Experiments concerning the cosmological constant problem (CCP), quantum entanglement (QE) and the Dirac monopole (DM) seem to prevent a unified theory of all forces [1] - [3] . A fractal zeta universe as a cosmic-ray-charge-cloud-superfluid (FZU) resolves CCP, QE and DM by treating spacetime as a bifurcating process [4] . The origin of charge and mass in FZU is explained by Feigenbaum renormalization extending Hieb’s hypothesis [5] [6] . Hieb’ conjecture $2\pi {\delta }_F^2\simeq {\alpha }_f^{-1}$ already has accuracy 9.12 × 10⁻⁴ with Feigenbaum constant δ_F and fine structure constant α_f which is refinable on entropy-surface-area $4\pi R^2\simeq g_1^{{⋮}^{g_n}}$ for $g_1+\cdots +g_n$ [7] [8] . Similarly, FZU introduces dimensionless coupling constants G_w on w-spherical shells of a general Riemann surface enveloped by a general sphere where g_i enter as number theoretic generators which oversee 10³ orders of magnitude. Like a black hole a given fractal point in space consists of w-coordinate spheres encapsulating a universe of buds, shoots, leaves of a tree and of bizarre plants as k-components which are period-doublings. The enveloping sphere represents matter as Einsteinian elastic spacetime. Whereas the Feigenbaum constant δ_F is transformation degree dependent the fine structure constant α_f is energy- and time-dependent (e.g. α_f at 91GeV ≃ 127.5) [9] . This would question any algebraic approach to δ_F and α_f [9] . FZU coupling constants G_w (h_t) create a cubic van-der-Waals-like potential minimum in dependence on topological entropy h_t: At high density of chaotic k-components quadratic and quartic mass terms are higher than the linear rest mass term which creates matter from zeros of the zeta function. A pseudo-congruence $2^{2^k}=G_w^{-1}$ between k-components on the maximal general Riemann surface is called QE and defines a charge [10] . This surprising result up to k = 10 reproduces the static limit of α_f which is also the infinite-energy limit of α_f and solves CCP [10] . A pseudo-congruence $2^{2^{10}}\simeq 1$ of optimal k-components in ℂ⁵ of a quadratic map around nontrivial zeros of the Riemann zeta function ζ (z) is called charge. Complex-generator-cycles of optimal k-components $2^{2^k}$ in ℂ^w for w = 1, 2, 3, 4, 5 are surrounded by vanishing Gaussian periods. For a highly-composite congruent number field $2^{2^k}=g^{2^k}=g^{n^2}$ with generator g as a root of unity a vanishing series of periods can occur for components k = 2, 4, 6, 8, 10 below k ≤ 10. If generator 2^k is a square, i.e. for k = 2, 4, 6, 8, 10 five interactions w = 1, 2, 3, 4, 5 are in accordance with the Weber-Schottky conditions [11] . Matter as spacetime cavity cycles or buds of cyclic generators is inseparably connected with k pseudo-congruence. This form of QE yields air ionisation caused by cosmic rays (≡bifurcated spacetime) not only in the exosphere but also near plants and conductivity plateaus whenever a zeta function zero is iterated. FZU sees experimental support in measurements in vegetation areas [12] . Moreover, within FZU problems with cosmological constant, Dirac monopole and quantum entanglement can be overcome. The first one consists in a 50 - 200 orders of magnitude too large vacuum energy density Λ_c (which is the factor $2^{2^9}$), the second in charges as a fusilli-like cloud of magnets (a ball of k-components) and the third in a spooky action at a distance confirmed by quantum experiment (k-correlation). The paper aims to discuss these problems by k-itineraries for a bifurcated spacetime curvature. Extending the concept of vibrations of fractal strings a spacetime point is defined by congruent and incongruent k-components of quadratic maps of complex curvature [13] . Charge quanta are set as component-correlated nontrivial zeros of the Riemann zeta function and the Dirichlet L-function [10] [14] . A unified vacuum is a non-dissipative, non-radiative highly k-component correlated liquid [10] [14] . Elliptic curves as attractors are already known as an exactly solvable chaos [15] [16] . Iterates of doubly-periodic lattices as chaotic period-doublings due to lattices of algebraic units have been rarely investigated. In this paper iterated cyclic periods ν_Sh of intervals according to the theorem of Sharkovskii belong to pseudo-random and pseudo-congruent lattices of fundamental units. Period-doubling is discussed as a hyperelliptic-elliptic addition step supported also by the Friedmann solution for time [17]

$ct={\displaystyle \int \frac{\sqrt{R_u}\text{d}{R}_u}{\sqrt{{\varphi }_3\left(R_u\right)}}}$ (1)

in dependence on universe radius. Scalar curvature R

$R=\frac{c^2}{R_u^2}$

of a spherically symmetric universe differs from R_u in Equation (1) by an arbitrary quadratic map which is extended to complex plane by γ (ϕ₃). A map $z_{k+1}\leftarrow z_k^2+c$ is written as the quartic polynomial $\left(R_{\mu \nu }^4+2ℱR_{\mu \nu }^2-{\mathcal{G}}^2\right)=0$for an arbitrary matrix $R_{\mu \upsilon }=\mathrm{Re}{z}_k$ and $ℱ=\frac{1}{2}\mathrm{Re}\left(c-z_{k+1}\right)$, $\mathcal{G}=\frac{1}{2}\mathrm{Im}\left(c-z_{k+1}\right)$ invariant with respect to γ (ϕ₃). Under γ (ϕ₃) Rez_k is viewed as complex curvature R_μυ or complex universe radius R_u. The quartic R_μν scales $z_k\to G_w{z}_k$ with renormalized charge G_w = e is known from quantum electrodynamics [18] . With $ℱ=\frac{1}{2}\mathrm{Re}\left(c-z_{k+1}\right)$, $\mathcal{G}=\frac{1}{2}\mathrm{Im}\left(c-z_{k+1}\right)$ on complex plane oriented in space $z=\sqrt{ℱ+i\mathcal{G}}\simeq E+iB$ is a complex unified field if the normal of the complex plane of z_k is oriented in space. Similarly, R_μυ is a Kepler- or Coulomb field singularity ≃1/r² subjected to a quadratic map. A Mandelbrot map is part of a Hermite map

$F\left(t,z\right)=\gamma \left({\varphi }_3\left(t\right)\right)\circ z\simeq {\varphi }_3\left(t\right)/\left(t-z\right)-\frac{1}{3}{{\varphi }^{\prime }}_3\left(t\right)$

for subsequent cubic roots e_i of ϕ₃ from subsequent quartic roots x_q of ϕ₄. Under γ (ϕ₃) the polynomial ϕ₃ in Equation (1) gets a modular invariant where its argument can be written as an algebraic unit. Complex cubic roots $z\simeq E\to E_{i,q}\to E_{i,s}$ as algebraic units E and quartic roots x with one quartic root shifted to ±∞, ±i∞ transmit to a 3⋅4 degrees of freedom i, q ≃ s. An iterated solution for the 4⋅4 matrix R_μυ is searched in extended μ, s space with field densities $F+{\gamma }_{5}G=-2^{-5}{F}^2$ with $F={\left[{\gamma }_{\mu },{\gamma }_{\nu }\right]}_{-}{R}_{\mu \upsilon }$, ${\gamma }_5=i{\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4$ and ${\gamma }_5^2=-1$. Invariants in ϕ₃ can be written in terms of complex conjugated units of the bicubic field. Under period-doubling in γ (ϕ₃) $ℱ$ and $\mathcal{G}$ get differences of ℘(ω)-functions as a factor of the Legendre module λ which depends on unit E_q = e^l on four possible Feigenbaum stability axes q. Rotated cardioid planes $z\simeq e^F=chz+sh\left(z\right)F/z$ depend on the quartic polynomial for R = R_μν, z = |z| with unit determinant [19] . Iterating $\exp \left(i{g}_1^{g_l}{\gamma }_{s{s}^{\prime }}^{\mu }{x}_{\mu }\right)f_{s^{\prime }}$ in Equation (13) 1/2w (w − 1) parameters and 3w − 3 equations yield an underdetermined system of maximal w = 1, …, 5 independent complex planes [20] . The importance of the linear map γ consists in identifying $\delta x_q\simeq G_{s{s}^{\prime }}$, $\delta e_i\simeq G_{s{s}^{\prime }}{G}_{s{s}^{\prime }}$ where

$F\left(t,z\right)\simeq A_{\mu }{G}_{s{s}^{″}}{\gamma }_{s^{″}{s}^{‴}}^{\mu }{G}_{s^{‴}{s}^{\prime }}-\frac{1}{3}\frac{\delta A_{\mu }}{\delta \left(G_{s{s}^{″}}{\gamma }_{s^{″}{s}^{‴}}^{\mu }{G}_{s^{‴}{s}^{\prime }}\right)}$ (2)

A congruence ${\varphi }_3\left(z\right)\mathrm{mod}{\varphi }_4\left(x\right)=A_{\mu }\simeq a_{\mu }$ of polynomial ${\Phi }_n\left(t\right)={\displaystyle {\sum }_{i=0}^n{a}_i{t}^{n-i}}$ for $\delta e_i\simeq {\left(\delta x_q\right)}^2$ and quadruples $\delta x_q\simeq {\psi }_q\simeq {\psi }_s$ is viewed as a potential like z ≃ ℘(u). The coordinate index μ = 1, 2, 3, 4 denotes a self-consistent Feigenbaum stability axis from iterates around inflection tangents. The current $j_{\mu }={\bar{\psi }}_s{\gamma }_{s{s}^{\prime }}^{\mu }{\psi }_{s^{\prime }}$ is quadratic in δx_q where $q=\left\{k,k+1,k+2,k+3\right\}$ denotes a quadruple of simplest cycles of iterated intervals and ψ_s has a bicubic norm. A point as irreducible quadruple q ≃ 1, 2, 1'2' permeates FZU as matter-anti-matter, tidal forces and dark non-radiative exchange scattering.

A quadratic map $z_{\kappa +1}\leftarrow z_{\kappa }^2+c$ is written in extensions of a pure bicubic field 𝕂 [∂] and 𝕂 [∂^1/2]. Variable z is a complex algebraic unit living in doubly-periodic lattices ω. General relativity can be written as a vanishing discriminant Δ_F [ϕ₃] in Equation (1)

${\Delta }_F=\frac{3^4{c}_l^4}{{\Lambda }_c^2}\left(-\frac{4c_l^2}{3{\Lambda }_c}+3A^2\right)$ (3)

with velocity of light c_l, cosmological constant Λ_c, and arbitrary constant A yielding real coordinates and vacuum energy ${\Lambda }_c=4c_l^2/3^2{A}^2$. A Minkowski-bound of Δ_F for cyclic extensions $\mathbb{K}\left[\partial ,g_1^{\cdots g_l}\right]$ and rational Δ_F = 0 induce a highly nonlinear L-function (4). This singular case displays elastic continua δ_kℒ = 0 as minima of action ℒ for a discrete sequence of steps k→∞. Chaotic itineraries of a quadratic map of curvature R and universe radius R_u yield finite non-equilibrium values δ_kℒ, δ_kδ_kℒ, δ_kR_μν, δ_kδ_kR_μν , δ_kT_μν, δ_kδ_kT_μν of curvature R_μυ and stress-energy T_μν. For a Hermite-Tschirnhaus substitution γ (ϕ₃) the discriminant scales as ${\Delta }_F\to {\varphi }_3^2{\Delta }_F$. Therefore, the complex invariant f (ω) in γ (ϕ₃(f (ω))) is like R_μυ and T_μν of fundamental significance as iterates over tensile forces. The resulting dense lattice of algebraic units {l} generates spacetime superimposed by fluctuating elliptic lattices {𝕃} with Poncelet triangles of inscribed and circumscribed cones as tidal-like forces. Discrete iterates of γ (ϕ₃) are points and segments as wave packets. Laps l_ω = l_γω are orbits of an assembled shift

${\delta }_k{\displaystyle \prod {\delta }_{l_{\omega }}}$

of k-components where modular units $g\left(a\omega \right)=g\left(a\gamma \omega \right)$ are inert for lap number $\#l_{\omega }\to \infty$. The map γ ( $\det \gamma \ne 1$, $z_k{}_{+1}=\gamma z_k$) forces complex multiplication (CM) of points and segments of curves by period-doubling k-component orbits which relate to Lorentz-transformations γ_L. Treating each k-component as an excited particle quantum statistics overestimates vacuum density Λ_c by a factor $2^{2^9}$ [10] [14] . Mathematically the Euclidean norm ${\sum }_{\left(q\right)}{E}_q^{-2}={\sum }_{\left(s\right)}{\psi }_s{\bar{\psi }}_s$ in Equation (11) is formally equivalent to quantum statistics. However, ${\sum }_{\left(s\right)}{\psi }_s{\bar{\psi }}_s=1$ in CCP overestimates the binary tree of k-components in QE by factor $2^{2^9}$ because the real algebraic unit in the cyclotomic L-function decreases.

A Riemann zeta function ζ (z) allows to start from holomorphic function ξ (z) and holomorphic Dirichlet L-function [13] .

$\frac{\zeta \left(z,\mathbb{K}\right)}{\zeta \left(z\right)}=\frac{\Gamma \left(z/2\right)z\left(z-1\right)\zeta \left(z,\mathbb{K}\right)}{2{\pi }^{z/2}\xi \left(z\right)}=L\left(z,\chi \right)$ (4)

which can be scanned by γ (ϕ₃) in the Dirichlet L-function with character χ of a cubic field extension 𝕂 [∂]. The z→1 limit $L\left(1,\chi \right)\simeq H_{\Delta }{R}_{\Delta }$ is proportional to a regulator $R_{\Delta }={\ln }_{b}E$ with fundamental unit E, for base b, class number H_Δ of a cubic normal field with discriminant Δ. The Riemann zeta function ζ (z) is related to the quantum statistical scattering amplitude $\mathbb{A}\left(s\right)={\displaystyle {\sum }_{\left(n\right)}\frac{1}{s-m_n^2}}$ by the entire function

$\xi \left(z\right)=\left(\begin{array}{c}z\\ 2\end{array}\right){\pi }^{-\frac{z}{2}}\Gamma \left(\frac{z}{2}\right)\zeta \left(z\right)={\text{e}}^{{\displaystyle \int \text{d}s\mathbb{A}\left(s\right)}}$ (5)

for a quadratic map between s and z. Both ξ (z) and γξ (γz) satisfy a hyperbolic Laplace equation. A fractional substitution γ (ϕ₃) is capable to scan masses m_n in nontrivial zeros $z_{nt}=1/2+i{m}_n$ of ξ (z) and ζ (z). A Mandelstam plane $s,t,u\simeq {\delta }_{k}e={\Lambda }_{\omega }^2\left(\lambda ,1,{\lambda }^{\prime }=1-\lambda \right)$ with $s+t+u=0$ and $m_n\simeq \sqrt{s}\simeq \lambda$ are differences of cubic roots where a scaling ${\Lambda }_{\omega }=\left(2K/{\omega }_1\right)\simeq 1$ (ultraviolet cutoff) is consistent with a quadratic map ${\lambda }_k\simeq 1/2+i\sqrt{{\lambda }_{k+1}}$ for a process $s\simeq \lambda$ and $\sqrt{s}\simeq \lambda$. Iterates λ = λ_m/m + 1/2 are treated as Dirac-like currents where ${\left({\lambda }_m/m\right)}^2-1/4=2^4/f^{24}$ and ${\lambda }_m={\bar{\psi }}_q{\lambda }_{mq{q}^{\prime }}{\psi }_{q^{\prime }}$ couple to the Weber invariant f (ω). A certain quadratic map γ_n of cubic roots $s={\gamma }_n\circ s$ relates ξ-zeros z_nt (≡ 𝔸(s)-poles, charged excitations) to the single pole of ζ (z) at z = 1 (collective excitations, Kepler singularity). Equation (2) for γ_n couples the function G_ss' to a collective potential A_μ. Then the L-function enables a Dedekind zeta function

$\left(z-1\right)\zeta \left(z,\mathbb{K}\right)\simeq \sum \frac{z-1}{N{m}^z\left(f\left(g_1\right)\right)}\simeq H_{\Delta }{R}_{\Delta }$ (6)

to relate masses m_n ≃ z_nt to regulator index R_Δ and class number H_Δ.

FZU regards every point self-similarly enveloped by w-shells of a most general Riemann surface regardless of whether it is a charge, a universe or a plant. Coupling constants are derived from the L-function in particular from the regulator of a cyclic number field. A cyclic number field e.g. with periods ν_Sh where series of motions vanish can be multiplied by a coupling constant G_w. This Lagrange normal base with Gaussian periods is called interaction w = 1, 2, 3, 4, 5 [11] . Within a dimensionless FZU all physical fields are quadruples of simplest cycles ψ_s of algebraic units E which are information currents where the L-function is a statistical sum. Their assignment to forces, (strong weak, em, grav, dark) results from the pre-factor G_w. which differs by up to 10² orders of magnitude. This independence on laboratory dimension allows a self-similar view to five interactions. All forces are treated uniquely by Feynman diagrams with Euclidean norm ${\sum }_{\left(q\right)}{E}_q^{-2}={\sum }_{\left(s\right)}{\psi }_s{\bar{\psi }}_s$ where the bi spinor ψ_s is viewed as spacetime curvature R_μν ≃ F_μν ≃ E, B of a simplest cycle quadruple of a bicubic field. A decreasing circulant regulator index $R_{\Delta }=\det \ln {\epsilon }_{ij}\left({\omega }_k\right)=\det \ln {\epsilon }_i{\bar{\epsilon }}_j\left[{\nu }_{Sh}\right]$ behaves like a minimum of an action functional. The regulator index R_Δ for a bicubic field of class number H_k = 1 and R_Δ = logE ≃1 of fundamental unit E ≃ 1 whereas the lower limit of the regulator R_Δ for a dense lattice of units and imaginary fields with $n=r+s-1\to \infty$ [21] [22] .

$R_{\Delta }>{2^{3-n}{\text{e}}^{n/2}{\pi }^{n/2-1}{n}^{-2}l{n}^{n-1}\left(2\right)|}_{n\to \infty }\to \infty$ (7)

can tend to infinity. In case of cyclotomic units

$E\left(g,g_{\infty }\right)=\sqrt{\left(\frac{\epsilon \left(g,g_{\infty }\right)}{\epsilon \left(1,g_{\infty }\right)}\right)\left(\frac{\epsilon \left(-g,g_{\infty }\right)}{\epsilon \left(-1,g_{\infty }\right)}\right)}=\frac{1-Z^g}{\left(1-Z\right)Z^{\left(g-1\right)/2}}$ (8)

with complex units $\epsilon \left(g,g_{\infty }\right)=\left(1-1^{g/g_{\infty }}\right)$ and generator $Z=1^{1/g_{\infty }}$ one has [22]

$R_{\Delta }\to -\frac{1}{2}\left(g-1\right)\ln Z-\ln \left(1-Z\right)-\ln \left(1-Z^g\right)\to -\frac{1}{2}\left(g-1\right)\ln Z+Z+Z^g$ (9)

Possible is $R_{\Delta }\to \infty$ but also for $g_{\infty }\to \infty$ a vanishing $R_{\Delta }\to \ln Z+Z\to 0$. This is because small and large generator values enter the cyclotomic norm. The lower limit in Equation (7) results from a minimum of a quadratic form $\sum \left(l^2+l\right)$ for a given number field with r + s − 1 units forcing upper and lower limits of the regulator R_Δ for r real and s complex conjugated pairs of roots of unity [21] [22] . Forcing $R_{\Delta }\to \ln Z+Z\to 0$ requires a process for a tower of number field extension as a feasible process of optimal units $E=b^{{\Omega }_w-ℒ}$ leading to a tower $b^{b^{\mathrm{...}{\Omega }_w-ℒ}}$.

An additional term in the quadratic form of logarithms $l={\ln }_{b}E$ of fundamental units

$\sum \left({\mu }_1{l}^2+{\mu }_{2}l+{\mu }_3{b}^{2l}\zeta \left(l_s,m_s,l\right)\right)$ (11)

ensures a feasible solution for regulator index for base b [23] [24] . An q = r dimensional Euclidean norm ${\sum }_{\left(q\right)}{E}_q^{-2}$ is replaced by a norm-quadruple of finite periods of intervals multiplied by the geometric zeta function $\zeta \left(l_s,m_s,z\right)={\displaystyle {\sum }_{j\in \mathbb{N}}{m}_s^j{l}_s^{-\left(j+1\right)z}}$

$\zeta \left(l_s,m_s,z\right)={\frac{1}{m_s}\frac{1}{{\text{e}}^s-1}|}_{s=z\ln l_s-\ln m_s}={\frac{1}{m_s}\frac{1}{{\text{e}}^{\beta \nu }-1}|}_{\nu =z/H\left(l_s,m_s\right)-1,\beta =1/\ln m_s}.$ (12)

String length l_s = 3 and multiplicity m_s = 2 describe Hausdorff dimension

$H\left(l_s,m_s\right)={\ln }_{l_s}{m}_s$

of a Cantor set. Subseqent feasible solutions $E=b^{{\Omega }_w-ℒ}$ yield a tower $b^{b^{\mathrm{...}{\Omega }_w-ℒ}}$ of degree higher than 2^k of a subsequent map γ (ϕ₃). Base b = 2 congruences is expected at the third level for $2^{2^9}\simeq G_5^{-1}$ which supports a Fermat number transform in [25] . In this subsequent process the regulator index R_Δ can be lowered a system of equations for a power tower of generators g_i as roots of unity where cyclic periods υ_Sh are related to pseudo-congruences mod (g_i-1) in the tower $g_1^{\cdots g_l}\simeq g_1{}^{g_2{}^{g_3}}$. Periods υ_Sh are mapped to doubly-periodic lattices ω where period-doubling ${\omega }_k\to {\omega }_{k+1}+{\omega }_{k+2}$ is viewed as a subsequent generation of new lattices. Topological entropy $h_t\left(f\right)$ is generated by cyclic extensions $\mathbb{K}\left[\partial ,g_1^{\cdots g_l}\right]$. A cyclic field for a minimum of $R_{\Delta }={\Omega }_w-ℒ$ has the highest information densities. If the exponent of a generator g₁ is a square vanishing Gaussian periods are possible where $R_{\Delta }\simeq g_1{}^{g_2{}^{g_3}}$. For $2^{2^k}$-components with k = 2, 4, 6, 8, 10 one has $R_{\Delta }\simeq {\Omega }_w-ℒ$ for neighbouring interaction layers (w, w + 1) = (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), respectively. A Dirichlet L-function oscillating like a local Lovelock-like Lagrangian near $ℒ\simeq {ℒ}_0$ defines a particle. Oscillations of $z=\log E$ in ζ (l_s, m_s, z) occur on a circle of radius $\left|z\right|=H\left(l_s,m_s\right)$. A former real unit E gets complex by substituting γE which justifies to replace the quantity $l=\ln E$ by the tensorial object ${\text{e}}^R=\exp \left(R_{\mu \upsilon }{\left[{\gamma }_{\mu },{\gamma }_{\nu }\right]}_{-}\right)$ where the skew tensor R_μυ depends on a three-component complex z ≃ l, i.e. units E and lnE oscillate. Subsequent γ (ϕ₃)-maps are addition steps on each iterated curve with universal covering u,v,u±v. Period-doubling ${\omega }_k\to {\omega }_{k+1}+{\omega }_{k+2}$ yields in case of ω→2ω for the nome $q^{i\pi \omega }={\text{e}}^{-\pi K^{\prime }/K}$ the exact result $4q^{2^{k-1}}={\lambda }_k$. Accordingly, the Legendre module acts as a generator $4q^{2^{k-1}}={\lambda }_k$ of cyclic fields. For simplest cycle quadruples q triples of invariants f_k, f_k+1, f_k+2 weave a global metrical texture that is perceived as mass relating quadruple indices q to spin indices s. Periods ν_Sh in $\mathbb{K}\left[\partial ,g_1^{\cdots g_l}\right]$ with a tower $g_1^{\cdots g_l}\simeq g_1{}^{g_2{}^{g_3}}$ give a generator g₁ e.g. for a g_∞^th root of unity $g_1^{g_{\infty }}=1$ on complex plane for four-component complex roots

$i{\gamma }^{\mu }\frac{\partial f_s}{\partial x_{\mu }}\simeq g_1^{g_l}{f}_s$ (13)

with symbolic solution $\exp \left(i{g}_1^{g_l}{\gamma }_{s{s}^{\prime }}^{\mu }{x}_{\mu }\right)f_{s^{\prime }}$ which relates congruences in $g_1^{g_l}$ to a Dirac-like mass. The bi spinor f_s ≃ ψ_s is a cyclic quadruple $q\simeq s=\left\{1,2,3,4\right\}$ in a field $\mathbb{K}\left[\partial ,g_1^{\cdots g_l}\right]$ projected onto complex plane. L-functions with circulant regulator determinant $\varphi \left(g_1\right)={\displaystyle {\sum }_{i=0,\cdots ,g_{\infty }-2}{a}_i{g}_1{}^{g_2^i}}$ yield a series expansion $\left({\sum }_{\left(q\right)}{E}_q^{-2}\right)\zeta \left(l_s,m_s,\log E_q\right)$ in the amplitude 𝔸. The norm $E{f}^{\prime }{f}^{″}=1$ with bicubic complex conjugates $f^{\prime }$ and $f^{″}$ of component ψ_s ≃ R_μυ is a cyclic complex curvature pending between flat, closed and open universes. A linear-dependence between z_k, z_{k + 1} and z _{k + 2} gives Equation (2) as a renormalized map

${\gamma }^{\left(ren\right)}=\gamma +\gamma \circ {\Gamma }^{\left(ren\right)}\circ {\gamma }^{\left(ren\right)}$ (14)

with vertex Γ^(ren). In the limit k→∞ one gets the Feigenbaum equation

${\gamma }^{\left(ren\right)}\left(z\right)=-{\alpha }_F{\gamma }^{\left(ren\right)}\circ {\gamma }^{\left(ren\right)}\left(-z/{\alpha }_F\right)$ (15)

with k-component generator g^k = α_F. A particle peels out from a $2^{2^k}$ polar ball originating from a non-trivial zero z_nt. Simplest cycles of f (ω) have $\mathrm{deg}{\gamma }^{\left(ren\right)}=2^k$ whereas $z_{nt}\left(1-z_{nt}\right)=2^4/f^{24}$ has degree (4⋅3)^k. Both congruences are consistent for a k-component-Fermat number transform which is invertible in $2^{2^{10}}$ for the first four prime number.

Processing L-functions and regulators for various iterates the action functional 𝓛 in process (11) and (18) consist of a stationary bifurcating term μ₁ (outside zeros z_nt), a count rate μ₂ and a scattering term μ_3. These terms are called holomorphic potential or conductivity plateau, air ionization or net rate and, cosmic rays or bifurcating trees. Transitions between plateaus generate a net rate (18) with occupation number (12) due unstable orbits of bifurcation. A formerly iterated unit $z=E=\exp \left(l\right)\simeq \lambda$ undergoes additional optimizing as Feigenbaum renormalization or a regulator process $z^{\left(ren\right)}=E^{\left(ren\right)}=\exp \left(l^{\left(ren\right)}\right)\simeq {\lambda }^{\left(ren\right)}$ via terms μ₁, μ₂, μ₃ in Equation (18). Supposing that nontrivial zeros z_nt of ζ (z) describe masses m_n charge quanta are definable by entire, holomorphic ξ (z) and L (z,χ) which satisfy a hyperbolic Laplacian ${\Delta }_h\xi \left(z\right)=0$ with ${\Delta }_h=y^2{\Delta }_{xy}=\mathrm{Im}{\lambda }^2{\Delta }_{xy}$. For ξ (z_nt) = 0 and λ = z_nt one gets the screened Poisson equation

${\Delta }_{xy}\left(L\left(z,\chi \right)\xi \left(z\right)\right)+{\mu }_{s}L\left(z,\chi \right)\xi \left(z\right)={\mu }_c\left(\mathrm{Im}\lambda -m_n\right)$ (16)

for m_n slices with $\mathrm{Im}{z}_{nt}=\mathrm{Im}\lambda =m_n$ with Lagrange parameter μ_s and μ_c for conditions ${\Delta }_h\xi \left(z\right)=0$ and $\mathrm{Im}\lambda =\mathrm{Im}z=m_n$. Equating z ≃ λ and ξ (z) ≃ j for module λ and complex current j a plateau $\xi \simeq E\left(z\right)={\chi }^{-1}\left(\nabla V,\nabla T\right)$ of conductivity χ and complex electric field E (z) is equivalent to the existence of a holomorphic function ξ (z). Equivalently for field $\nabla V$ and temperature gradient $\nabla T$ a plateau denotes a holomorphic global potential

$V_{T_{global}}={\displaystyle \int \nabla V_{T_{cloud}}\text{d}{l}_{xy}}$ (17)

which describes a non-radiative, non-dissipative superfluid of discontinuous self-similar segments dl_xy. Traversed Xi-function zeros give a holomorphic current j ≃ E with divj = divE = 0. Equation (16) is invariant for subsequent chaotic γ (ϕ₃ (f (ω))-maps. The algorithm is completed by an optimal step for a regulator index minimum for base b of the quadratic form (11) giving

$2{\mu }_{1}l+2{\mu }_2{E}_q^{-2}\zeta \left(l_s,m_s,l\right)+{\mu }_3\left({\displaystyle {\sum }_q{E}_q^{-2}}\right){\zeta }^{\prime }\left(l_s,m_s,l\right)=0$ (18)

with Lagrange multiplier μ₁, μ₂, μ₃. Generators g_i in algebraic units E_wqi cause stable laps l_ω of orbits felt as masses. The number of stable 2^k orbits is called a lap l_ω whereas a k-component is an unstable orbit of bifurcation. Simplest cycles quadruples $q:\left\{k+3\in \left\{k,k+1,k+2\right\}\right\}$ are stable against laps l_ω. For $z_{k+1}=F\left(t,z_k\right)=\gamma \left({\varphi }_3\right)\circ z_k$ entropy $H={\sum }_{\left(i\right)}{P}_i\ln P_i$ can be calculated by ${\text{e}}^P={\displaystyle {\prod }_{l,k}{\text{e}}^{P_{l,k}/\left|F^{\prime }\left(t,z_{l,k}\right)\right|}}$ in terms of the probability density P (z) for finding an orbit at z. Like Lyapunov exponents highest information densities are expected at critical points $F^{\prime }\left(t,z\right)=0$. The thermodynamic variable ${\Omega }_w=1/2{\mu }_1$ in $l={\ln }_{b}E={\Omega }_w-ℒ$ depends on topological entropy

$h_t\left(f\right)=\underset{n\to \infty }{\lim }\frac{1}{n}\ln l_{\omega }\left(f^n\right)$ (19)

with lap number l_ω. Then the action reads $ℒ={\mu }_2{E}_q^{-2}\zeta \left(l_s,m_s,l\right)+\frac{1}{2}{\mu }_3\left({\displaystyle {\sum }_{\left(q\right)}{E}_q^{-2}}\right){\zeta }^{\prime }\left(l_s,m_s,l\right)$. Like the Weierstrass ℘-function the cubic ${\wp }^3-g_2\wp -g_3\simeq {\varphi }_3\left(f\right)$ is regarded as a potential energy Ω_w. For all interactions w = 1, 2, 3, 4, 5 the complex logarithm of algebraic units $l=\ln E$ is viewed as curvature tensor which oscillates on a circle of radius Hausdorff dimension H (l_s, m_s). A circulant regulator $l_i\to R_{k,ij}=1^{ij/g_{\infty }}\cdot \text{diag}\left(\ln E\left(g_{\infty }^i\right)\right)\cdot 1^{-ij/g_{\infty }}$ near H (l_s, m_s) leads to $\prod E=G_w\prod {\text{e}}^R=G_w\prod \left(chz+sh\left(z\right)R/z\right)\simeq G_{w}R$. where $\ln G_w=w!2^w{H}^w\left(3,2\right)$. The thermodynamic potential ${\Omega }_w={\displaystyle \int \text{d}{\sigma }_5{j}_w{A}_w}$ in units $E=b^{{\Omega }_w-ℒ}$ gets a complex finite generation count rate. Interaction with the next neighbour in vector potential $G_{w-1}{A}_{w-1}+\kappa A_w$ dominates for Born-Oppenheimer parameter $\kappa ={\left(G_w/G_{w-1}\right)}^{1/4}>G_{w-1}$. Therefore, this interaction requires $w>5\ln 3/2\ln 2$ or w > 3,9524 which is realized for gravity and dark matter. A physical interaction is defined as a minimum of the L-function $H_{\Delta }{\ln }_{b}E\simeq H_{\Delta }\left({\Omega }_w-ℒ\right)$ in accordance with elastic spacetime. Because $\lambda {\lambda }^{\prime }=2^4/f^{24}$ depends on the algebraic units E, one has ${\ln }_b\left(\lambda {\lambda }^{\prime }\right)=-4{\ln }_{b}2+8H_{\Delta }{\ln }_{b}E=-4{\ln }_{b}2+8H_{\Delta }\left({\Omega }_w-ℒ\right)$ reducing action $ℒ$ by topological entropy 4ln_b2. Holomorphic plateau-like information currents describe dissipation-less doubly-periodic waves of temperature and entropy. A bifurcation of curvature on a complex surface is viewed as branches and leaves of trees and bizarre plants as shown in Figure 1 .

Figure 1. (left) Fractal zeta zeros in nature and nanostructure laboratory: Plant, green trees under blue sky [26] (right) Plateaus in quantized Hall conductivity [27] .

The charge condition in Equation (16) goes via holomorphic ξ (z) plateaus-like where a plateau denotes a neutral chaotic quadrupolar motion between ±1/2 ± im_n with e.g. coupling constant $2\pi {\delta }_F^2\simeq {\alpha }_f^{-1}$. A charge ‘e’ forms out for congruences $2^{2^k}=G_w^{-1}$ on ℂ^w. Then a 1meV↔10²⁰eV congruence justifies to multiply the global potential (16) by the charge ‘e’ and to replace fractional ζ (l_s, m_s, l)-segments in Equation (17) by a differential dl_xy. A maximal five-layer gyro-twist surface ℂ^w contains k-component period-doubling

${\delta }_k{\displaystyle \prod {\delta }_{l_{\omega }}}$

The tower $2^{2^k}$ is regarded as tree of particles on fractal length dl_xy of leaves of the tree in Figure 1 . Beyond resolving a 10²⁰eV energy into a 1meV potential the cosmic microwave background above GZK cutoff is simply first periods ν_Sh at low k. It is claimed that measured microwave emission near conductivity plateaus proves this conjecture [28] [29] . Vanishing Gaussian periods in interfaces are viewed as self-contained intermediate layer between e.g. five atmospheric layer ℂ⁵. Accordingly, the module $2^{2^{10}}$ explains CCP, the definition of quantum statistics as well QE in the universe [10] . Oscillations of L-function near vanishing Gaussian periods are particles. Topological entropy h_t is like stirring a non-dissipative highly-correlated solid-fluid-gaseous slushy with a whisk creating periods ν_Sh. This k-component process is universal in micro space and macro space and creates plateaus as simple zeta zeros

$\xi \left(z\simeq z_{nt}\right)={\displaystyle {\prod }_{{z^{\prime }}_{nt}}\left(1-z/{z^{\prime }}_{nt}\right)}\simeq M\left(1-z/z_{nt}\right)$

which can be regarded as a giant mass

$M\simeq {\displaystyle {\prod }_{{z^{\prime }}_{nt}}\left(1-z_{nt}/{z^{\prime }}_{nt}\right)}.$

The γ-correlated k-component tree ${\delta }_k\xi ={\gamma }^{\left(ren\right)}\circ {\delta }_{k}z=\left(\gamma +\gamma \circ {\Gamma }^{\left(ren\right)}\circ {\gamma }^{\left(ren\right)}\right)\circ {\delta }_{k}z$ of bifurcating spacetime ranges from plants or conductivity plateaus in semiconductor layer laboratory up to higher atmospheric layers [12] [30] .

Besides ergodic time-reversible laps non-ergodic time-irreversible components exist. Every point is surrounded by $2^{2^k}$ components which means that matter as cosmic rays is created for large M e.g. at seasonal growth of plants. Like cosmic rays enhanced air ionization rate near plants as well a seasonal behavior of cosmic rays is expected. Both predictions are measured e.g. in vegetation areas in Figure 2 and seasonal variations of cosmic ray intensity [30] .

The minimal and maximal nontrivial case is taking spacetime as discrete dynamics on elliptic curves. This iterates quadratically a complex curvature of spacetime. Complex curvature opens a spacetime bud or cavity of closed lines of complex numbers composed by roots of unity. Number-theoretically period-doubling steps k increase the lattice dimension k of algebraic units as information density which self-consistently minimizes a regulator index of cyclic extensions of number fields. Like organic growth period-doubling creates buds, shoots, leaves of a tree. Matter as correlated spacetime cavities or buds with cyclic generators is inseparably connected with k pseudo-congruent iterated complex curvatures. This highly correlated non radiative non-dissipative potential defines an upper velocity of quantum entanglement in spacetime which underlies causal interactions. Examples are air ionisation at plant growth due to cosmic-ray-like shower k-components which are highly correlated up to the exosphere and, thus, are a stabilizing environment of every matter. Like holomorphic surfaces of plants, a conductivity plateau displays a path-independent holomorphic potential between two points on a two-dimensional surface. The transition between plateaus is connected with large mass changes M and an emission rate of a k-component tree proportional to the geometric zeta function (12).