Models that link and suggest data about elementary particles, dark matter, and the cosmos

We suggest progress regarding the following six physics opportunities. List all elementary particles. Describe dark matter. Explain ratios of dark matter to ordinary matter. Explain eras in the history of the universe. Link properties of objects. Interrelate physics models. We use models based on Diophantine equations. 5 associates with some elementary fermions and with color charge. We posit that the jay (or, 1J) boson associates with notions of Pauli repulsion. Pauli repulsion associates with the notion that two fermions (whether elementary fermions or not elementary fermions) cannot occupy the same state. Pauli repulsion associates with repulsive aspects of the residual strong force. We suggest the possibility (but do not necessarily require) that - with respect to the Standard Model notion that SU (3) symmetry associates with the strong interaction - the jay boson might (for some modeling, perhaps yet to be determined) associate with the identity operator that SU (3) representations for the gluons (which transmit color charge) exclude. We suggest that mutual - for gluons and jay bosons - association with the strong force associates with the notion that the 1J PROP solution equals the 1U PROP solution. We suggest that the jay boson can interact with each fermion elementary particle, regardless of whether the notion of color charge pertains for the elementary fermion. We suggest that the jay boson can interact with each fermion non-elementary-particle object. In each one of the leftmost two columns in the table, two uses of a pair of the symbol † points to dual use of a solution.


Overview -physics results
This essay pursues the following two challenges.
Describe new elementary particles and dark matter.
Use descriptions of elementary particles and dark matter to explain astrophysics data and cosmology data.
Our explanations regarding large-scale data might help validate our descriptions of possible new elementary particles and our description of dark matter. We suggest explanations for known data for which -seemingly -other modeling does not oer explanations. We suggest data -about aspects of the universe -that people might be able to verify or refute.    , preview table 19 and table   20.) People have observed galaxies that associate with each one of the suggested approximate ratios -one to zero-plus, ve-plus to one, four to one, and zero-plus to one.    LRI solutions come in pairs. For example, regarding electromagnetism, one so-called PROP solution associates with the property of charge. That PROP solution has a so-called CURR partner solution that associates with a current of charge. We  suggested new elementary particles. (The number S in a symbol SΦ, associates with elementary-particle spin in units of ℏ.) 0.5M associates with three spin-one-half heavy neutrinos. 2G associates with a spin-two graviton. 3G associates with a spin-three relative of the photon and the graviton. 4G associates with a spin-four relative of the photon and the graviton.

Overview -research and results
0I associates with a spin-zero inaton. 0.5R associates with three spin-one-half zero-charge analogs to quarks. 1J associates with a spin-one zero-charge boson that associates with Pauli repulsion. Our modeling explains the ratio. Regarding some specic depletion of cosmic microwave background radiation (or, CMB), people have observed the ratio, some people speculate that the eects that people might not attribute to ordinary matter are eects of dark matter, and our modeling suggests that non-ordinary-matter eects are eects of dark matter. We use the two-word term early galaxies to include galaxies observed at redshifts of at least (and possibly somewhat less than) seven. Most relevant data about later galaxies pertains to galaxies observed at redshifts considerably less than seven. The three-word term dark matter galaxy pertains to a galaxy for which the DM:OM ratio is one to zero-plus. Possibly, current techniques are not adequately sensitive to detect early dark matter galaxies. Figure 4: Dark matter density of the universe and ordinary matter density of the universe. The DM (or, dark matter) relative densities sum to approximately 5.38 times the OM (or, ordinary matter) relative density. Across isomers, the masses of similar elementary particles are identical. However, for charged leptons, associations between avour and mass are not necessarily identical. Dierences in associations between charged-lepton avours and charged-lepton masses lead to dierences in the evolution of stu that associates with each isomer. The stu that associates with at least four DM isomers of elementary particles evolves so that the associated IGM (or, intergalactic medium) does not interact electromagnetically much with itself, compared to the interactivity of OM IGM. The lack -across at least four DM isomers -of much IGM electromagnetic self-interaction might associate with observations regarding the Bullet Cluster collision of two galaxy clusters.  aspects of the era. The rightmost four columns associate with a noteworthy cause for the era. Generally, the noteworthy cause gains prominence before the era starts. Our work proposes the rst two eras to which the image alludes. Other work and our work suggest the era of ination. Other work and our work model aspects of the two multi-billion-years eras.
Our work might explain seeming diculties that other work seems to exhibit regarding modeling aspects of the current multi-billion-years era of increasing rate of separation. Figure 6: Suggested eras and suggested DM:OM ratios for galaxies. The stage that associates with a row in the table precedes stages that associate with subsequent rows in the table. For each row, the leftmost two columns associate with aspects of the stage. The rightmost four columns associate with a noteworthy cause for the stage. The noteworthy cause might gain prominence before the stage starts. Some galaxies do not transit beyond some stages. Our work points to possible propensities for nature to form galaxies with DM:OM ratios of approximately one to zero-plus (that is, dark matter galaxies), ve-plus to one, four to one, and zero-plus to one. Galaxies that both had more than one original clump and had three original-clump isomers might tend to cease star formation earlier than do some other galaxies.   and-charge-current PROP-and-CURR pair is one.
The reach that associates with the charge-dipoleand-related-current PROP-and-CURR pair is one.
Each one of the STUI has, in eect, its own instance of each of these two electromagnetic-centric PROP-and-CURR pairs. Each STUI does not interact with any other STUI via either of these two electromagnetic-centric PROP-and-CURR pairs.
The notion of six isomers and the notion of instances of LRI PROP-and-CURR pairs seem to sufce to explain ratios of dark matter to ordinary matter. (Perhaps, preview table 20.) The two notions might suce to explain the size of the recent multi-billion-years era increases in the rate of expansion of the universe. We discuss prospectively some aspects, assuming that our work gains attention.
We discuss neutrino masses and oscillations.
Reference [12] discusses modeling and data about neutrino masses and oscillations.
We suggest neutrino masses. (Perhaps, preview We discuss gravitation.
Reference [13] discusses experimental tests of theories of gravity.
We suggest eects -associating with isomers of elementary particles and with reaches of components of gravity -that suggest that other modeling regarding gravity would not be adequately accurate for some circumstances. This essay discusses some such circumstances. We are uncertain as to the extent to which aspects that reference [13] or reference [14] discuss would tend to validate or refute aspects of our modeling that pertains to gravitation.
We use modeling -regarding gravity -that has some similarities to models that people associate with the term gravitoelectromagnetism.
(References [1] and [2] discuss gravitoelectromagnetism.) Our modeling regarding gravity has some similarities to models that use classical physics perturbations regarding Newtonian gravity. (Reference [15] deploys modeling that associates with non-spherical distributions of mass.) We discuss physics constants and properties.
Our work seems to interrelate some physics constants. (Perhaps, preview table 11 and table 13.) Our work seems to interrelate some properties, including via modeling that catalogs physics properties. (Perhaps, preview table 3 and table 9 We discuss phenomena during and after the lead-up to the current multi-billion-years era of increases in the rate of expansion of the universe. Various people suggest that concordance cosmology underestimates increases in the rate of expansion. (References [18], [19], [20], [21], and [22] discuss relevant notions.) We think that we point to a basis for the un- We discuss observations and models regarding galaxy formation.
Reference [24] discusses galaxy formation and evolution, plus contexts in which galaxies form and evolve. Reference [24] discusses parameters by which people classify and describe galaxies.
We suggest that -regarding galaxies -observations of ratios of dark matter to ordinary matter might tend to cluster near some specic ratios.
(Perhaps, preview table 20.) Our modeling seems to explain such ratios.
Our modeling suggests that ratios of dark matter to ordinary matter might reect fundamental aspects -of nature -that concordance cosmology modeling does not include. Here, a key aspect is that of isomers. (Perhaps, preview table 20.) Reference [24] seems not to preclude galaxies that have few ordinary matter stars. Reference [24] seems not to preclude galaxies that have little ordinary matter.
We think that dark matter to ordinary matter ratios that our modeling suggests are not necessarily incompatible with veried concordance cosmology modeling.
We discuss observations and models regarding interactions between galaxies.
Reference [25] suggests that concordance cosmology modeling might not adequately explain gravitational interactions between neighboring galaxies. We suggest that notions pertaining to reaches and isomers might help to bridge the gap between observations and concordance cosmology modeling.
We think that our work points to a possible opportunity to study harmony between results based on established kinematics models and results based on our notions of components of gravity.

Methods
This We imagine two non-moving objects -object A and object B -that are located a distance r from each other. Each object has non-zero charge and non-zero mass. We consider the impacts of eldssuch as electromagnetism or gravity -generated by object A on object B.
The electric potential that aects object B varies as r −1 . The gravitational potential that aects object B varies as r −1 .
We imagine hypothetical eects that associate with hypothetical interactions by object B with a hypothetical combination -produced by object Aof electric eld and gravitational eld. We imagine that the potential that associates with these interactions varies as r −1 times r −1 , which equals r −2 .
We discuss aspects of hypothetical particles that might intermediate interactions between object A and object B. We use the two-item term object C to denote such a hypothetical particle. We imagine that objects C traverse straight-line trajectories from object A to object B. We use the word axis to associate with the straight line.
We imagine objects C that have some similarities to and some dierences from either an atom or a solar system. One or more components of an object C orbit a point that is central to object C.
An object C exhibits orbitals. We imagine that, with respect to the axis that runs from object A to object B, each orbital associates with a unique magnitude l o ℏ of orbital angular momentum. Here, l o is a positive integer. Up to one entity can associate with (or, occupy) an orbital. The integer l max denotes the maximum value of l 0 that associates with an occupied orbital. Relative to the axis that runs from object A to object B, the angular momentum that associates with an occupied orbital is one of −l o ℏ and +l o ℏ. (We exclude -for the occupied orbital that associates with l o -values of l for which −l o < l < +l o .) The angular momentum that associates with an unoccupied orbital is 0ℏ. (Regarding considering the object to be atom-like, the following notions pertain. The nucleus has zero spin. Entities that occupy orbitals have zero spin. Entities that occupy orbitals do not interact with each other.) Relative to the axis, the total angular momentum that associates with an object C is the sum -over the occupied orbitals -of the respective ±l o ℏ.
Regarding modeling that we discuss below, the following notions pertain. We use the symbol Γ to denote an ascendingorder list of the relevant o ∈ O. Within a list Γ, we separate values of l o by using the symbol`. The symbol l max denotes the maximum value of l o in Γ.
For example, Γ = 1'3 associates with l max = 3 and with 1 ∈ O, 2 / ∈ O, and 3 ∈ O. We dene l Σ to be the sum of the various values of ±l o . The expression l Σ = o∈O (±l o ) pertains.
We dene Σ to be the absolute value of the sum of the various values of ±l o . The equation Σ = |l Σ | = | o∈O (±l o )| pertains. We associate the word solution with the notion of Σ = |l Σ | = | o∈O (±l o )|. The term two-word Diophantine equations associates with the modeling that we pursue. Table 1 alludes to all l Σ = o∈O (±l o ) expressions for which 1 ≤ l o ≤ l max ≤ 4 and no two values of l o are the same. The rightmost ve columns dis- We use the symbol ΣgΓ to denote the combination of a list Γ and a relevant value of Σ. The letter g anticipates an association with electromagnetism and an association with gravity. (Perhaps, think of g as in gamma rays and g as in gravity. Perhaps, anticipate that 1gΓ associates with electromagnetism and that 2gΓ associates with gravity.) We associate the symbol Σg with solutions of the form ΣgΓ. We associate the symbol Σg' with Σg solutions for which Σ ∈ Γ. We associate the symbol Σg with Σg solutions for which Σ / ∈ Γ.

We develop modeling that associates with intrinsic electromagnetic and gravitational
properties of objects and with aspects of electromagnetic and gravitational elds.
We explore the notion that some solutions that  We explore the notion that some solutions that We posit the following associations. 1g associates with electromagnetism. 2g associates with gravitation. Each ΣgΓ solution (or, Σ = |l Σ | = | o∈O (±l o )| solution) associates with two l Σ = o∈O (±l o ) expressions. We associate l Σ < 0 with left-circular polarization. We associate l Σ > 0 with right-circular polarization. Table 2 discusses interpretations -regarding properties of an object -regarding Σg' solutions for which 1 ≤ Σ ≤ 2 and 1 ≤ l max ≤ 4. Table 2 suggests two uses for the words monopole, dipole, quadrupole, and octupole. One use associates with mathematics and with table 1.
One use associates with physics and with the dependence of potentials that associate with the modeling of components of LRI-centric interactions (or, LRI forces).
We posit that a solution associates with a socalled RDP of the form Ξ −nΣgΓ . RDP stands for radial dependence of potential. Here, we consider Newtonian modeling for potentials (as in potential energy) that associate with elds (such as the electromagnetic eld and the gravitational eld) that an object produces. For a solution other than a monopole solution, the potential can (and generally does) vary based on angular coordinates (as well as based on a radial coordinate). We posit that Ξ −1 = r −1 , in which r is the spatial distance from the object. (We provide a cautionary note regarding terminology. Per table 2, we associate the solution for which Σ is one and Γ is 1`2`4 with each one of the following: Ξ −3 and hence mathematical quadrupole, r −3 and hence a behavior of potential that associates with a notion of quadrupole, and a physics object that associates with a magnetic dipole that rotates around an axis that does not equal the axis that associates with the magnetic dipole. One way to think about the seeming tension between quadrupole and dipole is to associate the factor Ξ −1 that associates with l o = 4 with (ct) −1 instead of with r −1 . Here, c denotes the speed of light and t denotes the time that light takes to go from the magnetic-dipole object to the distance r from the object. This interpretation has consistency with the notion that the relevant quadrupole component of the electromagnetic eld associates with an object that people might characterize as having the properties of a magnetic dipole.) 2.1.4. We extend our modeling to include extrinsic properties of objects.
We deploy the symbol PROP to associate with ΣgΓ solutions that we associate with intrinsic properties of objects. We deploy the symbol CURR to associate with ΣgΓ solutions that we associate with currents of properties.
We anticipate extending the notions of PROP and CURR to apply widely regarding modeling regarding LRI. We anticipate that, for each LRI PROP solution, there is an LRI CURR solution.
Notions of three degrees of freedom seem to pertain regarding solutions that table 2 shows.
The following examples -of three degrees of freedom -pertain regarding 1g' solutions. Regarding Table 1: Σ = |l Σ | = | o∈O (±lo)| solutions, assuming that 1 ≤ lo ≤ lmax ≤ 4 and that no two values of lo are the same. The columns labeled l 1 through l 4 show contributions toward expressions l Σ = o∈O (±lo). In those four columns, the symbol 0 is a placeholder for an unused pair, −lo and +lo, of values. The symbol n o0 denotes the number of times the symbol 0 associates with an lo for which 1 ≤ lo ≤ lmax. The symbol n Γ denotes the number of elements in the list Γ. For each row, there are 2 n Γ possible ways to assign signs regarding the set of n Γ terms. There are 2 n Γ expressions of the form l Σ = o∈O (±lo). Thus, there are 2 n Γ −1 solutions Σ = |l Σ | = | o∈O (±lo)|. The Σ column shows values of Σ that associate with solutions. For example, for lmax = 2 and Γ = 1'2, the two solutions feature, respectively, Σ = 1 (as in 1=| − 1 + 2|) and Σ = 3 (as in 3=| + 1 + 2|). The number n ΣgΓ equals 2 n Γ −1 and states the number of solutions. The column for which the one-word label is notion refers to the number of solutions. For monopole, one solution pertains. For dipole, two solutions pertain. For quadrupole, four solutions pertain. For octupole, eight solutions pertain. For the case of octupole, each one of Σ = 2 and Σ = 4 associates with two solutions.  Table 2: Interpretations -regarding properties of an object -regarding Σg' solutions for which 1 ≤ Σ ≤ 2 and 1 ≤ lmax ≤ 4.
We suggest the following notions. 1g1 associates with a component -of the electromagnetic eld that the object produces -that associates with the object's charge. The word scalar associates with this solution. 1g1`2 associates with the object's magnetic eld. An axis associates with that eld. The one-element term 3-vector associates with this solution. (For a bar magnet, the notions of charge and rotation do not necessarily pertain.) 1g1`2`4 associates with a combination of magnetic eld and rotation (over time) of the axis of the magnetic eld. (The Earth is an object for which the axis of rotation does not equal the axis of the magnetic eld.) The one-element term 3-vector associates with that rotation. 2g2 associates with the object's mass. The word scalar associates with this solution. 2g2`4 associates with rotation of the object's mass. An axis associates with that rotation. The one-element term 3-vector associates with this solution. (Regarding general relativity, this solution associates with aspects of rotational frame dragging.) 2g1`2`3 associates with a non-spherically symmetric distribution of mass. 2g1`2`3`4v associates with rotation (of a non-spherically symmetric distribution of mass) around a minor axis of moment of inertia. The one-element term 3-vector associates with that rotation. 2g1`2`3`4w associates with rotation (of a non-spherically symmetric distribution of mass) around a major axis of moment of inertia. The one-element term 3-vector associates with that rotation. (Regarding general relativity, each of 2g1`2`3`4v and 2g1`2`3`4w might associate with aspects of rotational frame dragging.) For gravity produced by an object like the Sun, 2g' solutions other than 2g2 associate with adjustments with respect to the gravity that associates with 2g2. Regarding large-scale gravitation, 2g' solutions other than 2g2 can associate with gravitational eects that dominate gravitational eects that associate with 2g2.
1g1`2, three degrees of freedom pertain. Two degrees of freedom associate with the orientation of the magnetic moment 3-vector. One degree of freedom associates with the magnitude of the magnetic moment 3-vector. Compared to 1g1`2, 1g1`2`4 has three more degrees of freedom. Two degrees of freedom associate with the orientation of the angular velocity 3-vector. One degree of freedom associates with the magnitude of the angular velocity 3-vector.
Regarding each of the solutions that table 2 shows, l o = 4 seems to associate -regarding rotation -with three degrees of freedom.
We suggest that -for some aspects of our modeling -three degrees of freedom, mathematics associating with two one-dimensional harmonic oscillators, and mathematics associating with the group SU (2) associate with each other. (For integers l such that l ≥ 2, reference [26] interrelates mathematics associating with l one-dimensional harmonic oscillators and mathematics associating with the group SU (l).) Here, we consider that one oscilla- Regarding CURR solutions that associate with PROP modeling for which l max ≤ 4, we posit that 8 ∈ Γ associates with velocity.
The notion that 8 ∈ Γ associates with velocity extends a series that seems to pertain regarding properties. The series includes charge, mass, intrinsic angular velocity, and velocity. For some of our modeling regarding electromagnetism, 1 ∈ Γ associates with charge. For some of our modeling regarding gravity, 2 ∈ Γ associates with mass. For some of our modeling, 4 ∈ Γ associates with intrinsic angular velocity (or -associating with some modeling -with intrinsic angular momentum). For some of our modeling for which l max = 8, 8 ∈ Γ associates with extrinsic velocity.
Each l o in the series 1, 2, 4, and 8 associates with the notion that log 2 (l o ) is an integer. We explore notions regarding other values of l o .
We discuss notions regarding l o = 3. 1g1`2 is a PROP solution that associates with intrinsic (nominal) magnetic moment. (See table 2 (Perhaps, preview table 3, table 7, and   table 8.) Along with charge, mass, internal angular velocity (or internal angular momentum), and velocity, people measure or infer -regarding objects -energy. Per discussion above regarding a series of l o for which log 2 (l o ) is an integer, we posit that -for PROP solutions -l o = 16 ∈ Γ associates with energy. We posit that -for modeling that comports with special relativity -l o = 32 ∈ Γ associates with momentum.
2.1.6. We list and discuss some solutions that associate with l max ≤ 32. had maintained a lack of natural oblateness.
One case associates with the notion of the socalled natural oblateness that we just posited.
The last two cases illustrate the notion that, for modeling based on 2g`PROP solutions, 4 ∈ Γ associates with reducing gravitational eects and thereby associates with gravitational repulsion.
Specically, for a 2gΓ b and a 2gΓ a for which 2 ∈ Γ a , 2 ∈ Γ b , 4 / ∈ Γ a , and Γ b equals (in eect) Γ a '4,  Table 3 extends  table 2 so as to include -for each PROP Σg' solution -a CURR Σg' solution. For example 1g7`8 satises Σ = 1 and associates with -for an object that produces an electromagnetic eld -the charge-current 3-vector that complements the scalar charge that associates with PROP and with 1g1. No one ΣgΓ solution associates with both PROP and CURR. The symbol n Γ,PROP denotes the number of elements in the Γ that associates with PROP. Table 3 lists an RDF -or radial dependence of force -for each PROP solution. For a CURR ΣgΓ solution, the RDP (or, radial dependence of potential) equals Ξ −1 times the RDP for the associated PROP ΣgΓ solution. For each one of PROP and CURR, the RDF equals Ξ −1 times the RDP. For example, for each of 1g1 and 2g2, the RDF is Ξ −2 , which is r −2 . Table 3 shows properties -of objects that produce ΣgΓ components of Σg -that associate with the PROP solution. The and internal angular velocity Anomalous magnetic moments Anomalous magnetic moments the following notion pertains. Eects that associate with 2gΓ b decrease the gravitational attraction that associates with 2gΓ a .
Regarding 3g',        exclude. We suggest that mutual -for gluons and jay bosons -association with the strong force associates with the notion that the 1J PROP solution equals the 1U PROP solution. We suggest that the jay boson can interact with each fermion elementary particle, regardless of whether the notion of color charge pertains for the elementary fermion. We suggest that the jay boson can interact with each fermion non-elementary-particle object. In each one of the leftmost two columns in the table, two uses of a pair of the symbol † points to dual use of a solution. 0 = . . ., re 0gΓ for PROP 0 = . . ., re 0gΓ for CURR n Γ,PROP Family Bosons n EP 1J Jay 1   We use a two-word phrase isomer number to denote one isomer. Here, number can be any one of zero, one, . . ., and ve. We associate the two-word term isomer zero with the isomer that includes ordinary matter. We use the two-word phrase alt isomer to denote any one of the ve isomers that does not associate with ordinary matter.

2.2.3.
We discuss modeling -regarding simple elementary particles -that might associate with the notion of six isomers of simple elementary particles. All six isomers produce and interact with a common notion of gravity. We suggest that one instance of 2g2 mediates interactions between all six isomers.
We say that one instance of 2g2 has a reach of six, as in six isomers. We suggest that each isomer associates with its own instance of 1g1 and its own instance of 1g1`2. We say that each instance of 1g1 has a reach of one, as in one isomer. Each instance of 1g1`2 has a reach of one. Each isomer -including the ordinary matter isomer -scarcely interacts with any other isomer via electromagnetism.
We address the topic of reach for each ΣgΓ to which table 1 alludes. Based on the reach of 1g1 and the reach of 1g1`2, we suggest that n o0 = 0 associates with a reach of one. Based on the reach of 2g2, we suggest that n o0 = 1 associates with a reach of six. We posit that, for n o0 ≥ 1, the reach (of one instance of a relevant PROP ΣgΓ) equals the number of generators of the group SU (7) divided by the number of generators of the group SU (2n o0 +1).
For an integer l that is at least two, the number of generators of the group SU (l) is l 2 − 1. The reach that associates with n o0 = 2 is two. The reach that associates with n o0 = 3 is one. The number of instances of a PROP ΣgΓ component of a ΣG elementary particle is six divided by the reach that associates with the PROP ΣgΓ solution.
We assume that the reach of a CURR counterpart solution to a PROP ΣgΓ solution is the same as the reach of the PROP ΣgΓ solution.
We address the reach of the 2g1`2`3`8`16 PROP solution, which table 3 lists and table 1 does not list. For 2g1`2`3`8`16, each of 1, 2, and 3 appears in Γ and 4 does not appear in Γ. We assume that n o0 = 1. The reach for 2g1`2`3`8`16 is six. Energy-and-momentum 4-somes can associate with stusuch as a galaxy cluster -that associates with all of the six isomers. Energy-and-momentum 4-somes can associate with stu -such as a simple elementary particle -that associates with less than all of the six isomers.
The reach for 2g1`2`3`4`8`16 is one. Energy-decayand-momentum-change 4-somes might -if modeling based on them has physics relevance -associate with single-isomer stu -such as an atomic nucleus. Table 9 shows the reach (ρ I ) for -and other information about -each one of some solutions that Regarding the notion of a reach, ρ I , of two, there are three instances of the PROP solution. We number the isomers so that one instance of the 1g3`4 solution intermediates interactions between isomer zero and isomer three. One instance of the 1g3`4 solution intermediates interactions between isomer one and isomer four. One instance of the 1g3`4 solution intermediates interactions between isomer two and isomer ve.
We use notation of the form Σ(ρ I )gΓ to denote a ΣgΓ solution and the reach ρ I that associates with one modeling use that features an instance of the solution. For example, 2(2)g2`4 pertains regarding 2g2`4. We extend use of such notation to non-LRI elementary particles. For non-LRI elementary particles, the reach is one and notation of the form S(1)Φ pertains.
We posit that -for each Σ(2)gΓ solution -one instance of the solution intermediates interactions between isomer zero and isomer three.    Our work shows that a mass -so-called m(18, 3) -seems to have meaning beyond the notion that -for the mass m(18, 3) -gravitational attraction between two Q = 1 identical elementary fermions would be three-quarters of the electrostatic repul- have entries in the form of a name of one particle or a name of a set of more than one particle, followed (in parentheses) by a number of particles, followed by a symbol for the family of particles. NYN denotes not yet named. NYD denotes not yet detected. One might assert that people know of some NYD particles, at least indirectly. The word free associates with modeling that features PROP solutions for which 16 / ∈ Γ. The word entwined associates with modeling that features PROP solutions for which 16 ∈ Γ. Each ΣG particle for which Σ ≤ 3 associates with more than one PROP solution. For 1G, some modeling (for example, regarding light in a laser cavity) might associate with entwined. (     can vary based on charge, but not based on mass. We explore the notion that one can express a cl , the anomalous magnetic moment for the cl charged lepton, via the expression a 7 +a 6 t cl . Here, a 7 might vary only with charge and would be a constant with respect to a choice between cl = e (for the electron), cl = µ (for the muon), and cl = τ (for the tau). Here, a 6 might vary only with mass. We assume that t cl is (log(m cl /m e )) 2 . (Perhaps, compare with table 12 and with aspects -that comport with squares of properties -of table 13. The notion of squares of properties might associate with notions of self-interactions.) Based on data that reference [8] provides regarding the electron and the muon, we calculate a 7 and a 6 . Then, we calculate a value, a τ,PM , for a τ . Here, PM denotes the two-word term proposed modeling. PM associates with our work.
Reference [27] provides, based on Standard Model modeling techniques, a rst-order result -which we call a τ,SM -for a τ . Here, SM denotes the two-word term Standard Model. The value of a τ,PM results in a value of (a τ,PM − a τ,SM )/a τ,SM of approximately −0.00228. Each of a τ,PM and a τ,SM comports with experimental data that reference [8] that underlies aspects of table 12, table 13, and table 14.) Table 12 and table 13  We discuss possible masses for heavy neutrinos.
For purposes of estimating or calculating masses, neutrinos associate with a value of l m for which −6 ≤ l m ≤ −3. Charged leptons associate with 0 ≤ l m ≤ 3. If heavy neutrinos associate with 6 ≤ l m ≤ 9, a lower bound on rest energies for heavy neutrinos might be m(6, 3)c 2 ∼ 6 × 10 3 GeV, which might be large enough to comport with limits that associate with observations. (References [28] and [29] discuss limits that observations may set. People have not detected 0.5M particles.) To the extent the lower bound associates with m(6, 3/2)c 2 , the lower bound would be ∼ 2.5 × 10 9 GeV.
3.1.5. We discuss a possible limit regarding the spins of elementary particles that intermediate long-range interactions. Table 15 suggests the possibility that -for LRI elementary particles ΣG -Σ might be no greater than four. Table 12: Values of log 10 (m particle /me) for known charged elementary fermions. Regarding avour, this table generalizes, based on terminology that associates with charged leptons and neutrinos. For example, people use the term electron-neutrino. The symbol l f numbers the three avours. The l f (0.5C 1 ) terms pertain for fermions in the 0.5C 1 family. The symbol 0.5Q >0 denotes the pair 0.5Q 1/3 and 0.5Q 2/3 . The l f (0.5Q >0 ) terms pertain for quarks (or, elementary particles in the two families 0.5Q 2/3 and 0.5Q 1/3 ). lm is an integer parameter. The domain −6 ≤ lm ≤ 18 might have relevance regarding modeling. Q denotes the magnitude of charge, in units of |qe|. The family 0.5C 1 associates with Q = 1. The family 0.5Q 2/3 associates with Q = 2/3. The family 0.5Q 1/3 associates with Q = 1/3. Regarding the rightmost four columns, items show log 10 (m particle /me) and -for particles that nature includes -the name of an elementary fermion. For each † case, no particle pertains. Each number in the column with label Q = 1/2 equals the average of the number in the Q = 2/3 column and the number in the Q = 1/3 column. The notion of geometric mean pertains regarding the mass of the Q = 2/3 particle and the mass of the Q = 1/3 particle. Regarding each † case, a formula for m(lm, lq) calculates this number. Regarding the formula, the domain 0 ≤ lq ≤ 3 pertains. Regarding table 12, lq = 3Q pertains. Table 13 shows the formula.

Main calculation
These calculations produce numbers that table 12 shows.

Neutrinos
We suggest masses for the three 0.5N neutrinos.
People suggest -based on observations -that the sum of the three neutrino rest energies is at least approximately 0.06eV/c 2 and not more than approximately 0.12eV/c 2 . People suggest that astrophysics data suggests that at least two distinct masses pertain. We oer two possibilities.

Monopole properties
A force strength factor of 4 seems to associate with 1g1 and a force strength factor of 3 seems to associate with 2g2. (See, above, the equation (4/3) × (β 2 ) 6 = ((q e ) 2 /(4πε 0 ))/(G N (m e ) 2 ).) Possibly, other force strength factors would be 2 for 3g3, 1 for 4g4, and 0 (or, zero) for 5g5. Possibly, the notion of zero force strength regarding 5g5 associates with a lack of relevance for (and a lack of monopole properties that would associate with) solutions ΣgΣ for which Σ ≥ 5 and with a lack of LRI elementary particles ΣG for which Σ ≥ 5.

Dark matter
This unit suggests specications for dark matter.
Regarding each l that is at least one, we assume that the elementary particles in isomer l matchwith respect to mass -the elementary particles in isomer zero.
For 0 ≤ l ≤ 5, we associate the quarks in isomer l with three values of l m . (See table 12 and table  13.) The values are 3l + 0, 3l + 1, and 3l + 2. Across the six isomers, quarks associate with each value of l m that is in the range 0 ≤ l m ≤ 17. Regarding quarks and avours, we assume that -within isomer l -avour 1 associates with l m = 3l, avour 2 associates with l m = 3l + 1, and avour 3 associates with l m = 3l + 2.
Aspects of table 12 and table 13 point to the possibility that means for matching avours and masses for charged leptons do not match means for matching avours and masses for quarks. For charged leptons, isomer zero does not have a charged lepton that associates with l m = 1 and does have a charged lepton that associates with l m = 3. We assume that -for each l -a charged lepton associates with each of l m = 3l + 0, l m = 3l + 2, and l m = 3l + 3.
We assume that -for each isomer l such that 1 ≤ l ≤ 5 -the charged-lepton avour that associates with l m = 3(l) + 0 equals the avour that associates with the isomer l − 1 charged lepton that associates with the same value of l m and -thuswith l m = 3(l − 1) + 3. We assume that across the six isomers, one cyclical order pertains regarding avours for charged leptons. Table 16 shows, for isomers of charged elementary fermions, matches between masses and avours.
Beyond the topic of avours, the topic of handedness exists. Ordinary matter associates with lefthandedness. Our modeling suggests the possibility that isomers 0, 2, and 4 associate with lefthandedness and that isomers 1, 3, and 5 associate with right-handedness. 3.2.2. We prepare to discuss the evolution of stu that associates with each isomer. DMAI associates with the notion that -regarding isomer zero -these particles measure as being dark matter and do not measure as being ordinary matter.
We use the three-element term isomer number Our work does not necessarily suggest that a two-orthree-hadron hadron-like particle can include both at least one quark and at least one arc.
0.5M particles model as free. (See table 8.) Regarding each one of the six isomers, we suggest that stu made from DMAI behaves within bounds for dark matter that associate with concordance cosmology.
3.2.3. We discuss -for each dark matter isomerthe evolution of stu that associates with that isomer.
Here, we use the two-word term alt isomer to designate an isomer other than isomer zero and isomer three.
A charged baryon that includes exactly three avour 3 quarks is more massive than the counterpart zero-charge baryon that includes exactly three   Here -and nowhere else in this essay -the letter g associates with gluons. Here -and nowhere else in this essay -the symbol γ associates with the photon.
We discuss the evolution of isomer three OMSE stu.
The following possibilities pertain. The evolution of isomer three OMSE stu parallels the evolution of ordinary matter (or, isomer zero OMSE stu ). The evolution of isomer three OMSE stu does not parallel the evolution of ordinary matter (or, isomer zero OMSE stu ). The second possibility might associate with -for example -a dierence in handedness -with respect to charged leptons or with respect to W bosons -between isomer three and isomer zero. (See discussion related to table 3.3.2. We provide perspective regarding long-range interactions between objects. As two objects move away from each other, the relative eect of an RDF Ξ −(k+1) component decreases compared to the eect of an RDF Ξ −k component. One might associate the two-word phrase time period with a time range in which an RDF Ξ −l component provides dominant eects. Assuming that objects move away from each other and that one time period associates with Ξ −(k+1) and another time period associates with Ξ −k , the time period that associates with Ξ −(k+1) comes before the time period that associates with Ξ −k . Two smaller objects (such as galaxies) transit similar time periods more quickly than do two larger objects (such as galaxy clusters).
3.3.3. We discuss known and suggested eras in the history of the universe.  [30] and [10].
For data and discussion about the two multi-billionyears eras, see references [31], [32], [33], and [34].) Table 18 suggests details regarding eras to which   table 17 alludes. Before ination, boson PROP solutions for which Σ ≥ 2 and 8 ∈ Γ associate with dominant long-range eects. The word entwined associates with those PROP solutions. After ination, compared to boson PROP solutions for which Σ ≥ 2 and 8 / ∈ Γ, boson PROP solutions for which Σ ≥ 2 and 8 ∈ Γ do not associate with signicant long-range eects. Boson PROP solutions for which Σ ≥ 2 and 8 ∈ Γ continue to associate with relevant eects, but just on small (distance) scales. The word free associates with PROP solutions for which Σ ≥ 2 and 8 / ∈ Γ. Perhaps, a notion of a phase changefor the universe -pertains regarding times around ination. Figure 10 interrelates isomers of elementary particles, components of gravity, eras in the evolution of the universe, and eras in the evolution of galaxies.
(Regarding galaxies, perhaps preview discussion related to table 19.)

Formation and evolution of galaxies
This unit suggests that our notions regarding long-range interactions and our specications for dark matter combine to provide insight regarding galaxy formation and galaxy evolution.

We suggest aspects regarding events leading
to the formation of a galaxy.
Reference [35] suggests that galaxies form around early clumps of stu. The reference associates the word halo with such clumps. -which is attractive might contribute to the formation of smaller-scale clumps. The reach that associates with 2g1`2`3 is one.
We suggest that each one of many early halos associates with one isomer. We associate with such early halos the three-element term one-isomer original clump. We know of no reason why the six isomers would not form such clumps approximately equally.
(Concordance cosmology suggests that known elementary fermions form early in the era in which eects that associate with 2g1`2`3 dominate regarding large-scale phenomena. Per remarks above, we suggest that that era starts after the formation of halos. Also, we suggest that our scenario does not depend on whether or when 0.5M particles rst form.) Table 19 discusses suggestions regarding the formation and early evolution of a galaxy for which a notion of a one-isomer original clump pertains. Table 17: Eras regarding the rate of separating of large clumps. The rightmost two columns suggest eras. (Table 18 discusses aspects that associate with each of some eras.) Subsequent rows associate with later eras. The word ination names the era that associates with the third row in the table. Regarding eras that would precede ination, our modeling points to the possibility for the two eras that the table discusses. Concordance cosmology suggests ination and the next two eras. Regarding ination, people hypothesize this era. People suggest that the inationary era started about 10 −36 seconds after the Big Bang. People suggest that the inationary era ended between 10 −33 seconds after the Big Bang and 10 −32 seconds after the Big Bang. Possibly, no direct evidence exists for this era. Observations support the notions of the two billions-of-years eras. TBD denotes to be determined. The symbol † denotes a possible association between the relevant era and the notion of a Big Bang. The leftmost four columns describe phenomena that our modeling suggests as noteworthy causes for the eras. (Regarding phenomena that associate with gravitation, table 17 echoes aspects -including aspects regarding attraction and repulsion -that table 5 and table 9 show.) An RDF associates with the PROP solution.
Generally, a noteworthy cause associates with notions of acceleration. Generally, an era associates with a range of velocities.
A noteworthy cause may gain prominence before an era starts.

Force
Would decrease - Table 18: Details regarding eras regarding the rate of separating of large clumps. Table 17 discusses the eras. Table 18 de-emphasizes the notion that 0.5M elementary fermions might form before the beginning of the rst multi-billion-years era. Each of the symbols 2g1`2`3`4x and 2g1`2`3`4y denotes either or both of 2g1`2`3`4v and 2g1`2`3`4w.

Rate of separating Note
Is negative Possibility: 2g1`2`3`8`16 and its compacting of some form of energy lead to conditions suitable for the universe to form and evolve.
Possibility: The value of six for ρ I associates with setting up a system for which roughly equal creation of isomers pertains.
Possibility: Isomers of 0.5R and 1J form.
Possibility: The following interactions might characterize this era. For each interaction, the net circular polarization for each of before and after the interaction might be zero. Presumably, the formation of gluons (or, 1(1)U) could associate with the formation of arcs (or, 0.5(1)R)).
Possibility: The six isomers of 0.5R populate approximately equally.
Possibility: Some clumps of 0.5R stu serve -eventually -as seeds for galaxies.
Turns positive 0g1`3`4`8 associates with the 1J (or, jay) boson. The jay boson associates with the notion of Pauli repulsion.
Possibility: 1J bosons stop the implosion of stu that is signicantly 0.5R particles.
The following interaction might characterize this era. Here, the net circular polarization for each of before and after the interaction might be two.
Possibility: The six isomers of 0I populate approximately equally.
Possibility: Aspects of this era associate with notions of a Big Bang.
Increases rapidly Some concordance cosmology modeling suggests that inatons provide the major component of stu.
Decreases Some concordance cosmology modeling suggests that the rst signicant appearance of most known elementary particles occurs early in this era.

Increases -
Would decrease This essay does not try to explore the possibility that (or to estimate a time at which) a transition -for the largest observable objects -from repulsion based on 2g2`4 to attraction based on 2g2 might occur. Figure 10: Isomers of elementary particles, components of gravity, eras in the evolution of the universe, and eras in the evolution of galaxies. Some current galaxies did not transit beyond the rst era regarding the evolution of galaxies.
The original clump repels (some) stu that associates with the isomer that associates with the original clump and (most) stu that associates with one other isomer.
Attractive 2g2 The original clump attracts stu that associates with the four not-repelled isomers and stu that associates with the isomer that associates with the original clump.
Presumably, some galaxies form based on two or more clumps, for which all of the clumps associate with just one isomer. Presumably, some galaxies form based on two or more clumps, for which some clumps associate with isomers that are not the same as the isomers that associate with some other clumps.
3.4.2. We suggest aspects regarding the evolution of galaxies.
We suggest two eras regarding the evolution of galaxies. The rst era associates with the rst two rows in table 19. The second era associates with the 2g2 attractive force that associates with the third row in table 19.
Some galaxies do not exit the rst era and do not collide with other galaxies.
Many galaxies result from aspects associating with the 2g2 attractive force that associates with the third row in table 19. We discuss three cases.
(Mixed cases and other cases might pertain.) Each of some era one galaxies does not collide with other galaxies. Such a galaxy accumulates (via 2g2 attraction) stu associating with various isomers that have representation in nearby IGM (or, intergalactic medium).
The galaxy becomes an era two galaxy. The galaxy might include stu that signicantly associates with as many as ve isomers.
Each of some era two galaxies merges (via 2g2 attraction) mainly just with galaxies that feature the same ve isomers. The galaxy that merged, in eect, loses it status of being a galaxy. The resulting larger object is an era two galaxy. The galaxy might include stu that signicantly associates with as many as ve isomers.
Each of some era one or era two galaxies merges (via 2g2 attraction) with other galaxies. The galaxy that merged, in eect, loses its status of being a galaxy. The resulting larger object is an era two galaxy. The galaxy might include stu that signicantly associates with as many as six isomers.
3.4.3. We suggest an explanation for the quenching of star formation within some galaxies and the stopping of the accrual of matter by some galaxies.
People report the notion that some galaxies seem to stop forming stars. (See reference [36] and reference [37].) Such so-called quenching might take place within three billion years after the Big Bang, might associate with a relative lack of hydrogen atoms, and might pertain to half of a certain type of galaxy. (See reference [37].) Reference [38] discusses a galaxy that seems to have stopped accruing both ordinary matter and dark matter about four billion years after the Big Bang.
We suggest that the quenching and the stopping of accruing nearby matter might associate with repulsion that associates with 2(2)g2`4. Quenching might associate with galaxies for which original clumps featured isomer zero stu or isomer three stu. The galaxy that reference [38]  references [39] and [40].) We suggest that the undetected object might be a clump of dark matter.

Ratios of dark matter eects to ordinary matter eects
This unit shows that our specication for dark matter seems to explain observed ratios of dark matter eects to ordinary matter eects. Table 20 provides explanations for observed ratios of dark matter eects to ordinary matter effects. (For data and discussion regarding densities of the universe, see reference [8].
For data and discussion regarding galaxy clusters, see references [41], [42], [43], and [44]. For data and discussion regarding absorption of CMB, see references [45], [46], and [47]. For data and discussion regarding observed early galaxies, see references [48] and [49]. Reference [48] inuenced our choice of a time range to associate with the word early. For data and discussion regarding the combination of 0 + :1 and later, see references [50], [51], [52], [53], [54], and [55]. For data and discussion regarding observed dark matter galaxies, see references [35], [56], and [57]. Current techniques might not be capable of observing early dark matter galaxies. References [58] and [59] suggest, regarding galaxy clusters, the existence of clumps of dark matter that might be individual galaxies. Extrapolating from results that references [35] and [60] discuss regarding ultrafaint dwarf galaxies that orbit the Milky Way galaxy might suggest that the universe contains many DM:OM 1 : 0 + later galaxies. For data and discussion regarding galaxies for which ratios of ∼4:1 pertain, see references [61] and [62]. For data Many later galaxies 5 + : 1 Over time, galaxies collide. Collisions tend to result in the formation of larger galaxies that include much stu from smaller galaxies. A later galaxy that results from enough collisions is likely to associate with somewhat similar -across the six isomers -amounts of stu from originally one-(or few-) isomer original clump galaxies. and discussion regarding later galaxies for which ratios of 5 + :1 pertain, see reference [35]. References [63] and [64] provide data about collisions of galax- We consider interactions in which two jay bosons move in parallel, interact, and produce one aye boson plus something else. Here, we assume that conservation of angular momentum pertains and that one can de-emphasize orbital angular momentum.
We consider two cases. In the rst case, the two jay bosons have the same (one of either right or left) circular polarization. Conservation of angular momentum allows an outgoing combination of one 2G particle and one 0I particle. Conservation of angular momentum precludes producing one 1G particle and one 0I particle. In the second case, one jay boson has left circular polarization and the other jay boson has right circular polarization. Conservation of angular momentum allows the production of two 0I particles and prohibits the production of one 1G particle and one 0I particle.  People suggest that concordance cosmology modeling underestimates -for the second multibillion-years era -increases in the rate of expansion of the universe. (See references [19], [20], [21], [22], [72], [73], and [74].) We suggest the following explanation for such underestimates.
When using modeling based on general relativity, people might try to extend the use of an equation of state (or use of a cosmological constant) that works well regarding early in the rst multi-billionyears era. Regarding that time, our modeling suggests dominance by attractive eects that associate with the 2g1`2`3 component of gravity. The notion of a reach of one pertains. The symbol 2(1)g1`2`3 pertains. Our modeling suggests that -later in the rst multi-billion-years era -repulsive eects that associate with 2(2)g2`4 become signicant. Dominance by 2(2)g2`4 pertains by the time the second multi-billion-years era starts. However, people's use of an equation of state that has roots in the time period in which 2(1)g1`2`3 dominates would -at bestextrapolate based on a notion of 2(1)g2`4 (and not a notion of 2(2)g2`4). That modeling would underestimate the strength of the key driver -of expansion -by a factor of two.
We point -conceptually -to the following possible remedy.    We explore one way that our modeling might point to such symmetries.
Extant modeling associates a U (1) symmetry with the photon. U (1) associates with mathematics that associates with a one-dimensional harmonic oscillator. 2G associates with one elementary particle.    4.6.2. We discuss aspects related to the value of two for reach (or, ρ I ). This essay suggests that ρ I = 2 pertains for some components of long-range interactions (or, LRI). This essay suggests that the notion of ρ I = 2 might have importance regarding explaining data regarding the following -some depletion of CMB, large-scale clumping, the recent multi-billion-years era of increases regarding the rate of separation of large clumps, gravitational interactions between neighboring galaxies, and galaxy formation.

Conclusions
This unit summarizes aspects of our work and suggests perspective about our work.

Our modeling
Our modeling features two bases.
One basis unies and decomposes aspects of electromagnetism and gravity. For each of electromagnetism and gravity, the decomposition seems to associate well with properties -of objects -that people can measure and that extant modeling features. We suggest the possibility that the notion that our work suggests specications and data that extant modeling does not suggest points to possible usefulness for our work. Our suggestions include a specication for dark matter, specications for new elementary particles, and more (than current measurements provide) accurate masses for neutrinos and some other known elementary particles.
We suggest that the small set of bases for our modeling, the breadth of seemingly coherent scope of our modeling, the simplicity of relevant Diophantine equations, and the possible ease of integrating our modeling and extant modeling point to possible usefulness for our work.

Our work
Our work suggests augmentations -to physics modeling -that produce results that may provide progress regarding the following physics opportunities. Complete the list of elementary particles. Describe dark matter. Explain ratios of dark matter to ordinary matter. Explain eras in the history of the universe. Link properties of objects. Interrelate physics models. Table 23: Approximate relationships between modeling that can deploy elementary-particle properties and aspects of our modeling. n I denotes a number -one or six -of isomers. Extant modeling associates with n I = 1. Each one of some of the items in the symbol column does not associate with an extant modeling symbol. The symbol NNR denotes the three-word phrase not necessarily relevant. Regarding NEW, the symbol NNR associates with the notion that mass does not vary with velocity. Regarding CNC, 1g1 associates with charge and 1g7`8 associates with current. No other components have relevance. CNC associates with charge-current 4-vectors and with Maxwell's equations. Compared to CNC, QED adds associations with magnetic elds created by other than charge currents and associates with anomalous magnetic moments.
The symbol PEF associates with the three-word phrase Pauli exclusion force. We suggest that PEF associates with 1J, each 0.5Φ family, and fermions that are not elementary particles. WIP associates with 1W 1 and 1Z. We use our modeling to suggest explanations for data that other modeling seems not to explain.

Modeling
We use our modeling to suggest results regarding data that people have yet to gather.
The breadth and depth of the matched data might suce to justify using our modeling.
The breadth and unity -within itself and with physics modeling that people use successfully -of our modeling might support the usefulness of our Phys. Rev. X, 11(4):041050,