In the terms of Evariste Galois' mathematical "Group Theory", the "Tetrahedron Model" is a qualitative description of the symmetry group of light, or free electromagnetic energy.

Overview

Regarding the question of continuous symmetries in the "Tetrahedron Model":

1) From unknown origins (?inflation via the Multiverse?), the Universe begins in a state of perfect symmetry with equal parts of matter and antimatter (the "Big Bang" or "Creation Event"). Matter-antimatter symmetry is the primordial energetic state of the Cosmos; due to continual and instantaneous annihilation reactions, this primordial symmetric state is equivalent to pure high energy light (electromagnetic radiation). Total energy and charge is zero since the contribution of both from antimatter is negative. This primordial electromagnetic symmetry must somehow be broken to produce our asymmetric Cosmos of bound and free electromagnetic energy (matter plus light).

2) Maintaining the primordial symmetry of light is the role of electric charge, as the opposite electric charges of matter vs antimatter attract each other and motivate annihilation reactions. Light is a perfectly symmetric energy form, carrying no charges, having no time dimension, and having no "location" in space (since in its own reference frame light is everywhere simultaneously). Light is a "non-local" 2-dimensional transverse wave having no spatial dimension in the direction of propagation nor any temporal dimension. Light has no mass, produces no gravitational field, and travels with an "intrinsic motion c" which creates an expanding and cooling domain - the entropic spatial conservation arena of free electromagnetic radiation. This is the primordial symmetric energy state of light which electric charge protects and conserves via matter-antimatter annihilation reactions. Light exhibits a continuous "non-local" symmetry - like space itself, it is globally symmetric everywhere, and any photon (or wave of electromagnetic radiation) can be swapped for any other. "Velocity c" is best conceived not as an actual velocity, but as the gauge of the non-local symmetric energetic state of free electromagnetic energy, including the metric symmetry of space (created by light's intrinsic, entropic motion) in which time, gravitation, and mass are banished ("mass" is a bound form of electromagnetic energy with no intrinsic spatial motion; instead, mass has intrinsic temporal motion, producing a "gravitational field"). Light's "velocity" in its own reference frame is essentially infinite, as it has forever to go nowhere (time stops and meter sticks shrink to zero at "velocity c"). Light protects its own symmetric energy state via matter-antimatter annihilations motivated by electric charge (the photon is the field vector of electric charge). Electric charge is "discrete" but the symmetry it protects is continuous; we know in any case that electric charge is absolutely conserved.

3) Breaking the primordial electromagnetic symmetric energy state is the combined role of the strong and weak nuclear forces, which early-on work in concert to produce our "matter only" cosmos.

4) Restoring the symmetric energy state of light is the patient and inexorable role of gravity. Light's symmetric energy state is broken (due to the asymmetric activity of the two nuclear forces) by the creation (from light) of massive, local, temporal, and immobile matter. Gravity converts mass back to light in stars, quasars, and, via Hawking's "quantum radiance", in black holes. Gravity and its "location" charge are continuous to the extent spacetime is continuous. We also know the gravitational "location" charge is conserved, as no massive particle has ever been found without one; nor can a gravitational charge be destroyed except by total conversion of mass to light, which completely repays the gravitational symmetry debt or "location" charge. The gravitational "location" charge (Gm) is conserved to the same extent that mass (m) is conserved. 

Because the gravitational metric of the black hole causes all things, regardless of mass, to move at velocity c, it is apparently just as symmetric as the original electromagnetic metric of light, which it replaces. "The extremes meet": perhaps this is why gravity drives so relentlessly toward the formation of black holes - they represent the ultimate symmetric energy state of matter. In any case, we expect black holes to be full of nothing but light, due to proton decay in the interior, perhaps in the extreme pressure of the central "singularity". However, the entropy drive of black holes, being contractile one-way time/gravity, is less symmetric than the entropy drive of space/light, which is expansive "all-way". Hence in accord with Noether's Theorem, black holes should eventually convert their one-way entropy drives to light's all-way entropy drive, and evaporate (Hawking's "quantum radiance").

(See: "A Description of Gravity"; see also: "The Conversion of Space to Time"; see also: "Nodes of the Gravitational Metric".)

Symmetry Breaking

3a) Breaking the primordial symmetry of light via the combined asymmetric action of the nuclear forces: the strong and weak forces act together. (See: "The Particle Table".)

In the bound energy (atomic matter) section of the "Tetrahedron Model", we enter a more "discrete" area of physical theory, involving the "identity" and "color" charges of the weak and strong forces. Both these charges, however, conserve and protect continuous symmetries that eventually refer back to spacetime and light, and the primordial electromagnetic symmetry between matter and antimatter. When considering "discrete" particles or properties of matter, we should remember Heisenberg's Uncertainty Principle, deBroglie's wave/particle relation, and Born's probabilistic interpretation of Schrodinger's collapsed wave function - all of which suggest links between so-called "discrete" particles and the continuous realm of waves, spacetime, and light.

Symmetry-breaking begins when a primordial super-massive lepton - a "leptoquark", the heaviest member of the leptonic spectrum of elementary particles, breaks into three parts (quarks) under the self-repulsion of its own electric charge, seeking a lower energy distribution of its mass and electric charge. This self-disintegration places a natural upper limit on the mass of the elementary particle (leptonic) spectrum. Such a disintegration is possible and permitted (lawful) only because the strong (color) force is available to hold the fractured particle together into a quark combination featuring a whole leptonic unit of electric charge (the reassembled original charge) - such as a proton, or else no charge at all (the quark combination of a neutron), provided in the latter case the original electric charge can be transferred to and conserved by some other member of the leptonic spectrum (such as an electron). 

The strong force "gluon" field springs naturally from the splitting of the massive body of the leptoquark - gluons seem to be versions of split photons (photons are the field vectors of electric charge), apparently determined to hold the original charge together as a whole quantum unit, at least as seen by the outside world. (See: "The Strong Force: Two Expressions".) Gluons attract each other and have been compared to "sticky light", and exhibit "asymptotic freedom", attracting each other more strongly as the quarks attempt to separate, less strongly as they crowd together.  (See: Science 15 Oct. 2004 Vol. 306 p. 400) It all makes sense from the perspective of symmetry conservation, as these whole unit charges are necessary for cancelling, neutralizing, or carrying the charges of other members of the leptonic elementary particle spectrum, or annihilating the opposite charges of antimatter. The strong force protects the global and continuous symmetry of whole quantum units of (electric) charge. Throughout the spectrum of electrically charged leptons and hadrons, every charge is the equivalent of every other charge, able to replace, neutralize, or even annihilate its opposite number as the opportunity arises. Charge conservation is symmetry conservation by proxy. The charges of matter are the symmetry debts of light.

Because of the three-family structure of the quark spectrum, many (15) electrically neutral quark combinations are possible - heavy analogs of the neutron. These electrically neutral leptoquarks are subject to leptonic weak force decays in which a leptoquark neutrino (conserving "lepton number charge") and a neutral pion (conserving energy) replace the leptoquark (as mediated by the heavy "X" IVB). (See: "The Origin of Matter and Information".) During the early micro-moments of the "Big Bang", an asymmetry of unknown origin produced slightly more of these weak-force decays in the anti-leptoquark population (about one extra per billion decays), leaving our universe with an excess of matter leptoquarks - which subsequently decay to protons and neutrons. This may provide an "anthropic" explanation for the otherwise mysterious existence of the three quark-lepton families - they are necessary to create sufficient numbers of neutral leptoquarks for the weak-force asymmetric decays that produced our "matter only" cosmos.

The role of the strong force is to maintain the uniformity (symmetry) of quantized whole charge units throughout the spectrum of elementary particles - whether they be elementary leptons or composite hadrons comprising quark triplets or doublets (the strong force requires the electric charges of baryons and mesons to conform to those of the pre-existing elementary leptonic spectrum - from which the leptoquarks are derived). By means of this symmetry, all electrically charged fermions (baryons, mesons, leptons) can carry, cancel, or neutralize each others' electrical charges, and annihilate the electric charges of their corresponding antiparticles.

The strong force facilitates the breaking of the primordial electromagnetic symmetry between matter and antimatter by enabling the creation of nuclear sub-units (quarks) and hence the formation of electrically neutral leptoquarks. Symmetry-breaking occurs via the asymmetric weak-force decay of electrically neutral composite quark combinations (electrically neutral leptoquarks) - decaying to the matter-only barons of our cosmos (protons, neutrons, hyperons).

The "gluon" field of the strong force also exhibits a symmetry between its "color" charges that allows any quark to bond with any other quark,  providing the electrical charges of the quark combination sum to whole quantum (leptonic) charges (including zero). Color charges themselves must sum to "white" (red/green/blue combinations, as in baryons) or neutralize each other (color/anticolor combinations, as in mesons). The symmetry of the gluon color field is continuous in the sense that all color charges are equivalent (and hence can be swapped among themselves), just as all electrical charges are equivalent and can be traded or shared between particles as a common currency (as in the universal proton-electron atomic combination). Like electric charge, color charge is strictly conserved - as we should expect, if the latter is actually a subdivision of the former. Because gluons are composed of color-anticolor charges in every combination, the gluon field as a whole must sum to zero. This final symmetry of the color force leads to the expectation of proton decay at very high energy (perhaps via the massive "X" IVB, and/or at the central singularity of black holes).

The Weak Force: (See: "The "W" IVBs and the Weak Force Mechanism".)

Whereas the strong force controls the global (universal) symmetry of electric charge among the leptons and hadrons (protecting whole quantum unit charges), the weak force controls the global or universal symmetry of mass between all particles of a given species.  Electrons, for example, are created with exactly the same mass wherever and whenever they are "born"; an electron created in the decay of a neutron today could be swapped with an electron created aeons ago during the "Big Bang", and nobody would know the difference.  This global or universal symmetry of mass (rest-mass energy) within any elementary particle species, lepton or quark, is due to the activity of the weak force and the mediation of its "Intermediate Vector Bosons" (IVBs). Not only does this symmetry allow one member of a particular species to be seamlessly swapped for another of its kind, it allows the annihilation reactions of particle-antiparticle pairs to proceed in a timely and orderly fashion - conserving light's symmetry (always the " bottom line").

The weak force is the only force capable of creating, destroying, and transforming single elementary particles (quarks and leptons) - as opposed to particle-antiparticle pairs. How does it manage to create them all with perfectly identical masses (within type)? The heavy masses of the weak force IVBs (~80 and ~90 proton masses respectively for the "W" and "Z" IVBs) simply recreate the primordial energetic conditions under which these particles were originally created. Within type, they are all stamped out of the original mold, as it were. In turn, the Higgs boson gauges (determines) the masses of the IVBs, selecting the force-unification era of the early universe in which these particles were first created (the electroweak force-unity era, in the case of our ordinary particles). (See: "The Higgs Boson and the Weak Force IVBs".)

Of the charges of the four forces, the "identity" charge of the weak force may be the most interesting. Neutrinos are the explicit form of this charge, one for each of the three known elementary leptons, the electron, muon, and tau (and another set for their antiparticles). I presume there exists a 4th neutrino, the identity charge of the leptoquark. If it does exist, it may be quite heavy and be the source of the mysterious "dark matter". The neutrinos are mysterious particles, and much remains to learn about them. (See: Scientific American May 2010 p. 38 "Through Neutrino Eyes" Gelmini et. al.)

The "identity" charge is the symmetry debt of the photon's state of "anonymity": every photon is identical to every other photon (hence "anonymous"), but the massive elementary leptons are distinctly different from the photon and from one another - the muon is about 200 times heavier than the electron, and the tau is about 3600 times heavier (the photon is massless). While neutrinos carry leptonic "identity" charge in explicit form, the massive leptons themselves carry it in "hidden" form, where it is recorded simply as "lepton number" charge. The "hidden" form is necessary because the identity charge of the neutrinos involves "parity" (left- or right-handed spin), and massive particles cannot conserve this charge. Ordinary neutrinos are almost massless and travel at very nearly velocity "c", so they do an almost perfect job of conserving this charge. All neutrinos are left-handed, and all anti-neutrinos are right-handed, neatly distinguishing matter from antimatter, and leading to the notion that our universe is in some sense "left-handed". The weak force reactions of matter are therefore said to be left-handed, while those of antimatter are right-handed - in terms of spin or "parity". Parity is conserved when the universe and anti-universe are considered jointly.

In all the Universe there are only three (perhaps 4) species of elementary particles - the leptonic spectrum. Only elementary particles have associated neutrino "identity" charges - quarks have none as they are sub-elementary. (I presume the leptoquark neutrino carries the identity charge for the quarks and baryons). The neutrino is in effect a "certificate of authenticity" that guarantees "this is the genuine article" - an elementary particle that is correct in all its conserved parameters - charge, spin, mass, etc. - and hence can be "swapped out" for any other of its kind ever made, and (of special importance) can annihilate with its antiparticle, thereby returning both to the original symmetric energy state of light from which they were created.

To create a new (single) electron however, is a monumental task, as it must be the same as any other electron ever made (or that ever will be made). This requires the mediation of the very massive "W" IVB (Intermediate Vector Boson: ~80 proton masses), to re-create the original energy density of the early micro-moments of the "Big Bang" in which electrons were first created. This is in fact the energy density of the electroweak force-unity era, the time and energy at which the electromagnetic and weak forces were unified. The characteristic of this unification energy is that the massive leptons and their neutrinos are all equivalent, swapping specific identities amongst themselves as if they were all members of a single genus (the "leptonic genus"), rather than distinct species or particles. Similarly, the quarks are likewise swapping specific identities ("flavors") amongst themselves as if they were all equal members of a single "hadron genus" (there are no neutrinos for the quarks; quarks are present as quark-antiquark pairs of all "flavors"). At the electroweak force-unity energy level, transformations of lepton/lepton or quark/quark identity or flavor are effortlessly accomplished. However, quarks and leptons do not mix at the electroweak energy level. (See: "The Higgs Boson and the Weak Force IVBs".)

The leptonic spectrum of elementary particles is clearly some sort of resonant series. In this case it appears to be a resonant series of the combined electromagnetic and weak forces. The leptonic series identifies the mass-energy at which the electromagnetic/photon and weak force/neutrino frequencies are in sympathetic vibration - the leptonic particle series delineates the nodes of sympathetic vibration or resonance between these two forces at the electroweak energy level. The electromagnetic force can probably produce particles of any rest-mass energy, but it is only at the nodes of the resonance series where these two forces are in sympathetic vibration that massive particles can be paired with neutrino identity charges. This joining of forces is necessary to produce particles that can be conserved in the sense that they can be exactly reproduced at any time and place, matching up exactly with others of their kind, including in annihilation reactions with their antiparticles.

While the electromagnetic and weak forces are "in resonance" at the electroweak force-unity energy level, the massive leptons of the electromagnetic force (such as the electron) and its neutrino, or "identity charge" of the weak force, are joined together at a "generic identity" level, in which they exchange identities without restriction. It is during this period of exchange, and because of it, that the massive lepton (electron) acquires its "hidden" identity charge, or is in some way prepared to acquire and carry one. Such is also the case for the entire "leptonic spectrum" - the electron, muon, tau, and presumably the leptoquark also. Of (infinitely?) many possible rest-mass energies that can be produced by the electromagnetic force, just these four are compatible with the weak force to the degree that instead of a massive electromagnetic particle-antiparticle pair being produced (such as an electron-positron pair), in the electroweak "resonance" a mixed pair is produced instead - the electron-neutrino pair (actually a positron neutrino, balancing identity charge). In the decay of a neutron to a proton, it is just this electron-positron neutrino pair which we find accompanying the proton as products of the decay (the electron and proton balancing each others' electric charges).


We see the massive IVBs as energetic "keys" which open the door to a conservation domain of massive particles - the leptonic spectrum. These keys are quantized pulses of energy which pluck the strings of the cosmic lyre of spacetime with exactly the right energy to establish a sympathetic resonance between the electromagnetic and weak forces, producing massive particles of bound electromagnetic energy which are conserved by weak force identity charges -symmetry debts of light's "anonymity".

The "identity" charge of the weak force is the most significant of the charges of the four forces. It is the beginning of Information, and from this humble beginning proceeds the entire manifest universe.

While the huge masses of the IVBs "steal the show" in the weak force, what is important is the uniformity of the leptons produced in the process, and this depends upon the neutrino identity charges. It is somewhat similar to the massive human body which must be used to create the next generation via a minute quantity of genetic material; the body is impressive but it is the microscopic genes which count. Imagine what would happen if no identity charge were required for the creation of (single) leptons - electrons of every conceivable mass could be produced with untold consequences for the formation of atoms and for annihilation reactions with antiparticles.

 Elementary particles are special forms of bound energy and the Universe is prepared to produce and conserve only a very few types. The major requirement is that they must (as leptons) be able to pair with a neutrino identity charge, which ensures they can be absolutely conserved and accurately reproduced. As for the quarks, they are held to the "neutrino standard" only at the higher energy of the leptoquark (the GUT or "Grand Unified Theory" force-unity energy level), but not at the lower energy of the electroweak force. However, they appear only as quark-antiquark pairs (mesons) in the electroweak "fluid", so their match-up with antiparticles is in any case never an issue.

Hence we have the long-range pair of "spacetime" forces - the electromagnetic and gravitational forces - responsible for creating and conserving the continuous symmetries of space and time; and we have the short-range pair of "particle" forces - the strong and weak forces, responsible for creating and conserving the continuous symmetries of whole quantum units of electric charge, and the invariant "rest" mass (within type) of elementary particles. All four forces work together to produce the charged and massive particles, planets, stars, and galaxies of our Universe. Noether's Theorem unites them all: The charges of matter are the symmetry debts of light.