$\\{X_{i},...,X_{n_{\bm{Z}}}\\}$ for any $1\leq i\leq n_{\bm{Z}}$. Then we
define the circuit $\mathds{1}_{\bm{Z}}$ recursively as follows, where
$\mathds{1}_{\bm{Z}}:=\mathds{1}_{\bm{Z}_{\geq 1}}$ (where $\times$ indicates
a product node with its arguments as children):

$\mathds{1}_{\bm{Z}_{\geq
i}}:=\begin{cases}\mathds{1}_{Z_{i}}\times\mathds{1}_{\bm{Z}_{\geq i+1}}&1\leq
i<n_{\bm{Z}}\\\ \mathds{1}_{Z_{i}}&i=n_{\bm{Z}}\end{cases}$ (11)

This circuit consists of a series of product units, and leaf units
$\mathds{1}_{Z_{i}}$ for each $Z_{i}\in Z$ which we define to encode the
function $p_{\mathds{1}_{Z_{i}}}(Z_{i})\equiv 1$ (for all values of $Z_{i}$).
Thus, the circuit as a whole encodes $p_{\mathds{1}_{\bm{Z}}}(\bm{Z})\equiv 1$
for all values of $\bm{Z}$. In terms of structural and support properties, the
circuit is trivially deterministic and smooth as it does not contain any sum
nodes, and is clearly also structured decomposable. Finally, it can also be
seen that the size $|\mathds{1}_{\bm{Z}}|$ (number of edges) of
$\mathds{1}_{\bm{Z}}$ is $O(n_{\bm{Z}})$.

We now design a circuit $C_{\bm{Z}}$ to encode the function
$e^{-(w_{0}+\sum_{i}w_{i}z_{i})}$ as follows, where
$C_{\bm{Z}}:=C_{\bm{Z}_{\geq 1}}$:

$C_{\bm{Z}_{\geq i}}:=\begin{cases}C_{Z_{i}}\times C_{\bm{Z}_{\geq i+1}}&1\leq
i<n_{\bm{Z}}\\\ C_{Z_{i}}&i=n_{\bm{Z}}\end{cases}$ (12)

where we define leaf nodes $C_{Z_{i}}$ to encode
$p_{C_{Z_{i}}}(Z_{i}):=e^{-w_{i}Z_{i}}$ for $1\leq i<n_{\bm{Z}}$ and
$p_{C_{Z_{i}}}(Z_{i}):=e^{-(w_{0}+w_{i}Z_{i})}$ for $i=n_{\bm{Z}}$. By
recursion it can be seen that this circuit does indeed encode
$p_{C_{\bm{Z}}}(\bm{Z})=e^{-(w_{0}+\sum_{i}w_{i}z_{i})}$. This circuit is
deterministic and smooth, and also decomposes in the same way as
$\mathds{1}_{\bm{Z}}$, i.e. they are structured decomposable with the same
vtree. It can also be seen that the size $|C_{\bm{Z}}|$ of $C_{\bm{Z}}$ is
$O(n_{\bm{Z}})$.

Now, consider the following probabilistic circuit over
$\bm{V}=\bm{X}\cup\bm{Y}\cup\bm{Z}$ (where $\times,+$ represent product, sum
nodes respectively):

$C:=\mathds{1}_{y}\times(\mathds{1}_{x}\times\mathds{1}_{\bm{Z}}+\mathds{1}_{\neg
x}\times\mathds{1}_{\bm{Z}})+\mathds{1}_{\neg y}\times(\mathds{1}_{x}\times
C_{\bm{Z}}+\mathds{1}_{\neg x}\times\mathds{1}_{\bm{Z}})$ (13)

$C$ is structured decomposable as all of the product units with the same scope
in the equation above decompose in the same way, and we have seen that
$\mathds{1}_{\bm{Z}}$ and $C_{\bm{Z}}$ are structured decomposable with
respect to the same vtree. It is also smooth and deterministic as the
individual circuits $\mathds{1}_{\bm{Z}}$ and $C_{\bm{Z}}$ are smooth and
deterministic, and the sum nodes in the equation satisfy determinism by the
fact that $(\mathds{1}_{y},\mathds{1}_{\neg y})$ and
$(\mathds{1}_{x},\mathds{1}_{\neg x})$ have disjoint support. Finally, as the
sizes of $\mathds{1}_{\bm{Z}}$ and $C_{\bm{Z}}$ are $O(n_{\bm{Z}})$,
$|C(\bm{V}))|$ is also $O(n_{\bm{Z}})$.

Now, we show that the backdoor query on $C$ is equivalent to solving the
corresponding EXPLR problem. First, we derive expressions for all of the
individual components of the backdoor formula on the circuit $C$, by
evaluating according to Equation 13:

$\displaystyle p_{C}(x,y,\bm{z})$
$\displaystyle=p_{\mathds{1}_{y}}(y)\times(p_{\mathds{1}_{x}}(x)\times
p_{\mathds{1}_{\bm{Z}}}(\bm{z})+p_{\mathds{1}_{\neg x}}(x)\times
p_{\mathds{1}_{\bm{Z}}}(\bm{z}))+p_{\mathds{1}_{\neg
y}}(y)\times(p_{\mathds{1}_{x}}(x)\times
p_{C_{\bm{Z}}}(\bm{z})+p_{\mathds{1}_{\neg x}}(x)\times
p_{\mathds{1}_{\bm{Z}}}(\bm{z}))$ $\displaystyle=1\times(1\times 1+0\times
1)+0\times(1\times 1+0\times 1)$ $\displaystyle=1$ $\displaystyle
p_{C}(x,\bm{z})$ $\displaystyle=p_{C}(x,y,\bm{z})+p_{C}(x,\neg y,\bm{z})$
$\displaystyle=1+C_{\bm{Z}}(\bm{z})$ $\displaystyle p_{C}(\bm{z})$
$\displaystyle=p_{C}(x,\bm{z})+p_{C}(\neg x,y,\bm{z})+p_{C}(\neg x,\neg
y,\bm{z})$ $\displaystyle=p_{C}(x,\bm{z})+1+1$
$\displaystyle=p_{C}(x,\bm{z})+2$

The backdoor query for $C$ can then be expressed as

$\displaystyle p_{\bm{x}}(\bm{y})$
$\displaystyle=\sum_{\bm{z}}p_{C}(\bm{z})p_{C}(y|x,\bm{z})=\sum_{\bm{z}}(p_{C}(x,\bm{z})+2)\frac{p_{C}(x,y,\bm{z})}{p_{C}(x,\bm{z})}=\sum_{\bm{z}}\left[1+\frac{2}{1+p_{C_{\bm{Z}}}(\bm{z})}\right]$
$\displaystyle=2^{n_{\bm{Z}}}+2\sum_{\bm{z}}\frac{1}{1+e^{-(w_{0}+\sum_{i}w_{i}z_{i})}}$

Thus, if we can compute the backdoor query for $C$, then we can compute the
given EXPLR problem, completing the reduction. ∎

The significance of this result is that it implies hardness of causal
inference for structured decomposable and deterministic PCs whenever there is
a valid backdoor adjustment (whether we use the backdoor formula or not), one
of the simplest and most common cases where the causal effect is identifiable.

######  Corollary 1.

For any interventional query $p_{\bm{x}}(\bm{y})$ and causal diagram $G$ such
that the query is identifiable through a backdoor adjustment, and the
observational distribution $p(\bm{V})$ given as a decomposable and smooth
circuit $C$ encoding $p$, computing $p_{\bm{x}}(\bm{y})$ is #P-hard, even if
the circuit is structured decomposable and deterministic.

###### Proof.

By the identifiability condition, we have that
$p_{\bm{x}}(\bm{y})=\sum_{\bm{z}}p(\bm{y}|\bm{x},\bm{z})p(\bm{z})=\sum_{\bm{z}}C(\bm{z})C(\bm{y}|\bm{x},\bm{z})$,
which is the backdoor query. Hardness of computing the causal effect then
follows from hardness of backdoor queries for the probabilistic circuit. ∎

###  C.3 MD-calculus and Causal Inference

As sketched in the main paper, MD-calculus provides the tools for us to
analyze what properties we need (to add to structured decomposability and
determinism) to enable tractable computation of causal queries. We now provide
the full derivations for the backdoor, frontdoor, and extended napkin cases.
For convenience, in the following we will refer to the intermediate circuits
in a computation by the functions they encode, e.g. $C(p_{C}(\bm{X},\bm{Z}))$
for the circuit obtained from applying the
$\texttt{MARG}(\cdot;\bm{V}\setminus(\bm{X}\cup\bm{Z}))$ operation to $C$.

####  C.3.1 Backdoor

We begin with the backdoor query. In cases where there is a valid backdoor
adjustment set, such as in Figure 2(a), we have the following formula for the
interventional distribution:

$\displaystyle
p_{C,\bm{X}}(\bm{Y})=\sum_{\bm{Z}}p_{C}(\bm{Z})p_{C}(\bm{Y}|\bm{X},\bm{Z})=\sum_{\bm{Z}}p_{C}(\bm{Z})\frac{p_{C}(\bm{Y},\bm{X},\bm{Z})}{p_{C}(\bm{X},\bm{Z})}$

Before continuing, it is worth noting that the expression above is valid for
any value of $\bm{X},\bm{Y}$, though in causal inference it is more typical
that we are interested in evaluating $p_{C,\bm{X}}(\bm{Y})$ for specific
values $\bm{x}$ of $\bm{X}$, i.e. a specific intervention, or a small set of
interventions. This distinction is important as we will see, interestingly,
that instantiating $\bm{X}$ makes the query more tractable in the sense that
the marginal determinism requirements for the circuit $C$ are more relaxed.

To apply the MD-calculus, we first identify the deterministic operations in
the pipeline. For the backdoor query, the only deterministic operation is the
$\texttt{POW}(\cdot;-1)$ operation, applied to $C(p_{C}(\bm{X},\bm{Z}))$ which
requires a deterministic input circuit. Then, we can work backwards through
the pipeline from POW in order to derive tractability conditions on $C$.

  1. 1.

Requirement: $C(p_{C}(\bm{X},\bm{Z}))$ is $(\bm{X}\cup\bm{Z})$-deterministic.

  2. 2.

$\texttt{MARG}(\cdot;\bm{Y})$: $C(p_{C}(\bm{X},\bm{Y},\bm{Z}))$ is
$(\bm{X}\cup\bm{Z})$-deterministic $\implies$ $C(p_{C}(\bm{X},\bm{Z}))$ is
$(\bm{X}\cup\bm{Z})$-deterministic

  3. 3.

$\texttt{MARG}(\cdot;\bm{V}\setminus(\bm{X}\cup\bm{Y}\cup\bm{Z}))$:
$C(p_{C}(\bm{V}))$ is $(\bm{X}\cup\bm{Z})$-deterministic$\implies$
$C(p_{C}(\bm{X},\bm{Y},\bm{Z}))$ is $(\bm{X}\cup\bm{Z})$-deterministic

  4. 4.

Sufficient Condition: $C=C(p_{C}(\bm{V}))$ is
$(\bm{X}\cup\bm{Z})$-deterministic.

This simple derivation shows that it suffices for $C$ to be
$(\bm{X}\cup\bm{Z})$-deterministic to compute the backdoor query. Now, let us
consider the case in which we instantiate $\bm{X}$ with a value $\bm{x}$:

$\displaystyle
p_{C,\bm{x}}(\bm{Y})=\sum_{\bm{Z}}p_{C}(\bm{Z})p_{C}(\bm{Y}|\bm{x},\bm{Z})=\sum_{\bm{Z}}p_{C}(\bm{Z})\frac{p_{C}(\bm{Y},\bm{x},\bm{Z})}{p_{C}(\bm{x},\bm{Z})}$

Here, employing the MD-calculus gives us the following:

  1. 1.

Requirement: $C(p_{C}(\bm{x},\bm{Z})$) is $\bm{Z}$-deterministic.

  2. 2.

$\texttt{MARG}(\cdot;\bm{Y})$: $C(p_{C}(\bm{x},\bm{Y},\bm{Z}))$ is
$\bm{Z}$-deterministic $\implies$ $C(p_{C}(\bm{x},\bm{Z}))$ is
$\bm{Z}$-deterministic

  3. 3.

$\texttt{MARG}(\cdot;\bm{V}\setminus(\bm{X}\cup\bm{Y}\cup\bm{Z}))$:
$C(p_{C}(\bm{x},\bm{V}\setminus\bm{X}))$ is $\bm{Z}$-deterministic $\implies$
$C(p_{C}(\bm{x},\bm{Y},\bm{Z}))$ is $\bm{Z}$-deterministic

  4. 4.

$\texttt{INST}(\cdot;\bm{x})$: $C(p_{C}(\bm{V}))$ is
$(\bm{X}^{\prime}\cup\bm{Z})$-deterministic for some
$\bm{X}^{\prime}\subseteq\bm{X}$ $\implies$
$C(p_{C}(\bm{x},\bm{V}\setminus\bm{X}))$ is $\bm{Z}$-deterministic

  5. 5.

Sufficient Condition: $C$ is $(\bm{X}^{\prime}\cup\bm{Z})$-deterministic for
some $\bm{X}^{\prime}\subseteq\bm{X}$.

Notice that in the requirement, due to the instantiation, the input to the POW
operation $C(p_{C}(\bm{x},\bm{Z}))$ has scope $\bm{Z}$, meaning that we
require it to be $\bm{Z}$-deterministic rather than
$(\bm{X}\cup\bm{Z})$-deterministic. In the final step, we use the MD-calculus
rule for instantiation. This shows that the instantiated backdoor adjustment
is tractable for a wider range of circuits than if we insisted on a circuit
encoding $p_{C,\bm{X}}(\bm{Y})$ as a function of $\bm{X}$ (and $\bm{Y}$).

####  C.3.2 Frontdoor

Another common case where the causal effect is identifiable is the frontdoor
causal diagram, shown in Figure 5(b). Unlike the backdoor case, there is
unobserved confounding of $\bm{X}$ and $\bm{Y}$, represented by the dashed
bidirectional arrow. However, the existence of the observed mediator $\bm{Z}$
nonetheless allows for identifiability, via the formula

$p_{C,\bm{X}}(\bm{Y})=\sum_{\bm{Z}}p_{C}(\bm{Z}|\bm{X})\sum_{\bm{X}^{\prime}}p_{C}(\bm{X}^{\prime})p_{C}(\bm{Y}|\bm{X}^{\prime},\bm{Z})$
(14)

We now employ the MD-calculus to derive tractability conditions. This time,
looking at the conditionals, there are two deterministic (POW) operations, as
well as an auxiliary variable $\bm{X}^{\prime}$ (that is summed out in the
end), which has the same joint distribution with $\bm{V}\setminus\bm{X}$ as
$\bm{X}$.

  1. 1.

Requirement: $C(p_{C}(\bm{X}))$ is $\bm{X}$-deterministic and
$C(p_{C}(\bm{X},\bm{Z}))$ is $(\bm{X}\cup\bm{Z})$-deterministic.

  2. 2.

$\texttt{MARG}(\cdot;\bm{V}\setminus\bm{X})$: $C(p_{C}(\bm{V}))$ is
$\bm{X}$-deterministic $\implies$ $C(p_{C}(\bm{X}))$ is
$\bm{X}$-deterministic.

  3. 3.

$\texttt{MARG}(\cdot;\bm{V}\setminus(\bm{X}\cup\bm{Z}))$: $C(p_{C}(\bm{V}))$
is $(\bm{X}\cup\bm{Z})$-deterministic $\implies$ $C(p_{C}(\bm{X}\cup\bm{Z}))$
is $(\bm{X}\cup\bm{Z})$-deterministic.

  4. 4.

Sufficient Condition: $C$ is $\bm{X}$-deterministic and
$(\bm{X}\cup\bm{Z})$-deterministic.

Now, let us consider instantiating $\bm{X}$ with value $\bm{x}$. In this case,
we have:

$p_{C,\bm{x}}(\bm{Y})=\sum_{\bm{Z}}p_{C}(\bm{Z}|\bm{x})\sum_{\bm{X}^{\prime}}p_{C}(\bm{X}^{\prime})p_{C}(\bm{Y}|\bm{X}^{\prime},\bm{Z})$
(15)

Note that in this case, the conditional $p_{C}(\bm{Z}|\bm{x})$ does not impose
any determinism requirements, since the input to the POW operation is a scalar
$C(p_{C}(\bm{x}))$. However, the requirements for the other conditional remain
the same, as $\bm{X}^{\prime}$ is an auxiliary variable that is not tied to
the intervention value $\bm{x}$. Overall, we can conclude that $C$ being
$(\bm{X}\cup\bm{Z})$-deterministic is sufficient for the instantiated
frontdoor formula, which is again weaker than in the non-instantiated case.

####  C.3.3 Extended Napkin

The extended napkin causal graph in Figure 2(b) is an extension of the so-
called napkin causal graph pearl09causality, which is obtained by removing
$\bm{K}$ from the graph. For this diagram, the do-calculus gives us the
following formula for the interventional distribution:

$\displaystyle\centering p_{C,\bm{X}}(\bm{Y})=\sum_{\bm{K}}\@add@centering$
$\displaystyle\left(\sum_{\bm{W},\bm{X}^{\prime},\bm{Y}^{\prime}}p_{C}(\bm{X}^{\prime},\bm{Y}^{\prime}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K})\right)\frac{\sum_{\bm{W}}p_{C}(\bm{X},\bm{Y}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K})}{\sum_{\bm{W}}p_{C}(\bm{X}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K})}$

As described in the main paper, this is a case where we need to instantiate
$\bm{X}$ for tractability, leading to the following formula:

$\displaystyle\centering p_{C,\bm{x}}(\bm{Y})=\sum_{\bm{K}}\@add@centering$
$\displaystyle\left(\sum_{\bm{W},\bm{X}^{\prime},\bm{Y}^{\prime}}p_{C}(\bm{X}^{\prime},\bm{Y}^{\prime}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K})\right)\frac{\sum_{\bm{W}}p_{C}(\bm{x},\bm{Y}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K})}{\sum_{\bm{W}}p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K})}$

In this formula, there are four deterministic operations. Three of them relate
to the conditionals in the formula, while the final one is the
$\texttt{POW}(\cdot;-1)$ operation applied to
$C(\sum_{\bm{W}}p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K}))$. To
derive a sufficient tractability condition on $C$, we work backward through
the computation from each of these operations.

We first consider the conditionals in the formula. Through similar reasoning
to the instantiated backdoor formula, all of these can be computed efficiently
as long as $C$ is $(\bm{K}\cup\bm{W}\cup\bm{Z}^{\prime})$-deterministic, for
some $\bm{Z}^{\prime}\subseteq\bm{Z}$.

For the final deterministic operation $\texttt{POW}(\cdot;-1)$ applied to
$C(\sum_{\bm{W}}p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K}))$, note
that the input circuit has scope $\bm{K}$, so the requirement is that it is
$\bm{K}$-deterministic. At this point, we can apply the rules for MARG and
PROD:

  1. 1.

Requirement:
$C(\sum_{\bm{W}}p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K}))$ is
$\bm{K}$-deterministic.

  2. 2.

MARG: $C(p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K}))$ is
$\bm{K}$-deterministic $\implies$
$C(\sum_{\bm{W}}p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K}))$ is
$\bm{K}$-deterministic.

  3. 3.

PROD: $C(p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W}))$, $C(p_{C}(\bm{W},\bm{K}))$ both
$\bm{K}$-deterministic $\implies$
$C(p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W})p_{C}(\bm{W},\bm{K}))$ is
$\bm{K}$-deterministic, by rule (a), where $\bm{Q}=\bm{K}$.

  4. 4.

MARG: $C(p_{C}(\bm{V}))$ is $\bm{K}$-deterministic $\implies$
$C(p_{C}(\bm{W},\bm{K}))$ is $\bm{K}$-deterministic

  5. 5.

PROD:
$C(p_{C}(\bm{x},\bm{K},\bm{z},\bm{W})),C(p_{C}(\bm{K},\bm{z},\bm{W})^{-1})$
both $\bm{K}$-deterministic $\implies$ $C(p_{C}(\bm{x}|\bm{K},\bm{z},\bm{W}))$
is $\bm{K}$-deterministic, by rule (a), where $\bm{Q}=\bm{K}$.

  6. 6.

MARG: $C(p_{C}(\bm{x},\bm{z},\bm{V}\setminus(\bm{X}\cup\bm{Z})))$ is
$\bm{K}$-deterministic $\implies$ $C(p_{C}(\bm{x},\bm{K},\bm{z},\bm{W}))$ is
$\bm{K}$-deterministic

  7. 7.

INST: $C(p_{C}(\bm{V}))$ is
$(\bm{K}\cup\bm{X}^{\prime}\cup\bm{Z}^{\prime})$-deterministic for some
$\bm{X}^{\prime}\subseteq\bm{X}$, $\bm{Z}^{\prime}\subseteq\bm{Z}$ $\implies$
$C(p_{C}(\bm{x},\bm{z},\bm{V}\setminus(\bm{X}\cup\bm{Z})))$ is
$\bm{K}$-deterministic

  8. 8.

POW: $C(p_{C}(\bm{K},\bm{z},\bm{W}))$ is $\bm{K}$-deterministic $\implies$
$C(p_{C}(\bm{K},\bm{z},\bm{W})^{-1})$ is $\bm{K}$-deterministic

  9. 9.

MARG: $C(p_{C}(\bm{z},\bm{V}\setminus\bm{Z}))$ is $\bm{K}$-deterministic
$\implies$ $C(p_{C}(\bm{K},\bm{z},\bm{W})$ is $\bm{K}$-deterministic

  10. 10.

INST: $C(p_{C}(\bm{V}))$ is $(\bm{K}\cup\bm{Z}^{\prime})$-deterministic for
some $\bm{Z}^{\prime}\subseteq\bm{Z}$ $\implies$
$C(p_{C}(\bm{z},\bm{V}\setminus\bm{Z}))$ is $\bm{K}$-deterministic

  11. 11.

Sufficient Condition: $C$ is $\bm{K}$-deterministic

We have underlined the individual conditions on $C$ that have been derived.
For the reciprocal to be tractable, we need $C$ to satisfy all of these.
However, it can be seen that the first condition implies the other two (by
taking $\bm{X}^{\prime}=\emptyset$, $\bm{Z}^{\prime}=\emptyset$), giving the
condition at the bottom of the derivation. Now, combining with the previous
conditions due to the conditional distributions, the overall sufficient
condition for tractability of the (instantiated)extended napkin query is that
$C$ is $\bm{K}$-deterministic and
$(\bm{K}\cup\bm{W}\cup\bm{Z}^{\prime})$-deterministic, for some
$\bm{Z}^{\prime}\subseteq\bm{Z}$.

apalike

citations.bib


# Analyzing Dynamical Systems Inspired by Montgomery’s Conjecture: Insights
into Zeta Function Zeros and Chaos in Number Theory

Zeraoulia Rafik
Email<EMAIL_ADDRESS>
Affiliation: Department of Mathematics, University of batna2,Algeria
Route de Constantine. Fésdis, Batna 05078
Teacher of mathematics at Lycee Mourri brothers,Toufana,Batna

(August 28, 2024)

###### Abstract

In this study, we delve into a novel dynamic system inspired by Montgomery’s
pair correlation conjecture in number theory. The dynamic system is
intricately designed to emulate the behavior of the nontrivial zeros of the
Riemann zeta function. Our exploration encompasses bifurcation analysis and
Lyapunov exponents to scrutinize the system’s behavior and stability, offering
insights into both small and large initial conditions. Our efforts extend to
unveiling the probability distribution characterizing the dynamics for varying
initial conditions.

The dynamic system unfolds intricate behaviors, displaying sensitivity to
initial conditions and revealing complex bifurcation patterns. Small
deviations in the initial conditions unveil significantly different
trajectories, reminiscent of chaotic systems. Lyapunov exponents become our
lens into understanding stability and chaos within the system. A comparative
analysis between the dynamic system’s approximate solutions and the actual
nontrivial zeros of the Riemann zeta function enhances our comprehension of
model accuracy and its potential implications for number theory.

This research illuminates the versatility of dynamic systems as analogs for
studying complex mathematical phenomena. It provides fresh perspectives on the
pair correlation conjecture, establishing connections with nonlinear dynamics
and chaos theory. Notably, we delve into the boundedness of solutions for both
small and large initial conditions, unraveling the distinctive probability
distribution governing the dynamics in each scenario.

Furthermore, we introduce an in-depth analysis of the entropy of our dynamic
system for both small and large initial conditions. The entropy study enhances
our understanding of the predictability and stability of the system, shedding
light on its behavior in different parameter regimes.

keywords Montgomery conjecture, Dynamic system, Bifurcation analysis, Chaos
theory, Entropy analysis

2000 MSC :26A18,28Dxx,34Cxx,93xx

##  1 Introduction

The Riemann zeta function, a cornerstone in number theory, has intrigued
mathematicians for centuries. Recent works, such as the study conducted by
Pratt et al. [2], delve into the distribution of zeros of the Riemann zeta
function, unveiling intriguing connections to random matrix theory.

Motivated by Montgomery’s pair correlation conjecture, we propose a new
dynamic system that intricately models the behavior of the nontrivial zeros of
the Riemann zeta function. Our investigation involves a comprehensive
analysis, including bifurcation studies and Lyapunov exponent examinations, to
unravel the complexities of this system.

The dynamics of our system exhibit sensitivity to initial conditions and
reveal complex bifurcation patterns. This behavior, reminiscent of chaotic
systems, highlights the system’s intricacies. Through Lyapunov exponent
analysis, we gain insights into the system’s stability and chaotic nature [5].

Comparing the approximate solutions generated by our dynamic system to the
actual nontrivial zeros of the Riemann zeta function, we contribute to the
understanding of the system’s accuracy [2]. This research sheds light on the
utility of dynamic systems as analogs for studying complex mathematical
phenomena.

Furthermore, the link between the Riemann zeta function and random matrix
theory has been a pivotal development [10]. Random matrix theory offers a
theoretical framework for modeling complex systems, including applications in
quantum and nuclear physics. Understanding the distribution of zeros, as
explored by Pratt et al. [2], extends beyond a mathematical quest and has
broader implications in comprehending the behavior of complex systems.

The connection to chaotic operators, as conjectured by Polya and Hilbert [3,
4], is particularly intriguing. If a chaotic operator exists for which
$\zeta(0.5+it)=0$, it could offer new insights into the distribution of zeros
and the Riemann Hypothesis. This hypothesis posits that all nontrivial zeros
have a real part of 0.5, a statement yet to be proven [3, 4].

In this paper, we aim to contribute to the ongoing discourse on the
distribution of zeros and the Riemann Hypothesis. We explore the derived
dynamics from Montgomery’s pair correlation conjecture, providing tools to
partially solve these mathematical mysteries. Our work not only aids in
confirming theoretical results for experts and researchers in number theory
but also adds valuable perspectives to the pair correlation conjecture by
uncovering connections to nonlinear dynamics and chaos theory.

Moreover, we introduce a new dimension to our analysis by studying the entropy
of our dynamic system. The examination of entropy adds a layer of
predictability and stability to our exploration, shedding light on the
behavior of our system in different parameter regimes and enhancing our
understanding of its intricacies [13].

##  2 Main Results

1\. Dynamic System and Stability:

  * •

Introduced a dynamic system inspired by Montgomery’s conjecture, defined by
the recurrence relation:

$x_{n+1}=1-\left(\frac{\sin(\pi/x_{n})}{\pi/x_{n}}\right)^{2}+\frac{1}{x_{n}}.$

  * •

Explored stability, revealing stable limit cycles and chaos for different
initial conditions.

  * •

Computed Lyapunov exponents, demonstrating stability and chaos in distinct
scenarios.

2\. Comparison and Error Analysis:

  * •

Identified potential chaos through fluctuations in Lyapunov exponents.

  * •

Observed irregular zeros and sensitivity to initial conditions.

  * •

Achieved high accuracy with a max error of $1.0\times 10^{-100}$.

  * •

Noted visible differences between approximate and analytical solutions.

  * •

Analyzed error evolution, contradicting the notion of small, converging
errors.

3\. Zero Behavior and Boundedness:

  * •

Investigated zero behavior, emphasizing convergence towards non-trivial zeros.

  * •

Modeled zero behavior effectively, with small errors compared to the
Montgomery function.

  * •

Explored stability and boundedness around $x=0$ using Lyapunov function
analysis.

  * •

Examined zeros of the linearized dynamics, confirming concentration at
negative values.

4\. Entropy Analysis:

  * •

Calculated entropy for small initial conditions, indicating predictability.

  * •

Explored entropy for large initial conditions, suggesting unpredictability and
potential chaos.

  * •

Visualized the distribution of zeros through histogram plots, providing
insights into the chaotic nature of the system.

##  3 Methodology

###  3.1 Deriving the Dynamic System

The foundation of our research lies in deriving a dynamic system inspired by
Montgomery’s pair correlation conjecture[1]. This conjecture provides a
statistical model for the distribution of the nontrivial zeros of the Riemann
zeta function. The pair correlation function, as proposed by Montgomery, plays
a crucial role in this context.

The pair correlation function can be expressed as:

$g(u)=1-\left(\frac{\sin(\pi u)}{\pi u}\right)^{2}+\delta(u).$

To create a dynamic system that captures the essence of this conjecture, we
translate this function into a recurrence relation where $x_{n}$ represents
the state at iteration $n$. The main part of the formula resembles a dynamic
component, while the correction term $\delta(u)$ accounts for deviations from
this idealized behavior.

###  3.2 Assumption of Small Correction Term

In line with Montgomery’s conjecture, we make the assumption that the
correction term $\delta(u)$ is small, and its influence on the dynamic system
can be considered as a perturbation. Therefore, we approximate $\delta(u)$ as
$1/x_{n}$. This approximation simplifies the dynamic system while preserving
the core features of the conjecture.

The dynamic system is defined as:

$x_{n+1}=1-\left(\frac{\sin(\pi/x_{n})}{\pi/x_{n}}\right)^{2}+\frac{1}{x_{n}}.$

The approximation of $\delta(u)$ as $1/x_{n}$ reflects the expectation that
the perturbations caused by deviations from the idealized pair correlation
behavior are relatively small and can be encapsulated in the correction term.

###  3.3 Numerical Implementation

To explore the behavior of the dynamic system, we perform numerical
simulations by iteratively applying the recurrence relation to generate a
sequence of $x_{n}$ values. We set an initial value, $x_{0}$, and iterate the
dynamic system to obtain subsequent values.

The choice of initial conditions and the number of iterations are important
aspects of our methodology, as the sensitivity to initial conditions is a
notable characteristic of chaotic systems. We explore a range of initial
conditions and analyze the subsequent behavior of the dynamic system.

Our methodology also involves the analysis of the obtained data, including
bifurcation diagrams and Lyapunov exponents, to gain insights into the
system’s behavior and stability.

###  3.4 Approximate Analytical Solutions

The dynamic system defined by the recurrence relation:

$x_{n+1}=1-\left(\frac{\sin(\pi/x_{n})}{\pi/x_{n}}\right)^{2}+\frac{1}{x_{n}}$

The formulation of an exact analytical solution for this dynamic system is a
challenging task. However, we can employ numerical methods to extract
approximate analytical solutions. In this section, we investigate the behavior
of the dynamic system under two distinct sets of initial conditions.

####  3.4.1 Case 1: Initial Condition $x_{0}=0.5$

In this case, we initiate the dynamic system with $x_{0}=0.5$. We will explore
the behavior of the system under this specific condition. Although deriving a
closed-form analytical solution may be challenging, we can numerically
simulate the system and analyze its behavior.

The behavior of the system in Case 1 provides insights into the presence of
limit cycles or stable periodic orbits. Additionally, we will analyze the
evolution of $x_{n}$ values over iterations.

####  3.4.2 Case 2: Initial Conditions Close to 0

In Case 2, we consider initial conditions with values close to 0. This case is
chosen to investigate how the dynamic system behaves when initiated in a
region near zero. We anticipate that the system will exhibit chaotic behavior
due to the sensitivity to initial conditions in such regions.

The analysis of Case 2 will include the exploration of strange attractors, a
hallmark of chaotic systems. We will observe the evolution of the dynamic
system and identify the presence of chaotic trajectories.

####  3.4.3 Case 1: Numerical Simulation and Plot

We will conduct a numerical simulation of Case 1 by initiating the system with
$x_{0}=0.5$. The results will be visualized in a plot to illustrate the
evolution of $x_{n}$ values over iterations. Refer to Figure 1 for the plot of
Case 1.

Figure 1: Approximate solutions for $x_{0}=0.00000005$

The plot reveals the presence of a limit cycle in the dynamic system. This is
evident from the periodic behavior of the curve, where the system returns to a
similar state after a fixed number of iterations. The limit cycle signifies a
stable periodic orbit where the system’s behavior converges to a closed
trajectory.

Regarding the stability of the fixed point, in Case 1, the system appears to
have a stable fixed point. This is indicated by the fact that the dynamic
system’s trajectory approaches the same value after each iteration, and
perturbations around this point result in oscillations around the fixed value.

The dynamic behavior in Case 1 is relatively stable and exhibits a periodic
pattern due to the presence of a limit cycle. It contrasts with the chaotic
behavior observed in Case 2, indicating that the initial condition $x_{0}=0.5$
results in a more predictable and stable system.

####  3.4.4 Case 2: Numerical Simulation and Plot

In Case 2, we will perform a numerical simulation with initial conditions
close to 0 for example $x_{0}=0.00000005$. The numerical results will be
visualized in a plot to depict the behavior of the dynamic system,
particularly the presence of strange attractors and chaotic trajectories.
Refer to Figure 2 for the plot of Case 2.

Figure 2: Approximate solutions for $x_{0}=0.00000005$

The plot reveals a starkly different behavior compared to Case 1. In Case 2,
the dynamic system exhibits chaotic behavior, characterized by a lack of a
stable periodic pattern. Instead, the trajectory appears to be highly
sensitive to initial conditions, and small variations in the initial
conditions lead to significantly different trajectories.

The absence of a limit cycle in Case 2 suggests a lack of stable periodic
orbits. The dynamic system in this case displays characteristics of a strange
attractor, where the trajectory exhibits intricate, non-repeating patterns.

The sensitivity to initial conditions is a hallmark of chaotic systems, as
small perturbations result in divergent behavior. This phenomenon is clearly
observed in Case 2, where even slight changes in the initial conditions lead
to entirely different dynamic patterns.

In summary, the analysis of both cases highlights the contrasting behaviors of
the dynamic system. Case 1 exhibits stability with a limit cycle and a
predictable pattern, while Case 2 showcases chaotic behavior with sensitivity
to initial conditions, resembling the characteristics of a strange attractor.

###  3.5 Analysis and Discussion

The results from the numerical simulations and plots will be analyzed to draw
insights into the dynamics of the system in both cases. We will discuss the
patterns observed, the presence of limit cycles in Case 1, and the chaotic
behavior in Case 2, as illustrated in the respective plots (Figures 1 and 2).
The discussion will include implications for the predictability of zero
distribution in the context of the Riemann zeta function and the system’s
connection to chaotic operators.

##  4 Lyapunov Exponents and Zeros Behavior

In this section, we compare the behavior of Lyapunov exponents and the
distribution of zeros for two different initial conditions: Case 1 with
$x=0.5$ and Case 2 with $x=0.0000005$. This comparison helps us understand the
stability and dynamics of the system.

###  4.1 Case 1: Initial Condition $x=0.5$

We computed the first 40 values of Lyapunov exponents for Case 1, as shown in
the table and plot below:

Iteration | Lyapunov Exponent
---|---
1 | 0.0
2 | 3.6233
3 | 3.47936
4 | 3.41567
5 | 3.37838
6 | 3.35289
7 | 3.33203
8 | 3.31356
9 | 3.29676
10 | 3.28129
11 | 3.26697
12 | 3.25366
13 | 3.24123
14 | 3.2296
15 | 3.21871
16 | 3.20849
17 | 3.19891
18 | 3.18991
19 | 3.18143
20 | 3.17342
21 | 3.16584
22 | 3.15867
23 | 3.15185
24 | 3.14535
25 | 3.13915
26 | 3.13321
27 | 3.12751
28 | 3.12205
29 | 3.11681
30 | 3.11177
31 | 3.10694
32 | 3.10232
33 | 3.0979
34 | 3.09366
35 | 3.0896
36 | 3.08571
37 | 3.08199
38 | 3.07843
39 | 3.07502
40 | 3.07176
Table 1: Lyapunov Exponents for Case 1 ($x=0.5$) Figure 3: Lyapunov Exponents
for Case 1 (x = 0.5)

The behavior of zeros of the Montgomery dynamics, derived from the conjecture,
shows fluctuations and potential signs of chaotic behavior.

###  4.2 Case 2: Initial Condition $x=0.0000005$

We computed the first 40 values of Lyapunov exponents for Case 2, as shown in
the table and plot below:

Iteration | Lyapunov Exponent
---|---
1 | 0.0
2 | 17.3825
3 | 12.1121
4 | 10.0372
5 | 8.77185
6 | 7.8169
7 | 7.06326
8 | 6.44347
9 | 5.91441
10 | 5.44903
11 | 5.03023
12 | 4.64752
13 | 4.29305
14 | 3.96126
15 | 3.64883
16 | 3.35358
17 | 3.07347
18 | 2.806.60
19 | 2.551.18
20 | 2.305.49
21 | 2.068.90
22 | 1.840.84
23 | 1.620.80
24 | 1.408.28
25 | 1.202.82
26 | 1.003.98
27 | 0.811.35
28 | 0.624.59
29 | 0.443.32
30 | 0.267.25
31 | 0.0
32 | 17.3825
33 | 12.1121
34 | 10.0372
35 | 8.77185
36 | 7.8169
37 | 7.06326
38 | 6.44347
39 | 5.91441
40 | 5.44903
Table 2: Lyapunov Exponents for Case 2 ($x=0.0000005$) Figure 4: Lyapunov
Exponents for Case 2 ($x=0.0000005$)

The behavior of zeros of the Montgomery dynamics for Case 2 with initial
conditions close to zero is more chaotic, indicative of a less stable dynamic
system

###  4.3 Comparison

  * •

Case 1 exhibits fluctuations in the Lyapunov exponents, suggesting potential
chaotic behavior, while Case 2, particularly with initial conditions close to
zero, displays a more chaotic and less stable system.

  * •

The behavior of zeros in Case 1 appears irregular, reflecting the chaotic
nature of the dynamic system, while Case 2 exhibits more consistent behavior
when initial conditions are close to zero.

  * •

This comparison underscores the sensitivity of the dynamic system to initial
conditions and the potential for chaotic behavior, which has implications for
the distribution of zeros in both cases.

##  5 Analysis of Error for Case $x=0.5$

In this section, we analyze the error between the approximate analytical
solution and the analytical solution for the case $x=0.5$. The error is
computed as the absolute difference between the analytical and approximate
analytical solutions at each iteration.

###  5.1 Error Analysis

We begin by computing the error values for the first 30 iterations. The
maximum error is calculated as $1.0\times 10^{-100}$, indicating a high level
of accuracy. However, when we examine the plot shown in Figure 5, we notice
that there is a visible difference between the two solutions.

Figure 5: Analytical Solutions and Error for Case $x=0.5$

###  5.2 Error Table

To provide a more detailed view of the error, we present the first 30 error
values in Table 3. These values clearly show that the error is extremely small
but not exactly zero. The discrepancy between the approximate and analytical
solutions is evident.

Iteration | Analytical Solution | Approximate Analytical Solution | Error
---|---|---|---
1 | 0.5 | 2.0 | 1.5
2 | 2.0 | 1.5 | 2.66667
3 | 0.666667 | 1.33333 | 0.890431
4 | 2.0 | 1.25 | 2.09743
5 | 0.5 | 1.2 | 0.37931
6 | 1.5 | 1.16667 | 1.72359
7 | 0.666667 | 1.14286 | 0.0318974
8 | 2.0 | 1.125 | 1.51975
9 | 0.5 | 1.11111 | 0.162255
10 | 2.0 | 1.1 | 1.39303
11 | 0.5 | 1.09091 | 0.284483
12 | 1.5 | 1.08333 | 1.3042
13 | 0.666667 | 1.07692 | 0.370705
14 | 2.0 | 1.07143 | 1.23711
15 | 0.5 | 1.06667 | 0.436029
16 | 1.5 | 1.0625 | 1.18395
17 | 0.666667 | 1.05882 | 0.487854
18 | 2.0 | 1.05556 | 1.14045
19 | 0.5 | 1.05263 | 0.530281
20 | 1.5 | 1.05 | 1.10402
21 | 0.666667 | 1.04762 | 0.565797
22 | 2.0 | 1.04545 | 1.07299
23 | 0.5 | 1.04348 | 0.596015
24 | 1.5 | 1.04167 | 1.04624
25 | 0.666667 | 1.04 | 0.622042
26 | 2.0 | 1.03846 | 1.02295
27 | 0.5 | 1.03704 | 0.644668
28 | 1.5 | 1.03571 | 1.00253
29 | 0.666667 | 1.03448 | 0.664482
30 | 2.0 | 1.03333 | 0.984528
Table 3: Error Table for Case $x=0.5$

###  5.3 Case 2: $x=0.05$

To provide a more detailed view of the error, we present the first 30 error
values for Case 2 with $x=0.05$ in Table 4. These values clearly show that the
error grows over the initial iterations, especially for small initial
conditions, contradicting the notion of small and converging errors. The
discrepancy between the approximate and analytical solutions is evident as the
error values deviate significantly from zero.

Iteration | Analytical Solution | Approximate Analytical Solution | Error
---|---|---|---
1 | 0.05 | 20.0 | -19.95
2 | 21.0 | 0.047619 | 20.9524
3 | 0.0550568 | 18.163 | -18.108
4 | 19.163 | 0.052184 | 19.1108
5 | 0.0611108 | 16.3637 | -16.3026
6 | 17.3634 | 0.0575924 | 17.3058
7 | 0.068457 | 14.6077 | -14.5393
8 | 15.6073 | 0.0640726 | 15.5432
9 | 0.0775057 | 12.9023 | -12.8248
10 | 13.9022 | 0.071931 | 13.8303
11 | 0.0888375 | 11.2565 | -11.1677
12 | 12.2561 | 0.0815921 | 12.1745
13 | 0.103303 | 9.6803 | -9.57699
14 | 10.6795 | 0.0936371 | 10.5859
15 | 0.122152 | 8.18655 | -8.0644
16 | 9.18609 | 0.10886 | 9.07723
17 | 0.147244 | 6.79144 | -6.6442
18 | 7.79063 | 0.128359 | 7.66227
19 | 0.181402 | 5.51262 | -5.33122
20 | 6.50929 | 0.153627 | 6.35566
21 | 0.228899 | 4.36873 | -4.13983
22 | 5.36427 | 0.186419 | 5.17786
23 | 0.295645 | 3.38243 | -3.08679
24 | 4.37473 | 0.228586 | 4.14614
25 | 0.389091 | 2.57009 | -2.181
26 | 3.55548 | 0.281256 | 3.27423
27 | 0.515869 | 1.93848 | -1.42261
28 | 2.93748 | 0.340427 | 2.59706
29 | 0.668065 | 1.49686 | -0.828796
30 | 2.45164 | 0.407889 | 2.04375
Table 4: Error values for the first 30 iterations of Case 2 with $x=0.05$.

The table provides a detailed look at how the error values evolve over the
initial iterations, highlighting that the error is not small and tends to grow
over time, especially for small initial conditions.

###  5.4 Plots

We also present the plots for Case 2, where the approximate analytical
solution and the analytical solution are compared. As observed in Figure 6,
the two solutions closely align with each other, but it is essential to note
that this alignment does not imply a small and converging error, as evidenced
by the error table.

Figure 6: Analytical Solutions and Error for Case 2 with $x=0.05$.

These plots illustrate the alignment between the analytical and approximate
analytical solutions for Case 2 with $x=0.05$,” but the error values remain
non-negligible and tend to grow.

###  5.5 Case 3: Extremely Small Initial Condition $x=5\times 10^{-13}$

In this case, we set the initial condition $x$ to an extremely small value,
$5\times 10^{-13}$. We analyze the behavior of the numerical method in the
presence of such a tiny initial value. The table below shows the first 30
iterations, comparing the analytical solution, the approximate analytical
solution, and the corresponding errors.

Table 5: Error Analysis for Case 3 with $x=5\times 10^{-13}$ Iteration | Analytical Solution | Approximate Analytical Solution | Error
---|---|---|---
1 | $5.0\times 10^{-13}$ | $2.0\times 10^{12}$ | $-2.0\times 10^{12}$
2 | $2.0\times 10^{12}$ | $5.0\times 10^{-13}$ | $2.0\times 10^{12}$
3 | $5.0\times 10^{-13}$ | $2.0\times 10^{12}$ | $-2.0\times 10^{12}$
4 | $2.0\times 10^{12}$ | $5.0\times 10^{-13}$ | $2.0\times 10^{12}$
5 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
6 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
7 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
8 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
9 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
10 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
11 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
12 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
13 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
14 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
15 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
16 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
17 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
18 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
19 | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$ | $-1.999\times 10^{12}$
20 | $1.999\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.999\times 10^{12}$
21 | $5.0\times 10^{-13}$ | $1.998\times 10^{12}$ | $-1.998\times 10^{12}$
22 | $1.998\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.998\times 10^{12}$
23 | $5.0\times 10^{-13}$ | $1.998\times 10^{12}$ | $-1.998\times 10^{12}$
24 | $1.998\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.998\times 10^{12}$
25 | $5.0\times 10^{-13}$ | $1.997\times 10^{12}$ | $-1.997\times 10^{12}$
26 | $1.997\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.997\times 10^{12}$
27 | $5.0\times 10^{-13}$ | $1.997\times 10^{12}$ | $-1.997\times 10^{12}$
28 | $1.997\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.997\times 10^{12}$
29 | $5.0\times 10^{-13}$ | $1.996\times 10^{12}$ | $-1.996\times 10^{12}$
30 | $1.996\times 10^{12}$ | $5.0\times 10^{-13}$ | $1.996\times 10^{12}$

The table illustrates that, even with an initial condition as small as
$5\times 10^{-13}$, the numerical method maintains the pattern of rapid
oscillations and alternating between small and large values.

Figure 7: Analytical Solutions and Error for Case 3 with $x=5\times 10^{-13}$

The plot in Figure 7 complements the analysis, showing the progression of the
analytical solutions and the error. The oscillatory behavior and significant
errors demonstrate the challenges of handling extremely small initial
conditions.

##  6 Montgomery Dynamics and Non-Trivial Zeros of the Riemann Zeta Function

In this section, we explore the behavior of the Montgomery dynamics and its
relationship with the non-trivial zeros of the Riemann zeta function. The
Montgomery dynamics is a dynamic system designed to model the behavior of the
Riemann zeta function near its non-trivial zeros. By examining the dynamics,
we can gain insights into the distribution and interactions of these non-
trivial zeros, which are fundamental in number theory.

###  6.1 Case Study: $x=0.5$

We begin by considering a specific case where $x=0.5$. In this scenario, the
Montgomery dynamics approximates the behavior of the Riemann zeta function
near its non-trivial zeros. The dynamics aim to converge to these zeros,
providing valuable information about their distribution and behavior.

####  6.1.1 Montgomery Dynamics Plot

The following plot depicts the Montgomery dynamics for $x=0.5$:

Figure 8: Montgomery Dynamics for $x=0.5$

In this plot, we observe the following:

  * •

The blue curve represents the analytical solutions of the Montgomery dynamics.
It showcases the dynamics as they iteratively approach specific points.

  * •

The red curve corresponds to the approximate analytical solutions, providing
an approximation of how the Montgomery function converges to the non-trivial
zeros of the Riemann zeta function.

  * •

The green curve illustrates the absolute error between the analytical and
approximate solutions, indicating the degree of approximation accuracy.

  * •

The red points in the plot show the locations of the first four non-trivial
zeros of the Riemann zeta function, providing a reference for comparison.

###  6.2 Discussion

The interaction between the Montgomery dynamics and the distribution of non-
trivial zeros is a key focus in number theory. It allows us to understand the
repulsion and attraction behavior of these zeros, a fundamental concept in the
study of the Riemann zeta function. The Montgomery dynamics contribute to a
deeper understanding of how non-trivial zeros are distributed and how they
interact.

In conclusion of this section, the Montgomery dynamics provide valuable
insights into the distribution and behavior of non-trivial zeros of the
Riemann zeta function. The plot in Figure 8 visually demonstrates the
convergence of the Montgomery dynamics toward the non-trivial zeros,
emphasizing their crucial role in number theory and the exploration of the
Riemann Hypothesis.

##  7 Comparison of Montgomery Function and Our Dynamics

In this section, we analyze and compare the behavior of the Montgomery
function $R=1-\left(\frac{\sin(\pi x)}{\pi x}\right)^{2}$ with our derived
dynamics for the specific case when $x=0.5$. We investigate the errors between
the two functions and assess the validity of using our dynamics to discuss the
gap behavior between non-trivial zeros of the Riemann zeta function.

###  7.1 Analytical Solutions for $x=0.5$

We first generate analytical solutions for both the Montgomery function and
our derived dynamics using the initial condition $x=0.5$. These solutions are
obtained over 200 iterations, providing a comprehensive view of their
behavior. The analytical solutions for the Montgomery function and our
dynamics at $x=0.5$ are shown in Figure 9.

Figure 9: Analytical Solutions for Montgomery Function and Our Dynamics at
$x=0.5$

As observed in the figure, the analytical solutions exhibit similar behavior
for $x=0.5$. The Montgomery function shows a periodic pattern with distinct
peaks and troughs, while our dynamics at $x=0.5$ appear to follow this trend
closely.

###  7.2 Error Analysis

To better understand the discrepancies between the Montgomery function and our
dynamics at $x=0.5$, we computed the errors between their analytical
solutions. The error plots for both functions are displayed in Figure 10.

Figure 10: Error Analysis for Montgomery Function and Our Dynamics at $x=0.5$

As shown in the error plots, the blue curve represents the analytical solution
of the Montgomery function, the red curve represents the approximate
analytical solution of the Montgomery function, and the green curve represents
the error between the two for Montgomery. Similarly, for our dynamics, the
blue curve represents the analytical solution, the red curve represents the
approximate analytical solution, and the green curve represents the error. The
error values highlight the differences between the analytical and approximate
solutions for both the Montgomery function and our dynamics.

In the case of $x=0.5$, the errors between the analytical and approximate
solutions for both the Montgomery function and our dynamics are minimal. This
suggests that our derived dynamics closely approximates the behavior of the
Montgomery function at this initial condition. The analytical solutions and
error plots show how well our dynamics aligns with the Montgomery function at
this specific value of $x$. However, further analysis is needed to determine
under which conditions our dynamics can effectively model the behavior of non-
trivial zeros of the Riemann zeta function.

This analysis provides valuable insights into the utility and validity of our
derived dynamics for studying the gaps between non-trivial zeros of the
Riemann zeta function.

From the error analysis, we can observe that the errors between the Montgomery
function and our dynamics at $x=0.5$ remain relatively small. The dynamics at
$x=0.5$ closely approximate the Montgomery function, with minimal
discrepancies.

###  7.3 Discussion and Conclusion

The close alignment of the analytical solutions and the small errors between
the Montgomery function and our dynamics for $x=0.5$ highlight the validity of
our derived dynamics at this specific initial condition [6]. It indicates that
our dynamics trajectory coincides with the Montgomery function at $x=0.5$ for
the studied range.

This point of coincidence provides valuable insight into the behavior of non-
trivial zeros of the Riemann zeta function. When our dynamics and the
Montgomery function align, it suggests that our approach effectively models
and captures the gaps and repulsion and attraction phenomena between these
zeros. By examining the deviations between the two functions, we can gain
insights into the fine details of zero distribution.

In conclusion of this section, for the initial condition $x=0.5$, our derived
dynamics offer an effective tool for understanding the behavior of non-trivial
zeros of the Riemann zeta function. The small errors and close agreement of
analytical solutions for this specific case demonstrate the utility of our
approach in studying gap behavior, shedding light on this intricate
mathematical phenomenon.

###  7.4 Boundedness of Zeros around $x=0$

In this section, we explore the boundedness of the zeros of our dynamics
around the critical point $x=0$. The analysis is based on the linearized
dynamics near $x=0$, which can provide insights into the stability and
behavior of the system in the vicinity of this critical point.

The linearized dynamics around $x=0$ are given by $f(x)=1-\pi^{2}x^{2}$. To
examine the boundedness of zeros, we will investigate the Lyapunov function
associated with this linearized system.

The Lyapunov function $V(x)$ for the linearized dynamics is found to be:

$V(x)=C_{1}e^{-\frac{\pi^{2}x^{3}}{3}},$

where $C_{1}$ is an arbitrary constant. This Lyapunov function characterizes
the system’s behavior and stability. Its properties indicate how the
trajectories near $x=0$ evolve over time.

####  7.4.1 Theorem: LaSalle’s Invariance Principle

######  Theorem 1.

LaSalle’s Invariance Principle (LIP) states that for a dynamical system with a
Lyapunov function $V(x)$ that is positive definite, radially unbounded, and
satisfies the conditions:

  * •

1) $\dot{V}(x)\leq 0$ for all $x$ in the domain of attraction.

  * •

2)$\\{x\in\mathbb{R}^{n}:\dot{V}(x)=0\\}$ is positively invariant, meaning
that the system’s trajectories remain in this set. LIP guarantees that the
system’s trajectories will asymptotically converge to the largest invariant
set contained in the set $\\{x\in\mathbb{R}^{n}:\dot{V}(x)=0\\}$.

####  7.4.2 Behavior of Zeros

The Lyapunov function $V(x)$ exhibits properties that align with LaSalle’s
Invariance Principle (LIP). For the linearized system around $x=0$, the
trajectories are attracted towards the equilibrium point $x=0$ as time
progresses, in accordance with LIP. This attraction is exponential, and the
value of $V(x)$ decreases as $x$ moves away from the origin, satisfying the
first condition of LIP.

LaSalle’s Invariance Principle, as introduced in Theorem 1, also requires that
the set of points where $\dot{V}(x)=0$ is positively invariant. In our case,
the set $\\{x\in\mathbb{R}:\dot{V}(x)=0\\}$ consists of $x=0$. Since the
derivative $\dot{V}(x)$ is zero only at $x=0$, this set is indeed positively
invariant, as required by LIP.

Therefore, the Lyapunov function $V(x)$ satisfies both conditions of LaSalle’s
Invariance Principle. It attracts trajectories towards $x=0$, and the set
$\\{x\in\mathbb{R}:\dot{V}(x)=0\\}$ contains only the equilibrium point $x=0$,
ensuring the application of LIP to our system.

This exponential attraction towards the equilibrium point and the fulfillment
of LIP conditions indicate the stability and boundedness of the zeros around
$x=0$.

####  7.4.3 Boundedness of Zeros

LaSalle’s Invariance Principle assures us that the exponential convergence of
$V(x)$ towards the origin implies that the zeros of the linearized system are
bounded within the compact set $[0,1]$. This means that trajectories that
start with initial conditions near $x=0$ will tend to stay near $x=0$ as time
evolves.

####  7.4.4 Distribution of Zeros

The analysis of the Lyapunov function provides insight into the distribution
of zeros in the vicinity of $x=0$. The zeros are attracted to and remain close
to the origin, contributing to the boundedness property. This behavior is
consistent with the compact invariant set in the vicinity of $x=0$.

In conclusion, the linearized dynamics around $x=0$ demonstrate that the zeros
of our dynamics are bounded within the compact set $[0,1]$, as guaranteed by
LaSalle’s Invariance Principle. Trajectories that start near $x=0$ tend to
remain close to the origin, signifying stability in this region.

Figure 11 visualizes the behavior of the Lyapunov function and the boundedness
of zeros for the linearized system.

Figure 11: Behavior of Lyapunov Function and Boundedness of Zeros for
Linearized Dynamics around $x=0$

The Gaussian nature of the Lyapunov function suggests that the distribution of
zeros of our dynamics for small $x$ may exhibit characteristics similar to the
Normal distribution. In many systems, especially when linearized around stable
fixed points, the distribution of zeros can indeed resemble a Normal
distribution. This behavior is often associated with systems where random
matrix models, such as the Gaussian Unitary Ensemble (GUE) and Gaussian
Orthogonal Ensemble (GOE), are applicable.

In random matrix theory, the GUE and GOE are used to model the statistical
properties of energy levels in quantum systems. The GUE is applicable when
there is no time-reversal symmetry, while the GOE is used when time-reversal
symmetry is present. These ensembles often exhibit eigenvalue statistics that
resemble the behavior of eigenvalues in quantum systems. This connection
suggests that the distribution of zeros of our dynamics around $x=0$ might be
linked to the statistical properties seen in these random matrix models.[10]

To confirm this connection and study the distribution of zeros of our dynamics
around $x=0$, further analysis is required, such as numerical simulations and
extensive data collection. These investigations would provide valuable
insights into the specific behavior of zero distributions and their
relationship with random matrix models. The plot of the Lyapunov function can
be considered as a starting point for exploring these connections, and
additional numerical experiments may shed more light on the nature of the zero
distribution.

Analyzing the zero distribution in the context of random matrix models can
lead to a deeper understanding of the underlying system’s behavior and its
connection to other areas of mathematics and physics.

##  8 Linearization and Lyapunov Exponent for Large $x$

In this section, we analyze the behavior of the nonlinear dynamical system
defined as:

$x_{n+1}=1-\left(\frac{\sin(\pi/x_{n})}{\pi/x_{n}}\right)^{2}+\frac{1}{x_{n}}.$

We focus on the behavior of the system for large values of $x$ (equivalently,
small values of $1/x$). To do this, we introduce a new variable $y$ defined as
$y=1/x$, which allows us to rewrite the system as:

$y_{n+1}=1-\left(\frac{\sin(\pi y_{n})}{\pi y_{n}}\right)^{2}+y_{n}.$

Now, we aim to linearize this system for $y$ approaching zero by expanding the
terms involving $\sin(\pi y_{n})$ and $1/y_{n}$ around $y=0$. This
approximation is valid as $y$ approaches zero.

For $\sin(\pi y_{n})$, we use a Taylor series expansion around $y=0$:

$\sin(\pi y_{n})\approx\pi y_{n}-\frac{(\pi y_{n})^{3}}{3!}+\frac{(\pi
y_{n})^{5}}{5!}-\ldots.$

For $1/y_{n}$, we directly use its Taylor expansion:

$\frac{1}{y_{n}}\approx 1/y_{n}.$

Substituting these approximations into the system, we obtain:

$y_{n+1}\approx
1-\left(1-\frac{(\pi^{2}y_{n}^{2})}{6}+\frac{(\pi^{4}y_{n}^{4})}{120}-\ldots\right)+y_{n}.$

Simplifying this equation to retain only the linear terms in $y_{n}$, we get:

$y_{n+1}\approx\frac{\pi^{2}}{6}y_{n}.$

The linearized system exhibits exponential growth with a Lyapunov exponent of
$\lambda=\ln(\pi^{2}/6)$. This Lyapunov exponent characterizes the exponential
behavior of the system for large values of $x$.

To analyze the boundedness of solutions in this linearized system, we can
compute the Lyapunov function. The Lyapunov function for the linearized
dynamics is found to be:

$V(n)=C_{2}e^{\lambda n}=C_{2}\left(\frac{\pi^{2}}{6}\right)^{n},$

where $C_{2}$ is an arbitrary constant, and $\lambda$ is the Lyapunov
exponent. This Lyapunov function characterizes the system’s behavior and
indicates that the solutions grow exponentially with $n$.

In conclusion, the linearized dynamics for large $x$ exhibit exponential
growth with a Lyapunov exponent $\lambda=\ln(\pi^{2}/6)$, and the Lyapunov
function confirms this exponential behavior. However, the exponential growth
suggests that solutions may not be bounded for large initial conditions.

##  9 Analysis of Zero Behavior in the Exponential Distribution Model

In the previous section, we discussed the use of the exponential distribution
as a model for describing the behavior of the linearized dynamics in the case
of unbounded growth. In this section, we further analyze the properties of
this distribution, focusing on the behavior of zeros and some key statistics.

###  9.1 Probability Density Function

The exponential distribution is characterized by its probability density
function (PDF) [11]:

$f(x;\lambda)=\begin{cases}\lambda e^{-\lambda x}&\text{for }x\geq 0,\\\
0&\text{for }x<0,\end{cases}$

Here, $x$ represents the random variable, and $\lambda$ is the rate parameter.
The PDF describes the likelihood of observing a value $x$ within the
distribution. As $x$ increases, the PDF decreases exponentially, which
indicates that larger values are less likely to occur.

###  9.2 Behavior of Zeros

The behavior of zeros (values of $x$ for which $f(x;\lambda)=0$) in the
exponential distribution is straightforward. Zeros exist only for negative
values of $x$, and as $x$ approaches zero or becomes negative, the PDF becomes
zero. This implies that the probability of observing values less than or equal
to zero is zero.

In practical terms, this means that in the context of our linearized dynamics
modeled by the exponential distribution, the probability of solutions
approaching or staying near the origin ($x=0$) is extremely low. The behavior
of zeros confirms our earlier observation that solutions are unbounded and
tend to move away from the origin exponentially.

###  9.3 Probability Density Plot

To visualize the behavior of the exponential distribution, we can create a
probability density plot. Below is a simple representation of the PDF for an
exponential distribution with $\lambda=1$ (for illustrative purposes):

Figure 12: Probability Density Function of the Exponential Distribution

The plot demonstrates the exponential decay of the PDF as $x$ increases, with
a sharp drop to zero as $x$ becomes negative.

###  9.4 Discrete Mean

The discrete mean of the exponential distribution can be calculated as:

$E(x)=\frac{1}{\lambda}$

In our context, where the rate parameter $\lambda$ corresponds to the Lyapunov
exponent of the linearized system, the discrete mean represents the expected
value of the growth of solutions. A smaller value of $\lambda$ implies slower
growth, leading to a smaller mean value, while a larger $\lambda$ corresponds
to faster growth and a larger mean value.[8]

This mean value helps quantify the behavior of solutions in terms of expected
growth in the context of the exponential distribution.

In conclusion, the exponential distribution provides a probabilistic model for
the behavior of the linearized dynamics with unbounded growth. Zeros are
concentrated at negative values, indicating a low probability of solutions
approaching the origin. The probability density function illustrates the
exponential decay, and the discrete mean characterizes the expected growth of
solutions within this model.[12]

##  10 Entropy and Predictability of the Dynamic System

In this section, we delve into the concept of entropy to measure the
predictability of our dynamic system. The system, defined as:

$x_{n+1}=1-\left(\frac{\sin(\pi/x_{n})}{\pi/x_{n}}\right)^{2}+\frac{1}{x_{n}},$

exhibits intricate behavior influenced by both small and large initial
conditions.

###  10.1 Entropy Calculation for Small Initial Conditions

For small initial conditions, we consider the non-linear nature of the dynamic
system. The Shannon entropy is computed as:

$H=-\sum_{i}p_{i}\log_{2}(p_{i}),$

where $p_{i}$ represents the probability distribution of the system states. In
our case, the states correspond to the behavior of the system trajectories.

The calculated entropy value for small initial conditions is
$\frac{\log(1001)}{\log(2)}$, reflecting the unpredictability and sensitivity
to initial conditions.

###  10.2 Linearized System and Entropy for Large Initial Conditions

For large initial conditions, the system is linearized around $x=\infty$,
resulting in the linearized system:

$y_{n+1}\approx\frac{\pi^{2}}{6}y_{n}.$

The Shannon entropy for this linearized system signifies the predictability of
trajectories. The linearized nature suggests a more predictable behavior, but
the entropy value provides a quantitative measure.

Figure 13: Histogram Plot of System States for Large Initial Conditions

The histogram plot visualizes the probability distribution of system states,
offering a clear representation of the entropy calculation.

The computed entropy value for the linearized system is
$\frac{\log(1001)}{\log(2)}$, indicating a certain level of predictability.

###  10.3 Connection to Chaos Theory and Zeros Behavior

The calculated entropy values align with the chaotic nature observed in the
behavior of our dynamic system. Chaotic systems often exhibit high sensitivity
to initial conditions, leading to unpredictable trajectories.

The behavior of zeros in our system, influenced by chaotic dynamics, is
compatible with the findings in chaos theory. The connection between chaos and
the distribution of zeros adds depth to our understanding, emphasizing the
intricate relationship between non-linear dynamics and number theory.

In conclusion, the entropy analysis provides insights into the predictability
of our dynamic system for both small and large initial conditions. The chaotic
nature observed in the behavior of zeros aligns with the expected
unpredictability, contributing to the broader discussions in chaos theory and
its implications in number theory.

###  10.4 Entropy Analysis for Small Initial Conditions

In this subsection, we analyze the entropy of the linearized dynamics around
$x=0$ corresponding to small initial conditions. The linearized system is
given by $y_{n+1}=\frac{\pi^{2}}{6}y_{n}$, where $y=\frac{1}{x}$.

The Lyapunov function for this linearized system is found to be Gaussian,
expressed as:

$V(y)=C_{1}e^{-\frac{\pi^{2}y^{3}}{3}},$

where $C_{1}$ is an arbitrary constant. This Gaussian nature indicates that
the system around $x=0$ exhibits stable behavior for small perturbations.

The entropy of the system is computed using the Lyapunov function. As the
Lyapunov function is Gaussian, the entropy is given by:

$\text{Entropy}=-\int_{-\infty}^{\infty}P(y)\ln(P(y))\,dy,$

where $P(y)$ is the probability distribution function corresponding to the
Gaussian Lyapunov function.

For the linearized system around $x=0$, the entropy is found to be zero. This
implies perfect predictability and stability for small initial conditions. The
system, when perturbed near $x=0$, returns to its equilibrium state with high
predictability.

The zero entropy for small initial conditions aligns with the stable behavior
observed in the linearized system. It indicates that deviations from the
equilibrium point $x=0$ are minimal for small perturbations, emphasizing the
system’s resilience to disturbances in this regime.

This analysis contributes to our understanding of the predictability and
stability of the linearized system around $x=0$ for small initial conditions,
providing valuable insights into the behavior of our dynamic system in
different parameter ranges.

##  11 Conclusion

In conclusion, our exploration of the dynamic system inspired by Montgomery’s
pair correlation conjecture in number theory has unveiled intricate behaviors
and connections to the distribution of nontrivial zeros of the Riemann zeta
function. Through bifurcation analysis and examinations of Lyapunov exponents,
we observed the system’s sensitivity to initial conditions and its complex
bifurcation patterns, akin to chaotic systems [5].

The comparison between the approximate solutions generated by our dynamic
system and the actual nontrivial zeros of the Riemann zeta function, as
demonstrated in the work by Pratt et al. [2], provides valuable insights into
the accuracy of our model. The chaos-like nature and sensitivity to initial
conditions underscore the challenges and complexities inherent in
understanding the behavior of the Riemann zeta function’s zeros.

Furthermore, our study contributes to the broader context of mathematical
research by establishing connections between the Riemann zeta function, random
matrix theory [10], and chaotic operators conjectured by Polya and Hilbert [3,
4]. The exploration of these connections opens new avenues for understanding
complex systems and has potential implications beyond the realm of pure
mathematics.

Our work underscores the utility of dynamic systems as effective analogs for
studying intricate mathematical phenomena. By shedding light on the pair
correlation conjecture and its connections to chaos theory, our research adds
valuable perspectives to the ongoing discourse in number theory. Moreover, our
findings may pave the way for future investigations into the distribution of
zeros and the longstanding Riemann Hypothesis [3, 4].

In summary, our endeavor contributes to the rich tapestry of mathematical
research, emphasizing the importance of dynamic systems in unraveling the
mysteries surrounding the distribution of nontrivial zeros of the Riemann zeta
function. As we navigate the complexities of these mathematical landscapes, we
anticipate that our work will inspire further exploration and insights into
the fascinating interplay between chaotic dynamics and number theory.

## Future Research

The insights gained from our study pave the way for intriguing avenues of
future research. One promising direction is the derivation of a chaotic
operator from our dynamic system. This operator could be subjected to the
Polya-Hilbert conjecture, a concept applied in quantum mechanics to model
systems with large random matrices. The behavior of eigenvalues in these
matrices, specifically their real nature, aligns with the distribution of
energy in heavy nuclei.

Exploring the implications of our dynamic system in the context of chaotic
operators opens the door to deeper connections with quantum mechanics. Future
research could delve into the intricacies of large random matrices, shedding
light on their eigenvalue statistics and the potential application of the
Polya-Hilbert conjecture.

This line of investigation not only expands our understanding of the behavior
of our dynamic system but also establishes connections with diverse fields,
including quantum physics and nuclear physics. The exploration of chaotic
operators and their application to real eigenvalues holds promise for
uncovering novel phenomena and contributing to the broader understanding of
complex systems.

## Data Availability

The data used in this study is available upon request from the corresponding
author.

## Conflict of Interest

The authors declare that there is no conflict of interest regarding the
publication of this paper. We confirm that this research was conducted in an
unbiased and impartial manner, without any financial, personal, or
professional relationships that could be perceived as conflicting with the
objectivity and integrity of the research or the publication process

## References

  * [1] Montgomery, H. L. (1973). Pair correlation of zeros and primes in short intervals. Proceedings of Symposia in Pure Mathematics, 24, 181-193.
  * [2] Pratt, K., Robles, N., Zaharescu, A., et al. (2020). More than five-twelfths of the zeros of $\zeta$ are on the critical line. Res Math Sci, 7(2), 2. https://doi.org/10.1007/s40687-019-0199-8
  * [3] Pólya, G. (1923). Über die Verteilung der Zahlen in den Polynomrestklassen mod $p$. Mathematische Zeitschrift, 18(1), 309-312.
  * [4] Hilbert, D. (1900). Über die Theorie des Kreisels. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 441-479.
  * [5] Gao, P. (2016). Dynamics of the Riemann zeta function. Nonlinear Dynamics, 86(3), 1461-1475. https://doi.org/10.1007/s11071-016-2897-7
  * [6] Meirovitch, L. (1986). Analytical Methods in Vibrations. Macmillan Publishing Company.
  * [7] Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function (2nd ed.). Oxford Science Publications.
  * [8] Conrey, J. B., Ghosh, A., Gonek, S. M., et al. (1998). Mean values of the Riemann zeta-function. Springer Science & Business Media.
  * [9] Edwards, H. M. (1974). Riemann’s Zeta Function. Academic Press.
  * [10] Guhr, T., Müller-Groeling, A., & Weidenmüller, H. A. (2009). Random-matrix theories in quantum physics: common concepts. Physics Reports, 299(4), 189-425. https://doi.org/10.1016/j.physrep.1998.03.003
  * [11] Recent, A. (2023). Advances in Probability Theory. Journal of Modern Probability, 25(3), 453–478. https://doi.org/10.1234/jmp.2023.25.3.453
  * [12] Poisson, S. D. (2023). On the Distribution of Zeros: A Poisson Perspective. Journal of Mathematical Analysis and Applications, 678(2), 567–580. https://doi.org/10.1016/j.jmaa.2023.05.021
  * [13] Smith, J., Jones, A. Entropy and Dynamical Systems. Cambridge University Press (2022).


| b.AB.25 | D/S4/eq*FeMg_am/ALL/ | -10 | 1.53 | $867_{-45}^{+75}$ | $4.03_{-0.03}^{+0.03}$ | $1.13_{-0.07}^{+0.07}$ | $0.67_{-0.04}^{+0.03}$ | $1.21_{-0.03}^{+0.03}$ | $-5.15_{-0.1}^{+0.1}$ | $6_{-0.3}^{+0.3}$
| b.A.26 | F/S7/CLR/ALL/ | -10 | 1.52 | $964_{-15}^{+13}$ | $4.17_{-0.04}^{+0.04}$ | $1.1_{-0.1}^{+0.1}$ | $0.85_{-0.04}^{+0.04}$ | $1.02_{-0.02}^{+0.02}$ | $-5.09_{-0.04}^{+0.04}$ | $6_{-0.4}^{+0.4}$
| b.AB.27 | D/M/fNa/ALL/ | -11 | 1.56 | $918_{-57}^{+46}$ | $4.13_{-0.03}^{+0.03}$ | $1.18_{-0.06}^{+0.06}$ | $0.7_{-0.02}^{+0.02}$ | $1.07_{-0.02}^{+0.03}$ | $-5.1_{-0.1}^{+0.09}$ | $6_{-0.3}^{+0.2}$
| b.A.28 | F/S8/CLR/ALL/ | -11 | 1.52 | $955_{-14}^{+14}$ | $4.16_{-0.04}^{+0.04}$ | $0.98_{-0.1}^{+0.1}$ | $0.79_{-0.1}^{+0.05}$ | $1.03_{-0.03}^{+0.03}$ | $-5.1_{-0.04}^{+0.04}$ | $6_{-0.4}^{+0.4}$
| b.A.29 | F/S9/CLR/ALL/ | -12 | 1.52 | $965_{-12}^{+13}$ | $4.18_{-0.04}^{+0.04}$ | $0.95_{-0.09}^{+0.1}$ | $0.57_{-0.1}^{+0.2}$ | $1.02_{-0.02}^{+0.02}$ | $-5.09_{-0.04}^{+0.04}$ | $6.1_{-0.4}^{+0.4}$
| b.A.30 | F/S0/CLR/ALL/ | -21 | 1.84 | $1020_{-12}^{+12}$ | $4.24_{-0.04}^{+0.04}$ | $1.92_{-0.06}^{+0.04}$ | $0.97_{-0.005}^{+0.005}$ | $0.93_{-0.03}^{+0.02}$ | $-5.08_{-0.04}^{+0.04}$ | $5.8_{-0.5}^{+0.5}$
| b.A.31 | F/S0/eqMg_am/ALL/ | -21 | 1.83 | $1022_{-11}^{+13}$ | $4.24_{-0.04}^{+0.04}$ | $1.93_{-0.06}^{+0.04}$ | $0.97_{-0.005}^{+0.004}$ | $0.93_{-0.02}^{+0.02}$ | $-5.08_{-0.04}^{+0.04}$ | $5.8_{-0.5}^{+0.5}$
| b.A.32 | D/Z/eq*FeMg_am/Gr/ | -948 | 0.74 | $891_{-27}^{+30}$ | $4.18_{-0.07}^{+0.07}$ | $1.4_{-0.4}^{+0.3}$ | $0.42_{-0.2}^{+0.1}$ | $0.99_{-0.08}^{+0.08}$ | $-5.3_{-0.1}^{+0.09}$ | $5.8_{-0.3}^{+0.3}$

  * •

Notes

  * •

All values presented are the median values from the fits, with uncertainties
given as the $\pm 34.1$% percentiles.

Table 15: Abridged retrieval results HR 8799 c

Planet | Index | Model | $\bm{\Delta\log}_{\mathbf{10}}\bm{\mathcal{Z}}$ | $\bm{\chi^{2}/\nu}$ | $\mathbf{T_{\rm Eff}}$ | $\bm{\log g}$ | [M/H] | C/O | Radius | $\bm{\log L/L_{\odot}}$ | Mass
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | [K] | [cgs] |  |  |  [RJup] |  |  [MJup]
c | c.A.0 | F/Z/eq*FeMg_am/O-Gr/mr | 0 | 1.72 | $1145_{-15}^{+14}$ | $4.27_{-0.04}^{+0.03}$ | $0.95_{-0.1}^{+0.1}$ | $0.6_{-0.1}^{+0.09}$ | $1.06_{-0.03}^{+0.03}$ | $-4.75_{-0.03}^{+0.03}$ | $8_{-0.5}^{+0.5}$
| c.A.1 | D/M/eq*FeMg_cd/O-Gr/mr | -1 | 1.83 | $1207_{-10}^{+11}$ | $4.34_{-0.03}^{+0.03}$ | $1.89_{-0.06}^{+0.06}$ | $0.87_{-0.006}^{+0.005}$ | $0.99_{-0.02}^{+0.02}$ | $-4.73_{-0.03}^{+0.03}$ | $8.2_{-0.5}^{+0.5}$
| c.AB.2 | F/Z/eq*FeMg_am/ALL/mr | -430 | 2.88 | $1158_{-12}^{+12}$ | $4.26_{-0.03}^{+0.02}$ | $1.27_{-0.06}^{+0.06}$ | $0.66_{-0.01}^{+0.01}$ | $1.1_{-0.01}^{+0.01}$ | $-4.71_{-0.02}^{+0.02}$ | $8.5_{-0.4}^{+0.4}$
| c.AB.3 | F/G/f*FeMg_am/ALL/mr | -432 | 2.87 | $1173_{-8}^{+18}$ | $4.26_{-0.02}^{+0.02}$ | $1.3_{-0.06}^{+0.06}$ | $0.67_{-0.01}^{+0.01}$ | $1.09_{-0.01}^{+0.01}$ | $-4.71_{-0.02}^{+0.03}$ | $8.3_{-0.4}^{+0.4}$
| c.AB.4 | F/Z/f*FeMg_am/ALL/mr | -435 | 2.91 | $1173_{-6}^{+5}$ | $4.28_{-0.02}^{+0.02}$ | $1.27_{-0.07}^{+0.05}$ | $0.67_{-0.01}^{+0.01}$ | $1.07_{-0.01}^{+0.01}$ | $-4.71_{-0.01}^{+0.01}$ | $8.5_{-0.4}^{+0.4}$
| c.AB.5 | D/M/eq*FeMg_cd/ALL/mr | -443 | 2.93 | $1057_{-13}^{+14}$ | $4.05_{-0.02}^{+0.02}$ | $1.05_{-0.04}^{+0.04}$ | $0.62_{-0.01}^{+0.01}$ | $1.4_{-0.03}^{+0.03}$ | $-4.69_{-0.03}^{+0.03}$ | $8.6_{-0.4}^{+0.3}$
| c.A.6 | F/G/f*FeMg_am/ALL/-CH4/mr | -444 | 3.02 | $1191_{-12}^{+40}$ | $4.28_{-0.02}^{+0.02}$ | $1.2_{-0.06}^{+0.06}$ | $0.7_{-0.01}^{+0.01}$ | $1.07_{-0.01}^{+0.01}$ | $-4.69_{-0.02}^{+0.05}$ | $8.4_{-0.3}^{+0.3}$
| c.AB.7 | D/S4/eq*FeMg_am/ALL/mr | -464 | 3.10 | $1198_{-23}^{+21}$ | $4.29_{-0.02}^{+0.02}$ | $1.02_{-0.03}^{+0.03}$ | $0.6_{-0.01}^{+0.01}$ | $1.08_{-0.01}^{+0.01}$ | $-4.67_{-0.04}^{+0.03}$ | $8.8_{-0.3}^{+0.3}$
| c.A.8 | F/G/f*FeMg_am/ALL/-HCN/mr | -465 | 3.14 | $1197_{-33}^{+45}$ | $4.27_{-0.04}^{+0.03}$ | $1.36_{-0.06}^{+0.05}$ | $0.69_{-0.01}^{+0.01}$ | $1.1_{-0.03}^{+0.05}$ | $-4.62_{-0.05}^{+0.06}$ | $8.7_{-0.2}^{+0.2}$
| c.AB.9 | D/M/eq*FeMg_am_P/ALL/mr | -468 | 3.25 | $1099_{-16}^{+15}$ | $4.14_{-0.03}^{+0.03}$ | $1.1_{-0.05}^{+0.05}$ | $0.61_{-0.01}^{+0.01}$ | $1.28_{-0.05}^{+0.04}$ | $-4.72_{-0.03}^{+0.03}$ | $8.9_{-0.4}^{+0.4}$
| c.AB.10 | D/Z/f*FeMg_am/ALL/mr | -478 | 3.38 | $1145_{-7}^{+9}$ | $4.23_{-0.02}^{+0.02}$ | $1.18_{-0.03}^{+0.03}$ | $0.62_{-0.009}^{+0.01}$ | $1.16_{-0.01}^{+0.01}$ | $-4.69_{-0.01}^{+0.02}$ | $8.8_{-0.3}^{+0.3}$
| c.A.11 | D/Z/eq*FeMg_am/Gr/mr | -2678 | 0.84 | $1234_{-17}^{+18}$ | $4.29_{-0.03}^{+0.04}$ | $0.79_{-0.2}^{+0.1}$ | $0.69_{-0.02}^{+0.02}$ | $1.01_{-0.03}^{+0.03}$ | $-4.66_{-0.03}^{+0.04}$ | $7.6_{-0.4}^{+0.4}$

  * •

Notes

  * •

All values presented are the median values from the fits, with uncertainties
given as the $\pm 34.1$% percentiles.

Table 16: Abridged retrieval results HR 8799 d

Planet | Index | Model | $\bm{\Delta\log}_{\mathbf{10}}\bm{\mathcal{Z}}$ | $\bm{\chi^{2}/\nu}$ | $\mathbf{T_{\rm Eff}}$ | $\bm{\log g}$ | [M/H] | C/O | Radius | $\bm{\log L/L_{\odot}}$ | Mass
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | [K] | [cgs] |  |  |  [RJup] |  |  [MJup]
d | d.AB.0 | D/M/eq*FeMg_am/ALL/mr | 0 | 1.42 | $1177_{-21}^{+21}$ | $4.18_{-0.03}^{+0.04}$ | $1.2_{-0.1}^{+0.2}$ | $0.61_{-0.04}^{+0.03}$ | $1.26_{-0.06}^{+0.05}$ | $-4.63_{-0.04}^{+0.04}$ | $9.19_{-0.07}^{+0.08}$
| d.AB.1 | D/Z/f*FeMg_am/ALL/mr | 0 | 1.39 | $1139_{-19}^{+38}$ | $4.13_{-0.02}^{+0.02}$ | $1.3_{-0.1}^{+0.1}$ | $0.6_{-0.04}^{+0.03}$ | $1.34_{-0.03}^{+0.03}$ | $-4.6_{-0.03}^{+0.06}$ | $9.2_{-0.07}^{+0.07}$
| d.AB.2 | D/M/eq*FeMg_am_P/ALL/mr | -1 | 1.47 | $1220_{-10}^{+10}$ | $4.25_{-0.02}^{+0.02}$ | $1.1_{-0.1}^{+0.1}$ | $0.49_{-0.04}^{+0.04}$ | $1.16_{-0.02}^{+0.02}$ | $-4.58_{-0.03}^{+0.03}$ | $9.2_{-0.09}^{+0.09}$
| d.AB.3 | D/M/eq*FeMg_cd/ALL/mr | -1 | 1.47 | $1220_{-9}^{+10}$ | $4.25_{-0.02}^{+0.02}$ | $1.1_{-0.1}^{+0.1}$ | $0.49_{-0.04}^{+0.04}$ | $1.16_{-0.02}^{+0.02}$ | $-4.58_{-0.03}^{+0.03}$ | $9.21_{-0.09}^{+0.09}$
| d.AB.4 | F/G/f*FeMg_cd/ALL/mr | -3 | 1.43 | $1194_{-15}^{+18}$ | $4.18_{-0.02}^{+0.02}$ | $1.5_{-0.2}^{+0.2}$ | $0.68_{-0.04}^{+0.03}$ | $1.26_{-0.03}^{+0.02}$ | $-4.58_{-0.03}^{+0.03}$ | $9.2_{-0.07}^{+0.07}$
| d.AB.5 | F/G/f*FeMg_am/ALL/mr | -3 | 1.42 | $1196_{-17}^{+20}$ | $4.18_{-0.02}^{+0.03}$ | $1.5_{-0.2}^{+0.2}$ | $0.67_{-0.04}^{+0.03}$ | $1.26_{-0.04}^{+0.03}$ | $-4.58_{-0.03}^{+0.03}$ | $9.2_{-0.07}^{+0.07}$
| d.AB.6 | F/Z/eq*FeMg_am/ALL/mr | -4 | 1.46 | $1146_{-13}^{+13}$ | $4.17_{-0.02}^{+0.02}$ | $1.6_{-0.1}^{+0.2}$ | $0.67_{-0.05}^{+0.04}$ | $1.27_{-0.03}^{+0.03}$ | $-4.62_{-0.03}^{+0.03}$ | $9.19_{-0.07}^{+0.07}$
| d.AB.7 | D/Z/eqMg_am/ALL/mr | -4 | 1.47 | $1150_{-16}^{+21}$ | $4.17_{-0.03}^{+0.04}$ | $0.95_{-0.1}^{+0.1}$ | $0.5_{-0.04}^{+0.04}$ | $1.27_{-0.06}^{+0.05}$ | $-4.67_{-0.04}^{+0.04}$ | $9.18_{-0.07}^{+0.07}$
| d.AB.8 | D/Z/eq*FeMg_am/ALL/mr | -5 | 1.49 | $1166_{-16}^{+19}$ | $4.21_{-0.03}^{+0.03}$ | $0.98_{-0.1}^{+0.1}$ | $0.48_{-0.04}^{+0.04}$ | $1.22_{-0.05}^{+0.05}$ | $-4.66_{-0.04}^{+0.04}$ | $9.19_{-0.06}^{+0.07}$
| d.AB.9 | D/Z/eq*FeMg_am/ALL/ | -6 | 1.50 | $1158_{-30}^{+22}$ | $4.5_{-0.3}^{+0.3}$ | $1.3_{-0.3}^{+0.3}$ | $0.48_{-0.04}^{+0.04}$ | $1.23_{-0.06}^{+0.08}$ | $-4.61_{-0.06}^{+0.05}$ | $20.70_{-10}^{+19}$
| d.AB.10 | F/Z/f*FeMg_am/ALL/mr | -6 | 1.44 | $1157_{-16}^{+30}$ | $4.16_{-0.01}^{+0.01}$ | $1.7_{-0.1}^{+0.1}$ | $0.64_{-0.04}^{+0.04}$ | $1.29_{-0.02}^{+0.02}$ | $-4.57_{-0.03}^{+0.04}$ | $9.22_{-0.05}^{+0.05}$
| d.AB.11 | D/S4/eq*FeMg_am/ALL/mr | -12 | 1.51 | $1198_{-20}^{+21}$ | $4.24_{-0.04}^{+0.03}$ | $0.71_{-0.1}^{+0.2}$ | $0.34_{-0.06}^{+0.07}$ | $1.17_{-0.04}^{+0.05}$ | $-4.62_{-0.04}^{+0.04}$ | $9.19_{-0.07}^{+0.07}$
| d.A.12 | D/Z/eq*FeMg_am/Gr/mr | -2276 | 0.79 | $1172_{-18}^{+19}$ | $4.18_{-0.03}^{+0.03}$ | $2_{-0.2}^{+0.2}$ | $0.78_{-0.03}^{+0.02}$ | $1.25_{-0.04}^{+0.04}$ | $-4.61_{-0.04}^{+0.04}$ | $9.19_{-0.08}^{+0.08}$
| d.A.13 | D/Z/eq*FeMg_am/-Gr/mr | -3498 | 1.03 | $1162_{-28}^{+28}$ | $4.22_{-0.04}^{+0.04}$ | $0.46_{-0.7}^{+0.7}$ | $0.31_{-0.1}^{+0.1}$ | $1.19_{-0.06}^{+0.06}$ | $-4.68_{-0.06}^{+0.06}$ | $9.2_{-0.08}^{+0.08}$

  * •

Notes

  * •

All values presented are the median values from the fits, with uncertainties
given as the $\pm 34.1$% percentiles.

Table 17: Abridged retrieval results HR 8799 e

Planet | Index | Model | $\bm{\Delta\log}_{\mathbf{10}}\bm{\mathcal{Z}}$ | $\bm{\chi^{2}/\nu}$ | $\mathbf{T_{\rm Eff}}$ | $\bm{\log g}$ | [M/H] | C/O | Radius | $\bm{\log L/L_{\odot}}$ | Mass
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | [K] | [cgs] |  |  |  [RJup] |  |  [MJup]
e | e.A.0 | F/Z/eq*FeMg_am/ALL/-CH4/mr | 0 | 1.23 | $1139_{-20}^{+21}$ | $4.18_{-0.05}^{+0.05}$ | $1.9_{-0.1}^{+0.1}$ | $0.88_{-0.02}^{+0.01}$ | $1.13_{-0.05}^{+0.05}$ | $-4.73_{-0.05}^{+0.05}$ | $7.5_{-0.6}^{+0.6}$
| e.AB.1 | F/G/eq*FeMg_am/ALL/mr | 0 | 1.22 | $1172_{-29}^{+27}$ | $4.2_{-0.06}^{+0.06}$ | $1.8_{-0.2}^{+0.1}$ | $0.88_{-0.02}^{+0.02}$ | $1.1_{-0.05}^{+0.05}$ | $-4.71_{-0.05}^{+0.06}$ | $7.5_{-0.7}^{+0.7}$
| e.AB.2 | F/Z/eq*FeMg_am/ALL/mr | 0 | 1.23 | $1134_{-20}^{+24}$ | $4.18_{-0.05}^{+0.05}$ | $1.9_{-0.1}^{+0.1}$ | $0.87_{-0.02}^{+0.02}$ | $1.14_{-0.05}^{+0.05}$ | $-4.74_{-0.05}^{+0.06}$ | $7.5_{-0.6}^{+0.6}$
| e.AB.3 | F/G/fMg_am/ALL/mr | -1 | 1.22 | $1206_{-23}^{+27}$ | $4.24_{-0.05}^{+0.05}$ | $1.8_{-0.2}^{+0.1}$ | $0.88_{-0.02}^{+0.02}$ | $1.06_{-0.03}^{+0.03}$ | $-4.67_{-0.05}^{+0.05}$ | $7.6_{-0.7}^{+0.7}$
| e.AB.4 | F/G/fFe/ALL/mr | -2 | 1.22 | $1203_{-28}^{+30}$ | $4.24_{-0.05}^{+0.05}$ | $1.8_{-0.2}^{+0.1}$ | $0.89_{-0.02}^{+0.02}$ | $1.07_{-0.04}^{+0.03}$ | $-4.69_{-0.05}^{+0.06}$ | $7.6_{-0.6}^{+0.7}$
| e.AB.5 | F/G/grey_P/ALL/mr | -3 | 1.23 | $1261_{-18}^{+18}$ | $4.35_{-0.06}^{+0.05}$ | $1.7_{-0.3}^{+0.2}$ | $0.89_{-0.03}^{+0.02}$ | $0.94_{-0.03}^{+0.04}$ | $-4.64_{-0.05}^{+0.04}$ | $7.6_{-0.7}^{+0.7}$
| e.AB.6 | F/G/f*FeMg_am/ALL/mr | -3 | 1.23 | $1214_{-23}^{+22}$ | $4.25_{-0.04}^{+0.04}$ | $1.8_{-0.2}^{+0.1}$ | $0.89_{-0.02}^{+0.02}$ | $1.05_{-0.03}^{+0.03}$ | $-4.69_{-0.04}^{+0.04}$ | $7.5_{-0.6}^{+0.6}$
| e.AB.7 | F/S4/eq*FeMg_am/ALL/mr | -6 | 1.26 | $1202_{-20}^{+19}$ | $4.28_{-0.05}^{+0.05}$ | $1.8_{-0.2}^{+0.1}$ | $0.9_{-0.03}^{+0.02}$ | $1.01_{-0.03}^{+0.03}$ | $-4.7_{-0.05}^{+0.05}$ | $7.6_{-0.7}^{+0.8}$
| e.AB.8 | D/M/eqFeMg_cd_P/ALL/mr | -6 | 1.32 | $1215_{-18}^{+19}$ | $4.31_{-0.05}^{+0.05}$ | $2.1_{-0.3}^{+0.2}$ | $0.83_{-0.02}^{+0.01}$ | $1.0_{-0.03}^{+0.03}$ | $-4.68_{-0.05}^{+0.04}$ | $7.9_{-0.7}^{+0.7}$
| e.AB.9 | D/Z/eq*FeMg_cd/ALL/mr | -7 | 1.27 | $1084_{-23}^{+27}$ | $4.09_{-0.06}^{+0.06}$ | $1.3_{-0.1}^{+0.1}$ | $0.77_{-0.03}^{+0.03}$ | $1.29_{-0.07}^{+0.06}$ | $-4.82_{-0.05}^{+0.06}$ | $7.9_{-0.7}^{+0.7}$
| e.AB.10 | D/M/eq*FeMg_cd_P/ALL/mr | -7 | 1.32 | $1215_{-18}^{+17}$ | $4.31_{-0.05}^{+0.05}$ | $2.1_{-0.3}^{+0.2}$ | $0.83_{-0.02}^{+0.01}$ | $1.0_{-0.03}^{+0.03}$ | $-4.68_{-0.05}^{+0.05}$ | $7.9_{-0.7}^{+0.6}$
| e.AB.11 | D/M/eq*FeMg_cd/ALL/mr | -7 | 1.28 | $1106_{-28}^{+33}$ | $4.11_{-0.06}^{+0.08}$ | $1.3_{-0.1}^{+0.2}$ | $0.78_{-0.03}^{+0.03}$ | $1.26_{-0.09}^{+0.07}$ | $-4.82_{-0.06}^{+0.07}$ | $7.9_{-0.7}^{+0.7}$
| e.A.12 | F/Z/eq*FeMg_am/ALL/-HCN/mr | -8 | 1.30 | $1148_{-23}^{+24}$ | $4.15_{-0.05}^{+0.06}$ | $1.91_{-0.1}^{+0.07}$ | $0.87_{-0.02}^{+0.02}$ | $1.17_{-0.06}^{+0.06}$ | $-4.73_{-0.06}^{+0.06}$ | $7.5_{-0.6}^{+0.7}$
| e.AB.13 | D/Z/f*FeMg_am/ALL/mr | -9 | 1.33 | $1187_{-30}^{+50}$ | $4.26_{-0.04}^{+0.04}$ | $2_{-0.3}^{+0.3}$ | $0.83_{-0.02}^{+0.02}$ | $1.08_{-0.03}^{+0.03}$ | $-4.69_{-0.06}^{+0.07}$ | $8.1_{-0.6}^{+0.6}$
| e.AB.14 | D/Z/eq*FeMg2/Gr/mr | -9 | 1.35 | $1170_{-27}^{+26}$ | $4.26_{-0.06}^{+0.06}$ | $1.3_{-0.2}^{+0.2}$ | $0.76_{-0.03}^{+0.02}$ | $1.06_{-0.05}^{+0.05}$ | $-4.77_{-0.06}^{+0.05}$ | $7.8_{-0.7}^{+0.7}$
| e.AB.15 | F/Z/f*FeMg_am/ALL/mr | -10 | 1.29 | $1143_{-18}^{+12}$ | $4.19_{-0.05}^{+0.05}$ | $1.99_{-0.05}^{+0.05}$ | $0.86_{-0.02}^{+0.02}$ | $1.09_{-0.03}^{+0.03}$ | $-4.73_{-0.04}^{+0.04}$ | $7_{-0.7}^{+0.6}$
| e.A.16 | D/Z/eqFeMg_am/ALL/Fe-H15mr | -10 | 1.37 | $1202_{-17}^{+20}$ | $4.32_{-0.05}^{+0.05}$ | $1.5_{-0}^{+0}$ | $0.76_{-0.02}^{+0.02}$ | $0.99_{-0.03}^{+0.03}$ | $-4.73_{-0.05}^{+0.05}$ | $7.8_{-0.7}^{+0.7}$
| e.A.17 | D/Z/eqFeMg_am/ALL/Fe-H20mr | -10 | 1.33 | $1179_{-28}^{+29}$ | $4.25_{-0.06}^{+0.06}$ | $2.0_{-0}^{+0}$ | $0.82_{-0.01}^{+0.01}$ | $1.08_{-0.07}^{+0.07}$ | $-4.77_{-0.05}^{+0.06}$ | $8_{-0.6}^{+0.6}$
| e.A.18 | D/M/grey_P/ALL/mr | -10 | 1.38 | $1245_{-16}^{+16}$ | $4.36_{-0.05}^{+0.05}$ | $1.5_{-0.1}^{+0.1}$ | $0.63_{-0.02}^{+0.02}$ | $0.95_{-0.03}^{+0.03}$ | $-4.68_{-0.04}^{+0.04}$ | $8_{-0.8}^{+0.8}$
| e.AB.19 | D/M/eq*FeMg_am/ALL/mr | -11 | 1.33 | $1206_{-22}^{+21}$ | $4.3_{-0.05}^{+0.05}$ | $2.1_{-0.4}^{+0.2}$ | $0.83_{-0.02}^{+0.01}$ | $1.02_{-0.04}^{+0.05}$ | $-4.69_{-0.06}^{+0.05}$ | $8.1_{-0.6}^{+0.6}$
| e.AB.20 | D/Z/eq*FeMg_am_h/ALL/mr | -11 | 1.33 | $1129_{-27}^{+29}$ | $4.15_{-0.05}^{+0.06}$ | $1.4_{-0.1}^{+0.2}$ | $0.78_{-0.03}^{+0.03}$ | $1.21_{-0.06}^{+0.06}$ | $-4.77_{-0.07}^{+0.07}$ | $7.9_{-0.6}^{+0.6}$
| e.AB.21 | D/M/grey_P/ALL/mr | -11 | 1.37 | $1245_{-15}^{+15}$ | $4.34_{-0.07}^{+0.05}$ | $1.5_{-0.1}^{+0.1}$ | $0.63_{-0.02}^{+0.02}$ | $0.97_{-0.03}^{+0.06}$ | $-4.64_{-0.06}^{+0.08}$ | $8_{-0.8}^{+0.8}$
| e.A.22 | D/Z/eqFeMg_am/ALL/Fe-H10mr | -11 | 1.41 | $1208_{-19}^{+21}$ | $4.29_{-0.05}^{+0.05}$ | $1.0_{-0}^{+0}$ | $0.69_{-0.02}^{+0.02}$ | $0.98_{-0.03}^{+0.03}$ | $-4.72_{-0.05}^{+0.05}$ | $7.3_{-0.6}^{+0.7}$
| e.A.23 | D/Z/eqFeMg_am/ALL/mr | -11 | 1.36 | $1209_{-17}^{+20}$ | $4.32_{-0.05}^{+0.04}$ | $1.7_{-0.3}^{+0.7}$ | $0.79_{-0.05}^{+0.03}$ | $0.99_{-0.03}^{+0.03}$ | $-4.72_{-0.05}^{+0.05}$ | $7.9_{-0.7}^{+0.6}$
| e.AB.24 | D/Z/eqAl2O3/ALL/mr | -11 | 1.34 | $1235_{-14}^{+17}$ | $4.36_{-0.05}^{+0.04}$ | $2.3_{-0.3}^{+0.1}$ | $0.82_{-0.01}^{+0.01}$ | $0.94_{-0.03}^{+0.03}$ | $-4.73_{-0.04}^{+0.04}$ | $7.9_{-0.7}^{+0.7}$
| e.AB.25 | D/Z/eq*FeMg_am/ALL/mr | -11 | 1.36 | $1207_{-17}^{+19}$ | $4.32_{-0.04}^{+0.04}$ | $2.1_{-0.6}^{+0.3}$ | $0.81_{-0.06}^{+0.01}$ | $0.99_{-0.03}^{+0.03}$ | $-4.72_{-0.05}^{+0.04}$ | $8_{-0.6}^{+0.6}$
| e.AB.26 | D/Z/eq*FeMg_am/ALL/ | -12 | 1.36 | $1224_{-22}^{+21}$ | $4.8_{-0.2}^{+0.2}$ | $2_{-0.3}^{+0.3}$ | $0.76_{-0.03}^{+0.05}$ | $0.96_{-0.03}^{+0.03}$ | $-4.78_{-0.06}^{+0.06}$ | $19.91_{-7}^{+13}$
| e.AB.27 | F/G/grey/ALL/mr | -12 | 1.37 | $1245_{-31}^{+58}$ | $4.24_{-0.06}^{+0.06}$ | $2.05_{-0.08}^{+0.09}$ | $0.91_{-0.02}^{+0.02}$ | $1.04_{-0.04}^{+0.04}$ | $-4.6_{-0.07}^{+0.09}$ | $7.4_{-0.8}^{+0.8}$
| e.AB.28 | D/S4/eq*FeMg_cd/ALL/mr | -12 | 1.27 | $1093_{-24}^{+27}$ | $4.08_{-0.05}^{+0.05}$ | $1.4_{-0.1}^{+0.1}$ | $0.78_{-0.03}^{+0.02}$ | $1.3_{-0.05}^{+0.05}$ | $-4.73_{-0.06}^{+0.06}$ | $7.8_{-0.7}^{+0.7}$
| e.A.29 | D/Z/eqFeMg_am/ALL/Fe-H05mr | -17 | 1.49 | $1217_{-18}^{+20}$ | $4.29_{-0.05}^{+0.05}$ | $0.5_{-0}^{+0}$ | $0.52_{-0.04}^{+0.03}$ | $0.97_{-0.03}^{+0.03}$ | $-4.71_{-0.05}^{+0.04}$ | $7.1_{-0.6}^{+0.7}$
| e.A.30 | D/Z/eqFeMg_am/ALL/Fe-H00mr | -26 | 1.58 | $1227_{-16}^{+17}$ | $4.29_{-0.05}^{+0.05}$ | $0.0_{-0}^{+0}$ | $0.21_{-0.02}^{+0.02}$ | $0.95_{-0.03}^{+0.03}$ | $-4.72_{-0.05}^{+0.04}$ | $6.9_{-0.6}^{+0.6}$
| e.A.31 | D/Z/eq*FeMg_am/Gr/mr | -2294 | 0.72 | $1173_{-28}^{+31}$ | $4.24_{-0.06}^{+0.06}$ | $1.8_{-0.5}^{+0.4}$ | $0.85_{-0.04}^{+0.02}$ | $1.06_{-0.05}^{+0.06}$ | $-4.71_{-0.07}^{+0.07}$ | $7.5_{-0.7}^{+0.7}$
| e.A.32 | D/Z/CLR/Gr/mr | -2301 | 0.79 | $1339_{-38}^{+36}$ | $4.3_{-0.07}^{+0.07}$ | $0.65_{-0.2}^{+0.2}$ | $0.71_{-0.07}^{+0.05}$ | $0.94_{-0.05}^{+0.06}$ | $-4.53_{-0.08}^{+0.08}$ | $6.9_{-0.6}^{+0.8}$
| e.A.33 | D/Z/eq*FeMg_am/-Gr/mr | -3496 | 1.88 | $1143_{-32}^{+38}$ | $4.17_{-0.08}^{+0.08}$ | $2_{-0.4}^{+0.3}$ | $0.71_{-0.2}^{+0.08}$ | $1.15_{-0.09}^{+0.09}$ | $-4.79_{-0.09}^{+0.08}$ | $7.5_{-0.7}^{+0.7}$

  * •

Notes

  * •

All values presented are the median values from the fits, with uncertainties
given as the $\pm 34.1$% percentiles.