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The following are some important advantages of this theory:

- There is a small set of fundamental assumptions with very few independent parameters. This implies that almost all of the parameters can be derived from first principles. This has already been done for the fine structure constant.
- In this theory, quantum properties are a consequence of the dynamics of the branched manifold. We show that the discrete behavior of the branched manifold can be modeled using a continuous approximation, which is a path integral. Because the underlying system is discrete and finite, the pathological infinities of the path integral never appear.
- From the fundamental assumptions, we show that the manifold can spontaneously produce pairs of knots of the form \( \mathbb{R}^3 \# (S^1 \times P^2).\) We show that these knots have the properties of the elementary fermions. Different embedding types of these knots explain the different generations of fermions. We therefore find that the theory spontaneously produces the elementary particles without additional assumptions. Furthermore, the theory tightly constrains the types of knots that can be created, and we therefore can mathematically constrain the types of possible particles.
- The spacetime manifold is constrained by our assumptions, but it is under-constrained. We therefore expect the manifold to maximize entropy. Using this principle of entropic maximization, we can derive a description of the dynamics of the spacetime manifold. We find that the manifold behavior is best described by a Lagrangian. In that Lagrangian, we find terms for gravity (scalar curvature \(R\)) and electromagnetism (the term \(F^{\mu\nu} F_{\mu\nu}.\) Including the effects of particle geometry and topology produces electroweak and strong forces.

**We assume a Minkowski 6-space \(\Omega.\)**The metric on \(\Omega\) is \({\eta_{\mu\nu}=\mathrm{diag}(1, -1,-1,-1,-1,-1)}.\) The coordinates are \(x^\nu.\) The dimension of the Minkowski space is chosen to allow the spacetime manifold to form knots.**We assume a branched 4-manifold \(M\) embedded in \(\Omega.\)**A*branch*of \(M\) is any closed unbranched 4-manifold \(B\) without boundary that is contained in \(M.\) The manifold is branched so that it can produce quantum properties. The metric \(\bar{\eta}_{\mu\nu}\) on \(M\) is inherited from \(\Omega.\) For convenience of coordinates we assume that, if \(M\) is flat, then \(M\) is in the subspace spanned by \(x^0, x^1, x^2, x^3.\)**We assume non-self-intersection of each branch of \(M.\)**For any branch \(B,\) the branch \(B\) cannot intersect itself. This is necessary to prevent knots from spontaneously untying.**We assume a vector field \(A^\nu.\)**The field satisfies \(\det(A_{\alpha, \mu}A^\alpha_{\hspace{1mm},\nu})=-1.\)**We assume a conformal weight \(\rho.\)**Then we define the metric \({g_{\mu \nu}=\rho^2 A_{\alpha, \mu}A^\alpha_{\hspace{1mm},\nu}}\) and a Ricci curvature \(\hat{R}^{\mu\nu}\) based on \(g_{\mu\nu}.\)**We assume a constraint on \(g_{\mu\nu}\) relative to \(\eta_{\mu\nu}.\)**The metrics \(g_{\mu\nu}\) and \(\eta_{\mu\nu}\) define sets \(g^+\) and \(\eta^+,\) and we assume that \(g^+\) must intersect \(\eta^+.\) For a point \(p,\) the sets \(g^+\) and \(\eta^+\) are the sets of points at positive distance from \(p,\) with respect to the metrics \(g\) and \(\eta.\) (In the Minkowski space, the set \(\eta^+\) is the future light cone of the Minkowski metric \(\eta.\)) This constraint on the metrics produces an important constraint on the electromagnetic potential.**We assume Ricci flatness \(\hat{R}^{\mu\nu}=0\) for \(g_{\mu\nu}.\)**This assumption, and the next two assumptions, prevent multiple kinds of divergent behavior. For example, they prevent the manifold from expanding to arbitrarily large size in the Minkowski space, and they prevent the manifold from branching an infinite number of times.**We assume that the weight \(w=(-\det(g))^{1/2}=\rho^4\) is conserved at branching.****We assume a lower limit \(w\ge1.\)**This implies that the manifold can branch only a finite number of times.

In this theory, we assume a few constraints on the spacetime manifold. One important result
of
those constraints is that the ways that the manifold can change topology are tightly
constrained. If particles are knots on the manifold, then every production of
particle/anti-particle pair is a change of topology of the manifold. Therefore a particular
topology is only a viable particle topology if it can be created on the spacetime manifold
subject to the constraining assumptions.

We prove that it is possible to produce pairs of knots of the form \( \mathbb{R}^3 \# (S^1 \times P^2)\) subject to our constraining assumptions. The type \( \mathbb{R}^3 \# (S^1 \times P^2)\) is relatively simple, analogous to a twist in the manifold. We then go on to show that those knots have the same properties as the elementary fermions. It remains to be shown that no other topology type is possible. It may be that there are other possible particle topologies and the only reason we see just the type \( \mathbb{R}^3 \# (S^1 \times P^2)\) is that it has the lowest energy. While it is possible to mathematically eliminate many classes of particle topology, a complete list of possible particle topologies has not yet been determined.

We prove that it is possible to produce pairs of knots of the form \( \mathbb{R}^3 \# (S^1 \times P^2)\) subject to our constraining assumptions. The type \( \mathbb{R}^3 \# (S^1 \times P^2)\) is relatively simple, analogous to a twist in the manifold. We then go on to show that those knots have the same properties as the elementary fermions. It remains to be shown that no other topology type is possible. It may be that there are other possible particle topologies and the only reason we see just the type \( \mathbb{R}^3 \# (S^1 \times P^2)\) is that it has the lowest energy. While it is possible to mathematically eliminate many classes of particle topology, a complete list of possible particle topologies has not yet been determined.

They are not quite the same. The mathematical theory of knot theory has a very specific
meaning for the word "knot." A
knot is an embeddding of \(S^n\) in \(S^{n+2}.\) The constraint on the dimensions
(an \(n\)-manifold in an \(n\)+2-manifold) is a tight constraint. For any other
dimensionality,
the
embedding would not be knotted. For example, a knot of the form \(S^n\) in \(S^{n+3}\) would
spontaneously untie.

Every elementary fermion in this theory has topology \( \mathbb{R}^3 \# (S^1 \times P^2).\) This is not a knot in the sense of knot theory. Also, this 3-dimensional topology can be embedded in \( \mathbb{R}^n\) for \(n \geq 5.\) A hadron, for example a proton, contains linked copies of \( \mathbb{R}^3 \# (S^1 \times P^2),\) each of which is a quark. Those quarks will remain linked to each other only if the 3 spatial dimensions of the spacetime manifold are embedded in a 5-dimensional space. For that reason, we see that the dimension of the embedding space is perfectly constrained by the particle topology of hadrons.

In this theory, we need a word to describe particle topology. There are two reasons that we use the word "knot". The first reason is that the word is convenient and approximately describes the particle topology. The second reason is that, for hadrons, the 3-dimensional particle topology must be embedded in an 5-dimensional space, which is the same constraint (\(n=3\) and \(n+2=5\)) that applies in the mathematical theory of knot theory.

Every elementary fermion in this theory has topology \( \mathbb{R}^3 \# (S^1 \times P^2).\) This is not a knot in the sense of knot theory. Also, this 3-dimensional topology can be embedded in \( \mathbb{R}^n\) for \(n \geq 5.\) A hadron, for example a proton, contains linked copies of \( \mathbb{R}^3 \# (S^1 \times P^2),\) each of which is a quark. Those quarks will remain linked to each other only if the 3 spatial dimensions of the spacetime manifold are embedded in a 5-dimensional space. For that reason, we see that the dimension of the embedding space is perfectly constrained by the particle topology of hadrons.

In this theory, we need a word to describe particle topology. There are two reasons that we use the word "knot". The first reason is that the word is convenient and approximately describes the particle topology. The second reason is that, for hadrons, the 3-dimensional particle topology must be embedded in an 5-dimensional space, which is the same constraint (\(n=3\) and \(n+2=5\)) that applies in the mathematical theory of knot theory.

In this theory, all elementary fermions are knots in the spacetime manifold that have
topology \( \mathbb{R}^3 \# (S^1 \times P^2).\) Leptons are elementary fermions and
therefore
they also have this topology. In this theory, we assume that the spacetime manifold is
constrained by Ricci flatness, and we find that Ricci flatness strongly constrains the
geometry
of the particles. In particular, as charged leptons approach other particles, Ricci flatness
forces the lepton radius to shrink. As the distance of approach goes to zero, the lepton
radius
also goes to zero. For this reason, leptons appear to be pointlike in collisions.

This description allows us to describe the spin angular momentum of leptons in familiar terms. For the case of pointlike leptons, spin angular momentum must be considered to be an inherent quantity that is not obtained in the normal sense of \( p \times r,\) because the calculation would be meaningless if the lepton radius is \(r=0.\) In Knot Physics, the radius is non-zero and the spin angular momentum can be calculated as a function of field strength from first principles.

This description allows us to describe the spin angular momentum of leptons in familiar terms. For the case of pointlike leptons, spin angular momentum must be considered to be an inherent quantity that is not obtained in the normal sense of \( p \times r,\) because the calculation would be meaningless if the lepton radius is \(r=0.\) In Knot Physics, the radius is non-zero and the spin angular momentum can be calculated as a function of field strength from first principles.

The description of quantum mechanics here is very different from the usual description. We
assume that the spacetime manifold is branched and we get all of our quantumness from that
property. The viewpoint we adopt is that the path integral of quantum mechanics is useful
because it is a convenient simplification of the complex dynamical system, which is the
branched manifold. This is similar to saying that the heat equation is a convenient
simplification of the transfer of heat energy through the vibrating molecules of an atomic
lattice. The heat equation has certain pathological properties, for example energy is
transferred faster than light speed. The path integral has certain pathological properties,
for example various infinities that are difficult to tame. We hypothesize that the pathology
results from modeling a discrete system as a continuous one. With the branched manifold, a
single branch of the manifold is analogous to a single atom of the atomic lattice, and we do
not try to know its random contribution. We just try to model the ensemble statistically. If
we model the ensemble using a continuous approximation, the result reproduces the path
integral of quantum mechanics.

So far all the key properties of physics (all four forces, quantum mechanics, and the known
particles) have been shown to follow without requiring additional assumptions. There is a
very short list of assumptions, even in comparison to the Standard Model, and the
consequences of those assumptions have been adequate to describe every physical phenomenon
that has been attempted. Interestingly, over the course of the development of the theory,
the set of assumptions has become ever smaller as previously distinct descriptions were
shown to be unified by ever simpler models.

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## Learn More

### Theory Summary

An overview of the entire theory, from simple assumptions about the spacetime manifold through particles, quantum mechanics, and forces

Learn more

### Theory Summary

An overview of the entire theory, from simple assumptions about the spacetime manifold through particles, quantum mechanics, and forces

Learn more