# Mechanical tuning of a rectangular axion haloscope operating around 8.4 GHz

Jessica Golm 1,2,∗, Jose María García-Barceló 3,∗, Sergio Calatroni 1, Walter
Wuensch 1, Babette Döbrich 3

###### Abstract

The axion haloscope is the currently most sensitive method to probe the
vanishingly small coupling of this prominent Dark Matter candidate to photons.
To scan a sizeable axion Dark Matter parameter space, the cavities that make
up the haloscope need to be tuned efficiently. In this article, we describe a
novel technique to tune axion haloscopes around $8.4$ GHz in a purely
mechanical manner without the use of dielectrics. We achieve tuning by
introducing a gap along the cavity geometry. Losses are added due to the
leaking of the field out of the structure only if the gap becomes too large
concerning the total width. A tuning range of around $600$ MHz is achieved,
depending on the environmental conditions. We present the results of a
corresponding prototype and outline prospects to further develop this
technique.

##  1 Introduction

An axion haloscope is an experimental apparatus used in the search for axions,
hypothetical particles that could make up Dark Matter in the Universe as
discussed in Peccei and Quinn (1977b, a). In recent years, there has been a
large increase in the number of experiments and approaches that aim to find or
exclude the axion as Dark Matter, see e.g. here Irastorza and Redondo (2018).

Axions are very light, neutral particles that are expected to interact very
weakly with ordinary matter and electromagnetic fields. Axion haloscopes are
designed to detect axions by converting them into detectable electromagnetic
signals (photons) in the presence of a strong magnetic field by the inverse
Primakoff effect, cf. Primakoff (1951). To improve this conversion process,
haloscopes use high-quality factor resonators, such as microwave cavities, see
e.g. here Sikivie (1983). The conversion of axions into photons occurs when
the mass of the axion matches approximately the resonant frequency of the
apparatus.

The primary goals of an effective axion detection system involve maximising
the power generated from the axion-photon interaction and expanding the range
of analysed axion masses (i.e. the range of frequencies scanned). The detected
radio frequency (RF) power ($P_{d}$) relies on inherent axion properties and
experimental cavity characteristics, as outlined in Díaz-Morcillo et al.
(2022):

$P_{d}\,=\,\kappa\,g^{2}_{a\gamma}\,\frac{\rho_{a}}{m_{a}}\,B_{e}^{2}\,C\,V\,Q_{l},$
(1)

Here, $\kappa$ signifies the coupling to the external receiver ($\kappa=0.5$
for critical coupling operation), $g_{a\gamma}$ represents the unknown axion-
photon coupling, $\rho_{a}$ denotes Dark Matter density, $m_{a}$ stands for
axion mass, $B_{e}$ represents the external static magnetic field (dependent
on the magnet used in the experiment), $C$ is a label for the so-called ‘form
factor’, $V$ is the cavity volume, and $Q_{l}$ represents the loaded quality
factor of the cavity. In contrast to $Q_{l}$, the unloaded quality factor
$Q_{0}$ is a common metric in characterising resonant cavities as it is
independent of external coupling, see e.g. here Pozar (2012); Rezaee et al.
(2012). The form factor ($C$), measuring the coupling between the external
magnetostatic field and the RF electric field induced by axion-photon
conversion, is expressed as:

$C\,=\,\frac{|\int_{V}\,\vec{E}\cdot\vec{B}_{e}\,dV|^{2}}{\int_{V}\,|\vec{B}_{e}|^{2}\,dV\int_{V}\,\varepsilon_{r}\,|\vec{E}|^{2}\,dV},$
(2)

Here, $\varepsilon_{r}$ represents the relative electrical permittivity within
the cavity of volume $V$. The sensitivity measure of the haloscope is the
axion-photon coupling detectable for a given signal-to-noise ratio
($\frac{S}{N}$), which can be calculated using:

$g_{a\gamma}\,=\,\left(\frac{\frac{S}{N}\,k_{B}\,T_{sys}}{\kappa\,\rho_{a}\,C\,V\,Q_{l}}\right)^{\frac{1}{2}}\frac{1}{B_{e}}\left(\frac{m_{a}^{3}}{Q_{a}\,\Delta
t}\right)^{\frac{1}{4}},$ (3)

In this equation, $k_{B}$ represents the Boltzmann constant, $T_{sys}$ is the
noise temperature of the system, $Q_{a}$ represents the quality factor of the
axion resonance, and $\Delta t$ is the data-taking time window, cf. Díaz-
Morcillo et al. (2022). To summarise, the controllable parameters in a given
experimental set-up encompass $\kappa$, $C$, $V$, and $Q_{l}$.

On the other hand, the mass of axions, if they exist, is not known. To
maximise the chances of detecting axions, the haloscope needs to be tuned to
the resonant frequency that corresponds to the expected mass range of axions
being searched for. Tuning ensures that the cavity is effectively only
sensitive to the specific axion masses under investigation, improving the
chances of successful detection. Thus, the scanning rate $\frac{dm_{a}}{dt}$
is commonly employed to assess the performance of a haloscope, a value derived
from Equation 3, cf. Kim et al. (2020):

$\frac{dm_{a}}{dt}\,=\,Q_{a}\,Q_{l}\,\kappa^{2}\,g_{a\gamma}^{4}\,\left(\frac{\rho_{a}}{m_{a}}\right)^{2}\,B_{e}^{4}\,C^{2}\,V^{2}\,\left(\frac{S}{N}\,k_{B}\,T_{sys}\right)^{-2}.$
(4)

Also, $\kappa$ is usually rewritten in terms of the coupling coefficient
$\beta$, which can be extracted from the expression

$\beta=\frac{\kappa}{1-\kappa}.$ (5)

A tuning step must not require the opening of the cryostat as this is very
time and energy-consuming. Until today, most of the adjustment techniques
applied in haloscope experiments for frequency tuning have relied on
mechanical systems. Various collaborations such as ADMX (cf. Boutan et al.
(2018)), HAYSTAC (cf. Zhong et al. (2018)), and IBS/CAPP (cf. Choi et al.
(2021)) utilise cylindrical cavities with metallic rods. These rods are
rotated inside the cavities, altering the electromagnetic field distribution
and, consequently, the resonant frequency of the operating mode. This rotation
is achieved either through several gears connected to a motor or by employing
piezoelectric elements. A slightly different technique was, for example,
employed by the QUAX collaboration (cf. Alesini et al. (2020)), by using
adjustable sapphire shells to modify the resonant frequency of the cavity. In
a different example, the CAST-CAPP/IBS group uses two movable dielectric
sapphire plates positioned symmetrically at the cavity sides (see e.g. here
Miceli (2015)). In Kuo (2021) tunable conic shell-cavities have been
investigated, providing good tuning results for large cavities operating
around $7.5$ and $20$ GHz. Table 1 gives an examination of some current tuning
systems for frequencies between $4$ and $11$ GHz, in comparison with the
outcomes obtained in this work.

Experiment | Freq. (GHz) | Tuning ($\%$) | $Q_{0}/10^{4}$ | $C$ | References
---|---|---|---|---|---
CAST-CAPP | $4.774-5.174$ | $7.7$ | $\sim 4$ | $\sim 0.53$ | Adair et al. (2022)
ADMX Sidecar - Run B | $5.086-5.799$ | $12.3$ | $0.22\times(1+\beta)$ | $0.44-0.61$ | Boutan et al. (2018)
HAYSTAC | $5.6-5.8$ | $3.45$ | $1.8$ | $\sim 0.5$ | Zhong et al. (2018)
ADMX Sidecar - Run C | $7.173-7.203$ | $0.42$ | $0.23\times(1+\beta)$ | $0.040-0.046$ | Boutan et al. (2018)
RADES - Vertical Cut | $7.753-8.420$ | $9.52$ | $1.5-2.7$ | $\sim 0.66$ | this work
QUAX | $10.1-10.3$ | $2$ | $\sim 9$ | $\sim 0.11/V$ | Braggio et al. (2023)

Table 1:  Overview of some existing tunable haloscopes operating at
frequencies between $4$ and $11$ GHz. For the last row, the form factor is
given in terms of volume ($C/V$), where $V$ is in L. Also, for the second and
fourth rows, the unloaded quality factor parameter has been left dependent on
the $\beta$ coupling employed for each corresponding experiment.

The percentage value of tuning has been calculated according to the following
equation

$\mathrm{Tuning}=\frac{f_{2}-f_{1}}{f_{2}}\times 100\quad[\%],$ (6)

where $f_{2}$ is the upper limit in the frequency tuning range and $f_{1}$ is
the lowest frequency. It is worth mentioning that the ADMX Sidecar experiment
has conducted axion data taking campaigns using two different resonance modes.
For Run B, this group has employed the $TM_{010}$ mode, while for Run C they
have used $TM_{020}$, to explore higher frequencies, where significantly lower
form factor values are achieved. On the other hand, the CADEx collaboration
proposed in Aja et al. (2022) the use of a sliding or tunable wall mechanism
in rectangular cavities. The ORGAN group, in McAllister et al. (2023), also
describes this type of mechanical tuning in detail, comparing it with the
tuning technique using metal rods usually employed in the axion community. In
addition, the QUAX group has explored in Braggio et al. (2023) opening
techniques in a cylindrical cavity using angular misalignment on the
longitudinal axis, achieving satisfactory results at frequencies above $10$
GHz, using a so-called ‘clamshell’ cavity.

It should be mentioned that there is also the possibility of broad-band
search, which removes the need of tuning, but typically cannot achieve a
signal amplification comparable to the resonant haloscopes. Examples of such
experiments are BRASS (magnetic mirror), see Bajjali et al. (2023), MADMAX
(with a set of movable dielectric disks in Brun et al. (2019)), ALPHA (using
metamaterials), cf.Gelmini et al. (2020), or BREAD, see Knirck et al. (2023).

Over the past five years, the RADES (Relic Axion Dark-Matter Exploratory
Setup) research group has extensively explored haloscopes, engaging in the
conceptualisation, production, and physics analysis of axion detectors in the
pursuit of detecting Dark Matter axions with masses approximately around $1$
and $34\,\mu$eV , cf. Álvarez Melcón et al. (2018, 2020, 2021); Díaz-Morcillo
et al. (2022); Ahyoune et al. (2023). This team has developed a mechanical
tuning technique where the haloscope volume is increased by mechanically
moving the cavity halves split symmetrically (cf. Arguedas-Cuendis et al.
(2020)), operating at X-band frequencies. This set-up can work similarly to
the clamshell technique (cf. Braggio et al. (2023)), but a symmetric opening
is more favourable, giving a bigger frequency tuning range, as shown below.
Also, the RADES group is currently exploring other methods based on electrical
tuning by the use of ferroelectric materials (see e.g. here García Barceló et
al. (2023)).

In this work, our goal is to build a tuning mechanism that can tune the
resonant frequency of the first RADES cavity (cf. Álvarez Melcón et al.
(2018)) at $4$ K by using the tuning technique presented in Arguedas-Cuendis
et al. (2020). This haloscope is based on an array of five subcavities
interconnected by waveguide irises to increase the total volume while
maintaining a high resonant frequency for operation in dipole magnets. The
frequency tuning approach presented in this study relies on the separation of
two symmetrical halves until RF leakage through the aperture becomes the
primary factor affecting $Q_{0}$. In the case of small apertures in this
haloscope, the effect of a vertical cut along the longitudinal direction is
negligible, as the microwave currents of the fundamental mode $TE_{101}$ are
parallel to this plane. This movement effectively increases the width of the
cavity, thereby altering the search range for axion frequencies. According to
Arguedas-Cuendis et al. (2020), the theoretical tuning range value for this
haloscope is around $10$ $\%$. Similar values are also obtained in García-
Barceló et al. (2023) for this type of tuning at X-band frequencies in other
cavities. For this study, experiments are carried out at cryogenic
temperatures, and simulations using the finite element method are employed to
comprehend how the cavity behaves at various openings.

##  2 Cavity design for mechanical tuning

Fig. 1 shows the schematics of the mechanical tuning idea. By introducing a
gap between two cavity halves the geometry can be widened which shifts the
resonant frequency to lower frequencies111In García-Barceló et al. (2023),
preliminary investigations have been conducted with this technique for
rectangular long single cavities, and 1D and 2D multicavity structures
operating at X-band frequencies. A similar research is expected for the long
single cavities, and 1D and 2D haloscopes with cylindrical shapes explored in
García-Barceló et al. (2023)..

Figure 1: Schematics of the tuning concept. The left side shows the cavity in
the closed position with no gap and at its highest resonant frequency. On the
right side, a gap was introduced between the two cavity halves, tuning the
cavity to a lower frequency depending on the gap size.

###  2.1 Cavity design

The tuning mechanism will be exemplified using multi-cell cavities, which were
developed by the RADES team, as described in Álvarez Melcón et al. (2018).
This design allows an increase in cavity volume while maintaining a high
resonant frequency as a single element of the cavity mostly determines the
frequency. A prototype resonating at $8.4$ GHz was built by joining five cells
with all-inductive irises (vertical windows in waveguide technology). While
the tuning will be demonstrated with this prototype, it works in principle for
other designs as well.

Fig. 2 shows the electric field distribution of the five configuration modes.

Figure 2: Electric field pattern for the five electromagnetic modes of the
cavity. Red colour refers to a maximum and blue colour to a minimum in the
electric field. Taken from Álvarez Melcón et al. (2018).

The coherence between the cavities is maintained only in the first mode which
is the only mode that couples to the axion. To tune this type of cavity, it
was cut along the symmetry plane as indicated in Fig. 3 (left).

Figure 3: Left: Mode pattern with symmetry plane. The fabrication is done in
two halves defined by this symmetry plane. 3D model housing of the vertical
cut inductive irises cavity to be manufactured (right). This piece is one of
the two symmetrical halves, which must be parallel. SMA coaxial ports are
situated at the $\varnothing 4.1$ holes. Dimensions given in Table 2.

The surface currents do not cross this plane of symmetry (they are parallel to
this plane); therefore, it is possible to cut the cavity without intercepting
the current. Thus, one can expect to retain a high $Q_{0}$ value despite the
introduction of the manufacturing cut and even when separating both halves.
This paper aims to study and measure to what extent this is possible.

To enable vertical cut tuning, small changes in the design of the 5-cell
cavity (with manufacturing horizontal cut), described in Álvarez Melcón et al.
(2018), were necessary. This cut can be used to introduce a gap that increases
the volume and moves the resonant modes to lower frequencies, as shown in Fig.
1, without a significant loss of quality factor, as will be shown in the
following. For this purpose, the gap should be small compared to the cavity
dimensions ($<10$ $\%$ of the width, a value extracted from García-Barceló et
al. (2023), where a similar tuning mechanism is employed). Fig. 3 (right)222In
the fabrication of these parts, alignment pins were included in the $x-$axis,
in the thickness $t_{z}$ of the housing at both ends in length. In addition,
screw holes were implemented both in the port sections ($y-$axis) and in the
$t_{z}$-thickness ($x-$axis). The latter to test the behaviour of the
completely closed and screwed-together structure. depicts the 3D model and
dimensions of the analysed haloscope. In Table 2, the dimension values are
shown.

Dimensions | Values (in mm)
---|---
$a$ | $22.86$
$b$ | $10.16$
$d_{1}$ | $26.68$
$d_{2}$ | $25$
$a_{i}$ | $8$
$t_{i}$ | $2$
$t_{x}$ | $2.5$
$t_{y}$ | $2.5$
$t_{z}$ | $6.5$
Table 2:  Dimension values of the vertical cut haloscope for cryogenic
conditions analysed in this study.

###  2.2 Simulations of losses introduced due to mechanical tuning

####  2.2.1 Parallel cavity halves

Before the manufacturing and measurements of this haloscope, a study was
carried out using the Computer Simulation Technology Studio Suite
software(2023) (CST), to study the behaviour of this structure in terms of
$Q_{0}$ and C factors against different openings that can cause radiation
losses.

Fig. 4 depicts the results obtained with a sweep of vertical cut gaps from $0$
to $4$ mm.

Figure 4: Results from CST simulations for the vertical cut tuning study
applied in the haloscope depicted in Figure 3 (right): frequency and tuning
versus gap (left), unloaded quality and form factors versus gap (centre), and
volume and figure of merit ($Q_{0}V^{2}C^{2}$) versus gap (right).

For these simulations, an electrical conductivity value in the copper cavity
of $\sigma_{c}=10^{9}$ S/m was used, as this is a realistic value according to
Golm et al. (2022) where the same coating and frequency were applied. In
addition, coaxial antenna lengths have been chosen to be critically coupled
($\beta=1$) for Port $1$ and very undercoupled ($\beta\simeq 0$) for Port $2$,
in the closed scenario (gap$=0$ mm). The antenna lengths have been maintained
for the study of all the gap values.

As can be seen, the obtained form factor follows a relatively constant value,
close to $C=0.66$, for the whole range. However, the unloaded quality factor
suffers a small detriment when the gap increases. For the full range of gaps
analysed, the figure of merit $Q_{0}V^{2}C^{2}$ remains almost constant at
around $11.5$ L, so it can be guaranteed that the tuning range is adequate,
obtaining a frequency tuning range of $1.07$ GHz (or $12.6$ $\%$), which is a
very satisfactory tuning compared to the results of other experimental groups
shown in Table 1, and even more so for operating frequencies close to $8.4$
GHz.

In addition, since this structure has a tuning system that connects to the
outside environment, a study has been made of possible signals that could
enter its interior. After several simulations, it has been verified that there
are no resonance echoes of a potential dipole magnet with a bore diameter of
about $50$ mm where this haloscope could be installed.

####  2.2.2 Misalignment studies

A simulation study has been conducted to consider possible misalignment
effects when separating the two halves. Due to the position of the coaxial
ports, the number of possible main misalignments is reduced to the following:
angular $y-$axis, angular $z-$axis, and lineal $y-$axis. Fig. 5 shows an
example of misalignment for each scenario.

Figure 5: Misalignment effect in the vertical cut haloscope for three
scenarios: angular $y-$axis (top left), angular $z-$axis (top right), and
lineal $y-$axis (bottom).

In addition, to reduce complexity, this tuning technique is based on leaving
the coaxial ports attached to one of the halves of the haloscope. This
introduces a small asymmetry between the two halves which can be neglected
(see Appendix A).

Also, to simplify this investigation of misalignments, a gap of $1$ mm has
been chosen. Fig. 6 depicts the results obtained for each case as a function
of the misalignment variables: $\theta_{y}$ for the angular $y-$axis,
$\theta_{z}$ for the angular $z-$axis, and $g_{y}$ for the linear $y-$axis.

Figure 6: Results obtained in the simulation of the misalignment study in the
angular $y-$axis (first column), in the angular $z-$axis (second column), and
in the lineal $y-$axis (third column). The first row of graphs shows the
variation of the quality and form factors versus the variation of the
misalignment variable. The second row plots the variation in frequency and
figure of merit $Q_{0}V^{2}C^{2}$ versus misalignment. The quality factor and
figure of merit parameters are given in terms of the percentage change from
the aligned scenarios ($\theta_{y}=\theta_{z}=g_{y}=0$).

As can be seen, the results show a degradation in the form factor value of
$\sim 40$ $\%$ in the case of angular misalignment in the $y-$axis. The main
reason for this reduction is because the individual resonant frequencies of
the subcavities are very sensitive to their widths, which are modified with
$\theta_{y}$, in turn changing the optimised condition for the operating mode
(see more on this concept in Álvarez Melcón et al. (2018)). Furthermore, for
this type of misalignment, the resonant frequency changes considerably (around
$1.2$ $\%$). However, the unloaded quality factor remains relatively stable.

On the other hand, for angular $z-$axis misalignment, the form factor remains
very high, although the quality factor suffers a reduction of more than $7$
$\%$ for negative values of $\theta_{z}$. However, as the form factor becomes
more important here for the figure of merit of the experiment, the latter
remains relatively high. In the case of frequency, it is also maintained
around stable values.

Finally, in the case of linear misalignment in the $y-$axis, it is seen that
this causes the largest decrease of the unloaded quality factor (more than
$80$ $\%$ for values of $g_{y}=0.5$ mm), while the form factor remains high
for the entire range, and this in turn reduces the figure of merit the most of
the three cases analysed. Fortunately, the outcomes show a resonant frequency
that is relatively stable for the entire range.

##  3 Proof of principle

The tuning idea was first implemented by separating the cavity halves of a
prototype with spacers that do not penetrate the cavity hollow volume or
disturb the electromagnetic field pattern. The prototype was made of non-
magnetic stainless steel (316 LN) coated with a $30$ $\mu$m copper layer. The
reason for choosing stainless steel as a base material for the cavity and not
producing it from pure copper is the forces the cavity can be exposed to
during a quench of the magnet (eddy currents). Cavities for axion Dark Matter
searches are usually installed in multitesla fields.

A photograph of the prototype and spacers is shown in Fig. 7. The top picture
shows the disassembled cavity halves and the spacers made of brass. These
spacers generate a gap of around $1.25$ mm and they are mounted on the screws
that connect the two cavity halves. The bottom picture depicts the assembled
cavity with spacers made of stainless steel. These spacers are commercially
available washers mounted on the screws that keep the cavity halves together.
The thickness of one washer is about $0.5$ mm. Using several washers in a
stack, gap sizes up to $2.5$ mm were realised to characterise the cavity
prototype at ambient conditions for different gap openings. After a
characterisation of the tuning properties of the cavity at room temperature,
the cavity was installed in a cryostat, and the unloaded quality factor
$Q_{0}$ was measured between $7$ and $300$ K for a gap size of $0$ and $1.25$
mm. The brass spacers were used during the cryogenic characterisation. In none
of the described mounting situations, the spacers entered the cavity
structure. The alignment of the two halves of the cavity with each other
depended only on the two spacers and the force with which the screws were
tightened on each side. Both cavity ports were kept on one cavity half when a
gap was applied.

Figure 7: Photograph of the vertical cut haloscope coated with copper before
assembly and brass spacers (top) and after assembly with a gap between the two
halves introduced with stainless steel washer spacer (bottom). Note that for
the characterisation both ports were attached to one cavity half unlike shown
on the picture in which each port is attached to another cavity half.

Fig. 8 (left) shows measurements of the tuning range and quality factor at
ambient conditions for a gap opening step size of $0.5$ mm. A stack of
stainless spacers was used to create the different gap sizes. There is a
linear dependence between gap size and frequency, as expected, and the total
tuning range at ambient conditions is $676$ MHz for a gap opening of $2.5$ mm.
For bigger gap opening a decrease in quality factor up to $35$ $\%$ for a
$2.5$ mm gap was observed during this test. From simulations in section 2.2.2
we know that these losses can be attributed to misalignment. As not all
spacers have exactly the same thickness, this uncertainty adds up when adding
more spacers for a wider gap.

Figure 8: Unloaded quality factor $Q_{0}$ and resonant frequency for different
gap sizes which were generated by spacers from washers (left) and $Q_{0}$ of
the cavity between $7$ and $300$ K for gap sizes of $0$ (red) and $1.25$ mm
(blue) using brass spacers (right).

For the characterisation at low temperature, the cavity was installed in a
vacuum chamber which was introduced in a helium cryostat. The cavity was
suspended in the cryostat by clamping it in a support structure on one
extremity and subsequently attached to a long rod. It was measured in two
configurations during cool-down for a gap size of $0$ and $1.25$ mm (half the
maximum possible opening within the holding structure mechanics) separated by
the brass spacers shown in Fig. 7 (top). The results from the cryogenic
measurements of the quality factor with spacers are shown in Fig. 8 (right).
For the cavity installed in the cryostat, the quality factors of $Q_{0}$ (gap
$=0$ mm) $=$ $6300$ $\pm$ $200$ and $Q_{0}$ (gap $=1.25$ mm) $=$ $5950$ $\pm$
$130$ were measured at ambient conditions for the respective openings. The
quality factor of the cavity installed in the cryostat and with no gap is
lower than measured for the same cavity outside the cryostat ($Q_{0}=7880$
$\pm$ $20$). These losses may be caused by some stresses applied to the cavity
by the holding structure which can result in a misalignment of the cavity.
Simulations showed that misalignments of $0.2$ mm in $y-$axis direction can
already cause a decrease in quality factor by more than $40$ $\%$, cf. section
2.2.2. During the cool-down, the cavity showed for both gap sizes the expected
behaviour. The resonant frequency changed about $25$ MHz between $300$ and
$40$ K attributed to the thermal contraction of stainless steel. Below $40$ K
the resonant frequency is stable which is in good agreement with the behaviour
of stainless steel regarding its thermal expansion at low temperatures.

Fig. 8 (right) shows that the quality factor for both configurations saturates
around $40-50$ K. The copper coating of this cavity was applied by DC (Direct
Current) galvanic plating with an organic brightener and the RRR (residual-
resistance ratio) is expected to be $38$ $\pm$ $2$. The electrical
conductivity for copper of this quality usually only saturates between $20$
and $30$ K, see Calatroni (2020b). The reason for these earlier saturations is
the Anomalous Skin Effect (ASE). The cavity operates at frequencies above $8$
GHz for the specified gap sizes. At those frequencies, the conductivity
saturates once the mean free path of the electrons becomes bigger than the
skin depth. Therefore, the electrical conductivity cannot increase further
even if the mean free path increases. Changes below this temperature are
attributed to other effects. The unloaded quality factor at low temperatures
was measured to $Q_{0}$ (gap$=0$ mm, $8.3$ K) $=$ $23200$ $\pm$ $800$ and
$Q_{0}$ (gap $=1.25$ mm, $7.3$ K) $=$ $27200$ $\pm$ $1500$. For the cavity
without a gap a higher quality factor was expected from simulations, see Fig.
4 (centre). But as mentioned previously the quality factor of the cavity after
installation in the cryostat was already less than expected from measurements
before the installation attributed probably to misalignment. The quality
factor of the cavity with a gap was smaller at room temperature than for the
configuration without a gap. During cool-down this ratio changed below $100$
K. Due to the forces from the thermal contraction, the two cavity halves may
have become better aligned and give values that are close to the values
simulated, see Fig. 4 (centre), and in good agreement with the conductivity
for copper considering ASE, see Calatroni (2020a).

##  4 Tuning mechanism for cryogenic environment

###  4.1 Set-up and sliding mechanism

After demonstrating that the tuning concept is viable in principle, a gear
system was designed to allow for translating the movement in a cryogenic
environment. For data-taking, haloscope axion cavities will be installed in
dipole magnets where the space for tuning is typically limited by small bore
diameters. Furthermore, the measurement takes place in a vacuum or liquid
helium environment at temperatures of $4.2$ K.

Figure 9: Cavity tuning test stand in the CERN Central Cryogenic laboratory
with adjusting screw and rod (red), RF cables (green), temperature sensor
(yellow), and the cavity (grey).

These conditions make it challenging to realise cavity tuning during data-
taking. Therefore, two possible solutions were deliberated. One option is to
install a piezo actuator in the bore to generate the movement. However, two
challenges have to be overcome: first, the space in the bore does not allow
for a larger actuator. Secondly, an actuator with sufficient force to move the
cavity halves is needed. Considering the size of the currently existing
actuators that can move our cavity halves, a gear system to generate the
movement would be necessary. Another possibility is to generate the movement
from the outside of the magnet by a long rod connected via gears to the two
cavity halves. This solution is also not without problems as complex
feedthroughs are required to propagate the rotational movement from the $300$
K to the $4$ K environment.

To study the gear-based tuning mechanism, a test stand was built in the CERN
Central Cryogenic Laboratory, and different aspects of this tuning were
investigated. Fig. 9 shows a schematic drawing of the test stand. The cavity
and the RF cables were installed in a vacuum chamber immersed in liquid
helium. To characterise the cavity two RF cables (green) coming from the
cavity were connected to a feedthrough in the vacuum flange at ambient
temperature, to which a Vector Network Analyser (Keysight N9918A) was
connected for quality factor measurements. To tune and change the gap size of
the cavity a long rod (about $1.5$ m long, $2$ mm diameter, stainless steel)
(red) was connected by a gear system to the cavity at the bottom of the vacuum
chamber. The rod had an adjusting screw at the end and by turning it the
cavity opened or closed, depending on the direction in which the gear is
turned. The movement of the rod for this test was generated by hand, but the
adjusting screw could be easily attached to a small motor which would allow
more precision in movements. A gas inlet on the cryostat provided the
possibility of measuring in a helium gas environment instead of in vacuum.

Photographs of the cavity installed in the holding system with gears to
transmit the movement can be seen in Fig. 10.

Figure 10: Photographs of the cavity halves installed in the tuning holding
structure with gears. Top view with RF ports (left) and bottom view (right).
The bottom view displays the gear mechanism that moves the cavity halves to
each other.

In Fig. 11, the individual parts of the tuning mechanism are shown. In (a),
the holding pieces for the cavities are illustrated. The cavity halves can
slide into these holding pieces, which are held by positioning screws. The
holding piece has on both sides sliding bars. In Fig. 11 (b), the sliding
structure is shown. The assembly consists of two of those structures that will
be connected by long stainless steel bars, which have on one side long holes
for easier positioning. The sliding bars shown in (a) have to be inserted in
the sliding rails shown in (b). Fig. 11 (c) shows the assembly of (a) and (b)
without cavities, and (d) shows the complete assembly with the cavity halves
installed. The assembly shows the two gear racks that are screwed to holding
piece (a) and a tuning rod with two gears glued to it. Thus, by turning this
tuning rod outside of the cryostat, the rotation will be transmitted to the
gears on the cavity inside the cryostat and converted into a linear, parallel
translation of one cavity half.

Figure 11: Drawings of the cavity support structure with (a) holding piece for
the cavity halves, (b) the sliding structure for alignment, (c) the assembly
of the sliding structure, and (d) the complete assembly with the gear system.

To guarantee a smooth movement of the cavity halves with the described tuning
mechanism at cryogenic temperatures and in vacuum, the materials have to be
chosen carefully. In vacuum, parts of the same material can easily adhere to
each other because there are no thin layers of other elements between the
parts. Under mechanical pressure, e.g. thermal contraction, they can bond.
Also, the friction coefficients between the same materials, i.e. aluminium on
aluminium (friction coefficient $1.6-2.2$, c.f. Deulin et al. (2010)) or
stainless steel on stainless steel (friction coefficient $2.9$, see Deulin et
al. (2010)), in vacuum are very high. Thus, it was decided to produce the
holding pieces for the cavity and the holding structure out of different
materials. The holding piece for the cavity was made of stainless steel (see
Fig. 11 (a)) and the structure on which the cavity slides back and forth
inside was made of aluminium (friction coefficient $0.3$, see Deulin et al.
(2010), and Fig. 11 (b)). The gear and gear rack were made of PTFE
(Polytetrafluoroethylene) which has a very low friction coefficient of $0.04$
for PTFE on PTFE, cf. Deulin et al. (2010).

After a first functionality test of the described mechanism and material, some
optimisations have been made. The sliding structure (see Fig. 11 (a)) was re-
fabricated using brass as it has a more similar expansion coefficient to
aluminium than stainless steel. Furthermore, the holding pieces for the cavity
were optimised by replacing the sliding bars with sliding pins, which have
less surface area touching the sliding rail, resulting in reduced friction and
less material deforming during cool-down.

###  4.2 Tuning results with the sliding mechanism

The tuning mechanism described in the previous section was tested on
functionality at $77$ K by immersing the set-up in liquid nitrogen. For
testing a tuning mechanism where pieces are made of materials with different
thermal expansion coefficients, it is crucial to wait for thermalisation.
After a 5-minute thermalisation time in the liquid bath, an opening and
closing of the cavity halves were demonstrated. The cavity spectrum and the
tuning range were measured at ambient conditions, see Fig. 12 (left), and in
liquid nitrogen, see Fig. 12 (right). The spectrum of the cavity shows five
peaks, one for each of the small subcavities, while the axion only couples to
the first mode. For explanations, see Álvarez Melcón et al. (2018). The
difference in frequency between the first peak of each spectrum (and thus the
tuning range) in the open (gap$=2.5$ mm) and closed (gap$=0$ mm) positions of
the cavity halves was measured to be $670$ MHz at ambient condition and $540$
MHz in liquid nitrogen at $77$ K. The tuning range in liquid nitrogen is
smaller compared to air because the higher dielectric constant of
$\textrm{LN}_{2}$ decreases the resonant frequency by a factor of
$1/\sqrt{\varepsilon}$. The dielectric constant of liquid nitrogen is $1.431$,
see Mathes (1967), and therefore a tuning range of $540$ MHz between $6.48$
and $7.02$ GHz corresponds to a tuning range in vacuum of about $650$ MHz
between $7.75$ and $8.40$ GHz.

In Fig. 12 (right) some disturbances in the second peak of the blue spectrum
are visible. Those have their origin in gaseous Nitrogen bubbles that enter
the cavity from time to time. Those bubbles are the result of the boiling of
the liquid nitrogen and change the spectra while entering the cavity.

Figure 12: Tuning range measured with the cavity embedded in the sliding
structure at ambient conditions (left) and in liquid nitrogen (right) with a
maximum gap size of $2.5$ mm (blue) and no gap (red). At $77$ K disturbances
of the spectra due to the boiling of liquid nitrogen are visible, most clearly
for the second cavity peak in the spectra of the cavity with a $2.5$ mm gap.

After demonstrating that the mechanism worked in a $77$ K environment, the
whole set-up was implemented in a cryostat with a $2$ mm thick tuning rod of
about $1.5$ m length going from $4$ to $300$ K. The cavity and all elements
were placed in a vacuum chamber inside the cryostat that was filled with
liquid helium. A schematic of the experimental set-up is shown in Fig. 9. It
was possible to reach temperatures around $20$ K in the vacuum chamber with
this setup. The reason for not reaching lower temperatures is the RF cables
which are made of copper, a very good thermal conductor, as well as the tuning
rod of $1.5$ m length (stainless steel, $2$ mm diameter) conducting heat from
the outside of the cryostat at ambient temperature to the cold vacuum chamber.
For measurements with spacers between the cavity halves, described in section
3, a minimum temperature of $6$ K could be reached. However, this was only
possible due to a complex thermalisation of the RF cables to intercept the
conducted heat into the vacuum chamber. Those cables need to be moveable
during tuning so as not to obstruct the mechanism, therefore they were not
thermalised. Instead, thinner, more flexible cables were used with the
disadvantage of higher RF losses.

The same results in terms of tuning are expected at $4$ and $20$ K, as
previous measurements showed that the resonant frequency (= thermal
contraction of the cavity) and quality factor (= electrical conductivity) for
the cavity are constant below $30$ K, see Fig. 8 (right). To use the cavity as
a haloscope in an axion search, the temperature should be as low as possible.
One possibility to solve the issue of not reaching the minimum temperature is
to let a small amount of exchange gas (about $1$ mbar of helium) in the vacuum
chamber, allowing convective heat transfer from the wall of the vacuum chamber
to the cavity. By adding the exchange gas, the cavity properties remain the
same, and the gas will not affect the haloscope data taking.

Figure 13: Tuning range measured with the cavity embedded in the sliding
structure at $20$ K with maximum gap size (blue) and without gap (red) (left).
Zoom in on the first cavity peak (axion mode) for a selected set of cavity
openings, demonstrating the minimum step size achieved for this measurement.

Fig. 13 shows the tuning results at $20$ K. It was demonstrated that the
cavity can be tuned in a cryogenic environment without performing a warm-up
achieving a tuning range of $667$ MHz. This result confirms the tuning range
predicted from the measurement in liquid nitrogen described before. Besides
the tuning range, a high scanning resolution is important to investigate all
possible frequencies while a haloscope is data-taking. Therefore, frequency
steps in the range of the 3 dB points of the resonant frequency should be
realised. The frequency step size required to achieve this depends on the
cavity properties: the unloaded quality factor ($Q_{0}$) and the coupling of
the port. Assuming critical coupling and a $Q_{0}$ of $27000$ (as measured for
this specific cavity, see section 3) we aim for a step size of $0.6$ MHz
($8.4$ GHz$/13500$) per step.

The described set-up reached a minimum step size of $3$ MHz while the rotation
was applied, turning an adjustment screw by hand; see Fig. 13 (right). Using a
motor turning the adjustment screw at ambient conditions will lead to higher
precision. Furthermore, high-precision gears and gear racks can help to
decrease the frequency step size. To maximise the detection rate during
tuning, over-coupling (i.e. $\beta=2$) instead of critical coupling is more
advantageous, cf.Kim et al. (2020). In that case, the minimum required step
size required for our cavity increases to $0.9$ MHz.

The cavity must maintain a high-quality factor during tuning. Therefore,
$Q_{0}$ of the cavity was observed while changing the resonant frequency with
the tuning mechanism and compared with the results obtained with the spacers,
described in section 3. The result of the unloaded quality factor
determination for different resonant frequencies of the cavity is illustrated
in Fig. 14.

Figure 14: Unloaded quality factor ($Q_{0}$) of the cavity for different gap
sizes generated by the tuning mechanism (yellow) and by spacers (blue)
measured at cryogenic temperatures below $20$ K.

From the simulation in section 2.2.1 we expect a quality factor of about
$27000$ at cryogenic temperatures which does not degrade too much for a gap of
$1.25$ mm, cf. Fig. 4 (centre). Fig. 14 shows that this is hard to achieve.
The blue points mark the measurements done with a spacer (gap$=1.25$ mm) or
the cavity bolted together with screws (gap $=0$ mm). The measurements of
those configurations were already partially discussed in section 3, where the
quality factor for no gap turned out smaller than with a gap of $1.25$ mm,
most probably due to misalignment while installation. Both measurements were
repeated, and different quality factors for the same gap size/resonant
frequency were obtained, as shown in the graph. Only considering the maximum
$Q_{0}$ measured for both spacer configurations (gap$=0$ and $1.25$ mm) the
results are in good agreement with the simulations from section 2.2.1 for
parallel cavities. The lower values for $Q_{0}$ for the same spacer
configurations are attributed to misalignments. The simulations of section
2.2.2 showed how sensitive the cavity is to certain misalignment directions.
The results underline the importance of a good alignment mechanism for the
tuning, as a good-aligned setup is already hard to achieve with spacers.

The yellow triangles are the results obtained when changing the resonant
frequency with the tuning mechanism described in section 4.1. The quality
factor is for all measurements below the values obtained with spacers and
varies for the same resonant frequency. The maximum decrease in quality factor
measured with the tuning mechanism compared to its optimal values for no gap
($Q_{0}$ about $27000$) is about $45$ $\%$. These losses are also attributed
to misalignments of the two cavity halves, which leads to the conclusion that
the alignment mechanism still needs to be improved and tolerances for the
production of future assemblies need to be reduced.

##  5 Conclusion and outlook

We have presented a concept to tune rectangular cavities for haloscope
searches mechanically. The concept foresees cutting cavities along a symmetry
plane of the field lines. We have validated this mechanism in practice and
simulation. Assuming that the cavities are kept parallel, the simulations in
section 2.2.1 predict a tuning range of up to $1.1$ GHz ($12.6$ $\%$) with
nearly no degradation in the figure of merit ($Q_{0}V^{2}C^{2}$) over the
whole range.

The cavity could be tuned in a range of $667$ MHz $\sim 9.5$ $\%$ at $20$ K
using the sliding mechanism outlined in section 4.1. The available bore size
limited the maximum gap size ($2.5$ mm) and hence the tuning range for the
presented configuration. The obtained unloaded quality factor varied between
($1.5-2.1$) $\times 10^{4}$ while tuning. From the simulations in section
2.2.1 higher quality factors were expected. As can be observed from the
simulations in section 2.2.2, the misalignment of the two cavity halves is the
source of the losses (up to $45$ $\%$) in this assembly. Previous measurements
in section 1 showed that greater values of $Q_{0}$ ($2.7\times 10^{4}$ at $7$
K, gap$=1.25$ mm) are possible for parallel cavity-halves (separated by
spacers). This number is consistent with the parallel cavity-half simulations
shown in section 2.2.1.

Moreover, misalignment can cause a $60$ $\%$ decrease in form factor C; see
Fig. 6. This is particularly relevant for cavity designs with subcavities
connected by irises, as the misalignment alters the ideal operating mode
conditions. Examining the same misalignment for a long single cavity, merely a
minor deterioration of the form factor ($0.66$ to $0.63$) is discernible.

The minimum step size achieved for this mechanism was $3$ MHz. To have
continuous tuning steps overlapping in their $3$ dB range a step size of less
than $0.9$ MHz ($3$ $\mu$m) is required. To improve the step size of the
assembly, high-precision gears and a motor-driven adjustment of the tuning rod
are required.

The state of the art of various current tuning methods for different axion
groups and the outcomes of this effort were displayed in Table 1 of section 1.
The mechanical tuning presented in this work is a competitive way to tune an
axion haloscope because it can achieve the same tuning percentage as the ADMX
Sidecar - Run B while also having a larger form factor and quality factor.
However, it is necessary to make improvements to the current tuning design to
produce a smaller step size and handle the misalignment more effectively.
Further studies using an improved gear mechanism, combined with a mechanism
that moves the coupler are ongoing in the group.

## APPENDIX

### A. Misalignment due to coaxial ports

In this work, an additional misalignment study has been carried out on the
haloscope with vertical cut tuning. The results obtained in section 2.2 are
based on a system in which the coaxial ports are screwed to one of the two
halves, causing a small misalignment or asymmetry in the total system. In this
case, the behaviour of the structure has been analysed in simulation by
positioning the coaxial ports centred in the middle of the opening of the
structure (gap$/2$).

In Fig. 15 the results obtained in quality factor with a range of openings
from gap$=0$ to $2.5$ mm are depicted.

Figure 15: (Left) 3D model of the vertical cut structure with gap$=2$ mm
applying symmetry with the coaxial ports laying in the middle of the gap
(centred at gap$/2=1$ mm), and (Right) quality factor results from CST
simulations for this structure without (blue solid line) and with (red dashed
line) centring the coaxial ports.

We can therefore conclude that this misalignment has practically no effect on
this parameter and can be ignored in our system.

### B. Misalignment in a long single cavity

In addition, the simulation of an individual cavity with the same dimensions
of width, height and total length as the vertical cut multicavity (simply
eliminating the inductive irises inside) has been carried out to observe the
behaviour in form factor with angular misalignment in the $y-$axis (this is
$\theta_{y}$), which is the most relevant effect studied for this parameter
(see Figure 6 (top-left corner plot)). The width of the single cavity is
$22.86$ mm, the width is $10.16$ mm, and the length is $136.36$ mm.

Applying a misalignment of $\theta_{y}=0^{o}$ to $0.6^{o}$, a range of
detriment form factor values from $0.659$ to $0.634$ was obtained, confirming
that the effect is not high for individual cavities. Thus, we can conclude
that the low form factor results obtained for the vertical cut haloscope with
angular $y-$axis misalignment are due to the asynchrony of the individual
frequencies of the subcavities of which it is composed.

##  6 Keywords:

axion, haloscope, tuning, cryogenics, dark matter

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

JG carried out the measurements, performed data analyses, interpreted the
results and contributed to the text. JMGB performed the cavity simulations and
wrote the corresponding section. BD wrote parts of the text, acquired the
funding to carry out the experiment, supervised the measurements, and
contributed to interpreting the results. The initial idea of tuning with a
vertical cut came from WW. He and SC also co-supervised the measurements and
contributed to interpreting the results.

## Funding

This work was funded by the European Research Council under grant
ERC-2018-StG-802836 (AxScale project). JG is also funded by the ‘Physics
Beyond Collider’ initiative. This work has also been funded by
MCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”, under
grants PID2019-108122GB-C33 and PID2022-137268NB-C53. JMGB thanks the grant
FPI BES-2017-079787, funded by MCIN/AEI/10.13039/501100011033 and by ”ESF
Investing in your future”.

## Acknowledgments

This work was performed in the RADES group and we acknowledge the contribution
and invaluable help of its members. Additionally, we would like to thank the
entire CERN cryolab team, especially Agostino Vacca for helping to modify the
mechanical components of the tuning mechanism.

## Data Availability Statement

Simulation data (3D model of the vertically cut cavity in “.stp” format) as
well as plot data for the misalignment studies are available at the MPCDF
MetaStore. Further data is available upon request.

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#  Proof the Collatz Conjecture by a new view of natural numbers

FENG Jishe  E-mail<EMAIL_ADDRESS>(School of Mathematics and Information
Engineering, Longdong University, Qingyang, Gansu, 745000, China.)

###### Abstract

The two main strategies used in this study are the binary representation and
the decomposition of a natural number into many compound functions of odd
function and even function. The Collatz conjecture regarding odd even numbers
in number theory can be examined and discussed using them in this way: The
sequence created by the finite iterations of the Collatz function becomes the
ultimately periodic sequence if any natural number is the beginning value,
proving the conjecture that has been held for 85 years.

###### keywords:

binary representation; composite function; Collatz conjecture; ultimately
periodic sequence; chaos

00footnotetext: Received date: 2023-9-11.00footnotetext: Foundation item:
Educational technology innovation project of Gansu Province (No. 2022A-133)

##  1  Introduction

In the study of number theory, odd and even numbers are a fundamental pair of
ideas. Natural number sets come in two different varieties. Numerological
theory frequently examines the connections between various numbers. There are
numerous conjectures that attempt to generalize the law of different sorts of
natural numbers discovered in a restricted range to the entire infinite set of
natural numbers. This article will examine the famous Collatz conjecture,
which states that for each natural number $n$, if it is even, divide by 2, if
it is odd, multiply by 3, add 1, and so on, the result must finally reach 1.
It is also referred to as the $3n+1$ conjecture and was put forth in 1937 by
Lothar Collatz, also known as the $3n+1$ problem. The mathematician Paul Erdos
once said of this conjecture: ”Mathematics may not be ready for such
problems”[1,2].

The inconsistencies between the finite and the infinite, as well as the
relation between various kinds, present difficulties in the study of number
theory problems. We are talking about the connection between two different
mathematical ideas: the iterated sequence is a ultimately periodic sequence
whether the initial value is odd or even.

The finite and the infinite can be connected by the useful mathematical
construct known as a function, and the resulting outcomes will also be finite
and infinite. A special function that has particular significance in discrete
mathematics is the piecewise function. Compound functions and piecewise
functions combined is a highly clever mathematical trick. Particularly in
number theory, which is the most fundamental idea and mathematical expression,
the sequence of numbers is a close connection between functions and finite and
infinite. Numerous conjectures are obtained through restricted iteration of an
iterative algorithm, which is a widely used method in number theory, but
people typically lack the means to demonstrate the accuracy and reasoned
nature of conjectures. A fresh technique or new knowledge is frequently used
to support a hypothesis. For the Collatz conjecture, we can describe it as a
function:

$T(n)=\left\\{\begin{array}[]{cc}3n+1,&\text{if }n\text{ is odd number,}\\\
\frac{n}{2}&\text{if }n\text{ is even number.}\end{array}\right.$ (1.1)

The following sequence is obtained via the composite function (iteration):

$a=\\{n,T(n),T(T(n)),T(T(T(n))),\cdots\\}$$=\\{n,T(n),T^{2}(n),T^{3}(n),\cdots\\}$.

Consequently, the Collatz conjecture can be stated as follows: The sequence
$a$ always leads to the integer $1$, regardless of where you start with the
natural number n, namely $T^{m}(n)=1$. The series $a$ is an infinite sequence
of ultimately period [9,10]: the preperiod $\eta(n)$ varies with the initial
value $n$, but the ultimately period is always $\\{4,2,1\\}$.

##  2  Composition of odd and even functions and binary representation of
natural numbers

If a natural number can be divided by 2, it is said to be an even number;
otherwise, it is said to be odd number. The Peano’s Axiom states that 1 is the
smallest natural number. The set of natural numbers $N=\\{1,2,3,\cdots\\}$ may
be separated into odd and even sets, we will utilize the standard definition
of natural numbers in this work.

$\\{\textbf{natural number}\\}=\\{\textbf{odd number}\\}\bigcup\\{\textbf{even
number}\\}$.

In the set of natural numbers where 1 is the smallest odd number and 2 is the
smallest even number, we can use the expression $n=2k-1$ to indicate that it
is an odd, and the expression $n=2k$ to indicate that it is an even, where $k$
is any natural number.

We introduce two functions $O(x)=2x+1$ to express odd numbers greater than 1,
and $E(x)=2x$ to express even numbers.

Definition 1 A natural number $n$ is obtained by composition of the odd
function $O(x)=2x+1$ and the even function $E(x)=2x$ several times, namely
$n=f(f(\cdots f(1)))=f^{k}(1)$, the function $f$ is either odd function $O(x)$
or even function $E(x)$.

For example,
$7=2*3+1=2*(2*1+1+1)=O(O(1)),189=2*2*(2*4+1)=2*(2*(2*(2*1))+1)=E(O(E(E(1))))$.
In order to more clearly express the odd-even composition process of a natural
number, we use binary representation of a natural number $n$, for example:
$7=(111)_{2}$, $18=(10010)_{2}$.

Similarly, we can have a piecewise function

$f^{-1}(n)=\left\\{\begin{array}[]{cc}\frac{n+1}{2},&\text{if }n\text{ is odd
number,}\\\ \frac{n}{2},&\text{if }n\text{ is even number.}\end{array}\right.$
(2.1)

If $n=f(f(\cdots f(1)))=f^{k}(1)$, then the inverse function is $f^{-k}(n)=1$.

For generalization, we give the definitions of three natural numbers:

Definition 2(i) By applying the odd function $O(x)$ $m$ compositions, a
natural number, namely $O^{m}(1)=2^{m}-1=2^{m-1}+2^{m-2}+\cdots+2+1=(11\cdots
1)_{2}$, is obtained. such as
$3=(11)_{2},7=(111)_{2},15=(1111)_{2},31=(11111)_{2},63=(111111)_{2},\cdots$,
which we call it as pure odd number;

(ii) By applying the even function $E(x)$ $m$ compositions, a natural number,
namely $E^{m}(1)=2^{m}=(10\cdots 0)_{2}$, is obtained, such as,
$2=(10)_{2},4=(100)_{2},8=(1000)_{2},16=(10000)_{2},32=(100000)_{2},64=(1000000)_{2},\cdots$,
which we call it as pure even number;

(iii) The natural number obtained by the composition of odd function $O(x)$
and even function $E(x)$, we call it mixed number. Such as,
$18=(10010)_{2},28=(11100)_{2},67=(1000011)_{2},309=(100110101)_{2}$.

In particular, the natural numbers obtained by the finite alternately
composition of the odd function $O(x)$ and the even function $E(x)$, namely,
$[E(O(1))]^{m}=(101\cdots 101)_{2}$. Such as
$5=(101)_{2},21=(10101)_{2},85=(1010101)_{2},341=(101010101)_{2},1365=(10101010101)_{2},5461=(1010101010101)_{2},\cdots$,
which we call hard number.

Definition 3 The binary string of a natural number is a representation of its
odd-even composite function, where the 1 in the $i(i>0)$-bit from right to
left is the $i(i>0)$ sub-odd function, and 0 is the corresponding sub-even
function.

For a natural number $n$, if its binary string has $k$ bits, then the degree
of composite function is $k-1$. The binary string of a pure odd number is made
of all 1; and the binary string of a pure even number is all 0 aside from one
1 in left; the binary string of a mixed number is made of many 0 and 1. Fig.1
shows the decomposition of the composite function (inverse of the composite
function) of 60.

Figure 2.1:  Natural number 60 is obtained from 1 through the composition of
five even and odd functions, its binary string is $(111100)_{2}$.

In this way, we can classify the set of natural numbers in another way,

Property 4 The set of natural numbers can be divided into three different
sets:

$\ \textbf{\\{\text{natural number} \\}}=\\{\text{pure odd
number}\\}\bigcup\\{\text{pure even number}\\}\bigcup\\{\text{mixed
number}\\}$

The pure odd number is an odd number, the pure even number is an even number,
and mixed numbers can be either an odd number or an even number when compared
to the conventional classification. The last bit of the binary string, which
is either 0 or 1, indicates whether it is even or odd. The entire binary
string implicitly indicates one of the following three types: all bits are 0s
with the exception of one 1 in the left implicitly pure even, all bits are 1s
implicitly pure odd, and there are 0 and 1 in any number of bits of the binary
string implicitly mixed.

Example 1 (1)60, 97 are mixed numbers.

(2)64,1180591620717411303424 are pure even numbers.

(3)63,1180591620717411303423 are pure odd numbers.

$60=2*30=2*(2*15)=2*(2*(2*7+1)))=2*(2*(2*(2*(2*3)+1)+1)))=2*(2*(2*(2*(2*(2*1+1)+1)+1)))=E(E(O(O(O(1))))=(111100)_{2}$;

$97=2*48+1=2*(2*24)+1=2*(2*(2*12))+1=2*(2*(2*(2*6)))+1=2*(2*(2*(2*(2*3)))+1=2*(2*(2*(2*(2*(2*1+1))))+1=O(E(E(E(E(O(1))))))=(1100001)_{2}$.

$64=2^{6}=(1000000)_{2},1180591620717411303424=2^{70}=(10000\cdots 0)_{2}$ are
pure even numbers;

$63=2^{6}-1=(111111)_{2},1180591620717411303423=2^{70}-1=(11\cdots 1)_{2}$ are
pure odd numbers.

This composite function has special significance, for example,

Example 2 Let $[x]$ be $x$ the integer part, that is, $[x]=max\\{y\in Z:y\leq
x\\}$, $\\{x\\}$ be fractional part, namely $\\{x\\}=x-[x]$. Assuming
$g(x)=\frac{1}{[x]-\\{x\\}+1}$, is rational set $Q^{+}$ and rational set $Q$
can be expressed as respectively

$Q^{+}=\\{g(0),g^{2}(0),g^{3}(0),\cdots\\}$,

$Q=\\{0,g(0),-g(0),g^{2}(0),-g^{2}(0),g^{3}(0),-g^{3}(0),\cdots\\}$ .

See the references[5] for proof.

##  3  Tabular and algebraic expressions

For the Collatz conjecture, if expressed by the function $T(n)$, we find that
$T^{k}(n)=1$.

For the sake of discussion, we combine the compound function (iterative
relation) to get the reduced Collatz function

$RT(n)=\left\\{\begin{array}[]{cc}\frac{3n+1}{2^{m}},&\text{if }n\text{ is odd
number,}\\\ \frac{n}{2^{r}}&\text{if }n\text{ is even
number.}\end{array}\right.$ (3.1)

The result is an odd number. We introduce tabular form and algebraic
expression to express the reduced Collatz function $RT(n)$.

For example, for $n=67$ we use the formula (1.1) and iteration, get the
following table. Where the last column is its algebraic expression:.

67$\rightarrow$ | 202 $\rightarrow$ | 101 |  |  | $\frac{3^{8}}{2^{19}}\cdot 67$ | $\frac{3^{7}}{2^{19}}$
---|---|---|---|---|---|---
101$\rightarrow$ | 304$\rightarrow$ | 152$\rightarrow$ | 76$\rightarrow$ | 38$\rightarrow$ | 19 | $\frac{3^{6}}{2^{18}}$
19$\rightarrow$ | 58$\rightarrow$ | 29 |  |  |  | $\frac{3^{5}}{2^{14}}$
29$\rightarrow$ | 88$\rightarrow$ | 44$\rightarrow$ | 22$\rightarrow$ | 11 |  | $\frac{3^{4}}{2^{13}}$
11$\rightarrow$ | 34$\rightarrow$ | 17 |  |  |  | $\frac{3^{3}}{2^{10}}$
17$\rightarrow$ | 52$\rightarrow$ | 26$\rightarrow$ | 13 |  |  | $\frac{3^{2}}{2^{9}}$
13$\rightarrow$ | 40$\rightarrow$ | 20$\rightarrow$ | 10$\rightarrow$ | 5 |  | $\frac{3}{2^{7}}$
5$\rightarrow$ | 16$\rightarrow$ | 8$\rightarrow$ | 4$\rightarrow$ | 2$\rightarrow$ | 1 | $\frac{1}{2^{4}}$
1 |  |  |  |  |  |

Representing the numbers in the table by its binary string to get

1000011$\rightarrow$ | 11001010$\rightarrow$ | 1100101 |  |  | $\frac{3^{8}}{2^{19}}\cdot 67$ | $\frac{3^{7}}{2^{19}}$
---|---|---|---|---|---|---
1100101$\rightarrow$ | 100110000$\rightarrow$ | 10011000$\rightarrow$ | 1001100$\rightarrow$ | 100110$\rightarrow$ | 10011 | $\frac{3^{6}}{2^{18}}$
10011$\rightarrow$ | 111010$\rightarrow$ | 11101 |  |  |  | $\frac{3^{5}}{2^{14}}$
11101$\rightarrow$ | 1011000$\rightarrow$ | 101100$\rightarrow$ | 10110$\rightarrow$ | 1011 |  | $\frac{3^{4}}{2^{13}}$
1011$\rightarrow$ | 100010$\rightarrow$ | 10001 |  |  |  | $\frac{3^{3}}{2^{10}}$
10001$\rightarrow$ | 110100$\rightarrow$ | 11010$\rightarrow$ | 1101 |  |  | $\frac{3^{2}}{2^{9}}$
1101$\rightarrow$ | 101000$\rightarrow$ | 10100$\rightarrow$ | 1010$\rightarrow$ | 101 |  | $\frac{3}{2^{7}}$
101$\rightarrow$ | 10000$\rightarrow$ | 1000$\rightarrow$ | 100$\rightarrow$ | 10$\rightarrow$ | 1 | $\frac{1}{2^{4}}$
1 |  |  |  |  |  |

The above table is simplified with the help of the reduced Collatz function,
and the simplified binary string table obtained is as follows, and we will use
this form as the default table in the rest of this article.

1000011$\rightarrow$ | 11001010$\rightarrow$ | 1100101 | $\frac{3^{8}}{2^{19}}\cdot 67$ | $\frac{3^{7}}{2^{19}}$
---|---|---|---|---
1100101$\rightarrow$ | 100110000$\rightarrow$ | 10011 |  | $\frac{3^{6}}{2^{18}}$
10011$\rightarrow$ | 111010$\rightarrow$ | 11101 |  | $\frac{3^{5}}{2^{14}}$
11101$\rightarrow$ | 1011000$\rightarrow$ | 1011 |  | $\frac{3^{4}}{2^{13}}$
1011$\rightarrow$ | 100010$\rightarrow$ | 10001 |  | $\frac{3^{3}}{2^{10}}$
10001$\rightarrow$ | 110100$\rightarrow$ | 1101 |  | $\frac{3^{2}}{2^{9}}$
1101$\rightarrow$ | 101000$\rightarrow$ | 101 |  | $\frac{3}{2^{7}}$
101$\rightarrow$ | 10000$\rightarrow$ | 1 |  | $\frac{1}{2^{4}}$
1 |  |  |  |

In the future, we will use this table as a research tool to write the
algebraic expression of the last column in the table:

$T^{27}(67)=T(8,19,67)=\frac{1}{2^{4}}+\frac{3}{2^{7}}+\frac{3^{2}}{2^{9}}+\frac{3^{3}}{2^{10}}+\frac{3^{4}}{2^{13}}+\frac{3^{5}}{2^{14}}+\frac{3^{6}}{2^{18}}+\frac{3^{7}}{2^{19}}+\frac{3^{8}}{2^{19}}\cdot
67=1$

In general, the algebraic expressions are obtained: Starting from the last row
of the table and going up to the binary corresponding to the initial value n
of the first row, the numerator is $3^{k},k=0,1,2,3,\cdots$, The denominator
is $2^{r_{k}}$, $r_{0}=m_{0},r_{k}=r_{k-1}+m_{k},k=1,2,3,\cdots$, $m_{k}$ here
for the before k lines at the end of the second column of binary string number
0. The details are expressed in the last line of the corresponding table, and
we write out the algebraic expression of them as follows:

$\displaystyle T^{106}(31)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{4}}+\frac{3}{2^{9}}+\frac{3^{2}}{2^{10}}+\frac{3^{3}}{2^{11}}+\frac{3^{4}}{2^{14}}+\frac{3^{5}}{2^{18}}+\frac{3^{6}}{2^{20}}+\frac{3^{7}}{2^{22}}+\frac{3^{8}}{2^{26}}+\frac{3^{9}}{2^{27}}$
$\displaystyle+\frac{3^{10}}{2^{28}}+\frac{3^{11}}{2^{29}}+\frac{3^{12}}{2^{32}}+\frac{3^{13}}{2^{33}}+\frac{3^{14}}{2^{34}}+\frac{3^{15}}{2^{35}}+\frac{3^{16}}{2^{36}}+\frac{3^{17}}{2^{37}}+\frac{3^{18}}{2^{39}}$
$\displaystyle+\frac{3^{19}}{2^{40}}+\frac{3^{20}}{2^{42}}+\frac{3^{21}}{2^{43}}+\frac{3^{22}}{2^{44}}+\frac{3^{23}}{2^{47}}+\frac{3^{24}}{2^{49}}+\frac{3^{25}}{2^{50}}+\frac{3^{26}}{2^{51}}+\frac{3^{27}}{2^{52}}$
$\displaystyle+\frac{3^{28}}{2^{54}}+\frac{3^{29}}{2^{55}}+\frac{3^{30}}{2^{56}}+\frac{3^{31}}{2^{58}}+\frac{3^{32}}{2^{59}}+\frac{3^{33}}{2^{61}}+\frac{3^{34}}{2^{63}}+\frac{3^{35}}{2^{64}}+\frac{3^{36}}{2^{65}}$
$\displaystyle+\frac{3^{37}}{2^{66}}+\frac{3^{38}}{2^{67}}+\frac{3^{39}}{2^{67}}\cdot
31$ $\displaystyle=$ $\displaystyle 1$

$\displaystyle T^{107}(63)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{4}}+\frac{3}{2^{9}}+\frac{3^{2}}{2^{10}}+\frac{3^{3}}{2^{11}}+\frac{3^{4}}{2^{14}}+\frac{3^{5}}{2^{18}}+\frac{3^{6}}{2^{20}}+\frac{3^{7}}{2^{22}}+\frac{3^{8}}{2^{26}}+\frac{3^{9}}{2^{27}}$
$\displaystyle+\frac{3^{10}}{2^{28}}+\frac{3^{11}}{2^{29}}+\frac{3^{12}}{2^{32}}+\frac{3^{13}}{2^{33}}+\frac{3^{14}}{2^{34}}+\frac{3^{15}}{2^{35}}+\frac{3^{16}}{2^{36}}+\frac{3^{17}}{2^{37}}+\frac{3^{18}}{2^{39}}$
$\displaystyle+\frac{3^{19}}{2^{40}}+\frac{3^{20}}{2^{42}}+\frac{3^{21}}{2^{43}}+\frac{3^{22}}{2^{44}}+\frac{3^{23}}{2^{47}}+\frac{3^{24}}{2^{49}}+\frac{3^{25}}{2^{50}}+\frac{3^{26}}{2^{51}}+\frac{3^{27}}{2^{52}}$
$\displaystyle+\frac{3^{28}}{2^{54}}+\frac{3^{29}}{2^{55}}+\frac{3^{30}}{2^{56}}+\frac{3^{31}}{2^{58}}+\frac{3^{32}}{2^{59}}+\frac{3^{33}}{2^{63}}+\frac{3^{34}}{2^{64}}+\frac{3^{35}}{2^{65}}+\frac{3^{36}}{2^{66}}$
$\displaystyle+\frac{3^{37}}{2^{67}}+\frac{3^{38}}{2^{68}}+\frac{3^{39}}{2^{68}}\cdot
63$ $\displaystyle=$ $\displaystyle 1$

$\displaystyle T^{118}(97)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{4}}+\frac{3}{2^{9}}+\frac{3^{2}}{2^{10}}+\frac{3^{3}}{2^{11}}+\frac{3^{4}}{2^{14}}+\frac{3^{5}}{2^{18}}+\frac{3^{6}}{2^{20}}+\frac{3^{7}}{2^{22}}+\frac{3^{8}}{2^{26}}+\frac{3^{9}}{2^{27}}$
$\displaystyle+\frac{3^{10}}{2^{28}}+\frac{3^{11}}{2^{29}}+\frac{3^{12}}{2^{32}}+\frac{3^{13}}{2^{33}}+\frac{3^{14}}{2^{34}}+\frac{3^{15}}{2^{35}}+\frac{3^{16}}{2^{36}}+\frac{3^{17}}{2^{37}}+\frac{3^{18}}{2^{39}}$
$\displaystyle+\frac{3^{19}}{2^{40}}+\frac{3^{20}}{2^{42}}+\frac{3^{21}}{2^{43}}+\frac{3^{22}}{2^{44}}+\frac{3^{23}}{2^{47}}+\frac{3^{24}}{2^{49}}+\frac{3^{25}}{2^{50}}+\frac{3^{26}}{2^{51}}+\frac{3^{27}}{2^{52}}$
$\displaystyle+\frac{3^{28}}{2^{54}}+\frac{3^{29}}{2^{55}}+\frac{3^{30}}{2^{56}}+\frac{3^{31}}{2^{58}}+\frac{3^{32}}{2^{59}}+\frac{3^{33}}{2^{61}}+\frac{3^{34}}{2^{63}}+\frac{3^{35}}{2^{64}}+\frac{3^{36}}{2^{65}}$
$\displaystyle+\frac{3^{37}}{2^{66}}+\frac{3^{38}}{2^{69}}+\frac{3^{39}}{2^{70}}+\frac{3^{40}}{2^{71}}+\frac{3^{41}}{2^{73}}+\frac{3^{42}}{2^{75}}+\frac{3^{43}}{2^{75}}\cdot
97$ $\displaystyle=$ $\displaystyle 1$

$\displaystyle T^{91}(10027)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{4}}+\frac{3}{2^{9}}+\frac{3^{2}}{2^{10}}+\frac{3^{3}}{2^{11}}+\frac{3^{4}}{2^{14}}+\frac{3^{5}}{2^{18}}+\frac{3^{6}}{2^{20}}+\frac{3^{7}}{2^{22}}+\frac{3^{8}}{2^{26}}+\frac{3^{9}}{2^{27}}$
$\displaystyle+\frac{3^{10}}{2^{28}}+\frac{3^{11}}{2^{29}}+\frac{3^{12}}{2^{32}}+\frac{3^{13}}{2^{33}}+\frac{3^{14}}{2^{34}}+\frac{3^{15}}{2^{35}}+\frac{3^{16}}{2^{36}}+\frac{3^{17}}{2^{37}}+\frac{3^{18}}{2^{39}}$
$\displaystyle+\frac{3^{19}}{2^{42}}+\frac{3^{20}}{2^{43}}+\frac{3^{21}}{2^{44}}+\frac{3^{22}}{2^{45}}+\frac{3^{23}}{2^{48}}+\frac{3^{24}}{2^{49}}+\frac{3^{25}}{2^{53}}+\frac{3^{26}}{2^{56}}+\frac{3^{27}}{2^{58}}$
$\displaystyle+\frac{3^{28}}{2^{60}}+\frac{3^{29}}{2^{61}}+\frac{3^{30}}{2^{61}}\cdot
10027$ $\displaystyle=$ $\displaystyle 1$

##  4 The characteristics of sequence of binary string of the Collatz function
iteration

The Collatz function is expressed in binary form as

$T(n)=\left\\{\begin{array}[]{cc}(11)_{2}\cdot(1\times\cdots\times
1)_{2}+1=(1\times\times\times 10\cdots 0)_{2},&\text{if }n\text{ is odd
number,}\\\ \frac{(1\times\cdots\times 10\cdots
00)_{2}}{(10)_{2}}=(1\times\cdots\times 10\cdots 0)_{2},&\text{if }n\text{ is
even number.}\end{array}\right.$ (4.1)

The characteristics of the first and last parts are represented by the figure
2.

Figure 4.1: Binary representation of the Collatz function, add up at most two
digits from the left side of the binary string,subtract at least one digit
from the right side of the binary string.

By observing the binary strings of iterated Collatz functions with initial
values of 31,63 (see the tabulation procedure in reference [7]) and 97,10027
as the follows, we get some laws of pure even, pure odd and mixed numbers with
initial values. In the following, we will analyze and discuss the table of the
iterative process of the Collatz functions from the two dimensions of row and
column of the tables.

|  |  | $\frac{3^{43}}{2^{75}}\cdot 97$
---|---|---|---
$97=(1100001)_{2}$ $\rightarrow$ |  $(100100100)_{2}$ $\rightarrow$ | $(1001001)_{2}$ | $\frac{3^{42}}{2^{75}}$
$73=(1001001)_{2}$ $\rightarrow$ |  $(11011100)_{2}$ $\rightarrow$ | $(110111)_{2}$ | $\frac{3^{41}}{2^{73}}$
$55=(110111)_{2}$ $\rightarrow$ |  $(10100110)_{2}$ $\rightarrow$ | $(1010011)_{2}$ | $\frac{3^{40}}{2^{71}}$
$83=(1010011)_{2}$ $\rightarrow$ |  $(11111010)_{2}$$\rightarrow$ | $(1111101)_{2}$ | $\frac{3^{39}}{2^{70}}$
$125=(1111101)_{2}$$\rightarrow$ |  $(101111000)_{2}$$\rightarrow$ | $(101111)_{2}$ | $\frac{3^{38}}{2^{69}}$
$47=(101111)_{2}$ $\rightarrow$ |  $(10001110)_{2}$ $\rightarrow$ | $(1000111)_{2}$ | $\frac{3^{37}}{2^{66}}$
$71=(1000111)_{2}$ $\rightarrow$ |  $(11010110)_{2}$$\rightarrow$ | $(1101011)_{2}$ | $\frac{3^{36}}{2^{65}}$
$107=(1101011)_{2}$$\rightarrow$ |  $(101000010)_{2}$$\rightarrow$ | $(10100001)_{2}$ | $\frac{3^{35}}{2^{64}}$
$161=(10100001)_{2}$$\rightarrow$ |  $(111100100)_{2}$$\rightarrow$ | $(1111001)_{2}$ | $\frac{3^{34}}{2^{63}}$
$121=(1111001)_{2}$$\rightarrow$ |  $(101101100)_{2}$$\rightarrow$ | $(1011011)_{2}$ | $\frac{3^{33}}{2^{61}}$
$91=(1011011)_{2}$$\rightarrow$ |  $(100010010)_{2}$$\rightarrow$ | $(10001001)_{2}$ | $\frac{3^{32}}{2^{59}}$
$137=(10001001)_{2}$$\rightarrow$ |  $(110011100)_{2}$$\rightarrow$ | $(1100111)_{2}$ | $\frac{3^{31}}{2^{58}}$
$103=(1100111)_{2}$$\rightarrow$ |  $(100110110)_{2}$$\rightarrow$ | $(10011011)_{2}$ | $\frac{3^{30}}{2^{56}}$
$155=(10011011)_{2}$$\rightarrow$ |  $(111010010)_{2}$$\rightarrow$ | $(11101001)_{2}$ | $\frac{3^{29}}{2^{55}}$
$233=(11101001)_{2}$$\rightarrow$ |  $(1010111100)_{2}$$\rightarrow$ | $(10101111)_{2}$ | $\frac{3^{28}}{2^{54}}$
$175=(10101111)_{2}$$\rightarrow$ |  $(1000001110)_{2}$$\rightarrow$ | $(100000111)_{2}$ | $\frac{3^{27}}{2^{52}}$
$263=(100000111)_{2}$$\rightarrow$ |  $(1100010110)_{2}$$\rightarrow$ | $(110001011)_{2}$ | $\frac{3^{26}}{2^{51}}$
$395=(110001011)_{2}$$\rightarrow$ |  $(10010100010)_{2}$$\rightarrow$ | $(1001010001)_{2}$ | $\frac{3^{25}}{2^{50}}$
$593=(1001010001)_{2}$$\rightarrow$ |  $(11011110100)_{2}$$\rightarrow$ | $(110111101)_{2}$ | $\frac{3^{24}}{2^{49}}$
$445=(110111101)_{2}$$\rightarrow$ |  $(10100111000)_{2}$$\rightarrow$ | $(10100111)_{2}$ | $\frac{3^{23}}{2^{47}}$
$167=(10100111)_{2}$$\rightarrow$ |  $(111110110)_{2}$$\rightarrow$ | $(11111011)_{2}$ | $\frac{3^{22}}{2^{44}}$
$251=(11111011)_{2}$$\rightarrow$ |  $(1011110010)_{2}$$\rightarrow$ | $(101111001)_{2}$ | $\frac{3^{21}}{2^{43}}$
$377=(101111001)_{2}$$\rightarrow$ |  $(10001101100)_{2}$$\rightarrow$ | $(100011011)_{2}$ | $\frac{3^{20}}{2^{42}}$
$283=(100011011)_{2}$$\rightarrow$ |  $(1101010010)_{2}$$\rightarrow$ | $(110101001)_{2}$ | $\frac{3^{19}}{2^{40}}$
$425=(110101001)_{2}$$\rightarrow$ |  $(10011111100)_{2}$$\rightarrow$ | $(100111111)_{2}$ | $\frac{3^{18}}{2^{39}}$
$319=(100111111)_{2}$$\rightarrow$ |  $(1110111110)_{2}$$\rightarrow$ | $(111011111)_{2}$ | $\frac{3^{17}}{2^{37}}$
$479=(111011111)_{2}$$\rightarrow$ |  $(10110011110)_{2}$$\rightarrow$ | $(1011001111)_{2}$ | $\frac{3^{16}}{2^{36}}$
$719=(1011001111)_{2}$$\rightarrow$ |  $(100001101110)_{2}$$\rightarrow$ | $(10000110111)_{2}$ | $\frac{3^{15}}{2^{35}}$
$1079=(10000110111)_{2}$$\rightarrow$ |  $(110010100110)_{2}$$\rightarrow$ | $(11001010011)_{2}$ | $\frac{3^{14}}{2^{34}}$
$1619=(11001010011)_{2}$$\rightarrow$ |  $(1001011111010)_{2}$$\rightarrow$ | $(100101111101)_{2}$ | $\frac{3^{13}}{2^{33}}$
$2429=(100101111101)_{2}$$\rightarrow$ |  $(1110001111000)_{2}$$\rightarrow$ | $(1110001111)_{2}$ | $\frac{3^{12}}{2^{32}}$
$911=(1110001111)_{2}$$\rightarrow$ |  $(101010101110)_{2}$$\rightarrow$ | $(10101010111)_{2}$ | $\frac{3^{11}}{2^{29}}$
$1367=(10101010111)_{2}$ | $(1000000000110)_{2}$ | $(100000000011)_{2}$ | $\frac{3^{10}}{2^{28}}$
$2051=(100000000011)_{2}$$\rightarrow$ |  $(1100000001010)_{2}$$\rightarrow$ | $(110000000101)_{2}$ | $\frac{3^{9}}{2^{27}}$
$3077=(110000000101)_{2}$$\rightarrow$ |  $(10010000010000)_{2}$$\rightarrow$ | $(1001000001)_{2}$ | $\frac{3^{8}}{2^{26}}$
$577=(1001000001)_{2}$$\rightarrow$ |  $(11011000100)_{2}$$\rightarrow$ | $(110110001)_{2}$ | $\frac{3^{7}}{2^{22}}$
$433=(110110001)_{2}$$\rightarrow$ |  $(10100010100)_{2}$$\rightarrow$ | $(101000101)_{2}$ | $\frac{3^{6}}{2^{20}}$
$325=(101000101)_{2}$$\rightarrow$ |  $(1111010000)_{2}$$\rightarrow$ | $(111101)_{2}$ | $\frac{3^{5}}{2^{18}}$
$61=(111101)_{2}$$\rightarrow$ |  $(10111000)_{2}$$\rightarrow$ | $(10111)_{2}$ | $\frac{3^{4}}{2^{14}}$
$23=(10111)_{2}$$\rightarrow$ |  $(1000110)_{2}$$\rightarrow$ | $(100011)_{2}$ | $\frac{3^{3}}{2^{11}}$
$35=(100011)_{2}$$\rightarrow$ |  $(1101010)_{2}$$\rightarrow$ | $(110101)_{2}$ | $\frac{3^{2}}{2^{10}}$
$53=(110101)_{2}$ | $(10100000)_{2}$ | $(101)_{2}$ | $\frac{3}{2^{9}}$
$5=(101)_{2}$ | $(10000)_{2}$ | $(1)_{2}$ | $\frac{1}{2^{4}}$

|  |  |  $\frac{3^{30}}{2^{61}}\cdot$10027
---|---|---|---
10027=(10011100101011)${}_{2}\rightarrow$ | (111010110000010)${}_{2}\rightarrow$ | (11101011000001)2 | $\frac{3^{29}}{2^{61}}$
15041=(11101011000001)${}_{2}\rightarrow$ | (1011000001000100)${}_{2}\rightarrow$ | (10110000010001)2 | $\frac{3^{28}}{2^{60}}$
11281=(10110000010001)${}_{2}\rightarrow$ | (1000010000110100)${}_{2}\rightarrow$ | (10000100001101)2 | $\frac{3^{27}}{2^{58}}$
8461=(10000100001101)${}_{2}\rightarrow$ | (110001100101000)${}_{2}\rightarrow$ | (110001100101)2 | $\frac{3^{26}}{2^{56}}$
3173=(110001100101)${}_{2}\rightarrow$ | (10010100110000)${}_{2}\rightarrow$ | (1001010011)2 | $\frac{3^{25}}{2^{53}}$
595=(1001010011)${}_{2}\rightarrow$ | (11011111010)${}_{2}\rightarrow$ | (1101111101)2 | $\frac{3^{24}}{2^{49}}$
893=(1101111101)${}_{2}\rightarrow$ | (101001111000)${}_{2}\rightarrow$ | (101001111)2 | $\frac{3^{23}}{2^{48}}$
335=(101001111)${}_{2}\rightarrow$ | (1111101110)${}_{2}\rightarrow$ | (111110111)2 | $\frac{3^{22}}{2^{45}}$
503=(111110111)${}_{2}\rightarrow$ | (10111100110)${}_{2}\rightarrow$ | (1011110011)2 | $\frac{3^{21}}{2^{44}}$
755=(1011110011)${}_{2}\rightarrow$ | (100011011010)${}_{2}\rightarrow$ | (10001101101)2 | $\frac{3^{20}}{2^{43}}$
1133=(10001101101)${}_{2}\rightarrow$ | (110101001000)${}_{2}\rightarrow$ | (110101001)2 | $\frac{3^{19}}{2^{42}}$
425=(110101001)${}_{2}\rightarrow$ | (10011111100)${}_{2}\rightarrow$ | (100111111)2 | $\frac{3^{18}}{2^{39}}$
319=(100111111)${}_{2}\rightarrow$ | (1110111110)${}_{2}\rightarrow$ | (111011111)2 | $\frac{3^{17}}{2^{37}}$
479=(111011111)${}_{2}\rightarrow$ | (10110011110)${}_{2}\rightarrow$ | (1011001111)2 | $\frac{3^{16}}{2^{36}}$
719=(1011001111)${}_{2}\rightarrow$ | (100001101110)${}_{2}\rightarrow$ | (10000110111)2 | $\frac{3^{15}}{2^{35}}$
1079=(10000110111)${}_{2}\rightarrow$ | (110010100110)${}_{2}\rightarrow$ | (11001010011)2 | $\frac{3^{14}}{2^{34}}$
1619=(11001010011)${}_{2}\rightarrow$ | (1001011111010)${}_{2}\rightarrow$ | (100101111101)2 | $\frac{3^{13}}{2^{33}}$
2429=(100101111101)${}_{2}\rightarrow$ | (1110001111000)${}_{2}\rightarrow$ | (1110001111)2 | $\frac{3^{12}}{2^{32}}$
911=(1110001111)${}_{2}\rightarrow$ | (101010101110)${}_{2}\rightarrow$ | (10101010111)2 | $\frac{3^{11}}{2^{29}}$
1367=(10101010111)${}_{2}\rightarrow$ | (1000000000110)${}_{2}\rightarrow$ | (100000000011)2 | $\frac{3^{10}}{2^{28}}$
2051=(100000000011)${}_{2}\rightarrow$ | (1100000001010)${}_{2}\rightarrow$ | (110000000101)2 | $\frac{3^{9}}{2^{27}}$
3077=(110000000101)${}_{2}\rightarrow$ | (10010000010000)${}_{2}\rightarrow$ | (1001000001)2 | $\frac{3^{8}}{2^{26}}$
577=(1001000001)${}_{2}\rightarrow$ | (11011000100)${}_{2}\rightarrow$ | (110110001)2 | $\frac{3^{7}}{2^{22}}$
433=(110110001)${}_{2}\rightarrow$ | (10100010100)${}_{2}\rightarrow$ | (101000101)2 | $\frac{3^{6}}{2^{20}}$
325=(101000101)${}_{2}\rightarrow$ | (1111010000)${}_{2}\rightarrow$ | (111101)2 | $\frac{3^{5}}{2^{18}}$
61=(111101)${}_{2}\rightarrow$ | (10111000)${}_{2}\rightarrow$ | (10111)2 | $\frac{3^{4}}{2^{14}}$
23=(10111)${}_{2}\rightarrow$ | (1000110)${}_{2}\rightarrow$ | (100011)2 | $\frac{3^{3}}{2^{11}}$
35=(100011)${}_{2}\rightarrow$ | (1101010)${}_{2}\rightarrow$ | (110101)2 | $\frac{3^{2}}{2^{10}}$
53=(110101)${}_{2}\rightarrow$ | (10100000)${}_{2}\rightarrow$ | (101)2 | $\frac{3}{2^{9}}$
5=(101)${}_{2}\rightarrow$ | (10000)${}_{2}\rightarrow$ |  (1)2 | $\frac{1}{2^{4}}$

###  4.1 Row characteristic

1) In each row, the first column is always odd (empty when the initial value
is even), and the last bit of its binary string must be 1,

The second column must be even, its binary string must end with at least one
0, the number of subsequent even numbers must be as many as the number of
zeros at the end of the second column’s binary string, and the number of zeros
at the end of each even number to the right is one less than the previous one,
until all zeros are deleted to become the last odd number in the row.

The last column must be odd, and the last bit of its binary string must be 1.

2) When there are only three numbers in a row, that is, only one even number,
the last odd number must be greater than the first odd number (the first
column); When there are more than three numbers in a row, that is, more than
two even numbers, the last odd number must be smaller than the first odd
number.

3) The preceding binary string is identical from the second column to the last
column in one line, except for the all 0 at the end.

###  4.2  Column characteristic

From top to bottom, the binary string in the first column of two adjacent rows
has the following two characteristics:

1) If the number of bits of 1 in the last substring of the previous row is
greater than 1, the number of bits of 1 in the last substring of the next row
is reduced by one, and the momentum of this reduction of one bit remains
unchanged until it finally becomes only one; The corresponding number is
greater than the number in the previous row;

2) If the last substring of the binary string in a row contains only one bit
of 1, then the last substring of the binary string in the next row contains
either one bit of 1 or many bits of 1, and the corresponding number is smaller
than the number in the previous row;

3) Within each line, the number of bits in the first binary string is
increased by 1 bit with the number in the second binary string, but for hard
numbers, the number is increased by 2 bits. The reduction is at least 1 bit,
and when it is a hard number of $m$ bits, the number is reduced by $m+1$.

###  4.3 Comprehensive characteristic

1) The substring at the end of the binary (right) is observed that when the
substring is 1, the units digit of the corresponding decimal number can be any
one of $1,3,5,7,9$. For instance, $161=(10100001)_{2}$, $433=(110110001)_{2}$,
$325=(101000101)_{2}$, $577=(1001000001)_{2}$, $2429=(100101111101)_{2}$.

2) The number ending in decimal is 9 and the corresponding binary number can
end in any digits of 1, for instance: $319=(100111111)_{2}$,
$479=(111011111)_{2}$,$719=(1011001111)_{2}$, $1079=(10000110111)_{2}$,
$1619=(11001010011)_{2}$, $2429=(100101111101)_{2}$.

3) When the last substring of binary is reduced by one bit from many, the
corresponding decimal number’s units are always reciprocated within the three
groups of numbers: $1,7$, or $3,5$, and unit bit is always $9$.

###  4.4 Discussion according three sets

The following is a discussion of the natural numbers according to our
classification, that is, when the initial value of the iteration process of
the Collatz function is pure even, pure odd and mixed numbers, it always
reaches the smallest natural number 1, thus proving the establishment of
Collatz conjecture.

(1)For a pure even $n=2^{k}=(10\cdots 0)_{2}$, it requires only an iteration
of the $k$ times Collatz function to reach the smallest natural number 1, that
is, the conjecture holds.

(2)For a special class of mixed numbers, if $a_{k}$ in the iteration sequence
$a$ of the Collatz function is a special class of mixed numbers - the hard
number $\frac{4^{k}-1}{3}=(101\cdots 101)_{2}$, then its sequent iteration
result is

$a_{k}=\frac{4^{k}-1}{3}=(101\cdots 101\cdots
101)_{2},a_{k+1}=4^{k}=2^{2k}=(10\cdots 0)_{2},\cdots,a_{3k+1}=1.$

This means that the Collatz conjecture is valid at this point. For ordinary
mixed numbers, this is a complicated process, which is carried out in
conjunction with the discussion of pure odd numbers in follows.

(3) For a pure odd number $n=2^{k}-1=(11\cdots 1)_{2}$, if the Collatz
function is iterated in binary form, that is, the second column in the
preceding table becomes a mixed number, which can be observed by several
examples. Research papers on this topic are [6,7,8]. By checking the change
law of the last substring in the binary string and the change law of the total
length of the binary string, we show that the Collatz function always reaches
the minimum natural number $1$ in the finite iteration value, thus proving the
correctness of Collatz conjecture.

In view of the general, a pure odd $n=2^{r}-1$, there is a mathematical
formula: $2^{r}-1=2^{r-1}+2^{r-2}+2^{r-3}+\cdots+2+1$. Two numbers $r$ and $m$
in the following tables are the values we verify for the Collatz conjecture by
iterating $T^{m}(2^{r}-1)=1$, $r=1,2,3,\cdots,203$ in the Maple program below:

restart;

$i:=2^{r}-1;$

$convert(i,binary,decimal);$

$jj:=0;$

$while~{}~{}i\neq 1~{}~{}do$

$i:=piecewise(irem(i,2)=0,(1/2)\dot{i},3\dot{i}+1);$

$convert(i,binary,decimal);$

$jj:=jj+1$

$enddo$

By verifying the first $203$ pure odd number $2^{r}-1$, we can get the
following rule:

1) Odd number $r$ greater than 1, the corresponding test Collatz function
value has the following relation:

if $T^{m}(2^{r}-1)=1$, then $T^{m+1}(2^{r+1}-1)=1$.

2) $T^{856}(2^{57}-1)=T^{856}(2^{65}-1)=1$

3) $T^{857}(2^{58}-1)=T^{857}(2^{66}-1)=1$

4) $T^{1451}(2^{86}-1)=T^{1451}(2^{117}-1)=1$

5)$T^{1455}(2^{90}-1)=T^{1455}(2^{121}-1)=1$

6)$T^{1456}(2^{91}-1)=T^{1456}(2^{122}-1)=1$

7)$T^{1457}(2^{92}-1)=T^{1457}(2^{123}-1)=1$

8)$T^{1458}(2^{93}-1)=T^{1458}(2^{124}-1)=1$

r | m | r | m | r | m | r | m | r | m | r | m | r | m
---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 0 | 30 | 449 | 59 | 858 | 88 | 1360 | 117 | 1451 | 146 | 2010 | 175 | 2039
2 | 7 | 31 | 450 | 60 | 859 | 89 | 1454 | 118 | 1452 | 147 | 2011 | 176 | 2040
3 | 16 | 32 | 451 | 61 | 860 | 90 | 1455 | 119 | 1484 | 148 | 2012 | 177 | 2041
4 | 17 | 33 | 527 | 62 | 861 | 91 | 1456 | 120 | 1485 | 149 | 2013 | 178 | 2042
5 | 106 | 34 | 528 | 63 | 862 | 92 | 1457 | 121 | 1455 | 150 | 2014 | 179 | 2061
6 | 107 | 35 | 529 | 64 | 863 | 93 | 1458 | 122 | 1456 | 151 | 2015 | 180 | 2062
7 | 46 | 36 | 530 | 65 | 856 | 94 | 1459 | 123 | 1457 | 152 | 2016 | 181 | 2063
8 | 47 | 37 | 531 | 66 | 857 | 95 | 1460 | 124 | 1458 | 153 | 2017 | 182 | 2064
9 | 61 | 38 | 532 | 67 | 729 | 96 | 1461 | 125 | 1658 | 154 | 2018 | 183 | 2047
10 | 62 | 39 | 533 | 68 | 730 | 97 | 1462 | 126 | 1659 | 155 | 2019 | 184 | 2048
11 | 156 | 40 | 534 | 69 | 930 | 98 | 1463 | 127 | 1660 | 156 | 2020 | 185 | 2049
12 | 157 | 41 | 535 | 70 | 931 | 99 | 1464 | 128 | 1661 | 157 | 2021 | 186 | 2050
13 | 158 | 42 | 536 | 71 | 932 | 100 | 1465 | 129 | 1662 | 158 | 2022 | 187 | 2069
14 | 159 | 43 | 586 | 72 | 933 | 101 | 1466 | 130 | 1663 | 159 | 2023 | 188 | 2070
15 | 129 | 44 | 587 | 73 | 934 | 102 | 1467 | 131 | 1995 | 160 | 2024 | 189 | 2071
16 | 130 | 45 | 588 | 74 | 935 | 103 | 1437 | 132 | 1996 | 161 | 2025 | 190 | 2072
17 | 224 | 46 | 589 | 75 | 1073 | 104 | 1438 | 133 | 1604 | 162 | 2026 | 191 | 2073
18 | 225 | 47 | 590 | 76 | 1074 | 105 | 1439 | 134 | 1605 | 163 | 2027 | 192 | 2074
19 | 177 | 48 | 591 | 77 | 938 | 106 | 1440 | 135 | 1606 | 164 | 2028 | 193 | 2972
20 | 178 | 49 | 592 | 78 | 939 | 107 | 1441 | 136 | 1607 | 165 | 2029 | 194 | 2973
21 | 303 | 50 | 593 | 79 | 940 | 108 | 1442 | 137 | 1608 | 166 | 2030 | 195 | 2974
22 | 304 | 51 | 594 | 80 | 941 | 109 | 1474 | 138 | 1609 | 167 | 2031 | 196 | 2975
23 | 473 | 52 | 595 | 81 | 1446 | 110 | 1475 | 139 | 2003 | 168 | 2032 | 197 | 2728
24 | 474 | 53 | 852 | 82 | 1447 | 111 | 1476 | 140 | 2004 | 169 | 2033 | 198 | 2729
25 | 444 | 54 | 853 | 83 | 1448 | 112 | 1477 | 141 | 1961 | 170 | 2034 | 199 | 2730
26 | 445 | 55 | 598 | 84 | 1449 | 113 | 1646 | 142 | 1962 | 171 | 2035 | 200 | 2731
27 | 384 | 56 | 599 | 85 | 1450 | 114 | 1647 | 143 | 2007 | 172 | 2036 | 201 | 2980
28 | 385 | 57 | 856 | 86 | 1451 | 115 | 1648 | 144 | 2008 | 173 | 2037 | 202 | 2981
29 | 448 | 58 | 857 | 87 | 1359 | 116 | 1649 | 145 | 2009 | 174 | 2038 | 203 | 2085

##  5 Comparison of two piecewise functions and the proof of Collatz
conjecture

Comparing the Collatz function $T(x)$ and the function $f^{-1}(x)$, if their
domain is defined as the set of natural numbers, we find that they have the
following relation:

1) The function $f^{-1}(x)$ is strictly monotonically decreasing,

2) When $x$ is purely even, the function $T(x)$ is only one case of the
functions, is strictly monotonically decreasing;

3) When $x$ is a pure or mixed odd number, the function $T(x)$ is wavy, which
is increasing, followed by one or more decreasing processes, that is,
”increase – decrease – increase”, or ”increase – decrease $\cdots$ decrease –
increase”. For example, Fig. 3 and Fig. 4 are the plots of the iterated
sequence of Collatz functions with initial values of pure odd
$255=2^{8}-1=(11111111)_{2}$ and mixed odd number $97=(1100001)_{2}$,
respectively.

Figure 5.1: Point plot of a sequence of 47 iterations of the Collatz function
for pure odd 255. Figure 5.2: Point plot of a sequence of 118 iterations of
the Collatz function for mixed number 97.

Due to the fact that an odd number can be either pure or mixed, when $x$ is
odd, the Collatz function $3x+1$ can be parted into two parts. i.e.,
$3x+1=2x+(x+1)$.

(i) when $x$ is a pure odd number, i.e., $x=2^{r}-1$, The binary string
$2x=2^{r+1}-2$ is even, with just one 0 at the end, and the other part
$x+1=2^{r}$ is pure even. As a result, $3x+1=2x+(x+1)=2^{r+1}+2^{r}-2$ is a
mixed even number with only one 0 in the last bit and $r-1$ bits 1 in the
second-to-last substring, corresponding $\frac{3x+1}{2}=2^{r}+2^{r-1}-1>x$, it
means function T(x) is increase.

(ii) When $x$ is a mixed odd number, it just has one 1 in the last binary
substring, the value $\frac{3x+1}{2^{r}}<x,(r>1)$ is smaller odd than $x$, it
means function T(x) is decrease. The last binary substring of number
$\frac{3x+1}{2^{r}},(r>1)$ has two kinds, which :

(a) There many 1 in the last binary substring. For example,
$893=(1101111101)_{2}$, $\frac{3\times 893+1}{2^{3}}=335=(101001111)_{2}=335$,
$335<893$.

(b) Only 1 in the last binary substring. For example, $17=(10001)_{2}$,
$\frac{3\times 17+1}{2^{2}}=(1101)_{2}=13$, $13<17$.

When the end of the binary substring is 1, using $\times$ denotes either 1 or
0, we discuss the changes of the last substring three digits and four digits
in the procedure of the Collatz function sequences:

$\times 001\rightarrow\times\times 100\rightarrow\times\times 1$, $\times
101\rightarrow\times 000\rightarrow\times$

$\times 0101\rightarrow\times\times 0000\rightarrow\times\times$, $\times
1101\rightarrow\times\times 1000\rightarrow\times\times 1$

$3x+1$ can adjust the structure of its binary substring, when the end of the
binary substring is 1, the value of Collatz function
$T(x)=\frac{3x+1}{2^{r}},r>1$ decrease, thus the number of binary string
digits decrease at least 2. This process continues several times, and
eventually you can reach the minimum value of 1. The Collatz function shows
that the Collatz conjecture holds.

Another Proof We give the statement ”period three implies chaos”[3] another
interpretation: for any positive integer n, the sequence of the Collatz is an
ultimately periodic sequence, its preperiod $\eta(n)$ is a related-to n
positive, and the least period $\\{4,2,1\\}$, $\rho(n)=3$.

We use the inversion of ”period three implies chaos”[3] is the Collatz
sequence of any initial value $n$. when the Collatz Conjecture is correct,
then the reverse order of sequence obtained for any natural initial n, the
sequence of the Collatz function is an ultimately periodic of period 3,
$\\{1,4,2\\}$.

##  6 Conclusion

From previous proof of the conjecture, it becomes a theorem.

Theorem For any natural number $n$, if it is even, divide by 2, if it is odd,
multiply by 3, add 1, and so on, the result must finally reach 1. Give another
statement: for any positive integer n, the sequence of the Collatz function is
an ultimately periodic sequence, its preperiod $\eta(n)$ is a related-to n
positive, and the least period $\\{4,2,1\\}$, $\rho(n)=3$.

##  7 References

## References

  * [1] Jeffrey C. Lagarias. ,  The 3x+1 Problem and Its Generalizations.  American Mathematical Monthly, Vol. 92, No. 1, 3-23(1985).
  * [2] Jeffrey C. Lagarias. ,  The 3x+1 Problem: An Overview.  arXiv: 2111.02635.
  * [3] Li, T., Yorke, J. A.,  Period three implies chaos.  Am. Mat. Monthly, 82, 985–992 (1975).
  * [4] TERENCE TAO,  Almost all orbits of the Collatz map attain almost bounded values.  arXiv:1909.03562v5, 2022,1,15.
  * [5] https://www.sohu.com/a/611510165_107944.
  * [6] Alf : Alf Kimms,  The structure of the 3x+1 problem.  Electronic Journal of Graph Theory and Applications. 9(1)(2021)157–174.
  * [7] Richard Kaufman,  A reduced forward Collatz algorithm: How binary strings change their length under 3x+1.  arXiv:2301.07466.
  * [8] Patrick Chisan Hew,  Collatz on the Dyadic rdtional in [0.5,1) with fractals: how bit strings change their length under 3x+1.  Experimental Mathematics, 2021, 30(4): 481-488.
  * [9] G. Ganesan,  Linear recurrences over a finite field with exactly two periods.  Advances In Applied Mathematics . 127(2021)102180.
  * [10] D. Quijada,  Periods of linearly recurring sequences.  Bachelor thesis, Washington and LeeUniversity, 2015.
  * [11] Leonel Sternberg,  Predictable trajectories of the reduced Collatz iteration and a possible pathway to the proof of the Collatz conjecture(Version 2).  arXiv:2209.14230.
  * [12] F. Jishe,  Proof the Collatz Conjecture in Binary Strings.  DOI: 10.20944/preprints202304.0931.v2.