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000001 | MATH.SEAM.ESERIES.03
000002 | 23 October 2026
000003 | 
000004 | Object class and markings
000005 | 
000006 | Let L be the existing even unimodular E8 lattice in standard R8, Phi its
000007 | 240 norm-two roots, and J the existing Gaussian lattice isometry
000008 | J(x0,x1,...) = (-x1,x0,...), J^2=-I. All slices below are actual root
000009 | subsystems in this fixed lattice chart. No quotient by a change of J is
000010 | taken. A chosen root or A2 subsystem is additional slice marking.
000011 | The standard D5+D3 coordinate split is retained only in the explicit flag.
000012 | No statement about a noncentral lattice-cocycle lift is made here.
000013 | 
000014 | 1. Root slices and E7
000015 | 
000016 | For alpha in Phi define Phi_alpha = Phi intersect alpha-perp. This is E7.
000017 | The root inner-product distribution is {-2:1,-1:56,0:126,1:56,2:1}.
000018 | At Lie-algebra level the associated root grading is
000019 |   e8 = 1[-2] + 56[-1] + (e7 + C)[0] + 56[1] + 1[2].
000020 | This is the standard E7+A1 branching, not a new physical identification.
000021 | The signs alpha and -alpha define the same E7 slice, giving120 slices.
000022 | 
000023 | Orthogonality and J^2=-I give (alpha,Jalpha)=0. Hence J interchanges
000024 | Phi_alpha and Phi_Jalpha. They differ since Jalpha belongs to the first
000025 | but not the second. Thus there are60 exchanged pairs and no J-fixed
000026 | root-normal E7 slice. Independently, no J-invariant E7 root subsystem
000027 | can exist: its real span has odd dimension7, while J^2=-I forces every
000028 | invariant real subspace to have even dimension.
000029 | 
000030 | Any orthogonal root pair in E8 is Weyl equivalent to e0+e1,e0-e1.
000031 | Indeed Weyl(E8) is transitive on roots, and the Weyl(E7) stabilizer of
000032 | the first root is transitive on the roots perpendicular to it. In this
000033 | chart the common perpendicular roots are exactly +/-ei +/-ej with
000034 | 2<=i<j<=7, giving D6 and60 roots. The normal plane contains only
000035 | the four roots +/-alpha,+/-Jalpha, hence A1+A1.
000036 | 
000037 | 2. Literal bridge to the existing60 Gaussian reflections
000038 | 
000039 | For norm-two alpha the real root reflection is
000040 |   s_alpha(v)=v-(v,alpha)alpha.
000041 | Since alpha and Jalpha are perpendicular,
000042 |   r_alpha=s_alpha s_Jalpha
000043 |          =I-alpha alpha^T-(Jalpha)(Jalpha)^T.
000044 | It is -I on the normal real plane and +I on its common perpendicular.
000045 | It commutes with J and is an order-two complex reflection on C4.
000046 | With h(v,alpha)=(v,alpha)+i(v,Jalpha), linear in the first argument,
000047 |   r_alpha(v)=v-2 h(v,alpha)/h(alpha,alpha) alpha.
000048 | This is term by term the original v634 S1 Gaussian reflection formula.
000049 | The identity of operators, rather than equality of group orders, identifies
000050 | the generated group with the existing G31 realization.
000051 | 
000052 | The following sets are in explicit bijection:
000053 |   Phi/<J>, the60 Gaussian root lines;
000054 |   the60 pairs {Phi_alpha,Phi_Jalpha};
000055 |   the60 common D6 root subsystems;
000056 |   the60 operators r_alpha.
000057 | Each plane has exactly4 roots, so there is no extra multiplicity. The
000058 | complex-linear reflections recover the plane as their -1 eigenspace;
000059 | the common D6 span recovers the plane by orthogonal complementation.
000060 | All240 root shells and all60 operators are checked in exact doubled
000061 | integer coordinates. Their faithful root permutations generate a group
000062 | of order46080, computed by SymPy stabilizer-chain group algorithms.