--- title: Constant Prediction Number in Set Theory url: https://www.emergentmind.com/topics/constant-prediction-number type: topic --- # Constant Prediction Number in Set Theory Searching arXiv for recent papers on the set-theoretic constant prediction number and related terminology. The **constant prediction number** is a predictor-based cardinal invariant, usually denoted \(\mathfrak v_n^{\mathrm{const}}\) and, in the binary case, \(\mathfrak v_2^\mathsf{const}\). It measures the least size of a family of predictors sufficient to predict every real in a bounded-gap sense: a predictor need not be eventually correct at every coordinate, but it must be correct at least once in each interval of some fixed finite length. In current work, the notion is paired with the **constant evasion number**, organized via relational systems, and connected to Tukey reducibility, ideals on product spaces, and forcing constructions that place these invariants alongside the classical characteristics in Cichoń’s diagram [2507.11339] [2503.24185]. ## 1. Core definition In the binary formulation, a predictor is a function \[ \pi:{}^{<\omega}2\to 2. \] For a real \(f\in{}^\omega 2\), the relation \[ f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr) \] means that \(\pi\) **predicts constantly** \(f\). Equivalently, there is a fixed finite window length \(k\) such that, for all sufficiently large \(i\), every interval \([i,i+k)\) contains some coordinate where the predictor correctly guesses \(f(j)\) from the initial segment \(f{\upharpoonright}j\) [2503.24185]. The constant prediction number is then \[ \mathfrak{v}_2^\mathsf{const} = \min\bigl\{|S| : S\subseteq \Pi_2 \text{ and } \forall x\in{}^\omega 2\ \exists \pi\in S\ (x\sqsubset^{pc}\pi)\bigr\}, \] where \(\Pi_2\) is the class of all binary predictors. Thus \(\mathfrak{v}_2^\mathsf{const}\) is the smallest size of a family of predictors that constantly predicts every binary real [2503.24185]. The broader framework replaces \(2\) by an arbitrary \(n\le \omega\). There a predictor is a function \[ \sigma:\,{}^{<\omega}n\to n, \] and constant prediction is written \[ f\ ^{cp}_{=}\ \sigma \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ \bigl(f(j)=\sigma(f{\upharpoonright}j)\bigr). \] This formulation makes explicit the intended strengthening of ordinary predictor correctness: the issue is not eventual pointwise correctness, but correctness recurring with uniformly bounded gaps [2507.11339]. ## 2. Duality with constant evasion The dual invariant is the **constant evasion number**. In the binary setting it is \[ \mathfrak{e}_2^\mathsf{const} = \min\bigl\{|F| : F\subseteq{}^\omega 2 \text{ and } \neg\exists \pi\in\Pi_2\ \forall f\in F\ (f\sqsubset^{pc}\pi)\bigr\}. \] A family \(F\) witnesses \(\mathfrak e_2^\mathsf{const}\) when no single predictor constantly predicts every member of \(F\) [2503.24185]. This duality is naturally expressed through a relational-system viewpoint. For constant prediction, the points are reals \(f\), the witnesses are predictors \(\pi\), and the relation is constant predictability. In that language, \(\mathfrak e_2^\mathsf{const}\) is the minimal size of an unbounded family of reals, while \(\mathfrak v_2^\mathsf{const}\) is the minimal size of a dominating family of predictors [2503.24185]. The Tukey-theoretic structure is already visible in the binary theory. The system \(E_2\) for constant prediction admits variants \(E_2^k\), and the comparison \[ E_2^k \le_T E_2 \] yields \[ e_2^\mathsf{const}=b(E_2) \le b(E_2^k),\qquad d(E_2^k)\le d(E_2)=\mathfrak{v}_2^\mathsf{const}. \] These inequalities are part of the preservation machinery used in forcing arguments, and they show that the bounded-gap relation is robust under natural finite-window refinements [2503.24185]. ## 3. Generalizations and variants The general theory extends constant prediction beyond equality on fixed finite alphabets. Given a sequence of binary relations \(\rel=(\rel_n)_{n\in\omega}\) with \(\rel_n\subseteq b(n)\times b(n)\), one defines \[ f^{cp}_{\rel}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ \bigl(f(j)\rel_j\sigma(f{\upharpoonright}j)\bigr). \] This makes the bounded-gap paradigm available for many kinds of local correctness conditions, not just exact equality [2507.11339]. The paper on generalization and variants isolates several important families of invariants: equality prediction on \(\prod b\), inequality prediction, the relation \(\le\), the finite-base versions \(e^{\mathrm{const}}_K\) and \(v^{\mathrm{const}}_K\), and the corresponding “eventual” analogues. In this setting the constant prediction number is the domination number of the corresponding constant-prediction relational system \(E^{cp}_{b,\rel}\) [2507.11339]. A key comparison is with eventual prediction: \[ f^{pr}_{\rel}\sigma \iff \forall^\infty j\ \bigl(f(j)\rel_j\sigma(f{\upharpoonright}j)\bigr). \] Since eventual prediction implies constant prediction, there is a Tukey reduction \[ E^{cp}_{b,\rel}\leq_T E^{pr}_{b,\rel}, \] and hence \[ e^\forall_{b,\rel}\le e^{\mathrm{const}}_{b,\rel}, \qquad v^{\mathrm{const}}_{b,\rel}\le v^\forall_{b,\rel}. \] The resulting picture places constant prediction between ordinary eventual prediction and more classical cardinal characteristics. Intuitively, it measures how many predictors are needed to cover all reals in a very strong local sense: every real must be hit by some predictor infinitely often with bounded gaps [2507.11339]. ## 4. Ideals and connections with classical invariants The general theory associates to each predictor \(\sigma\) and window length \(k\) a set \(A_{b,\rel}^{\sigma,k}\) of reals whose behavior is constantly predicted by \(\sigma\) with bound \(k\). From these sets one obtains ideals \(I^{\mathrm{const},0}_{b,\rel}\) and \(I^{\mathrm{const}}_{b,\rel}\) on \(\prod b\). The constant prediction and evasion numbers are exactly the uniformity and covering-type invariants of these ideals [2507.11339]. This ideal-theoretic description connects the invariants to the standard small-unboundedness framework. The paper states that \[ I^{\mathrm{const}}_{b,\rel}\subseteq M(\prod b), \] and, for finite alphabets, \[ I^{\mathrm{const}}_{K,\rel}\subseteq E(\omega^K), \] where \(E\) denotes the \(F_\sigma\)-measure-zero ideal. These inclusions provide the bridge from predictor-based combinatorics to the classical ideals that organize Cichoń-type diagrams [2507.11339]. A major structural theme is that many of the constant-prediction variants lie between the classical bounding and dominating numbers \(b,d\), the meager-ideal invariants \((M)\), and the null-ideal invariants \((N)\). In particular, the paper identifies inequality prediction as yielding a new combinatorial characterization of the meager ideal’s uniformity and covering numbers. This suggests that constant prediction is not merely a refinement of classical prediction notions, but a source of alternate presentations of familiar invariants [2507.11339]. ## 5. Forcing and consistency landscape The current forcing theory shows that constant prediction and evasion can be inserted into large constellations of cardinal characteristics. A central result is that \(\mathfrak e_2^\mathsf{const}\) and \(\mathfrak v_2^\mathsf{const}\) can be added to **Cichoń’s maximum with distinct values** by ccc forcing constructions [2503.24185]. The constructions use FS iterations, simple matrix iterations, ultrafilter limits, finitely additive measure limits in one alternative construction, and a \(<\theta\)-uf-extendable matrix iteration framework. The preservation tools include \(E_2^k\)-goodness, \(Lc^*\)-goodness, \(\Pr^2_{\bar n}(\lambda)\), and iteration lemmas ensuring that the intended inequalities persist while the forcing remains ccc [2503.24185]. The later generalization paper records a range of further consistency results. Among them are the statements that it is consistent that \[ b=\nu