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
the Companion Matrix

The associated Frobenius companion matrix of this quartic system is

companion matrix

This matrix acts as the step-forward operator of the corresponding linear recurrence: it advances the system by one discrete tick, encoding the quartic relation as a first-order evolution in a four-dimensional state space. The fundamental invariants of M are its eigenvalues and the symmetric relationships they encode through traces and determinants. The eigenvalues of M are the four roots {x1, x2, x3, x4} of T(x).

The trace of M is the sum of its eigenvalues, tr(M)=0, while the trace of higher powers encodes higher power sums; for example, tr(M^2)=-4π. These traces provide coordinate-free access to the quartic’s internal invariants.

The determinant of M is the product of its eigenvalues, 2π, and more generally, det(M^k) = (2π)^k for all integers k. By contrast, det(e^M) = e^{tr(M)} = 1, so the exponential e^M is a volume-preserving transformation in 4-dimensions. This reflects only the vanishing trace of M in the exponential map, not the volume rescaling induced by the discrete step itself: the discrete evolution rescales volume by a factor of 2π at each tick.

The spectral radius of M is determined by the modulus of the complex pair, zhe_r.

matrix properties

Since M is invertible, it admits a complex matrix logarithm on a chosen spectral branch. Choose one such logarithm A = log(M) on a chosen branch. By construction, e^A = M and therefore e^{At} = M^t. This furnishes a continuous interpolation of the discrete dynamics. Unlike e^M, the interpolating flow e^{At} does not preserve volume. Its determinant isdet(e^{At}) = e^{tr(A)t} = (2π)^tso volume scales continuously by a factor of (2π)^t, matching the discrete jumps (2π)^k at integer times. Thus, the algebraic structure is clear: the discrete step M rescales volume in quantized steps, while its logarithmic generator induces a smooth dilation whose integer-time restriction reproduces those jumps. This is the signature of a system whose geometry is continuous, but whose action is discrete.

The coefficient 2πa introduces an odd-degree asymmetry into the recurrence. Relative to the simplest bilinear structures, this breaks time-reversal symmetry and acts as the discrete analogue of a non-symmetric (and later, non-self-adjoint) contribution. Because the coefficients of T(x) are real, any nonreal eigenvalues occur in complex-conjugate pairs. In the parameter regime relevant to our hyperbolic partitions, two eigenvalues form a complex conjugate pair; together they span a real two-dimensional oscillatory plane, characterized by simultaneous rotation and radial dilation. The other two eigenvalues are real and generate a hyperbolic invariant subspace.

In real Jordan form, therefore, M decomposes into a two-dimensional oscillatory block and a two-dimensional hyperbolic block, yielding the invariant splittingℝ^4 ≅ ℝ_osc^2 ⨁ ℝ_hyp^2

Viewed this way, M acts as the discrete-time propagator of the system, while A = log(M) functions as its continuous generator. Together, the quartic T(x) and its companion matrix define a clockable computational architecture: a system whose geometry is continuous, but whose action advances in discrete, quantized steps.