the hyperbolic partition equation
Hyperbolic Partition Equation

The hyperbolic partition equation encodes how the two-layer system coherently partitions within its mass gap—the normalized Planck mass. Its 4 solutions zhe_1, zhe_2, zhe_3 and zhe_4 define the hyperbolic partition constants—which possess the following product, sum and quadrance (sum of squares).

Hyperbolic Partition Product
Hyperbolic Partition Sum
Hyperbolic Partition Quadrance
zhe_1.svg= 0.0854245431533304 ...
zhe_2.svg= 3.66756753485501 ...
zhe_3.svg= –1.87649603900417 ... + 4.06615262615972 ...i
zhe_4.svg= –1.87649603900417 ... – 4.06615262615972 ...i

The square of the first solution is the fine structure constant zhe_squared=alpha.

In polar coordinates zhe_3 and zhe_4 are expressed as zhe_3_polar and zhe_4_polar, where:

zhe_r= 4.47826244916751 ...
zhe_theta= 2.00316562310924 ...

In 1591, François Viète showed that the invariants (a, b, c, d) of the general monic quartic

x^4 + a x^3 + b x^2 + c x + d

with roots x₁, x₂, x₃, and x₄, are prescribed by its sum of roots, pairwise product sum of roots, triple product sum of roots, and its product of roots.

∑ x_i = −a∑ x_i x_j = b∑ x_i x_j x_k = −c∏ x_i = d

Let’s apply these insights to the hyperbolic partition equation, which converts into a monic depressed quartic

∑ x_i = −a

with component 1, component 2, component 3, and a_scale .

zhe big productroot sumzhe big quadrance
zhe inverse productbi-product root sumsum of cubes
inverse sumtriple product sum of rootssum of fourth powers
inverse biproduct suminverse triple product sum of rootsmodulus squared
hyperbolic partition product
hyperbolic partition sum
zhe bi product sum
zhe triple product sum
square modulus
hyperbolic partition quadrance
sum of cubes
sum of fourth powers
inverse product
inverse triple product sum
inverse biproduct sum
inverse sum
zhe -1zhe inversion
zhe_1 zhe_2 productzhe_3 zhe_4 product