Automatic Cantor Real Bases

• Automatic Cantor real bases are variable-base numeration systems defined by sequences of real numbers >1 that yield unique greedy expansions.
• They employ finite-state methods such as sofic shifts, transducer computations, and substitution dynamics to link symbolic representations with arithmetic properties.
• The framework supports gap coding in B-integers, decision procedures for admissibility, and reveals strong algebraic rigidity in alternate numeral systems.

Searching arXiv for recent and relevant papers on Cantor real bases, alternate bases, automaticity, substitutions, and transducers. Automatic Cantor real bases arise in the study of Cantor real numeration systems, where the base is a sequence of real numbers greater than $1$ rather than a single radix. In the one-sided setting, a Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$ with $\prod_{n\in\mathbb N}\beta_n=+\infty$ yields expansions of the form

$$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$

while in the bi-infinite setting a base $B=(\beta_n)_{n\in\mathbb Z}$ with divergence of the forward and backward products provides positional representations of all non-negative real numbers (Charlier et al., 2021, Charlier et al., 2023). In this literature, “automaticity” is not a single formal notion. It is studied through sofic shift spaces defined by quasi-greedy boundary words, through substitutive and $S$-adic codings of $B$-integers and their gaps, and through transducers that compute expansions uniformly across infinite families of bases (Charlier et al., 2021, Charlier et al., 2023, Charlier et al., 7 Jul 2025).

1. Foundational numeration framework

A Cantor real base is a sequence of real numbers $>1$ satisfying a divergence condition that guarantees well-defined positional expansions. For $x\in[0,1)$, the greedy $\boldsymbol{\beta}$-expansion is obtained by the recursion

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$0

and, for $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$1,

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$2

This process converges because $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$3, and the resulting digit sequence is lexicographically maximal among all representations of the same number (Charlier et al., 2021).

The constant-base case $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$4 recovers classical Rényi $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$5-numeration, whereas integer-valued sequences recover classical Cantor series. Periodic Cantor real bases, called alternate bases, satisfy $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$6 for some period $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$7. They form the main setting in which finite-state and substitutive phenomena become explicit, because periodicity couples the local digit constraints to a finite family of shifted boundary expansions (Charlier et al., 2021).

The bi-infinite theory extends the numeration system from $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$8 to all non-negative reals. For a two-way real Cantor base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$9, a digit sequence with a left tail of zeros has a positional value obtained from positive- and negative-index contributions, and the greedy representation $\prod_{n\in\mathbb N}\beta_n=+\infty$0 generalizes the usual $\prod_{n\in\mathbb N}\beta_n=+\infty$1-expansion. In this setting, the analogue of the integers is the set

$\prod_{n\in\mathbb N}\beta_n=+\infty$2

whose combinatorics drives much of the symbolic theory (Charlier et al., 2023).

2. Admissibility, boundary expansions, and sofic structure

The central symbolic object for greedy expansions is the family of quasi-greedy expansions of $\prod_{n\in\mathbb N}\beta_n=+\infty$3 in all shifts of the base. If $\prod_{n\in\mathbb N}\beta_n=+\infty$4, then the generalized Parry theorem states that an infinite word $\prod_{n\in\mathbb N}\beta_n=+\infty$5 belongs to the language $\prod_{n\in\mathbb N}\beta_n=+\infty$6 of greedy expansions of numbers in $\prod_{n\in\mathbb N}\beta_n=+\infty$7 if and only if

$\prod_{n\in\mathbb N}\beta_n=+\infty$8

Accordingly, the $\prod_{n\in\mathbb N}\beta_n=+\infty$9-shift

$$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$0

is characterized by the weak inequalities

$$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$1

(Charlier et al., 2021).

For alternate bases of period $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$2, the decisive finite-state criterion is ultimate periodicity of the $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$3 shifted boundary words. The $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$4-shift is sofic if and only if $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$5 is ultimately periodic for every $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$6. This is the periodic-base analogue of the classical Bertrand-Mathis criterion for ordinary $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$7-shifts, and the proof is constructive: when the quasi-greedy expansions are ultimately periodic, a finite automaton $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$8 is built that accepts the factor language $$x=\sum_{n\in\mathbb N}\frac{a_n}{\prod_{i=0}^n \beta_i},$$9 (Charlier et al., 2021).

The lazy theory is parallel rather than independent. When

$B=(\beta_n)_{n\in\mathbb Z}$0

is finite, lazy expansions exist on $B=(\beta_n)_{n\in\mathbb Z}$1. They are obtained from greedy expansions by the digitwise complement map

$B=(\beta_n)_{n\in\mathbb Z}$2

which satisfies

$B=(\beta_n)_{n\in\mathbb Z}$3

The lazy $B=(\beta_n)_{n\in\mathbb Z}$4-shift is sofic if and only if all quasi-lazy $B=(\beta_n)_{n\in\mathbb Z}$5-expansions of $B=(\beta_n)_{n\in\mathbb Z}$6 are ultimately periodic, giving a lazy Bertrand-Mathis theorem in the alternate-base setting (Cisternino, 2022).

The main notions of finite-state structure therefore organize around boundary words rather than around a single fixed automaton.

Structure	Criterion	Source
Greedy $B=(\beta_n)_{n\in\mathbb Z}$7-shift is sofic	all shifted quasi-greedy expansions of $B=(\beta_n)_{n\in\mathbb Z}$8 are ultimately periodic	(Charlier et al., 2021)
Lazy $B=(\beta_n)_{n\in\mathbb Z}$9-shift is sofic	all shifted quasi-lazy expansions of $S$0 are ultimately periodic	(Cisternino, 2022)
Finite gap alphabet for $S$1-integers	the alternate base is Parry	(Charlier et al., 2023)

A common misconception is to identify all of these phenomena with automaticity in the narrow Cobham sense. The papers instead treat a broader finite-state and symbolic-dynamical regime: automata recognize factor languages, boundary words control admissibility, and periodicity of those boundary words is the mechanism that produces soficity (Charlier et al., 2021).

3. $S$2-integers, gap codings, and substitutive structure

For the bi-infinite system, the analogue of the $S$3-integers is the discrete set $S$4 with $S$5, $S$6, and $S$7. If $S$8 and $S$9 are consecutive $B$0-integers and $B$1 is the largest index at which their expansions differ, then

$B$2

so the set of possible gaps is encoded by the indices $B$3 rather than directly by their real values (Charlier et al., 2023).

The associated coding word is

$B$4

In general, $B$5 is over a countably infinite alphabet. The shift on bases,

$B$6

induces substitutions

$B$7

and Proposition 3.5 gives the recursive identity

$B$8

Iterating along the orbit of $B$9 under $>1$0 yields

$>1$1

so the infinite sequence coding the gaps between consecutive $>1$2-integers is $>1$3-adic (Charlier et al., 2023).

The periodic case is substantially more rigid. An alternate base $>1$4 is Parry if, for every $>1$5, the sequence $>1$6 is eventually periodic. Proposition 4.2 proves that the set of distances between consecutive $>1$7-integers is finite if and only if $>1$8 is Parry. In that case, the infinite-alphabet coding $>1$9 can be collapsed to a finite-alphabet word $x\in[0,1)$0, and Theorem 4.4 shows that

$x\in[0,1)$1

where the composition is primitive. Thus, in the Parry alternate case, the gap structure is not merely $x\in[0,1)$2-adic; it is a fixed point of a primitive substitution obtained by composing the $x\in[0,1)$3 substitutions along one period (Charlier et al., 2023).

This substitutive description generalizes the classical situation for constant $x\in[0,1)$4, where Fabre and Burdík et al. associated primitive substitutions to $x\in[0,1)$5-integers for Parry numbers. In the variable-base setting, the distinction between arbitrary Cantor bases and periodic Parry alternate bases is exactly the distinction between general $x\in[0,1)$6-adicity and primitive finite-alphabet substitution dynamics (Charlier et al., 2023).

4. Arithmetic finiteness, rewriting systems, and normalization

A second axis of automatic behavior concerns arithmetic closure of finite expansions. For an alternate base $x\in[0,1)$7, the set

$x\in[0,1)$8

consists of numbers whose greedy expansion has finite support. Two closure properties are studied: positive finiteness (PF), which requires closure under addition when the result remains in $x\in[0,1)$9, and finiteness property (F), which adds the corresponding subtraction closure (Masáková et al., 2023).

The arithmetic constraints are strong. If $\boldsymbol{\beta}$0 has (PF), then

$\boldsymbol{\beta}$1

is a Pisot or Salem number and each $\boldsymbol{\beta}$2. If $\boldsymbol{\beta}$3 has (F), then $\boldsymbol{\beta}$4 must be a simple Parry alternate base, meaning each expansion $\boldsymbol{\beta}$5 is finite, and every non-identity embedding $\boldsymbol{\beta}$6 yields a vector $\boldsymbol{\beta}$7 that is not positive (Masáková et al., 2023).

The constructive sufficient criterion is the monotonicity chain

$\boldsymbol{\beta}$8

If this holds for every $\boldsymbol{\beta}$9, then $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$00 is Parry, $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$01 has (PF), and if $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$02 is also simple Parry then it has (F). The proof is based on rewriting rules for finite digit strings. A finite string is admissible if each suffix is lexicographically smaller than the shifted expansion of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$03, and non-admissible strings contain bad patterns from a set $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$04. Each $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$05 can be rewritten into a lexicographically larger finite string $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$06 with the same value: $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$07 Repeated rewriting normalizes digitwise sums $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$08 to the greedy expansion $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$09, thereby providing a constructive addition algorithm for alternate bases (Masáková et al., 2023).

Termination is ensured by a weight function

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$10

with positive integer weights satisfying periodic invariance $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$11 and monotonicity under rewriting $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$12 for all $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$13. Since rewriting strictly increases lexicographic order without increasing weight, infinite rewriting is impossible (Masáková et al., 2023).

These arithmetic results interact with periodicity in a precise way. For alternate bases, eventual periodicity of all rational numbers in $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$14 already forces the same algebraic conclusion that had previously been known under a stronger assumption involving all shifts: $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$15 For pure periodicity, Property (PP) requires that every rational in some interval $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$16 have a purely periodic $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$17-expansion. If $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$18 has (PP), then $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$19 is a Pisot or Salem unit, each $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$20, and the same positivity obstruction under nontrivial embeddings appears. Conversely, if $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$21 has Property (F) and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$22 is a Pisot unit, then every shift $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$23 has Property (PP) (Masáková et al., 2024).

5. Single-transducer computation and decision procedures

A distinct and explicitly automata-theoretic viewpoint appears in the transducer model for variable bases. Instead of fixing a numeration system and varying the represented number, one fixes a real number $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$24 and allows the Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$25 to vary. For a finite alphabet $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$26, the greedy transducer $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$27 has state set $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$28, input alphabet $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$29, output alphabet $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$30, and transitions

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$31

where $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$32. Reading an input base sequence $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$33 from state $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$34 outputs exactly the greedy expansion $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$35. A quasi-greedy transducer $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$36 is defined analogously using $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$37 and outputs $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$38 (Charlier et al., 7 Jul 2025).

The main finiteness theorem states that if $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$39 is an algebraic integer of degree $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$40, $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$41 is a finite alphabet of Pisot numbers of degree $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$42 all lying in $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$43, and

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$44

then the reachable subtransducers $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$45 and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$46 are finite. The proof represents reachable states by polynomials in $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$47, controls all Galois conjugates using the Pisot property, and concludes finiteness from the lattice structure of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$48 (Charlier et al., 7 Jul 2025).

This uniform finite-state model has direct automatic and morphic consequences. If $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$49 consists of positive powers of a Pisot number $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$50, the same finiteness conclusion holds. If $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$51 is $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$52-automatic or morphic, then for $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$53, both $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$54 and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$55 are also $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$56-automatic or morphic, respectively (Charlier et al., 7 Jul 2025).

Periodicity is characterized graph-theoretically by the 2-walk property. Under the same algebraic hypotheses, the bases $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$57 for which $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$58 is finite or ultimately periodic are exactly the ultimately alternate bases if and only if the corresponding transducer has no two distinct closed walks from the same state with distinct inputs and identical output. The paper gives a decision procedure for this obstruction in

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$59

time, where $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$60 is the finite state set of the reachable transducer (Charlier et al., 7 Jul 2025).

The same framework yields admissibility decision procedures for automatic bases. If $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$61 is $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$62-automatic and the predicates

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$63

are first-order definable in $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$64, then it is decidable whether a given $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$65-automatic sequence is $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$66-admissible. More generally, admissibility is decidable when each shift boundary word has the form

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$67

for a finite language $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$68, with $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$69 and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$70 $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$71-definable (Charlier et al., 7 Jul 2025).

6. Algebraic rigidity, Parry data, and classification of alternate bases

The automatic and substitutive theories of alternate bases are accompanied by strong algebraic rigidity. One form appears in the Solomyak-type results for periodic Cantor real expansions on $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$72. If $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$73 is an alternate base and the expansion $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$74 or $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$75 of a suitable $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$76 is eventually periodic, with

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$77

and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$78, then $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$79 is algebraic over $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$80. Moreover, if $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$81 is a nontrivial conjugate of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$82 over that field and

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$83

is fixed by the conjugation, then

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$84

where $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$85 and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$86 is the minimal-modulus real solution of an explicit equation. In the special case $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$87 and $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$88, one obtains

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$89

hence

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$90

This extends the classical conjugate bound from one real base to periodic Cantor real bases (Caalim et al., 6 May 2025).

A second form of rigidity concerns the inverse problem: whether a finite collection of quasi-greedy expansion sequences determines the alternate base. For a period $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$91, let

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$92

and impose the Parry condition

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$93

Every alternate real base of period $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$94 yields such a $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$95-tuple of sequences. The open problem was whether the converse base is unique. The solution uses matrices with cyclically monotone rows: if every diagonal element is strictly larger than all other elements in its row, then the matrix is regular. This regularity yields injectivity of a map $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$96 built from alternate power series, and injectivity of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$97 implies uniqueness of the alternate base corresponding to a Parry-admissible $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$98-tuple (Šťovíček et al., 24 Apr 2026).

The outcome is a full one-to-one correspondence between periodic alternate Cantor real bases of period $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb N}$99 and $\prod_{n\in\mathbb N}\beta_n=+\infty$00-tuples of nonnegative integer sequences satisfying the Parry condition. Existence had been proved earlier, and uniqueness now holds in full generality. A plausible implication is that, in the periodic setting, the boundary data governing soficity, substitution dynamics, and admissibility are not merely descriptive invariants; they determine the base itself (Šťovíček et al., 24 Apr 2026).

Taken together, these developments show that automatic Cantor real bases are best understood as a convergence of three theories. The first is symbolic-dynamical, where admissibility is controlled by quasi-greedy or quasi-lazy boundary words and finite automata appear through soficity. The second is substitutive, where $\prod_{n\in\mathbb N}\beta_n=+\infty$01-integers and gap sequences are $\prod_{n\in\mathbb N}\beta_n=+\infty$02-adic in general and primitive-substitutive in the Parry alternate case. The third is automata-theoretic in the stronger computational sense, where a single finite reachable transducer can compute expansions of a fixed number across infinitely many bases under Pisot hypotheses. The algebraic theorems on periodicity, conjugates, and uniqueness show that these finite-state descriptions are coupled to strong arithmetic constraints rather than being purely combinatorial (Charlier et al., 2021, Charlier et al., 2023, Charlier et al., 7 Jul 2025, Masáková et al., 2024, Caalim et al., 6 May 2025, Šťovíček et al., 24 Apr 2026).