--- title: 'Pokémon Theorem: Fairness, Automata & Graphs' url: https://www.emergentmind.com/topics/pokemon-theorem type: topic --- # Pokémon Theorem: Fairness, Automata & Graphs The expression **Pokémon theorem** is used for several unrelated results in contemporary mathematical and theoretical-computer-science literature. In fairness research, it denotes an RKHS impossibility theorem showing that, under unequal base rates, any finite collection of linear mean-fairness criteria leaves a residual group discrepancy witnessed by the maximum mean discrepancy (MMD) [2605.09221]. In automata theory, it denotes the statement that a disjunctive input stream eventually wins any deterministic finite-state game with a single absorbing win state reachable from every state [2512.23143]. In graph theory and combinatorics, it is also used as a nickname for the Graham–Pollak theorem, which states that the biclique partition number of the complete graph \(K_n\) is \(n-1\) [1007.1553]. ## 1. Terminological scope The three principal uses of the term are distinct in content, assumptions, and mathematical apparatus. | Usage | Core statement | Source | |---|---|---| | Fairness impossibility | Finite linear mean-fairness audits cannot exhaust group discrepancy under unequal base rates | [2605.09221] | | Finite-state games | A disjunctive input eventually contains a synchronizing win word | [2512.23143] | | Biclique partitions | Any edge-partition of \(K_n\) into complete bipartite graphs needs at least \(n-1\) parts | [1007.1553] | A recurrent source of confusion is that the term does **not** designate a single canonical theorem. The fairness result is formulated in RKHS geometry; the automata result concerns synchronizing words and disjunctive sequences; the graph-theoretic usage is a popular nickname for a theorem about biclique partitions. The common label is therefore historical or expository rather than structural. ## 2. RKHS fairness impossibility and the modern Pokémon theorem In the fairness literature, the setup is as follows. The protected attribute is \(A \in \mathcal{G}\), typically \(\mathcal{G}=\{a,b\}\); the outcome is \(Y\in\{0,1\}\); the features are \(X\in\mathcal{X}\); and a score takes the form \(S=s(X)=\langle w,\phi(X)\rangle\) for some \(w\in\mathcal{H}\), where \(k\) is a positive-definite kernel on \(\mathcal{X}\) with RKHS \(\mathcal{H}\) and feature map \(\phi\). Base rates are \(\pi_g:=\mathbb{P}(Y=1\mid A=g)\), and unequal base rates mean \(\pi_a\neq \pi_b\). The kernel mean embedding of a distribution \(P\) is \[ \mu_P:=\mathbb{E}_{X\sim P}[\phi(X)]\in\mathcal{H}, \qquad \langle f,\mu_P\rangle=\mathbb{E}_P[f(X)]. \] The relevant conditional mean embeddings are \[ \mu_g:=\mathbb{E}[\phi(X)\mid A=g],\qquad \mu_{y,g}:=\mathbb{E}[\phi(X)\mid Y=y,A=g], \] with group-difference vectors \[ \delta:=\mu_a-\mu_b,\qquad \delta_y:=\mu_{y,a}-\mu_{y,b}. \] If \(k\) is characteristic, then \[ \mathrm{MMD}_k(P_a,P_b)=\|\mu_a-\mu_b\|_{\mathcal H}=\|\delta\|_{\mathcal H}, \] and equality of mean embeddings implies equality of distributions [2605.09221]. The central observation is that several scalar fairness criteria become **linear constraints on conditional mean embeddings**. For \(S=\langle w,\phi(X)\rangle\), demographic parity is \[ \mathbb{E}[S\mid A=a]=\mathbb{E}[S\mid A=b] \iff \langle w,\delta\rangle=0. \] Equalized odds, or class-conditional balance within groups, is \[ \mathbb{E}[S\mid Y=y,A=a]=\mathbb{E}[S\mid Y=y,A=b] \iff \langle w,\delta_y\rangle=0,\quad y\in\{0,1\}. \] Group-conditional unbiasedness, \[ \mathbb{E}[S\mid A=g]=\pi_g, \] is likewise a first-moment condition. By contrast, calibration or sufficiency, \(Y\perp A\mid S\), is distributional rather than a single linear CME constraint, and therefore lies outside the paper’s “linear mean-fairness” scope. The geometric source of incompatibility is the law of total expectation in CME form: \[ \mu_{S\mid A=g} = \pi_g\,\mu_{S\mid Y=1,A=g}+(1-\pi_g)\,\mu_{S\mid Y=0,A=g}. \] If one imposes both group-level parity and class-conditional equality while \(\pi_a\neq\pi_b\), the different coefficients \(\pi_g\) overdetermine the mean constraints. This suggests that the impossibility is not merely a collection of ad hoc scalar incompatibilities, but a single Hilbert-space phenomenon induced by unequal base rates. ## 3. Residual witnesses, Kolmogorov widths, and the strengthened KMR dichotomy The qualitative Pokémon theorem is formulated for a characteristic kernel with \(P_a\neq P_b\), hence \(\delta\neq 0\), and a finite audit subspace \[ V=\mathrm{span}\{v_1,\dots,v_m\}\subset\mathcal H \] satisfying \(\langle v_i,\delta\rangle=0\) for \(i=1,\dots,m\). Its conclusion is that a residual violation always survives outside \(V\). A canonical witness is the normalized MMD direction \[ v_{m+1}:=\frac{\delta}{\|\delta\|_{\mathcal H}}\in V^\perp, \qquad \langle v_{m+1},\delta\rangle=\|\delta\|_{\mathcal H} =\mathrm{MMD}_k(P_a,P_b)>0. \] Thus, satisfying any finite collection of linear mean-fairness criteria does not exhaust the discrepancy between distinct groups. The quantitative theorem introduces the pooled covariance operator \[ \Sigma=\mathbb{E}_{X\sim Q}\big[(\phi(X)-\mu)\otimes(\phi(X)-\mu)\big], \qquad Q:=\tfrac12(P_a+P_b), \] with Mercer decomposition \(\Sigma=\sum_{j\ge 1}\lambda_j e_j\otimes e_j\). Under polynomial eigendecay \(c_1j^{-\alpha}\le \lambda_j\le c_2j^{-\alpha}\) for some \(\alpha>1\), and a source condition \(\delta=\Sigma^r u\) with \(\|u\|_{\mathcal H}\le R\), the worst-case residual over the source ellipsoid \[ B_r(R):=\{\Sigma^r u:\|u\|_{\mathcal H}\le R\} \] obeys \[ \inf_{\substack{V\subset\mathcal H\\ \dim V\le m}} \sup_{\delta\in B_r(R)} \|P_{V^\perp}\delta\|_{\mathcal H}^2 = R^2\lambda_{m+1}^{2r} = \Theta\!\big(R^2(m+1)^{-2\alpha r}\big). \] The optimal \(m\)-dimensional audit is the top-\(m\) Mercer eigenspace \(V=\mathrm{span}\{e_1,\dots,e_m\}\). In the approximate case, if an orthonormal family \(\{v_i\}\) satisfies \(|\langle v_i,\delta\rangle|\le \varepsilon_i\), then \[ \|P_{V^\perp}\delta\|^2 \ge \mathrm{MMD}_k(P_a,P_b)^2-\sum_{i=1}^m\varepsilon_i^2. \] Accordingly, the residual fairness violation decays only at the Kolmogorov \(m\)-width rate permitted by the pooled spectrum [2605.09221]. The same framework strengthens the Kleinberg–Mullainathan–Raghavan dichotomy. If \(S\in[0,1]\) satisfies group-conditional unbiasedness, \[ \mathbb{E}[S\mid A=g]=\pi_g, \] together with positive-class balance and negative-class balance, \[ \mathbb{E}[S\mid Y=1,A=a]=\mathbb{E}[S\mid Y=1,A=b],\qquad \mathbb{E}[S\mid Y=0,A=a]=\mathbb{E}[S\mid Y=0,A=b], \] then either \(\pi_a=\pi_b\) or \(S=Y\) almost surely. The decisive identity is \[ \pi_g=\pi_g\mu^+ + (1-\pi_g)\mu^-, \qquad (\pi_a-\pi_b)\bigl(1-(\mu^+-\mu^-)\bigr)=0. \] This replaces full calibration with the weaker first-moment condition of group-conditional unbiasedness. The paper also gives a bridge from directional class-balance control to near-perfection: if class-balance holds on an audited subspace \(V_m\), \(S=\langle w,\phi(X)\rangle\in[0,1]\) with \(\|w\|\le W\), and \(\Delta p\neq 0\), then for all \(t\in(0,1]\), \[ \mathbb{P}(|S-Y|>t)\le \frac{W\,\rho_m}{|\Delta p|\,t}, \] where \[ \rho_m:=\max_{g\in\{a,b\}} \left\|\,p_{\bar g}\,\delta_1+(1-p_{\bar g})\,\delta_0\,\right\|_{\mathcal H}. \] Under the same spectral regularity, \(\rho_m\) inherits the \(\Theta((m+1)^{-\alpha r})\) decay. ## 4. Fair representation learning, approximate frontiers, and empirical patterns The representation-level impossibility theorem considers a measurable encoder \(\Phi:\mathcal X\to\mathcal Z\), a characteristic kernel \(k_Z\) on \(\mathcal Z\), and the representation embeddings \[ \mu_{\Phi,g}:=\mathbb{E}[\phi_Z(\Phi(X))\mid A=g],\qquad \mu_{\Phi,y,g}:=\mathbb{E}[\phi_Z(\Phi(X))\mid Y=y,A=g]. \] Parity in representation space is \(\mu_{\Phi,a}=\mu_{\Phi,b}\), and class-conditional separation is \(\mu_{\Phi,y,a}=\mu_{\Phi,y,b}\) for \(y\in\{0,1\}\). If \(\pi_a\neq\pi_b\) and both properties hold, then \[ \mu_{\Phi,1}=\mu_{\Phi,0} \quad\text{and}\quad \Phi(X)\perp Y. \] The CME identity \[ \mu_{\Phi,g}=\pi_g\,\mu_{\Phi,1}+(1-\pi_g)\,\mu_{\Phi,0} \] forces class collapse when the base rates differ. The approximate relaxation gives the signal frontier: if \[ \|\mu_{\Phi,a}-\mu_{\Phi,b}\|_{\mathcal H_Z}\le \varepsilon,\qquad \|\mu_{\Phi,y,a}-\mu_{\Phi,y,b}\|_{\mathcal H_Z}\le \rho, \] then for \(g\in\{a,b\}\), \[ \|\mu_{\Phi,1,g}-\mu_{\Phi,0,g}\|_{\mathcal H_Z} \le \frac{\varepsilon+\rho}{|\pi_a-\pi_b|}. \] For binary classifiers under exact separation, with \[ \mathrm{DP\_gap} = \bigl|\mathbb{P}(\widehat Y=1\mid A=a)-\mathbb{P}(\widehat Y=1\mid A=b)\bigr|, \] the paper derives \[ \mathrm{DP\_gap}=|\Delta p|\,|TPR-FPR| \] and \[ \mathrm{error}(\widehat Y)\ge \min(p,1-p)\Bigl(1-\frac{\mathrm{DP\_gap}}{|\Delta p|}\Bigr), \qquad p:=\mathbb{P}(Y=1). \] The empirical study uses Adult Income (sex groups), \(n\approx 45K\), estimated \(|\Delta p|\approx 0.20\); COMPAS (race restricted to African-American vs. Caucasian), \(n\approx 5.3K\), \(|\Delta p|\approx 0.13\); and an ACS PUMS subsample (California 2018, race/sex), \(n=20K\), \(|\Delta p|\approx 0.085\). The residual fraction \[ \|P_{V_m^\perp}\hat\delta\|^2/\|\hat\delta\|^2 \] decays roughly polynomially on log–log axes when \(V_m\) is the top-\(m\) eigenspace of the empirical pooled covariance. The reported numbers of criteria needed to reach approximately \(1\%\) residual are Adult \(\approx 35\), COMPAS \(4\), and ACS \(\approx 8\). For separation-enforcing methods, specifically Hardt post-processing and ExponentiatedGradient with equalized-odds, the scatter of \((\mathrm{DP\_gap},\mathrm{error})\) lies on or above the Pareto bound within bootstrap uncertainty. For representational methods, specifically LFR, Fair-VAE, and adversarial debiasing, the plane \[ \bigl(\|\hat\mu_{\Phi,a}-\hat\mu_{\Phi,b}\|,\; \max_g\|\hat\mu_{\Phi,1,g}-\hat\mu_{\Phi,0,g}\|\bigr) \] respects the linear constraint suggested by the signal frontier, and the upper-left corner—small parity gap but large class signal—is empirically empty [2605.09221]. These results motivate a specific interpretation of fairness auditing. Finite checklists of moment-style criteria, including demographic parity and equalized odds, do not certify group equality when \(P_a\neq P_b\). The paper therefore treats MMD as a data-dependent fairness “budget,” with top-eigen allocations optimizing worst-case audit coverage. A plausible implication is that the practical question is less whether one can eliminate all residual mean-level discrepancy and more how rapidly the residual decays under increasing audit dimension. ## 5. The automata-theoretic Pokémon theorem In automata theory, a game is modeled as a deterministic finite-state, edge-labeled digraph \(G=(V,E)\), or equivalently as a DFA or finite-state transducer \[ G=(Q,\Sigma,\delta,q_0,F), \] where \(\Sigma\) is the finite alphabet of button inputs, \(\delta:Q\times\Sigma\to Q\) is the transition function, and \(F=\{\tilde q\}\) is a single absorbing winning state. Determinism is preserved even for games using a pseudorandom number generator by including the internal seed or pRNG state in \(Q\). The key assumption is **win-reducibility**: from every state \(q\in Q\), there exists some finite word \(w\in\Sigma^\ast\) with \(\delta(q,w)=\tilde q\) [2512.23143]. The core lemma is a synchronizing-to-win statement: there exists a synchronizing word \(W\) mapping every state to the absorbing win state, with \[ |W|\le |G|^2. \] The construction concatenates shortest state-specific winning words. Once such a \(W\) exists, the Pokémon theorem follows immediately from disjunctiveness. If \[ x=0.d_1d_2d_3\ldots \] is disjunctive in base \(b\), meaning that every finite word over \(\{0,\dots,b-1\}\) appears somewhere as a contiguous block in the digit expansion of \(x\), then \(W\) appears somewhere in the input stream generated by \(x\). At that moment the game reaches the absorbing win state and remains there. Formally, for any deterministic finite-state game with one absorbing win state \(\tilde v\) such that every state can reach \(\tilde v\), any disjunctive \(x\) eventually wins. The measure-theoretic consequence is that, for any fixed base \(b\ge 2\), Lebesgue almost every real number is normal in base \(b\), hence disjunctive in base \(b\), and therefore almost every \(x\) eventually wins any such game. The theorem does **not** show that \(\pi\) wins: the normality or disjunctiveness of \(\pi\) is unknown. This corrects a common overstatement sometimes associated with “\(\pi\) plays Pokémon.” The paper also records a true-RNG variant: if the RNG has full support over finite words and \(x\) is disjunctive, then \(x\) wins with probability \(1\). Quantitatively, a universal worst-case sequence can be obtained by concatenating all words of length \(|G|^2\), implying a coarse upper bound of \[ |G|^2\cdot b^{|G|^2} \] button presses before a synchronizing win word must appear. For Pokémon Sapphire, using \(|G|\le 8\times 10^6\) and \(b=8\), the resulting estimate is \[ (8\times 10^6)^2\cdot 8^{(8\times 10^6)^2} \approx 2^{1.9\times 10^{14}} \] button presses. The theorem is therefore existential rather than practically algorithmic. The paper further notes that finding a shortest synchronizing word is NP-hard. ## 6. The graph-theoretic Pokémon theorem and the Graham–Pollak result In combinatorics, “Pokémon theorem” is used as a nickname for the Graham–Pollak theorem. Let \(K_n\) be the complete graph on \(n\) labeled vertices. An edge-partition of \(K_n\) into complete bipartite graphs is a family of bicliques \(\{(L_i,R_i)\}_{i=1}^m\) whose edge sets are pairwise disjoint and whose union is \(E(K_n)\). The theorem states that the biclique partition number of \(K_n\) equals \(n-1\); equivalently, \[ E(K_n)=\bigsqcup_{i=1}^m E\bigl(K_{|L_i|,|R_i|}\bigr) \quad\Rightarrow\quad m\ge n-1. \] This lower bound is tight. A star decomposition with \[ L_i=\{i\},\qquad R_i=\{i+1,i+2,\dots,n\},\qquad i=1,\dots,n-1, \] partitions \(E(K_n)\) into \(n-1\) bicliques. The counting proof in [1007.1553] replaces the usual linear-algebraic argument by a pigeonhole argument on vertex labelings. Assuming \(m\le n-2\), one considers all labelings \(\sigma:[n]\to[k]\) and the corresponding pattern vector \(P(\sigma)\in\mathbb Z^{n-1}\), where the first \(m\) coordinates are the sums over the left sides \(L_i\) and the last coordinate is the total sum over all vertices. For sufficiently large \(k\), two distinct labelings \(\sigma_1\neq \sigma_2\) must have the same pattern. Writing \(\tau=\sigma_1-\sigma_2\), one obtains \[ \sum_{v=1}^n \tau(v)=0 \quad\text{and}\quad \sum_{v\in L_i}\tau(v)=0\ \text{ for all }i. \] The identity \[ \Bigl(\sum_{v=1}^n\tau(v)\Bigr)^2 = \sum_{v=1}^n\tau(v)^2 + 2\sum_{1\le i