The hyperbola and hyperbolic

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.
As

one has for any hyperbolic angle that the point

satisfies the equation

which is the equation of a hyperbola relative its canonical coordinate system.

This passage is from Wiki. All it means, from a dimesnional point of view, is that the Lucas numbers have reached their limit and the system is no longer hyperbolic, so it will be at disequiibrium, not adiabatic in its motion. Hence solar systems are still packing sphere, but the planets are not flexible so the eccentricity is very narrow, 3 degrees instead of 90 degrees. Hence the center emits light and protons. The protons emitted are just a shift in the Wythoff Array distributed to restore sphere packing. Eentually the star equilibriates, migrating back to the optimum dimensions, trying to get back to the Lucas numbers.

When the imbalance is great enough, the center acts like a quasar for some time, the accretion angle is near the fine structure, the effect is to rebuild the baryons. This likely happens when protons near the end of their lifetime. This would be a cycling universe, naturally rebuilding the very vacuum itself. In other words, the system is bound by the optimum sphere packing dimension, likely 12 or 15, and that dimensionality has to be maintained so the span across which the system moves up and down the Wythoff array is limited by the sphere packing dynamics.  The black hole is the extreme and Hawking radiation is simply the most extreme eccentricity allowed, and at that point protons will recycle through the baryon process.

Markov numbers and the Wythoff array are likely related, and there is a bounded region in which the configuration of sphere packing must be contained. The universe is stuck, everywhere it can only get Pi computed to the dimensional optimum, other wise additive combinatorics do not work and causality is erased. The answer is 'because', the because why game has a finite terminal.