# All in Action

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. The Natural Law of Maximal Energy Dispersal

_{t}= ∂/∂t) which is equivalent to the steepest directional (i.e., velocity

**v**) gradient (D =

**v**·∇). Along these paths the flows of energy are most voluminous and the free-energy minimum state will be attained in least time. In other words, the natural process directs itself along the resultant of forces. The statistical measure for the natural process is entropy S = k

_{B}lnP which is the logarithmic probability and hence an additive measure [15]. Since the most voluminous flows of energy will direct along the steepest directional descents, entropy will not only be increasing but it will be increasing in least time. This universal imperative is recognized in the maximum entropy production principle [16,17]. The maximal consumption of free energy implies also the maximum power principle [18]. Since the natural process follows the least-time path, it can be also formulated by the principle of least action.

_{B}to the process generator L = Tđ

_{t}Q in accordance with classical thermodynamics [8]. The probability in motion d

_{t}P = LP (Equation 1) is expressed using energy densities [19] so that each energy density ϕ

_{j}, present in indistinguishable numbers N

_{j}of j-entities, is assigned with ϕ

_{j}= N

_{j}exp(G

_{j}/k

_{B}T) where G

_{j}is relative to the average energy k

_{B}T of the system per entity. According to the scale-free formalism [20,21] each j-entity itself is regarded as a system of diverse k-entities. Thus each population N

_{k}is associated with ϕ

_{k}= N

_{k}exp(G

_{k}/k

_{B}T). These are bound forms of energy. When energy flows from one bound form to another, energy is in a free form.

_{j}, is a product ΠN

_{k}of embedded k-entities, each distinct type available in numbers N

_{k}. The energy difference between entities j and k is ΔG

_{jk}= G

_{j}− G

_{k}or g

_{jk}ΔG

_{jk}= G

_{j}− g

_{jk}G

_{k}when the j-entity is formed from indistinguishable (symmetrical) k-entities in degenerate numbers g

_{jk}. The change in vector potential, i.e., radiation ΔQ

_{jk}couples to the jk-transformation orthogonal, as indicated by i, to the scalar potential difference. The notation emphasizes, for example, that when an electron falls along the electric potential gradient, it will radiate light orthogonally to its path. Likewise, electromagnetic radiation as a transverse wave will induce a varying voltage in an antenna perpendicular to the direction of propagation. The notation that explicitly distinguishes scalar and vector potentials, is of no apparent value in equilibrium thermodynamics since in the thermodynamic steady state there is no net flux of energy between the system and its surroundings. Moreover, since the statistical mechanics of closed systems aims only to reveal the stationary-state partition, there is no need to include explicitly the vector potential because at the equilibrium the net flux vanishes. However, here when describing evolutionary processes as evolving landscapes, it is useful to make the explicit distinction between bound and free forms of energy.

_{B}:

_{jk}= Δμ

_{jk}− iΔQ

_{jk}, also known as affinity [24], is the motive force that directs the transforming flow d

_{t}N

_{j}from N

_{k}to N

_{j}by its scalar (chemical) potential Δμ

_{jk}= μ

_{j}− Σμ

_{k}= k

_{B}T(lnϕ

_{j}− Σlnϕ

_{k}) and vector ΔQ

_{jk}potential differences. The adopted approximation lnN

_{j}! ≈ N

_{j}lnN

_{j}− N

_{j}implies that lnP

_{j}is a sufficient statistic [25] for k

_{B}T when the j-system is allowed absorb or emit quanta without a marked change in its average energy content, i.e., A

_{jk}/k

_{B}T << 1. In its diverse populations N

_{j}and in the free energy terms A

_{jk}the system holds capacity C = TdS/dT to resist the changes in its average energy imposed by its surroundings at a different temperature.

**Figure 1.**A self-similar energy level diagram describes the nested hierarchy of nature where each j-system (a composite solid in blue color) is regarded as a system within systems that are all being ultimately composed of multiple elementary constituents (blue solids). All systems evolve via step-by-step jk-transformations toward more probable states by consuming mutual energy density differences contained in chemical potentials Δμ

_{jk}(horizontal arrows) and those in vector potentials ΔQ

_{jk}(vertical wavy arrows) relative to the surroundings. During this natural process, the entities will distribute on the energy levels so that the free energy is consumed in least time. Therefore at any given time the probability distribution P

_{j}closely outlines the maximum entropy partition of a statistical system. Exchange of entities at the same level (bow arrows) does not introduce a change in energy. Thus these reversible processes do not affect the average energy k

_{B}T.

_{j}will change at once, e.g., from fifty-fifty indeterminism to full certainty. Moreover, when interactions are insufficient to establish common k

_{B}T over a given period τ of time, the entities fail to form a system but remain as constituents that surround statistical systems at a lower level of hierarchy where interactions are more frequent and intense [27,28,29].

_{jk}≠ 0 [14,30]. In view of that a net non-dissipative system is stationary. The maximum entropy state is stable so that an internal perturbation δN

_{j}away from the steady-state population N

_{j}

^{ss}will induce returning forces and opposing flows, i.e., dS(δN

_{j}) < 0 and d

_{t}S(δN

_{j}) > 0 [31]. In contrast any change in surroundings will compel the system to move toward a new steady state. In other words the equilibrium will shift to counteract the imposed change so that a new equilibrium will be established [32].

_{j}/∂N

_{j})(dN

_{j}/dt) [13,14]:

_{B}T. It means that the steps of evolution are small in energy compared to the total energy of the system. However, the actual jk-transformations do advance in quantized steps of ΔQ

_{jk}during Δt since each state associates with a group of symmetry [4] and a breaking one symmetry to another is a discontinuous event.

## 3. The Evolving Energy Landscape

_{B}T to give continuity for the flows of energy [14,30]:

_{j}= ∂

_{Nj}U

_{jk}and d

_{t}N

_{j}= v

_{j}∂

_{x}N

_{j}and v

_{j}= d

_{t}x

_{j}the three-term formula is transcribed to a convenient continuum approximation:

_{jk}= −x

_{j}m

_{jk}a

_{k}is consumed during dt, the power ∂

_{t}Q

_{jk}= v

_{j}∂

_{t}m

_{jk}v

_{k}= v

_{j}∂

_{t}E

_{jk}v

_{k}/c

^{2}is dissipated and the balance is maintained by the change in the kinetic energy 2K

_{jk}= v

_{j}m

_{jk}v

_{k}(Figure 2). This conversion of energy from one form to another has been conjectured already a long time ago [33,34]. Of course the change in mass (dm) is often very small, but conceptually important because it accompanies a change of state. For example, in a chemical reaction the change of mass per molecule is only a very small fraction of the mass of electron, nevertheless to recognize dm is essential because it signifies the transition from one state to another.

_{j}m

_{jk}x

_{k}and by the invariant mass m

_{jk}. According to m = E/c

^{2}the mass defines the jk-system’s energy content in terms of a radiation equivalent which can be dissipated, i.e., absorbed into the surrounding energy density, the free space known as the vacuum. It makes sense to give the energy content in terms of photon equivalents because heat, i.e., electromagnetic radiation is the lowest form of energy. The systemic energy is related to its surrounding radiation by the index n

_{jk}= c

^{2}/v

_{j}v

_{k}(isotropic v

_{j}= v

_{k}). For example, energy in radiation, when it is spatially confined to a standing wave, can be given in terms of a mass equivalent. Thus the conservation of energy is respected in the transformation between the bound and free forms of energy.

**Figure 2.**A step of evolution is a transformation from a closed action coordinated at x

_{k}to another bound state at x

_{j}so that the change in kinetic energy d

_{t}2K balances the changes in the scalar v

_{x}∂

_{x}U and vector potentials ∂

_{t}Q. The dissipation to the surroundings results from the concomitant change in mass Δm = ΔE/c

^{2}. In the stationary state, net fluxes vanish so that the closed least-action trajectory d

_{t}2K = 0 can be integrated over the motional period τ to yield the steady-state balance 2K + U = 0.

_{k}toward the density at x

_{j}during t is described to channel along an arc s

_{jk}. The affine connection along a continuous curve x = x(t) between the two spatial density loci spans a length s = ∫(

**F**·

**v**)

^{½}dt = ∫(d

_{t}2K)

^{½}dt [35]. When the landscape is stationary, the curve is integrable. In contrast when the landscape is evolving, the flows of energy are non-integrable because their paths keep changing as the landscape is changing due to the flows themselves. At branching points where paths diverge, differentials đ

_{t}are inexact. A small flow will not perturb much a statistical system and its trajectories can be estimated rather well. However, even a small flow of energy will be sufficient to move a microscopic system substantially. Eventually the step of change from an initial state may be so dramatic that the system at a final state is beyond recognition.

_{x}and temporal ∂

_{t}gradients as a 4-vector [36,37]:

_{μ}acts on the scalar U and vector

**Q**potentials which, in turn, are given as the free energy 4-vector potential in the one-form space-time basis:

**F**= d

**A**is obtained. It is represented by the covariant antisymmetric rank 2 tensor:

_{t}

**p**=

**F**= −∇U + ∂

_{t}

**Q**/c and

**R**= ∇ ×

**Q**. When the components of scalar ϕ = U/q and vector

**A**= −

**Q**/qc potentials are divided by charge q, F

_{μν}is the electromagnetic force. The change d

_{t}

**p**, i.e.,

**F**relates to the change in the angular momentum d

_{t}

**L**, i.e., torque

**τ**=

**r**×

**F**normalized by radius of curvature

**r**(Figure 2). The continuity is preserved by the invariants F

_{μν}F

^{μν}= 2(

**F**

^{2}−

**R**

^{2}) and ∗F

_{μν}F

^{μν}= 4

**F**·

**R**and ∗F

_{μν}∗F

^{μν}= 2(

**F**

^{2}+

**R**

^{2}).

**F**=

**J**so that the conserved energy density, i.e., the invariant mass, orbits with phase velocity d

_{t}φ = ω exactly once in a period τ = ω

^{−1}on the closed least-action path according to

**p**×

**ω**= −∇U. In the thermodynamic steady state there is no net emission from the system or net absorption to the surroundings over the motional period. Then the exterior algebra of the system’s dual d

**F**= 0 says that the surrounding is a flat landscape with respect to the system and thus exerts no forces. In other words, the system’s average energy density k

_{B}T exactly matches that of the surroundings. In the stationary state there is no net force and light propagates straight.

**F**= d

**A**of some principal high-symmetry bundle, is leveling off via symmetry-breaking transformations from one state to another more probable one. Bound forms of energy, i.e., fermions open up to output free forms of energy, i.e., bosons that dissipate to the surroundings. Since the system and its surroundings share a common interface for the flows of energy, the entropy of both the system and its surroundings will increase when mutual energy density differences decrease. The conservation of energy including both the system and its dual is respected by the differential geometry. The star operator transforms an oriented inner product density contained in an element of space, e.g., given in the Cartesian base, *(dx ∧ dy ∧ dz) = dt to radiated energy density that is carried away by an element of time.

_{j}= ∂

_{Nj}U

_{jk}and μ

_{k}= ∂

_{Nk}U

_{jk}adjust to accommodate or discard the vector potential that couples the system to its surroundings via the jk-transformations. The diminishing curvature of a differentiable landscape, i.e., the force can be represented by a vector field gradient. The non-vanishing Lie’s derivative [45] means that the change

**v**= d

_{t}

**x**in the coordinate and the change

**F**= d

_{t}

**p**in the momentum, are not collinear due to the net energy flux ∂

_{t}Q over dt to the system from the surroundings or vice versa. Therefore operators in [$\widehat{p}$, $\widehat{x}$] = −iħ do not commute by the minimum amount of action in a change of state. This uncertainty of at least one quantum in determining the state of the system is inherent in any measurement because the detection requires that at least one quantum is either emitted from the system to the detector or vice versa but that very quantum will change the state of the system. Consequently the systems that are undergoing changes of state contain open actions. These evolving systems cannot be ranked with respect to each other because their energy is changing.

_{μ}= (−c, v

_{x}, v

_{y}, v

_{z}). The flow tensor contracts to the 0-form d

_{t}2K = Σd

_{t}2K

_{μν}= −

**v**· ∇U + ∂

_{t}

**Q**+

**v**×

**R**where the change in the kinetic energy balances the changes in the scalar potential due to matter flows as well as the changes in the vector potential due to radiation fluxes. When the system communicates with its surroundings exclusively via radiation, Equation 10 is familiar from the theory of electromagnetism [46] where the radiation

**Q**at the speed of light c dissipates orthogonal to the source moving down along –∇U at velocity

**v**(Figure 2) [14]. Conversely, when the system is stationary ∂

_{t}

**Q**+

**v**×

**R**= 0, its stable orbits are governed by ∂

_{t}2K +

**v**· ∇U = 0 which is integrable to the steady-state balance 2K + U = 0 or differentiable to the equation for standing waves.

_{j}, which defines its spatial locus x

_{j}, to ϕ

_{k}, which in turn defines x

_{k}, is identified as the flow of time [14]. Thus the notion of time presupposes the notion of space [47]. The motion down along the spatial gradient is irreversible when emitted quanta escape forever and reversible when quanta are reabsorbed. Emission will change the coordinate of the source relative to the sink, and absorption will change the coordinate of sink relative to the source (or vice versa). Conversely, when the system is stationary, so are its surroundings. It is familiar from unitary transformations that the steady phase velocity d

_{t}φ = ω does not suffice to distinguish the systemic motions from the surrounding motions.

## 4. The Preon Action

_{j}will in fact cause the action to step in momentum p and in length x = vt from one closed orbit to another. The constant of action ħ, as the absolutely least action, can be considered as a physical entity (Figure 3). The most elementary fermion is the energy density of the bound geodesic given by the geometric product

**L**=

**px**[48] which has a specific handedness, usually referred to as spin ±½.

**Figure 3.**(A) The basic element of space is the most elementary fermion, the neutrino. The confined circulation of energy exists in two chiral forms ν and ν

^{*}corresponding to the opposite senses of circulation. The vertical bars denote the respective scalar potentials −U = 2K and U = −2K. (B) The basic element of time is the most elementary boson, the photon. The open flow of energy exists in two forms γ and γ

^{*}of opposite polarizations that correspond to the opposite (color-coded) phases iQ and −iQ (the figures were drawn with Mathematica 7 that was appended with CurvesGraphics6 written by G. Gorni.)

**L**associated with the kinetic energy within the orbital period 2Kτ = ħ. Thus ħ can be regarded as the fundamental element of space. The most elementary boson carries the energy along the open directed geodesic

**pv**t which has a specific handedness, usually referred to as polarization ±1. This oriented element of time is equal to the absolutely least action 2Kt = h that contains the energy carried within the wave’s period. This open action that equals Planck’s constant can be regarded as the fundamental element of time. It takes two of these actions to reverse the polarization from +1 to −1. The first will interfere destructively (head-on) with the original handedness and the other will create the mirror hand. In the following we will refer to the fundamental oriented element of time as the photon γ and its opposite sense of polarization as the antiphoton γ

^{∗}. Thus the photon is considered as its own antiparticle. Accordingly, we will refer to the fundamental oriented element of space as the neutrino ν and its opposite sense of circulation as the antineutrino ν

^{∗}. According to the physical portrayal of nature, the constant of action is the most elementary action which is abbreviated here as the preon [49].

## 5. Multiple Actions

^{−}is described as a least action path where preons coil to a closed torus having the electron neutrino ν

_{e}chirality (Figure 4). Electron’s steady-state characteristics are obtained from d

_{t}L = 0. This resolves to a constant 2K = ∫ρ

**v**·

**E**dt = ∫ρ

**E**·d

**x**= e

^{2}/4πεx where the density ρ distributes on the torus’ path length x so that the conserved quantity, known as the elementary charge e = ρx, sums from the current in the field

**E**according to Gauss’s law. The invariant fine structure identifies by integration to the normalized constant L/ħ = ∫2Κdt/ħ = e

^{2}(μ/ε)

^{½}/2h = α where the squared impedance Z

^{2}= μ/ε = (cε)

^{−2}, in turn, characterizes the stationary-state density that satisfies the invariant condition ∂

_{t}ϕ + c

^{2}∇·

**A**= 0 [51]. Moreover, the circulation gives rise to the magnetic moment

**μ**

_{e}= ∫

**r**×ρ

**v**dx = e

**r**×

**p**/m

_{e}= e

**L**/m

_{e}. Its anomaly α/2π [52], i.e., the excess of μ

_{e}over μ

_{B}= eħ/m

_{e}results from the helical pitch. The rise of coiling contributes to μ

_{e}= eIω/m

_{e}= ex

^{2}/t along the torus path length x beyond the plain multiples of ħ (as if the path were without pitch) where inertia I = m

_{e}x

^{2}and

**L**= I

**ω**(Figure 4).

^{+}is a torus but with antineutrino ν

_{e}

^{∗}handedness. In energy-sparse surroundings e

^{+}and e

^{-}interfere almost completely destructively. Only the helical-pitch modulation does not cancel. Thus, the annihilation bursts out anti-parallel rays of photons γ and γ

^{∗}so that each is equivalent to the comparatively low mass m

_{e}= 511 keV/c

^{2}that is characteristic of the elementary charge (Figure 4). In topological terms, the low mass means that the winding number of the elementary charge about the torroid’s center is low. When dissipation is normalized by the quantum of action, the characteristic trembling frequency ω = 2m

_{e}c

^{2}/ħ [53], is obtained.

^{−}boson that mediates the weak force is regarded as an open-ended helix of ν

_{e}-chirality. The helical line is of high energy and thus in energy-sparse surroundings it is not the least-action path but decays as W

^{−}→ e

^{−}+ ν

_{e}

^{*}. The pitch-accumulated lag-phase φ = 2π is absorbed at the torus closure by the antineutrino. Since the torus itself is a loop, the elementary charge is conserved in the decay process. Likewise, the W

^{+}boson is an open-ended helix of ν

_{e}

^{*}-chirality that processes as W

^{+}→ e

^{+}+ ν

_{e}. The neutral Z

^{0}boson is also an open-ended path where a γ-linker joins two helices one having ν

_{e}and the other ν

_{e}

^{∗}-chirality. Energy-sparse surroundings drive the decay Z

^{0}→ e

^{+}+ e

^{−}. The weak bosons display extraordinary high masses in comparison to their closed form fermion-antifermion counterparts, because these open paths have comparatively little topological self-screening via intrinsic phase cancellation, i.e., their winding numbers are high.

**Figure 4.**(A) Electron e

^{−}is a closed least-action torus which is composed of multiple preon actions having neutrino sense of chirality. The helical pitch is seen from the roll of the arrow heads. The lag-phase due to each winding totals φ = 2π around the torus. (The pitch is exaggerated for clarity). (B) Conversely, positron e

^{+}is a closed torus of multiple preons having antineutrino sense of chirality. (C) When the surrounding energy density is high, e

^{−}will convert by breaking its closed chiral symmetry to the open W

^{−}boson and electron neutrino ν

_{e}. (D) Conversely, e

^{+}will convert to W

^{+}boson and electron antineutrino ν

_{e}

^{*}. The actions having opposite sense of circulations will annihilate so that streams of γ and γ

^{*}stem only from the lag-phase modulation which accrued along the opposite helical pitches.

^{+}is portrayed as a least-action path where two

^{2}/

_{3}-helices of ν

_{e}

^{∗}-chirality, known as up-quarks u, and one

^{1}/

_{3}-helix of ν

_{e}-chiralily, i.e., down-quark d, join via three γ-linkers, known as gluons g. Along each u the helical path accumulates φ =

^{4π}/

_{3}and likewise d accrues φ =

^{2π}/

_{3}so that a gluon, as an open preon, links any two quarks at the angles that the faces of an equilateral tetrahedron make with each other (Figure 5). The front-end ⊙ of one u links via g to the back-end ⊗ of the other u, and then further ⊙-u links via g to ⊗-d and finally ⊙-d links via g to ⊗-u to close the path. In the tripod constellation each quark as a directional element of the closed path is distinguishable from any another which is the essence of quantum chromodynamics.

^{+}transforms to neutron n via electron capture where e

^{−}breaks, when attracted to u, to W

^{−}which makes a snug fit at u so that a partial annihilation will commence and yield d (Figure 5). Likewise, when n is free, i.e., in energy-sparse surroundings, the reverse process begins when W

^{+}is attracted to make a snug fit at d and the partial W

^{+}d-annihilation will yield u. The provided “wire frame models” are illustrative but perhaps puzzling since p

^{+}by the perimeter is longer than n, yet the mass of the neutron is slightly bigger than that of a proton. However as clarified earlier above, the mass is a measure of energy in terms of net dissipation. The topological self-screening of u in p

^{+}and d in n are nearly the same. The small difference is accrued from the incomplete cancellation of opposite phases of pitch. The significant difference in proton and neutron magnetic moments, in turn, is understood to stem from the substantial differences in the oriented areas that are closed by the respective currents. The wire frames (Figure 5) allow an easy imagination of various natural process such as a pion decay π

^{+}: ud

^{*}→ W

^{+}→ e

^{+}+ ν

_{e}

^{*}, where the ud

^{*}-ring opens so that d

^{*}(oppositely wound d) will resettle via the high-mass intermediate W

^{+}integrally to the low-mass e

^{+}-torus.

**Figure 5.**(A) Proton p

^{+}is a closed circulation where each up-quark u (red) is

^{2}/

_{3}of torus of ν

_{e}

^{*}-chirality and the down-quark d (blue) is

^{1}/

_{3}of torus of ν

_{e}-chirality. The lag-phase φ =

^{4π}/

_{3}accrued along each u and φ =

^{2π}/

_{3}along d due to the helical pitch, define the relative angles of quarks, i.e., symmetry of the closure linked by gluons g (black arrows). (B) Electron capture p

^{+}+ e

^{−}+ ν

_{e}

^{*}→ n is intermediated by W

^{-}which as an open action will initiate a partial annihilation at an exposed end of u (blow-up) that will yield d. (C) The resulting neutron n is a three-gluon-linked closed action udd.

^{+}→ e

^{+}+ π

_{0}[54] is seen in terms of actions so that the X-boson (balanced by e

^{+}+ d

^{*}) would be attracted to p

^{+}to make a snug fit along u-⊗-g-⊙-u. However, such an annihilation process is unlikely to take place in low-density surroundings (Equation 1) because it would yield disjoint d and d

^{*}which are high-energy by-products. The process does not yield the anticipated pion π

_{0}which would indeed have only moderate mass since in the meson, the circulations of linked quark and antiquark pair cancel apart from the lag-phase accrued along their opposite pitches. Thus, the violation of color confinement would require extraordinarily high-energy surroundings to provide the asymptotic freedom [55] for quark-gluon plasma, or the violation of baryon number conservation would require a blazing transformation [56,57].

## 6. Fundamental Forces

^{*}between p

^{+}and e

^{−}(Figure 6).

**Figure 6.**(A) Electromagnetic interaction arises from the density difference between charged fermions and their surroundings. (B) Sparse surroundings cause attraction by accepting full wavelength modules |∞| of photons from the bound actions that pair opposite charges q

^{−}and q

^{+}. For clarity only the action along the shortest electromagnetic field line (dashed) is decorated with density modules. Beyond the bound pair of charges energy density propagates at the speed c

^{2}= 1/μ

_{o}ε

_{o}. The energy density remains finite also long the dipole axis where the anti-phase waves do not couple to the antenna. (C) Like charges generate alike-polarized, open actions that cannot pair to form a bound action. (D) Therefore in sparse surroundings repulsion remains the mechanism to dilute the density between q

^{+}and q

^{+}.

_{o}ε

_{o}= c

^{−2}and μ

_{o}/ε

_{o}= Z

^{2}, accept actions that are released in transitions from one standing-wave orbit to another (Figure 7).

**Figure 7.**(A) Gravitational interaction arises from the energy density difference between net neutral bodies and their surroundings. (B) Sparse surroundings attract by accepting doubly paired density modules ||∞|| of gravitons from the bound actions that pair the bodies along gravitational lines. (C) Irradiative combustion of high-density actions is a powerful mechanism to diminish the density difference between a body and its surroundings. (D) The burning star gives away density, hence the conservation of energy (Equation 6) requires for a planet to advance its perimeter from one modular orbit to another.

**r**

_{12}is obtained, as before, from the steady-state condition, i.e., the least-action which is equivalent to the steady-state balance 2K + U = 0 as follows:

_{i}r

_{i}r

_{i}= Σm

_{i}m

_{j}r

_{ij}

^{2}/Σm

_{i}= Σm

_{i}m

_{j}r

_{ij}

^{2}/M and period τ

^{2}= 1/ω

^{2}= t

^{2}r

^{3}/R

^{3}. The inertia, when normalized by the total mass M of the Universe, can be seen as an expression of the principle 2πGρt

^{2}= 1 where the mass density ρ is within the radius R = ct [61], so that any given density is coupled to every other density. The numerous (decay) paths to disperse energy span the affine energy landscape, the universal buoyancy where the density flows level off any density difference in the least time. Thus the cosmological principle, i.e., homogeneity at the largest scale can be seen as a consequence of a natural selection for the maximal dispersal.

^{−27}kg/m

^{3}[62,63] corresponds to a tiny curvature over a titanic radius R = ct [64,65,66] and sets the gravitational constant G = 1/2πρt

^{2}to its current value. The minute but non-negligible acceleration a

_{t}= GM/R

^{2}= 1/ε

_{o}μ

_{o}R = c

^{2}/R = c/t = cH drives the on-going expansion at the rate H [67,68] that is changing as d

_{t}H = −H

^{2}= −2πρG. On the basis of ρ or equivalently of H the dilution factor n = L/ħ is on the order of 10

^{120}. This factor is the well-known discrepancy between reality and the calculated vacuum energy density (as if no expansion had taken place). At any given time the total mass of the Universe in its dissipative equivalent Mc

^{2}= Ma

_{t}R = R∂

_{r}U = −U matches the scalar potential. This innate equivalence is known also as the zero-energy principle [6]. The balance is also reflected in the quantization of the electromagnetic spectrum. The cosmic microwave background radiation profiles according to the Planck’s law a discrete quasi-stationary entropy partition of fermions (Figure 1). The unfolding Universe is stepping from one mode to lower and lower harmonics [69] along its least action path toward the perfect, i.e., torsion-free flatness d

_{t}L = τ = 0. The acceleration, which is proportional to the winding number θ ∝ R

^{−2}of those folded actions, will be gradually limiting toward zero. When all densities have vanished, no differences, i.e., forces will exist either.

## 7. The Mass Gap

_{ν}> 0, i.e., the conserved quantity whereas the open preon associated with U(1) is without mass m

_{γ}= 0. The most elementary symmetry cannot be broken any further. For that reason there is no primitive root of unity, i.e., a standing-wave solution corresponding to a mass. It follows that there is a mass gap which is a finite difference in energy between the lowest bound state and the free, open state.

_{t}(ρv

^{2}) = −

**v**∇·u − ε

_{ο}c

^{2}∇·(

**E×B**), balances the change in the scalar potential density u due to the average density ρ of matter with the change in the electromagnetic radiation density. The vector μ

_{ο}

^{−1}

**E°B**[46] embodies energy of free space which is unique up to a phase and invariant under the Poincaré group in accordance with Maxwell equation d

**F**= 0 in the absence of charges and currents.

_{t}Q/c) along the flow of another vector field (

**v**·∇U). The smooth, differentiable manifold, such as the gauge group U(1)×SU(2)×SU(3) of the Standard Model, is admittedly suitable for mathematical manipulation, but it would be appropriate only for a stationary system because it fails to describe the discontinuous event of breaking symmetry.

**Figure 8.**Level diagram depicts an array of actions (colored) that are classified according to unitary groups of symmetry. Each stationary action equals the kinetic energy 2K integrated over the period τ along a directed path. It associates via Noether’s theorem with a conserved quantity m. A mass gap Δm ≠ 0 exists between the closed action of the most elementary fermions, the neutrinos defined by the next lowest symmetry group U(2) and the open action of the propagating photons defined by the most elementary group U(1). The radiated energy ΔΕ = Δmc

^{2}in the transformation from U(2) down to U(1) amounts from the photons that disperse in the surrounding free space. The energy density of the free space is non-zero as is apparent from the finite speed of light. Only one of the two chiral forms of U(2) fermions can be detected via emissive transformations to U(1) whereas a detection by absorption would transform the U(2) particle to another one of a higher symmetry group.

**Theorem**: For any-one compact simple gauge group G, there exists no quantum Yang-Mills theory on R

^{4}that has a mass gap.

**F**denotes curvature via a gauge-covariant extension of the exterior derivative from the one-form

**A**of the G gauge connection and g

^{2}is the determinant of the metric tensor of the spacetime. It follows that G governs the symmetry of the action, which is the integral of L. Since the quadratic form that is constructed from

**F**and its Hodge dual *

**F**on the Lie algebra of G on R

^{4}is invariant, so also L is invariant and so is its integral action invariant. The invariant action relates via the Noether’s theorem to a quantity m whose value is conserved. It then follows that a difference between any two values m

_{G}and m

_{H}of a conserved quantity cannot be compared within any-one theory based only on a single gauge group. Specifically, in a Yang–Mills theory that is based exclusively on one gauge group G describes only a single, invariant state. When the theory is based on U(1), it describes the vacuum state (the free space). When the theory is based on another Lie group, it describes another state. The difference Δm = m

_{H}− m

_{G}≠ 0 in the conserved quantity between U(1) and the next lowest symmetry group is referred to as the mass gap. However, the gap is not defined, i.e., it does not exist within any given theory based exclusively on a single gauge group.

**Corollary**: For any-two distinct gauge groups G and H, there exists a theory on R

^{4}that has a mass gap.

^{4}via the Noether’s theorem to a quantity m whose value is conserved. Conversely, the two distinct gauge groups relate two distinct symmetries of invariant actions to two different values m

_{G}and m

_{H}of the conserved quantity. Thus there exists a difference Δm = m

_{H}− m

_{G}≠ 0 in the conserved quantity, i.e., the mass gap, between the vacuum state energy E

_{G}governed by the most elementary G = U(1) symmetry group and the lowest excited state energy E

_{H}governed by the next most elementary symmetry group H.

**x**of a field will be extending with diminishing momentum

**p**, their product

**px**remains invariant so that associated observables remain countable. In other words, a theory based on action is self-similar but does not require renormalization since it is not troubled by infinities and singularities related to energy and time as well as to momentum and length. Yet, it may appear for some to be unconvincing to claim, as above, that SU(2) which is by determinant isomorphic to U(1), does have a distinct property, i.e., mass. However, a unitary group U(n) is non-Abelian for n > 1 whereas U(1) is Abelian. The sense of circulation distinguishes the closed action from a point, and also from its open form because when the chiral, closed action is mapped on the polarized open action, the two open termini are distinct from the integrally closed and connected path. To bridge a topological inequivalence requires a dissipative transformation process. The mass gap is between the photon and the neutrino. Moreover, the Euler-Lagrange integral of a bound trajectory is well-defined, i.e.,countable whereas Maupertuis’ action as an open path is ambiguous, i.e.,uncountable. The ambiguity is reflected in the assignment of a value to the free energy of the vacuum. Instead the surrounding vacuum characteristics c

^{2}= 1/ε

_{o}μ

_{o}are defined via the closure according to the Stokes theorem.

## 8. Algebraic and Non-algebraic Varieties

_{B}Tt = nħ (see Equation 2). The stationary state (d

_{t}P = 0) measure lnP = ΣN

_{j}is a sum over all closed graphs, as is familiar from index theory [72] and from the linked cluster theorem for reversible, i.e., non-propagating processes [73].

^{k}(M) > H

^{n−k}(M). The Poincaré duality theorem states that for M the k

^{th}cohomology group of M is isomorphic to the (n − k)

^{th}homology group of M, for all integers k.

^{n}(M,

**X**) = ⊕ H

^{p,q}(M) where p + q = n and H

^{p,q}(M) is the subgroup of cohomology classes that are represented by harmonic forms of type (p, q). It is instructive to consider the (twice) differentiable and connected M, because an n-form ω, i.e., the Lagrangian, can be paired by integration, i.e., via the action ∫ω = 〈ω, [M]〉 with the homology class [M], i.e., the fundamental class whose the top relative homology group is infinite cyclic H

^{n}(M, Z) > Z. The action ∫ω over M depends only on the cohomology class of ω. The physical insight to the mathematical problem of non-countability is consistent with the fact that the Hodge conjecture holds for sufficiently general and simple Abelian varieties such as for products of elliptic curves. Moreover, the stationary least action is consistent with the combination of two theorems of Lefschetz that prove the Hodge conjecture true when the manifold has dimension at most three which is required for the stability of the free-energy minimum varieties.

^{k,k}(M) form from the complex subvarieties? Of course, it has been found mathematically that there are also non-algebraic varieties. For example, when the variety has complex multiplication by an imaginary quadratic field, then the Hodge class is not generated by products of divisor classes. These troublesome varieties correspond to the open actions. They are without norm, hence non-modular and indivisible. The n-form ω of a non-modular curve over a finite field is non-intergrable because it is without bounds. Although the natural processes terminate at the irreducible open preon, the path is open because one photon after another leaves the system. The manifold keeps contracting, mathematically without a bound, but physically the landscape ceases to exist when the last fermion opens up and transforms to boson that leaves forever. Also singularities associated with the squared operators, e.g., the square of an exterior derivate, are troublesome abstractions. At a singular point the algebraic variety is not flat which means it is non-stationary and uncountable. Thus the proof of the conjecture that Hodge cycles are rational linear combinations of algebraic cycles, hinges on excluding non-integrable classes, i.e., those without divisors. This depends on the definition of manifold.

**Conjecture**: Let M be a projective complex manifold. Then every Hodge class on M is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of M.

^{n}of the dimension of M is one-to-one. The homeomorphism excludes any open curve, i.e., any polynomial without at least one root. Thus it follows that Hodge cycles are rational linear combinations of algebraic cycles on a projective complex manifold. Since the projective complex manifold has no class without a divisor, the conjecture is true.

## 9. Discussion

## Acknowledgments

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