# The Regular Element Property in Constructive Mathematics

Thierry Coquand

(Computer Science Department, University of Gothenburg)

## Introduction

The goal of this note is to present Kaplansky’s proof of the Regular Element
Property [5] and to explain how this argument can be adapted to the case of a
coherent, strongly discrete and Noetherian (with an inductive definition of
Noetherian [4]) rings in a constructive setting. We thus get, in this setting,
an algorithm which given a f.g. regular ideal $I={\langle
a_{1},\dots,a_{n}\rangle}$, build a regular element in $I$. This complements
in a way the previous treatment by Richman [6].

One maybe interesting feature of the constructive reformulation is that it
uses the technique of point-free open induction [2, 1]. (A previous inductive
proof of the nilregular element property [3] used a simpler form of
induction.)

##  1 Classical Proof

We recall that incomparable prime ideals $P_{1},\dots,P_{m}$ are independent:
we cannot have $\cap_{i\neq k}P_{i}\subseteq P_{k}$ since this would imply
that we have $i\neq k$ such that $P_{i}\subseteq P_{k}$.

Let ${v}^{\perp}$ be the ideal of elements $x$ such that $vx=0$. An ideal of
the from ${v}^{\perp}$ for $v\neq 0$ is called orthogonal ideal.

Each maximal orthogonal ideal ${v}^{\perp}$ is a prime ideal: if $ab$ in
${v}^{\perp}$ then $abv=0$ and $av\neq 0$ then ${(av)}^{\perp}={v}^{\perp}$ by
maximality and hence $b$ in ${v}^{\perp}$.

So if we have $v_{1},\dots,v_{m}$ non zero elements such that each
${v_{i}}^{\perp}$ is orthogonal and maximal, they are necessarily incomparable
and prime and so the ideals ${v_{i}}^{\perp}$ are independent: $\cap_{i\neq
k}{v_{i}}^{\perp}$ is not a subset of ${v_{k}}^{\perp}$.

Using Noetherianity, we see that the set of maximal orthogonal ideals has to
be finite: if $v_{m+1}$ is in ${\langle v_{1},\dots,v_{m}\rangle}$ then
$\cap_{i\leqslant m}{v_{i}}^{\perp}$ is a subset of ${v_{m+1}}^{\perp}$ and
then $v_{1},\dots,v_{m},v_{m+1}$ cannot be independent.

We consider then the finite set ${v_{1}}^{\perp},\dots,{v_{m}}^{\perp}$ of all
maximal orthogonal ideals. It is an independent set. So, for each $k$, we have
$u_{k}$ such that $u_{k}v_{k}\neq 0$ and $u_{k}v_{i}=0$ if $i\neq k$.

Let $I={\langle a_{1},\dots,a_{n}\rangle}$ be a regular ideal. For each $k$ we
have $i_{k}$ such that $u_{k}v_{k}a_{i_{k}}\neq 0$ since $u_{k}v_{k}\neq 0$.
Let $a=\Sigma_{k}u_{k}a_{i_{k}}$ element of $I$. We have
$av_{k}=u_{k}v_{k}a_{i_{k}}\neq 0$. I claim that $a$ is regular. Indeed if
$va=0$ and $v\neq 0$ we have $a$ in ${v}^{\perp}$ and $a$ is not in any
${v_{1}}^{\perp},\dots,{v_{m}}^{\perp}$ which contradicts the fact that this
is a list of all maximal orthogonal ideals.

##  2 Constructive Version

We are going to see that the classical argument of the first section can be
read as an algorithm. The idea is to proceed dynamically: we use a finite set
${v_{1}}^{\perp},\dots,{v_{m}}^{\perp}$ which is an approximation of the set
of all maximal orthogonal ideals to compute a candidate for a regular element
in a given regular ideal. If it is not regular, we use this to compute a
better approximation. The problem is then reduced to showing that this process
will end eventually.

If $S$ is a relation on a set $X$, we say that $x$ is $S$-accessible if, and
only if, all $y$ such that $S(x,y)$ are $S$-accessible. The relation $S$ is
well-founded if, and only if, all elements are accessible.

We assume now that the ring $R$ is coherent and strongly discrete. Each ideal
${v}^{\perp}$ is then a f.g. ideal.

We can then decide inclusion between f.g. ideals. We write $I<J$ for the fact
that $I$ is a strict subset of $J$.

We assume that $<$ is well-founded.

We consider then following relation: $v_{1},\dots,v_{n}\prec
w_{1},\dots,w_{m}$ (for $m\leqslant n$) if, and only if, $v_{i}=w_{i}$ for
$i<m$ and ${v_{m}}^{\perp}<{w_{m}}^{\perp}$.

A list $v_{1},\dots,v_{n}$ is called independent if the corresponding ideals
${v_{i}}^{\perp}$ are independent. It is non decreasing if
${v_{i}}^{\perp}\subseteq{v_{j}}^{\perp}$ implies $i\leqslant j$. Clearly any
independent list is non decreasing.

If $\alpha,v$ is independent then so is $\alpha$. So the independent lists
form a tree, and the branching of the tree is given by a decidable subset of
the set $R$.

######  Lemma 2.0.1.

The relation $\prec$ is well-founded.

###### Proof.

We show by induction on $v$ and the length of $\alpha$ that any element
$v\alpha$ if $\prec$-accessible. Indeed, we have $v\alpha\prec\beta$ if, and
only if, $\beta=w$ for some $w>v$ or $\beta=v\gamma$ with $\alpha\prec\gamma$.
∎

######  Corollary 2.0.1.

If $\alpha$ is a non decreasing list, there exists an independent $\beta$ such
that $\alpha\prec^{*}\beta$.

###### Proof.

Indeed if $\alpha$ non decreasing is not independent, we find $\beta$ non
decreasing such that $\alpha\prec\beta$. We conclude by Lemma 2.0.1. ∎

We now consider the lexicographic ordering $\alpha<\beta$ if, and only if,
$\beta$ is an extension of some $\gamma$ such that $\alpha\prec\gamma$ or a
strict extension of $\alpha$.

######  Lemma 2.0.2.

The ordering $<$ is well-founded on independent lists.

###### Proof.

We use the point-free technique of open induction [2].

If $\alpha$ is a list, let ${\langle\alpha\rangle}$ the f.g. ideal it
generates.

Let $M$ be the set of independent $\alpha$ such that $\alpha\prec\beta$
implies that $\beta$ is $<$-accessible. If $\alpha\rightarrow\alpha v$ and
$\alpha v$ is in $M$ then so is $\alpha$ since $\alpha\prec\beta$ implies
$\alpha v\prec\beta$. So $M$ defines a subtree in the tree of independent
sequences.

This tree has a complex branching structure, since the branching is now given
by the predicate of being $\prec$-accessible111This is what is interesting
from a constructive point of view, the argument is a priori in ID2 and not
ID1..

The relation of adding one element $\alpha\rightarrow\alpha v$ is well-founded
on the set $M$, since we then have ${\langle\alpha\rangle}<{\langle\alpha
v\rangle}$ and the ring $R$ is Noetherian.

This means that the following induction principle holds: is $\psi(\alpha)$ is
such that $\psi(\alpha)$ holds whenever $\psi(\alpha v)$ holds for all $\alpha
v$ in $M$, then $\psi$ holds for the empty sequence.

We apply this for the case where $\psi(\alpha)$ expresses that $\alpha$ is
accessible for the lexicographic ordering.

We take $\alpha$ in $M$ and assume that $\alpha v$ in $M$ implies $\alpha v$
accessible.

Then we show that $\alpha$ is accessible. For this we show that $\alpha v$ is
accessible whenever $\alpha v$ is independent by induction on $v$.

Let $v$ be such that $\alpha w$ is accessible for each $w>v$. We show that
$\alpha v$ is accessible.

It is enough to show that $\alpha v$ is in $M$. Assume $\alpha v\prec\beta$
then either $\alpha\prec\beta$ (and then $\beta$ is accessible since $\alpha$
is in $M$) or $\beta=\alpha w$ with $v<w$, and it is accessible by hypothesis.

By the induction principle on $M$, it follows that the empty sequence is
accessible. ∎

We can also extract from the classical section the following algorithm, which
holds for any ring with a decidable equality. Given $I={\langle
a_{1},\dots,a_{m}\rangle}$ which is regular.

######  Lemma 2.0.3.

If $\alpha=v_{1},\dots,v_{n}$ if an independent sequence, then we can build
$a\neq 0$ in $I$ such that $av_{i}\neq 0$ for each $i$.

We deduce from all this the following algorithm for computing a regular
element in a coherent, strongly discrete and inductive Noetherian rings.

We start from the empty sequence $\alpha_{0}=()$. Since $I$ is regular we can
find $u_{0}$ in $I$ which is $\neq 0$. Then either $u_{0}$ is regular or we
find $\alpha_{1}>\alpha_{0}$ such that $\alpha_{1}$ is independent. Using
Lemma 2.0.3, we compute $u_{1}\neq 0$ in $I$ such that $u_{1}v\neq 0$ for each
$v$ in $\alpha_{1}$. Either $u_{1}$ is regular, or we find $v\neq 0$ such that
$u_{1}v=0$ and $\alpha_{2}>\alpha_{1}v$ which is independent, using Corollary
of Lemma 2.0.1, and so on.

This process should eventually produces a regular element in $I$ using Lemma
2.0.2.

## References

  * [1] U. Berger.  A Computational Interpretation of Open Induction.  LICS 2004: 326, 2003.
  * [2] Th. Coquand.  Constructive topology and combinatorics.  In: Myers, J.P., O’Donnell, M.J. (eds) Constructivity in Computer Science. Lecture Notes in Computer Science, vol 613, 1991.
  * [3] Th. Coquand, H. Lombardi and P. Schuster.  A nilregular element property.  Archiv der Mathematik, p. 49-54, 2005.
  * [4] C. Jacobsson and C. Löfwall.  Standard bases for general coefficient rings and a new constructive proof of Hilbert’s basis theorem.  J. Symb. Comput. 12, 337-371, 1991.
  * [5] I. Kaplansky.  Commutative rings. Allyn and Bacon, 1970.
  * [6] F. Richman.  The Regular Element Property.  Proceedings of the AMS, Vo. 126, 2123-2129, 1998.


# A low-inclination neutral Trans-Neptunian Object in a extreme orbit

Ying-Tung Chen ( 陳英同) Institute of Astronomy and Astrophysics, Academia
Sinica, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan  Marielle R. Eduardo
Institute of Astronomy and Astrophysics, Academia Sinica, No.1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan  Marco A. Muñoz-Gutiérrez Institute of
Astronomy and Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd,
Taipei 10617, Taiwan  Shiang-Yu Wang ( 王祥宇) Institute of Astronomy and
Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd, Taipei 10617,
Taiwan  Matthew J. Lehner Institute of Astronomy and Astrophysics, Academia
Sinica, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan  Department of
Physics and Astronomy, University of Pennsylvania, 209 S. 33rd St.,
Philadelphia, PA 19125, USA  Chan-Kao Chang ( 章展誥) Institute of Astronomy and
Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd, Taipei 10617,
Taiwan

###### Abstract

We present photometric observations and numerical simulations of 2016 SD106, a
low inclination ($i=4.8\arcdeg$) extreme trans-Neptunian Object with a large
semi-major axis ($a=350$ au) and perihelion ($q=42.6$ au). This object
possesses a peculiar neutral color of $g-r=0.45\pm 0.05$ and $g-i=0.72\pm
0.06$, in comparison with other distant trans-Neptunian objects, all of which
have moderate-red to ultra-red colors. A numerical integration based on
orbital fitting on astrometric data covering eight years of arc confirms that
2016 SD106 is a metastable object without significant scattering evolution.
Each of the clones survived at the end of the 1 Gyr simulation. However, very
few neutral objects with inclinations $<5\arcdeg$ have been found in the outer
solar system, even in the main Kuiper belt. Furthermore, most mechanisms which
lift perihelion distances are expected to produce a very low number of extreme
objects with inclinations $<5\arcdeg$. We thus explored the possibility that a
hypothetical distant planet could increase the production of such objects. Our
simulations show that no 2016 SD106-like orbits can be produced from three
Kuiper belt populations tested (i.e. plutinos, twotinos, and Haumea Family)
without the presence of an hypothetical planet, while a few similar orbits can
be obtained with it; however, the presence of the additional planet produces a
wide range of large semimajor-axis / large perihelion objects, in apparent
contradiction with the observed scarcity of objects in those regions of phase
space. Future studies may determine if there is a connection between the
existence of a perihelion gap and a particular orbital configuration of an
hypothetical distant planet.

celestial mechanics — surveys — Kuiper belt objects: individual (2016 SD106)

††facilities: CFHT (MegaCam), Subaru (Hyper Suprime-Cam)††software: orbfit
(Bernstein & Khushalani, 2000), TRIPPy (Fraser et al., 2016), MERCURY
(Chambers, 1999), and NUMPY (Harris et al., 2020)

##  1 Introduction

Trans-Neptunian Objects (TNOs) are not simply a group of planetesimals that
survived after the early evolution of the solar system, but are also a
complicated combination of remnants transferred from different locations of
the solar system during the planetary migration. TNOs can be broadly
classified into dynamically cold and hot populations. The dynamically cold
objects (the so-called cold classical Kuiper Belt objects), which
observationally congregate at inclinations $i<5\arcdeg$ and around semi-major
axes $a$ between 39.4 and 47.7 au, are thought to be formed in situ and have
survived after the migration of Neptune, whereas many of the dynamically hot
objects that spread over a wide range of semi-major axes from 30 to many
hundreds au were perturbed/scattered by Neptune, i.e., hot classical Kuiper
Belt objects, objects in mean-motion resonances (MMRs) with Neptune and
scattered disc objects (SDOs). Additionally, detached objects (DOs) are not in
resonance with Neptune, and typically have semimajor axes $a>47.4$ au,
eccentricities $e>0.24$, and perihelion $q\geq 38$ au. Because the exact
boundary between SDOs and DOs are still being investigated, we refer to $q>40$
and $a>250$ au TNOs as “extreme” TNOs. DOs are not influenced gravitationally
by Neptune or the other currently known giant planets (see the dynamical
reviews by Saillenfest (2020) and Gladman & Volk (2021)), and the physical
mechanisms that placed the DOs into their current orbits remain undetermined.
Such objects likely originated from the SDO population, but they are now
totally disconnected from it because their perihelia have been lifted in the
past by external perturbers or other unknown mechanisms. Several scenarios to
lift $q$ have been proposed: (1) gravitational influence of nearby stars in
the Sun’s birth cluster (Brasser et al., 2006, 2012), (2) a passing star
(Morbidelli & Levison, 2004; Kenyon & Bromley, 2004), (3) undetected planet(s)
in the past (Gladman & Chan, 2006; Silsbee & Tremaine, 2018), or at present
(Trujillo & Sheppard, 2014; Batygin & Brown, 2016; Batygin et al., 2021).
However, current data are insufficient to identify the dominant mechanism.

Surface composition plays another key role in the investigation of the origin
of solar system objects. A correlation between color and inclination was found
for the population of classical TNOs (Trujillo & Brown, 2002). The dynamically
cold objects, i.e. cold classical Kuiper Belt objects, which have redder
colors, are representative of the primordial bodies formed beyond Neptune’s
orbit. In contrast, the dynamically hot objects, including hot classical
objects, MMRs, SDOs, and DOs, were formed in different regions of the solar
system before being excited during Neptune’s migration, and thus have a wider
color distribution. For example, the color distributions of observed plutinos
and twotinos (3:2 and 2:1 MMR with Neptune) range from neutral to ultra red.
We note that the MMRs and SDOs often experience resonance sticking or Kozai
interactions, which change the semi-major axes, eccentricities, and
inclinations of TNOs and/or switch them to other resonances, but both
interactions are rarely seen for objects with $a>250$ au. In addition, the
Haumea family, the well-known collisional population, has a unique neutral
color distribution ($g-r<0.5$), which is different from the color of other
populations. Their neutral surfaces possibly originate from their past
collisional history which exposed underlying material (Trujillo et al., 2007;
Rabinowitz et al., 2008). Subsequent discoveries with similar surface colors
and inclinations support this hypothesis (Brown et al., 2007). To date, very
few neutral objects in the outer solar system have been confirmed with
$i<5\arcdeg$.

The Hyper Suprime-Cam Subaru Strategic Program (HSC-
SSP)111https://hsc.mtk.nao.ac.jp/ssp/ is a deep multi-band imaging survey of
1400 deg.2 of the sky with the 8-m Subaru telescope (Aihara et al., 2018).
Although the main science goals are cosmology and galaxy evolution, its depth
and field coverage provide a unprecedented dataset to detect distant/faint
objects beyond Neptune. With this dataset, we have discovered 178 TNOs in
Deep/UDeep fields (Eduardo et al., in preparation) and 231 TNOs in Wide fields
(Chen et al., 2018) including an extremely neutral DO. Because the cadence of
HSC-SSP was not designed for color observations of TNOs, corrections for
rotational and phase angle effects could not be made. We thus made additional
observations to obtain precise multi-color measurements.

In this study, we present the color estimation of this extreme neutral object,
along with the results of simulations of 2016 SD106, (with and without a
hypothetical additional planet) to explore its stability and possible origin.

##  2 Observations

The Dark Energy Survey (DES) team discovered 2016 SD106 and first submitted
the observations to the Minor Planet Center (MPC) (Bernardinelli et al.,
2022). However, 2016 SD106 was also independently discovered in the HSC-SSP
DEEP2-3 Deep field. Data from the HSC-SSP Deep fields include repeated multi-
band ($grizy$) observations at different epochs. The number of measurements in
$g$, $r$, and $i$ are 10, 10, and 14, respectively. The HSC-SSP typically
finished all observations for a single filter for Deep/UDeep fields within one
night, then observations for other filters for the same field usually were
executed in the same dark run or later runs. The color estimation of 2016
SD106 from HSC-SSP observations shows a neutral color with a very large
uncertainty ($g-r=0.50\pm 0.24$ and $g-i=0.70\pm 0.23$). However, most
moderate/small size TNOs are known to be non-spherical with rotation periods
in the range of several hours. The observed cross-sections thus change with
time, causing inaccurate color estimates if observations using different
filters are not nearly simultaneous. Therefore, we requested and received CFHT
Director’s Discretionary Time (DDT) to confirm the color estimate of 2016
SD106. Consecutive observations for color measurements were completed using
MegaCam (Boulade et al., 2003) on CFHT within a span of 1.8 hours on 2021
November 2. The observations were conducted using a sequence of filters of the
form $r$-$g$-$r$-$i$-$i$-$r$-$z$-$r$-$g$-$r$-$i$-$i$-$r$-$z$-$r$, with all
observations performed using sidereal tracking. Assuming a uniform surface
color and linear variation, this sequence would remove any rotation effects.
The astrometry and photometry of the CFHT DDT and HSC-SSP data were calibrated
using the Pan-STARRS1 catalog (Schlafly et al., 2012; Tonry et al., 2012;
Magnier et al., 2013). The photometry was re-measured with the moving-object
photometry package TRIPPy (Fraser et al., 2016) using a point spread function
of reference stars to generate trailing apertures according to object motion.

Upon completion of the analysis, we measured the color of 2016 SD106 to be
$g-r=0.45\pm 0.05$ and $g-i=0.72\pm 0.06$, indicating an extreme neutral
surface (see Table 1) similar to those of neutral resonant objects (2004 EW95,
90482 Orcus, and 2004 TV357), SDOs (1996 TL66), the solar $g-r=0.45\pm 0.02$
(Holmberg et al., 2006), and the Haumea Family member (1996 TO66) $g-r\sim
0.46\pm 0.02$ (see Figure 1). Comparing with the color database in Minor
Bodies in the Outer Solar System (MBOSS, Hainaut et al., 2012), 2016 SD106 is
the bluest object among all known SDOs/DOs with color measurements. If the
peculiar color of 2016 SD106 implies a high albedo of $p\simeq 0.51$, as the
Haumea Family (Ortiz et al., 2017), the estimation of size is $\sim 80$ km;
the estimation of size is $\sim 250$ km with general low albedo of $p=0.05$.
Although we do not know the variability of the phase-curve and light curve,
the absolute magnitude could be roughly estimated as $H_{r}=6.46$. We note
that the $g-r$ (0.45) agrees to the other known neutral objects, but $g-i$
(0.71) is higher than typical value ($\sim 0.6$) due to the brighter magnitude
on $i$-band. Following the same method in Sheppard (2010), the spectral
gradient (SG) based on $g-r$ and $g-i$ is 1.1 and 6.1 percent of reddening per
100 nm, respectively. (We are not reporting the $z$-band photometry because
there are currently too few DOs with such measurements for useful
comparisons.)

To minimize the orbital uncertainties, we combine the observations from HSC-
SSP, CFHT DDT and
DES222https://minorplanetcenter.net/db_search/show_object?object_id=2016+SD106
for orbital determination. Based on the observations spanning over 8
oppositions, we determined the orbit using the orbit fitting code of Bernstein
& Khushalani (2000). The resulting uncertainties in heliocentric orbital
elements are all small: $a=354.23\pm 0.49$ au, $e=0.87973\pm 0.00014$,
$i=4.8080\pm 0.0001\arcdeg$, longitude of ascending node $\Omega=219.494\pm
0.002\arcdeg$, and argument of pericenter $\omega=163.032\pm 0.005$.
Pericenter passage will occur at $2464675.363\pm 0.84$.

Table 1: Orbital elements and colors of selected extreme TNOs and
representative objects with neutral color in the outer solar system.

Designations | $q$ | $a$ | $e$ | $i$ | $H$ | Arc | $g-r$ | $g-i$ | Orbit |
---|---|---|---|---|---|---|---|---|---|---
| (au) | (au) |  | ($\arcdeg$) |  | (days) |  |  | Class |
2016 SD106 | 42.65 | 354.7 $\pm$ 0.4 | 0.87 | 4.80 | 6.8 | 2977 | 0.45 $\pm$ 0.05 | 0.72 $\pm$ 0.06 | extreme |
90377 Sedna | 76.36 | 510.3 $\pm$ 0.1 | 0.85 | 11.93 | 1.55 | 11418 | 0.85 $\pm$ 0.03 | 1.31 $\pm$ 0.04aafootnotemark:  | extreme |
474640 Alicanto | 47.29 | 344.4 $\pm$ 1.1 | 0.86 | 25.53 | 6.46 | 7736 | 0.69 $\pm$ 0.06 | 1.01 $\pm$ 0.06aafootnotemark:  | extreme |
2012 VP113 | 80.43 | 269.2 $\pm$ 0.5 | 0.70 | 21.07 | 4.09 | 3028 | 0.70 $\pm$ 0.05 | 1.02 $\pm$ 0.06bbfootnotemark:  | extreme |
2013 SY99 | 50.08 | 824 $\pm$ 32 | 0.93 | 4.21 | 6.7 | 1156 | 0.64 $\pm$ 0.06 | 1.10 $\pm$ 0.11ccfootnotemark:  | extreme |
19308 (1996 TO66) | 38.49 | 43.528 $\pm$ 0.001 | 0.11 | 27.33 | 4.88 | 13933 | 0.46 $\pm$ 0.02 | 0.58 $\pm$ 0.02aafootnotemark:  | Haumea |
120216 (2004 EW95) | 26.97 | 39.429 $\pm$0.008 | 0.35 | 29.31 | 6.61 | 7035 | 0.47 $\pm$ 0.06 | 0.53 $\pm$ 0.09d*d*footnotemark:  | 3:2 |
90482 Orcus | 30.13 | 39.0971 $\pm$0.0004 | 0.23 | 20.57 | 2.19 | 25690 | 0.46 $\pm$ 0.04 | 0.64 $\pm$ 0.06f*f*footnotemark:  | 3:2 |
2004 TV357 | 34.50 | 47.715 $\pm$ 0.002 | 0.27 | 9.76 | 6.9 | 5607 | 0.47 $\pm$ 0.02 | 0.60 $\pm$ 0.02eefootnotemark:  | 2:1 |
15874 (1996 TL66) | 34.97 | 83.733 $\pm$ 0.003 | 0.58 | 23.95 | 5.46 | 9222 | 0.48 $\pm$ 0.07 | 0.67 $\pm$ 0.10f*f*footnotemark:  | SDO |

††footnotetext: Note: Except 2016 SD106, all orbital elements were taken from:
https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html. For DOs, we only selected
$q>40$ objects which have $gri$ color measurements in the literature. Adapting
the transformation equations between $BVRI$ and $gri$ in Smith et al. (2002),
the $BVRI$ colors of 2016 SD106 are given by $B-V=0.66$ , $V-R=0.37$, and
$R-I=0.47$. The * symbol indicates the measurements were transferred from
$BVRI$ data.

††footnotetext: References: $a$ = Sheppard (2010), $b$ = Trujillo & Sheppard
(2014), $c$ = Bannister et al. (2017), $d$ = Perna et al. (2013), $e$ =
Sheppard (2012), $f$ = Hainaut et al. (2012)

Figure 1: Left panel: $gri$ colors of 2016 SD106, selected extreme TNOs, and
objects from Sheppard (2010) and Sheppard (2012), which include most of the
dynamically excited TNO populations. The colors of high-perihelion objects
mentioned in Bannister et al. (2017) are indicated by triangles, and the solar
color is indicated by the yellow star. The dashed lines roughly indicate the
different color regions. Right panel: $BVRI$ colors of 2016 SD106, the
particularly neutral plutino 2004 EW95 (Perna et al., 2013), and plutinos,
SDOs, and DOs from MBOSS (Hainaut et al., 2012). The $BVRI$ color of 2016
SD106 was transformed from the original Sloan colors using the equations from
Smith et al. (2002).

##  3 Numerical Integration and Analysis

With the available astrometric data over eight oppositions, we can accurately
estimate the orbital parameters of 2016 SD106 and determine its long-term
stability. Using the covariance matrix output from the orbit fitting code of
Bernstein & Khushalani (2000), we generated 1,000 clones within 3$\sigma$ of
2016 SD106 best-fit orbit. We then performed 1 Gyr long numerical simulations
using the symplectic integrator from the MERCURY package (Chambers, 1999),
considering the four giant planets, a 180 day initial time-step, and a 10,000
au ejection distance. We found that all clones survived after 1 Gyr, with
91.7% of them keeping $a<1,000$ au. The orbital distributions of the clones at
three different times (0, 0.1, and 1.0 Gyr) are illustrated in Figure 2. Our
results indicate that a diffusive behavior in $a$ dominates the evolution of
2016 SD106, which is similar to the behavior of 2013 SY99 (Bannister et al.,
2017). The semi-major axes diffused both inward and outward due to planetary
perturbations. The variations in inclination were small on a Gyr timescale,
with an oscillating period of $\sim$100 Myr between $4\arcdeg$ and $7\arcdeg$.
The change of perihelion distances was very small over time ($\Delta q<1$ au),
compared, e.g. with the few au change found for the semi-stable extreme object
2000 CR105 (Gladman et al., 2002).

Figure 2: The orbital distribution of 1,000 clones of 2016 SD106 after 0.1 Gyr
(orange dots) and 1 Gyr simulation (blue dots). The red dots indicate the
initial conditions. The clone orbits are much more stable than typical
Centaurs/SDOs, with $\Delta q<1$ au over time. Only a few percent of clones
evolve to $a>1,000$ au.

The semi-major axis of 2016 SD106 is quite close to that of the hypothetical
planet (HP) of Brown & Batygin (2021), and the existence of this HP could
dramatically change the orbital elements of distant TNOs (Shankman et al.,
2017; Bannister et al., 2017). In order to explore possible origins of low-$i$
neutral DOs, we implemented experimental simulations for four dynamical
populations with/without the HP from Brown & Batygin (2021): (1) 2016 SD106
clones in this study, (2) plutinos, (3) twotinos, and (4) Haumea family
members, as these populations contain known neutral objects. Plutino and
twotino particle distributions were taken from the L7 model of Petit et al.
(2011). We used the objects identified as resonant (those showing libration in
a 10 Myr interval) in Muñoz-Gutiérrez et al. (2019), totalling 3340 plutinos
and 870 twotinos. For the Haumea family, we generated a random set of 2000
objects, within the boundaries 42 au $<a<44.5$ au, $0.1<e<0.2$, and
$24\arcdeg<i<28.5\arcdeg$, following Lykawka et al. (2012). The arguments of
perihelion, longitudes of ascending node, and mean anomalies, were randomly
chosen between $0\arcdeg$ and $360\arcdeg$ in all cases. We used MERCURY, with
an increased initial timestep of 400 d (in order to speed up integrations
without loss of precision) and an ejection distance of 20,000 au, to integrate
our systems for 1 Gyr. We considered the evolution of particles under the
influence of the giant planets, with and without the presence of the HP
($a=380$ au, $e=0.21$, $i=16\arcdeg$, $\omega=150^{\circ}$,
$\Omega=100^{\circ}$, M$\;=6.2$ M⊕) (Figures 3 and 4). We note that the
longitude of perihelion of 2016 SD106 ($\varpi\sim 22\arcdeg$) lies within the
predicted cluster of $\Delta\varpi$, given the orbital configuration of the HP
used here.

With and without the HP, over 85% of the surviving plutinos and twotinos
remain within the range of $\Delta a_{0}\pm 0.7$ au after 1 Gyr, and a similar
fraction of the surviving Haumea family members remain in the range
$42<a<44.5$ au. The 2016 SD106 clones with no HP show very little change, but
the HP clearly has a significant influence on the evolution of the 2016 SD106
clone orbits. The $i$ and $q$ of particles from the other dynamical
populations evolved and spread evenly between $3\arcdeg$ to $40\arcdeg$ and 30
to 380 au by the end of the simulation. We note that a population with large
$i$ and large $q$ would be significantly harder to detect since the objects
spend most of their time at high latitude and larger barycentric distances.
Therefore a large population of such objects would be required to result in
the detection of 2016 SD106. Our results show that the HP does make it
possible for particles in various TNO populations to diffuse towards 2016
SD106-like orbits. However, such diffusion produces a significant population
of large $a$ / large $q$ objects, in apparent contradiction with an identified
observational scarcity of objects in this region of the solar system phase-
space, i.e. the so-called perihelion gap (Oldroyd & Trujillo, 2021).

Figure 3: Heatmaps of the orbital evolution of 2016 SD106 clones and Haumea
Family members in simulations with and without the HP. The white cross and
green dot show the initial orbit of 2016 SD106 and HP, respectively. The
initial orbits in each simulation are overplotted as black dots, while light
blue dots show the final distribution of surviving particles after 1 Gyr. The
observational $q$-gap suggested by Oldroyd & Trujillo (2021) is outlined with
a red square. The black curve shows the stability limit for distant SDOs as
derived by Batygin et al. (2021); above this line in the $a$-$q$ plane,
particles are quickly destabilized and ejected from the solar system.

Figure 4: Same as Figure 3 except for plutinos and twotinos.

##  4 Discussion

The existence of low-$i$ neutral extreme objects has significant implications.
The DOs have $a$ and $i$ distributions similar to those of SDOs, except the
values of $q$ are large enough to avoid the strong perturbations from Neptune.
Scattering processes cannot vary $i$ much since $q$ and the Tisserand
parameter relative to Neptune, $T_{\mathrm{N}}$, are almost constant during
the scattering process (Saillenfest, 2020). Both SDOs and DOs have similar
color distributions as shown in Hainaut et al. (2012) and Figure 1. The
orbital boundary between SDOs and DOs is not clearly defined, as it may vary
by different classification criteria, e.g. Khain et al. (2020), Gladman & Volk
(2021) or a stability validation. For example, the well-known neutral SDO,
136199 Eris was classified as DO in MBOSS. However, even if we neglect the
classification definition, TNOs still seem to follow the color-inclination
correlation. The color-inclination correlation demonstrated in Peixinho et al.
(2015) shows a lack of objects with ($i<5\arcdeg$, $B-R\sim 1.0$), and remains
a weaker correlation for all TNOs. The only known object in this study close
to these conditions is the plutino 612029 (1995 HM5), with $i=4.8\arcdeg$ and
$B-R=1.01\pm 0.2$. Marsset et al. (2019) combined the OSSOS dataset with
previously published measurements (including samples in Peixinho et al.
(2015)), and investigated the spectral slope (SS) of dynamical hot populations
($i>5\arcdeg$). We note that the definition of the spectral slope (SS) here is
different from the spectral gradient (SG) we evaluated for 2016 SD106,
although the two values are comparable. In the detached group, only 2003 FZ129
($a=61.4$ au, $e=0.38$, $i=5.8\arcdeg$, $q=38.0$ au) shows a moderate-red-to-
neutral slope (9.88) and an inclination close to $5\arcdeg$, although it shows
a moderate-red color $B-R=1.32\pm 0.04$. The inclinations of other DOs are
larger than $17\arcdeg$. The few neutral objects with SS $<5$ in both SDOs/DOs
all show medium to high inclinations (1996 TL66: $23\arcdeg$, 2000 YC2:
$18\arcdeg$, 2000 PE30: $18\arcdeg$). 2016 SD106 is indeed an unique object
among the known SDOs and DOs. Although the inclination distribution of the
distant TNOs is poorly known due to relatively small numbers and survey
biases, most known SDOs/DOs do not cluster on the ecliptic plane, considering
that most outer solar system surveys have a bias towards detecting low-$i$
objects. So the existence of 2016 SD106 constrains the mechanisms by which
extreme objects are produced, and may imply that a particular dynamical
evolution occurred with a few extreme objects, producing the special
surfaces/orbits which don’t follow the color-inclination correlation found in
TNOs.

The following are possible scenarios related to the origin of 2016 SD106:

Kozai resonance in MMR

Considering the Kozai resonance as a mechanism to raise perihelia, Kozai
interactions must be coupled with increases of both ${q}$ and ${i}$. It is
difficult to explain low-i DOs with large $q$. Results in Section 3 show that
the possibility of 2016 SD106 coming from a high order MMRs with Neptune is
extremely low. The closest possible MMRs to 2016 SD106 are 1:40 ($\sim 351$
au) and 1:41 ($\sim 357$ au), which are too weak to preserve long-lived
resonant objects. Some literature also indicate that the resonant mechanism is
too weak to produce DOs beyond 250 au, especially for low-$i$ objects
(Gallardo et al., 2012; Brasser & Schwamb, 2015). Thus the Kozai resonance
cannot explain the low $i$ and high $q$ of 2016 SD106. Recent studies also
indicate that theoretical models of planetary migration are unlikely to
produce high-perihelion ($q>40$), low-$i$ DOs (Kaib & Sheppard, 2016; Pike &
Lawler, 2017). In addition, the major MMRs beyond Neptune, i.e. plutinos and
twotinos, are very stable at present, and Kozai resonance on these major MMRs
is unlikely to produce DOs with only the four known giant planets (see Figure
4).

Object diffused from the inner fringe of the Oort cloud

The orbital elements indicate that 2016 SD106 is in the diffusive region and
away from the strong scattering region (Duncan et al., 1987; Bannister et al.,
2017). 2013 SY99, with a larger semi-major axis $a=824$ au, could be explained
with a diffuse behavior of the “inner fringe” of the Oort cloud in a long-time
integration (Bannister et al., 2017) for $45<q<50$ au and $1000<a<2500$ au.
The diffusion mechanism along with planetary perturbations could produce the
population of 2013 SY99, initiating from part of SDOs without an additional
source of mass. The efficiency of this scenario is still unknown, which is
critical to understand if the production of extreme neutral objects from SDOs
is possible.

Analogues to the Haumea family

The Haumea family is a collisional population which share similar orbital
parameters and surface colors. The possible orbital spaces of this family,
assuming a reasonable range of impact velocities, don’t cover the region of
the detached population (Lykawka et al., 2012), and most of survived particles
have $i=20\sim 35\arcdeg$ and $a=35\sim 55$ au. The currently known/possible
Haumea family members have moderate inclinations ($i\sim 28\arcdeg$), rather
than low inclinations. If an extremely rare collision event happened in the DO
region, then neutral objects could be formed in situ, in which case we should
expect more neutral DOs to be detected.

Hypothetical Planet

Our simulations with a HP showed the possibility to provide a path to bring
TNOs to the detached region. This pathway also fills up the observational
$q$-gap ($50\lesssim q\lesssim 65$ au and $0.65\lesssim e<1$) suggested by
Oldroyd & Trujillo (2021), who also point out that the presence of a HP would
produce observable objects within the $q$-gap. Different q-gap outcomes are
produced with different orbital configurations of the HP, which suggests that
a relationship between various dynamical populations, the observational q-gap
depth, and an HP may exist. Some plutinos, twotinos, and Haumea family members
can be influenced by the HP and approach the detached region and inner Oort
Cloud. The $i$ and $q$ evolution of these clones lowers their detectability
dramatically, like 2013 SY99 in Bannister et al. (2017), as the clones spend
almost all of their time at higher inclinations and larger perihelia.

2004 EW95

Although the high-$i$ plutino hints at a different dynamical evolution, 2004
EW95 ($i=29\arcdeg$) is the first TNO confirmed with a neutral surface
possibly composed by more carbon and silicates (ferric oxides and
phyllosilicates) than typical TNOs (Seccull et al., 2018). 2004 EW95 is
thought to have originated closer to the Sun during planetary migration. This
object indeed supports the model prediction that a small fraction of objects
with carbonaceous surfaces could be resident in the outer solar system. Future
spectral observations of 2016 SD106 in optical and IR are essential to confirm
its surface composition and albedo.

A search of the MPC Database for objects with $a>250$ and $q>40$ found that
2013 RA109 and 2014 WB556 are potentially neutral color objects as well,
although their inclinations are moderately high ($12.4\arcdeg$ and
$24.2\arcdeg$). We emphasize that there may be no significant discrepancy
between the neutral color of 2016 SD106 and the color-inclination correlation,
since this correlation has not yet been fully measured over the different
dynamical populations. While the color and orbital elements of 2016 SD106
provide valuable tracers of DO formation, more systematic color observations
of DOs discovered in future surveys (e.g. LSST) could provide more clues to
understand their formation.

##  5 Summary

Our observations show the low-$i$ extreme TNO 2016 SD106 has an unusual
neutral color of $g-r=0.45\pm 0.05$ and $g-i=0.72\pm 0.06$. A numerical
integration of clones shows that its orbital evolution over 1 Gyr is stable,
with some diffusion behavior. Current dynamical populations with neutral
surface members could not easily produce 2016 SD106-like objects when only
accounting for perturbations due to known planets. Although a HP could provide
the possibility for plutinos, twotinos or Haumea family members to migrate
towards the extreme region, this mechanism would require a vast population to
produce the detection of 2016 SD106. This rare neutral object with large-$a$,
large-$q$, and low-$i$ provides an additional tracer to understand the origin
of extreme objects.

Based on data collected at the Subaru Telescope and retrieved from the HSC
data archive system, which is operated by the Subaru Telescope and Astronomy
Data Center (ADC) at NAOJ. Also based on observations obtained with
MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-
France-Hawaii Telescope (CFHT) which is operated by the National Research
Council (NRC) of Canada, the Institut National des Science de l’Univers of the
Centre National de la Recherche Scientifique (CNRS) of France, and the
University of Hawaii. The observations at the Canada-France-Hawaii Telescope
were performed with care and respect from the summit of Maunakea which is a
significant cultural and historic site. This publication uses data generated
via the Zooniverse.org platform, development of is which funded by generous
support, including a Global Impact Award from Google, and by a grant from the
Alfred P. Sloan Foundation. The Hyper Suprime-Cam (HSC) collaboration includes
the astronomical communities of Japan and Taiwan, and Princeton University.
The HSC instrumentation and software were developed by the National
Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics
and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the
High Energy Accelerator Research Organization (KEK), the Academia Sinica
Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton
University. Funding was contributed by the FIRST program from the Japanese
Cabinet Office, the Ministry of Education, Culture, Sports, Science and
Technology (MEXT), the Japan Society for the Promotion of Science (JSPS),
Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ,
Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of
software developed for Vera C. Rubin Observatory. We thank the Rubin
Observatory for making their code available as free software at
http://pipelines.lsst.io/. The Pan-STARRS1 Surveys (PS1) and the PS1 public
science archive have been made possible through contributions by the Institute
for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the
Max Planck Society and its participating institutes, the Max Planck Institute
for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial
Physics, Garching, The Johns Hopkins University, Durham University, the
University of Edinburgh, the Queen’s University Belfast, the Harvard-
Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global
Telescope Network Incorporated, the National Central University of Taiwan, the
Space Telescope Science Institute, the National Aeronautics and Space
Administration under grant No. NNX08AR22G issued through the Planetary Science
Division of the NASA Science Mission Directorate, the National Science
Foundation grant No. AST-1238877, the University of Maryland, Eotvos Lorand
University (ELTE), the Los Alamos National Laboratory, and the Gordon and
Betty Moore Foundation. .

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