So we are packing a big sphere

Why would the Lagrange numbers mark degrees of freedom? Are the possible exchanges by the vacuum bound by geometric lattice considerations?

The ratio, P/Q that defines the est Phi ratio for simple gaussian nulls is set by a change in bubble size, the exchanging vacuum elements alter curvature by optimizing bandwidth, assume they can change their relative size. By adjusting size and lengthening the recursive order, they get twice the possible exchanges. A gain from quantization. That gain would be represented by look back for any of the constituents looking for exchange. Geometrically there must be another degree of symmetry in the ideal arrangement.

When the vacuum exchange constituents adapt, the spectrum of exchange is optimally matched to the spectrum of the unit circle. Noise is a maximally orthogonal group of one order above the unit circle. So the simple gaussian balls have square motion separating their probability of appearance. In a curved space, these fuzzys collide causing a sampling error, and putting spheres out of place. The disruptive noise cause a unstable change in the spectrum of motion, and the vacuum constituents change shape to quantize, and it has trapped the second degree of motion. That is how they do the finite log.

Under the pressure of vacuum curvature, collisions cause mixing of degree two quants.  With the current degrees of freedom, the unit sphere is shaped to avoid lower order emissions. So the mixing skips the missing quant levels and shifts the chain back once again, increasing the polynomial degree to level three for noise. The wave constituents add another dimension to unit sphere curvature, and in the process, lock in a cube root.

So that makes the unit sphere adding up the finite log from the high frequency to one, and kinetic motion adding up to the  rest from one to the lowest frequency.

Bandlimited exchange rate:
That means all exchanges are locally determined, up to some degree of reach. If the process is locally additive then they can be treated as exponents. Exponents are quantized because of the  band limit of the exchange and minimum redundancy.

So I find this, Vector Hyperbolic Equations which finds the set of third order polynomials on the sphere.  Not I am stuck reading, but I want the integer solutions.