The cosmological constant problem (CCP) questions unique vacuum stress-energy density due to discrepancies up to hundreds of orders of magnitude for various interactions [1]. Field equations require unique vacuum energy density ρ v a c and a real Langrangian. Thermodynamic laws ρ v a c concern a volume vol ( ℳ ) of a closed system in Minkowski spacetime ℳ with well-defined temperature. Models on pulsating or oscillating universes propose that space and matter are dynamic, oscillating, or pulsating fields [2]-[5]. Spacetime can be explained by additions on elliptic curves and chaotic one-dimensional quadratic maps [6][7]. The fine structure constant can be explained by Feigenbaum renormalization [8]. Covariant coordinates can be defined by rational triangles [6]. The paper continues definition of ℳ , mass and charge by the hyperbolic border between the Julia set and the Fatou set J ( N q ) − ℱ ( N q ) of a quadratic map [7][9]. Rotations by ± π in interval [ 0 , 1 ] are capable to explain wave vectors k μ on Mandelstam plane M s t u and Feigenbaum renormalization on complex plane [7]. A quadruple q of algorithmic period-3 steps q ≃ k + 3 ∈ k , k + 1 , k + 2 is under constrained with respect to two steps k and k + 1 . Section 2 introduces a vacuum state which covers quantum statistics (QS) and general relativity (GR). Section 2, 3, 4, 5, 6 show that ground states are paired superfluid unified respiratory-vacuum states with quadratic in-mass fluctuations and doubly-periodic non-Hermitian interactions. Elliptic symmetry explains that lower energies ρ v a c can be obtained in QS and GR. Addition on elliptic curves obeys high complexity used in cryptography which would indicate a low statistical weight of cyclic orbits. However, equivalent elliptic curves obey symmetries by permutations of quartic roots x i . Two-component ± rotations create 2 2 k wave vectors k μ for cyclotomic units ζ ( 2 ) and ζ ( 3 ) with mod 2 and mod 3 congruences explaining matter as a quasi-continuum of k -steps.

The partition function ζ ( z , Z [ g 2 , g 3 ] ) for a complex Lagrangian ℒ is investigated for logarithmic singularities of an elliptic integral of the third kind. Symplectic structures of ℒ are suspected only in the immediate vicinity of zeros for entire transcendent function ϕ ( ∞ ) ( z ) . This algorithmically accessible region on complex plane requires a unique factorization domain (UFD). The non-Hermitian Lagrangian ℒ restricts to a quadratic root finding process N q ( z ) of q ( z ) = ( z − z 1 ) ( z − z 2 ) where z ≃ ϕ ( ∞ ) ( z ) ≃ ζ ( z ) ≃ Z compares to zeta function ζ ( z ) and partition function Z which is denoted by ζ ( z , Z ) . Starting from homographies in projective space ℙ 3 induced currents are defined by the Legendre module λ = ( x i x j ) ( x k x l ) / ( x i x k ) ( x j x l ) as cross-ratios of four points x i which are projectible to complex plane ℂ . The reduced vector of points i , j

consists of cross-ratios λ i j = M i j ( z ) = z − z i z − z j . λ i j pairs i , j are pinned to planar rational triangles T ( z q ) as ℳ -invariant currents on complex plane ℂ where

The aim of the present paper is to show that (1) is a vacuum state which allows equivalent doubly-periodic states explaining the CCP. Processing two simultaneous currents is a unified respiratory vacuum in a breathing pulsating universe. N q ( z ) is conjugate to z 2 by a Moebius substitution M i j ( z k ) = M i j ( z k + 1 ) 2 [10]. One-periodic vacuum states ϕ v a c ( λ i j = 0 , 1 ) enable rational z -values in

T ( z q ) . Homogeneous quartic roots ( x i x j ) = 1 2 i ψ i ∧ ψ ¯ j depend on a four-component complex ψ i = x i 1 + i x i 2 on Gaussian plane which are 4∙4 component Grassmann variable pairs ( c 1 , c ¯ 2 ) and ( c - 1 , c 2 ) ( c ¯ 1 , c 2 ) in a forthcoming paper. The product of (1) with ( x i x k ) ( x i x l ) yields the paired state (4) of a renormalization group (RG) flow where the R Δ derivative d d ln ( z − e i ) is the fermion number

Operator N Δ . In case λ i j = a + a = 0 , 1 functions are rewritable in terms of Fermion creation a + and annihilation a operator subjected to a 2 ⋅ 2 substitution in a forthcoming paper. Doubly-periodic functions ϕ v a c ( λ i j ) consist of k -components which are viewed as particle pairs. A particle is an invariant image reflected by epipolar, trifocal and quadrifocal matrices. Their two-dimensional minors are binary invariant on Poincare’ upper half plane ℋ 2 . The entire transcendent polynomial ϕ ( ∞ ) ( z )

is regarded an eigenvalue equation of a non-Hermitian matrix H ∞ ( z ) n n ′ = δ n n ′ z n t − z 0 which only requires a Hilbert-Schmidt norm N ( ∞ ) = tr H ( ∞ ) H ( ∞ ) + . The state ϕ v a c ( λ i j ) of triangles z 0 + z i + z j = 0 approximates only a definite zero z n t , n . The determination of a specific transcendent value of z n t , n is left open. Accordingly, the zeta function is a partition function

where ζ ( z , Z ) is created by non-Hermitian Lagrangian ℒ ( ∞ ) . The present paper investigates only an encircling of z n t , n by fractional substitutions of γ ( 3 ) ∘ z [6]

This quadratic map is a permutation of quartic roots [11]. However, γ ( 3 ) maps the complex plane to a cubic number field with half-differentials for λ -invariant λ i j states in a Newtonian root finding process

The rational substation (18) implies a symbolic cubic power integral base w k → w k by { w } = { w 0 = 1 , w 1 , w 2 } . The algorithmic advantage is that (18) obeys a transvectant giving the invariant equation ϕ ( 3 ) ( z ) → 4 z 3 − g 2 z − g 3 = 0 with elliptic invariants g 2 , g 3 . Accordingly, phases on complex plane L ( w , z ) = ln ( w − z ) allow to define binary invariant Green’s function G ( w , z ) = ∂ z L ( w , z ) = 1 z k + 1 − z k by using a singularity δ ( φ q − φ q ′ ) in interval [ 0 , 1 ] . The physical origin of the Dirac delta function δ ( φ q − φ q ′ ) is a source in heat equation because the string (11) enters theta constants via cubic roots. A time-interval average 〈 ( t + Δ t ) 2 〉 ≃ t Δ t connects the heat equation with the wave

where a γ iterated quadratic form q ( z ) has different representations

which are compared to a holomorphic

for contours C n around z n t . Various representations of q ( z ) -iterates apply to mean values in QS or as a product of inverse functions G − 1 in GR in the vicinity of a definite ξ ( z ) zero. Mathematically, the exponent in (18) indicates a relation to poles of L -functions and a proportionality to the regulator R Δ of algebraic units. For r -dimensional units in (15) R Δ is determined by feasible units ε which maximize the ε - density. These optimal states φ q 2 ≃ ( φ q + ρ 0 ) 2 ≃ e φ q are realizable by mod 2 and mod 3 congruent cyclotomic units ζ r ( 2 ) and ζ r ( 3 ) in definite circles of radius ρ 0 on complex plane [7]. mod 2 congruences are used to define fermions. The inverse Green’s function G 0 − 1 measures the distance ρ 0 from a definite point on complex plane. Optimal states are expansion series of self-energy Σ ≃ G G into the Green’s function on complex plane for a one-dimensional singularity δ ( φ q − φ ′ q ) of ± π rotations on real interval [ 0 , 1 ] . Therefore, optimal (feasible) units ε are realized for discrete mod 2 and mod 3 congruences in G ( z ) ≃ ln ( G 0 − 1 − Σ ) . Zeros of (8) are also zeros of Z [ g 2 , g 3 ] which are discussed in conjunction with topological phase transitions [13]. Topological phase transitions induced by permutations of quartic roots x i are plausible on a circle q ( z ) travelling around a circulating string in ℛ L . This logarithmic singularity in a non-Hermitian ℒ is a complex non-dissipative state as an exact elliptic integral of the third kind. For rational z the envelope polynomial of (1) is ϕ ( 3 ) ( z ) = z 3 − g 2 z − g 3 because a rational substitution z → γ ( n ) ∘ z yields invariants g 2 , g 3 resulting in z ≃ f ( ω ) = e − i π ω 24 ∏ n = 1 ∞ ( 1 + e i π ω ( 2 n − 1 ) ) . For invariant z k + 1 ← F ( 3 ) ( w , z k ) = w 0 ( 4 z k 2 + 2 3 g 2 ) + w 1 z k a Mandelbrot map with c = 2 3 g 2 is very close to a field density. Introducing field strength D μ ν = 2 ℜ e z k → [ k μ , k ν ] − , current density I μ ν = 2 J m z k , field densities ℱ = 1 2 ℛ ( c − z k + 1 ) and G = 1 2 J ( c − z k + 1 ) the map (5) reads D μ ν 4 + 2 ℱ D μ ν 2 − G 2 = 0 . This can be rewritten as a density ℱ + i G = D μ ν 2 + 2 i D μ ν I μ ν − I μ ν 2 in the presence of a current. The invariant g 2 = 3 ( ℱ + i G ) + 3 2 z k + 1 in (5) is like an energy density writing D μ ν + D μ ν * = X = ℱ + i G ≃ E + i B for an electromagnetic field [4][14].

Minkowski spacetime ℳ is seen as a subset of rational coordinates of a under constrained complex Riemann surface ℛ L of bifurcating elliptic curve fragments. Iterates (4) and (5) create an under constrained Riemann surface ℛ L in

the immediate vicinity of a nontrivial zero z n t = 1 2 + i m n of a transcendent entire

polynomial. Under conformal transformations γ ∘ z creates a half-differential d z on ℛ L [15]. For invariant quadruples q an expansion up to the discretized second derivative is sufficient. One has z k + 2 − z k + 1 = δ F ( z k + 1 − z k ) with

Feigenbaum constant δ F . The set J ( N q ) − ℱ ( N q ) are discrete additions on elliptic curves E λ . Under γ the diameter ( Δ z ) depends on q -invariants with half-differentials d z . The adiabatic approximation ( g 2 , g 3 = c o n s t ) for ϕ ( 3 ) ( z ) consists of two concentric balls L 1 = L 2 = L 3 in an infinite string texture L 4 → ∞ in ℛ L . Their Hausdorff measure is the string volume vol q ( ℛ L ) connected with the Lebesgue measure of d z ≃ ( Δ z ) 2 by a ball of volume vol k , k + 1 ( ℛ L )

Wave vectors k μ are simply discrete φ q sequences. Two φ k pairs constitute a quadruple φ q of pairs with ± π rotations leading to a coordinate-spin-quadruple φ q with D = D μ ν [ γ μ , γ ν ] with Dirac matrices γ ν . The diameter of (9) is

with complex angle φ 0 defined by Δ = g 2 3 sin 2 φ 0 . Chaotic dynamics is on circles of radius L q with discrete angles

are chosen centered around z n t on ℛ L . k -components of γ correspond to cubic roots

leading to the 12-component string φ q . A φ q vector winds into the half-differentials d z on ℛ L

with Lagrangian ℒ ( z k , z k + 1 ) two-step density. φ q ≃ φ μ coefficients are wave vectors k μ ≃ 2 π / L q . The surface ℛ L is defined by covariant substitutions k μ = e μ ν k ν . The vierbein e μ ν introduces differentials ∂ μ = e ν μ ∂ ν . Shifting a triangle T ( z q ) by d z yields again a median of triangle T ( z q ) with four squares to ensure rationality. The phase d φ q is the arc length d φ q ≃ d s of a circle around T ( z q ) . From one-periodic wave vectors k μ one can conclude to rational coordinates x μ . An optimal phase φ q 2 ≃ e φ q requires to expand complex angles i φ q + L q around self-consistent circles for mod 2 and mod 3 congruences. It is claimed that the square of the arc length is metric in ℳ

with tensor g μ ν = e μ α e ν β η α β and signature η μ ν = ( − 1 , 1 , 1 , 1 ) due to L 1 = L 2 = L 3 ≠ L 4 with L 4 → ∞ in the adiabatic approximation. This model uses degrees of freedom by x i - permutations (4). This period-fluctuating cubic field can also be described by a real unit ε with ε = ∏ r ζ r ( 2 ) ζ r ( 3 ) ρ r mod 2 and mod 3 congruent cyclotomic units ζ r ( 2 ) and ζ r ( 3 ) . Another representation of cubic roots

and discriminant [16]

is conceivable through fluctuating units ρ l in the region of phases φ . The volume vol k , k + 1 ( R L ) of the ball is calculated by the density of ideals of units which is the limit of the number of ideals T in (17) per its norm t [16]. This proves that the half-differential dz vicinity of z n t depends on a Lagrangian ℒ ( z k , z k + 1 ) defined by the circulant regulator index R Δ = ln ε [17]. A semi-quantitative calculation up to constants would yield [18]

Summarizing, the first non- trivial case of a quadratic q ( z ) = ( z − z 1 ) ( z − z 2 ) ≃ G − 1 ( z ) G − 1 ( z ) yields a product of inverse Green’s functions G − 1 ( z ) whereas QS is linear in G − 1 . For k → ∞ (4) and (5) describe two hyperbolic regions with focal points z 1 , z 2 provided it is a UFD. An expansion into ∂ z N q converges for ∂ z N q < 2 into 1 + 1 2 ∂ z N q . A UFD allows to relate ∂ z to Δ z and vice versa. Rational coordinates require a UFD for rational z ∈ Q with Δ z [ ∂ z ] and ∂ z [ Δ z ] . Rational iterated variable z

are cubic roots concentric around L q . The adiabatic approach with four quartic roots spins a 4-component thread ψ q of 4-component ribbons shown in Figure 1 including the spectator root at infinity around f ( ω ) ≃ ζ ( 12 ) e − i π ω 24 . It is noted that the UFD derivative

fluctuates with z around a cubic base component w 1 . Accordingly, the expansion of q ( z ) is into the vertex Γ [ D μ ν ] .

Figure 1. A ball of strings ( φ k ) T exp ( D ) φ k + 1 ≃ ( Δ z ) 2 . Pair of ± π rotations in [ 0 , 1 ] (solid), spectator root in x i = { ± ∞ + ± i ∞ , e i } (dotted), infinite shift (dash-dotted).

The estimated phase volume det ln ε Δ is a determinant of a circulant matrix giving det ln ε = ∏ j = 0 r − 1 ϕ ( r ) ( ζ ( r ) j ) with coefficients ε j in polynomial ϕ ( m ) for various cyclotomic generators ζ ( r ) , r = 2 , 3 , 4 , 6 , 12 , 24 . The under constrained Riemann surface ℛ L is thought as bifurcating line bundles which are regular and invertible for a UFD. This constrains the minimum of det ln ε Δ to a limited number of fundamental units ε . However, the complex phase L ( w , z ) in ℛ L

fluctuates in degrees of freedom of discriminant (13) and (14). The cubic field has cylindrical orbits of complex phase l = ln ε = ( ρ , φ ) . Feigenbaum renormalization z → g ( z ) with respect to variable φ is set in context to direct and exchange diagrams in QS with respect to the vertex Γ. The one-dimensional renormalized function g ( φ ) or g ( Γ ) produces a single maximum or twin peak maxima. It is claimed that a quadratic expansion in φ or Γ with three conformal steps is sufficient for γ -invariance [15]. In this approximation the Lagrangian is quadratic in N Δ [17]

Then γ -invariance of orbits at the minimum of the circulant ∏ j = 0 r − 1 ϕ ( r ) ( ζ ( r ) j ) yields a linear and a quadratic condition

where ln ε ≃ ∫ 0 z G ( z ) d z → ∮ d ν ( φ q + 1 2 ( ln g 2 − ln 3 ) ) . This proves a relation between the mod 2 field ζ ( 2 ) and the Green’s function G ( z ) . Locally the linear term in (20) with (13) and (18) reads for φ q 2 ≃ ρ e φ q , Δ φ q 2 ≃ Δ ρ e φ q ≃ g μ ν d x μ d x ν and normal vector d x μ d x ν

where − 2 δ ( Δ ρ ρ ) δ g μ ν = − R μ ν is a curvature tensor. The second term is γ -invariant

and should be written in terms of the Schwarzian derivative { F , z } = F ⃛ F ˙ − 3 2 ( F ¨ F ˙ ) 2 where F ˙ = ∂ z F

The singularity δ ( φ q − φ ′ q ) yields the RG flow stress-energy { z μ , z ν } = γ ν ∂ μ G s s ′ ± .

Further investigation is needed to show that radial and tangential derivatives on a circle on complex plane become curvature tensor and stress-energy.

For ϕ v a c ( λ i j = 0 , 1 ) the discriminant Δ vanishes which yields for frequency ν 0 = 9 g 3 / 2 g 2 .

a one-periodic behavior of ℒ and energy (21) [19]. The period rectangle ω changes into a line. For arbitrary g 2 , g 3 the Lagrangian

can be written in terms of changes of topological entropy h t due to additions on equivalent elliptic curves. Topological entropy

is defined by the specific cardinality C ( N , g 2 , g 3 ) for indistinguishable orbits [20]. On universal covering on complex plane C ( N , g 2 , g 3 ) can be related to n t h order functions [19]

defined by Weierstrass σ -functions and the i t h derivative of the Weierstrass function ℘ ( i ) . Then the cardinality C ( N , g 2 , g 3 ) is an elliptic integral of the third kind. This integral is related to the two-dimensional Green’s function

which is based on the one-dimension Dirac delta function δ ( φ q − φ ′ q ) . Due to (19) a phase L ( w , z ) is reformulable as an integral over a vertex Γ

or an integral over the electro-chemical potential μ with fugacity RG flow Γ = e μ ≃ z . The squared Dirac equation for ψ s ≃ ψ q

m n 2 operates on the Mandelstam plane M s t u with s , t , u ≃ φ q 2 ≃ ( k μ + k ν ) 2 ≃ m n 2 . As known (31) implies negative mass densities ρ v a c . Next it is shown that the logarithmic singularity in (25)-(29) yields equivalent ζ ( z , Z [ g 2 , g 3 ] ) minima

Constrained one-periodic systems of length L have energy

ρ v a c ≃ ∑ n ω n ≃ − ∑ n 2 π L n ≃ − 2 π L ζ ( − 1 ) which can be related by renormalization

(Casimir effect) to the zeta function at argument z = − 1 [21]. Confinement replaces a unit volume by the lower mean of a standing wave sin 2 ( ω n ) with real ω n which is a small correction of ρ v a c . Doubly-periodic ω imply a self-consistent confinement by means of non-dissipative damped exponential tails. Whereas in QS a complex energy induces damping the non-Hermitian ℒ of (1) induces a superfluid pairing of charges [22][23]. Invariant quadratic root finding (5) creates the state (1) and induces a complex period ω . It is shown that this simple model simulates phase transitions at zeros of ζ ( z , Z [ g 2 , g 3 ] ) where iterates z k become periods with complex multiplication. In the cubic case z k is a transvectant which tends to the Weber invariant f ( ω ) with f f 1 f 2 = 2 [11]

where bars denote conjugated units ε ~ f 1 / f 2 , ε ¯ ~ f / f 2 of a cubic normal field K ⋅ K ′ ⋅ K ″ of discriminant Δ and Dedekind eta function η ( ω ) . Logarithmic singularities in σ ( u − u i , ω k )

are proportional to Δ ρ ρ ≃ ∏ i , k σ ( u − u i , ω k ) ≃ e L ( w , u ) ≃ ∑ ε ω . Nontrivial zeros become approximated by mean values of periods

The vacuum state (1) is capable to resolve an algorithmic step quadruple q whereas ℳ is not. Processing in particle accelerators, fusion reactors and artificial photosynthesis is mainly sequential steps of one-periodic interactions. This classifies unique vacuum energy, binding energy, inverse temperature β as a mean thermodynamic energy Ω which reads in QS

without having logarithmic singularities [24]. Doubly-periodic processing consists in infinitely many simultaneous changes of at least two different parameters like a breathing process. Vacuum polarization in QS is one-periodic virtual scattering and a one-periodic chemical potential ν with occupation number N Δ for λ i j = a + a = 0 , 1 in ϕ v a c ( λ i j = 0 , 1 ) . Accordingly, Feynman diagrams sum direct and exchange scattering. QS proves this behavior by Γ-linear and Γ- quadratic scattering amplitudes. Both terms are statistically equally weighted over smoothed out ℒ - singularities. The QS time interval of the measurement is large as compared to internal frequencies. The Feigenbaum renormalized ζ z , Z [ g 2 , g 3 ] receives either a single maximum or two maxima. Accordingly, the logarithmic singularity in

is a complex chemical potential of an eternal process of pair creation and topological phase transition. This process traverses a zero of the partition function for ϕ v a c ( λ i j ) with arbitrary λ i j around a phase transition on a circle quadratic in two complex masses and two complex curvatures of spacetime which is felt as a drift-diffusion process with two velocities of light c l . It is argued that in a spacetime volume vol ( ℳ ) both processes are averaged. The standard spacetime is sequences of k -component states with lower energy in dependence on γ -processing.

The doubly-periodic paired vacuum state (1) is a quasi-stationary state which encounters phase transitions by travelling in the neighborhood of zeros of zeta functions and partition functions. Therefore, the unique vacuum state of a real Lagrangian e.g. with ζ ( z = − 1 ) at the Casimir effect can be undercut by quasi-stationary continued fractions γ ∘ z . A physical realization would be smart technology by correlated one-periodic processing. This not exceptional process is a precursor for stable spacetime ℳ . A quadratic amplitude amplified Carnot cycle is proposed for changing correlated both topological entropy h t and temperature β − 1 . This replaces a rectangular entropy-temperature cycle h t , β − 1 by a circular-like cycle of an open system where β − 1 is not well defined. Whereas for closed systems temperature is well defined, open systems depend on temperature fluctuations. The dimensionless interaction state (1) should hold for all physical interactions. A forthcoming work aims to show that permutations (4) on complex plane relate the shifted ground state ( z − e i ) M i j ( z ) to a paired superfluid state comparable to a BCS-state with non-Hermitian Lagrangian of a renormalization group flow [22][25]. Invariant quadratic root finding on complex plane is used as a precondition for covariance which results in two different curvatures and two masses in each spacetime point [26].