A Collatz Conjecture Proof

Robert H. Nichols, Jr.
The Labry School of Science, Technology, and Business
Cumberland University
Lebanon, TN 37087
USA
<EMAIL_ADDRESS>

###### Abstract

We represent the generalized Collatz function with the recursive ruler
function $r(2n)=r(n)+1$ and $r(2n+1)=1$. We generate even-only and odd-only
Collatz subsequences that contain significantly fewer elements term by term,
to $2$ and $1$, respectively, than are present in the original $3n+1$ and the
Terras-modified Collatz sequences. We show that a nonlinear, coupled system of
difference equations yields a complete acyclic (except for the trivial cycle)
Collatz tree in odds not divisible by $3$ with root vertex $1$. We construct a
complete Collatz tree with the axiom of choice and prove the Collatz
conjecture.

##  1 Introduction: The Collatz sequence and the Collatz conjecture

We address a long–standing and still–open mathematical problem regarding
integer sequences attibuted to Lothar Collatz in the 1930s (cf. [1]). We
represent the non-negative integers with $\mathbb{N}$ and the positive
integers with $\mathbb{N}_{>0}$. We define a $sequence$ and a $subsequence$ of
positive integers.

######  Definition 1.1 (Sequence).

For $n\in\mathbb{N}$ and $x_{n}\in\mathbb{N}_{>0}$ a sequence $X$ in
$\mathbb{N}_{>0}$ is an ordered set of positive integers denoted by X =
$X(x_{0})=(x_{n})_{n\geq 0}=(x_{0},x_{1},\ldots)$.

######  Definition 1.2 (Subsequence).

For $n\in\mathbb{N}$ and $x_{n}\in\mathbb{N}_{>0}$ let $(x_{n})_{n\geq 0}$ be
a sequence. A subsequence of $(x_{n})_{n\geq 0}$ is a sequence
$(x_{n_{i}})_{i\geq 0}$ with $n_{i}<n_{i+1}$ for all $i\geq 0$.

If a sequence $A$ is a subsequence of $B$, then we say that $B$ covers $A$ and
denote the covering relation by $A\lessdot B$. For a sequence $A$ the
cardinality $|A|$ is the number of elements in $A$. Definitions 1.1 and 1.2
allow sequences, for example, to have infinite cardinality, to be non-
decreasing, or to contain cycles of repeating elements. If we choose to
terminate a sequence $A$ at the first occurrence of a particular number, say
$f\in\mathbb{N}$, then we denote the finite sequence as $A_{f}$ with
cardinality $|A_{f}|$. The smallest $n$ such that $a_{n}=f$ is the stopping
time of $A(a_{0})$ and is denoted by $A_{S}$. Thus, if we have a finite
sequence $A_{f}$ then we have that $|A_{f}(a_{0})|=A_{S}(a_{0})+1$.

For $n\in\mathbb{N}$ and $c_{n}\in\mathbb{N}_{>0}$ we define the Collatz
sequence $C(c_{0})=(c_{0},c_{1},\ldots)$ with the recursive function

$c_{n+1}=\begin{cases}c_{n}/2&\text{if }c_{n}\equiv 0\ \mbox{\rm(mod}\
2\mbox{\rm)}\\\ 3c_{n}+1&\text{if }c_{n}\equiv 1\ \mbox{\rm(mod}\
2\mbox{\rm)}\\\ \end{cases}$ (1)

(cf. [1, 2], A014682 [3], A070165 [3]). The Collatz conjecture (CC) is that
$1\in C(n)$ for all $n\in\mathbb{N}_{>0}$ (cf. [1, 2]). Other formulations of
the CC exist (cf. [1, 2]). The CC is equivalent to the statements that
$|C_{1}(n)|$ is finite for all $n\in\mathbb{N}_{>0}$ [4] and that the
repeating subsequence $(4,2,1)$ is the only cycle in $C$. In addition, if
“sufficient” subsequences of the integers satisfy the CC then the CC is true
(see [1, 2], e.g. [10]).

##  2 The Terras-modified Collatz sequence

Terras (1976) [4] defines a subsequence of $C$ (1) to address the Collatz
conjecture. For $n\in\mathbb{N}$ and $t_{n}\in\mathbb{N}_{>0}$ the Terras-
modified Collatz sequence $T(t_{0})=(t_{0},t_{1},\ldots)$ is

$t_{n+1}=\begin{cases}t_{n}/2&\text{if }t_{n}\equiv 0\ \mbox{\rm(mod}\
2\mbox{\rm)}\\\ (3t_{n}+1)/2&\text{if }t_{n}\equiv 1\ \mbox{\rm(mod}\
2\mbox{\rm)}\\\ \end{cases}$ (2)

([4, 5], cf. [1, 2], A070168 [3]). For $n\in\mathbb{N}_{>0}$ we have by
construction that $T(n)$ (2) is a subsequence of $C(n)$ (1). Further, by
construction as well, we have that if sequences $T$ satisfy the Collatz
conjecture, then the Collatz conjecture is true and vice versa.

##  3 A generalized alternating parity Terras-modified Collatz sequence

For $n\in\mathbb{N}$, $g_{n}\in\mathbb{N}_{>0}$, odd $a\in\mathbb{N}$, and odd
$b\in\mathbb{N}$ we define a generalized Terras-modified Collatz sequence
$G(g_{0})=(g_{0},g_{1},\ldots)$ with the recursive function

$g_{n+1}=\begin{cases}g_{n}/2&\text{if }g_{n}\equiv 0\ \mbox{\rm(mod}\
2\mbox{\rm)}\\\ (ag_{n}+b)/2&\text{if }g_{n}\equiv 1\ \mbox{\rm(mod}\
2\mbox{\rm)}.\\\ \end{cases}$ (3)

We note that Conway ([6], cf. [1, 7]) has addressed some similar
generalizations of the Collatz problem.

We now solve the difference equations that generate sequences $G$ (3). We
first define the $parity$ of an integer as its property of being $even$ or
$odd$. We present these results in the following theorem.

######  Theorem 3.1.

Let $\mu(g_{0})\in\mathbb{N}$ and $\nu(g_{0})\in\mathbb{N}$ be the number of
iterations of (3) for even and odd $g_{0}\in G$ to change parity to odd
$g_{\mu(g_{0})}$ and even $g_{\nu(g_{0})}$, respectively. For even $g_{0}$ the
odd solution to (3) is

$g_{\mu(g_{0})}=\bigg{(}\dfrac{1}{2}\bigg{)}^{\mu(g_{0})}g_{0}.$ (4)

For odd $g_{0}$ the even solution to (3) is

$g_{\nu(g_{0})}=\bigg{(}\dfrac{a}{2}\bigg{)}^{\nu(g_{0})}\bigg{(}g_{0}+\dfrac{b}{a-2}\bigg{)}-\dfrac{b}{a-2}$
(5)

with $b\equiv 0$ mod ($a-2$).

###### Proof.

First, we consider the initial condition that $g_{0}\equiv 0$ (mod $2$) in
$\eqref{Gseq}$. The associated first-order, linear, homogeneous difference
equation is

$2g_{n+1}-g_{n}=0.$ (6)

We assume a solution to (6) of the form $g_{n}=m\lambda^{n}$ with
$m\in\mathbb{N}$, $n\in\mathbb{N}$, and $\lambda\in\mathbb{R}$. We find that
$m=g_{0}$ and $\lambda=1/2$, yielding

$g_{n}=\bigg{(}\dfrac{1}{2}\bigg{)}^{n}g_{0}.$ (7)

For even $g_{0}$ in (7) we have that $g_{n}\equiv 1$ (mod $2$) when
$n=\mu(g_{0})$, yielding (4). Second, we consider the initial condition that
$g_{0}\equiv 1$ (mod $2$) in (3). The associated first-order, linear,
inhomogeneous difference equation is

$2g_{n+1}-ag_{n}=b.$ (8)

To find homogeneous and particular solutions to (8) we iterate (3) over odd
elements $g_{n-1}$ to $g_{0}$. We have that

$\displaystyle g_{n}$ $\displaystyle=\dfrac{a}{2}g_{n-1}+\dfrac{b}{2}$
$\displaystyle=\bigg{(}\frac{a}{2}\bigg{)}\bigg{(}\dfrac{a}{2}g_{n-2}+\dfrac{b}{2}\bigg{)}+\dfrac{b}{2}$
$\displaystyle=\bigg{(}\frac{a}{2}\bigg{)}\bigg{(}\dfrac{a}{2}\Big{(}\dfrac{a}{2}g_{n-3}+\dfrac{b}{2}\Big{)}+\dfrac{b}{2}\bigg{)}+\dfrac{b}{2}$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
$\displaystyle=\bigg{(}\frac{a}{2}\bigg{)}^{n}g_{0}+\dfrac{b}{2}\Bigg{(}\bigg{(}\dfrac{a}{2}\bigg{)}^{n-1}+\bigg{(}\dfrac{a}{2}\bigg{)}^{n-2}+\ldots+\bigg{(}\dfrac{a}{2}\bigg{)}^{2}+\bigg{(}\dfrac{a}{2}\bigg{)}^{1}+1\Bigg{)}$
$\displaystyle=\bigg{(}\frac{a}{2}\bigg{)}^{n}g_{0}+\dfrac{b}{2}\Bigg{(}\dfrac{1-\big{(}a/2\big{)}^{n}}{1-\big{(}a/2\big{)}}\Bigg{)}\quad\text{by
the geometric series identity.}$

The general solution to (8) for odd $g_{0}$ is thus

$g_{n}=\bigg{(}\frac{a}{2}\bigg{)}^{n}\bigg{(}g_{0}+\frac{b}{a-2}\bigg{)}-\frac{b}{a-2}.$
(9)

Given that $g_{0}$ is an odd integer we now impose the constraint that
$b\equiv 0$ mod ($a-2$) so that the factor $g_{0}+b/(a-2)$ in (9) is an even
integer. For odd $g_{0}$ we have that $g_{n}\equiv 0$ (mod $2$) when
$n=\nu(g_{0})$, yielding (5). ∎

######  Remark 3.1.

Even integers $g_{n}$ in (9) may exist for odd $a$ and odd $b\not\equiv 0$ mod
($a-2$), but we do not investigate those sequences.

The iteration exponents $\mu(g_{0})$ and $\nu(g_{0})$ in (4) and (5) are
related to the $ruler$ function, which we now define.

######  Definition 3.1.

For $n\in\mathbb{N}_{>0}$ the ruler function $r(n)$ is the exponent of the
highest power of two that divides $2n$ (A001511 [3]) and satisfies the
recursive equation

$\displaystyle r(2n)$ $\displaystyle=r(n)+1$ (10) $\displaystyle r(2n+1)$
$\displaystyle=1.$

By Definition 3.1 we have that $\mu(g_{0})\in\mathbb{N}_{>0}$ and
$\nu(g_{0})\in\mathbb{N}_{>0}$ in (4) and (5), respectively, satisfy

$\displaystyle\mu(g_{0})$ $\displaystyle=r(g_{0}/2)$ $\displaystyle\text{if
}g_{0}\equiv 0\ \mbox{\rm(mod}\ 2\mbox{\rm)}$ (11) $\displaystyle\nu(g_{0})$
$\displaystyle=r((g_{0}+b/(a-2))/2)$ $\displaystyle\text{if }g_{0}\equiv 1\
\mbox{\rm(mod}\ 2\mbox{\rm)}.$

We now present a generalized, alternating parity, Terras-modified Collatz
function with the recursive ruler function $r(2n)=r(n)+1$ and $r(2n+1)=1$. We
denote this function as the $GAPT$ Collatz function.

######  Definition 3.2 (GAPT Collatz Function).

We let $c=b/(a-2)$ with odd $a\in\mathbb{N}$ and odd $b\in\mathbb{N}$ in (5)
such that $c$ is a positive odd integer. For $n\in\mathbb{N}$,
$x(n)\in\mathbb{N}_{>0}$ (11), $y(n)\in\mathbb{N}_{>0}$ (11), and
$h_{n}\in\mathbb{N}_{>0}$ we define the generalized alternating parity Terras-
modified (GAPT) Collatz sequence $H(h_{0})=(h_{0},h_{1},\ldots)$ with the
recursive function

$h_{n+1}=\begin{cases}\bigg{(}\dfrac{1}{2}\bigg{)}^{r(h_{n}/2)}h_{n}&\text{if
}h_{n}\equiv 0\ \mbox{\rm(mod}\ 2\mbox{\rm)}\\\
\bigg{(}\dfrac{a}{2}\bigg{)}^{r((h_{n}+c)/2)}(h_{n}+c)-c&\text{if }h_{n}\equiv
1\ \mbox{\rm(mod}\ 2\mbox{\rm)}.\end{cases}$ (12)

As far as we know a similar ruler function representation for odd Collatz
iterations first appears in the work of Crandall [8], however, an explicit
functional form for $r(n)$ is not provided. We have found a similar
(unpublished) approach [9].

######  Theorem 3.2.

Subsequences of consecutive even or odd elements in the generalized sequence
$G$ (3) are finite in length.

###### Proof.

For generalized sequences $G$ (3) and $H$ (12) we have by construction that
$H\lessdot G$. The exponents in the alternating parity sequence $H$ (12) exist
and are finite by Definition 3.1. Thus, subsequences of consecutive even or
odd elements in $G$ (3) are finite in length. ∎

##  4 Alternating Parity Terras-modified (APT) Collatz sequences

Henceforth we only consider the original $3n+1$ Collatz conjecture with $a=3$,
$b=1$, and $c=1$ in (12). We now define the alternating parity Terras-modified
(APT) Collatz function.

######  Definition 4.1 (APT Collatz Sequences).

For $n\in\mathbb{N}$ and $a_{n}\in\mathbb{N}_{>0}$ we define the alternating
parity Terras-modified (APT) Collatz sequence $A(a_{0})=(a_{0},a_{1},\ldots)$
as

$a_{n+1}=\begin{cases}\bigg{(}\dfrac{1}{2}\bigg{)}^{r(a_{n}/2)}a_{n}&\text{if
}a_{n}\equiv 0\ \mbox{\rm(mod}\ 2\mbox{\rm)}\\\
\bigg{(}\dfrac{3}{2}\bigg{)}^{r((a_{n}+1)/2)}(a_{n}+1)-1&\text{if }a_{n}\equiv
1\ \mbox{\rm(mod}\ 2\mbox{\rm)}\end{cases}$ (13)

with $r(n)$ given by Definition 3.1.

######  Corollary 4.1.

For $n\in\mathbb{N}_{>0}$ subsequences of consecutive even or odd elements in
$C(n)$ (1) and $T(n)$ (2) are finite in length.

###### Proof.

Corollary 4.1 follows from Theorem 3.2 with $a=3$, $b=1$, and $c=1$ in (12). ∎

######  Theorem 4.1.

For $n\in\mathbb{N}_{>0}$ the sequences $C(n)$ (1), $T(n)$ (2), and $A(n)$
(13) satisfy the following statements.

  1. a)

$A(n)\lessdot T(n)\lessdot C(n)$.

  2. b)

$1\in C(n)$ if and only if $1\in A(n)$.

  3. c)

If $1\in C(n)$ then $|A_{1}(n)|\leq|T_{1}(n)|\leq|C_{1}(n)|$.

###### Proof.

These statements follow directly by construction of $A$ (13) from $T$ (2) and
of $T$ (2) from $C$ (1) by reducing the number of iterations in $C$ and $T$. ∎

######  Conjecture 4.1.

For $n\in\mathbb{N}_{>0}$ the APT Collatz sequences $A(n)$ (13) satisfy $1\in
A(n)$, implying that the CC is true.

##  5 Modified APT (MAPT) Collatz sequences

We now address the ruler function exponents in (13). We first define two
functions that are related to the ruler function to simplify (perhaps) this
analysis.

######  Definition 5.1.

For $n\in\mathbb{N}$ we define the recursive function
$p:\mathbb{N}\rightarrow\mathbb{N}$ as

$\displaystyle p(2n)$ $\displaystyle=n$ (14) $\displaystyle p(2n+1)$
$\displaystyle=p(n)$

with $p(0)=0$ (A0025480 [3]). Herein we also use the notation that
$p_{n}=p(n)$.

######  Definition 5.2.

For $n\in\mathbb{N}$ we define the recursive function
$q:\mathbb{N}\rightarrow\mathbb{N}$ in terms of the ruler function $r(n)$ (10)
(A001511 [3]) as

$q(n)=r(n+1),$

implying that

$\displaystyle q(2n)$ $\displaystyle=1$ (15) $\displaystyle q(2n+1)$
$\displaystyle=q(n)+1$

with $q(0)=1$. Herein we also use the notation that $q_{n}=q(n)$.

We now represent even and odd integers in terms of $p_{n}$ (14) and $q_{n}$
(15), although alternative representations are possible.

######  Lemma 5.1.

For $n\in\mathbb{N}$, $p_{n}\in\mathbb{N}_{>0}$ and $q_{n}\in\mathbb{N}_{>0}$,
the $n^{\text{\tiny th}}$ elements of the nondecreasing sequences of even
positive integers $e_{n}=2(n+1)$ and odd positive integers $o_{n}=2n+1$ are

$e_{n}(p_{n},q_{n})=(2p_{n}+1)2^{q_{n}}$ (16)

and

$o_{n}(p_{n},q_{n})=(2p_{n}+1)2^{q_{n}}-1,$ (17)

respectively.

###### Proof.

For $m,n\in\mathbb{N}$ we prove by induction. For $n=0$ the base case for (16)
is

$\displaystyle e_{0}$ $\displaystyle=(2p_{0}+1)2^{q_{0}}$
$\displaystyle=(2(0)+1)2^{1}$ $\displaystyle=2.$

For the inductive step for (16) we assume that there exists an $m$ such that
$e_{m}=(2p_{m}+1)2^{q_{m}}=2(m+1)$ and show that $e_{m+1}=2(m+2)$. For even
$m$ we have odd $m+1=2j+1$ for some $j=m/2\in\mathbb{N}$ yielding

$\displaystyle e_{m+1}$ $\displaystyle=(2p_{m+1}+1)2^{q_{m+1}}$
$\displaystyle=(2p_{2j+1}+1)2^{q_{2j+1}}$
$\displaystyle=(2p_{j}+1)2^{q_{j}+1}$ $\displaystyle=(2(j+1))2^{1}$ by the
inductive hypothesis $\displaystyle=(2(m/2+1))2$ $\displaystyle=2(m+2).$

For odd $m$ we have even $m+1=2j$ for some $j=(m+1)/2\in\mathbb{N}$ yielding

$\displaystyle e_{m+1}$ $\displaystyle=(2p_{m+1}+1)2^{q_{m+1}}$
$\displaystyle=(2p_{2j}+1)2^{q_{2j}}$ $\displaystyle=(2j+1)2^{1}$
$\displaystyle=(2((m+1)/2)+1)2^{1}$ $\displaystyle=2(m+2).$

For $n=0$ the base case for (17) is

$\displaystyle o_{0}$ $\displaystyle=(2p_{0}+1)2^{q_{0}}-1$
$\displaystyle=(2(0)+1)2^{1}-1$ $\displaystyle=1.$

For the inductive step for (17) we assume that there exists an $m$ such that
$o_{m}=(2p_{m}+1)2^{q_{m}}-1=2m+1$ and show that $o_{m+1}=2m+3$. For even $m$
we have odd $m+1=2j+1$ for some $j=m/2\in\mathbb{N}$ yielding

$\displaystyle o_{m+1}$ $\displaystyle=(2p_{m+1}+1)2^{q_{m+1}}-1$
$\displaystyle=(2p_{2j+1}+1)2^{q_{2j+1}}-1$
$\displaystyle=(2p_{j}+1)2^{q_{j}+1}-1$ $\displaystyle=(2(j+1))2^{1}-1$ by the
inductive hypothesis $\displaystyle=(m+2)2^{1}-1$ $\displaystyle=2m+3.$

For odd $m$ we have even $m+1=2j$ for some $j=(m+1)/2\in\mathbb{N}$ yielding

$\displaystyle o_{m+1}$ $\displaystyle=(2p_{m+1}+1)2^{q_{m+1}}-1$
$\displaystyle=(2p_{2j}+1)2^{q_{2j}}-1$ $\displaystyle=(2j+1)2^{1}-1$
$\displaystyle=(m+2)2^{1}-1$ $\displaystyle=2m+3.$

∎

We substitute $e_{n}$ (16) and $o_{n}$ (17) for even and odd $a_{n}$,
respectively, into the ruler function arguments in (13) and obtain the
following identities.

######  Lemma 5.2.

For even and odd integers $e_{n}$ (16) and $o_{n}$ (17), respectively, the
ruler function (10) satisfies the identities

$r(e_{n}(p_{n},q_{n})/2)=q_{n}$ (18)

and

$r((o_{n}(p_{n},q_{n})+1)/2)=q_{n}.$ (19)

###### Proof.

For $n\in\mathbb{N}$ and $q_{n}\geq 1$ we recurse the argument of the ruler
function (10) and have that

$\displaystyle r(e_{n}(p_{n},q_{n})/2)$
$\displaystyle=r((2p_{n}+1)2^{{q_{n}}-1})$
$\displaystyle=r((2p_{n}+1)2^{q_{n}-2})+1$
$\displaystyle=r((2p_{n}+1)2^{q_{n}-3})+2$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
$\displaystyle=r((2p_{n}+1)2^{q_{n}-q_{n}})+q_{n}-1$
$\displaystyle=r(2p_{n}+1)+q_{n}-1$ $\displaystyle=q_{n}$

and, thus, that

$\displaystyle r((o_{n}(p_{n},q_{n})+1)/2)$
$\displaystyle=r((2p_{n}+1)2^{q_{n}-1})$ $\displaystyle=q_{m}.$

∎

Lemmas 5.1 and 5.2 applied to the APT Collatz sequences (13) yield a modified
representation of (13). We denote this modification as the $MAPT$ Collatz
function.

######  Theorem 5.1 (MAPT Collatz Function).

For $i,j,n\in\mathbb{N}$ and $p_{i}$ and $q_{i}$ in Definitions 5.1 and 5.2,
respectively, the APT Collatz function (13) is equivalent to the maps of an
even $a_{n}(i)$ to an odd $a_{n+1}(i)$

$\displaystyle a_{n}(i)$ $\displaystyle=(2p_{i}+1)2^{q_{i}},$
$\displaystyle(a_{n}$ $\displaystyle\equiv 0\ \mbox{\rm(mod}\ 2\mbox{\rm)})$
(20) $\displaystyle a_{n+1}(i)$ $\displaystyle=(2p_{i}+1),$
$\displaystyle(a_{n+1}$ $\displaystyle\equiv 1\ \mbox{\rm(mod}\ 2\mbox{\rm)})$

and an odd $a_{n}(j)$ to an even $a_{n+1}(j)$

$\displaystyle a_{n}(j)$ $\displaystyle=(2p_{j}+1)2^{q_{j}}-1,$
$\displaystyle(a_{n}$ $\displaystyle\equiv 1\ \mbox{\rm(mod}\ 2\mbox{\rm)})$
(21) $\displaystyle a_{n+1}(j)$ $\displaystyle=(2p_{j}+1)3^{q_{j}}-1,$
$\displaystyle(a_{n+1}$ $\displaystyle\equiv 0\ \mbox{\rm(mod}\
2\mbox{\rm)}).$

###### Proof.

For $n,a_{n}\in\mathbb{N}$ and for some $i,j\in\mathbb{N}$ in (13) we let even
$a_{n}(i)=e_{i}(p_{i},q_{i})$ (16) and odd $a_{n}(j)=o_{i}(p_{j},q_{j})$ (17)
from Lemma 5.1, and we use the ruler function identities (18) and (19) in
Lemma 5.2. For even $a_{n}(i)$ we have odd $a_{n+1}(i)$

$\displaystyle a_{n+1}(i)$
$\displaystyle=\bigg{(}\dfrac{1}{2}\bigg{)}^{r(a_{n}(i)/2)}a_{n}(i)$
$\displaystyle=\bigg{(}\dfrac{1}{2}\bigg{)}^{r(e_{i}(p_{i},q_{i})/2)}e_{i}(p_{i},q_{i})$
$\displaystyle=\bigg{(}\dfrac{1}{2}\bigg{)}^{q_{i}}(2p_{i}+1)2^{q_{i}}$
$\displaystyle=2p_{i}+1.$

For odd $a_{n}(j)$ we have even $a_{n+1}(j)$

$\displaystyle a_{n+1}(j)$
$\displaystyle=\bigg{(}\dfrac{3}{2}\bigg{)}^{r((a_{n}(j)+1)/2)}(a_{n}(j)+1)-1$
$\displaystyle=\bigg{(}\dfrac{3}{2}\bigg{)}^{r((o_{j}(p_{j},q_{j})+1)/2)}(o_{j}(p_{j},q_{j})+1)-1$
$\displaystyle=\bigg{(}\dfrac{3}{2}\bigg{)}^{q_{j}}((2p_{j}+1)2^{q_{j}}-1+1)-1$
$\displaystyle=(2p_{j}+1)3^{q_{j}}-1.$

∎

##  6 Even (EMAPT) and Odd (OMAPT) MAPT Collatz sequences

We now generate even-only and odd-only $MAPT$ Collatz subsequences of $A$ (13)
as represented in Equations (20) and (21) in Theorem 5.1, respectively.

######  Theorem 6.1.

The subsequences of even elements of $A$ (13) are

$a_{n+2}=\bigg{(}2p\bigg{(}p\big{(}(a_{n}-2)/2\big{)}\bigg{)}+1\bigg{)}3^{q\big{(}p\big{(}(a_{n}-2)/2\big{)}\big{)}}-1\\\
$ (22)

with $a_{n}=2m$ for $m\in\mathbb{N}$, and the subsequences of odd elements of
$A$ (13) are

$a_{n+3}=2p\bigg{(}\bigg{(}\Big{(}2p\big{(}(a_{n+1}-1)/2\big{)}+1\Big{)}3^{q\big{(}(a_{n+1}-1)/2\big{)}}-3\bigg{)}/2\bigg{)}+1\\\
$ (23)

with $a_{n+1}=2m+1$ for $m\in\mathbb{N}$.

###### Proof.

We prove algebraically. We recurse Equations (20) and (21) in Theorem 5.1 and
use the results of Lemma 5.1. For $i,j,k,\ldots\in\mathbb{N}$ we start with
some $i$ and even $a_{n}(i)$ yielding

$\displaystyle a_{n}(i)$ $\displaystyle=(2p_{i}+1)2^{q_{i}}$
$\displaystyle=2(i+1)$ $\displaystyle a_{n+1}(i)$ $\displaystyle=2p_{i}+1$
$\displaystyle=a_{n+1}(j)$ $\displaystyle a_{n+1}(j)$
$\displaystyle=(2p_{j}+1)2^{q_{j}}-1$ $\displaystyle=2j+1$ $\displaystyle
a_{n+2}(j)$ $\displaystyle=(2p_{j}+1)3^{q_{j}}-1$ $\displaystyle=a_{n+2}(k)$
$\displaystyle a_{n+2}(k)$ $\displaystyle=(2p_{k}+1)2^{q_{k}}$
$\displaystyle=2(k+1)$ $\displaystyle a_{n+3}(k)$ $\displaystyle=2p_{k}+1$
$\displaystyle=\ldots.$

For even $a_{n}(i)$ we have that $i=(a_{n}-2)/2$ and $j=p_{i}$. We substitute
these values for $i$ and $j$ into $a_{n+2}(j)=(2p_{j}+1)3^{q_{j}}-1$ yielding
(22). For odd $a_{n+1}(j)$ we have that $j=(a_{n+1}-1)/2$ and
$k=((2p_{j}+1)3^{q_{j}}-3)/2$. We substitute these values for $j$ and $k$ into
$a_{n+3}(k)=2p_{k}+1$ yielding (23). ∎

For even $n\in\mathbb{N}$ we define the even-only elements of sequences $A(n)$
by sequences $U(n)$.

######  Definition 6.1 (EMAPT Collatz Function).

By Theorem 6.1 we denote the sequences of even positive integers of $A$ (22)
by $U$ (Even MAPT Collatz function) with even $u_{n}=a_{n}$ and even
$u_{n+1}=a_{n+2}$ yielding

$u_{n+1}=\big{(}2p\big{(}p\big{(}(u_{n}-2)/2\big{)}\big{)}+1\big{)}3^{q\big{(}p\big{(}(u_{n}-2)/2\big{)}\big{)}}-1\\\
$ (24)

with $u_{0}=2m$ for $m\in\mathbb{N}$.

For odd $n\in\mathbb{N}$ we define the odd-only elements of $A(n)$ by
sequences $V(n)$.

######  Definition 6.2 (OMAPT Collatz Function).

By Theorem 6.1 we denote the sequences of odd positive integers of $A$ (23) by
$V$ (Odd MAPT Collatz function) with odd $v_{n}=a_{n+1}$ and odd
$v_{n+1}=a_{n+3}$ yielding

$v_{n+1}=2p\big{(}\big{(}\big{(}2p\big{(}(v_{n}-1)/2\big{)}+1\big{)}3^{q\big{(}(v_{n}-1)/2\big{)}}-3\big{)}/2\big{)}+1\\\
$ (25)

with $v_{0}=2m+1$ for $m\in\mathbb{N}$.

For even $u_{n}$ and odd $v_{n}$ elements of sequences $U$ (24) and $V$ (25),
respectively, we have that

$v_{n}=\frac{2u_{n}}{2^{r(u_{n})}}$ (26)

by $A$ (13). The sequences $U$ and $V$ satisfy the covering relations
$U(n)\lessdot A(n)$ and $V(n)\lessdot A(n)$ for even and odd $n\in\mathbb{N}$,
respectively. For even $n\in\mathbb{N}_{>0}$ if sequences $U(n)$ (24) satisfy
$2\in U(n)$, then the CC is true. Equivalently, by (26) and for odd
$n\in\mathbb{N}_{>0}$, if sequences $V(n)$ (25) satisfy $1\in V(n)$, then the
CC is true.

##  7 EMAPT Collatz sequences: $U(6n+2)$

We now constrain the set of even integers $u_{0}$ in sequences $U(u_{0})$ (24)
sufficient to prove the CC.

######  Theorem 7.1.

For $n\in\mathbb{N}$ and even $u_{n}\in\mathbb{N}_{>0}$ Equation (24) has the
form $u_{n+1}=6m+2$ for some $m\in\mathbb{N}$.

###### Proof.

For $n\in\mathbb{N}$ we define the functions $f_{1}(n)\in\mathbb{N}_{>0}$ and
$f_{2}(n)\in\mathbb{N}_{>0}$ as

$f_{1}(n)=p\big{(}p\big{(}(u_{n}-2)/2\big{)}\big{)}$

and

$f_{2}(n)=q\big{(}p\big{(}(u_{n}-2)/2\big{)}\big{)}-1.$

For some $m\in\mathbb{N}$ Equation (24) has the form

$\displaystyle u_{n+1}$ $\displaystyle=(2f_{1}(u_{n})+1)3^{f_{2}(u_{n})+1}-1$
$\displaystyle=6\left(3^{f_{2}(u_{n})}f_{1}(u_{n})+\dfrac{3^{f_{2}(u_{n})}-1}{2}\right)+2$
$\displaystyle=6m+2.$

∎

By Theorem 7.1 we only need to show that $2\in U$ for even numbers of the form
$u_{0}=6m+2$ to satisfy the CC. A similar result appears in the unpublished
work [9]. We state this result.

######  Theorem 7.2.

For $n\in\mathbb{N}$ if the EMAPT Collatz sequences $U(6n+2)$ (24) satisfy
$2\in U$ then the CC is true.

###### Proof.

This theorem follows from Theorem 7.1. ∎

##  8 The EMAPT Collatz sequences beyond $U(6n+2)$

For $n\in\mathbb{N}$, $\alpha\in\mathbb{N}$, and $\beta\in\mathbb{N}$ we now
further constrain the set of the integer arguments of $U(\alpha n+\beta)$ (24)
sufficient to prove the CC. Indeed, Monks (2006) [10] shows that the set of
arguments $\\{\alpha n+\beta\\}$ is sufficient to prove the CC for any
nonnegative integers $\alpha$ and $\beta$ with $\alpha\neq 0$.

For $n\in\mathbb{N}$ we consider the elements of $U$ (24) and reproduce here
as

$u_{n+1}=\big{(}2p\big{(}p\big{(}(u_{n}-2)/2\big{)}\big{)}+1\big{)}3^{q\big{(}p\big{(}(u_{n}-2)/2\big{)}\big{)}}-1.$

For index $x_{n}\in\mathbb{N}$ we let $u_{n}=6x_{n}+2$ and
$u_{n+1}=6x_{n+1}+2$. Theorem 7.1 implies that

$6x_{n+1}+2=\Big{(}2\,p\big{(}p\big{(}3x_{n}\big{)}\big{)}+1\Big{)}\,3^{q\big{(}p\big{(}3x_{n}\big{)}\big{)}}-1,$

and yields the sequence $X$ of indices of $U$

$x_{n+1}=\frac{1}{2}\left(\Big{(}2\,p\big{(}p\big{(}3x_{n}\big{)}\big{)}+1\bigg{)}\,3^{q\big{(}p\big{(}3x_{n}\big{)}\big{)}-1}-1\right).$
(27)

We may thus state the following theorem.

######  Theorem 8.1.

For $n\in\mathbb{N}$ and $x_{n}\in\mathbb{N}$ in (27) if sequences $X$ satisfy
$0\in X$ then the CC is true.

###### Proof.

For $n\in\mathbb{N}$ and $x_{n}\in\mathbb{N}$ if $x_{n}=0$, then $2\in U$. ∎

We can strengthen Theorems 7.2 and 8.1. For $n\in\mathbb{N}$,
$x_{n}\in\mathbb{N}$, and $m\in\mathbb{N}$, we observe computationally that
$x_{n+1}\neq 3m+2$. Before proving that statement, we need the following
lemma.

######  Lemma 8.1.

The function $p:\mathbb{N}\rightarrow\mathbb{N}$ with $p(2n)=n$ and
$p(2n+1)=p(n)$ (14) satisfies $p(3n)\equiv 0$ (mod $3$) or $p(3n)\equiv 2$
(mod $3$) for all $n\in\mathbb{N}$.

###### Proof.

We use strong induction. For the base case we have that $p(0)=0$. For the
inductive step we assume for some arbitrary $n\in\mathbb{N}$ and for
$m\in\mathbb{N}$ that $p(3m)\equiv 0$ (mod $3$) or that $p(3m)\equiv 2$ (mod
$3$) for $m\in\\{0,1,2,\ldots,n\\}$.

An integer $n\in\mathbb{N}$ is even or odd. Thus, an integer $a\in\mathbb{N}$
exists such that $n=2a$ or $n=2a+1$. For even $n=2a$ we have that

$p(3(n+1))=p(3(2a+1))=p(6a+3)=p(2(3a+1)+1)=p(3a+1).$

Similarly, an integer $b\in\mathbb{N}$ exists such that $a=2b$ or $a=2b+1$.
For even $a=2b$ we have that

$p(3(n+1))=p(3(2b)+1)=p(3b)\equiv 0\,\,\text{or }2\pmod{3}$

since $n=2a=2(2b)$ and $b=n/4\in\\{0,1,\ldots,n\\}$. For odd $a=2b+1$ we have
that

$p(3(n+1))=p(3(2b+1)+1)=p(2(3b+2))=3b+2\equiv 2\pmod{3}.$

For odd $n=2a+1$ we have that

$p(3(n+1))=p(3((2a+1)+1))=p(6a+6)=p(2(3a+3))=3a+3\equiv 0\pmod{3}.$

∎

We now proceed constraining indices $x_{n}$ in sequence $X$ (27).

######  Theorem 8.2.

For $x_{n}\in\mathbb{N}$ elements $x_{n+1}$ in (27) satisfy $x_{n+1}\neq 3m+2$
for $m\in\mathbb{N}$; equivalently, $x_{n+1}\equiv 0$ (mod  $3$) or
$x_{n+1}\equiv 1$ (mod $3$).

###### Proof.

We prove by contradiction. For $x_{n}\in\mathbb{N}$ we assume that
$x_{n+1}=3m+2$ for some $m\in\mathbb{N}$ in (27) yielding

$3m+2=\frac{1}{2}\left(\Big{(}2\,p\big{(}p\big{(}3x_{n}\big{)}\big{)}+1\Big{)}\,3^{q\big{(}p\big{(}3x_{n}\big{)}\big{)}-1}-1\right)$

and, thus,

$6m+5=\Big{(}2\,p\big{(}p\big{(}3x_{n}\big{)}\big{)}+1\Big{)}\,3^{q\big{(}p\big{(}3x_{n}\big{)}\big{)}-1}.$

By Equations (14) and (15) we have that $q(p(3x_{n}))\geq 1$. If
$q(p(3x_{n}))>1$ then we have a contradiction because $3\nmid(6m+5)$. If
$q(p(3x_{n}))=1$ then we have even $p(3x_{n})$ and $p(p(3x_{n}))=p(3x_{n})/2$.
We thus have that

$6m+4=p(3x_{n}).$

By Lemma 8.1 for some $i\in\mathbb{N}$ we have that $p(3x_{n})=3i$ or
$p(3x_{n})=3i+2$, implying that $6m+4=3i$ or $6m+4=3i+2$, respectively. We
have contradictions in both cases because $3\nmid(6m+4)$ and $3\nmid(6m+2)$,
respectively. ∎

We can thus state the one of the main results of this work.

######  Theorem 8.3.

For $u_{0}=18n+2$ or $u_{0}=18n+8$ in $U$ (24) if $2\in U$, then the CC is
true.

###### Proof.

The proof follows directly from Theorem 8.2. ∎

##  9 The EMAPT Collatz sequence $U(n)$ in terms of $r(n)$

For $m\in\mathbb{N}$ we now express the recursive terms $p(p(m))$ and
$q(p(m))$ in (24) as a function of the ruler function $r(m)$ (10). For
$i\in\mathbb{N}$ and $j\in\mathbb{N}$ we uniquely represent the integers $m$
in terms of the two-tuples $(i,j)$ as

$m=(2i+1)2^{j}-1.$

For $k\in\mathbb{N}$ and $l\in\mathbb{N}$ we represent the integers $i$ in
terms of the two-tuples $(k,l)$ as

$i=(2k+1)2^{l}-1.$

Thus, we have the three-tuple $(j,k,l)$ representation of $m$

$m=(2((2k+1)2^{l}-1)+1)2^{j}-1.$

We now recurse $p(p(m))$ and $q(p(m))$ using the definitions for $p(m)$ (14)
and $q(m)$ (15). We obtain that

$\displaystyle p(p(m))$ $\displaystyle=k$ $\displaystyle q(p(m))$
$\displaystyle=l+1.$

Thus, we have for any integer $(u_{n}-2)/2$ that

$(u_{n}(j,k,l)-2)/2=\left(2((2k+1)2^{l}-1)+1\right)2^{j}-1$

and, thus, that

$u_{n}(j,k,l)=\left((2k+1)2^{l+1}-1\right)2^{j+1}.$

By (24) we have that

$u_{n+1}(k,l)=(2k+1)\,3^{l+1}-1.$

We eliminate the common factor $2k+1$ and have that

$u_{n+1}(j,l)=\left(\dfrac{u_{n}}{2^{j+1}}+1\right)\left(\dfrac{3}{2}\right)^{l+1}-1.$
(28)

The maximum allowed tuples $(j,l)$ are the $2$-adic valuations of factors in
(28). We have that

$\displaystyle j$ $\displaystyle=r(u_{n})-2$ $\displaystyle l$
$\displaystyle=r\left(\dfrac{u_{n}}{2^{r(u_{n})-1}}+1\right)-2.$

Thus, a primary result of this work is that

$u_{n+1}=\left(\dfrac{2u_{n}}{2^{r(u_{n})}}+1\right)\left(\dfrac{3}{2}\right)^{r\left(\dfrac{2u_{n}}{2^{r(u_{n})}}+1\right)-1}-1$
(29)

with (26)

$v_{n}=\frac{2u_{n}}{2^{r(u_{n})}}.$

For even $u_{0}\in\mathbb{N}_{>0}$ in $U(u_{0})$ (29) all subsequent elements
$u_{n}$ in sequence $U(u_{0})$ are of the form $18m+2$ or $18m+8$. (We note
that for odd $u_{0}\in\mathbb{N}_{>0}$ in $U(u_{0})$ (29) all subsequent
elements $u_{n}$ beyond $u_{1}$ in sequence $U(u_{0})$ appear to be of the
form $18m+2$ or $18m+8$.) We conjecture that $2\in U(u_{0})$, which would
imply that the $CC$ is true.

If we substitute $v_{n}$ (26) into (29), then we more clearly have a fixed
rational root for the relation between adjacent elements in the sequences $U$
and $V$

$\left(\dfrac{u_{n+1}+1}{v_{n}+1}\right)^{1/r((v_{n}+1)/2)}=\dfrac{3}{2}.$
(30)

##  10 $Z$: Reverse Map of $U$

For $n\in\mathbb{N}$ and $\alpha_{n},\beta_{n}\in\mathbb{N}_{>0}$ we let
$\alpha_{n}=r(v_{n}+1)-1$ and $\beta_{n}=r(u_{n})-1$. The infinite linear
system of equations (29) for adjacent elements of $U$ satisfies

$2^{\alpha_{n}+\beta_{n}}u_{n+1}=3^{\alpha_{n}}u_{n}+2^{\beta_{n}}(3^{\alpha_{n}}-2^{\alpha_{n}}).$
(31)

We now consider the linear system of equations for the reverse map
$u_{n+1}\mapsto u_{n}$ in (31)

$u_{n}=2^{\alpha_{n}+\beta_{n}}3^{-\alpha_{n}}u_{n+1}-2^{\beta_{n}}3^{-\alpha_{n}}(3^{\alpha_{n}}-2^{\alpha_{n}}).$
(32)

In (32) we map $u_{n}\mapsto z_{n+1}$ and $u_{n+1}\mapsto z_{n}$ yielding

$z_{n+1}=m_{n+1}z_{n}+b_{n+1}$ (33)

with rational slopes

$m_{n}=2^{\alpha_{n}+\beta_{n}}3^{-\alpha_{n}}$

and rational intercepts

$b_{n}=-2^{\beta_{n}}3^{-\alpha_{n}}(3^{\alpha_{n}}-2^{\alpha_{n}}).$

Analogous to the $U$ and $V$ sequences, we define the odd only $W$ sequences
$w_{n}\neq 0\pmod{3}$ by

$w_{n}=\frac{2z_{n}}{2^{r(z_{n})}}$ (34)

and

$\dfrac{z_{n}+1}{w_{n+1}+1}=\left(\dfrac{3}{2}\right)^{r((w_{n+1}+1)/2)}$ (35)

with $\alpha_{n}=r(w_{n+1}+1)-1=r((w_{n+1}+1)/2)$ and $\beta_{n}=r(z_{n})-1$.

######  Theorem 10.1.

($z_{0}$-cycle; Steiner [11]) If $z_{1}=z_{0}$ in (37) then $z_{0}=2$.

###### Proof.

In the original $CC$ sequence the only 1-cycle is the trivial sequence
$(2,1,2,1,\ldots)$ [11]. We assume that $z_{1}=z_{0}$. We have by (33) that

$\displaystyle z_{0}$ $\displaystyle=\dfrac{b_{1}}{1-m_{1}}$ (36)
$\displaystyle=\dfrac{-2^{\beta_{1}}3^{-\alpha_{1}}(3^{\alpha_{1}}-2^{\alpha_{1}})}{1-2^{\alpha_{1}+\beta_{1}}3^{-\alpha_{1}}}$
$\displaystyle=\dfrac{-2^{\beta_{1}}(3^{\alpha_{1}}-2^{\alpha_{1}})}{3^{\alpha_{1}}-2^{\alpha_{1}+\beta_{1}}}.$

The only 2-tuples that satisfy (36) with $z_{0}\in\mathbb{N}$ are
$(\alpha_{1},\beta_{1})=(1,1)$, and we have that $z_{0}=2$. ∎

We can select an initial $z_{0}$ and recurse (33) yielding

$z_{n}=\left(\displaystyle\prod_{i=1}^{n}m_{i}\right)z_{0}+b_{n}+\displaystyle\sum_{i=1}^{n-1}\left(b_{i}\displaystyle\prod_{j=i+1}^{n}m_{j}\right).$
(37)

We can let

$M_{n}(z_{n})=\left(\displaystyle\prod_{i=1}^{n}m_{i}\right)$

and

$B_{n}(z_{n})=b_{n}+\displaystyle\sum_{i=1}^{n-1}\left(b_{i}\displaystyle\prod_{j=i+1}^{n}m_{j}\right).$

We thus have that

$z_{n}=M_{n}(z_{n})z_{0}+B_{n}(z_{n})$ (38)

with real 2-tuples $(M_{n}(z_{n}),B_{n}(z_{n}))$ for an infinite number of
$z_{n}$ and possibly unique $w_{n}$ with $z_{0}=2$. Rather than attempting to
show that (38) has $complete$ solutions in the evens with $z_{0}=2$, we now
focus on (35) to prove the Collatz conjecture.

##  11 The $WZ$ Tree

Herein we assume a fundamental knowledge of graphs and trees. We first
illustrate some properties of the $WZ$ sequence tree, partially shown in Fig.
1 with root vertex $(w_{1},z_{0})=(1,2)$. Each 2-tuple $(w_{n},z_{n})$
connected by a directed edge satisfies (34). Each 2-tuple vertex
$(w_{n+1},z_{n})$ satisfies (35). Each vertex has an infinite number of
children; clearly, but see (11.2) below. Each level $n$ is ordered from left
to right in terms of increasing unique $w_{n}$ terms and extends infinitely to
the right; clearly again, but see (11.2) below.

Figure 1: Partial (upper left) WZ Tree

For $m\in\mathbb{N}_{>0}$ the ordered sequence of all possible values for
$w_{n+1}$ in (35) is $(w_{n+1}(m))=((6m+(-1)^{m}-3)/2)$ and is complete in the
odds not divisible by $3$ (A007310 [3]). For each element in $(w_{n+1})$ we
generate a $z_{n}$ (35), a $w_{n}$ (34), and the map $w_{n}\mapsto w_{n+1}$

$\displaystyle(w_{n+1})=($ $\displaystyle
1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59,\ldots)$ (39)
$\displaystyle(w_{n})=($ $\displaystyle
1,1,13,13,5,13,11,5,19,11,121,5,7,31,49,121,37,5,47,67,\ldots).$

We now state two Lemmas.

######  Lemma 11.1.

The only cycle in the Collatz tree is the trivial $1\mapsto 1$.

###### Proof.

The $w_{n+1}$ values are unique in $(w_{n+1})$ (39), indicating that each
vertex generated in the forward (down) direction in the Fig. 1 is unique
except for the trivial cycle $1\mapsto 1$ (See Theorem 10.1). ∎

Given a $w_{n+1}$ the associated $w_{n}$ values are not uniquely determined,
cover the set of integers $\\{w_{n+1}\\}$, and occur an infinite number of
times.

######  Lemma 11.2.

Each integer in the sequence of unique odds $(w_{n+1})$ occurs an infinite
number of times in the sequence $(w_{n})$.

###### Proof.

For $m,n\in\mathbb{N}$ and a chosen $(z_{n},w_{n})$ 2-tuple, if $z_{m}\neq
z_{n}$ and $w_{m}=w_{n}$ then, by (34), we have that

$\frac{2z_{m}}{2^{r(z_{m})}}=\frac{2z_{n}}{2^{r(z_{n})}}$

and thus that

$z_{m}=2^{k}z_{n}$

for $k\in\mathbb{N}$ satisfying

$k=2^{r(z_{m})-r(z_{n})}.$

The sequence $(r(z_{m}))$ for all $z_{m}$ contains all positive integers an
infinite number of times by Definition 3.1. Thus, we have an infinite number
of $w_{m}=w_{n}$. ∎

##  12 A Collatz Conjecture Proof

We now present our last result.

######  Theorem 12.1.

The Collatz Conjecture is true.

###### Proof.

We have a well–ordered, countable infinite set of unique $w_{n+1}$ vertices
(39) that is complete in the odds not divisible by $3$. Except for the trivial
cycle we also have a countable infinite set of unique $(w_{n+1},w_{n})$
acyclic edges ((11.1) and (11.2)). By Zorn’s lemma, the well–ordering theorem
or the axiom of choice (which are equivalent [12]) we can construct an acyclic
tree that spans [13] our desired vertices, i.e., all odd numbers not divisible
by $3$ with root vertex $1$. The existence of this tree with the covering
relation $W(n)\lessdot C(n)$ is sufficient to complete the proof. ∎

##  13 Acknowledgements

Thank you.

## References

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# Effects of incompressibility $K_{0}$ on the symmetry energy observables in
heavy-ion collisions at intermediate energies

Xiao-Xiao Long  School of Physics and Electronic Science, Guizhou Normal
University, Guiyang 550025, China  Gao-Feng Wei<EMAIL_ADDRESS>School of Physics and Electronic Science, Guizhou Normal University, Guiyang
550025, China  Guizhou Provincial Key Laboratory of Radio Astronomy and Data
Processing, Guizhou Normal University, Guiyang 550025, China

###### Abstract

Within the possible least uncertainty on the nuclear incompressibility
$K_{0}$, we examine effects of $K_{0}$ on the symmetry energy observables in
heavy-ion collisions at intermediate energies. Based on simulations of Au + Au
collision at 400 MeV/nucleon using an isospin- and momentum-dependent
transport model, we find that the incompressibility $K_{0}$ indeed affects
significantly the attainable density in central regions, and thus the possible
symmetry energy observables, e.g., nucleon rapidity distributions and yields
of charged pions. Nevertheless, through examining the combined observables,
e.g., the free neutron over proton ratios $n/p$, the neutron-proton
differential transverse and directed flows as well as the charged pion ratio
$\pi^{-}/\pi^{+}$ and its kinetic energy distribution, we find that the
combined observables are less affected by the uncertainty of $K_{0}$. We also
compare and discuss our results with the corresponding data.

##  I introduction

As one of central issues in isospin nuclear physics, the symmetry energy
$E_{sym}(\rho)$ at suprasaturation densities has been a long-standing interest
due to its importance in understanding the properties of radioactive nuclei
and evolution of supernova and neutron stars Ste05 ; Lat12 ; Hor14 ; Heb15 ;
Bal16 ; Oer17 ; Tew18 . In terrestrial laboratories, heavy-ion collisions
(HICs) with rare isotopes provide a unique opportunity to generate directly
the isospin asymmetric nuclear matter at high densities, and thus enable one
to extract the information about the $E_{sym}(\rho)$ at high densities through
comparing the theoretical simulations of isospin observables with the
corresponding data Bar05 ; Kol05 ; li08 ; Lyn09 .

The incompressibility $K_{0}$ of nuclear matter at the saturation density
$\rho_{0}$, as an important input of most microscopic and/or phenomenological
heavy-ion transport models, affects the attainable density in collision
regions and thus the particle productions and/or distributions at final
states. Naturally, the accuracy of $K_{0}$ affects the quantitative extraction
of the symmetry energy using HIC models. However, the possible tightest
constraint or least uncertainty one currently obtains on the $K_{0}$ is
$230\pm 30$ MeV Sturm01 ; Fuch01 ; Dan02 ; Hart06 ; Pan93 ; Zhang94 . This
uncertainty on the value of $K_{0}$ naturally prevents one from quantitatively
extracting the high density symmetry energy information using isospin
observables. Actually, some literatures have already involved in
investigations of the uncertainty of $K_{0}$ in HICs and other aspects. For
example, using a Tübingen quantum molecular dynamics model, Ref. Cozma18
discussed effects of $K_{0}$ within a range from 210 to 280 MeV on the nucleon
elliptic flows, and Ref. Zan12  examined effects of $K_{0}$ within a range
from 195 to 225 MeV on the nucleus-nucleus dynamic potential in fusion
reactions, and Ref. Bon18  studied effects of $K_{0}$ within a range from 210
to 240 MeV on the giant resonances in 40,48Ca, 68Ni, 90Zr, 116Sn, 144Sm and
208Pb. Nevertheless, a systematical study related to the effects of $K_{0}$ on
the symmetry energy observables is rarely reported. On the other hand, it is
well known that central heavy-ion reactions at intermediate energies play a
special role in determination of the symmetry energy especially above twice
times saturation density. Therefore, it is naturally necessary to study
effects of $K_{0}$ on the symmetry energy observables in central heavy-ion
reactions. To this end, we perform a central Au + Au collision at 400 MeV to
study effects of $K_{0}$ within the possible tightest constraint, i.e.,
$K_{0}=230\pm 30$ MeV, on the pion and nucleon observables. It is shown that
the $K_{0}$ indeed affects significantly the attainable density in collision
regions, and thus the possible symmetry energy observables, e.g., charged pion
multiplicities etc. However, we find that combining individual observables
could reduce significantly the effects of uncertainties of $K_{0}$ and thus
show more sensitivities to the high density behavior of symmetry energy.

In the following, we first describe briefly the used isospin- and momentum-
dependent Boltzmann-Uehling-Uhlenbeck transport model (IBUU) IBUU1 ; IBUU2  in
Sec. II. We then discuss our results in Sec. III. A summary will be given in
Sec. IV.

##  II The Model

Figure 1: (Color online)Density dependence of the $E_{sym}(\rho)$ with
different $K_{0}$ and $L$.

The present study is carried out within an updated version of IBUU transport
model Wei23 . In this version, we adopt a separate density-dependent scenario
Wei20  for a more delicate treatment of the in-medium many-body force effects
as in Refs. Xu10 ; Chen14 . Also, we introduce a parameter $z$ Wei22  as in
Ref. Xu15  to mimic the value of $E_{sym}(\rho)$ at $\rho_{0}$ and
$2\rho_{0}/3$ to meet the best knowledge of $E_{sym}(\rho)$ at the two
densities one has obtained so far, e.g., $E_{sym}(\rho_{0})=32.5\pm 2$ MeV
Cozma18 , $32.5\pm 3.2$ MeV Wang18 , $33.0^{+2}_{-1.8}$ MeV Ess21 , $38.3\pm
4.7$ MeV Ree21  and $E_{sym}(2\rho_{0}/3)=25.5\pm 1$ MeV Ree21 ; Est21 ,
$26.2\pm 1$ MeV Wang13 . As to the high density behavior of $E_{sym}(\rho)$ ,
we use the parameter $x$ to control the slope value
$L\equiv{3\rho({dE_{sym}}/d\rho})$ of $E_{sym}(\rho)$ at $\rho_{0}$ as in the
original IBUU model IBUU1 ; IBUU2 . All of these features have been
incorporated into the present used model, see Ref. Wei23  for details.
Specifically, the isospin and momentum dependent nuclear interaction (MDI)
used is expressed as:

$\displaystyle U(\rho,\delta,\vec{p},\tau)$ $\displaystyle=$ $\displaystyle
A_{u}\frac{\rho_{-\tau}}{\rho_{0}}+A_{l}\frac{\rho_{\tau}}{\rho_{0}}+\frac{B}{2}{\Big{(}}\frac{2\rho_{\tau}}{\rho_{0}}{\Big{)}}^{\sigma}(1-x)$
(1) $\displaystyle+$
$\displaystyle\frac{2B}{\sigma+1}{\Big{(}}\frac{\rho}{\rho_{0}}{\Big{)}}^{\sigma}(1+x)\frac{\rho_{-\tau}}{\rho}{\big{[}}1+(\sigma-1)\frac{\rho_{\tau}}{\rho}{\big{]}}$
$\displaystyle+$ $\displaystyle\frac{2C_{l}}{\rho_{0}}\int
d^{3}p^{\prime}\frac{f_{\tau}(\vec{p}^{\prime})}{1+(\vec{p}-\vec{p}^{\prime})^{2}/\Lambda^{2}}$
$\displaystyle+$ $\displaystyle\frac{2C_{u}}{\rho_{0}}\int
d^{3}p^{\prime}\frac{f_{-\tau}(\vec{p}^{\prime})}{1+(\vec{p}-\vec{p}^{\prime})^{2}/\Lambda^{2}},$

where $\tau=1$ for neutrons and $-1$ for protons, and $A_{u}$, $A_{l}$,
$C_{u}(\equiv C_{\tau,-\tau})$ and $C_{l}(\equiv C_{\tau,\tau})$ are expressed
as

$\displaystyle A_{l}$ $\displaystyle=$ $\displaystyle
A_{l0}+U_{sym}^{\infty}(\rho_{0})-\frac{2B}{\sigma+1}$ $\displaystyle\times$
$\displaystyle\Big{[}\frac{(1-x)}{4}\sigma(\sigma+1)-\frac{1+x}{2}\Big{]},$
$\displaystyle A_{u}$ $\displaystyle=$ $\displaystyle
A_{u0}-U_{sym}^{\infty}(\rho_{0})+\frac{2B}{\sigma+1}$ $\displaystyle\times$
$\displaystyle\Big{[}\frac{(1-x)}{4}\sigma(\sigma+1)-\frac{1+x}{2}\Big{]},$
$\displaystyle C_{l}$ $\displaystyle=$ $\displaystyle
C_{l0}-2\big{(}U_{sym}^{\infty}(\rho_{0})-2z\big{)}\frac{p_{f0}^{2}}{\Lambda^{2}\ln\big{[}(4p_{f0}^{2}+\Lambda^{2})/\Lambda^{2}\big{]}},$
$\displaystyle C_{u}$ $\displaystyle=$ $\displaystyle
C_{u0}+2\big{(}U_{sym}^{\infty}(\rho_{0})-2z\big{)}\frac{p_{f0}^{2}}{\Lambda^{2}\ln\big{[}(4p_{f0}^{2}+\Lambda^{2})/\Lambda^{2}\big{]}}.$

The eight parameters embedded in above expressions, i.e., $A_{l0}$, $A_{u0}$,
$B$, $\sigma$, $C_{l0}$, $C_{u0}$, $\Lambda$ and $z$, are determined by
fitting eight experimental and/or empirical constraints on properties of
nuclear matter at $\rho_{0}=0.16$ fm-3. Among them, the first seven are the
binding energy $-16$ MeV, the pressure $P_{0}=0$ MeV/fm3 , the isoscalar
effective mass $m^{*}_{s}=0.7m$, the isoscalar potential at infinitely large
nucleon momentum $U^{\infty}_{0}(\rho_{0})=75$ MeV, the isovector potential at
infinitely large nucleon momentum $U^{\infty}_{sym}(\rho_{0})=-100$ MeV, and
the $E_{sym}(\rho)$ at $\rho_{0}$ and $2\rho_{0}/3$. The eighth is $K_{0}$
that we are going to examine. To this end, we take three values for $K_{0}$
within the possible least uncertainty as aforementioned in this study, i.e.,
200, 230 and 260 MeV. For these different settings of $K_{0}$, the
corresponding values of $A_{l0}$, $A_{u0}$, $B$ and $\sigma$ are shown in
Table 1, and the values of $C_{l0}$, $C_{u0}$ and $\Lambda$ are
$C_{l0}=-60.486$ MeV, $C_{u0}=-99.702$ MeV and $\Lambda=2.424p_{f0}$ , where
$p_{f0}$ refers to the nucleon Fermi momentum in symmetric nuclear matter
(SNM) at $\rho_{0}$. Also, to make the symmetry energy observable more clearly
reflecting effects of $K_{0}$, we adjust the values of $x$ and $z$ for
different $K_{0}$ settings to ensure the identical slope $L$ of
$E_{sym}(\rho)$ at $\rho_{0}$ as shown in Fig. 1. In addition, for a certain
$K_{0}$, we also take four different settings for $L$, and to enable us to
compare effects on the symmetry energy observables from $K_{0}$ and $L$. The
specific values of $x$ and $z$ and the corresponding $L$ are shown in Table 2.

Table 1:  The values of $A_{l0}$, $A_{u0}$, $B$, $\sigma$ and the resulting $K_{0}$. $A_{l0}$=$A_{u0}$ (MeV) |  $B$ (MeV) | $\sigma$ |  ${K_{0}}$ (MeV)
---|---|---|---
$-455.726$ | $530.726$ | $1.0646$ | $200$
$-66.963$ | $141.963$ | $1.2652$ | $230$
$-12.992$ | $87.992$ | $1.4657$ | $260$
Table 2:  The values of $x$, $z$ and the resulting $K_{0}$ and $L$. ${K_{0}}$ (MeV) | $x$ |  $z$ (MeV) |  $L$ (MeV)
---|---|---|---
$200$ | $0.54$ | $-1.408$ | $33.06$
$200$ | $0.08$ | $0.474$ | $61.95$
$200$ | $-0.38$ | $3.066$ | $92.66$
$200$ | $-0.84$ | $5.8$ | $123.73$
$230$ | $0.6$ | $-1.482$ | $33.06$
$230$ | $0.2$ | $0.326$ | $61.95$
$230$ | $-0.2$ | $2.844$ | $92.66$
$230$ | $-0.6$ | $5.505$ | $123.73$
$260$ | $0.66$ | $-1.3$ | $33.06$
$260$ | $0.31$ | $0.418$ | $61.95$
$260$ | $-0.04$ | $2.845$ | $92.66$
$260$ | $-0.39$ | $5.415$ | $123.73$

##  III Results and Discussions

Figure 2: (Color online)Evolution of reduced average densities $\rho/\rho_{0}$
in central spherical regions with a radius of 2 fm in Au+Au collisions at 400
AMeV with different $K_{0}$ and $L$.

Now, we present the results of Au+Au collisions at 400MeV/nucleon with an
impact parameter of $b=0-2$ fm, corresponding to a typical reaction with a
reduced impact parameter of $b_{0}\leq 0.15$ at 400MeV/nucleon carried out at
the FOPI detector FOPI07 ; FOPI10 ; FOPI12 .

Figure 3: (Color online)Rapidity distributions of neutrons (a) and protons (b)
in Au+Au collisions at 400 AMeV with different combinations of $K_{0}$ and
$L$. Figure 4: (Color online) Rapidity (a) and kinetic energy (b)
distributions of $n/p$ ratios in Au+Au collisions at 400 AMeV with different
combinations of $K_{0}$ and $L$.

###  III.1 Nucleon observables

Shown in Fig. 2 are the evolutions of reduced average densities in central
spherical regions with a radius of 2 fm. To study how the $K_{0}$ and $L$
affect the attainable compression densities in the reactions, we take 6
different combinations of $K_{0}$ and $L$ in simulations of Au + Au
collisions. It is seen that, for a certain $L$, the compression density
reached in central regions is significantly larger with a smaller value of
$K_{0}$ than that with a larger one; while for a certain $K_{0}$, the
compression density reached is significantly larger with a soft symmetry
energy (i.e., a smaller slope value $L$) than that with a stiff one (i.e., a
larger slope value $L$). These observations are exactly the features belonging
to HICs at intermediate energies Bar05 ; li08  since the effect of symmetry
energy on the compression density is decreasing even to a negligible degree as
the beam energy increases up to approximate 1 GeV and above Wei23 . However,
if one compares semiquantitatively effects between $K_{0}$ and $L$ on the
compression density, e.g., varying $K_{0}$ from 260 to 200 MeV and $L$ from
92.66 to 33.06 MeV, their relative changes are
$(260-200)/[(260+200)/2]\times\%\approx 26\%$ and
$(92.66-33.06)/[(92.66+33.06)/2]\times\%\approx 94.8\%$, one can explicitly
find from Fig. 2 that the effects of $K_{0}$ on the compression density is
obviously dominant than that of $L$, reflecting the fact that the nuclear
compression is overall dominated by the bulk EoS of nuclear matter. Naturally,
one expects that these features could be reflected by the nucleon observables
at final states. To this end, we show in Fig. 3 the rapidity distributions of
free neutrons and protons at final states, where the criterion of free
nucleons is defined as the relative distances $\Delta_{R}>3.575$ fm or momenta
$\Delta_{p}>0.3$ GeV/c as in coalescence models Gyul83 ; Aich87 ; Koch90 ;
Nagl96 ; Sche99 . It is seen that, for a certain $L$, both the free neutrons
and protons are more with a larger $K_{0}$ than those with a smaller $K_{0}$
as shown in insets, since the isoscalar potentials have approximately
identical effects on neutrons and protons. While for a certain $K_{0}$, it is
obvious to see that the variation tendency of neutrons is completely opposite
to that of protons when varying the $L$, reflecting the fact that the symmetry
potential/energy has opposite effects on neutrons and protons at high
densities, i.e., repulsion on neutrons but attraction on protons. To these
observations, one naturally expects the ratios of free neutrons over protons
could reduce the isoscalar potential effects and enlarge the symmetry
potential/energy effects.

Figure 5: (Color online)Directed flows of protons (a) and neutrons (b) in
Au+Au collisions at 400MeV/nucleon with different combinations of $K_{0}$ and
$L$. Figure 6: (Color online)Free neutron-proton differential directed (a) and
transverse (b) flows in Au+Au collisions at 400MeV/nucleon with different
combinations of $K_{0}$ and $L$.

Shown in Fig. 4(a) are the rapidity distributions of neutrons over protons
$n/p$ with different combinations of $K_{0}$ and $L$. As one expected, the
$n/p$ ratios indeed reduce significantly effects of $K_{0}$ and enlarge those
of $L$. Moreover, because nucleons at midrapidities are mainly from early
emissions during the compression stage, and thus carry the information of
symmetry energy at high densities. Therefore, we can observe a larger $n/p$
ratio with a stiff symmetry energy than that with a soft one. Similarly, it is
seen that the kinetic energy distributions of $n/p$ ratios at midrapidities
are mainly sensitive to the high density behavior of symmetry energy as shown
in Fig. 4(b).

###  III.2 Flow observables

Collective motions of final state nucleons are the direct reflections of the
pressure created in HICs and thus are closely related to the equation of state
of dense nuclear matter. Therefore, in this subsection, we examine how the
$K_{0}$ and $L$ affect the collective motions of final state nucleons. In our
studied reactions, the main collective motions could be reflected by the
directed flows $v_{1}$ and/or transverse flows $p_{x}$. Shown in Fig. 5 are
the rapidity dependent directed flows of free neutrons and protons at final
states. The insets are local amplification to explicitly show effects of
$K_{0}$ and $L$. To compare with the corresponding FOPI data FOPI07 ; FOPI10 ;
FOPI12 , we use the same reduced rapidity as in Ref. FOPI07 ; FOPI10 ; FOPI12
, i.e., $y/y_{proj.}$. First, it is seen that our results are consistent with
the data. Second, similar to the observations in rapidity distributions of
neutrons and protons shown in Fig. 3, the effects of symmetry energy/potential
are completely opposite for neutrons and protons, while the effects of $K_{0}$
on neutrons and protons are approximately identical. Therefore, we turn to the
free neutron-proton differential directed and transverse flows defined as
Bar05 ; Wei22 ; li00 ; li02 ,

$\displaystyle v_{1}^{np}$ $\displaystyle=$
$\displaystyle\frac{N_{n}(y)}{N(y)}{\left\langle
v_{1}^{n}(y)\right\rangle}-\frac{N_{p}(y)}{N(y)}{\left\langle
v_{1}^{p}(y)\right\rangle},$ (2) $\displaystyle p_{x}^{np}$ $\displaystyle=$
$\displaystyle\frac{N_{n}(y)}{N(y)}{\left\langle
p_{x}^{n}(y)\right\rangle}-\frac{N_{p}(y)}{N(y)}{\left\langle
p_{x}^{p}(y)\right\rangle},$ (3)

where $N_{n}(y)$, $N_{p}(y)$, and $N(y)$ respectively represents the total
number of free neutrons, protons, and nucleons at rapidities $y$. Shown in
Fig. 6 are the corresponding simulations of free neutron-proton differential
directed $v_{1}^{np}$ and transverse $p_{x}^{np}$ flows. One can see that both
the $v_{1}^{np}$ and $p_{x}^{np}$ indeed reduce significantly effects of
$K_{0}$ and show more sensitivities to the symmetry energy/potential.

###  III.3 Pion observables

Figure 7: (Color online)Multiplicities of $\pi^{-}$ (a) and $\pi^{+}$ (b) in
Au+Au collisions at 400MeV/nucleon with different combinations of $K_{0}$ and
$L$. Figure 8: (Color online)Ratios $\pi^{-}/\pi^{+}$ as a function of $L$ (a)
and pion kinetic energy (b) in Au+Au collisions at 400MeV/nucleon with
different combinations of $K_{0}$ and $L$.

In HICs at intermediate energies, pions are produced mostly from the decay of
$􏰍\Delta(1232)$ resonances. Specifically, for the charged pions, the main
channels are $\Delta^{-}\leftrightarrow n+\pi^{-}$ and
$\Delta^{++}\leftrightarrow p+\pi^{+}$. On the other hand, the main channels
of producing $\Delta^{-}$ and $\Delta^{++}$ from nucleon-nucleon collisions at
high densities are $nn\rightarrow p+\Delta^{-}$ and $pp\rightarrow
n+\Delta^{++}$. Equivalently, one can view the production of $\pi^{-}$ as
mainly from inelastic $nn\rightarrow pn\pi^{-}$ channels while that of
$\pi^{+}$ mainly from $pp\rightarrow pn\pi^{+}$ channels. This is why the
ratio $\pi^{-}/\pi^{+}$ is sensitive to the high density behavior of nuclear
symmetry energy in HICs.

Shown in Fig. 7 are the multiplicities of $\pi^{-}$ and $\pi^{+}$ in Au+Au
collisions at 400MeV/nucleon with different combinations of $K_{0}$ and $L$.
First, it can be observed that, consistent with the previous observations
using most transport models, the yields of $\pi^{-}$ and $\pi^{+}$ are
sensitive to $L$, and the sensitivity of $\pi^{-}$ is greater than that of
$\pi^{+}$. Moreover, the variation tendency of $\pi^{-}$ with $L$ is opposite
to that of $\pi^{+}$ similar to that of nucleons. At the same time, we can
find that yields of both $\pi^{-}$ and $\pi^{+}$ are also sensitive to the
$K_{0}$. More specifically, both $\pi^{-}$ and $\pi^{+}$ are more produced in
collisions with a smaller $K_{0}$ due to a larger compression formed in
collisions as shown in Fig. 2. To this observation, one naturally expects that
the ratio $\pi^{-}/\pi^{+}$ could reduce significantly effects of $K_{0}$ and
thus show more sensitivities to the high density behavior of symmetry energy.
Indeed, this can be demonstrated by the total $\pi^{-}/\pi^{+}$ ratios and the
kinetic energy distributions as shown in Fig. 8. In addition, we also show the
corresponding data in Figs. 7 and 8. It can be found that extracting the
information of high density symmetry energy from multiplicities of both
$\pi^{-}$ and $\pi^{+}$ depends seriously on the used $K_{0}$, while that from
both total $\pi^{-}/\pi^{+}$ ratio and its kinetic energy distributions is
free of uncertainties of $K_{0}$.

##  IV Summary

In summary, we have studied the effects of incompressibility $K_{0}$ within
its possible least uncertainty on the symmetry energy observables, including
nucleon observables and their flows as well as pion observables. It is shown
that the $K_{0}$ indeed affects significantly the attainable density in
central collision regions, and thus the possible individual observables, e.g.,
nucleon rapidity distributions and yields of charged pions. However,
considering that the effects of $K_{0}$ on neutrons and protons are
approximately identical, we have examined and found that the combined
observables could reduce significantly effects of $K_{0}$ and thus show more
sensitivities to the symmetry energy. These findings indicate that extracting
the information of high density symmetry energy from individual observables
depends seriously on the uncertainty of $K_{0}$, while that from combined
observables is free of uncertainties of $K_{0}$.

This work is supported by the National Natural Science Foundation of China
under grant Nos.11965008, 11405128 and Guizhou Provincial Science and
Technology Foundation under Grant No.[2020]1Y034, and the PhD-funded project
of Guizhou Normal university (Grant No.GZNUD[2018]11).

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[]Per EKummervold []Javierde la Rosa []FreddyWetjen []Rolv-ArildBraaten []Per
ErikSolberg

# Whispering in Norwegian: Navigating Orthographic and Dialectic Challenges

###### Abstract

This article introduces NB-Whisper, an adaptation of OpenAI's Whisper,
specifically fine-tuned for Norwegian language Automatic Speech Recognition
(ASR). We highlight its key contributions and summarise the results achieved
in converting spoken Norwegian into written forms and translating other
languages into Norwegian. We show that we are able to improve the Norwegian
Bokmål transcription by OpenAI Whisper Large-v3 from a WER of 10.4 to 6.6 on
the Fleurs Dataset and from 6.8 to 2.2 on the NST dataset.

###### keywords:

speech recognition, language models, whisper

##  1 Introduction

Automatic Speech Recognition (ASR) holds the promise of changing the way we
interact with technology by enabling machines to process human speech.

Early Norwegian speech recognition research initially focused on limited
vocabularies suitable for telephone applications, confronting challenges like
compound words and varied pronunciations of numbers (Svendsen et al., 1989;
Paliwal, 1992). Funded by the European Union at the start of the century,
subsequent projects expanded the scope to more complex linguistic elements,
contributing to valuable datasets and technical advancements using hidden
Markov models and Mel Frequency Cepstral Coefficients (Amdal and Ljøen, 1995;
Höge et al., 1997). However, these systems, often utilising the Hidden Markov
Model Toolkit (Young, 1994), were limited in handling open-ended recognition
and struggled with out-of-vocabulary words or real conversations. The
introduction of newer datasets in recent years has led to systems with
improved performance, setting the stage for the Wav2Vec work described in De
la Rosa et al. (2023) that seeks to address these longstanding challenges in
Norwegian speech recognition.

OpenAI's Whisper (Radford et al., 2023) represents a significant deviation
from the traditional ASR models. Unlike the standard approach of unsupervised
pretraining followed by finetuning on a verbatim dataset mapping phonemes to
graphemes, Whisper is pretrained on a vast, loosely aligned corpus of
subtitles. This method has yielded impressive results, particularly in
English, by not only handling capitalization and punctuation in a single step
but also producing transcriptions that closely resemble written natural
language. This approach effectively translates the spoken language domain into
written text.

However, Whisper's effectiveness diminishes when dealing with Norwegian, a
language characterised by its rich dialectical diversity and two written
standards: Bokmål and Nynorsk. To address this limitation, we developed NB-
Whisper, a specialised adaptation that provides tailored recognition for both
Norwegian Bokmål and Nynorsk, as well as translations into English.

While our primary aim for the project is on enhancing ASR for Norwegian, the
insights and methodologies developed through NB-Whisper have broader
implications. Our work aims to contribute to the general body of knowledge in
ASR, providing strategies and frameworks that can be adapted for other
languages, especially those with similar linguistic complexities. By pushing
the boundaries in Norwegian ASR, we seek to pave the way for innovations and
improvements in speech recognition technologies globally.

##  2 Language Variability and ASR Challenges

Norwegian has two written standards with an equal official status: Bokmål and
Nynorsk. Bokmål is close to the Oslo dialect and historically influenced by
Danish. Nynorsk, written by about 13% of Norwegian pupils in 2023 (Statistics
Norway, 2023), is close to a number of rural dialects, particularly on the
west coast. The two written standards differ mostly in inflectional forms and
vocabulary, but there are also some syntactic differences. In addition, the
orthographic norm of each written standard allows for a significant degree of
variation compared to other European languages, mirroring the variation found
in dialects. For example, all feminine nouns in Bokmål such as “lua”, ‘the
hat’, can also be written in the masculine “luen”. In Nynorsk, infinitives can
end on -e or -a, as in “å elske” or “å elska”, ‘to love’.

The spoken dialects of Norwegian, while mutually intelligible, differ
substantially in grammar, pronunciation and vocabulary. For example, the
interrogative pronoun “who”, written “hvem” in Bokmål and “kven” in Nynorsk,
is recorded with 38 distinct pronunciations in the Nordic Dialect Corpus
(Johannessen et al., 2012). There is no official norm of spoken Norwegian, and
dialects are widely used, even in official settings such as in parliament and
on the news.

ASR systems for Norwegian need to be able to handle the substantial dialect
variation to be useful in real world applications. Also, they should be able
to transcribe in both written standards. Written transcripts such as meeting
notes, subtitles, or parliamentary interventions, are usually expected to
render the speech consistently in Nynorsk or Bokmål, regardless of the dialect
of the speaker, so the written standard should not depend on the speaker's
dialect. Moreover, ASR test sets need to contain transcriptions in both
written standards, from speakers of different dialects, and with metadata
about dialects in order to give a realistic picture of ASR performance.

##  3 Model Architecture and Training

Whisper utilises an encoder-decoder Transformer architecture, with up to 30
second audio chunks re-sampled to 16,000 Hz and transformed into an 80-channel
Mel spectrogram. For the latest Large-v3, this is expanded to a 128-channel
Mel spectrogram (OpenAI, 2023). Feature normalisation and a two-layer
convolutional encoder with GELU activation are applied, followed by sinusoidal
position embeddings and transformer blocks. The decoder mirrors the encoder's
structure, employing learned position embeddings. Byte-level BPE tokenization
from GPT-2 is adapted for multilingual support (Radford et al., 2023).

In our NB-Whisper project, we keep the core architecture of Whisper to be able
to initiate training from the released checkpoint. Although the original
PyTorch training code was not released by OpenAI, a collaborative effort with
HuggingFace led to an alternative implementation in the Transformers library.
This has also been adapted for Jax. The project participated in developing and
open-sourcing training scripts for TPU-v4-pods, enabling dynamic changes to
the training data during runtime (The National Library of Norway, 2024).

Our approach involves modifying some hyperparameters and training procedures.
We increased the batch size to 1024 for all model sizes and scaled the
learning rate accordingly. Warmup steps were slightly increased since we
initialised from pretrained weights. Inspired by the Distil-Whisper work in
Gandhi et al. (2023) and their regularization practices, we also adopted a
reduced setting for the weight decay with regards to the original OpenAI's
Whisper implementation. We found that adding BPE dropout (increased to 0.2)
was beneficial. Lastly, we did not implement the stochastic dropout used in
OpenAI's Large v2 and v3. Instead we used activation dropout as a substitute,
noting some but limited effect.

Model | OpenAI Whisper | NB-Whisper
---|---|---
Tiny | $1.5\times 10^{-3}$ | $6\times 10^{-4}$
Base | $1\times 10^{-3}$ | $4\times 10^{-4}$
Small | $5\times 10^{-4}$ | $2\times 10^{-4}$
Medium | $2.5\times 10^{-4}$ | $1\times 10^{-4}$
Large | $2\times 10^{-4}$ | $7\times 10^{-5}$
Table 1: Learning rates. Hyperparameters | OpenAI Whisper | OpenAI Whisper Large v3 | NB-Whisper
---|---|---|---
Updates | 1,048,576 | 655,360111Current numbers are for v2. It is reported that they have extended the dataset for v3 but OpenAI has to our knowledge not disclosed the number of trained steps. | 200,000 + 50,000
Batch Size | 256 | 1,024 | 1,024
Warmup Updates | 2,048 | 2,048 | 10,000 / 5,000
Max grad norm | 1 | 1 | 1
Optimizer | AdamW | AdamW | AdamW
$\beta_{1}$ | 0.9 | 0.9 | 0.9
$\beta_{2}$ | 0.98 | 0.98 | 0.98
$\epsilon$ | $10^{-6}$ | $10^{-6}$ | $10^{-6}$
Weight Decay | 0.1 | 0.1 | 0.01
Weight Init | Gaussian Fan-In | Gaussian Fan-In | OpenAI Whisper
Learning Rate Schedule | Linear Decay | Linear Decay | Linear Decay
BPE Dropout | 0 | 0.1 | 0.2
Stochastic Depth | 0 | 0 | 0
Activation Dropout | 0 | 0 | 0.1
Table 2: Hyperparameters.

The training comprises two phases: an initial 200,000-step training, followed
by dataset cleaning (see Section 4), and a final 50,000-step training. The
latter phase is done on a cleaner dataset, and the main purpose of this
training is to reduce hallucinations.

##  4 Dataset

We train on the following sources:

  1. (a)

NRK Subtitles. Subtitles from the Norwegian Broadcasting Corporation (NRK),
known for their non-verbatim style, present a challenge for verbatim speech
recognition due to significant deviations from spoken words. The subtitles are
aligned using the timestamps from the subtitles. The data contains consistent
notation for separating simultaneous speakers, continued sentences across
timestamps, whether it is a recording of simultaneous texting, as well as some
rarer notation for denoting the name of the speaker, the language spoken,
credit to staff, etc. We clean this to get only the spoken text and combine it
into longer segments. Due to the non-verbatim nature of the transcripts, a
word-to-word transcription would typically have a word error rate (WER) around
30% on this dataset. Additionally, we extract audio segments without speech
from the same recordings.

  2. (b)

Audio Books. Professional narrators read these audio books with minimal
deviations from the text. For aligning the text we transcribed the text with
NB-Wav2Vec2 1B111https://huggingface.co/NbAiLab/nb-wav2vec2-1b-bokmaal by De
la Rosa et al. (2023), and used the ideas from Ljubešić et al. (2022) for the
forced alignment. Due to the existence of multiple correct spellings for words
in Norwegian, even the best possible WER score hovers around 1-2%. Still,
using the words we found both in the NB-Wav2Vec2 transcription and the aligned
text segments, we were able to extract chunks of text combined with
corresponding sound clips.

  3. (c)

The NST dataset. This dataset was created by the now defunct Nordisk
Språkteknologi. The dataset, which is distributed by the Language Bank at the
National Library of Norway with an open licence, contains 540 hours of
recordings of close to 1000 speakers with different dialectal backgrounds. As
speakers tend to adapt their speech somewhat to the written language when they
are reading out loud, the dataset has less evidence of dialectal grammar and
vocabulary than datasets with spontaneous speech. When testing ASR systems on
the NST dataset, we assume that the WER score will be approximately similar to
audio books (The Norwegian Language Bank, 2023).

  4. (d)

The Stortinget Speech Corpus. This is an open speech dataset created by the
National Library of Norway in 2023 (Solberg et al., 2023). It consists of
around 5,200 hours of transcribed speech from Stortinget, the Norwegian
parliament. The transcriptions are extracted from the official proceedings of
Stortinget using a string matching algorithm adapted from Ljubešić et al.
(2022).

Dataset | Stage 1 (hours) | Stage 2 (hours)
---|---|---
NRK - Subtitles | 16,518 | 2,478
NRK - No caption | 715 | 312
Audio Books | 2,461 | 2,275
The NST Dataset | 260 | 490
The Stortinget Speech Corpus | 2,230 | 523
Total | 22,184 | 6,078
Table 3: Dataset size in hours. Augmentations (translations, timestamps etc)
comes in addition.

To enhance our dataset, we augment the transcriptions with English
translations. Most of the text originally in Norwegian Bokmål is translated
into English using Google Translate, retaining any timestamp information. We
observed that translation quality improves substantially with contextual
information, such as translating entire programs and then splitting them using
a designated token.

Our training process involves two stages. The first stage uses the complete
dataset, and results in functioning models improving significantly on the
OpenAI Whisper. We then use these models to run inference on the entire
dataset. We identify various errors in the datasets, including misalignments
and omissions. Instead of using a teacher-student approach where we train on
the new output, we use the output to clean the dataset following several
criteria:

  * •

Fuzzy Matching for First and Last Words. We ensure at least a partial match
(80% threshold) for the first and last words of each segment to filter out
alignment errors.

  * •

Identifying Insertions. We remove audio snippets where the target contains
insertions not present in the audio, such as non-spoken transcriber comments.
This is determined by finding target n-grams (longer than 3 words) that do not
exist in any model predictions.

  * •

Detecting Omissions. Snippets are deleted if they contain spoken phrases
missing from the target text, identified by the absence of common n-grams
(longer than 3 words) in all model predictions.

  * •

NER Analysis with BERT. Since names often are inserted or spelled out
especially in the parliament transcript, we perform Named Entity Recognition
(NER) analysis using a BERT model. We use the number of named entities as a
filter.

Any snippet that violates any of these filters, is deleted from the training
set. We do not try to fix errors in the dataset.

##  5 Experimental Setup and Evaluation

We evaluated our models using test sets from select datasets, specifically
focusing on the Norwegian components:

  * •

Fleurs Dataset: This Google-curated, out-of-domain dataset comprises Norwegian
Bokmål text snippets (Conneau et al., 2023). It provides a valuable context
for assessing model performance with unfamiliar content.

  * •

NST Dataset: As detailed in Section 4, we used the test portion of this
dataset distributed by The Norwegian Language Bank (2023). Care was taken to
avoid speaker overlap between this test set and our training and validation
data. We note that both the NB-Whisper-models and the

For evaluating model accuracy, we used the JiWER package, processing all texts
to lowercase and removing punctuation prior to WER calculation. Our reporting
of non-normalized scores aims to provide a clear and direct comparison across
different datasets.

##  6 Results and Discussion

Model | Bokmål | Nynorsk
---|---|---
| Fleurs | NST | Common Voice
NB-Wav2Vec2 1B | 10.7 | 3.0 | 26.5
OpenAI Large v1 | 13.2 | 12.7 | 51.7
OpenAI Large v2 | 11.6 | 10.3 | 49.6
OpenAI Large v3 | 10.4 | 6.8 | 30.0
NB-Whisper Large (Ours) | 6.6 | 2.2 | 12.6
Table 4: Model performance comparison. Model | OpenAI Whisper | NB-Whisper
---|---|---
Tiny | 76.4 | 15.2
Base | 56.8 | 11.5
Small | 29.6 | 8.3
Medium | 15.5 | 7.2
Large | 10.4 | 6.6
Table 5: Norwegian Fleurs - Bokmål. Model | OpenAI Whisper | NB-Whisper
---|---|---
Tiny | 73.7 | 8.1
Base | 56.9 | 4.9
Small | 27.2 | 3.1
Medium | 14.6 | 2.3
Large | 6.8 | 2.2
Table 6: NST - Bokmål. Model | OpenAI Whisper | NB-Whisper
---|---|---
Tiny |  $>$100 | 28.0
Base |  $>$100 | 23.2
Small |  $>$100 | 19.9
Medium | 60.2 | 17.0
Large | 30.0 | 12.6
Table 7: Common Voice - Nynorsk.

We are showing clear improvements from OpenAI Whisper for all the model sizes
on a variety of datasets.

Table 4 compares the performance of OpenAI and NB-Whisper models alongside NB-
Wav2Vec2 models for both Bokmål and Nynorsk, highlighting their differences in
architecture, size and approach. Whisper models, designed for robust
multilingual transcription and translation, effectively handle noisy or varied
speech data through extensive labeled data and context understanding
techniques. In contrast, Wav2Vec2 models Baevski et al. (2020) employ
unsupervised pre-training from raw audio, followed by fine-tuning on labeled
data. This makes them efficient for languages with scarce resources. However,
the written output is based word by word on the input, and lack certain
features typical for written language like capitalisation and punctuation. In
comparison, the output from a Whisper model tends to more closely resemble
written text in terms of quality.

Evaluating models quantitatively, such as by word-error-rate, presents certain
challenges. Our focus is on scripted and read aloud text, where model
comparability is highest. However, issues arise in normalization, like
standardizing outputs amid speech diversity and lacking a universal dataset,
complicating direct comparisons. Number normalization is notably difficult due
to varying expressions and language-specific rules. In this study we did
choose a light normalisation, mainly on punctuation and capitalization222A
stronger normalization was applied in De la Rosa et al. (2023) for the NB-
Wav2Vec2 models (e.g., spelling out numbers), hence the difference in scores..
Increasing the normalisation and standardisation, could have led to lower
scores.

Tables 6 and 7 detail the performance of OpenAI Whisper and NB-Whisper models
across various model sizes, comparing their effectiveness in recognizing
Bokmål and Nynorsk languages. In the context of NST Bokmål dataset, the
performance metrics indicate a clear trend: as the model size increases from
Tiny (39M parameters) to Large (1550M parameters), the accuracy improves
significantly for both OpenAI Whisper and NB-Whisper models, with NB-Whisper
generally outperforming its OpenAI counterpart across all sizes. Specifically,
the Tiny model shows a stark contrast in performance between the two, with NB-
Whisper Tiny achieving a 8.1% accuracy compared to OpenAI's Whisper's 73.7%.
This trend continues down to the Large model, where NB-Whisper achieves a 2.2%
word error rate, closely followed by OpenAI Whisper at 6.8%.

In the Common Voice Nynorsk evaluation, the discrepancy in performance is even
more pronounced, especially at smaller model sizes where OpenAI Whisper models
report word error rates exceeding 100%, indicating a significant challenge in
handling Nynorsk. However, NB-Whisper achieves a remarkable 28.0% WER score
even at the Tiny model size. As the model size increases, performance improves
dramatically, with the Large model achieving a 12.6% error rate NB-Whisper and
30.0% for OpenAI Whisper. This highlights the critical impact of model size on
the accuracy of speech recognition tasks in different languages, particularly
in less commonly supported languages like Nynorsk, where NB-Whisper models
demonstrate a better ability to understand and transcribe the language
accurately.

##  7 Challenges and Limitations

The evaluation of the model's handling of hallucinations and long text
transcriptions was conducted qualitatively, primarily due to a shortage of
Norwegian datasets specifically designed for such detailed quantitative
assessments. For further developing Norwegian ASR it would be vital to develop
such evaluation tools.

The model's architecture, while optimised for transcribing 30-second audio
clips, presents a notable limitation in live transcription scenarios. The
reliance on an autoregressive decoder is not ideal for doing real-time
transcription. This suggests that alternative architectures might be more apt
for applications requiring continuous, live transcription.

These challenges and limitations underscore the need for future advancements
in model architecture and broader linguistic evaluations.

##  8 Future Work

Future improvements can be made in dataset cleaning. Our multi-stage approach,
involving interim models, inherently risks discarding valuable data due to the
fine line between high-quality and faulty data. Re-evaluating the entire
dataset with improved tools developed during the project promises enhancements
in data quality.

The high degree of orthographic variation in Norwegian, which permits multiple
correct spellings, presents a substantial challenge. This variation can cause
inconsistent model outputs due to the different acceptable spellings found in
our corpus. Addressing this issue requires standardising the text within the
corpus. A practical solution involves fine-tuning large language models (LLMs)
to harmonise the corpus to a consistent style, significantly improving the
quality of the dataset. This approach is particularly important for languages
with a broad range of permissible orthographic variations, such as Norwegian.

##  9 Conclusion

The creation of NB-Whisper marks a step in advancing Automatic Speech
Recognition (ASR) for Norwegian, adapting OpenAI's Whisper to meet the
language's unique challenges. Our efforts in refining ASR capabilities have
shown promising improvements in handling Norwegian's diverse dialects and
written standards. However, we recognize that our project has limitations,
especially in evaluating certain aspects like hallucinations and long text
transcriptions. This is mainly because there are not enough detailed Norwegian
datasets to thoroughly test these areas.

Looking ahead, the project underscores the importance of continuous research
in ASR technology. While we have made strides in understanding and
transcribing Norwegian speech, there is a clear pathway for further
development, especially in model architecture and expanding linguistic
adaptability. A particular focus should be on increasing the consistency in
orthographic variation. This endeavour also opens possibilities for applying
our learnings to other languages with similar complexities, contributing to
the global progress of language technology.

##  10 Funding and Acknowledgement

Our research greatly benefited from the support provided by Google’s TPU
Research Cloud (TRC), which generously supplied us with Cloud TPUs essential
for our computational needs. We also extend our gratitude to Google Cloud for
their support through Google credits, enabling us to carry out the necessary
translations of our corpus into English. Special thanks are due to Sanchit
Gandhi at Hugging Face for his substantial assistance in developing the TPU
training scripts, a crucial component of our project. Thanks to Njaal Borch
for providing the code enabling the initial alignment of the texts from NRK.

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