Constant Prediction Number in Set Theory

• Constant prediction number is defined as the smallest size of a predictor family that ensures every real is predicted with some fixed bounded gap.
• It is paired with the constant evasion number, establishing a duality that links predictor efficacy and unavoidable prediction failure in set theory.
• Forcing constructions and ideal-theoretic approaches reveal that constant prediction invariants occupy a distinct position within cardinal characteristics frameworks.

Searching arXiv for 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 (Cardona et al., 15 Jul 2025, Cardona et al., 31 Mar 2025).

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 $\mathfrak v_2^\mathsf{const}$0 from the initial segment $\mathfrak v_2^\mathsf{const}$1 (Cardona et al., 31 Mar 2025).

The constant prediction number is then

$\mathfrak v_2^\mathsf{const}$2

where $\mathfrak v_2^\mathsf{const}$3 is the class of all binary predictors. Thus $\mathfrak v_2^\mathsf{const}$4 is the smallest size of a family of predictors that constantly predicts every binary real (Cardona et al., 31 Mar 2025).

The broader framework replaces $\mathfrak v_2^\mathsf{const}$5 by an arbitrary $\mathfrak v_2^\mathsf{const}$6. There a predictor is a function

$\mathfrak v_2^\mathsf{const}$7

and constant prediction is written

$\mathfrak v_2^\mathsf{const}$8

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 (Cardona et al., 15 Jul 2025).

2. Duality with constant evasion

The dual invariant is the constant evasion number. In the binary setting it is

$\mathfrak v_2^\mathsf{const}$9

A family $\pi:{}^{<\omega}2\to 2.$0 witnesses $\pi:{}^{<\omega}2\to 2.$1 when no single predictor constantly predicts every member of $\pi:{}^{<\omega}2\to 2.$2 (Cardona et al., 31 Mar 2025).

This duality is naturally expressed through a relational-system viewpoint. For constant prediction, the points are reals $\pi:{}^{<\omega}2\to 2.$3, the witnesses are predictors $\pi:{}^{<\omega}2\to 2.$4, and the relation is constant predictability. In that language, $\pi:{}^{<\omega}2\to 2.$5 is the minimal size of an unbounded family of reals, while $\pi:{}^{<\omega}2\to 2.$6 is the minimal size of a dominating family of predictors (Cardona et al., 31 Mar 2025).

The Tukey-theoretic structure is already visible in the binary theory. The system $\pi:{}^{<\omega}2\to 2.$7 for constant prediction admits variants $\pi:{}^{<\omega}2\to 2.$8, and the comparison

$\pi:{}^{<\omega}2\to 2.$9

yields

$f\in{}^\omega 2$0

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 (Cardona et al., 31 Mar 2025).

3. Generalizations and variants

The general theory extends constant prediction beyond equality on fixed finite alphabets. Given a sequence of binary relations $f\in{}^\omega 2$1 with $f\in{}^\omega 2$2, one defines

$f\in{}^\omega 2$3

This makes the bounded-gap paradigm available for many kinds of local correctness conditions, not just exact equality (Cardona et al., 15 Jul 2025).

The paper on generalization and variants isolates several important families of invariants: equality prediction on $f\in{}^\omega 2$4, inequality prediction, the relation $f\in{}^\omega 2$5, the finite-base versions $f\in{}^\omega 2$6 and $f\in{}^\omega 2$7, and the corresponding “eventual” analogues. In this setting the constant prediction number is the domination number of the corresponding constant-prediction relational system $f\in{}^\omega 2$8 (Cardona et al., 15 Jul 2025).

A key comparison is with eventual prediction: $f\in{}^\omega 2$9 Since eventual prediction implies constant prediction, there is a Tukey reduction

$$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$0

and hence

$$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$1

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 (Cardona et al., 15 Jul 2025).

4. Ideals and connections with classical invariants

The general theory associates to each predictor $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$2 and window length $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$3 a set $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$4 of reals whose behavior is constantly predicted by $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$5 with bound $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$6. From these sets one obtains ideals $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$7 and $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$8 on $$f\sqsubset^{pc}\pi \quad\Longleftrightarrow\quad \exists k\in\omega\ \forall^\infty i\ \exists j\in[i,i+k)\ \bigl(f(j)=\pi(f{\upharpoonright}j)\bigr)$$9. The constant prediction and evasion numbers are exactly the uniformity and covering-type invariants of these ideals (Cardona et al., 15 Jul 2025).

This ideal-theoretic description connects the invariants to the standard small-unboundedness framework. The paper states that

$\pi$0

and, for finite alphabets,

$\pi$1

where $\pi$2 denotes the $\pi$3-measure-zero ideal. These inclusions provide the bridge from predictor-based combinatorics to the classical ideals that organize Cichoń-type diagrams (Cardona et al., 15 Jul 2025).

A major structural theme is that many of the constant-prediction variants lie between the classical bounding and dominating numbers $\pi$4, the meager-ideal invariants $\pi$5, and the null-ideal invariants $\pi$6. 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 (Cardona et al., 15 Jul 2025).

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 $\pi$7 and $\pi$8 can be added to Cichoń’s maximum with distinct values by ccc forcing constructions (Cardona et al., 31 Mar 2025).

The constructions use FS iterations, simple matrix iterations, ultrafilter limits, finitely additive measure limits in one alternative construction, and a $\pi$9-uf-extendable matrix iteration framework. The preservation tools include $f$0-goodness, $f$1-goodness, $f$2, and iteration lemmas ensuring that the intended inequalities persist while the forcing remains ccc (Cardona et al., 31 Mar 2025).

The later generalization paper records a range of further consistency results. Among them are the statements that it is consistent that

$f$3

that

$f$4

that

$f$5

that

$f$6

and that

$f$7

It also proves models in which the generalized invariants $f$8 and $f$9 take prescribed regular values and are separated from $k$0 (Cardona et al., 15 Jul 2025).

These results make clear that constant prediction numbers are not pinned down by the classical diagram. They can be manipulated independently enough to occupy genuinely new positions in the forcing landscape, while still interacting tightly with the meager and null ideals and with the standard $k$1 and $k$2 invariants.

6. Terminological scope and unrelated uses

The phrase constant prediction number has acquired several unrelated technical uses. In the set-theoretic literature discussed above, it denotes the least size of a family of predictors that constantly predicts every real. In other literatures, however, similar wording refers to different objects.

Area	Meaning of the phrase or nearby phrase	Representative result
Set theory	Least size of a family of predictors that predicts every real with bounded gaps	$k$3 and its variants (Cardona et al., 31 Mar 2025, Cardona et al., 15 Jul 2025)
Several-steps-ahead prediction	Prediction horizon $k$4, with LLN when $k$5; the paper explicitly states its main theorem is not about a fixed constant prediction number	Quantitative LLN for $k$6-step-ahead forecasts (Vovk, 24 Aug 2025)
Card-guessing with feedback	“Constant prediction number” refers to the asymptotic advantage above the trivial baseline $k$7, governed by the explicit constant $k$8 multiplying $k$9	$i$0 (Ottolini et al., 2022)
Peer prediction	A constant number of tasks needed for exact dominant truthfulness	DMI-Mechanism is dominantly truthful when $i$1 (Kong, 2019)
Online sequence prediction	Horizon-independent regret achieved with a constant-sized prediction support	Constant regret with $i$2 and $i$3 under bounded exp-concave loss (Saad et al., 2022)

This terminological spread makes the set-theoretic definition especially important to state explicitly. In that setting, “constant” refers to bounded-gap correctness rather than to a fixed forecast horizon, a constant number of tasks, or a constant regret bound. The modern theory of $i$4 is therefore best understood as part of the general program of extracting new cardinal invariants from predictor relations and situating them within the forcing and ideal-theoretic structure surrounding Cichoń’s diagram.