, conservation, order decomposition QK=KN+1∑mQK(m)Q_K=K^N+1 _mQ^(m)_K (Lemmas S1–S3) every later step by hand S-A.2 QK=0Q_K1=0: the linear payoff is an exact null vector (Lemma 2) kernel statement, Step S-A.14 by hand S-A.3 ‖∑i∈T[ui]T‖2≤2∑i‖[ui]T‖2\| _i∈ T[u_i]_T\|^2≤ 2 _i\|[u_i]_T\|^2, hence QK(m)⪰0Q^(m)_K 0 for m≥3m≥ 3 (Proposition S5, Lemma S6) reduces the theorem to Q(1)+Q(2)⪰0Q^(1)+Q^(2) 0 by hand S-A.4 bond vectors D^k(f) D_k(f); first- and second-order components in bond form (Propositions S8, S9); Q(1)+Q(2)=N,K(D^(f))Q^(1)+Q^(2)= B_N,K( D(f)) (Proposition S10); injectivity on affine f (Lemma S11) Steps S-A.5–S-A.13 work on B alone by hand S-A.5 Hahn–Poincaré and spread identities; the sum-of-squares identity K2=∑kZk+Sq−NegK^2 B= _kZ_k+Sq-Neg (Theorem S14) Steps S-A.6–S-A.12 by hand S-A.6 Sq(D)=0Sq(D)=0 iff D∈span(⊗)D (1 1) (Lemma S15) kernel statement, Step S-A.14 by hand S-A.7 sharp level inequality with constant K−2K-2 (Lemma S16); the level pencil ϱ(μ) (μ) (Proposition S17) Steps S-A.8–S-A.11 by hand (Schur product) S-A.8 lower bound Sq≥SqlowSq _ low in the bond vectors alone; step coordinates (Propositions S19, S20) Step S-A.9 by hand S-A.9 Perron argument: ⪰0 B 0 if condition (T) holds on every bond (Lemmas S21, S22, Proposition S23) Steps S-A.10–S-A.11 by hand (M-matrix criterion) S-A.10 K≥5K≥ 5: (T) from Lemma S16 whenever (N−2)(K−4)≥4(N-2)(K-4)≥ 4 (Theorem S24) assembly by hand S-A.11 K=4K=4: root isolation of the cubic for ϱ(μ) (μ), ϱ≥2μν+5(μν)3 ≥ 2μν+5(μν)^3, hence (T) for N≥10N≥ 10 (Lemma S25, Theorem S26) assembly Sturm sequences; audit_K4.py (exact) S-A.12 K=3K=3: block representation, shear, C−γI⪰0C-γ I 0, Perron certificate for N≥17N≥ 17 (Lemmas S28–S31, Theorem S32) assembly computer algebra, symbolic in N: k3_representation_proof.py, s1_closed.py, verify_M_symbolic.py (exact) S-A.13 the remaining pairs (K,N)(K,N) by exact LDL⊤LDL certificates on the bond space or on its image (Table S2) assembly 2525 stored certificates; verify_certificates.py (exact) S-A.14 assembly; N=2N=2; the counterexample at K=2K=2, N=7N=7 Theorem 2 by hand; k2_witness.py (exact) S-A.1 Step 0: Hoeffding decomposition, the number operator and conservation This step fixes conventions and records, in the form used below, the facts behind Lemma 1 of the main text. Throughout, S=0,…,K−1NS=\0,…,K-1\^N carries the uniform probability measure and ‖g‖2=[g2]\|g\|^2=E[g^2] for functions on S; flows carry the counting inner product ⟨x,y⟩=∑ex(e)y(e) x,y = _ex(e)y(e). For a function g on S and a coordinate i, AiA_i denotes averaging over sis_i. The Hoeffding (ANOVA) decomposition is g=∑T⊆[N]gTg= _T [N]g_T with gT=∏j∈T(I−Aj)∏j∉TAjg_T= _j∈ T(I-A_j) _j∉ TA_j\,g; gTg_T depends only on sTs_T, has zero mean in each of its variables, and the gTg_T are mutually orthogonal. We write gm=∑|T|=mgTg_m= _|T|=mg_T. The number operator =∑i(I−Ai) N= _i(I-A_i) acts on gTg_T as multiplication by |T||T|. On the complete graph we abbreviate posi=posKN(i,s)pos_i=pos_K_N(i,s). Lemma S1 (Laplacian of the game graph). Let δ be the gradient and let δ∗δ^* be its adjoint. With the divergence convention (divx)(s)=∑eintosx(e)−∑eoutofsx(e)(divx)(s)= _e\,into\,sx(e)- _e\,out\ of\,sx(e) one has ⟨δϕ,x⟩=⟨ϕ,divx⟩ δφ,x = φ,divx , i.e. δ∗=divδ^*=div. Then δ∗δ=Kδ^*δ=K N; in particular δ∗δgT=K|T|gTδ^*δ g_T=K|T|\,g_T. For the game flow u of payoffs (ui)(u_i), (divu)(s)=K∑i[ui(s)−u¯i(s−i)],u¯i:=Aiui,(divu)(s)=K _i [u_i(s)- u_i(s_-i) ], u_i:=A_iu_i, (S1) and the potential energy is ‖δϕ⋆‖2=KN∑m=1N‖(divu)m‖2mK=KN+1∑m=1N‖gm‖2m,g:=∑i(ui−Aiui),\|δφ \|^2=K^N _m=1^N \|(divu)_m\|^2mK=K^N+1 _m=1^N \|g_m\|^2m, g:= _i(u_i-A_iu_i), (S2) which is the first identity of (12) in the main text. Proof. This is Lemma 1 of the main text, whose proof identifies divδ=Kdivδ=K N and divu=Kgdivu=Kg; the sum over the KNK^N profiles in ⟨ϕ⋆,divδϕ⋆⟩ φ ,divδφ supplies the factor KNK^N in (S2), and diagonalising N on the Hoeffding components gives the two forms. ∎ Lemma S2 (conservation). For every conflict graph and every payoff-from-rank function f, ∑iui(s) _iu_i(s) is independent of s on the complete graph, and ∑iposC(i,s)=|E| _ipos_C(i,s)=|E| on every graph. Consequently, on the complete graph, ∑i[ui]T=0 _i[u_i]_T=0 for every nonempty T, hence gT=∑i∈T[ui]T=−∑i∉T[u¯i]Tg_T= _i∈ T[u_i]_T=- _i∉ T[ u_i]_T. Proof. On KNK_N the map i↦posii _i is a permutation of 0,…,N−1\0,…,N-1\ (the order by (si,−i)(s_i,-i) is strict), so ∑if(posi)=∑rf(r) _if(pos_i)= _rf(r). On a general graph each conflict edge contributes exactly one unit to the position of whichever endpoint is served second. A constant has no nonconstant Hoeffding components, which gives ∑i[ui]T=0 _i[u_i]_T=0; finally [ui]T[u_i]_T with i∉Ti∉ T equals [u¯i]T[ u_i]_T because [ui]T[u_i]_T contains the factor AiA_i. ∎ Lemma S3 (order decomposition). For every f, ‖u‖2=KN+1∑i∑T∋i‖[ui]T‖2,‖Pu‖2=KN+1∑T≠∅1|T|‖∑i∈T[ui]T‖2.\|u\|^2=K^N+1 _i _T i\|[u_i]_T\|^2, \|Pu\|^2=K^N+1 _T≠ 1|T| \| _i∈ T[u_i]_T \|^2. (S3) Consequently F⊤QKF=KN+1∑m≥1QK(m)(f)F Q_KF=K^N+1 _m≥ 1Q^(m)_K(f) with QK(m)(f)=∑|T|=m[N−1N∑i∈T‖[ui]T‖2−1m‖∑i∈T[ui]T‖2],Q^(m)_K(f)= _|T|=m [ N-1N _i∈ T\|[u_i]_T\|^2- 1m \| _i∈ T[u_i]_T \|^2 ], (S4) which is the order decomposition used in stage 1) of the outline in the main text. Proof. Since I−AiI-A_i annihilates the Hoeffding components on sets not containing i and fixes the others, ui−Aiui=∑T∋i[ui]Tu_i-A_iu_i= _T i[u_i]_T, an orthogonal sum; this gives the first identity by Lemma 1. Summing over i and collecting by index set, g=∑i(ui−Aiui)g= _i(u_i-A_iu_i) has components gT=∑i∈T[ui]Tg_T= _i∈ T[u_i]_T, and the second identity is again Lemma 1. The last statement is the definition QK=N−1NMtot−MpotQ_K= N-1NM_tot-M_pot with F⊤MtotF=‖u‖2F M_totF=\|u\|^2 and F⊤MpotF=‖Pu‖2F M_potF=\|Pu\|^2 (flow norms). ∎ S-A.2 Step 1: proof of Lemma 2 (exact eigenvector) With the order decomposition in place, the first thing to establish is that the linear payoff is an exact null vector of QKQ_K at every finite K. The result is used twice: for the kernel statement of Theorem 2 (Step S-A.14), and inside the certificates of Step S-A.13, which need to know that the bond form annihilates the linear payoff. Lemma (2 of the main text). For every N≥2N≥ 2 and K≥2K≥ 2, QK=0Q_K1=0, i.e. Mpot=N−1NMtotM_pot1= N-1NM_tot1. Players are indexed 0,…,N−10,…,N-1, so player i has exactly i opponents of smaller index. Write ulinu lin for the flow of the linear game and ufu^f for that of f(KN,K)G_f(K_N,K). (1) Reduction. By definition of the Gram matrices, QK=0Q_K1=0 is equivalent to ⟨Pulin−N−1Nulin,uf⟩=0 Pu lin- N-1Nu lin,\,u^f =0 for all f. By Proposition 2 with di≡N−1d_i≡ N-1, Pulin=δϕPu lin=δφ with ϕ=N−1K∑isiφ= N-1K _is_i, which on the edges of player i is the flow of the payoff (N−1)ϕ1(si)(N-1) _1(s_i), ϕ1(a):=a−s¯K _1(a):= a- sK, s¯=K−12 s= K-12. Writing −posi=∑j≠iψij(si,sj)-pos_i= _j≠ i _ij(s_i,s_j) with ψij(a,b)=−[b>a] _ij(a,b)=-1[b>a] (j>ij>i) or −[b≥a]-1[b≥ a] (j<ij<i), the Hoeffding decomposition of each kernel is ψij=ψ¯ij+ϕ1(a)−ϕ1(b)+Γij(a,b) _ij= ψ_ij+ _1(a)- _1(b)+ _ij(a,b) (both kernels have the same first-order parts ϕ1(a) _1(a) and −ϕ1(b)- _1(b); this is checked directly: bψij(a,b)E_b _ij(a,b) equals −K−1−aK- K-1-aK or −K−aK- K-aK, both equal to a−s¯K a- sK up to a constant). Hence ulinu lin is the flow of (N−1)ϕ1(si)+Γi(N-1) _1(s_i)+ _i with Γi:=∑j≠iΓij(si,sj) _i:= _j≠ i _ij(s_i,s_j), and the harmonic part h=ulin−δϕh=u lin-δφ is the flow of Γi _i. For two payoff families (wi)(w_i), (vi)(v_i) the inner product of their flows is ⟨flow(w),flow(v)⟩=∑i∑s−i∑a<b(wi(b)−wi(a))(vi(b)−vi(a))=KN+1∑i[Cov(wi,vi∣s−i)], (w),flow(v) = _i _s_-i _a<b (w_i(b)-w_i(a) ) (v_i(b)-v_i(a) )=K^N+1 _iE [Cov(w_i,v_i s_-i) ], (S5) since ∑a<b(xb−xa)(yb−ya)=K2Cov(x,y) _a<b(x_b-x_a)(y_b-y_a)=K^2Cov(x,y) for the uniform law on 0,…,K−1\0,…,K-1\. Because Γi _i and ϕ1(si) _1(s_i) are centred in sis_i for every s−is_-i, the conditional covariances reduce to plain expectations, and the claim becomes ∑i[Γivi]=∑i[ϕ1(si)vi]for all f, _iE[ _i\,v_i]= _iE[ _1(s_i)\,v_i] all f, (S6) where vi=f(posi)v_i=f(pos_i). (2) Using conservation. Let Ξi=∑j≠i(−ϕ1(sj)) _i= _j≠ i(- _1(s_j)) be the second-variable first-order part, so that −posi=−posi+(N−1)ϕ1(si)+Ξi+Γi-pos_i=-Epos_i+ (N-1) _1(s_i)+ _i+ _i. By Lemma S2, ∑i[Ξivi]=−∑j[ϕ1(sj)∑i≠jvi]=−∑j[ϕ1(sj)(const−vj)]=∑j[ϕ1(sj)vj] _iE[ _iv_i]= - _jE [ _1(s_j) _i≠ jv_i ]=- _jE [ _1(s_j)(const-v_j) ]= _jE[ _1(s_j)v_j]. Therefore ∑i[Γivi]=∑i[(posi−posi)vi]−N∑i[ϕ1(si)vi] _iE[ _iv_i]= _iE[(Epos_i-pos_i)v_i]-N _iE[ _1(s_i)v_i], and (S6) is equivalent to ∑i[posi][f(posi)]−∑rf(r)=N+1K∑iCov(si,f(posi)), _iE[pos_i]\,E[f(pos_i)]- _rrf(r)= N+1K _iCov (s_i,f(pos_i) ), (S7) where we used ∑i[posif(posi)]=∑rf(r) _iE[pos_if(pos_i)]= _rrf(r) (Lemma S2). (3) Generating functions. Both sides of (S7) are linear in f, so it suffices to take f(r)=xrf(r)=x^r. Put y=x−1y=x-1. Given sis_i, write a^=K−1−si a=K-1-s_i; opponent j<ij<i is served before i with probability a^+1K a+1K and opponent j>ij>i with probability a^K aK, independently across j. Let a^=1+a^Ky B_ a=1+ aKy and a^=1+a^+1Ky=a^+1 A_ a=1+ a+1Ky= B_ a+1. Then Πi(x):=[xposi]=1K∑a^=0K−1a^ia^N−1−i,μi:=[posi]=(N−1)K−12+iK. _i(x):=E[x^pos_i]= 1K _ a=0^K-1 A_ a^\,i B_ a^\,N-1-i, _i:=E[pos_i]= (N-1) K-12+iK. With ∑i=0N−1iN−1−i=N−N− _i=0^N-1 A^i B^N-1-i= A^N- B^N A- B, a^−a^=yK A_ a- B_ a= yK, telescoping (a^=a^+1 A_ a= B_ a+1, 0=1 B_0=1, K=1+y B_K=1+y), :=∑a^=0Ka^N S:= _ a=0^K B_ a^N and G(x):=∑r=0N−1xr=(1+y)N−1yG(x):= _r=0^N-1x^r= (1+y)^N-1y: ∑iΠi _i _i =1y∑a^=0K−1(a^N−a^N)=G, = 1y _ a=0^K-1( A_ a^N- B_ a^N)=G, ∑iΠi _ii\, _i =1y∑a^=0K−1(Na^N+a^N)−K((1+y)N+1−1)y2 = 1y _ a=0^K-1(N A_ a^N+ B_ a^N)- K ((1+y)^N+1-1 )y^2 =(N+1)−N−(1+y)Ny−K((1+y)N+1−1)y2, = (N+1) S-N-(1+y)^Ny- K ((1+y)^N+1-1 )y^2, Z(x) Z(x) :=∑i[sixposi]=1y∑a^=0K−1(K−1−a^)(a^N−a^N) := _iE[s_ix^pos_i]= 1y _ a=0^K-1(K-1- a)( A_ a^N- B_ a^N) =−K−(1+y)Ny. = S-K-(1+y)^Ny. (The second line uses ∑iiN−1−i=∂N−N− _ii\, A^i B^N-1-i= A _ A A^N- B^N A- B; the third is a summation by parts: with a^=a^+1 A_ a= B_ a+1, ∑a^=0K−1(K−1−a^)(a^+1N−a^N)=∑a^=1K−1a^N−(K−1)0N=−0N−KN−(K−1) _ a=0^K-1(K-1- a)( B_ a+1^N- B_ a^N)= _ a=1^K-1 B_ a^N-(K-1) B_0^N= S- B_0^N- B_K^N-(K-1), and 0N=1 B_0^N=1, KN=(1+y)N B_K^N=(1+y)^N.) The generating-function form of (S7) is ∑iμiΠi−xG′=N+1K(Z−K−12G) _i _i _i-xG = N+1K (Z- K-12G ). Substituting, S and K cancel and the difference of the two sides equals NG+N+1y−(1+y)N+1−1y2−xG′ NG+ N+1y- (1+y)^N+1-1y^2-xG =Ny(1+y)N+y−(1+y)N+1+1y2 = Ny(1+y)^N+y-(1+y)^N+1+1y^2 −Ny(1+y)N−(1+y)N+1+1+y2=0. - Ny(1+y)^N-(1+y)^N+1+1+yy^2=0. This proves (S7), hence Lemma 2. ∎ Remark S4. The quantisation K disappears in part (3): the identity only uses that, conditionally on sis_i, the opponents are served before i independently, and that positions form a permutation. S-A.3 Step 2: proof of Proposition S5 (higher orders) The order decomposition of Lemma S3 splits QKQ_K into a sum over Hoeffding orders m. This step disposes of every order m≥3m≥ 3 at once, by an argument that uses nothing about f; what remains afterwards is the two-body form QK(1)+QK(2)Q^(1)_K+Q^(2)_K, which occupies Steps S-A.4–S-A.13. Proposition S5 (higher orders). For all N≥3N≥ 3, K≥2K≥ 2 and m≥3m≥ 3, QK(m)⪰0Q^(m)_K 0, where QK(m)(f)=N−1N∑|T|=m∑i∈T‖[ui]T‖2−1m∑|T|=m‖∑i∈T[ui]T‖2.Q^(m)_K(f)= N-1N _|T|=m _i∈ T\|[u_i]_T\|^2- 1m _|T|=m \| _i∈ T[u_i]_T \|^2. More precisely, for every T with |T|=m|T|=m, ‖∑i∈T[ui]T‖2≤2∑i∈T‖[ui]T‖2 \| _i∈ T[u_i]_T \|^2≤ 2 _i∈ T\|[u_i]_T\|^2, and the constant 22 cannot be improved at K=2K=2. Proof. Fix T and write Xi=[ui]T=(I−Ai)FiX_i=[u_i]_T=(I-A_i)F_i with Fi=Ψ(i)D(i)F_i= ^(i)D^(i) as in Lemma S6 below, whose proof does not use this proposition. Three facts. (i) Cross terms. For i≠ki≠ k in T, FkF_k is centred in sis_i (it contains the factor θ(si,⋅)−pθ(s_i;·)-p), so AiFk=0A_iF_k=0 and, AiA_i being an orthogonal projection, ⟨AiFi,Fk⟩=⟨AiFi,AiFk⟩=0 A_iF_i,F_k = A_iF_i,A_iF_k =0; similarly the other cross projections vanish, whence ⟨Xi,Xk⟩=⟨Fi,Fk⟩ X_i,X_k = F_i,F_k . (i) The cross kernel is entrywise nonpositive. For i<ki<k, conditioning on si=as_i=a, sk=bs_k=b and averaging over the remaining coordinates, ⟨Fi,Fk⟩=1K2∑a,bD(i)(a)D(k)(b)Ωik(a,b), F_i,F_k = 1K^2 _a,bD^(i)(a)D^(k)(b)\, _ik(a,b), Ωik(a,b)=−([b>a]−p(a))([b>a]−q(b))∏j≠i,kϰ(tj(i)(a),tj(k)(b)), _ik(a,b)=- (1[b>a]-p(a) ) (1[b>a]-q(b) ) _j≠ i,k (t^(i)_j(a),t^(k)_j(b) ), with q(b)=b/Kq(b)=b/K and ϰ(t,t′)=p(max(t,t′))−p(t)p(t′)≥0 (t,t )=p( (t,t ))-p(t)p(t )≥ 0. Indeed the factor of Ψ(i) ^(i) involving sks_k is [b>a]−p(a)1[b>a]-p(a) (as k>ik>i), and the factor of Ψ(k) ^(k) involving sis_i is [a≥b]−p(b−1)=−([b>a]−q(b))1[a≥ b]-p(b-1)=-(1[b>a]-q(b)) (as i<ki<k, p(b−1)=1−q(b)p(b-1)=1-q(b)): the two indicators are complementary, so the product is −(1−p)(1−q)≤0-(1-p)(1-q)≤ 0 if b>ab>a and −pq≤0-pq≤ 0 if b≤ab≤ a. The remaining factors are covariances of two increasing indicators of the same variable, hence nonnegative. (i) Diagonal terms. ‖Xi‖2=‖Fi‖2−‖AiFi‖2\|X_i\|^2=\|F_i\|^2-\|A_iF_i\|^2, where ‖Fi‖2\|F_i\|^2 depends on D(i)D^(i) only through D(i)(a)2D^(i)(a)^2, and ‖AiFi‖2=1K2∑a,a′D(i)(a)D(i)(a′)Ωid(a,a′)\|A_iF_i\|^2= 1K^2 _a,a D^(i)(a)D^(i)(a ) d_i(a,a ) with Ωid(a,a′)=∏j≠iϰ(tj(i)(a),tj(i)(a′))≥0 d_i(a,a )= _j≠ i (t^(i)_j(a),t^(i)_j(a ))≥ 0. Let D~(i)=|D(i)| D^(i)=|D^(i)| and F~i,X~i F_i, X_i the corresponding vectors. By (i)–(i), ∑i≠k⟨Xi,Xk⟩ _i≠ k X_i,X_k ≤1K2∑i≠k∑a,b|D(i)(a)||D(k)(b)||Ωik(a,b)|=−∑i≠k⟨F~i,F~k⟩ ≤ 1K^2 _i≠ k _a,b|D^(i)(a)|\,|D^(k)(b)|\,| _ik(a,b)|=- _i≠ k F_i, F_k =∑i‖X~i‖2−‖∑iX~i‖2≤∑i‖X~i‖2. = _i\| X_i\|^2- \| _i X_i \|^2≤ _i\| X_i\|^2. By (i), ‖X~i‖2≤‖Xi‖2\| X_i\|^2≤\|X_i\|^2. Therefore ‖∑iXi‖2=∑i‖Xi‖2+∑i≠k⟨Xi,Xk⟩≤2∑i‖Xi‖2. \| _iX_i \|^2= _i\|X_i\|^2+ _i≠ k X_i,X_k ≤ 2 _i\|X_i\|^2. Summing over T, ∑|T|=m‖gT‖2≤2∑|T|=m∑i∈T‖[ui]T‖2,soQK(m)≥(N−1N−2m)∑|T|=m∑i∈T‖[ui]T‖2≥0 _|T|=m\|g_T\|^2≤ 2 _|T|=m _i∈ T\|[u_i]_T\|^2, Q^(m)_K≥ ( N-1N- 2m ) _|T|=m _i∈ T\|[u_i]_T\|^2≥ 0 as soon as m≥2N−1m≥ 2NN-1, i.e. m≥3m≥ 3 for N≥3N≥ 3. For K=2K=2 the interior members of T have Xi=0X_i=0 (their Ψ(i) ^(i) vanishes at both levels) and the two extreme members are parallel; for f chosen so that they have equal norm and the same orientation the inequality is an equality, so the constant 22 cannot be improved. ∎ In Proposition S5, QK(m)Q^(m)_K is viewed as a quadratic form in the increments F of f; since F⊤QKF=KN+1∑mQK(m)(f)F Q_KF=K^N+1 _mQ^(m)_K(f) by Lemma S3, positive-semidefiniteness statements about the QK(m)Q^(m)_K transfer to QKQ_K unchanged. Lemma S6 (structure of the components). Let T=i1<⋯<imT=\i_1<…<i_m\ and i∈Ti∈ T. Put tj(i)(a)=at^(i)_j(a)=a for j>ij>i and a−1a-1 for j<ij<i, θ(x;t)=[x>t]θ(x;t)=1[x>t], p(t)=K−1−tKp(t)= K-1-tK (with p(−1)=1p(-1)=1). Then [ui]T=(I−Ai)Fi,Fi(sT)=Ψ(i)(sT)D(i)(si),Ψ(i)=∏j∈T∖i(θ(sj,tj(i)(si))−p(tj(i)(si))),[u_i]_T=(I-A_i)F_i, F_i(s_T)= ^(i)(s_T)\,D^(i)(s_i), ^(i)= _j∈ T i (θ(s_j;t^(i)_j(s_i))-p(t^(i)_j(s_i)) ), with D(i)(a)=[Δm−1f(B∉T(a))]D^(i)(a)=E [ ^m-1f(B^(a)_∉ T) ], where B∉T(a)B^(a)_∉ T is the number of players outside T served before i when si=as_i=a. Proof. We have posi=∑j∈T∖iθ(sj,tj(i)(si))+B∉Tpos_i= _j∈ T iθ(s_j;t^(i)_j(s_i))+B_∉ T. Averaging over the coordinates outside T gives f(∑j∈T∖iθj+B)Ef ( _j∈ T i _j+B ). For j∈T∖ij∈ T i and any function ζ, ζ(θj)=ζ(0)+θjΔζ(0)ζ( _j)=ζ(0)+ _j ζ(0) because θj∈0,1 _j∈\0,1\, so (I−Aj)ζ(θj)=(θj−pj)Δζ(0)(I-A_j)ζ( _j)=( _j-p_j) ζ(0); applying this successively to all j∈T∖ij∈ T i produces the product Ψ(i) ^(i) and the (m−1)(m-1)-fold difference, and (I−Ai)(I-A_i) is the last factor. ∎ S-A.4 Step 3: bond vectors and the two-body reduction (first part of Proposition S10) The two-body form QK(1)+QK(2)Q^(1)_K+Q^(2)_K is a quadratic form in the N−1N-1 increments of f. This step rewrites it as a quadratic form N,K B_N,K in N−1N-1 bond vectors D^k(f)∈ℝK D_k(f) ^K, one for each pair of adjacent positions, whose entries are averaged first differences of f. The point of the change of variables is that N,K B_N,K is defined on all of (ℝK)N−1(R^K)^N-1, not only on the image D^(f)\ D(f)\, and positivity on the larger space is what Steps S-A.5–S-A.12 prove; Lemma S11 at the end of this step brings the kernel back to f. Notation for the rest of the appendix. Priority levels are a∈0,…,K−1a∈\0,…,K-1\ and carry the uniform probability measure; VarVar and CovCov of functions of the level refer to it. Bonds are indexed by k∈0,…,N−2k∈\0,…,N-2\. For D∈ℝKD ^K we write (J1D)(a)=∑x≤aD(x),(J0D)(a)=∑x<aD(x),(J_1D)(a)= _x≤ aD(x), (J_0D)(a)= _x<aD(x), for the inclusive and exclusive cumulative sums, written without the operators in the main text; given bond vectors D0,…,DN−2∈ℝKD_0,…,D_N-2 ^K we put Gk=J1DkG_k=J_1D_k, Hk=J0DkH_k=J_0D_k, so that Gk−Hk=DkG_k-H_k=D_k, and define the vertex functions Ωj:=∑k<jHk+∑k≥jGk,j=0,…,N−1, _j:= _k<jH_k+ _k≥ jG_k, j=0,…,N-1, (S8) which satisfy Ωk−Ωk+1=Dk _k- _k+1=D_k. Further μk=N−1−kN _k= N-1-kN, νk=k+1N _k= k+1N (so that νk+μl=1−l−kN _k+ _l=1- l-kN for k≤lk≤ l; in particular μk+νk=1 _k+ _k=1), and (g)=∑y=1K−1y(K−y)K(g(y)−g(y−1))2,P(g):=(g)−Var(g),D(g)= _y=1^K-1 y(K-y)K (g(y)-g(y-1) )^2, P(g):=D(g)-Var(g), (S9) is the Hahn Dirichlet form. For a payoff-from-rank function f, with Δf(r)=f(r+1)−f(r) f(r)=f(r+1)-f(r), the bond vectors of f are D^k(a)=[Δf(Bin(k,K−aK)+Bin(N−2−k,K−1−aK))], D_k(a)=E [ f (Bin (k, K-aK )+Bin (N-2-k, K-1-aK ) ) ], (S10) the two binomials being independent. The bond form is the quadratic form on (ℝK)N−1(R^K)^N-1 K2N,K(D):=N−22∑k[μkP(Hk)+νkP(Gk)]⏟I+∑k≤lCov(Hk,Gl)⏟I−1N∑j=0N−1Var(Ωj)⏟I.K^2 B_N,K(D):= N-22 _k [ _kP(H_k)+ _kP(G_k) ]_I+ _k≤ lCov(H_k,G_l)_I- 1N _j=0^N-1Var( _j)_I. (S11) Lemma S7 (what an opponent sees). Conditionally on si=as_i=a, the events “j is served before i”, j≠ij≠ i, are independent; their probability is K−aK K-aK for j<ij<i (the event sj≥as_j≥ a) and K−1−aK K-1-aK for j>ij>i (the event sj>as_j>a). Moreover, for any random variable R with values in 0,…,N−2\0,…,N-2\, independent of the Bernoulli variables below, and any p∈[0,1−1K]p∈[0,1- 1K], f(R+Bern(p+1K))−f(R+Bern(p))=1KΔf(R).Ef (R+Bern(p+ 1K) )-Ef (R+Bern(p) )= 1K\,E\, f(R). (S12) Proof. The first statement is the definition (5) of the index protocol together with the independence of the levels. For the second, f(R+Bern(p))=f(R)+pΔf(R)Ef(R+Bern(p))=Ef(R)+p\,E f(R) is affine in p. ∎ Proposition S8 (first order). Let φj(a):=[uj∣sj=a] _j(a):=E[u_j s_j=a] and let Ωj _j be built from the bond vectors (S10) of f. Then φj=−1KΩj+const _j=- 1K _j+const, and QK(1)(f)=−1N∑j=0N−1Var(φj)=−1NK2∑j=0N−1Var(Ωj).Q^(1)_K(f)=- 1N _j=0^N-1Var( _j)=- 1NK^2 _j=0^N-1Var( _j). (S13) Proof. By Lemma S7, given sj=as_j=a the position of j is a sum of j independent Bernoulli variables of parameter K−aK K-aK (the opponents of smaller index) and N−1−jN-1-j of parameter K−1−aK K-1-aK. Passing from level a to level a−1a-1 raises every parameter by 1K 1K; we do this one opponent at a time, in two stages, applying (S12) at each flip with R the contribution of the other N−2N-2 opponents. Stage (i): raise the N−1−jN-1-j opponents of larger index from K−1−aK K-1-aK to K−aK K-aK. When m of the other N−2N-2 opponents are at K−aK K-aK and N−2−mN-2-m at K−1−aK K-1-aK, the increment is 1KD^m(a) 1K D_m(a) by (S10); here m runs from j to N−2N-2. Stage (i): all opponents are now at K−aK=K−1−(a−1)K K-aK= K-1-(a-1)K; raise the j opponents of smaller index to K−(a−1)K K-(a-1)K. When m of the others have been raised the increment is 1KD^m(a−1) 1K D_m(a-1), m=0,…,j−1m=0,…,j-1. Hence φj(a−1)−φj(a)=1K[∑m≥jD^m(a)+∑m<jD^m(a−1)]=1K[Ωj(a)−Ωj(a−1)], _j(a-1)- _j(a)= 1K [ _m≥ j D_m(a)+ _m<j D_m(a-1) ]= 1K [ _j(a)- _j(a-1) ], because Gm(a)−Gm(a−1)=D^m(a)G_m(a)-G_m(a-1)= D_m(a) and Hm(a)−Hm(a−1)=D^m(a−1)H_m(a)-H_m(a-1)= D_m(a-1). This gives the first claim. For (S13), [uj]j=φj−φj[u_j]_\j\= _j-E _j and, in (S4) with m=1m=1, N−1N‖[uj]j‖2−‖[uj]j‖2=−1NVar(φj) N-1N\|[u_j]_\j\\|^2-\|[u_j]_\j\\|^2=- 1NVar( _j). ∎ Proposition S9 (second order). Let i<ji<j, Xi:=[ui]i,jX_i:=[u_i]_\i,j\ and Xj:=[uj]i,jX_j:=[u_j]_\i,j\, and let Hk,GkH_k,G_k be built from the bond vectors of f. Then ‖Xi‖2=P(Hi)K2,‖Xj‖2=P(Gj−1)K2,⟨Xi,Xj⟩=−Cov(Hi,Gj−1)K2.\|X_i\|^2= P(H_i)K^2, \|X_j\|^2= P(G_j-1)K^2, X_i,X_j =- Cov(H_i,G_j-1)K^2. (S14) Proof. Write a=sia=s_i, b=sjb=s_j, let Aa,AbA_a,A_b be the averaging operators in these coordinates and Ca=I−AaC_a=I-A_a, Cb=I−AbC_b=I-A_b; they are commuting orthogonal projections. Put p(t)=K−1−tKp(t)= K-1-tK, so that b[b>t]=p(t)E_b1[b>t]=p(t) for −1≤t≤K−1-1≤ t≤ K-1. The component XiX_i. Given a and b, posi=[b>a]+Rpos_i=1[b>a]+R, where R is the contribution of the other N−2N-2 opponents: i of smaller index and N−2−iN-2-i of larger index, so by Lemma S7 [f(R+1)−f(R)∣a]=D^i(a)E[f(R+1)-f(R) a]= D_i(a) and [ui∣a,b]=χ(a)+[b>a]D^i(a)E[u_i a,b]=χ(a)+1[b>a] D_i(a) with χ(a)=[f(R)∣a]χ(a)=E[f(R) a]. Now Xi=CaCb[ui∣a,b]X_i=C_aC_bE[u_i a,b]; CbC_b annihilates χ(a)χ(a), and Cb[[b>a]D^i(a)]=([b>a]−p(a))D^i(a)=:F(a,b)C_b[1[b>a] D_i(a)]=(1[b>a]-p(a)) D_i(a)=:F(a,b). Thus Xi=CaFX_i=C_aF with F centred in b, and ‖Xi‖2=‖F‖2−‖AaF‖2\|X_i\|^2=\|F\|^2-\|A_aF\|^2. First, ‖F‖2=1K∑ap(a)(1−p(a))D^i(a)2=1K2∑a=0K−2(a+1)(K−1−a)KD^i(a)2=(Hi)K2,\|F\|^2= 1K _ap(a)(1-p(a)) D_i(a)^2= 1K^2 _a=0^K-2 (a+1)(K-1-a)K D_i(a)^2= D(H_i)K^2, since Hi(y)−Hi(y−1)=D^i(y−1)H_i(y)-H_i(y-1)= D_i(y-1). Second, ∑a[b>a]D^i(a)=Hi(b) _a1[b>a] D_i(a)=H_i(b) and ∑ap(a)D^i(a)=bHi(b) _ap(a) D_i(a)=E_bH_i(b), so (AaF)(b)=1K(Hi(b)−Hi)(A_aF)(b)= 1K(H_i(b)-EH_i) and ‖AaF‖2=Var(Hi)/K2\|A_aF\|^2=Var(H_i)/K^2. This proves the first identity. The component XjX_j. Symmetrically, given a,ba,b one has posj=[a≥b]+R′pos_j=1[a≥ b]+R , where the other opponents of j are j−1j-1 of smaller index and N−1−jN-1-j of larger index, so that the coefficient is D^j−1(b) D_j-1(b), and Xj=CbF~X_j=C_b F with F~(a,b)=([a≥b]−p(b−1))D^j−1(b) F(a,b)=(1[a≥ b]-p(b-1)) D_j-1(b), centred in a. Here p(b−1)(1−p(b−1))=b(K−b)K2p(b-1)(1-p(b-1))= b(K-b)K^2 and Gj−1(y)−Gj−1(y−1)=D^j−1(y)G_j-1(y)-G_j-1(y-1)= D_j-1(y) give ‖F~‖2=(Gj−1)/K2\| F\|^2=D(G_j-1)/K^2, while ∑b[a≥b]D^j−1(b)=Gj−1(a) _b1[a≥ b] D_j-1(b)=G_j-1(a) gives ‖AbF~‖2=Var(Gj−1)/K2\|A_b F\|^2=Var(G_j-1)/K^2. The cross term. Since CaC_a is self-adjoint, CaCb=CbCaC_aC_b=C_bC_a, CaF~=F~C_a F= F and CbF=FC_bF=F, ⟨CaF,CbF~⟩=⟨F,CbCaF~⟩=⟨F,CbF~⟩=⟨CbF,F~⟩=⟨F,F~⟩. C_aF,C_b F = F,C_bC_a F = F,C_b F = C_bF, F = F, F . The two indicators are complementary, [a≥b]=1−[b>a]1[a≥ b]=1-1[b>a] and p(b−1)=1−U[U≥b]p(b-1)=1-E_U1[U≥ b], whence ([b>a]−p(a))([a≥b]−p(b−1))=−([b>a]−p(a))([b>a]−bK). (1[b>a]-p(a) ) (1[a≥ b]-p(b-1) )=- (1[b>a]-p(a) ) (1[b>a]- bK ). Checking the cases b>ab>a and b≤ab≤ a separately, the right-hand side equals −ϰ(a,b−1)- (a,b-1), where ϰ(t,t′):=CovU([U>t],[U>t′])=p(max(t,t′))−p(t)p(t′)≥0 (t,t ):=Cov_U(1[U>t],1[U>t ])=p( (t,t ))-p(t)p(t )≥ 0 for U uniform on the levels. Therefore ⟨Xi,Xj⟩=−1K2∑a,bD^i(a)D^j−1(b)CovU([U>a],[U≥b])=−1K2Cov(Hi(U),Gj−1(U)).∎ X_i,X_j =- 1K^2 _a,b D_i(a) D_j-1(b)\,Cov_U (1[U>a],1[U≥ b] )=- 1K^2Cov (H_i(U),G_j-1(U) ). Proposition S10 (two-body reduction). For every N≥2N≥ 2, K≥2K≥ 2 and every payoff-from-rank function f, QK(1)(f)+QK(2)(f)=N,K(D^(f))Q^(1)_K(f)+Q^(2)_K(f)= B_N,K ( D(f) ). Proof. For m=2m=2, (S4) reads QK(2)=∑i<j[N−22N(‖Xi‖2+‖Xj‖2)−⟨Xi,Xj⟩]Q^(2)_K= _i<j [ N-22N(\|X_i\|^2+\|X_j\|^2)- X_i,X_j ], since N−1N−12=N−22N N-1N- 12= N-22N. By (S14), bond k occurs through P(Hk)P(H_k) in the N−1−kN-1-k pairs (k,j)(k,j), j>kj>k, and through P(Gk)P(G_k) in the k+1k+1 pairs (i,k+1)(i,k+1), i≤ki≤ k; this gives K−2IK^-2\,I. The map (i,j)↦(k,l)=(i,j−1)(i,j) (k,l)=(i,j-1) is a bijection from i<j\i<j\ onto k≤l\k≤ l\, so the cross terms give K−2IIK^-2\,I. Finally QK(1)=−K−2IIIQ^(1)_K=-K^-2\,I by (S13). ∎ Lemma S11 (kernel transfer). D^(f)∈span(⊗) D(f) (1 1), i.e. D^k(a)=c D_k(a)=c for all k and a, if and only if f is affine. Proof. If Δf≡c f≡ c then D^k≡c D_k≡ c by (S10). Conversely, at the top level a=K−1a=K-1 the second binomial in (S10) is degenerate at 00, so D^k(K−1)=Δf(Bin(k,1K)) D_k(K-1)=E\, f(Bin(k, 1K)), k=0,…,N−2k=0,…,N-2. This is a triangular linear system in Δf(0),…,Δf(N−2) f(0),…, f(N-2) with diagonal entries K−k>0K^-k>0; its unique solution with right-hand side c 1c\,1 is Δf≡c f≡ c. ∎ Proposition S10 uses only exchangeability within the two index groups, the one-flip identity and the orthogonality of the Hoeffding projections. It holds for every K≥2K≥ 2, including K=2K=2, where Theorem 2 fails: the hypothesis K≥3K≥ 3 enters only through Lemma S16 below. S-A.5 Step 4: the sum-of-squares identity The bond form (S11) has one negative term, II, of the same order as the two positive ones. This step rearranges the three terms into a sum of manifestly nonnegative pieces plus a single negative term of a different shape, −∑kckVar(Dk)- _kc_kVar(D_k), which involves the bond vectors only through their variances; every later step is a way of paying for that term. Two elementary identities are needed first. Lemma S12 (Hahn–Poincaré). For every function g of the level, (g)=∑m=1K−1m(m+1)Varm(g)D(g)= _m=1^K-1m(m+1)Var_m(g), where Varm(g)Var_m(g) is the squared norm of the component of g along the discrete Chebyshev polynomial of degree m on 0,…,K−1\0,…,K-1\. In particular (g)≥2Var(g)D(g)≥ 2Var(g), with equality iff g is affine. Proof. Summation by parts gives (g)=[g⋅Lg]D(g)=E[g· Lg] with (Lg)(y)=−[(y+1)(K−1−y)(g(y+1)−g(y))−y(K−y)(g(y)−g(y−1))](Lg)(y)=- [(y+1)(K-1-y)(g(y+1)-g(y))-y(K-y)(g(y)-g(y-1)) ], the second-order difference operator of the Hahn polynomials with parameters (0,0)(0,0), whose eigenfunctions are the discrete Chebyshev polynomials with eigenvalues m(m+1)m(m+1) [43, Sec. 9.5]. The components of degree m≥2m≥ 2 have eigenvalue at least 6>26>2. ∎ Lemma S13 (spread identity). For functions z0,…,zn−1z_0,…,z_n-1 of the level, n∑kVar(zk)−Var(∑kzk)=∑k<lVar(zk−zl)n _kVar(z_k)-Var ( _kz_k )= _k<lVar(z_k-z_l). Proof. Both sides equal (n−1)∑kVar(zk)−2∑k<lCov(zk,zl)(n-1) _kVar(z_k)-2 _k<lCov(z_k,z_l). ∎ Theorem S14 (sum-of-squares identity). For all bond vectors D0,…,DN−2∈ℝKD_0,…,D_N-2 ^K, K2N,K(D)=∑kZk+Sq(D)−∑kckVar(Dk),ck:=2Nμkνk−12,K^2 B_N,K(D)= _kZ_k+Sq(D)- _kc_kVar(D_k), c_k:=2N _k _k- 12, (S15) where Zk:=N−22[μk(−2Var)(Hk)+νk(−2Var)(Gk)]≥0Z_k:= N-22 [ _k(D-2Var)(H_k)+ _k(D-2Var)(G_k) ]≥ 0 and Sq(D):=12N∑k<l[(N−1−l)Var(Hk−Hl+Dl)+(k+1)Var(Gk−Gl+Dk)+(l−k)Var(Hk−Gl)]≥0.Sq(D):= 12N _k<l [(N-1-l)Var(H_k-H_l+D_l)+(k+1)Var(G_k-G_l+D_k)+(l-k)Var(H_k-G_l) ]≥ 0. (S16) Proof. 1) Vertex variables. For j∈0,…,N−1j∈\0,…,N-1\ and a bond k put Zjk:=HkZ_jk:=H_k if k<jk<j and Zjk:=GkZ_jk:=G_k if k≥jk≥ j, so that Ωj=∑kZjk _j= _kZ_jk, and let jE_j denote the average over j. For fixed k, Zjk=HkZ_jk=H_k for the N−1−kN-1-k values j>kj>k and Zjk=GkZ_jk=G_k for the k+1k+1 values j≤kj≤ k. Hence, in (S11), I=N−22j∑kP(Zjk),I=jVar(∑kZjk),∑kZk=N−22j∑k(−2Var)(Zjk).I= N-22\,E_j _kP(Z_jk), =E_jVar ( _kZ_jk ), _kZ_k= N-22\,E_j _k(D-2Var)(Z_jk). 2) Spreading. With n=N−1n=N-1 bonds, Lemma S13 gives, for each j, N−22∑kP(Zjk)=N−22∑k(−2Var)(Zjk)+12∑k<lVar(Zjk−Zjl)−12∑kVar(Zjk)+12Var(Ωj); N-22 _kP(Z_jk)= N-22 _k(D-2Var)(Z_jk)+ 12 _k<lVar(Z_jk-Z_jl)- 12 _kVar(Z_jk)+ 12Var( _j); indeed both sides equal N−22∑k(Zjk)−N−22∑kVar(Zjk) N-22 _kD(Z_jk)- N-22 _kVar(Z_jk). Averaging over j and inserting into (S11), K2=∑kZk+12j∑k<lVar(Zjk−Zjl)+Ξ,Ξ:=I−12j[Var(Ωj)+∑kVar(Zjk)].K^2 B= _kZ_k+ 12E_j _k<lVar(Z_jk-Z_jl)+ , :=I- 12E_j [Var( _j)+ _kVar(Z_jk) ]. (S17) 3) The pair term. For k<lk<l the pair (Zjk,Zjl)(Z_jk,Z_jl) equals (Gk,Gl)(G_k,G_l) for the k+1k+1 values j≤kj≤ k, (Hk,Gl)(H_k,G_l) for the l−kl-k values k<j≤lk<j≤ l, and (Hk,Hl)(H_k,H_l) for the N−1−lN-1-l values j>lj>l. Therefore 12j∑k<lVar(Zjk−Zjl)=12N∑k<l[(k+1)Var(Gk−Gl)+(l−k)Var(Hk−Gl)+(N−1−l)Var(Hk−Hl)]. 12E_j _k<lVar(Z_jk-Z_jl)= 12N _k<l [(k+1)Var(G_k-G_l)+(l-k)Var(H_k-G_l)+(N-1-l)Var(H_k-H_l) ]. (S18) 4) The residual. Since 12[Var(Ωj)+∑kVar(Zjk)]=∑kVar(Zjk)+∑k<lCov(Zjk,Zjl) 12[Var( _j)+ _kVar(Z_jk)]= _kVar(Z_jk)+ _k<lCov(Z_jk,Z_jl), the same count gives Ξ=∑k<l[(1−l−kN)Cov(Hk,Gl)−νkCov(Gk,Gl)−μlCov(Hk,Hl)]+∑k[Cov(Hk,Gk)−μkVar(Hk)−νkVar(Gk)]. = _k<l [ (1- l-kN )Cov(H_k,G_l)- _kCov(G_k,G_l)- _lCov(H_k,H_l) ]+ _k [Cov(H_k,G_k)- _kVar(H_k)- _kVar(G_k) ]. As 1−l−kN=νk+μl1- l-kN= _k+ _l and G=H+DG=H+D, the first bracket is μlCov(Hk,Dl)−νkCov(Gl,Dk) _lCov(H_k,D_l)- _kCov(G_l,D_k); as μk+νk=1 _k+ _k=1, the second is μkCov(Hk,Dk)−νkCov(Gk,Dk)=(μk−νk)Cov(Hk,Dk)−νkVar(Dk) _kCov(H_k,D_k)- _kCov(G_k,D_k)=( _k- _k)Cov(H_k,D_k)- _kVar(D_k). Thus Ξ=∑k<l[μlCov(Hk,Dl)−νkCov(Gl,Dk)]+∑k[(μk−νk)Cov(Hk,Dk)−νkVar(Dk)]. = _k<l [ _lCov(H_k,D_l)- _kCov(G_l,D_k) ]+ _k [( _k- _k)Cov(H_k,D_k)- _kVar(D_k) ]. (S19) 5) Absorption. Expanding the shifted variances and using 2Cov(H,D)=Var(G)−Var(H)−Var(D)2Cov(H,D)=Var(G)-Var(H)-Var(D), 12Var(Hk−Hl)+Cov(Hk,Dl) 12Var(H_k-H_l)+Cov(H_k,D_l) =12Var(Hk−Hl+Dl)+12[VarGl−VarHl]−Var(Dl), = 12Var(H_k-H_l+D_l)+ 12 [VarG_l-VarH_l ]-Var(D_l), (S20) 12Var(Gk−Gl)−Cov(Gl,Dk) 12Var(G_k-G_l)-Cov(G_l,D_k) =12Var(Gk−Gl+Dk)−12[VarGk−VarHk]−Var(Dk). = 12Var(G_k-G_l+D_k)- 12 [VarG_k-VarH_k ]-Var(D_k). (S21) The weights match: the (H,H)(H,H) pair carries N−1−l2N=μl2 N-1-l2N= _l2 in (S18) and μl _l in (S19), the (G,G)(G,G) pair νk2 _k2 and νk _k. Applying (S20)–(S21) converts (S18) plus the off-diagonal part of (S19) into Sq(D)Sq(D) plus multiples of 12(VarGk−VarHk) 12(VarG_k-VarH_k) and of Var(Dk)Var(D_k), which we now collect together with the diagonal part of (S19), written as (μk−νk)⋅12(VarGk−VarHk)−μk−νk2Var(Dk)−νkVar(Dk)( _k- _k)· 12(VarG_k-VarH_k)- _k- _k2Var(D_k)- _kVar(D_k). Coefficient of 12(VarGk−VarHk) 12(VarG_k-VarH_k). Bond k is the second index of k pairs, each contributing +μk+ _k by (S20), and the first index of N−2−kN-2-k pairs, each contributing −νk- _k by (S21); the diagonal contributes μk−νk _k- _k. The total is 1N[k(N−1−k)−(k+1)(N−2−k)+(N−2−2k)]=0 1N [k(N-1-k)-(k+1)(N-2-k)+(N-2-2k) ]=0. Coefficient of Var(Dk)Var(D_k). The same three sources give −kμk-k _k, −(N−2−k)νk-(N-2-k) _k and −μk−νk2−νk=−12- _k- _k2- _k=- 12, and k(N−1−k)+(k+1)(N−2−k)N+12=2(N−1−k)(k+1)N−12=ck k(N-1-k)+(k+1)(N-2-k)N+ 12= 2(N-1-k)(k+1)N- 12=c_k. 6) Signs. Zk≥0Z_k≥ 0 by Lemma S12, and Sq≥0Sq≥ 0 because its weights are nonnegative. ∎ S-A.6 Step 5: the kernel of SqSq The term SqSq of Theorem S14 is what pins down the kernel of N,K B_N,K: once positivity is known, (D)=0 B(D)=0 forces Sq(D)=0Sq(D)=0, and the following lemma turns that into D∈span(⊗)D (1 1). It is used in the kernel part of Proposition S23 and in the assembly (Step S-A.14). Lemma S15 (kernel of SqSq). Let N≥3N≥ 3, K≥2K≥ 2. If Sq(D)=0Sq(D)=0 then D∈span(⊗)D (1 1): there is c∈ℝc with Dk=c 1D_k=c\,1 for every bond k. Proof. On the range 0≤k<l≤N−20≤ k<l≤ N-2 the three weights in (S16) are at least 11. Hence Sq(D)=0Sq(D)=0 forces, for every pair k<lk<l, each of Hk−GlH_k-G_l, Hk−Hl+DlH_k-H_l+D_l and Gk−Gl+DkG_k-G_l+D_k to be constant in the level. Using H(a+1)−H(a)=D(a)H(a+1)-H(a)=D(a) and G(a+1)−G(a)=D(a+1)G(a+1)-G(a)=D(a+1) and taking first differences at a=0,…,K−2a=0,…,K-2: Dk(a)=Dl(a+1),Dk(a)−2Dl(a)+Dl(a+1)=0,2Dk(a+1)−Dl(a+1)−Dk(a)=0.D_k(a)=D_l(a+1), D_k(a)-2D_l(a)+D_l(a+1)=0, 2D_k(a+1)-D_l(a+1)-D_k(a)=0. Substituting the first relation into the second gives Dl(a+1)=Dl(a)D_l(a+1)=D_l(a), so DlD_l is constant whenever l occurs as a second index; substituting it into the third gives Dk(a+1)=Dk(a)D_k(a+1)=D_k(a), so DkD_k is constant whenever k occurs as a first index. Since N≥3N≥ 3 there are at least two bonds, and every bond is a first index (k≤N−3k≤ N-3) or a second index (l≥1l≥ 1); so Dk=ckD_k=c_k1 for all k, and the first relation gives ck=clc_k=c_l. ∎ All three groups of squares are needed: the first alone does not constrain D0(K−1)D_0(K-1) nor DN−2(0)D_N-2(0); the second removes the latter freedom and the third the former. S-A.7 Step 6: the sharp level inequality The terms ZkZ_k of Theorem S14 are nonnegative by Lemma S12, but that alone does not pay for −ckVar(Dk)-c_kVar(D_k). This step shows that ZkZ_k dominates Var(Dk)Var(D_k) with an explicit constant, and that the constant carries the factor K−2K-2: this is the one place where the hypothesis K≥3K≥ 3 enters. Lemma S16 (sharp level inequality). For every K≥3K≥ 3, μ∈[0,1]μ∈[0,1], ν=1−μν=1-μ and D∈ℝKD ^K, μ(−2Var)(J0D)+ν(−2Var)(J1D)≥(K−2)μνVar(D),μ(D-2Var)(J_0D)+ν(D-2Var)(J_1D)\ ≥\ (K-2)\,μν\,Var(D), (S22) and the constant K−2K-2 cannot be improved. Moreover, in the step coordinates cm=D(m)−D(m−1)c_m=D(m)-D(m-1), m=1,…,K−1m=1,…,K-1, the difference of the two sides of (S22) equals 2K2c⊤[(Kg)∘Φμ]c 2K^2\,c [(Kg) _μ ]c, where g(m,m′)=min(m,m′)(K−max(m,m′))/Kg(m,m )= (m,m )(K- (m,m ))/K is the Green’s function of the Dirichlet Laplacian of the path 1,…,K−1\1,…,K-1\ and Φμ=M~μ−Kμν2⊤ _μ= M_μ- Kμν211 is positive semidefinite, with M~μ(m,m′)=a~min(m,m′)b~max(m,m′),a~m=m−1+2μ2,b~m=K−m−1+2ν2. M_μ(m,m )= a_ (m,m ) b_ (m,m ), a_m= m-1+2μ2, b_m= K-m-1+2ν2. (S23) Proof. 1) Pair representation. Let Φ=J1D =J_1D and, for 1≤y≤y′≤K−11≤ y≤ y ≤ K-1, γ(y,y′):=yK(1−y′K)γ(y,y ):= yK (1- y K ). Since Var(Φ)=1K2∑a<b(∑a<y≤bD(y))2Var( )= 1K^2 _a<b ( _a<y≤ bD(y) )^2 under the uniform law on the levels, the coefficient of D(y)D(y′)D(y)D(y ), y≤y′y≤ y , in Var(Φ)Var( ) is 1K2#(a,b):a<y,b≥y′=γ(y,y′) 1K^2\#\(a,b):a<y,\ b≥ y \=γ(y,y ) (counted twice if y<y′y<y ). Writing D(y)D(y′)=12(D(y)2+D(y′)2)−12(D(y)−D(y′))2D(y)D(y )= 12(D(y)^2+D(y )^2)- 12(D(y)-D(y ))^2, Var(Φ)=∑y≥1D(y)2[γ(y,y)+∑y′≠yγ(min,max)]−∑1≤y<y′γ(y,y′)(D(y)−D(y′))2.Var( )= _y≥ 1D(y)^2 [γ(y,y)+ _y ≠ yγ( , ) ]- _1≤ y<y γ(y,y ) (D(y)-D(y ) )^2. A direct summation gives ∑y′≠yγ(min,max)=K−22γ(y,y) _y ≠ yγ( , )= K-22γ(y,y), while (Φ)=K∑yγ(y,y)D(y)2D( )=K _yγ(y,y)D(y)^2. Hence the diagonal terms cancel in −2VarD-2Var and (−2Var)(J1D)=2K2∑1≤y<y′≤K−1y(K−y′)(D(y)−D(y′))2.(D-2Var)(J_1D)= 2K^2 _1≤ y<y ≤ K-1y\,(K-y )\, (D(y)-D(y ) )^2. The same computation for J0DJ_0D, which is J1DJ_1D shifted by one level, gives (−2Var)(J0D)=2K2∑0≤y<y′≤K−2(y+1)(K−1−y′)(D(y)−D(y′))2(D-2Var)(J_0D)= 2K^2 _0≤ y<y ≤ K-2(y+1)(K-1-y )(D(y)-D(y ))^2. 2) The pencil. For 0≤y<y′≤K−10≤ y<y ≤ K-1 (both weights vanish outside the ranges above), μ(y+1)(K−1−y′)+νy(K−y′)=(y+μ)(K−μ−y′)−μν,μ(y+1)(K-1-y )+ν\,y(K-y )=(y+μ)(K-μ-y )-μν, an identity in (y,y′,μ)(y,y ,μ). Since Var(D)=1K2∑y<y′(D(y)−D(y′))2Var(D)= 1K^2 _y<y (D(y)-D(y ))^2, (S22) is equivalent to ∑0≤y<y′≤K−1[(y+μ)(K−μ−y′)−K2μν](D(y)−D(y′))2≥0. _0≤ y<y ≤ K-1 [(y+μ)(K-μ-y )- K2μν ] (D(y)-D(y ) )^2\ ≥ 0. (S24) 3) Step coordinates. With D(y′)−D(y)=∑y<m≤y′cmD(y )-D(y)= _y<m≤ y c_m, the left-hand side of (S24) is c⊤c Kc with, for m≤m′m≤ m , (m,m′)=∑y<m,y′≥m′[(y+μ)(K−μ−y′)−K2μν]=m(K−m′)[a~mb~m′−K2μν],K(m,m )= _y<m,\ y ≥ m [(y+μ)(K-μ-y )- K2μν ]=m(K-m ) [ a_m b_m - K2μν ], because ∑y<m(y+μ)=ma~m _y<m(y+μ)=m a_m and ∑y′≥m′(K−μ−y′)=(K−m′)b~m′ _y ≥ m (K-μ-y )=(K-m ) b_m . Thus =(Kg)∘ΦμK=(Kg) _μ. 4) Φμ⪰0 _μ 0. Let 0<μ<10<μ<1. The sequences a~ a and b~ b are positive, a~ a increasing and b~ b decreasing, with constant Wronskian a~m+1b~m−a~mb~m+1=12(a~m+b~m)=K4 a_m+1 b_m- a_m b_m+1= 12( a_m+ b_m)= K4. The inverse of such a one-pair (Green) matrix is tridiagonal [46, Ch. I, p. 78–79 and 103–104]: T~:=M~μ−1 T:= M_μ^-1 has off-diagonal entries −4K- 4K, interior diagonal entries 8K 8K, and corner entries 4K(1+12μ) 4K(1+ 12μ) and 4K(1+12ν) 4K(1+ 12ν). That is, T~=4K[Lpath+diag(12μ,0,…,0,12ν)]≻0 T= 4K [L_ path+diag( 12μ,0,…,0, 12ν) ] 0, with LpathL_ path the Neumann Laplacian of the path; in particular ⊤T~=4K(12μ+12ν)=2Kμν1 T1= 4K ( 12μ+ 12ν )= 2Kμν. For a positive definite M, c 11⊤⪰0c\,11 0 iff c 1⊤M−1≤1c\,1 M^-11≤ 1; with c=K2μνc= K2μν equality holds, so Φμ⪰0 _μ 0, with one-dimensional kernel. The cases μ∈0,1μ∈\0,1\ follow by continuity. 5) Schur product. g⪰0g 0 and Φμ⪰0 _μ 0, so =(Kg)∘Φμ⪰0K=(Kg) _μ 0 by the Schur product theorem. Sharpness of the constant K−2K-2: take D=e0D=e_0 and let μ→0μ→ 0. ∎ Proposition S17 (the level pencil). For 0<μ<10<μ<1 let ϱ(μ) (μ) be the smallest generalised eigenvalue of the pencil ((Kg)∘M~μ,Kg) ((Kg) M_μ,\,Kg ), and let c⋆(μ)c (μ) be the largest constant that can replace (K−2)μν(K-2)μν in (S22). Then ϱ(μ)=μν+12c⋆(μ) (μ)=μν+ 12c (μ), which is the definition used in the main text, and ϱ(μ)≥K2μν (μ)≥ K2μν. Proof. By part 3) of the preceding proof and Var(D)=K−2c⊤(Kg)cVar(D)=K^-2c (Kg)c, the left-hand side of (S22) equals 2K2c⊤[(Kg)∘M~μ]c−2μνVar(D) 2K^2c [(Kg) M_μ]c-2μ (D), so LHS≥λVar(D)LHS≥ (D) holds for all D iff (Kg)∘M~μ⪰(μν+λ2)Kg(Kg) M_μ (μν+ λ2)Kg; hence c⋆=2ϱ−2μνc =2 -2μν. Finally (Kg)∘M~μ−K2μν(Kg)=(Kg)∘Φμ⪰0(Kg) M_μ- K2μν\,(Kg)=(Kg) _μ 0, and Kg≻0Kg 0. ∎ S-A.8 Step 7: a lower bound in the bond vectors alone, and step coordinates The identity (S15) has a single negative term, −∑kckVar(Dk)- _kc_kVar(D_k), which involves the bond vectors only through their covariances. We bound the two positive terms from below by quantities of the same kind. Lemma S18 (minimising out a common part). Let w1,…,wr>0w_1,…,w_r>0 with ∑iwi=W _iw_i=W, and let Y1,…,YrY_1,…,Y_r be functions of the level. For every function X of the level, ∑iwiVar(X+Yi)≥∑iwiVar(Yi)−1WVar(∑iwiYi) _iw_iVar(X+Y_i)\ ≥\ _iw_iVar(Y_i)- 1WVar ( _iw_iY_i ). Proof. ∑iwiVar(X+Yi)=WVar(X)+2Cov(X,Y¯)+∑iwiVar(Yi) _iw_iVar(X+Y_i)=WVar(X)+2Cov(X, Y)+ _iw_iVar(Y_i) with Y¯=∑iwiYi Y= _iw_iY_i, and WVar(X)+2Cov(X,Y¯)=WVar(X+Y¯/W)−Var(Y¯)/W≥−Var(Y¯)/WVar(X)+2Cov(X, Y)=WVar(X+ Y/W)-Var( Y)/W≥-Var( Y)/W. ∎ Proposition S19 (the bond matrix). Let gb(k,l):=(min(k,l)+1)(N−1−max(k,l))Ng_b(k,l):= ( (k,l)+1)(N-1- (k,l))N, 0≤k,l≤N−20≤ k,l≤ N-2, and :=2(N−1)diag(μkνk)−2Ngb. P:=2(N-1)diag( _k _k)- 2N\,g_b. (S25) Then Sq(D)≥Sqlow(D):=∑k,lklCov(Dk,Dl)Sq(D)\ ≥\ Sq_ low(D):= _k,l P_klCov(D_k,D_l) for all bond vectors. Proof. Fix k<lk<l and put X=Hk−HlX=H_k-H_l, a=N−1−la=N-1-l, b=k+1b=k+1, c=l−kc=l-k, so that a+b+c=Na+b+c=N. Since G=H+DG=H+D, the three functions in (S16) are X+DlX+D_l, X+2Dk−DlX+2D_k-D_l and X−DlX-D_l, with weights a,b,ca,b,c. By Lemma S18, and because aDl+b(2Dk−Dl)−cDl=2bDk+(2a−N)DlaD_l+b(2D_k-D_l)-cD_l=2bD_k+(2a-N)D_l, the pair (k,l)(k,l) contributes to 2NSq2N\,Sq at least (a+c)Var(Dl)+bVar(2Dk−Dl)−1NVar(2bDk+(2a−N)Dl)=4a(N−a)NVar(Dl)+4b(N−b)NVar(Dk)−8abNCov(Dk,Dl),(a+c)Var(D_l)+bVar(2D_k-D_l)- 1NVar (2bD_k+(2a-N)D_l )= 4a(N-a)NVar(D_l)+ 4b(N-b)NVar(D_k)- 8abNCov(D_k,D_l), where we used N−(2a−N)2N=4a(N−a)N- (2a-N)^2N= 4a(N-a)N and −4b−4b(2a−N)N=−8abN-4b- 4b(2a-N)N=- 8abN. Now a=Nμla=N _l, N−a=NνlN-a=N _l, b=Nνkb=N _k, N−b=NμkN-b=N _k; dividing by 2N2N, the pair contributes at least 2μlνlVar(Dl)+2μkνkVar(Dk)−4νkμlCov(Dk,Dl)2 _l _lVar(D_l)+2 _k _kVar(D_k)-4 _k _lCov(D_k,D_l). Each bond belongs to N−2N-2 pairs, which gives the diagonal 2(N−2)μkνk=2(N−1)μkνk−2Ngb(k,k)2(N-2) _k _k=2(N-1) _k _k- 2Ng_b(k,k), and for k<lk<l the symmetric off-diagonal entry is −2νkμl=−2Ngb(k,l)-2 _k _l=- 2Ng_b(k,l). ∎ Proposition S20 (step coordinates). Let ck∈ℝK−1c_k ^K-1, ck(m)=Dk(m)−Dk(m−1)c_k(m)=D_k(m)-D_k(m-1), and let g, M~μ M_μ be as in Lemma S16. Then Cov(Dk,Dl)=K−2ck⊤(Kg)clCov(D_k,D_l)=K^-2c_k (Kg)c_l, and K2[∑kZk+Sqlow−∑kckVar(Dk)]=(N−2)∑kck⊤[(Kg)∘M~μk]ck+∑k,lΛklck⊤(Kg)cl,K^2 [ _kZ_k+Sq_ low- _kc_kVar(D_k) ]=(N-2) _kc_k [(Kg) M_ _k ]c_k+ _k,l _kl\,c_k (Kg)\,c_l, (S26) with Λ:=−diag((N−2)μkνk+ck) := P-diag ((N-2) _k _k+c_k ), where in the last expression ck=2Nμkνk−12c_k=2N _k _k- 12 is the scalar of (S15). Proof. Cov(D,D′)=1K2∑y<y′(D(y′)−D(y))(D′(y′)−D′(y))Cov(D,D )= 1K^2 _y<y (D(y )-D(y))(D (y )-D (y)), and the coefficient of c(m)c′(m′)c(m)c (m ), m≤m′m≤ m , in this sum is #y<m⋅#y′≥m′=m(K−m′)=Kg(m,m′)\#\y<m\·\#\y ≥ m \=m(K-m )=Kg(m,m ). By Lemma S16, μ(−2Var)(J0D)+ν(−2Var)(J1D)=2K2c⊤[(Kg)∘Φμ]c+(K−2)μνVar(D)μ(D-2Var)(J_0D)+ν(D-2Var)(J_1D)= 2K^2c [(Kg) _μ]c+(K-2)μ (D), and (Kg)∘Φμ=(Kg)∘M~μ−Kμν2Kg(Kg) _μ=(Kg) M_μ- Kμν2Kg; hence K2Zk=(N−2)ck⊤[(Kg)∘M~μk]ck−(N−2)μkνkck⊤(Kg)ck.K^2Z_k=(N-2)\,c_k [(Kg) M_ _k ]c_k-(N-2) _k _k\,c_k (Kg)c_k. Add K2Sqlow=∑k,lklck⊤(Kg)clK^2Sq_ low= _k,l P_klc_k (Kg)c_l and −K2ckVar(Dk)=−ckck⊤(Kg)ck-K^2c_kVar(D_k)=-c_k\,c_k (Kg)c_k. ∎ S-A.9 Step 8: a Perron argument in the space of bonds Proposition S20 has reduced positivity of N,K B_N,K to positivity of a block matrix on the space of bonds whose blocks are Hadamard products with the fixed positive definite matrix KgKg. This step shows that the whole matrix is positive as soon as a scalar condition, (T) below, holds on every bond; the condition is then checked branch by branch in Steps S-A.10–S-A.11. Let L be the Dirichlet Laplacian of the path with vertices 0,…,N−20,…,N-2 (diagonal 22, off-diagonal −1-1), and recall ϱ(μ) (μ) from Proposition S17. Lemma S21. gb=L−1g_b=L^-1, and the positive vector vk=(k+1)(N−1−k)=N2μkνkv_k=(k+1)(N-1-k)=N^2 _k _k satisfies Lv=2⋅Lv=2·1. Proof. gbg_b is the Green’s function of the Dirichlet problem on the path; v is quadratic in k with second difference −2-2 and vanishes at k=−1k=-1 and k=N−1k=N-1. ∎ Lemma S22 (symmetric Z-matrices). Let M be a real symmetric matrix with Mij≤0M_ij≤ 0 for i≠ji≠ j, and let v have positive entries. For every x, x⊤Mx=∑i(Mv)ivixi2+12∑i≠j(−Mij)vivj(xivi−xjvj)2.x Mx= _i (Mv)_iv_ix_i^2+ 12 _i≠ j(-M_ij)\,v_iv_j ( x_iv_i- x_jv_j )^2. (S27) Hence Mv≥0Mv≥ 0 implies M⪰0M 0; if moreover M is irreducible and Mv≠0Mv≠ 0, then M≻0M 0. Proof. Expand the square and use ∑jMijvj=(Mv)i _jM_ijv_j=(Mv)_i. If x⊤Mx=0x Mx=0 with M irreducible, the second sum forces x=tvx=t\,v, and then the first gives t2v⊤Mv=0t^2v Mv=0, so t=0t=0 when Mv≠0Mv≠ 0. ∎ Proposition S23 (condition (T)). Let N≥3N≥ 3, K≥3K≥ 3, and put θk:=(N−2)ϱ(μk)−Nμkνk+12 _k:=(N-2) ( _k)-N _k _k+ 12. If, for every bond k, (N−2)ϱ(μk)−2Nμkνk+12≥ 0,(N-2)\, ( _k)-2N _k _k+ 12\ ≥\ 0, (T) then N,K⪰0 B_N,K 0. If in addition (T) is strict for at least one bond, then kerN,K=span(⊗) B_N,K=span(1 1). Proof. By Theorem S14 and Propositions S19 and S20, K2=(Sq−Sqlow)+K−2ℱ(c)K^2 B=(Sq-Sq_ low)+K^-2F(c) with ℱF the right-hand side of (S26), and Sq−Sqlow≥0Sq-Sq_ low≥ 0. By definition of ϱ , (Kg)∘M~μk⪰ϱ(μk)Kg(Kg) M_ _k ( _k)\,Kg, so ℱ(c)≥∑k,lklck⊤(Kg)clF(c)≥ _k,l A_kl\,c_k (Kg)c_l with :=diag((N−2)ϱ(μk))+Λ A:=diag ((N-2) ( _k) )+ . Inserting (S25), =diag(θk)−2Ngb=diag(θk)−2NL−1. A=diag( _k)- 2N\,g_b=diag( _k)- 2N\,L^-1. Since Kg≻0Kg 0, the form ∑k,lklck⊤(Kg)cl=c⊤(⊗Kg)c _k,l A_klc_k (Kg)c_l=c ( A Kg)c is nonnegative iff ⪰0 A 0. Condition (T) says θk≥Nμkνk>0 _k≥ N _k _k>0. For Θ=diag(θk)≻0 =diag( _k) 0, Θ⪰2NL−1 2NL^-1 is equivalent to N2L⪰Θ−1 N2L ^-1, i.e. to :=NL−2Θ−1⪰0 M:=NL-2 ^-1 0. M is a symmetric Z-matrix and, by Lemma S21, (v)k=2N−2vkθk=2Nθk(θk−Nμkνk)≥0( Mv)_k=2N- 2v_k _k= 2N _k ( _k-N _k _k )≥ 0, so ⪰0 M 0 by Lemma S22, and ⪰0 B 0. For the kernel, let (D)=0 B(D)=0. If (T) is strict for some bond, then v≠0 Mv≠ 0; L is irreducible, so ≻0 M 0, hence ≻0 A 0 and ⊗Kg≻0 A Kg 0. As both Sq−SqlowSq-Sq_ low and ℱF are nonnegative, ℱ(c)=0F(c)=0, so c=0c=0: every bond vector is constant in the level, Dk=tkD_k=t_k1. Then Hk(a)=tkaH_k(a)=t_ka, Gk(a)=tk(a+1)G_k(a)=t_k(a+1), Sqlow(D)=0Sq_ low(D)=0 and Sq(D)=12N∑k<lN(tk−tl)2Var(a)Sq(D)= 12N _k<lN(t_k-t_l)^2Var(a), which vanishes only if all tkt_k coincide. Conversely (⊗)=0 B(1 1)=0 by Proposition S10 and Lemma 2 applied to the linear payoff, for which all QK(m)Q^(m)_K, m≥3m≥ 3, vanish. ∎ S-A.10 Step 9: the branch K≥5K≥ 5 For K≥5K≥ 5 the lower bound ϱ(μ)≥K2μν (μ)≥ K2μν of Proposition S17 verifies (T) for all but finitely many N. Theorem S24. Let K≥5K≥ 5 and N≥3N≥ 3 with (N−2)(K−4)≥4(N-2)(K-4)≥ 4. Then N,K⪰0 B_N,K 0 with kernel span(⊗)span(1 1). Explicitly this covers N≥6N≥ 6 for K=5K=5, N≥4N≥ 4 for K=6,7K=6,7, and N≥3N≥ 3 for K≥8K≥ 8. Proof. Write x=μkνk∈(0,14]x= _k _k∈(0, 14]. By Proposition S17, ϱ(μk)≥K2x ( _k)≥ K2x, so the left-hand side of (T) is at least xs+12x\,s+ 12 with s:=(N−2)K2−2Ns:= (N-2)K2-2N. If s≥0s≥ 0 this is positive. If s<0s<0 it is at least s4+12=18[(N−2)(K−4)−4]≥0 s4+ 12= 18 [(N-2)(K-4)-4 ]≥ 0, with strict inequality unless x=14x= 14. Since μkνk=14 _k _k= 14 for at most one bond and there are N−1≥2N-1≥ 2 bonds, (T) is strict for some bond, and Proposition S23 applies. ∎ Where the argument stops. At K=4K=4 the bracket is (N−2)⋅0−4<0(N-2)· 0-4<0 for every N: the bound ϱ≥K2μν ≥ K2μν is never sufficient, and the true value of ϱ is needed. At K=3K=3 even that fails (Step S-A.12). The three-branch case distinction is therefore forced by the problem, not by the method. S-A.11 Step 10: the branch K=4K=4 (computer-assisted root isolation) For K=4K=4 the pencil of Proposition S17 is 3×33× 3 and det((Kg)∘M~μ−ϱKg)=−4(ϱ,μ) ((Kg) M_μ- \,Kg )=-4\,P( ;μ) with (ϱ,μ)= ( ;μ)= 4ϱ3−10ϱ2+6ϱ+12μ4ϱ−24μ3ϱ+12μ2ϱ2−13μ2ϱ−12μϱ2+25μϱ 4 ^3-10 ^2+6 +12μ^4 -24μ^3 +12μ^2 ^2-13μ^2 -12μ ^2+25μ +4μ6−12μ5−3μ4+26μ3−3μ2−12μ. +4μ^6-12μ^5-3μ^4+26μ^3-3μ^2-12μ. Lemma S25. For every μ∈(0,1)μ∈(0,1), ϱ(μ) (μ) is the unique root of (⋅,μ)P(·;μ) in (0,710)(0, 710), and ϱ(μ)>ϱ0(μ):=2μν+5(μν)3. (μ)\ >\ _0(μ):=2μν+5(μν)^3. (S28) Proof. All three roots are positive, because both matrices of the pencil are positive definite. The following facts about polynomials in μ with rational coefficients are established by exact root counting (Sturm sequences; script audit_K4.py): (i) (0,μ)=μ(1−μ)q0(μ)P(0;μ)=μ(1-μ)q_0(μ) with degq0=4 q_0=4 and q0<0q_0<0 on [0,1][0,1]; (i) (710,μ)P( 710;μ), of degree 66, has no root in [0,1][0,1] and equals 51500>0 51500>0 at μ=12μ= 12; (i) (ϱ0(μ),μ)=[μ(1−μ)]2q1(μ)P( _0(μ);μ)=[μ(1-μ)]^2q_1(μ) with degq1=14 q_1=14 and q1<0q_1<0 on [0,1][0,1]. Moreover the three roots sum to 52+3μ(1−μ)≥52>3⋅710 52+3μ(1-μ)≥ 52>3· 710, so they are not all below 710 710. With the positive leading coefficient, (i) and (i) give exactly one root in (0,710)(0, 710) and two above; it is ϱ(μ) (μ). Since ϱ0≤3764<710 _0≤ 3764< 710 and <0P<0 on (0,ϱ(μ))(0, (μ)), (i) gives ϱ0(μ)<ϱ(μ) _0(μ)< (μ). ∎ The bound (S28) is nearly tight (minμ[ϱ−ϱ0]≈8×10−7 _μ[ - _0]≈ 8× 10^-7), which is why K=4K=4 needs it. Theorem S26. For K=4K=4 and N≥10N≥ 10, N,4⪰0 B_N,4 0 with kernel span(⊗)span(1 1). Proof. With x=μkνkx= _k _k and (S28), the left-hand side of (T) exceeds (N−2)(2x+5x3)−2Nx+12=5(N−2)x3−4x+12=:fN(x)(N-2)(2x+5x^3)-2Nx+ 12=5(N-2)x^3-4x+ 12=:f_N(x); the linear terms collapse to −4x-4x independently of N. On x>0x>0, fNf_N is minimised at x∗2=415(N−2)x_*^2= 415(N-2), where fN(x∗)=12−83x∗f_N(x_*)= 12- 83x_*. This is nonnegative iff x∗≤316x_*≤ 316, i.e. iff N−2≥1024135=7.58…N-2≥ 1024135=7.58…, i.e. iff N≥10N≥ 10. So (T) holds strictly for every bond, and Proposition S23 applies. ∎ The threshold is sharp for this route: for 3≤N≤93≤ N≤ 9 condition (T) with the exact ϱ fails at some bond. Those cases, and the finitely many cases left open by Theorem S24, are settled by exact certificates in Step S-A.13. S-A.12 Step 11: the branch K=3K=3 (computer-assisted) For K=3K=3 the lower bound Sq≥SqlowSq _ low is too lossy: the form (S26) is indefinite, with smallest eigenvalue of order −N/10-N/10. The part X=Hk−HlX=H_k-H_l discarded in Proposition S19 must be kept, and we work with the bond form itself. The argument has a fixed architecture, proved by hand, and a small number of closed forms for matrix entries which are polynomial identities in (N,i,j)(N,i,j) established by computer algebra. Representation. The level space is ℝ3R^3 with basis 1, t1=(−1,0,1)t_1=(-1,0,1), t2=(1,−2,1)t_2=(1,-2,1). Write Dk=uk+skt1+qkt2D_k=u_k1+s_kt_1+q_kt_2 and order the coordinates as (u,s,q)∈(ℝN−1)3(u,s,q)∈(R^N-1)^3, bonds being indexed by i=k+1∈1,…,N−1i=k+1∈\1,…,N-1\. Let T be the matrix of the quadratic form K2N,3K^2 B_N,3 in these coordinates, with blocks Tuu,Tur,TrrT_u,T_ur,T_r, r=(s,q)r=(s,q). Lemma S27 (block representation). Index bonds by k,l∈0,…,N−2k,l∈\0,…,N-2\ and write TklT_kl for the 3×33× 3 block of T coupling DkD_k and DlD_l in the basis (,t1,t2)(1,t_1,t_2). Then Tkl=12(Xkl+Xlk⊤)T_kl= 12(X_kl+X_lk ) with Xkl= X_kl= N−22Nδkl[(N−1−k)PH+(k+1)PG]+[k≤l]CHG N-22N\, _kl [(N-1-k)P_H+(k+1)P_G ]+1[k≤ l]\,C_HG −1N[(N−1−max(k,l))CHH+(min(k,l)+1)CGG+2max(l−k,0)CHG], - 1N [(N-1- (k,l))C_H+( (k,l)+1)C_G+2 (l-k,0)\,C_HG ], where the six rational 3×33× 3 matrices are 36PH=(24−12−12−1216−24−12−2496)36P_H= ( smallmatrix24&-12&-12\\ -12&16&-24\\ -12&-24&96 smallmatrix ), 36PG=(2412−12121624−122496)36P_G= ( smallmatrix24&12&-12\\ 12&16&24\\ -12&24&96 smallmatrix ), 36CHG=(2412−12−12−412−12−12−12)36C_HG= ( smallmatrix24&12&-12\\ -12&-4&12\\ -12&-12&-12 smallmatrix ), 36CHH=(24−12−12−1280−12024)36C_H= ( smallmatrix24&-12&-12\\ -12&8&0\\ -12&0&24 smallmatrix ), 36CGG=(2412−121280−12024)36C_G= ( smallmatrix24&12&-12\\ 12&8&0\\ -12&0&24 smallmatrix ): the Gram matrices, in the level basis, of P(J0⋅)P(J_0\,·), P(J1⋅)P(J_1\,·), Cov(J0⋅,J1⋅)Cov(J_0\,·,J_1\,·), Cov(J0⋅,J0⋅)Cov(J_0\,·,J_0\,·) and Cov(J1⋅,J1⋅)Cov(J_1\,·,J_1\,·) at K=3K=3. Proof. The three terms of (S11) contribute, respectively, the diagonal blocks N−22N[(N−1−k)PH+(k+1)PG] N-22N[(N-1-k)P_H+(k+1)P_G] (term I), the block CHGC_HG for k≤lk≤ l (term I), and, from −1N∑jVar(Ωj)- 1N _jVar( _j) with Ωj=∑k<jHk+∑k≥jGk _j= _k<jH_k+ _k≥ jG_k, the count of indices j for which the pair (Dk,Dl)(D_k,D_l) enters Ωj _j as (H,H)(H,H), (G,G)(G,G) or mixed: N−1−max(k,l)N-1- (k,l), min(k,l)+1 (k,l)+1 and |l−k||l-k| respectively. This gives the form of XklX_kl; what remains is to confirm the five level matrices and the bookkeeping. To this end note that, on each of the regions k<l\k<l\, k=l\k=l\, k>l\k>l\, every entry of NTklN\,T_kl is a polynomial of total degree at most 22 in (N,k,l)(N,k,l), and so is every entry of N times the Gram matrix of K2N,3K^2 B_N,3 computed directly from (S11) by polarisation. The difference vanishes identically for N=3,…,12N=3,…,12 (exact rational arithmetic, k3_representation_proof.py). A polynomial of total degree at most 22 in three variables that vanishes on the 3×3×33× 3× 3 grid N∈7,8,9N∈\7,8,9\, k∈0,1,2k∈\0,1,2\, l−k∈1,2,3l-k∈\1,2,3\ (which lies inside the region k<l\k<l\ for every such N) is zero, and likewise on the 3×33× 3 grid in (N,k)(N,k) for the diagonal region; the region k>l\k>l\ follows by symmetry. Hence the identity holds for all N≥3N≥ 3. ∎ Lemma S28 (constant mode). Tuu=13[(N−1)I−⊤]T_u= 13 [(N-1)I-11 ] and ⊤Tur=01 T_ur=0. Consequently N,3⪰0 B_N,3 0 iff S:=Trr−Tur⊤Tuu+Tur⪰0S:=T_r-T_ur T_u^+T_ur 0, where Tuu+=3N−1(I−1N−1⊤)T_u^+= 3N-1 (I- 1N-111 ); and if S≻0S 0 then kerN,3=span(⊗) B_N,3=span(1 1). Proof. For Dk=ukD_k=u_k1 one has Hk(a)=ukaH_k(a)=u_ka, Gk(a)=uk(a+1)G_k(a)=u_k(a+1), so Zk=0Z_k=0, Var(Dk)=0Var(D_k)=0 and Sq=12N∑k<lN(uk−ul)2Var(a)Sq= 12N _k<lN(u_k-u_l)^2Var(a) with Var(a)=23Var(a)= 23; this is 13[(N−1)‖u‖2−(⊤u)2] 13[(N-1)\|u\|^2-(1 u)^2]. The direction u=u=1 is ⊗∈ker1 1∈ B (proof of Proposition S23), hence is not coupled to r. The rest is the Schur complement on ⟂1 . ∎ Lemma S29 (closed forms; computer algebra). Let vi=i(N−i)v_i=i(N-i) and Γij=min(i,j)(N−max(i,j))/N _ij= (i,j)(N- (i,j))/N. Then S=(ABB⊤C)S= pmatrixA&B\\ B &C pmatrix with Aij=δij[5N−936+(N−2)2vi3N2(N−1)]−43NΓij−136,Bij=(2i−N)(5N−12)12Ni=j14−j3Ni<j112−j3Ni>j,Cij=5N−144i=j−34+|i−j|Ni≠j.A_ij= _ij [ 5N-936+ (N-2)^2v_i3N^2(N-1) ]- 43N _ij- 136, B_ij= cases (2i-N)(5N-12)12N&i=j\\[2.0pt] 14- j3N&i<j\\[2.0pt] 112- j3N&i>j cases, C_ij= cases 5N-144&i=j\\[2.0pt] - 34+ |i-j|N&i≠ j. cases Status. By Lemma S27 every entry of T is, on each of the regions i<j\i<j\, i=j\i=j\, i>j\i>j\, a polynomial of degree at most 22 in (N,i,j)(N,i,j); the Schur complement of Lemma S28 introduces only the sums ∑kTur(k,⋅)Tur(k,⋅) _kT_ur(k,·)T_ur(k,·), which s1_closed.py evaluates in closed form region by region by symbolic summation, giving the formulas above as polynomial identities in (N,i,j)(N,i,j). The shear. Put R=diag(2i−N3N)R=diag ( 2i-N3N ), γ:=N−42γ:= N-42 and Θ=(I−R0I) = pmatrixI&-R\\ 0&I pmatrix. Then ΘSΘ⊤=(00C−γI)+1γ(B~γI)(B~γI)⊤,A~=A−BR−RB⊤+RCR,B~=B−RC,:=A~−1γB~B~⊤, S = pmatrix M&0\\ 0&C-γ I pmatrix+ 1γ pmatrix B\\ γ I pmatrix pmatrix B\\ γ I pmatrix^\! , A=A-BR-RB +RCR, B=B-RC, M:= A- 1γ B B , (S29) which is an algebraic identity (complete the square against γIγ I). The shear is a congruence, not a rotation: it removes the O(N)O(N) diagonal of B, which couples sks_k and qkq_k on the same bond; B~ii=2i−N6N=O(1) B_i= 2i-N6N=O(1). Since γ>0γ>0 for N≥5N≥ 5, S≻0S 0 follows from ≻0 M 0 and C−γI⪰0C-γ I 0. Lemma S30 (C−γI⪰0C-γ I 0; by hand). Let zi=12−iNz_i= 12- iN and H:=2NΓ+2zz⊤+14⊤⪰0H:= 2N +2z + 1411 0. Then C=5N−114I−HC= 5N-114I-H and trH=3(N−1)4=5N−114−γtrH= 3(N-1)4= 5N-114-γ. Consequently, for any factorisation H=∑rhrhr⊤H= _rh_rh_r , C−γI=(trH)I−H=∑r(‖hr‖2I−hrhr⊤)⪰0.C-γ I=(trH)I-H= _r (\|h_r\|^2I-h_rh_r ) 0. Proof. Off the diagonal, 2NΓij=2min(i,j)N−2ijN2 2N _ij= 2 (i,j)N- 2ijN^2 and 2zizj=12−i+jN+2ijN22z_iz_j= 12- i+jN+ 2ijN^2, so Hij=34−|i−j|N=5N−114δij−CijH_ij= 34- |i-j|N= 5N-114 _ij-C_ij. On the diagonal the dependence on i cancels: Hii=2i(N−i)N2+2(12−iN)2+14=34H_i= 2i(N-i)N^2+2( 12- iN)^2+ 14= 34. Each matrix ‖h‖2I−hh⊤\|h\|^2I-h has eigenvalues ‖h‖2\|h\|^2 and 00. ∎ Lemma S31 ( M is a Z-matrix with positive Perron certificate; computer algebra). For i<ji<j, with d=j−id=j-i, ij=−i(N−j)54N4(N−4)[N2(73N−423)+8N2(N−i)+16d(N2−d2)+4N(i2+j2)+72Ni+120N(N−j)+48ij+8N+16d], M_ij=- i(N-j)54N^4(N-4) [N^2(73N-423)+8N^2(N-i)+16d(N^2-d^2)+4N(i^2+j^2)+72Ni+120N(N-j)+48ij+8N+16d ], which is negative for N≥6N≥ 6. Moreover ()i ( M1)_i =viℛ(N,vi)540N4(N−4)(N−1), = v_i\,R(N,v_i)540N^4(N-4)(N-1), ℛ =53N5−1338N4+5805N3−5380N2+384N−64 =53N^5-1338N^4+5805N^3-5380N^2+384N-64 +2(N−1)(29N2−120N+40)v+16(N−1)v2, +2(N-1)(29N^2-120N+40)v+16(N-1)v^2, and the substitution N=m+19N=m+19, v=u+(N−1)v=u+(N-1) turns ℛR into a polynomial in (m,u)(m,u) all of whose coefficients are positive (constant term 171000171000). Since vi−(N−1)=(i−1)(N−1−i)≥0v_i-(N-1)=(i-1)(N-1-i)≥ 0, >0 M1>0 for every N≥19N≥ 19. For N=17,18N=17,18 the vector wi=1+(2i−N)220N2w_i=1+ (2i-N)^220N^2 satisfies, in exact rational arithmetic, mini(w)i=78394731107488460 _i( Mw)_i= 78394731107488460 and 5964113446410440 5964113446410440 respectively. Status. The two entry formulas are polynomial identities proved by region-by-region symbolic summation (verify_M_symbolic.py, whose hard-coded block entries are checked against s1_closed.py by check_blocks.py); the positivity of the shifted coefficients and the two certificates for N=17,18N=17,18 are exact computations in the same script. Theorem S32. For K=3K=3 and every N≥17N≥ 17, N,3⪰0 B_N,3 0 with kernel span(⊗)span(1 1). Proof. By Lemma S31 and Lemma S22 (with v=v=1 or v=wv=w), ≻0 M 0; by Lemma S30, C−γI⪰0C-γ I 0. In (S29), if s≠0s≠ 0 the first block is positive on (s,q)(s,q), and if s=0s=0, q≠0q≠ 0 the square term contributes γ‖q‖2>0γ\|q\|^2>0. Hence ΘSΘ⊤≻0 S 0, so S≻0S 0, and Lemma S28 concludes. ∎ The threshold is genuine for the bond form: λmin(S) _ (S) is −0.003-0.003 at N=16N=16 and +0.018+0.018 at N=17N=17. For N≤16N≤ 16 the bond form is indefinite on the full bond space, but it is positive on the image D^(f)\ D(f)\, which is all that Theorem 2 needs; see Step S-A.13. The natural continuum limit of N,3 B_N,3 has spectral infimum 00, so positivity at K=3K=3 is a finite-N effect, which is why no continuum or Gershgorin shortcut is available. S-A.13 Step 12: the finitely many remaining cases (exact certificates) TABLE S2: Coverage of (K,N):K≥3,N≥3\(K,N):K≥ 3,\ N≥ 3\. No pair is left uncovered. The first column is analytic for K≥5K≥ 5 (Theorem S24) and rests on exact symbolic computation for K=4K=4 and K=3K=3 (Steps S-A.11–S-A.12). K all large N bond-space certificate image certificate direct 33 N≥17N≥ 17 (Thm. S32) – 4≤N≤164≤ N≤ 16 N=3N=3 44 N≥10N≥ 10 (Thm. S26) 5≤N≤95≤ N≤ 9 N=3,4N=3,4 – 55 N≥6N≥ 6 (Thm. S24) N=4,5N=4,5 N=3N=3 – 6,76,7 N≥4N≥ 4 (Thm. S24) N=3N=3 – – ≥8≥ 8 N≥3N≥ 3 (Thm. S24) – – – Theorems S24, S26 and S32 leave the pairs (K,N)(K,N) listed in Table S2. They are settled by two certificates, both in exact rational arithmetic (Python fractions), with no floating point. Bond-space certificate (full_space_certificate.py). The Gram matrix G of N,K B_N,K on ℝ(N−1)KR^(N-1)K is built by polarisation; the script checks that (⊗)=0 G(1 1)=0 exactly and that +(⊗)(⊗)⊤ G+(1 1)(1 1) has an LDL⊤LDL factorisation with positive pivots. This is equivalent to N,K⪰0 B_N,K 0 with kernel span(⊗)span(1 1). Image certificate (bond_certificate.py). Where the bond form is indefinite on the full space, the (N−1)×(N−1)(N-1)×(N-1) matrix of f↦N,K(D^(f))=QK(1)+QK(2)f B_N,K( D(f))=Q^(1)_K+Q^(2)_K is built exactly from (S10). It annihilates the linear payoff (Lemma 2 and Proposition S5), and the script certifies that it is positive definite on the complement spanned by the second differences ek−2ek+1+ek+2e_k-2e_k+1+e_k+2, k=0,…,N−3k=0,…,N-3, in payoff coordinates, again by LDL⊤LDL with positive pivots. Stored certificates and an independent checker. For each of the 2525 pairs the certified matrix A (that is, +(⊗)(⊗)⊤ G+(1 1)(1 1) , respectively the image matrix), the unit lower-triangular factor L and the pivots D are stored as exact rationals in certificates/*.json (make_certificates.py). The checker verify_certificates.py uses only the Python standard library and imports none of the code that produced them. For every file it (1) rebuilds A from (S11) and (S10) (an implementation independent of the generator, which evaluates the bond form through the bridge kernels of the original proof notes) and checks equality entry by entry; (2) checks that L is unit lower triangular, that all pivots are positive and that ⊤= L D L = A exactly; and (3) checks the kernel statement ((⊗)=0 G(1 1)=0, respectively that the bond vectors of the linear payoff pair to zero with everything). A reader who trusts rational arithmetic therefore needs to trust only this one short file, together with Proposition S10 (which identifies N,K(D^(f)) B_N,K( D(f)) with QK(1)+QK(2)Q^(1)_K+Q^(2)_K) and Proposition S5. All 2525 certificates pass. The pair (K,N)=(3,3)(K,N)=(3,3). Here the two-body form vanishes identically on the image, and we compute Q3Q_3 itself in exact arithmetic from the definition: Q3=83(1−1−11)Q_3= 83 pmatrix1&-1\\ -1&1 pmatrix, positive semidefinite with kernel span()span(1) (generalised eigenvalue 32/336=827 32/336= 827). The bond-space certificate fails at (K,N)=(3,3),(4,3),(4,4),(5,3)(K,N)=(3,3),(4,3),(4,4),(5,3), as full_space_certificate.py reports. This is expected: there the bond form is indefinite (or degenerate) on the full space, and the pairs are covered by the image certificate or, for (3,3)(3,3), directly (end_to_end.py). S-A.14 Step 13: assembly, the case N=2N=2, and K=2K=2 Everything is now in place: the orders m≥3m≥ 3 (Step S-A.3), the two-body form on the bond space (Steps S-A.4–S-A.13), and the kernel (Steps S-A.2 and S-A.6). The case N=2N=2 is trivial. Proof of Theorem 2. Let K≥3K≥ 3. N=2N=2. The increment space is one-dimensional and spanned by 1, so QK=0Q_K=0 by Lemma 2, and the statement is trivial (every nonconstant f is affine). N≥3N≥ 3. By Lemma S3, F⊤QKF=KN+1∑m≥1QK(m)(f)F Q_KF=K^N+1 _m≥ 1Q^(m)_K(f); by Proposition S5, QK(m)⪰0Q^(m)_K 0 for m≥3m≥ 3; by Proposition S10, QK(1)+QK(2)=N,K(D^(f))Q^(1)_K+Q^(2)_K= B_N,K( D(f)). By Theorems S24, S26, S32 and Table S2, this is nonnegative for every f, so QK⪰0Q_K 0. For the kernel, ∈kerQK1∈ Q_K by Lemma 2. Conversely let F⊤QKF=0F Q_KF=0. Every summand vanishes, in particular N,K(D^(f))=0 B_N,K( D(f))=0. In the large-N and bond-space cases kerN,K=span(⊗) B_N,K=span(1 1), so f is affine by Lemma S11; in the image-certified cases the form is positive definite on a complement of the linear payoff, so f is affine; and (3,3)(3,3) was computed directly. ∎ Where K≥3K≥ 3 enters, and the counterexample at K=2K=2. Proposition S10 and Theorem S14 hold for K=2K=2. The hypothesis enters through Lemma S16 alone: its margin (K−2)μνVar(D)(K-2)μ (D) vanishes identically for binary priorities, and every later step inherits this. Correspondingly the statement is false at K=2K=2: building Q2Q_2 at N=7N=7 from the definition in exact rational arithmetic (k2_witness.py), the strictly decreasing payoff with increments F=(9,8,6,4,2,1)F=(9,8,6,4,2,1) gives F⊤Q2F=−7237480<0,ρpot=5345427762354880>67,F Q_2F=- 7237480<0, _pot= 5345427762354880> 67, and the reversed increments give the same values. For N≤6N≤ 6, Q2⪰0Q_2 0. Table S3 records the margin N−1N−maxfρpot(f) N-1N- _f _pot(f) over non-affine f for small (K,N)(K,N): it is positive throughout K≥3K≥ 3, in agreement with Theorem 2, turns negative at K=2K=2 exactly at N=7N=7, and grows in magnitude with N. TABLE S3: Margin N−1N−maxfρpot(f) N-1N- _f _pot(f) over non-affine payoff-from-rank functions f on KNK_N, computed in floating point from the pencil (Mpot,Mtot)(M_pot,M_tot) (not part of any proof). When positive it is the gap λ1−λ2 _1- _2 between the two largest generalised eigenvalues, and as K→∞K→∞ it tends to λ1−λ2=2N+2 _1- _2= 2N+2 of Theorem 1. The entries at K=2K=2, N≥7N≥ 7 (bold) are negative, in agreement with the exact witness at N=7N=7; blank cells were not computed. N K=2K=2 K=3K=3 K=4K=4 K=5K=5 K=6K=6 K=7K=7 K=8K=8 3 +.167+.167 +.296+.296 +.342+.342 +.363+.363 +.374+.374 +.381+.381 +.385+.385 4 +.075+.075 +.216+.216 +.267+.267 +.291+.291 +.304+.304 +.311+.311 +.317+.317 5 +.033+.033 +.169+.169 +.219+.219 +.243+.243 +.256+.256 +.264+.264 +.269+.269 6 +.011+.011 +.138+.138 +.186+.186 +.209+.209 +.221+.221 +.229+.229 +.234+.234 7 −.001-.001 +.116+.116 +.161+.161 +.183+.183 +.195+.195 +.202+.202 +.207+.207 8 −.009-.009 +.100+.100 +.142+.142 +.163+.163 +.174+.174 +.181+.181 +.185+.185 9 −.014-.014 +.088+.088 +.127+.127 +.147+.147 +.157+.157 10 −.017-.017 +.078+.078 +.115+.115 11 −.019-.019 +.070+.070 12 −.020-.020 +.063+.063 S-A.15 The reciprocity law: proof of Proposition 7 This subsection is independent of Steps S-A.1–S-A.14. It does not give another proof of Theorem 2; it describes exactly how the finite-K forms deviate from the continuum ones. Setting. C=([N],E)C=([N],E) is a conflict graph and the protocol is given by a total order ≺ of the agents: j is served before its neighbour i iff sj>sis_j>s_i, or sj=sis_j=s_i and j≺ij i; ≺rev rev is the reversed order. For S⊆[N]S [N] put AS=∏j∈SAjA_S= _j∈ SA_j. With ui=f(posC(i,⋅))u_i=f(pos_C(i,·)), gi=(I−Ai)uig_i=(I-A_i)u_i and g=∑igig= _ig_i, the forms tot(f,f′)=∑i[gifgif′],pot(f,f′)=[gf−1gf′] m_ tot(f,f )= _iE [g_i^fg_i^f ], m_pot(f,f )=E [g^f\, N^-1g^f ] are K−(N+1)K^-(N+1) times the two energy forms, and K−(N+1)QK=N−1Ntot−potK^-(N+1)Q_K= N-1N m_ tot- m_pot on KNK_N. We write ≺(K) m (K) for either of them, as a function of K. Theorem (Proposition 7 of the main text). (a) ≺(K) m (K) is a polynomial in 1/K1/K of degree at most 2N−12N-1; its constant term is the corresponding form of the game with continuous priorities on C. (b) As polynomials in x=1/Kx=1/K, ≺(−x)=≺rev(x) m (-x)= m rev(x). (c) If C has an automorphism π with π(j)≺π(i)π(j) π(i) whenever i,j∈E\i,j\∈ E and i≺ji j, then ≺(K) m (K) is a polynomial in K−2K^-2 of degree at most N−1N-1 that vanishes at K=1K=1. (1) Reduction to counting. Let Uir=[posC(i,⋅)=r]U^r_i=1[pos_C(i,·)=r], so that ui=∑rf(r)Uiru_i= _rf(r)U_i^r. The AjA_j are commuting orthogonal projections, so ∏j∈T(I−Aj)∏j∉TAj=∑S⊇Tc(−1)|S∖Tc|AS _j∈ T(I-A_j) _j∉ TA_j= _S T^c(-1)^|S T^c|A_S, and −1 N^-1 acts on the range of this projection as 1/|T|1/|T| (Lemma 1). Hence both forms are linear combinations, with coefficients that do not depend on K, of F(K)=[Uir⋅ASUkr′],i,k∈[N],S⊆[N]F(K)=E [U_i^r· A_SU_k^r ], i,k∈[N],\ S [N] (the projections I−AiI-A_i in gig_i are absorbed into S, since ASAi=AS∪iA_SA_i=A_S∪\i\). Now ASUkr′(s)=′[Ukr′(s′)]A_SU_k^r (s)=E [U_k^r (s )], where sj′=sjs _j=s_j for j∉Sj∉ S and the sj′s _j, j∈Sj∈ S, are fresh independent uniform levels. With n=N+|S|≤2Nn=N+|S|≤ 2N, F(K)=K−n#(s,sS′)∈0,…,K−1n:pos(i,s)=r,pos(k,s′)=r′.F(K)=K^-n\,\# \(s,s _S)∈\0,…,K-1\^n:\ pos(i,s)=r,\ pos(k,s )=r \. (S30) (2) Lexicographic cells. Order the n coordinates by the agent they belong to, according to ≺ , and put sjs_j before its twin sj′s _j; call this total order of the coordinates ≺ again. The rule for twins is an auxiliary choice: any other rule gives another decomposition of the same set into cells, and (S31) below holds for each of them. Every point of the cube induces a strict total order σ of the coordinates: larger level first, ties broken by ≺ . The set in (S30) is a union of such cells: pos(i,s)pos(i,s) is the number of neighbours of i that precede sis_i in σ, and likewise for pos(k,s′)pos(k,s ); only coordinates of different agents are ever compared, so the convention for twins is immaterial. Let O be the set of orders σ for which the two conditions hold. It is defined without reference to the protocol. Lemma S33 (cells). For a strict order σ=(σ1,…,σn)σ=( _1,…, _n) let des≺(σ)=#t:σt+1≺σtdes_ (σ)=\#\t: _t+1 _t\. The number of points of 0,…,K−1n\0,…,K-1\^n whose lexicographic order is σ equals (K+n−1−des≺(σ)n) K+n-1-des_ (σ)n. Proof. The levels vtv_t of σt _t satisfy v1≥⋯≥vnv_1≥…≥ v_n, with vt>vt+1v_t>v_t+1 required exactly when σt+1≺σt _t+1 _t (a tie would have put σt+1 _t+1 first). Subtracting from vtv_t the number of forced strict steps at positions ≥t≥ t gives a bijection with the weakly decreasing sequences in 0,…,K−1−des≺(σ)\0,…,K-1-des_ (σ)\, of which there are (K−des≺(σ)+n−1n) K-des_ (σ)+n-1n. ∎ Therefore F≺(K)=K−n∑σ∈(K+n−1−des≺(σ)n).F (K)=K^-n _σ K+n-1-des_ (σ)n. (S31) Each binomial is a polynomial of degree n in K divisible by K, since 0≤n−1−des≤n−10≤ n-1-des≤ n-1. Hence F is a polynomial in 1/K1/K of degree at most n−1≤2N−1n-1≤ 2N-1, with constant term ||/n!|O|/n!, which is the probability that n i.i.d. continuous priorities fall in O, i.e. the value of F in the continuum game. This proves (a). (3) Reciprocity. For integers n≥1n≥ 1 and d, (−K+n−1−dn)=(−1)n(K+dn) -K+n-1-dn=(-1)^n K+dn as polynomials in K. Let ≺rev rev denote the reversal of the whole order of the n coordinates, twins included. Since ≺ is a total order, every adjacent pair of σ is either a descent or an ascent, so n−1−des≺(σ)=des≺rev(σ)n-1-des_ (σ)=des_ rev(σ). (Reversing the order of the agents but not that of the twins would break this identity whenever two twins are adjacent in σ.) On coordinates of different agents ≺rev rev is the reversed protocol, and by (2) its rule for twins is immaterial, so the right-hand side of (S31) written for ≺rev rev is F for the game with the reversed protocol. Substituting in (S31), F≺(−K)=(−K)−n(−1)n∑σ∈(K+des≺(σ)n)=K−n∑σ∈(K+n−1−des≺rev(σ)n)=F≺rev(K),F (-K)=(-K)^-n(-1)^n _σ K+des_ (σ)n=K^-n _σ K+n-1-des_ rev(σ)n=F rev(K), because O is the same set for both protocols. This is an instance of the reciprocity theorem for P-partitions [45, Sec. 3.15.3], equivalently of Ehrhart–Macdonald reciprocity for order polytopes: changing the sign of the dilation exchanges weak and strict inequalities, that is, it exchanges who wins a tie. Summing with the K-independent coefficients of (1) gives (b). (4) Symmetric protocols. If π is as in (c), relabelling the agents by π maps the game with protocol ≺rev rev onto the game with protocol ≺ on the same graph, and the forms, being sums over all agents, are invariant. Hence ≺rev=≺ m rev= m , and (b) shows that ≺ m is even in 1/K1/K; being of degree at most 2N−12N-1, it has degree at most 2N−22N-2. At K=1K=1 every gig_i vanishes, and (S31) holds for every integer K≥1K≥ 1, so the polynomial vanishes at K=1K=1. On KNK_N the reversal i↦N−1−i N-1-i is such an automorphism, which gives (26). ∎ What the expansion does not say. It concerns the forms, not their eigenvalues. At finite K the order forms do not commute, and the generalised eigenvalues are algebraic functions of y=K−2y=K^-2, not polynomials. Because the continuum eigenvalues λp _p are simple (Theorem 1), analytic perturbation theory applies at y=0y=0: the finite-K eigenvalues are analytic in y near 00, hence equal to λp+O(K−2) _p+O(K^-2), with no term of order K−1K^-1. Whether the coefficient of K−2K^-2 is non-zero for a given p is not claimed. Remark S34 (tests of the reciprocity law). By (b), when no automorphism reverses the protocol the expansions for ≺ and ≺rev rev have equal even and opposite odd coefficients: on the star with a favoured centre (N=4N=4, prize table (0,−1,−3,−4.5)(0,-1,-3,-4.5)) the term −0.5625K−1-0.5625K^-1 of pot m_pot changes sign under relabelling while 2.1043−2.5085K−22.1043-2.5085K^-2 does not, and the same holds on the path 00–11–33–22 and the paw graph to 10−710^-7; on K4K_4 the odd coefficients vanish. Remark S35 (the critical number of levels; what survives at finite K). The numbers K⋆(N)K (N) quoted after Proposition 7 of the main text are obtained by interpolating exact enumerations of QKQ_K in y=K−2y=K^-2 (for N≤6N≤ 6 the fit is over-determined, with residuals below 10−1310^-13). The same computations show, for N≤7N≤ 7, that in the discrete Chebyshev basis all forms split by the parity of the degree, that the order-m potential form vanishes on polynomials of degree less than m, and that the order forms do not commute at finite K: the order-by-order diagonalisation of Theorem 1 is a property of the continuum. The proof above requires a total-order protocol; orientations with directed cycles and randomised tie-breaking are not covered. Appendix S-B The harmonic component of the linear game: tie-pattern law and randomised tie-breaking Throughout this appendix the game is the linear rank-order game (C,K)G(C,K) on an arbitrary conflict graph C. By Proposition 3, along an edge (i,s→s′)(i,s→ s ) the harmonic flow is h(i,s→s′)=W−diℓ/Kh(i,s→ s )=W-d_i /K, the deviation of the number W of overtaken neighbours from its mean under uniformly random levels. Define the harmonic energy of a profile as H(s):=∑i∑si′>sih(i,s→s′)2,H(s):= _i _s _i>s_ih(i,s→ s )^2, so that ℰharm=∑sH(s)=KN[H]E_harm= _sH(s)=K^N\,E[H]: H assigns the energy of every edge of the game graph to its lower endpoint. H(s)H(s) depends on this convention and on the tie-breaking order, and should not be read as a dissipation at s. Let H¯ H be the average of H over a uniformly random total order of the agents used as the protocol. The deviation h is produced most strongly by ties, and the following theorem makes this exact: the conditional expectation of H¯ H given the tie pattern is an explicit linear function of the pattern. Theorem S36 (tie-pattern law). Let τjk=[sj=sk] _jk=1[s_j=s_k], ejk=[j,k∈E]e_jk=1[\j,k\∈ E], cjk=|(j)∩(k)|c_jk=|N(j) (k)|, and let H¯i H_i be the harmonic energy of player i averaged over a uniformly random tie-breaking order, H¯=∑iH¯i H= _i H_i. Then for every player i [H¯i∣τ]=K−16∑j,k⊆(i)τjk+K−212∑j∈(i)τij+i,i=di[K2−K+1−(di−1)(K−1)]12K,E[ H_i τ]= K-16 _\j,k\ (i) _jk+ K-212 _j (i) _ij+ c_i, c_i= d_i [K^2-K+1-(d_i-1)(K-1) ]12K, (S32) and summing over i, [H¯∣τ]=C(s)+CE[ H τ]=A_C(s)+ c_C, where C(s)=∑j<k[K−26ejk+K−16cjk]τjkA_C(s)= _j<k [ K-26e_jk+ K-16c_jk ] _jk is the weighted tie count with C=∑i=K2−112K∑idi−1K[K−26|E|+K−16∑i(di2)] c_C= _i c_i= K^2-112K _id_i- 1K [ K-26|E|+ K-16 _i d_i2 ]. Proof. (0) Localisation. By Proposition 3, h(i,s→s′)=W−diℓ/Kh(i,s→ s )=W-d_i /K depends on s only through s|i∪(i)s|_\i\ (i) and on the tie-breaking order only through its restriction to i∪(i)\i\ (i). Conditionally on the global tie pattern τ, the levels of the tie classes are a uniformly random injective assignment of 0,…,K−1\0,…,K-1\ to the classes; its restriction to i∪(i)\i\ (i) is again a uniformly random injective assignment to the local classes. Hence [H¯i∣τ]E[ H_i τ] depends only on τ|i∪(i)τ|_\i\ (i). Fix i, write b=sib=s_i, and let iT_i be the tie class of i (the players at level b). (1) Two-body form. For a target level a>ba>b and a neighbour j let Ij=[b<sj<a]+[sj=b,j≺i]+[sj=a,i≺j]I_j=1[b<s_j<a]+1[s_j=b,\,j i]+1[s_j=a,\,i j], where ≺ is the random order, so that W=∑jIjW= _jI_j. For uniform sjs_j and uniform order, Ij=ℓ/K=:ℓ¯EI_j= /K=: with ℓ=a−b =a-b, hence diℓ/K=∑jℓ¯d_i /K= _j and H¯i=∑a>b≺[(∑j(Ij−ℓ¯))2]=∑a>b[∑j≺(Ij−ℓ¯)2+∑j≠k≺(Ij−ℓ¯)(Ik−ℓ¯)]. H_i= _a>bE_ [ ( _j(I_j- ) )^2 ]= _a>b [ _jE_ (I_j- )^2+ _j≠ kE_ (I_j- )(I_k- ) ]. The kernels do not depend on did_i. (2) Types. For fixed b, ≺[⋅]E_ [·] depends on sj,sks_j,s_k only through their position relative to (b,a)(b,a) and on the order of i,j,k\i,j,k\. Taking the expectation over the levels given the local pattern, the one-body term depends only on whether j∼ij i (constants ϖ1t _1^t, ϖ1f _1^f), and the two-body term only on the pattern type of i,j,k\i,j,k\: all distinct (ϖdist _ dist), j∼k≁ij k i (ϖpair _ pair), exactly one of j,kj,k tied with i (ϖone _ one; the two cases are equal by symmetry), all tied (ϖall _ all). (3) Counting. Let nt=|i∩(i)|n_t=|T_i (i)| be the number of neighbours tied with i and nc=∑≠i(|∩(i)|2)n_c= _T _i |T (i)|2, the sum running over the other tie classes T, the number of tied neighbour pairs not involving i. Ordered neighbour pairs (j,k)(j,k) fall into the types with multiplicities nt(nt−1)n_t(n_t-1) (all tied), 2nt(di−nt)2n_t(d_i-n_t) (one tied with i), 2nc2n_c (j∼k≁ij k i) and (di−nt)(di−nt−1)−2nc(d_i-n_t)(d_i-n_t-1)-2n_c (distinct). Hence [H¯i∣τ] [ H_i τ] =ntϖ1t+(di−nt)ϖ1f+nt(nt−1)ϖall+2nt(di−nt)ϖone =n_t _1^t+(d_i-n_t) _1^f+n_t(n_t-1) _ all+2n_t(d_i-n_t) _ one +2ncϖpair+[(di−nt)(di−nt−1)−2nc]ϖdist +2n_c\, _ pair+ [(d_i-n_t)(d_i-n_t-1)-2n_c ] _ dist =2(ϖpair−ϖdist)nc+2(ϖall−2ϖone+ϖdist)(nt2) =2( _ pair- _ dist)\,n_c+2( _ all-2 _ one+ _ dist) n_t2 +[ϖ1t−ϖ1f+2(di−1)(ϖone−ϖdist)]nt + [ _1^t- _1^f+2(d_i-1)( _ one- _ dist) ]\,n_t +diϖ1f+di(di−1)ϖdist. +d_i _1^f+d_i(d_i-1) _ dist. Note ∑j,k⊆(i)τjk=(nt2)+nc _\j,k\ (i) _jk= n_t2+n_c and ∑j∈(i)τij=nt _j (i) _ij=n_t. (4) The six constants. Fix b=sib=s_i, a target a>ba>b, ℓ=a−b =a-b and ℓ¯=ℓ/K = /K. Given the levels of j and k, the random order enters only through the events j≺ij i, i≺ki k, etc., whose probabilities are 12 12 for one comparison and 13 13 (same side) or 16 16 (opposite sides) for two comparisons involving i. Hence, with Ij=[b<sj<a]+[sj=b][j≺i]+[sj=a][i≺j]I_j=1[b<s_j<a]+1[s_j=b]1[j i]+1[s_j=a]1[i j], ≺(Ij−ℓ¯)2=(1−ℓ¯)2b<sj<a,12[(1−ℓ¯)2+ℓ¯2]sj∈b,a,ℓ¯2otherwise,E_ (I_j- )^2= cases(1- )^2&b<s_j<a,\\[2.0pt] 12 [(1- )^2+ ^2 ]&s_j∈\b,a\,\\[2.0pt] ^2&otherwise, cases and, writing ϵj∈1,12,0 _j∈\1, 12,0\ for ≺IjE_ I_j according to whether sjs_j lies strictly between, on b,a\b,a\, or elsewhere, ≺(Ij−ℓ¯)(Ik−ℓ¯)=(ϵj−ℓ¯)(ϵk−ℓ¯)+ζjk,ζjk=112sj=sk∈b,a,−112sj,sk=b,a,0otherwiseE_ (I_j- )(I_k- )=( _j- )( _k- )+ _jk, _jk= cases 112&s_j=s_k∈\b,a\,\\[2.0pt] - 112&\s_j,s_k\=\b,a\,\\[2.0pt] 0&otherwise cases (the correction ζ is the covariance of the two order indicators: 13−14 13- 14 when both compare with i from the same side, 16−14 16- 14 from opposite sides). Each constant is now an average of these expressions over b uniform, a>ba>b, and the levels of the other players consistent with the type: a level tied with i is b; a free level is uniform on the K−1K-1 values ≠b≠ b; two free untied levels are uniform on distinct pairs. Counting the levels by region (b below, ℓ−1 -1 strictly between, one at a, K−1−aK-1-a above) turns each average into a polynomial in b,ℓb, which is then summed over 0≤b<a≤K−10≤ b<a≤ K-1; the summations are elementary (Faulhaber) and give ϖ1f=K2−K+112K,ϖ1t=(2K−1)(K−1)12K,ϖdist=ϖone=−K−112K,ϖpair=ϖall=(K−1)212K,so thatϖone−ϖdist=0,ϖpair−ϖdist=ϖall−ϖdist=K−112,ϖ1t−ϖ1f=K−212. gathered _1^f= K^2-K+112K, _1^t= (2K-1)(K-1)12K,\\[2.0pt] _ dist= _ one=- K-112K, _ pair= _ all= (K-1)^212K,\\[2.0pt] so that _ one- _ dist=0, _ pair- _ dist= _ all- _ dist= K-112, _1^t- _1^f= K-212. gathered (S33) For instance, ϖ1f _1^f is 1K(K−1)∑b<a[(ℓ−1)(1−ℓ¯)2+12((1−ℓ¯)2+ℓ¯2)+(K−1−ℓ)ℓ¯2] 1K(K-1) _b<a [( -1)(1- )^2+ 12 ((1- )^2+ ^2 )+(K-1- ) ^2 ] and ϖ1t=1K∑b<a12[(1−ℓ¯)2+ℓ¯2] _1^t= 1K _b<a 12 [(1- )^2+ ^2 ]. (4a) The four two-body constants, in full. The identity ϖone=ϖdist _ one= _ dist carries the theorem, so this part of the computation is given in detail. Fix b and a>ba>b, put ℓ=a−b =a-b, ℓ¯=ℓ/K = /K and δ:=12−ℓ¯δ:= 12- , and set, for a level v∈0,…,K−1v∈\0,…,K-1\, ϵ(v)=1,b<v<a,12,v∈b,a,0,otherwise,Φ(v):=ϵ(v)−ℓ¯,ε(v)= cases1,&b<v<a,\\[1.0pt] 12,&v∈\b,a\,\\[1.0pt] 0,&otherwise, cases (v):=ε(v)- , so that ≺Ij=ϵ(sj)E_ I_j=ε(s_j) and Φ(b)=Φ(a)=δ (b)= (a)=δ. Two identities drive the whole step. First, since ∑vϵ(v)=(ℓ−1)+12+12=ℓ=Kℓ¯ _vε(v)=( -1)+ 12+ 12= =K , ∑v=0K−1Φ(v)=0,hence∑v≠bΦ(v)=−δ. _v=0^K-1 (v)=0, _v≠ b (v)=-δ. (S34) Second, with E2:=∑vΦ(v)2=(ℓ−1)(1−ℓ¯)2+2δ2+(K−ℓ−1)ℓ¯2E_2:= _v (v)^2=( -1)(1- )^2+2δ^2+(K- -1) ^2, E2−Kδ2=2ℓ(K−ℓ)K−K+24.E_2-Kδ^2= 2 (K- )K- K+24. (S35) Indeed, writing m=K−ℓm=K- , (ℓ−1)m2+(m−1)ℓ2=Kℓm−(ℓ2+m2)=(K+2)ℓm−K2( -1)m^2+(m-1) ^2=K m-( ^2+m^2)=(K+2) m-K^2 and (K−2ℓ)2=(m−ℓ)2=K2−4ℓm(K-2 )^2=(m- )^2=K^2-4 m, so that K2(E2−Kδ2)=(K+2)ℓm−K2−K−24(K2−4ℓm)=2Kℓm−K+24K2K^2 (E_2-Kδ^2 )=(K+2) m-K^2- K-24 (K^2-4 m )=2K m- K+24K^2. Now average ≺(Ij−ℓ¯)(Ik−ℓ¯)=Φ(sj)Φ(sk)+ζjkE_ (I_j- )(I_k- )= (s_j) (s_k)+ _jk over the levels admitted by each type (a level tied with i equals b; a free level is uniform on the K−1K-1 values ≠b≠ b; two free untied levels are uniform on the ordered distinct pairs among those K−1K-1 values). The ζ term contributes +112+ 112 exactly when sj=sk∈b,as_j=s_k∈\b,a\ and −112- 112 exactly when sj,sk=b,a\s_j,s_k\=\b,a\; in the “one tied” type this requires sk=as_k=a, of probability 1K−1 1K-1, and in the “all distinct” type it cannot occur, since neither level equals b. Using (S34) and ∑v≠bΦ(v)2=E2−δ2 _v≠ b (v)^2=E_2-δ^2, all tied tied : : δ2+112, δ^2+ 112, (S36) exactly one tied with i one tied with i : : δK−1∑v≠bΦ(v)−112(K−1)=−δ2+112K−1, δK-1 _v≠ b (v)- 112(K-1)=- δ^2+ 112K-1, j∼k≁i j k i : : E2−δ2K−1+112(K−1), E_2-δ^2K-1+ 112(K-1), all distinct distinct : : (∑v≠bΦ(v))2−∑v≠bΦ(v)2(K−1)(K−2)=2δ2−E2(K−1)(K−2). ( _v≠ b (v) )^2- _v≠ b (v)^2(K-1)(K-2)= 2δ^2-E_2(K-1)(K-2). Each constant is 1K∑0≤b<a≤K−1 1K _0≤ b<a≤ K-1 of the corresponding line of (S36). Since δ=K−2ℓ2Kδ= K-2 2K and E2E_2 depend on (b,a)(b,a) only through ℓ , which occurs with multiplicity K−ℓK- , and since ∑ℓ=1K−1ℓ(K−ℓ)2=K2(K2−1)12 _ =1^K-1 (K- )^2= K^2(K^2-1)12, ∑b<a1=(K2),∑b<aδ2=(K−1)(K−2)24,∑b<a(E2−Kδ2)=K(K−1)(K−2)24, _b<a1= K2, _b<aδ^2= (K-1)(K-2)24, _b<a\! (E_2-Kδ^2 )= K(K-1)(K-2)24, (S37) the last by (S35), whence ∑b<aE2=K(K−1)(K−2)12 _b<aE_2= K(K-1)(K-2)12. Substituting (S37) into (S36) gives the four values in (S33). The two coincidences in (S33) are one identity, not two. Put Θ:=∑0≤b<a≤K−1[E2−Kδ2−K−212]. := _0≤ b<a≤ K-1 [E_2-Kδ^2- K-212 ]. Subtracting the lines of (S36) termwise, ϖone−ϖdist=ΘK(K−1)(K−2),ϖpair−ϖall=ΘK(K−1), _ one- _ dist= K(K-1)(K-2), _ pair- _ all= K(K-1), and by (S37), Θ=K(K−1)(K−2)24−K−212(K2)=0 = K(K-1)(K-2)24- K-212 K2=0. Equivalently, by (S35) the summand of Θ is 2ℓ(K−ℓ)K−K+13 2 (K- )K- K+13, and 2K∑ℓ(K−ℓ)2=K(K−1)(K+1)6=K+13(K2) 2K _ (K- )^2= K(K-1)(K+1)6= K+13 K2. Hence ϖone=ϖdist _ one= _ dist and ϖpair=ϖall _ pair= _ all: the first removes the did_i-dependence of the ntn_t coefficient in (3), the second merges (nt2) n_t2 and ncn_c into the single statistic ∑j,k⊆(i)τjk _\j,k\ (i) _jk. Substituting (S33) into (3): the coefficient of ncn_c is K−16 K-16; the coefficient of (nt2) n_t2 is K−16 K-16 as well, so the two combine into K−16∑j,k⊆(i)τjk K-16 _\j,k\ (i) _jk; the coefficient of ntn_t is ϖ1t−ϖ1f=K−212 _1^t- _1^f= K-212, since ϖone=ϖdist _ one= _ dist; the constant is i=diϖ1f+di(di−1)ϖdist c_i=d_i _1^f+d_i(d_i-1) _ dist. The identity ϖone=ϖdist _ one= _ dist is what removes the did_i-dependence of the ntn_t coefficient. This proves (S32). (5) Summation. A tied pair j,k\j,k\ contributes K−212 K-212 to H¯j H_j and to H¯k H_k if j,k∈E\j,k\∈ E, and K−16 K-16 to H¯i H_i for every common neighbour i. Finally [H¯i]=i+K−16(di2)1K+K−212diK=di(K2−1)12KE[ H_i]= c_i+ K-16 d_i2 1K+ K-212 d_iK= d_i(K^2-1)12K gives C=K2−112K∑idi−1K[K−26|E|+K−16∑i(di2)] c_C= K^2-112K _id_i- 1K [ K-26|E|+ K-16 _i d_i2 ]. ∎ Corollary S37 (complete graph). On KNK_N, with A(s)A(s) the number of tied pairs, [H∣A=a]=c1a+c0E[H A=a]=c_1a+c_0 for the index protocol, where c1=(K−1)N−K6c_1= (K-1)N-K6 and c0=N(N−1)[K2+K−1−(K−1)N]12Kc_0= N(N-1)[K^2+K-1-(K-1)N]12K. Proof. On KNK_N every pair is an edge with N−2N-2 common neighbours, so the weight of every tied pair is K−26+(N−2)K−16=(K−1)N−K6=c1 K-26+(N-2) K-16= (K-1)N-K6=c_1, and Theorem S36 gives [H¯∣τ]=c1A(s)+c0E[ H τ]=c_1A(s)+c_0 with A=∑j<kτjkA= _j<k _jk and c0=KN=N(N−1)[K2+K−1−(K−1)N]12Kc_0= c_K_N= N(N-1)[K^2+K-1-(K-1)N]12K (a direct simplification). Conditioning on A and using the tower property gives [H¯∣A=a]=c1a+c0E[ H A=a]=c_1a+c_0. It remains to replace H¯ H by the fixed-order energy H: for a permutation σ of the players, H(σs)H(σ s) is the energy of the game with tie-breaking order σ; since the law of s given A is exchangeable in the players, [H∣A]=[H∘σ∣A]E[H A]=E[H σ A] for every σ, hence [H∣A]=[H¯∣A]E[H A]=E[ H A]. ∎ Fig. S2: The tie-pattern law against exhaustive enumeration (floating point; not part of any proof). (a) Complete graph, index protocol: [H∣A]E[H A] against the number A of tied pairs (points) and the affine law of Corollary S37 (lines). (b) C4C_4, P4P_4, S4S_4, K=3K=3: the order-averaged energy [H¯∣C]E[ H _C] lies on the line of slope one given by (S32); conditioning on tied conflict edges alone (C4C_4, grey, shifted for display) is not affine. Theorem S36 was also checked against exhaustive enumeration on P4P_4, S4S_4, C4C_4, C5C_5 and random graphs with N≤6N≤ 6, K≤5K≤ 5 (Figure S2). The full pattern is needed: on C4C_4, P4P_4 and S4S_4 (K=3K=3, averaged over all 2424 orders) the conditional energy equals C(τ)+CA_C(τ)+ c_C to within 4×10−104× 10^-10, whereas conditioning on the number of tied conflict edges alone is not monotone on C4C_4. S-B.1 Randomised tie-breaking A deterministic protocol amplifies the harmonic component but does not create it. Proposition S38 (randomised tie-breaking). If ties are broken uniformly at random and payoffs are taken in expectation, ℰpotE_pot is as in Proposition 2 and ℰharmE_harm is multiplied by (K−2)/(K+1)(K-2)/(K+1). The game is an exact potential game iff K=2K=2. Proof. Now ψ(x,y)=−12−12sgn(y−x)ψ(x,y)=- 12- 12sgn(y-x) for every ordered pair, so again ψ(x,y)+ψ(y,x)≡−1ψ(x,y)+ψ(y,x)≡-1, ω+≡0ω^+≡ 0, and the own effect is unchanged. Moreover Varψ=K−14KVarψ= K-14K, so ‖ω‖2=K−14K−K2−16K2=(K−1)(K−2)12K2\|ω\|^2= K-14K- K^2-16K^2= (K-1)(K-2)12K^2, which is (K−2)/(K+1)(K-2)/(K+1) times the deterministic value K2−112K2 K^2-112K^2 of Proposition 2. Apply Proposition 1(c),(d). ∎ What creates the harmonic component is the antisymmetry of the comparison, ψ(x,y)+ψ(y,x)≡−1ψ(x,y)+ψ(y,x)≡-1; the deterministic protocol adds to it the resolution of ties, which is why the factor vanishes at K=2K=2, where every randomly resolved tie is a coin flip with no residual interaction. Appendix S-C The continuum game and the proof of Theorem 1 S-C.1 Setting Let N players choose si∈[0,1]s_i∈[0,1] i.i.d. uniformly, posi(s)=#j≠i:sj>sipos_i(s)=\#\j≠ i:s_j>s_i\ and ui=f(posi)u_i=f(pos_i). Flows are functions xi(t→t′;s−i)x_i(t→ t ;s_-i), antisymmetric in (t,t′)(t,t ), with ∥x∥2=∑is−i[12∬xi(t→t′;s−i)2dtdt′]\|x\|^2= _iE_s_-i [ 12 x_i(t→ t ;s_-i)^2\,dt\,dt ]; gradients are (δϕ)i=ϕ(t′,s−i)−ϕ(t,s−i)(δφ)_i=φ(t ,s_-i)-φ(t,s_-i), game flows are ui(t′,s−i)−ui(t,s−i)u_i(t ,s_-i)-u_i(t,s_-i), and P is the orthogonal projection onto the closure of imδimδ. For v,w∈L2([0,1]N)v,w∈ L^2([0,1]^N) one has ⟨δiv,δiw⟩i=[(v−Aiv)(w−Aiw)] _iv, _iw _i=E[(v-A_iv)(w-A_iw)] because 12∬(v(t′)−v(t))(w(t′)−w(t))tdt′=Covt(v,w) 12 (v(t )-v(t))(w(t )-w(t))\,dt\,dt =Cov_t(v,w). Proposition S39 (Hodge projection is the inverse number operator). With g=∑i(ui−Aiui)g= _i(u_i-A_iu_i), Pu=δϕ⋆Pu=δφ with ϕ⋆=−1gφ = N^-1g, and ‖Pu‖2=∑m≥1‖gm‖2m=∑T≠∅1|T|‖∑i∈T[ui]T‖2,‖u‖2=∑i‖ui−Aiui‖2=∑i∑T∋i‖[ui]T‖2.\|Pu\|^2= _m≥ 1 \|g_m\|^2m= _T≠ 1|T| \| _i∈ T[u_i]_T \|^2, \|u\|^2= _i\|u_i-A_iu_i\|^2= _i _T i\|[u_i]_T\|^2. (S38) Proof. ‖u−δϕ‖2=∑i‖(I−Ai)(ui−ϕ)‖2\|u-δφ\|^2= _i\|(I-A_i)(u_i-φ)\|^2 is minimised at the solution of ∑i(I−Ai)ϕ=∑i(I−Ai)ui _i(I-A_i)φ= _i(I-A_i)u_i, i.e. ϕ=g Nφ=g; N is invertible on mean-zero functions (it multiplies the order-m component by m), and ‖δϕ⋆‖2=⟨ϕ⋆,ϕ⋆⟩=⟨g,−1g⟩\|δφ \|^2= Nφ ,φ = g, N^-1g . The second expression for ‖u‖2\|u\|^2 is Parseval for the Hoeffding decomposition, since ui−Aiuiu_i-A_iu_i consists of the components with T∋iT i. ∎ S-C.2 The two energies (eq. (21) of the main text) Lemma S40 (Dirichlet variance identity). Var(ui)=Vark∼U0,…,N−1f(k)Var(u_i)=Var_k U\0,…,N-1\f(k) and Var(u¯i)=1N+1Var(ui)Var( u_i)= 1N+1Var(u_i); hence ‖u‖2=N2N+1Varkf(k)\|u\|^2= N^2N+1Var_kf(k). Proof. posipos_i is uniform on 0,…,N−1\0,…,N-1\ (positions form a uniformly random permutation). Given s−is_-i with order statistics z1>⋯>zN−1z_1>…>z_N-1 and spacings ςk=zk−zk+1 _k=z_k-z_k+1 (z0=1z_0=1, zN=0z_N=0), posi=kpos_i=k iff zk+1<si<zkz_k+1<s_i<z_k, so u¯i=∑kf(k)ςk u_i= _kf(k) _k. The spacings are Dirichlet(1,…,1)(1,…,1) with N cells: ςk=1NE _k= 1N, ςkςl=1+δklN(N+1)E _k _l= 1+ _klN(N+1). Hence u¯i2=1N(N+1)[(∑kfk)2+∑kfk2]E u_i^2= 1N(N+1) [( _kf_k)^2+ _kf_k^2 ] and Varu¯i=1N+1[1N∑fk2−1N2(∑fk)2]=1N+1VarkfVar u_i= 1N+1 [ 1NΣ f_k^2- 1N^2(Σ f_k)^2 ]= 1N+1Var_kf. Finally ‖u‖2=∑i[Varui−Varu¯i]=N⋅N+1Varkf\|u\|^2= _i[Varu_i-Var u_i]=N· NN+1Var_kf. ∎ Lemma S41 (Bernoulli-sum decomposition). Let Bt=#j≠i:sj>tB_t=\#\j≠ i:s_j>t\ and κ(t,t′)=min(t,t′)−tt′κ(t,t )= (t,t )-t . For T∌iT i with |T|=m|T|=m, [u¯i]T=∫01∏j∈T([sj>t]−(1−t))dm[f](1−t)dt,dm[f](q)=[Δmf(Bin(N−1−m,q))],[ u_i]_T= _0^1 _j∈ T (1[s_j>t]-(1-t) )\,d_m[f](1-t)\,dt, d_m[f](q)=E [ ^mf(Bin(N-1-m,q)) ], and this expression does not depend on i∉Ti∉ T. Consequently ‖gm‖2=(N−m)2Jm,Jm:=(Nm)∬κ(q,q′)mdm[f](q)dm[f](q′)qdq′,\|g_m\|^2=(N-m)^2J_m, J_m:= Nm κ(q,q )^m\,d_m[f](q)\,d_m[f](q )\,dq\,dq , (S39) and ‖Pu‖2=∑m(N−m)2mJm\|Pu\|^2= _m (N-m)^2mJ_m, ‖u‖2=N∑m(N−m)Jm\|u\|^2=N _m(N-m)J_m. Proof. u¯i(s−i)=∫01f(Bt)t u_i(s_-i)= _0^1f(B_t)\,dt, since posi=Bsipos_i=B_s_i and sis_i is uniform. For fixed t, BtB_t is a sum of i.i.d. Bernoulli(1−t)(1-t) variables ξj=[sj>t] _j=1[s_j>t]; for a function F of such a sum, the Hoeffding component on T is ∏j∈T(ξj−(1−t))⋅[Δ|T|F(Bt(Tc))] _j∈ T( _j-(1-t))·E[ ^|T|F(B^(T^c)_t)] (expand F(ξT+B(Tc))F( _T+B^(T^c)) in the Walsh basis ∏j∈T(ξj−(1−t)) _j∈ T( _j-(1-t)); each centred factor acts as a forward difference). Integrating over t gives the formula, whose right-hand side involves i only through the size N−1−mN-1-m of the complement. By Lemma S2, gT=−∑i∉T[u¯i]T=−(N−m)×g_T=- _i∉ T[ u_i]_T=-(N-m)× the common expression; its squared norm is computed using [([s>t]−(1−t))([s>t′]−(1−t′))]=κ(t,t′)E [(1[s>t]-(1-t))(1[s>t ]-(1-t )) ]=κ(t,t ) for each of the m coordinates, which gives (S39) after the substitution q=1−tq=1-t, under which κ is invariant. The formula for ‖Pu‖2\|Pu\|^2 is (S38). For ‖u‖2\|u\|^2: the same computation gives Var(u¯i)=∑m(N−1m)∬κmdmdmVar( u_i)= _m N-1m κ^md_md_m, and Lemma S40 yields ‖u‖2=N[Varui−Varu¯i]=N2Varu¯i=N2∑m(N−1m)∬κmdmdm=N∑m(N−m)Jm\|u\|^2=N [Varu_i-Var u_i ]=N^2Var u_i=N^2 _m N-1m κ^md_md_m=N _m(N-m)J_m, using N(N−1m)=(N−m)(Nm)N N-1m=(N-m) Nm. ∎ Proof of Corollary 3, with the equality case. Since ρpot(∞)(f)=‖Pu‖2/‖u‖2ρ^(∞)_pot(f)=\|Pu\|^2/\|u\|^2, it suffices to show that, by Lemma S41, N−1N‖u‖2−‖Pu‖2=∑m(N−m)Jm[(N−1)−N−m]=∑m≥1(N−m)JmN(m−1)m≥0, N-1N\|u\|^2-\|Pu\|^2= _m(N-m)J_m [(N-1)- N-mm ]= _m≥ 1(N-m)J_m N(m-1)m\ ≥ 0, since Jm≥0J_m≥ 0 (κmκ^m is a positive semidefinite kernel). Equality holds iff Jm=0J_m=0 for all m≥2m≥ 2, i.e. dm[f]≡0d_m[f]≡ 0; as q↦[Δmf(Bin(n,q))]q [ ^mf(Bin(n,q))] is the Bernstein polynomial of Δmf ^mf, this holds iff Δmf≡0 ^mf≡ 0 for all m≥2m≥ 2, i.e. iff f is affine. ∎ S-C.3 Proof of Theorem 1 Theorem (1 of the main text). Let tp(k)=∑r=0p(−1)r(pr)(p+r)kr¯(N−1)r¯t_p(k)= _r=0^p(-1)^r pr p+rr k r(N-1) r, p=1,…,N−1p=1,…,N-1, be the discrete Chebyshev polynomials, and Jm(f,g)=(Nm)∬κmdm[f](q)dm[g](q′)qdq′J_m(f,g)= Nm κ^m\,d_m[f](q)\,d_m[g](q )\,dq\,dq the order-m form, so that ℰpot(f,g):=∑m(N−m)2mJm(f,g)E_pot(f,g):= _m (N-m)^2mJ_m(f,g) and ℰtot(f,g):=N∑m(N−m)Jm(f,g)E_ tot(f,g):=N _m(N-m)J_m(f,g) polarise ‖Pu‖2\|Pu\|^2 and ‖u‖2\|u\|^2. (a) For every m and all p≠p′p≠ p , Jm(tp,tp′)=0J_m(t_p,t_p )=0; moreover Jm(tp,tp)=0J_m(t_p,t_p)=0 for m>pm>p and, for m≤pm≤ p, Jm(tp,tp)=(Nm)m(p+m)!p(p+1)(2p+1)(p−m)![(N−1)m¯]2.J_m(t_p,t_p)= Nm m\,(p+m)!p(p+1)(2p+1)\,(p-m)!\, [(N-1) m ]^2. (S40) (b) λp=ℰpot(tp,tp)ℰtot(tp,tp)=(N−p)(N+p+1)Np(p+1)[1−(Np)(N+p+1p)] _p= E_pot(t_p,t_p)E_ tot(t_p,t_p)= (N-p)(N+p+1)N\,p(p+1) [1- Np N+p+1p ], and λp+1<λp _p+1< _p for 1≤p≤N−21≤ p≤ N-2. Throughout, q=1−tq=1-t, κ=min(q,q′)−qq′κ= (q,q )-q and dm[f](q)=[Δmf(Bin(N−1−m,q))]d_m[f](q)=E[ ^mf(Bin(N-1-m,q))] as in Lemma S41; n:=N−1n:=N-1; P~p(q):=Pp(1−2q)=∑r=0p(−1)r(pr)(p+r)qr P_p(q):=P_p(1-2q)= _r=0^p(-1)^r pr p+rrq^r is the shifted Legendre polynomial, P~j(m)(q):=Pj(m,m)(1−2q) P^(m)_j(q):=P^(m,m)_j(1-2q) the shifted Jacobi (Gegenbauer) polynomial, and m(q):=[q(1−q)]m w_m(q):=[q(1-q)]^m. (i) From Chebyshev to Legendre and Jacobi polynomials. Let f(q):=f(Bin(n,q)) b_f(q):=Ef(Bin(n,q)) be the Bernstein polynomial of f. For f=kr¯f=k r one has [Bin(n,q)r¯]=nr¯qrE[Bin(n,q) r]=n r\,q^r and Δmkr¯=rm¯kr−m¯ ^mk r=r m\,k r-m, hence dm[kr¯](q)=rm¯(n−m)r−m¯qr−m=1nm¯dmdqm(nr¯qr)d_m[k r](q)=r m\,(n-m) r-m\,q^r-m= 1n m d^mdq^m (n r\,q^r ), because nr¯=nm¯(n−m)r−m¯n r=n m\,(n-m) r-m. By linearity, dm[f]=1nm¯f(m)for every f,d_m[f]= 1n m\, b_f^(m) every f, (S41) which is the classical derivative formula for Bernstein polynomials. Moreover tp(q)=∑r(−1)r(pr)(p+r)qr=P~p(q) b_t_p(q)= _r(-1)^r pr p+rrq^r= P_p(q): the binomial law maps the discrete Chebyshev polynomials to the Legendre polynomials. Since dmdxmPp(x)=(p+m)!2mp!Pp−m(m,m)(x) d^mdx^mP_p(x)= (p+m)!2^m\,p!P^(m,m)_p-m(x) and dq=−2dx ddq=-2 ddx for x=1−2qx=1-2q, dm[tp](q)=cp,mP~p−m(m)(q),cp,m:=(−1)m(p+m)!p!nm¯(m≤p),d_m[t_p](q)=c_p,m\, P^(m)_p-m(q), c_p,m:= (-1)^m(p+m)!p!\,n m (m≤ p), (S42) and dm[tp]=0d_m[t_p]=0 for m>pm>p. (i) The kernel κmκ^m carries the Jacobi weight. Let mg(q):=∫01κ(q,q′)mg(q′)dq′ K_mg(q):= _0^1κ(q,q )^mg(q )\,dq . Lemma S42. (a) If g is a polynomial of degree at most b, then mg=mR K_mg= w_m\,R with R a polynomial of degree at most b. (b) For every j≥0j≥ 0, mP~j(m)=m,jmP~j(m),m,j=m(j+m)(j+m+1). K_m P^(m)_j= k_m,j\, w_m\, P^(m)_j, k_m,j= m(j+m)(j+m+1). Proof. (a) It suffices to take g=qbg=q^b. Since κ(q,s)=s(1−q)κ(q,s)=s(1-q) for s≤qs≤ q and q(1−s)q(1-s) for s≥qs≥ q, mg(q)=(1−q)mqm+b+1m+b+1+qmG(q),G(q):=∫q1(1−s)msbs. K_mg(q)=(1-q)^m q^m+b+1m+b+1+q^m\,G(q), G(q):= _q^1(1-s)^ms^b\,ds. G is a polynomial of degree m+b+1m+b+1 with G(i)(1)=0G^(i)(1)=0 for i≤mi≤ m, so G=(1−q)m+1G1G=(1-q)^m+1G_1 with degG1=b G_1=b; the leading coefficient of G is −(−1)m/(m+b+1)-(-1)^m/(m+b+1), hence that of G1G_1 is 1/(m+b+1)1/(m+b+1). Thus mg=m[qb+1m+b+1+(1−q)G1] K_mg= w_m [ q^b+1m+b+1+(1-q)G_1 ], and in the bracket the coefficients of qb+1q^b+1 cancel. (b) m K_m is self-adjoint on L2(0,1)L^2(0,1). Write mP~j(m)=mR K_m P^(m)_j= w_mR with degR≤j R≤ j. For a polynomial S of degree <j<j, part (a) gives mS=mRS K_mS= w_mR_S with degRS<j R_S<j, so ∫SmR=∫(mS)P~j(m)=∫mRSP~j(m)=0 S\, w_mR= ( K_mS) P^(m)_j= w_mR_S\, P^(m)_j=0, because the P~j(m) P^(m)_j are orthogonal for the weight m w_m. Hence R is orthogonal, for that weight, to all polynomials of lower degree, i.e. R=P~j(m)R= k\, P^(m)_j. To find k let q→0q→ 0: the first term of the display above is O(qm+1)O(q^m+1), so mP~j(m)(q)=qm∫01(1−s)mP~j(m)(s)s+O(qm+1) K_m P^(m)_j(q)=q^m _0^1(1-s)^m P^(m)_j(s)\,ds+O(q^m+1), while mP~j(m)=qmP~j(m)(0)+O(qm+1) w_m P^(m)_j=q^m P^(m)_j(0)+O(q^m+1) with P~j(m)(0)=Pj(m,m)(1)=(j+mj) P^(m)_j(0)=P^(m,m)_j(1)= j+mj. The Rodrigues formula reads mP~j(m)=1j!djdsj[sj+m(1−s)j+m] w_m P^(m)_j= 1j! d^jds^j [s^j+m(1-s)^j+m ], and j integrations by parts (all boundary terms vanish) give ∫01(1−s)mP~j(m)s _0^1(1-s)^m P^(m)_j\,ds =1j!∫01s−mdjdsj[sj+m(1−s)j+m]s = 1j! _0^1s^-m d^jds^j [s^j+m(1-s)^j+m ]ds =m(m+1)⋯(m+j−1)j!∫01(1−s)j+ms=(m+j−1j)j+m+1. = m(m+1)·s(m+j-1)j! _0^1(1-s)^j+mds= m+j-1jj+m+1. Dividing by (j+mj) j+mj gives m,j=m(j+m)(j+m+1) k_m,j= m(j+m)(j+m+1). ∎ (i) Every order form is diagonal; the potential form. By (S42) and Lemma S42(b), for m≤min(p,p′)m≤ (p,p ), Jm(tp,tp′)=(Nm)cp,mcp′,mm,p′−m∫01mP~p−m(m)P~p′−m(m)q,J_m(t_p,t_p )= Nmc_p,mc_p ,m\, k_m,p -m _0^1 w_m\, P^(m)_p-m P^(m)_p -m\,dq, which vanishes for p≠p′p≠ p by Jacobi orthogonality; this is (a), the case m>min(p,p′)m> (p,p ) being trivial. For p=p′p=p , m,p−m=mp(p+1) k_m,p-m= mp(p+1) and the Jacobi norm ∫01m(P~p−m(m))2q=(p!)2(2p+1)(p+m)!(p−m)! _0^1 w_m ( P^(m)_p-m )^2dq= (p!)^2(2p+1)\,(p+m)!\,(p-m)! give (S40). In particular the tpt_p diagonalise ∑mwmJm _mw_mJ_m for any weights wmw_m, hence both ℰpotE_pot and ℰtotE_ tot (for the latter this is also clear from ℰtot(f,f′)=N+1∑k(fk−f¯)(fk′−f¯′)E_ tot(f,f )= NN+1 _k(f_k- f)(f _k- f ), Lemma S40). Finally, with (Nm)(N−m)2/[nm¯]2=N(N−m)!/[m!(N−1)!] Nm(N-m)^2/[n m]^2=N\,(N-m)!/[m!\,(N-1)!], ℰpot(tp,tp)=∑m=1p(N−m)2mJm(tp,tp)=N(N−1)!p(p+1)(2p+1)∑m=1p(N−m)!(p+m)!m!(p−m)!.E_pot(t_p,t_p)= _m=1^p (N-m)^2mJ_m(t_p,t_p)= N(N-1)!\,p(p+1)(2p+1) _m=1^p (N-m)!\,(p+m)!m!\,(p-m)!. In the same way N(N−m)Jm(tp,tp)∝(N−m−1)!(p+m)!/[(m−1)!(p−m)!]N(N-m)J_m(t_p,t_p) (N-m-1)!\,(p+m)!/[(m-1)!\,(p-m)!], which is the distribution ηp,m _p,m of the energy of tpt_p over the orders m=1,…,pm=1,…,p; the higher the order, the smaller its weight N−mN N-mmN in λp _p. (iv) Normalisation and closed form. The Hahn-polynomial norm (parameters (0,0)(0,0)) gives ∑k=0N−1tp(k)2=(N+p)!(N−1−p)!(2p+1)(N−1)!2 _k=0^N-1t_p(k)^2= (N+p)!\,(N-1-p)!(2p+1)\,(N-1)!^2, so ℰtot(tp,tp)=N+1⋅(N+p)!(N−1−p)!(2p+1)(N−1)!2E_ tot(t_p,t_p)= NN+1· (N+p)!(N-1-p)!(2p+1)(N-1)!^2 and λp=(N+1)!Np(p+1)(N−p−1)!(N+p)!∑m=1p(p+m)!(N−m)!m!(p−m)!. _p= (N+1)!N\,p(p+1)\,(N-p-1)!\,(N+p)! _m=1^p (p+m)!\,(N-m)!m!\,(p-m)!. Adding the m=0m=0 term N!N!, the sum becomes N!F12(−p,p+1,−N, 1)=(N+p+1)!(N−p)!(N+1)!N!\;_2F_1(-p,\,p+1;\,-N;\,1)= (N+p+1)!\,(N-p)!(N+1)! by the Chu--Vandermonde identity,11 1 The lower parameter −N-N is a nonpositive integer, so F12_2F_1 is meant here as the terminating series determined by the upper parameter −p-p. This is legitimate: the series stops after p+1≤Np+1≤ N terms, and (−N)m≠0(-N)_m≠ 0 for 0≤m≤p≤N−10≤ m≤ p≤ N-1, so every term is finite. Chu–Vandermonde in the form F12(−p,b,c,1)=(c−b)p(c)p_2F_1(-p,b;c;1)= (c-b)_p(c)_p is an identity between rational functions of c, and (c)p(c)_p does not vanish at c=−Nc=-N when p≤Np≤ N; it may therefore be evaluated at c=−Nc=-N. Explicitly, (−N)p=(−1)pN!(N−p)!(-N)_p=(-1)^p N!(N-p)! and (−N−p−1)p=(−1)p(N+p+1)!(N+1)!(-N-p-1)_p=(-1)^p (N+p+1)!(N+1)!. which yields the stated closed form with εp:=(Np)(N+p+1p)=∏i=0p−1N−iN+p+1−i. _p:= Np N+p+1p= _i=0^p-1 N-iN+p+1-i. For p=1p=1, ε1=N+2 _1= NN+2 and λ1=N−1N _1= N-1N. (v) Monotonicity. Put Dp=N(N+1)p(p+1)−1D_p= N(N+1)p(p+1)-1, so Nλp=Dp(1−εp)N _p=D_p(1- _p), Dp−Dp+1=2N(N+1)p(p+1)(p+2)D_p-D_p+1= 2N(N+1)p(p+1)(p+2) and εp+1=εpN−pN+p+2 _p+1= _p N-pN+p+2. Then λp>λp+1 _p> _p+1 reads Dp(1−εp)>Dp+1(1−εpN−pN+p+2)D_p(1- _p)>D_p+1 (1- _p N-pN+p+2 ); since Dp−Dp+1N−pN+p+2=(Dp−Dp+1)+Dp+12(p+1)N+p+2D_p-D_p+1 N-pN+p+2=(D_p-D_p+1)+D_p+1 2(p+1)N+p+2, this is (Dp−Dp+1)(1−εp)>εpDp+12(p+1)N+p+2(D_p-D_p+1)(1- _p)> _p\,D_p+1 2(p+1)N+p+2, i.e., with Dp+1=(N−p−1)(N+p+2)(p+1)(p+2)D_p+1= (N-p-1)(N+p+2)(p+1)(p+2), 1−εpεp>2(p+1)Dp+1(Dp−Dp+1)(N+p+2)=p(p+1)(N−p−1)N(N+1). 1- _p _p> 2(p+1)D_p+1(D_p-D_p+1)(N+p+2)= p(p+1)(N-p-1)N(N+1). Now εp−1=∏i=0p−1N+p+1−iN−i=∏i=0p−1(1+p+1N−i)≥1+p(p+1)N _p^-1= _i=0^p-1 N+p+1-iN-i= _i=0^p-1 (1+ p+1N-i )≥ 1+ p(p+1)N, so the left-hand side is at least p(p+1)N p(p+1)N, which exceeds the right-hand side because N−p−1<N+1N-p-1<N+1. ∎ Remark S43. Every step is audited in exact arithmetic by audits/check_spectrum.py ((S41), (S42), Lemma S42, (S40) against direct evaluation of the double integrals, and the closed forms of λp _p and of the total form for 3≤N≤93≤ N≤ 9). Appendix S-D Priced priorities This appendix proves the statement made after Corollary 2 of the main text: pricing can change behaviour drastically while leaving ℰharmE_harm fixed, and neither energy predicts the existence of pure equilibria. The protocol is the index protocol and C is an arbitrary conflict graph. Proposition S44 (priced priorities). Let ui(s)=−posC(i,s)−csiu_i(s)=-pos_C(i,s)-c\,s_i with c>0c>0, and bi:=#j∈(i):j<ib_i:=\#\j (i):j<i\. (a) ℰharmE_harm is as in Proposition 2 and ℰpot=cK∑i(di−cK)2E_pot=c_K _i(d_i-cK)^2. On a d-regular graph the game is purely harmonic at c=d/Kc=d/K, where marginal price equals marginal expected benefit, and uniform mixing is then a Nash equilibrium. (b) If 0<c<10<c<1 and E≠∅E≠ , there is no pure Nash equilibrium. (c) The all-zero profile is a Nash equilibrium iff c≥maxibic≥ _ib_i. Proof. (a) Pricing replaces the own effect i(x)=di(x−s¯)/K a_i(x)=d_i(x- s)/K of Proposition 2 by (di/K−c)(x−s¯)(d_i/K-c)(x- s) and nothing else; apply Proposition 1(c). If all other agents mix uniformly, [ui∣si=x]=const+x(di/K−c)E[u_i s_i=x]=const+x(d_i/K-c), constant in x at c=d/Kc=d/K. (b) Let s be a pure equilibrium. For an agent with si≥1s_i≥ 1 let Li∈ℤ≥0L_i _≥ 0 be the number of positions lost by lowering one level; no profitable deviation means Li≥c>0L_i≥ c>0, so Li≥1L_i≥ 1. For an agent with si≤K−2s_i≤ K-2 let WiW_i be the number of neighbours overtaken by raising one level; then Wi≤c<1W_i≤ c<1, so Wi=0W_i=0. In particular isolated agents sit at level 00. If all agents are at level 00, take an edge i,j\i,j\ with i<ji<j: raising j to level 11 overtakes i, so Wj≥1W_j≥ 1, a contradiction. Otherwise let smax≥1s_ ≥ 1 be the highest occupied level and m the largest-index agent at it. Since Lm≥1L_m≥ 1, some neighbour j is served before m after m lowers but not before; this requires either sj=smaxs_j=s_ with j>mj>m, excluded by the choice of m, or sj=smax−1s_j=s_ -1 with j<mj<m. In the latter case j, raising one level, ties m and is favoured by the protocol, so Wj≥1W_j≥ 1, a contradiction. (c) At s=0s=0 every agent is tied with all neighbours, so ui=−biu_i=-b_i. Raising to level t≥1t≥ 1 puts i ahead of every neighbour at payoff −ct-ct, best at t=1t=1. The deviation is strictly profitable iff c<bic<b_i. ∎ Thus pure equilibria exist at c=0c=0, vanish on all of (0,1)(0,1), and reappear for larger c, while ℰharmE_harm does not move. Part (b) uses the index protocol: on a triangle with the cyclic protocol and K=2K=2, the profile in which everybody bids 11 is an equilibrium for every c<1c<1. Exact computation on the intersection. Scanning c over 9191 values in [0,4.5][0,4.5] on the eight-vehicle intersection (K=3K=3), with the methods of Section VII of the main text, the potential share follows (a) to machine precision, from 4/54/5 at c=0c=0 through exactly 00 at c=d/K=4/3c=d/K=4/3 back to 0.960.96 at c=4.5c=4.5, while the projected harmonic energy is constant to 10−1410^-14 and the entropy production at βmax|Δu|=0.02β | u|=0.02 stays within 3×10−53× 10^-5 of the prediction of Proposition 6. Pure equilibria number three at c=0c=0, zero on (0,1)(0,1) as in (b), and one from c=maxibi=4c= _ib_i=4 on as in (c); in between they exist only at c=1c=1 and c=3c=3, a feature of this graph (a star whose centre has the largest index has one for every c∈(1,maxibi)c∈(1, _ib_i)). Appendix S-E Reproducibility and independent audit The proof of Theorem 2 is computer-assisted at three places only, namely the root isolation of Step S-A.11, the symbolic identities in N of Step S-A.12, and the certificates of Step S-A.13. Every other step, and all of Appendices S-B–S-D, is proved by hand. Table S4 lists the scripts of those three steps; they use exact arithmetic only (Python fractions or SymPy), and run_checks.sh runs them all in well under a minute. The floating-point computations of Section VII of the main text and of Appendix S-D enter no proof and are kept outside the package. TABLE S4: Scripts on which the proof of Theorem 2 depends, and the two independent checks. script step establishes verify_certificates.py + certificates/ (2525 files) S-A.13 the pairs of Table S2: rebuilds each certified matrix from Theorem S14 and Step S-A.4, checks LDL⊤=ALDL =A with positive pivots and the kernel; standard library only, imports nothing audit_K4.py S-A.11 the cubic of Lemma S25, exact root counting for ϱ≥ϱ0 ≥ _0, and (T) for all bonds iff N≥10N≥ 10 (Theorem S26); also audits the hand-proved closed forms of Steps S-A.8–S-A.9 k3_representation_proof.py, s1_closed.py, check_blocks.py, verify_M_symbolic.py S-A.12 Lemmas S27, S29 and S31: the block representation, the closed forms of its entries, the row sums M1, and the certificates for N=17,18N=17,18 k2_witness.py S-A.14 the K=2K=2, N=7N=7 witness from the definition of the Hodge projection; Q2=0Q_21=0; the sign change between N=6N=6 and N=7N=7 end_to_end.py – Theorem 2 for K=3K=3, N≤7N≤ 7; K=4K=4, N≤5N≤ 5; K=5K=5, N≤4N≤ 4; K=6,7K=6,7, N=3N=3, using no step of the proof (see below) An end-to-end check that bypasses the proof. end_to_end.py builds QK=N−1NMtot−MpotQ_K= N-1NM_ tot-M_pot directly from (19) (all KNK^N profiles, the flow of every increment indicator, and MpotM_pot from the Hoeffding decomposition of the divergence, Lemma S1) in exact rational arithmetic, and certifies QK=0Q_K1=0 and QK+⊤=LDL⊤Q_K+11 =LDL with positive pivots. This is Theorem 2 for that pair, obtained without the order decomposition, the bond variables or the sum-of-squares identity; it is feasible while KNK^N is a few thousand, and at K=2K=2 it reports positivity for N≤6N≤ 6 and failure at N=7N=7. Any other pair: python3 end_to_end.py K,N. The package also contains the certificate generator (bond_certificate.py, full_space_certificate.py, make_certificates.py), which the checker does not import, and audits/check_spectrum.py, a symbolic audit of the hand proof of Theorem 1.