Image cameras are measuring devices of high-data transfer which require software for handling epipolar constraints. Introducing rectangular projection matrices P ( m , n ) in projective space from ℙ m to ℙ n the joint image for several cameras results from 2 - 4 reconstruction equations X = P ( 4 , 3 ) x . The paper treats the map ℙ 3 to ℙ 2 by base points as observable reality (OR). Fixed points (base points) of pencils of conics in projective space are introduced as independent of images as existing matter. EM is claimed to be invariant rational base points. Fractional substitutions γ in two-dimensional minors as binary forms should be closely related to matter. The alternating construct ( X Y ) = X 1 Y 2 − X 2 Y 1 is a quadratic form ( θ ¯ 1 θ 2 ) γ ^ ( θ 1 θ ¯ 2 ) . The paper describes homogeneous variables by four variables θ 1 , θ 2 and its conjugates θ ¯ 1 , θ ¯ 2 . If γ is a rational transformation the quadratic form is called rational invariant. This paper emphasizes γ = γ ( 3 ) as a quadratic map of a cubic polynomial. A joint image ξ in projective plane ℙ 2 are epipolar constraints for world points X , Y in projective space ℙ 3 . Fundamental, trifocal and quadrifocal tensor F ^ , T ^ , Q ^ yield constraints [2]

where ε ( n ) is the Levi-Civita symbol in n dimensions. Here anti-symmetrized three-component epipoles e t x = 0 are denoted by a bracket. This 9(8), 27 and 81-point vector v e c F ^ , v e c T ^ , v e c Q ^ of a vectorized matrix denotes an over constrained system. An n t h order determinant K ^ vanishes if a number of ( r + 1 ) 2 , ( K = K ^ t ) , 1 2 ( r + 1 ) ( r + 2 ) , ( K ≠ K ^ t ) of its r t h minors is zero [9][10]. This is used to propose a fast algorithm by finding three vanishing 2∙2 minors to describe 9(6) vanishing 2∙2 minors of F ^ , T ^ , Q ^ . Equations (1), (2) constitute a window by up to four linear independent camera images in F ^ , T ^ , Q ^ in ℙ 2 which should hold for every measuring device for detecting world points.

The fundamental matrix is part of Kummer and Weddle surfaces K ( X ) and W ( Y ) as follows [9]. For a matrix M n , m of n rows and m columns K ( X ) is a vanishing M 4 , 4 determinant

Equation (3) is equivalent to a Grassmannian G ( 4 ) of four world points. The aim is to formulate G ( 4 ) invariant with respect to a fractional substitution γ of a given parameter θ . Four world points X = ( 1 , − θ , θ 2 , 1 ) and Y = ( 1 , − θ , θ 2 , − θ 3 ) are rational on a twisted cubic curve parametrized by parameter θ . Then the γ -invariant condition Y t K ^ Y = 0 reads

which requires three parameters θ k , θ k + 1 and θ k + 2 . Homogenous binary invariants ( X Y ) = X 1 Y 2 − X 2 Y 1 and polynomials ϕ ( n ) ( z ) = ∑ i = 0 , ⋯ , n ( k i ) a n − i z i in (4) are added by a five-dimensional Grassmannian G ( 5 ) ≃ ε ( 5 ) as a bordered determinant ( l l )

where ε ( n ) is the Levi-Civita symbol of size n . World points X μ = ( X , λ ) = P μ ( 5 , 4 ) X on world planes λ ∑ i l i X i = 0 of homogeneous l i = ( l 0 , − 3 l 1 , 3 l 2 , l i ) , l i = l 1 3 − i l 2 i yield a γ -invariant quadric of derivatives ℘ ± ± ± of the hyperelliptic ℘ -function on universal covering u ± . For det K = 0 , the Grassmannian G ( 5 ) is rationalizable due to the identity [9]

The joint image Grassmannian is

Next the map ℙ 3 to ℙ 2 is discussed algebraically. World points on K ( X ) are on hyperelliptic curves ϕ ( 6 ) ( θ ) generating six base points of a pencil of quadrics in ℙ 3 . Image points of F ^ , T ^ , Q ^ belong to a hyperelliptic curve ϕ ( 4 ) ( θ ) of four base points in a pencil of conics in ℙ 2 reducible to an elliptic curve ϕ ( 3 ) ( θ ) . Pencils of quadrics obey symmetric world matrices and image matrices. The epipolar constraint x F ^ x + x F ^ t x = C 1 ( x ) + ℘ ( u , ω ) C 2 ( x ) = 0 is equivalent to a pencil of conics C 1 ( x ) , C 2 ( x ) in ℙ 2 . Base points x ( θ ) depend on the Weierstrass function θ = ℘ ( u , ω ) . For x = x ( θ ) epipolar geometry x F ^ x + x F ^ t x = ϕ ( 4 ) ( θ ) includes numerous invariant ways reducing a quartic ϕ ( 4 ) ( θ ) to Weierstrass form ϕ ( 3 ) ( ℘ ( u , ω ) ) . Cubic modular invariant congruences mod ϕ ( 3 ) ( θ ) yield similar rationalized world points X and Y : X = ( x , 1 ) = ( 1 , − θ , θ 2 , 1 ) ≃ Y . Binary invariance is adding points on elliptic curves where ϕ ( 3 ) ( ℘ ( u , ω ) ) = det ( C ^ 1 + ℘ ( u , ω ) C ^ 2 ) → ϕ ( 4 ) ( θ ) [5]. Mean values of world points X in images x in pencils of quadrics in ℙ 3 and ℙ 2 allow a fast algorithm in terms of singular r t h minors. Therefore, three vanishing 2∙2 minors in a fundamental matrix F ^ entering the 3-way T ^ and the 4-way Q ^ are sufficient to describe 9(6) vanishing 2∙2 minors in K ^ . This window to world points in ℙ 3 consists of the roots of ϕ ( 4 ) ( θ ) in the Grassmannian (5) and (6). This window is described by up to four linear independent camera images in F ^ , T ^ , Q ^ in ℙ 2 which should hold for every measuring device. The present paper uses four fixed points (base points) of a pencil of camera images in order to describe regions around zeros of the partition function or zeta functions.

For vanishing binary forms four permuted variables constitute a group for both elliptic and hyperelliptic universal covering. A hyperelliptic Riemann surface splits into a product of elliptic curves if a hyperbolic alternating form α ( Π i , Π j ) vanishes for a complex base Π i , Π j of hyperelliptic tori of period Π = M 2 , 4 where Π i is a column of Π. Hyperelliptic periods with α ( Π i , Π j ) = 0 yield products of elliptic curves. Singular 2∙2 minors are equivalent to binary invariants ( X Y ) . 16 invariant alternating forms ( X Y ) are hyperelliptic theta functions [11]-[15]. Singular 2∙2 minors of matrices are equivalent to binary invariants ( X Y ) . Further on, a hyperelliptic Riemann surface splits into a product of elliptic curves if hyperbolic alternating forms α ( Π i , Π j ) vanish for a base Π i , Π j of hyperelliptic tori of period Π = M 2 , 4 [15]. Determinant (6) can be written as a Plücker relation det K ^ = ∏ i j k l M i j M k l where 2∙2 determinants M i j are invariant in orthogonal substitution g i j = G i i ′ j j ′ g i ′ g j ′ [16] which is used to define anti-commutating matrices below [17]. Matrix G i i ′ j j ′ = G i l i ′ G l j j ′ of 0 , ± 1 entries is bilinear in the 3-way G i j k

Bilinear composition g i j in hyperelliptic theta functions g i = ϑ [ g h ] ( u ± , Π ) with four-component indices ( i j k l ) from 16 hyperelliptic characteristics [ g h ] yield a quadrilinear invariance in the size 4 matrix K . The paper links ( X Y ) to α ( Π i , Π j ) mediated by invariant Weierstrass relations on universal covering m ± u and m 0 u for matrices

Matrices m 0 and m ± are second and third roots of unity with D F T 2 = ( 1 1 1 − 1 ) , stride-2 permutation matrix π 2 4 of size 4 and η m = diag ( − 1 , 1 , 1 , 1 ) . The parameter θ parametrizes X and Y which are ℘ -functions and its derivatives. Theta functions ϑ [ g h ] ( u ± , Π ) remain unknown while iterating ℘ . This algorithm assumes that base points x ( ℘ ) acquire a quadratic minimum iterating m ± u . Then ℘ and the Schwarzian derivative F ( 3 ) , z are nearly constant whereas the twofold integral of ϑ [ g h ] ( u ± , Π ) over u is quadratic in u . Peculiar is self-similarity relations between four products of ϑ [ g h ] ( u ± , Π ) and universal covering m ± u and m 0 u . Self-similarity between universal covering m 0 u ± , m ± u ± → u ± and alternating hyperbolic forms ( X Y ) is realized by finding and shuffling all possible permutations which constitute a logarithmic Riemann surface ℝ L . The paper claims that 2(3) vanishing forms ( X Y ) or α ( Π i , Π j ) support rational points as a product of elliptic curves on surface K ( X ) .

For arbitrary values of det γ the form ( X Y ) is invariant if ( X Y ) = 0 . The paper uses a two-component form of homogenous coordinates

and defines each component θ 1 , 2 four-component as a real permutation α q of quartic roots x i . For a complex cubic field x i get complex which requires to formulate the permutation matrix α q on complex plane. Accordingly, α q can be regarded as a symbolic operator. The binary form expression (4) simplifies for a cubic field ϕ 3 ( θ ) = 0 . For a cubic field, the rational fractional transformation γ is a quadratic map. The resulting vanishing invariants ( θ q θ q ′ ) in (4) enable complex multiplication (CM) where variables θ and periods ω transform similarly. The present paper shows that the reduction processes ϕ ( 6 ) ( θ ) → ϕ ( 4 ) ( θ ) → ϕ ( 3 ) ( θ ) splits a hyperelliptic Riemann surface by matrices α q { x i x j x k x l } → ± ∞ ± i ∞ , e j , e k , e l which permute four quartic roots x i into cubic roots e 1 , 2 , 3 . Numerous invariant transformations exist for the reduction ϕ ( 4 ) ( θ ) → ϕ ( 3 ) ( θ ) . This paper uses a permutation method. Permutation matrix α q forms a group of its products of order G 32 ( α q ) extending the Galois group S 4 of ϕ ( 4 ) ( α q θ ) of order 4! by the matrix −1. Permutation is restricted to two cubic roots in

with complex angle φ 0 defined by Δ = g 2 3 sin 2 φ 0 and Δ = g 2 3 − 27 g 3 2 . In homogeneous variables x i → ( x 1 i x 2 i ) ∈ ℂ 4 ⊗ ℙ 1 roots

are exponents. Roots for fundamental unit ε and regulator R Δ = ln ε

yield

Permutations of x i induce discrete dynamics from fractional substitutions θ k + N = γ ( 3 ) ∘ ⋯ ∘ γ ( 3 ) ∘ θ k . Living in the Galois group S 3 the quadratic map γ is viewed as permutation of roots of a cubic polynomial ϕ ( 3 ) ( x i ≃ z ) = ∏ β = j k l ( x i x β ) . The Hermite-Tschirnhaus substitution in the Galois group S 3

is rational quadratic in z . The fractional substitution γ ( 3 ) creates a z ∧ w transvectant [18]. This is equivalent to a Weierstrass form ϕ ( 3 ) ( z ) → 4 z 3 − γ 2 z − γ 3 with θ → z = f ( ω ) where f ( ω ) is a modular form of weight 0 for invariants γ 2 , 3 . However, the x i − z dualism ϕ ( 3 ) ( x i ) = ∏ β = j k l ( x i x β ) ≃ ϕ ( 3 ) ( z ) yields for permutations ( i j k l )

with M i j ( z ) = z − z i z − z j . The paper claims that the ground state is a vacuum consisting of products of elliptic segments (11). Permutation (15) leads to

and

which are different on a Riemann surface ℝ L of the complex logarithm and require a higher-dimensional space ℂ 4 ⊗ ℙ 1 to be conjugate to a permutation α q . The shuffle algorithm works with the symbolic derivative

γ ( 3 ) ( w ) is a fractional map, a quadratic map as well the permutation

Accordingly, ϕ ( 4 ) ( α q θ ) is determined mainly by permutations α q on a hyperelliptic surface. The surprising result of this paper is that period-3 rotations or chaotic simplest cycles γ 2 = γ obey a linearized theory based on interval [ 0 , 1 ] . A scaling

predicts for γ = γ the existence of a modular invariant of weight zero as well complex multiplication where z ≃ u .

It is claimed that the integer permutation matrix α q obeys also complex solutions which induces algebraic orbits around transcendental zeros e.g. of zeta functions [19]. Iterations on hyperelliptic surfaces K ( X ) and W ( Y ) are suspected to be related to a spinor concept [11][12][20]. The approach consists in an algorithm which shuffles singular 2∙2 in size four matrices as hyperbolic alternating forms in a determinant of size four. The commutative group G 32 ( K ) of satellite points at ± u ± on K ( X ) is claimed to be related to the fermion group G 32 ( α q ) which is introduced as follows. The paper links binary forms ( X Y ) to alternating forms α ( Π i , Π j ) = det ( Π i , Π j ) by invariant homogeneous variables γ ∘ θ = γ ^ ( θ 1 θ 2 ) → ( c 1 c ¯ 2 ) , ( c ¯ 1 c 2 ) . Earlier spinor formulations by means of alternating forms ( X Y ) describe quaternion dynamics [16]. The concept widens a scalar binary invariant as matrix form. The eigenstates of the matrix γ ^ ( 3 ) are taken 2∙4 component ( c ¯ 1 c 2 ) and ( c 1 c ¯ 2 ) which is capable to couple to a Green’s function approach.

Four roots in ϕ ( 4 ) ( θ ) yield an eight-component homogeneous vector to describe an eight-point algorithm of v e c F ^ . A permutation of two roots α q x i yields two matrices c 1 , c 2 and its conjugates c ¯ 1 , c ¯ 2 discussed in [17].

where Grassmann variables c 1 , c 2 , c ¯ 1 , c ¯ 2 : c i 2 = 0 , c ¯ i 2 = 0 constitute four permutation matrices α q where c 1 , c 2 , c ¯ 1 , c ¯ 2 , α q ∈ M 4 , 4 . Roots x i → ∞ shifted to infinity require homogeneous variables for 16 alternating forms α ( Π i , Π j ) → α ( α q Π i , α q Π j ) ∈ ℂ 4 ⊗ ℙ 1 . Grassmann variable c ≃ z − e i ≃ a + describes an annihilation process a + in an empty state. Grassmann variable c ≃ d d ( z − e i ) ≃ a describes a creation process a in an occupied state. The permutation process α q x i ∈ M 4 , 4 yields a square ( z − e i ) 2 which is a product of 8 Grassmann variables on the r.h.s of (11) which leads to ( z − e i ) 2 → 0 . Grassmann variables c 1 , c 2 , c ¯ 1 , c ¯ 2 are defined by sparse matrices permuting two-variables as a rotation by ± π in interval [ 0 , 1 ] . Homogeneous roots are defined by rows ( c ¯ 1 c 2 ) , columns ( c 1 c ¯ 2 ) and matrices ( b 0 0 b ¯ ) in unit binary forms α ( b , b ¯ ) = b b ¯ − b ¯ b = 1 . The general permutation

seems to be related to the concept of a paired state. The polynomial of the fractional substitution γ

is defined by Grassmann variables as its determinant [3]. Homogeneous

have nilpotent components. Grassmann variable c linearizes both polynomial ϕ ( 3 ) ( c z ) → − g 2 c z − g 3 and substitution

An information bit can be set finding a zero determinant in Section 2.1 which splits into the number of represented possible linear independent objects N o = 2 n of an n -dimensional vector. Vectorizations in Section 2.1 yield N o = 2 3 4 linear independent computer visioned images of 3 2 , 3 3 , 3 4 elements in epipolar geometry F ^ , T ^ , Q ^ which is about the third root of N e d d = 2 2 8 . The Eddington number N e d d is accepted as the number of fermions in the universe. World line variations X k + 1 − X k which live in the Kronecker product of vectorizations v e c   δ K ^ ≃ v e c K ^ ⊗ v e c K ^ are 2⁸-component which yields N e d d linear independent objects. For n → ∞ a vanishing Grassmannian G ( n ) of EM should correspond to transcendental roots. OR is estimated by a vectorized Grassmannian G ( 5 ) of dimension ( d i m v e c K ^ + d i m l ^ ) ⋅ ( d i m v e c P i ( 5 , 4 ) ) 5 ⋅ ( d i m v e c P i ( 4 , 3 ) ) 4 = 2 20 ⋅ 5 6 ⋅ 3 4 . This would yield to 2 2 20 ⋅ 5 6 ⋅ 3 4 linear independent objects as compared to 2 3 4 objects in OR.

The paper links topological entropy of world lines to base point iterations in projective reconstruction. World point hits of a 16-component vectorized form v e c K ^ are approximated by zeros of Grassmannian G ( 4 ) or G ( 5 ) . The vectorized matrix v e c δ K ^ of world line variations X k + 1 − X k is 2 8 -component which yields a number of bits 2 2 8 comparable to the Eddington number. Zeros of a general Grassmannian G ( ∞ ) are assumed to be transcendental. This paper discusses algorithmically accessible zeros of G ( ∞ ) by → G ( 4 ) of a reduced partition function ζ ( z , G ( 4 ) ) .

The paper approximates a Grassmannian G ( n ) for arbitrary world point shifts v e c K ^ ⊗ ⋯ ⊗ v e c K ^ with n → ∞ and transcendent roots by n = 4 . Quartic roots correspond to epipolar geometry and the algorithmically accessible group S 4 . Zeros for G ( ∞ ) → G ( 4 ) in a reduced partition function ζ ( z , G ( 4 ) ) orbit four conjugates of transcendental roots. A transcendental function ζ ( z , G ( ∞ ) ) → ζ ( z ) e.g. Riemann zeta and xi functions

with transcendental roots ρ i is formally equivalent to a fermion partition function of particles with energies

which proves equivalence found in [19]. Computationally roots of ζ ( z ) are suspected by secants between zero values 0+ and 0-. The present algorithm claims that a fast-shuffling algorithm of rational binary invariants creates orbits θ k + N = γ ( 3 ) ∘ ⋯ ∘ γ ( 3 ) ∘ θ k ∈ S 3 , S 4 of high lap number N [20]. These orbits are connected with finite values in (4)-(6) which are claimed to be statistical energies E i . A partition function

encircles zeros which are base points by orbits quadrilinear on complex plane. This emphasizes the role of Newtonian root finding for a complex quadric q ( z , z 1 ) . The quantity G i is defined below by the condition that near a suspected zero an information bit can be set. Approach (22) remembers Yang-Lee zeros and phase transitions [21][22]. Orbits around base points (11) enable an algebraic definition of topological entropy

by cardinality of orbits C ( N , u , ω k ) on universal covering [23]-[26]

with n t h order Weierstrass sigma functions and i t h derivatives of the Weierstrass ℘ -function. The c , c ¯ algorithm consists in finding two finite entries in an otherwise zero 4∙4 matrix with c 2 = 0 , c ¯ 2 = 0 which lefts open absolute values of iterated periods ω k and therefore, also of the Dedekind eta function η ( ω ) . Branches ℝ L contain a network of branching lines generated by γ ( 3 ) ∘ z . On definite points on complex plane ℂ hyperbolic forms α ( α q u , α q v ) exist with rational points on elliptic curves. Orthogonal substitutions g i j ∈ [ g h ] are self-similar to α ( α q Π i , α q Π ) [27]. The doubly-periodic motion is visualized by spheres in spheres on ℂ which is a self-similar precession of a spinning top which recovers astronomical motions. In this case, the partition function (22) gets polynomial ζ ( z , G ( 4 ) ) ≃ ϕ ( N ) ( z ) which corresponds to Yang-Lee zeros near impenetrable spheres [21]. Driven by permutations (15), the vacuum state

is a product of lines on elliptic curves with fluctuating zeros ( z − e i ) [ c , c ¯ ] . This approach concerns algorithmic orbits where transcendental zeros in (21) cannot be reached [20]. Valid periods ω k are found by Grassmann variables c , c ¯ for CM of periods ω of elliptic tori z ≃ ω . Self-similar universal covering m 0 u ± , m ± u ± → u ± and alternating hyperbolic forms ( X Y ) shuffles all possible permutations on ℝ L . From (11) one gets ϕ ( 3 ) ( θ ) ≃ ϕ ( 4 ) ( θ ) up to a quadratic pre-factor. Orthogonal substitutions g i j ∈ [ g h ] are bilinear in ϑ ( u ± ( i ) , Π ) . For q = k , k + 1 , k + 2 , k + 3 zeros in (22) transform as

Formal zeros in

are expected on products of elliptic tori for Grassmann variables (18) where

A Grassmann variable enters orthogonal substitutions g i j ∈ [ g h ] via the permutation matrix α q in distinction to [16][27]. Shifts by ± π / 3 in angle 1 3 ( φ 0 − ± π ) in (7)-(9) permutate hyperbolic forms α ( α q Π i , α q Π j ) ≃ α ( c Π i , c Π i ) . Permutations as ± π shifts can be mapped to rotations in interval [ 0 , 1 ] . Accordingly, a high statistical weight of simplest cycles of changes ± π 3 in interval [ 0 , 1 ] relates to the concept of spin. Then periods ω get statistical products of bases Π i , Π j and of Grassmann variables c , c ¯ . The partition function (22) gets polynomial as a product of sigma functions σ ( u , ω ) = ∏ ρ ( 1 − ρ ) e ρ + 1 2 ρ 2 . Over lattice sites ω one gets for an orthogonal substitution DFT 4 u i in ρ i = u i / ω in

if ∑ i = 1 , 2 , 3 , 4 ( ρ i + 1 2 ρ i 2 ) = 1 . Theta and sigma functions

of periods η i = ζ ( ω i / 2 , ω ) of Weierstrass zeta functions for discriminant Δ suggest an exponential

This algorithm determines periods ω via Grassmann variables c , c ¯ . Orbits in (22) and (25) yield a product permutations in

where z k + N = γ ( 3 ) ∘ ⋯ ∘ γ ( 3 ) ∘ z k . However, Dedekind eta functions η ( ω ) left undetermined. Certain powers η N ( ω ) ( N = 2 , 3 , 8 , 24 ) are defined due to symmetries, e.g. η 2 ( ω ) ≃ ω in CM. The statistical weight in (22) is estimated by the Grassmannian G ( 5 ) of maximal elements. Statistically a differential structure should emerge. First the logarithmic derivative of (11) yields [18].

independent on permutation α q . An introduced RG beta function

are compatible if z k + N = ln z k + 1 + ln z k + 2 . Conformal invariance demands three γ ( 3 ) -steps k , k + 1 , k + 2 or z k + 3 ∈ { z k → z k + 1 → z k + 2 } . Then simplest cycles are compatible with (32) for invariant half-order differential combinations [29][20]

which allow to define a metric line element d s 2 = y − 2 d z d z ¯ = − d z ln ( z − z ¯ ) d z ¯ ln ( z − z ¯ ) for complex conjugates w = z ¯ . A constant β ( z ) = g w 2 function is treated as a coupling constant g w for universal covering d u ≃ g w − 1 d z . A differential form of Grassmann variables c , c ¯ is c ¯ = z − e i and c = d d ( z − e i ) . Variable (11) permuted by x i → α q x i yields c , c ¯ ∈ G 32 ( α q ) which would correspond to ( z − e i ) 2 → 0 . Grassmann variables c , c ¯ and the γ -eigenstates ( c 1 c ¯ 2 ) , ( c ¯ 1 c 2 ) are understood as sparse M 4 , 4 -matrices permuting two x i components. Accordingly, permutations c , c ¯ are one-periodic base point variations in epipolar geometry x F ^ x on the border to elliptic curves. Projective space of fixed points of conics consists of segments of elliptic curves. Transcendent bifurcation lines z of base points are algorithmically accessible by cubic number fields. A regulator derivative N z = c ¯ c = d d R Δ confirms an optimal algorithm for a number operator with eigenvalues 0 and 1. The vacuum state (25) gets an eigenstate of partition function (20) written as

Generally, the regulator index R Δ is a quotient R Δ → R Δ / det ln ε r with circulant units ε r [30][31]. The fundamental unit of a cubic number field ε r is estimated from the class number formula [32]

of character χ i ≃ w ≃ w 0 , w 1 , w 2 of representations of the Galois group. Via e i → α q e i → c e i the logarithm R Δ = ln ε is given by a linear c -term in ln ( z − c e i ) . Unit forms α ( b , b ¯ ) in (17) yield a zero contribution in (36). The pulsating vacuum state (24) depends on whether the hyperelliptic surface of base points splits into a regulator index R Δ det ln ε r reducible to R Δ = ln ε ≃ ln ( z − e i ) or not. Permutation (17) suggests the importance of three-step simplest cycles where ϕ v a c ( z ) = ∏ ϕ ( 3 ) ( z ) gets a product of modular invariants. As a product of vertices c ¯ 1 b c 2 , c 1 b ¯ c ¯ 2 the base point vacuum ϕ v a c ( z ) is stable and noncritical for constant beta function (32). From (25), one gets