--- title: Constant Evasion Number in Cardinal Characteristics url: https://www.emergentmind.com/topics/constant-evasion-number type: topic --- # Constant Evasion Number in Cardinal Characteristics The **constant evasion number** is a cardinal characteristic of the continuum defined from a bounded-gap version of prediction on reals. For \(n\le \omega\), a predictor is a function \(\sigma:{}^{<\omega}n\to n\), and a real \(f\in{}^\omega n\) is **constantly predicted** by \(\sigma\) when there exists a fixed \(k\in\omega\) such that every interval of length \(k\) contains some coordinate at which the predictor is correct: \[ f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j). \] The constant evasion number \(e_n^{const}\) is the least size of a family of reals that no single predictor can constantly predict all at once, while the dual constant prediction number \(v_n^{const}\) is the least size of a family of predictors covering all reals under the same relation. Recent work places these invariants within the modern theory of relational systems, ideals, and forcing, and shows that the binary instances can be separated from the classical characteristics in Cichoń-style models [2507.11339] [2503.24185]. ## 1. Definition and bounded-gap prediction The defining feature of constant evasion is the replacement of ordinary eventual prediction by a **bounded-gap** success criterion. Instead of asking that a predictor be correct at all sufficiently large stages, one asks for a finite bound \(k\) such that every interval \([i,i+k)\) contains at least one correct prediction. In the notation used for the binary case, \[ f\sqsubset^{pc}\pi \] means \[ \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr), \] so the predictor need only hit the target repeatedly with bounded gaps rather than stabilize to correctness at every later coordinate [2503.24185]. For general \(n\le\omega\), the constant evasion number is \[ e_n^{const}:=\min\bigl\{|F|:\,F\subseteq{}^\omega n\ \neg\exists\sigma\in\Sigma_n\ \forall f\in F\colon f^{cp}_{=}\sigma\bigr\}, \] and the dual constant prediction number is \[ v_n^{const}:=\min\bigl\{|S|:\,S\subseteq\Sigma_n\ \forall f\in{}^\omega n\ \exists\sigma\in S\colon f^{cp}_{=}\sigma\bigr\}. \] Thus \(e_n^{const}\) measures how many reals are needed before every single predictor fails on at least one of them, whereas \(v_n^{const}\) measures how many predictors are needed to cover the whole space [2507.11339]. This formulation is a refinement of the classical prediction/evasion framework of Blass. A plausible implication is that constant evasion isolates a stronger obstruction than ordinary evasion: the predictor is not merely required to succeed infinitely often, but to do so with uniformly bounded waiting time between successes. ## 2. Relational-system and ideal-theoretic formulations The constant evasion number is naturally expressed as the bounding number of a relational system. For equality on base \(n\), the relevant system is \[ E_n^{cp}=\bigl({}^\omega n,\Sigma_n,{}^{cp}_{=}\bigr), \] with \[ e_n^{const}=b(E_n^{cp}),\qquad v_n^{const}=d(E_n^{cp}). \] This places constant evasion directly inside the Tukey-theoretic language used for many cardinal characteristics [2507.11339]. The framework extends beyond equality. Given \(b\in\omega^\omega\) with \(b(n)>1\) and a sequence of relations \(\rel=\langle \rel_n\rangle_{n\in\omega}\), one defines \[ f^{cp}_{\rel}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)\rel_j \sigma(f\upharpoonright j), \] and from this the invariants \(e^{const}_{b,\rel}\) and \(v^{const}_{b,\rel}\). The notational special cases include \(e_b^{const},v_b^{const}\) for equality, \(e_{b,\ne}^{const},v_{b,\ne}^{const}\) for inequality, and \(e_\le^{const},v_\le^{const}\) for the order relation \(\le\) [2507.11339]. A major structural result is the ideal representation. For \(\sigma\in\Sigma_b\) and \(k\in\omega\), \[ A_{b,\rel}^{\sigma,k}=\{x\in\prod b:\forall i\in\omega\ \exists j\in[i,i+k)\ x(j)\rel_j \sigma(x\upharpoonright j)\}, \] and the associated ideals \(I^{const,0}_{b,\rel}\) and \(I^{const}_{b,\rel}\) satisfy \[ \operatorname{add}(I^{const}_{b,\rel})=e^{const}_{b,\rel},\qquad \operatorname{cof}(I^{const}_{b,\rel})=v^{const}_{b,\rel}. \] The paper also proves \[ I^{const}_{b,\rel}\subseteq M(\prod b),\qquad I^{const}_{K,\rel}\subseteq E(\mathbb{R}^K), \] together with \[ I_b^{const}\subseteq I_{b,\ne}^{const},\qquad I^{const}\subseteq I^{const}_{\le}\subseteq I^{const}_{\ne}. \] These identifications show that constant evasion is not only a combinatorial parameter on predictors, but also an ideal-theoretic invariant with additivity and cofinality interpretations [2507.11339]. One especially notable consequence is \[ e_\ne^{const}=(M),\qquad v_\ne^{const}=(M), \] which gives a constant-prediction characterization of the meager ideal in the \(\ne\)-setting [2507.11339]. ## 3. Position among classical cardinal characteristics The constant evasion number belongs to the broader prediction/evasion hierarchy. In the classical setup, a predictor is a pair \[ \pi=(D,\{\pi_n:n\in D\}), \] where \(D\subseteq\omega\) is infinite and each \(\pi_n:\omega^n\to\omega\); the classical evasion number \(\mathfrak e\) is the least size of a family of reals defeating every predictor, and the corresponding prediction number is \(\mathfrak{pr}\) [2401.14600]. That earlier work showed that \(\mathfrak e\) can itself be inserted into Cichoń’s maximum as a distinct characteristic: \[ \aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathfrak{e}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}<\mathrm{non}(\mathcal{N})<\mathrm{cof}(\mathcal{N})<2^{\aleph_0}. \] This provides the immediate backdrop for the constant variants [2401.14600]. For the constant framework, the general inequalities recorded in the literature include \[ \min\{e^{const}_{ubd},b\}\le e^{const}\le e^{const}_{ubd}\le e_b^{const}\le e_{b'}^{const}\le e_2^{const}, \] \[ e^{const}\le e^{const}_\le\le e^{const}_\ne, \] and \[ \max\{e^{const},b\}\le e^{const}_\le\le \min\{d,(M)\}. \] Dually, \[ v_2^{const}\le v_b^{const}\le v_{ubd}^{const}\le v^{const}\le \max\{v_{ubd}^{const},d\}, \] \[ v_\ne^{const}\le v_\le^{const}\le v^{const}, \] and \[ \max\{(M),b\}\le v_\le^{const}\le \min\{v^{const},d\}. \] The same source recalls \[ e^{const}\le (M),\qquad (M)\le v^{const},\qquad v_2^{const}\ge (M), \] hence in particular \(v_2^{const}\ge e_2^{const}\) [2507.11339]. In the notation of the binary forcing paper, several further comparisons are emphasized: \[ b\leq \mathfrak{e}_2^{\mathsf{const}},\qquad \mathfrak{e}_2^{\mathsf{const}}\leq non(E),\qquad \mathfrak{v}_2^{\mathsf{const}}\geq cov(E), \] and the consistency statements \[ \mathfrak{e}_2^{\mathsf{const}}cof(N)\quad\text{is consistent}. \] These comparisons indicate that the constant invariants interact nontrivially with the null, meager, and \(E\)-ideal characteristics rather than collapsing to a single pre-existing value [2503.24185]. ## 4. Forcing constructions and Cichoń-style separation A central recent result is that the binary constant evasion and constant prediction invariants can be threaded into a full Cichoń-style configuration with pairwise distinct values. The forcing results produce ccc forcing notions realizing constellations in which the classical characteristics and the constant prediction/evasion invariants take prescribed regular values [2503.24185]. One displayed configuration yields \[ add(N)=\lambda_1\leq \mathfrak{e}_2^{\mathsf{const}}=\lambda_2\leq cov(N)=\lambda_3\leq b=\lambda_4\leq non(M)=\lambda_5\leq cov(M)=\lambda_6=c, \] together with the corresponding placement of \(\mathfrak{v}_2^{\mathsf{const}}\) on the right-hand side of the diagram. A second theorem realizes \[ \mathfrak{e}_2^{\mathsf{const}}=\lambda_1\leq add(N)=\lambda_2\leq cov(N)=\lambda_3\leq b=\lambda_4\leq non(M)=\lambda_5\leq cov(M)=\lambda_6=c, \] and a third gives \[ add(N)=\lambda_1\leq \mathfrak{e}_2^{\mathsf{const}}=\lambda_2\leq b=\lambda_3\leq cov(N)=\lambda_4\leq non(M)=\lambda_5\leq cov(M)=\lambda_6=c. \] The point of these theorems is that the constant invariants can be inserted into the “maximum” configuration rather than merely separated in isolated models [2503.24185]. The proofs combine several forcing technologies: **ultrafilter-limit iterations**, **finitely additive measure/fam-limit iterations**, **simple matrix iterations**, preservation of goodness for suitable relational systems, and the new combinatorial property \(\Pr^2_{\bar n}(\lambda)\) together with a related \(\lambda\)-\(\bar\rho\)-cc condition. A key forcing \(P^K\), for finite \(K\), generalizes Brendle’s forcing for constant prediction and adds a generic predictor that constantly predicts all ground-model reals in \({}^\omega K\); this poset is proved to be **uniformly \(\sigma\)-uf-lim-linked**. The paper also proves that \(P^\omega\) is \(\sigma\)-uf-linked [2503.24185]. Another major ingredient is the preservation theorem for \(\Pr^2_{\bar n}(\lambda)\): if each iterand in a finite-support iteration satisfies \(\Pr^2_{\bar n}(\lambda)\), then the whole iteration does too. This is used to preserve strongly unbounded families for the relational system \(Lc^*\), from which the authors derive \[ P_\pi \Vdash add(N)\leq \lambda \quad\text{and}\quad |\pi|\leq cof(N). \] The final separation step uses the standard submodel method: for a large ccc forcing \(P\), one intersects with a suitably chosen \(\sigma\)-closed elementary submodel \(N\preceq H_\chi\), and a preservation lemma transfers relations of the form \(P\Vdash K\,R\) or \(P\Vdash R\,K\) to \(P\cap N\) and \(K\cap N\) [2503.24185]. ## 5. Generalizations, dualities, and sensitivity to small changes The constant evasion number sits inside a large family of variants. The paper on generalizations treats equality, inequality, order, varying bases \(b\), “ubd” and “hyp” envelopes, and the eventual-prediction analogues \[ e_{b,\rel}^{\forall},\qquad v_{b,\rel}^{\forall}, \] defined from \[ f^{pr}_{\rel}\sigma \iff \forall^\infty j\ f(j)\rel_j\sigma(f\upharpoonright j). \] There is a direct Tukey comparison \[ E^{cp}_{b,\rel}\le_T E^{pr}_{b,\rel}, \] hence \[ e_{b,\rel}^{\forall}\le e^{const}_{b,\rel},\qquad v^{const}_{b,\rel}\le v_{b,\rel}^{\forall}. \] For equality, the paper also shows that the version based on limited predictors collapses to the unrestricted one: \[ e_{*b}^{const}=e_b^{const},\qquad v_{*b}^{const}=v_b^{const}. \] These results indicate that the bounded-gap relation defines a robust hierarchy rather than a single isolated invariant [2507.11339]. A complementary theme is sensitivity to small changes in the underlying relation. The paper on zero-prediction modifies the classical prediction system only on the countable set \[ 0=\{z\in\omega^\omega : z(n)=0\text{ for almost all }n\}, \] producing the invariant \(e_0\). It proves \[ e_0\le e,\qquad e_0\ge \min\{e,\mathfrak b\}, \] and shows that for any uncountable regular cardinals \(\kappa\le\lambda\), it is consistent that \[ \kappa=e_0(N), \] together with the corresponding dual problems [2507.11339]. The zero-prediction paper asks whether ZFC proves \[ e_0=\min\{e,\mathfrak b\}, \] equivalently whether \(e_0\le \mathfrak b\) must always hold [2605.26611]. The current state of the subject therefore presents the constant evasion number as a genuine cardinal characteristic with multiple realizations: it has a bounded-gap combinatorial definition, a relational-system formulation, an ideal-theoretic interpretation, a rich forcing theory, and a developing network of variants whose behavior is only partly understood. Source: https://www.emergentmind.com/topics/constant-evasion-number