0, 1, 2, 3, 4

It may be a good time revisit the post "24 Aspects of the Universe" where the quartet;

(P, V, n, T)

was mapped to the quartet;

(Energy, Time, Mass, Gravity) 

and each of these were extended by 3 more elements;

(Energy, Momentum, Angular Momentum, Spin)

(Time, Distance, Area, Volume)

(Mass, Electric Charge, Weak Charge, Strong Charge)

(Gravity, Electric Field, Weak Field, Strong Field)

And two more quartets were added to include the missing categories like Dark Matter;

(Higgs, ?, DE, DM)

(Atom, Planet, Star, Black Hole)

Thus all add up to 24=6*4=4! and the number of 2D square faces of 4D cube Tesseract. Perhaps the last two quartets are a bit tacked on but the 4 quartets linked to Ideal Gas are more natural. In the quartet (Time, Distance, Area, Volume), Distance and Area feel a bit out of place while Volume is already there, but it matches with (Energy, Momentum, Angular Momentum, Spin) since for example Momentum expresses in Distance. So I am pretty confident about the 4*4 matrix of fundamental categories but there are missing categories like Higgs, DM, DE and BH. So the 4*4 matrix based on Ideal Gas is not the whole story.

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How were the symmetries of the 4*4 matrix built and kept by the Universe? For example each quartet has the 2+2, 1+3 and 2*2 symmetries and this reflects on the 4*4 matrix; it has 4 2*2 corner matrices and also 4*1 and 4*3 matrices in various ways.

I think these symmetries were built in the past phases of the universe. They are still kept in the background and hold the structure of the universe together but they are not easily observable. The past phases are a part of modern cosmology already but many details are missing. Lets assume that the universe has gone through 5 phases and we live in the fifth one with index 4;

(0, 1, 2, 3, 4)

Each phase has categories in line with the square of it's index (Squared for interaction);

(0, 1, 4, 9, 16)

Adding up to 31 categories in total.

Phase 0;

The mother Universe. Nothing known to us yet exist.

Phase 1;

Boundary is formed and it gained unity. All categories are mixed in one state. The primordial Higgs field. Perhaps a Black Hole in the mother universe.

Phase 2;

4 new categories are created. Symmetry of the initial unity is broken but the phases 0 and 1 still exist in the background. Judging by the 4 corners of the 4*4 matrix;

(0, 0); ((Energy, Time), (Momentum, Distance)) ... Energy-like ... DarkEnergy.

(0, 1); ((Mass, Gravity), (Electric Charge, Electric Field)) ... Mass-like ... Dark Matter.

(1, 0); ((Spin, 3D Volume), (Angular Momentum, Area)) ... Spin-like.

(1, 1); ((Strong Charge, Strong Field), (Weak Charge, Weak Field)) ... Strong-charge-like.

The details don't yet exist and the 4 new categories are a mixture of the 4 categories it includes. Perhaps one dominates as noted. This could be the inflationary phase since Spin and Volume are created. Perhaps the phase 1 is the collapse into a BH and the phase 2 is the formation of a horizon.

Phase 3;

9 new categories are created. This is the phase that created the (1, 3) symmetry of the quartets of the phase 4.

Phase 4;

The current phase. The previous phases are still there but not that obvious since they are not made to interact with the past phases that easily.

In this model, plenty of capacity exists to include Higgs, DM, DE and BH. Higgs is the phase 1. DM and DE are phase 2. BH might be the phase 1. There is plenty of other places to fill in this model, so there must be many more categories yet unknown.

How are the past categories kept while building the new ones of a phase? Probably this is explained by the statistics of the thermodynamics of the phase.