TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/direct/THETA-E8-LATTICE-COVARIANCE.txt Original file SHA-256: 81132e7d111ea83bb5ac7e360839083433fc9efb3c2a1e74dab0e5ce1866424e Inclusive original lines: 1-200 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. 000001 | THETA FRAMES RECOVER THE ABSTRACT E8 LATTICE CLASS, WHILE THE ACTUAL SPIN QUARTER LIFT EXCHANGES TWO LATTICES 000002 | ============================================================================================================ 000003 | 27 October 2026 000004 | 000005 | 000006 | VERDICT 000007 | ======= 000008 | 000009 | The q norm completed second order genus two theta module contains more 000010 | integral information than a bare four dimensional complex vector space. In 000011 | a marked theta frame its sixty stabilizer rays admit a support weighted 000012 | Gaussian root normalization. 000013 | The resulting 240 vectors span an even unimodular real lattice of rank eight, 000014 | hence E8. A new exact construction, which does not supply E8 roots as input, 000015 | identifies its shell after the fact with the literal standard shell 000016 | 000017 | 112 D8 roots + 128 half roots. 000018 | 000019 | This reconstruction is canonical relative to a marked level two theta 000020 | structure, its Schrodinger polarization, a unit in the one dimensional q 000021 | norm factor, and the marked central Gaussian unit i. Changing the theta 000022 | frame preserves the abstract Euclidean lattice and its complex homothety 000023 | class. Changing the norm-line unit rotates the whole embedded lattice and 000024 | therefore preserves only that homothety class. The construction does not 000025 | select one embedded Gaussian lattice that is invariant under the actual 000026 | source quarter lift alpha. Theta 000027 | coordinate changes are projective gauge changes and may be phase corrected 000028 | into G31. The geometric alpha is fixed by alpha^2=iI and lies in the other 000029 | Clifford coset zeta8 G31. It necessarily exchanges 000030 | 000031 | Lambda <-> zeta8 Lambda. (0.1) 000032 | 000033 | Its square iI preserves each lattice. Rephasing alpha into G31 would change 000034 | alpha^2 from iI to I and would destroy the source spin relation. The exact 000035 | integral object carried by all accepted source marks is therefore a two 000036 | object E8 lattice local system, not one alpha invariant lattice. 000037 | 000038 | This is still a positive origin result. The auxiliary theta geometry 000039 | recovers the abstract integral E8 class without a physical P1 to C3 bulk 000040 | Dirac identification. What it does not recover is a single fixed lattice 000041 | VOA on which alpha acts as an automorphism. 000042 | 000043 | 000044 | 1. BASIS DEPENDENT FORMULA, BASIS INDEPENDENT ROOT CLASS 000045 | ======================================================= 000046 | 000047 | Separate the bare theta space from the completed source module: 000048 | 000049 | H_theta=H^0(Jac(C),O(2 Theta)), 000050 | W_q=H_theta tensor N_q^(-1). (1.0) 000051 | 000052 | The ordinary action on H_theta is neutral under the hyperelliptic spin 000053 | square beta. The selected inverse q norm line supplies beta=+i. Thus the 000054 | source relation alpha^2=i belongs to W_q, not to bare H_theta. 000055 | 000056 | Choose a level two theta structure and a unit nu in N_q^(-1). They identify 000057 | the Schrodinger frame of W_q 000058 | 000059 | with basis vectors e_x tensor nu indexed by x in F2^2. Put 000060 | 000061 | pi=1+i, zeta8=pi/sqrt(2). 000062 | 000063 | For an affine d plane A in F2^2, d=0,1,2, and the allowed stabilizer phase 000064 | function phi:A->mu4, define 000065 | 000066 | r(A,phi) 000067 | =2^(-1/2) pi^(2-d) 000068 | sum_(x in A) phi(x)(e_x tensor nu). (1.1) 000069 | 000070 | The exponent in (1.1) is forced only after imposing the Gaussian integral 000071 | recipe. A d plane has 2^d nonzero coordinates. Equal root norm two requires 000072 | the squared numerator norm to be 2^(2-d) before the common factor 000073 | 2^(-1/2). If that numerator is required to be a primitive Gaussian integer, 000074 | then, up to the already present central units mu4, it is pi^(2-d). Thus the 000075 | three weights pi^2, pi and 1 follow from support size, norm two and primitive 000076 | Gaussian integrality. Equal norm, support and the central i alone would 000077 | still permit arbitrary global phases on the rays. The exact covariance 000078 | below validates this integral recipe; it does not prove uniqueness among all 000079 | possible integral structures on the complex theta module. 000080 | 000081 | The ring Z[i] is not imported from a native E8 basis. It is the smallest 000082 | integer coefficient ring containing the already marked central mu4. The 000083 | additional content of the recipe is the primitive, rather than a divisible 000084 | or arbitrarily phased, integral normalization in that ring. 000085 | 000086 | The allowed phase functions are exactly the simultaneous eigenvectors of a 000087 | maximal commuting Pauli subgroup. Projectively they give all pure two qubit 000088 | stabilizer states. The support census is 000089 | 000090 | d=0: 4 rays, 000091 | d=1: 24 rays, 000092 | d=2: 32 rays. (1.2) 000093 | 000094 | For every vector in (1.1), 000095 | 000096 | ||r(A,phi)||^2 000097 | =2^(-1) |pi|^(4-2d) |A| 000098 | =2^(-1) 2^(2-d) 2^d 000099 | =2. (1.3) 000100 | 000101 | Multiplication by mu4 supplies four vectors on each ray. Thus 000102 | 000103 | R_theta={u r(A,phi):u in mu4} 000104 | 000105 | has 4(4+24+32)=240 norm two vectors. Define 000106 | 000107 | Lambda_theta=Span_Z R_theta. (1.4) 000108 | 000109 | The exact integral root check starts with (1.1), not with a native E8 shell. 000110 | In unscaled Gaussian integer coordinates z it finds 000111 | 000112 | roots from d=0: 16, 000113 | roots from d=1: 96, 000114 | roots from d=2: 128. (1.5) 000115 | 000116 | The physical real vector is z/sqrt(2). A Hermite normal form basis B of the 000117 | integer z span has 000118 | 000119 | |det B|=16, 000120 | G=B^T B/2 integral and even, 000121 | det G=1. (1.6) 000122 | 000123 | Thus 000124 | 000125 | Lambda_theta is isometric to E8. (1.7) 000126 | 000127 | There is also a literal shell comparison. On each Gaussian coordinate put 000128 | 000129 | a+ib |-> (a+b,a-b). (1.8) 000130 | 000131 | In doubled real coordinates (1.8) sends the generated shell to precisely 000132 | the standard 112 D8 integer roots and 128 E8 half roots. This comparison is 000133 | performed only after Lambda_theta has been generated. It is therefore a 000134 | certificate of the construction, not an E8 input hidden in the formula. 000135 | 000136 | This construction uses the support weights in (1.1). Taking arbitrary unit 000137 | representatives of the sixty rays would lose the Gaussian integral 000138 | relations. The factors pi^(2-d) are therefore structural, not a cosmetic 000139 | normalization. 000140 | 000141 | 000142 | 2. WHAT A THETA BASIS CHANGE ACTUALLY DOES 000143 | ========================================== 000144 | 000145 | The finite theta group is 000146 | 000147 | 1 -> mu4 -> N -> A[2] -> 1, 000148 | 000149 | and its Schrodinger representation is unique. Changing the theta structure 000150 | acts projectively through 000151 | 000152 | Sp(4,2) 000153 | 000154 | and linearly through the two qubit Clifford matrix group C2. In the exact 000155 | native chart, 000156 | 000157 | C2=G31 disjoint union zeta8 G31, (2.1) 000158 | G31/mu4=C2/mu8, 000159 | 000160 | and G31 is precisely the matrix group preserving the Gaussian E8 root shell. 000161 | The standard generators may be represented by 000162 | 000163 | X_1,Z_1,X_2,Z_2,S_1,S_2,CNOT, 000164 | Htilde_1=zeta8^(-1)H_1, 000165 | Htilde_2=zeta8^(-1)H_2, (2.2) 000166 | 000167 | up to the central mu4. The opposite sign zeta8 H_j gives the same G31 000168 | element up to i. 000169 | 000170 | Let chi(g) in F2 be the Hadamard, equivalently Q(zeta8)/Q(i) Galois, parity 000171 | of a Clifford frame change. Then 000172 | 000173 | g_corr=zeta8^(-chi(g)) g belongs to G31. (2.3) 000174 | 000175 | When two odd frame changes compose, the two possible evaluations of (2.3) 000176 | differ by zeta8^(-2)=-i, which is already in the central mu4. Hence (2.3) 000177 | is the correct projective metaplectic covariance rule. It proves 000178 | 000179 | g_corr Lambda_theta=Lambda_(g theta). (2.4) 000180 | 000181 | Using the uncorrected representative gives instead 000182 | 000183 | g Lambda_theta=zeta8^(chi(g)) Lambda_(g theta). (2.5) 000184 | 000185 | Thus a marked theta structure and a chosen unit nu determine an embedded 000186 | Lambda_theta. Replacing nu by lambda nu rotates it to lambda Lambda_theta. 000187 | The theta structure torsor therefore determines the isometry class of E8 and 000188 | the complex homothety class of the embedding without a preferred basis or 000189 | norm-line unit. A principally polarized abelian surface without a theta 000190 | structure does not canonically label the four Schrodinger coordinates or the 000191 | affine supports in (1.1). What descends without that frame is the isometry 000192 | or homothety class, not an embedded basis. Projective data also do not 000193 | distinguish the two representatives Lambda and zeta8 Lambda. 000194 | 000195 | No physical symmetry has been used in this paragraph. Equations (2.3) and 000196 | (2.4) are coordinate transition rules between theta frames. 000197 | 000198 | 000199 | 3. THE ACTUAL SOURCE ALPHA IS NOT A COORDINATE GAUGE 000200 | ====================================================