TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-equivariant-active-search-20261024/THEOREMS.md Original file SHA-256: 666123fe4881001a342ec762bbe42595033b74a015282f774ef17d05903c5f49 Inclusive original lines: 5-104 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000005 | ## 1. Fixed source and definitions 000006 | 000007 | Retain the original integral E8 basis, its ordered sign cocycle, and J from 000008 | `marked-seam-monodromy-20261023` and its upstream sources. Put 000009 | T0=diag(1,-1,1,-1,1,-1,1,-1). Write j and t0 for the generator normalized 000010 | field lifts. The original cocycle verifies j^4=t0^2=(t0*j)^2=1 on all lattice 000011 | generators. Equality on these generators determines these automorphisms of 000012 | the central lattice extension. These are specific lifts of this subgroup; 000013 | no splitting of the original global G31 extension follows. 000014 | 000015 | Let V=Hom(E8,F2), with its fixed basis, and D_c the output sign character. 000016 | Let A=(J^-1)^T and K=(T0^-1)^T on V. In the fixed coordinates their columns, 000017 | encoded as binary masks with coordinate zero in the least significant bit, 000018 | are 000019 | 000020 | ``` 000021 | A = [1,253,114,9,185,32,96,129] 000022 | K = [1,253,250,9,16,32,65,129] 000023 | ``` 000024 | 000025 | They obey A^2=K^2=I and AK=KA. The field automorphisms generated here have 000026 | faithful normal forms (c,k,e)=D_c j^k t0^e, with c in V, k modulo4, e modulo2. 000027 | Their product and inverse are 000028 | 000029 | ``` 000030 | (c,k,e)(d,l,f) = (c+A^k K^e d, k+(-1)^e l mod4, e+f mod2), 000031 | (c,k,e)^-1 = (A^k K^e c, (-1)^(e+1) k mod4, e). 000032 | ``` 000033 | 000034 | The character action is reduced modulo2 in the exponent of A. Projection 000035 | to the eight distinct dihedral lattice isometries and faithfulness of the 000036 | characters prove uniqueness. The group has order2048. Its subgroup Gamma 000037 | over has order1024 and is the previously studied group. 000038 | 000039 | Let X=CP1 minus(1,i,-1,-i), with basepoint0 and positive radial meridians 000040 | gamma_i in cyclic order, product1. Every lifted meridian is M_i=D_ci j. 000041 | For a reflection tau_s(z)=i^s conjugate(z), choose the fibre marking 000042 | T=D_t j^s t0 and require T^2=1 and 000043 | 000044 | ``` 000045 | T M_i T^-1 = M_((s-i) mod4)^-1. 000046 | ``` 000047 | 000048 | For rotation z->iz choose R=D_r j and require R^4=1 and 000049 | R M_i R^-1=M_(i+1 mod4). With both markings at s=0 also require 000050 | T R T^-1=R^-1. These are extra declared structures. They do not derive a 000051 | physical space, state, preferred dimension or physical dynamics from P1. 000052 | 000053 | Morphisms in the first count fix the projected operators and are simultaneous 000054 | conjugations by D_g. The second count forgets the additional lifts and counts 000055 | the image in the established meridian moduli modulo all Gamma conjugations. 000056 | These counts must not be interchanged. 000057 | 000058 | ## 2. Complete equations and finite classification 000059 | 000060 | Put H=A^s K. Normalizing the group words gives precisely these equations: 000061 | 000062 | ``` 000063 | all cases: c0+A*c1+c2+A*c3 = 0; 000064 | reflection: (I+H)*t = 0; 000065 | (I+A)*t+H*ci+A*c_((s-i) mod4) = 0, for every i; 000066 | rotation: (I+A)*r+A*ci+c_(i+1 mod4) = 0, for every i; 000067 | dihedral: (I+A)*t+(K+A)*r = 0, in addition to s=0 reflection and rotation. 000068 | ``` 000069 | 000070 | R^4 imposes no extra character equation because I+A+A^2+A^3=0. 000071 | Necessity and sufficiency follow by the unique normal forms: each original 000072 | relator has identity lattice projection and is exactly the character displayed. 000073 | There is no replacement by a dimension or charge count. 000074 | 000075 | | Marking | Binary variables | Constraint rank | Solution dimension | Character gauge rank | Marked classes modulo D_g | 000076 | |---|---:|---:|---:|---:|---:| 000077 | | Reflection s=0 | 40 | 20 | 20 | 4 | 65536 | 000078 | | Reflection s=1 | 40 | 20 | 20 | 4 | 65536 | 000079 | | Rotation | 40 | 24 | 16 | 4 | 4096 | 000080 | | Joint rotation and reflection s=0 | 48 | 30 | 18 | 4 | 16384 | 000081 | 000082 | For a projected operator p the gauge translation is (I+p)*g. Simultaneous 000083 | translations have rank4 in every row of the table and lie in the solution 000084 | space. Their stabilizer has order16. Thus every orbit has16 elements and 000085 | the number of classes is 2^(solution dimension-4). This is a complete finite 000086 | classification with normal forms obtained by reducing a solution by this 000087 | explicit translation subspace. 000088 | 000089 | On the image obtained by forgetting r,t, Gamma acts by uniform translations 000090 | (I+A)g on the four ci and the simultaneous involution A. The image has 000091 | dimensions16,16,12,10 in the table order. After translation reduction the 000092 | dimensions are12,12,8,6, and the A fixed dimensions are8,10,8,6. Burnside gives 000093 | 000094 | ``` 000095 | reflection s=0: (2^12+2^8)/2 = 2176 underlying Gamma classes; 000096 | reflection s=1: (2^12+2^10)/2 = 2560 underlying Gamma classes; 000097 | rotation: (2^8+2^8)/2 = 256; 000098 | joint: (2^6+2^6)/2 = 64. 000099 | ``` 000100 | 000101 | The full projected image and every Gamma orbit were also enumerated separately. 000102 | This verifies the fixed space calculation without using its Burnside formula. 000103 | These are images in the existing526336 classes; they are not counts of all 000104 | possible outer isometries, reflections or physical realizations.