TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-joint-source-20261023/JOINT-CLOCK.txt Original file SHA-256: 02a15050dc0d8f7ad69ad1d867318d7a9abfa4bee274e14f1f06a12cf16c9628 Inclusive original lines: 1-190 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 | MATH.SEAM.ESERIES.JOINTCLOCK.01 000002 | 23 October 2026 000003 | Exact joint action of the marked cover clock and the native full E8 lattice fields 000004 | 000005 | Verdict 000006 | 000007 | PASS in the stated mathematical scope. There is a canonical associated 000008 | lattice-VOA local system carrying the peripheral all-J monodromy, and the 000009 | forced C12 cover carries an exact marked order-twelve compiler action on that 000010 | same object. The peripheral J and compiler J are not identified as cover 000011 | automorphisms. They have the same fixed fibre lift Jhat but different base 000012 | components. 000013 | 000014 | [OMITTED: non-scientific conversation or execution metadata] 000015 | [OMITTED: non-scientific conversation or execution metadata] 000016 | [OMITTED: non-scientific conversation or execution metadata] 000017 | [OMITTED: non-scientific conversation or execution metadata] 000018 | [OMITTED: non-scientific conversation or execution metadata] 000019 | 000020 | 1. Fixed object and conventions 000021 | 000022 | Let 000023 | 000024 | B° = P1 minus {1,i,-1,-i,infinity}, 000025 | C_n : u_n^n=x^4-1, 000026 | Delta_n(x,u_n)=(x,zeta_n u_n), 000027 | Q_n(x,u_n)=(i x,u_n). 000028 | 000029 | The finite meridians are positively ordered. The original peripheral C4 000030 | cover is C4, and the forced common cover is C12. On C12 the marked compiler 000031 | cover clock is 000032 | 000033 | R12=Q12 Delta12^4, 000034 | R12^4=Delta12^4, 000035 | R12^9=Q12, 000036 | R12^12=1. 000037 | 000038 | Retain the original even E8 lattice L, the literal ordered cocycle epsilon, 000039 | and its lattice VOA 000040 | 000041 | V_L = M(1) tensor C_epsilon[L]. 000042 | 000043 | For an isometry g of L, S_g denotes the existing generator-normalized lift. 000044 | For a character a in A=Hom(L,F2), put 000045 | 000046 | D_a E_lambda=(-1)^a(lambda) E_lambda. 000047 | 000048 | Every lift over g is S_g D_a. On a full lattice field it acts by 000049 | 000050 | S_g D_a [h1(-n1)...hk(-nk) E_lambda] 000051 | =(g h1)(-n1)...(g hk)(-nk) 000052 | eta_g(lambda)(-1)^a(lambda) E_(g lambda), 000053 | 000054 | where eta_g is fixed by the unchanged epsilon and ordered basis. This is the 000055 | existing full lattice-field action, including charged sectors and every 000056 | oscillator descendant. No new carrier or product is introduced. 000057 | 000058 | The lattice operators are the existing commuting matrices 000059 | 000060 | J^4=1, sigma^3=1, c=J sigma, 000061 | c^4=sigma, c^9=J, c^12=1. 000062 | 000063 | Fix Jhat=S_J, the generator-normalized lift already used by the marked seam 000064 | study. 000065 | 000066 | 2. Canonical associated local system 000067 | 000068 | Use C12 as a principal refinement of C4 and let its deck generator act on the 000069 | fibre through Jhat. Define 000070 | 000071 | Vper = (C12 times V_L)/equivalence, 000072 | 000073 | with 000074 | 000075 | (Delta12 p,v) equivalent to (p,Jhat v). (2.1) 000076 | 000077 | Equivalently, the diagonal action is 000078 | 000079 | k.(p,v)=(Delta12^k p,Jhat^(-k)v). 000080 | 000081 | Because Jhat^4=1, this representation of the C12 deck group factors through 000082 | C4 and has kernel . Therefore (2.1) is canonically the same local 000083 | system as 000084 | 000085 | (C4 times V_L)/((Delta4 p,v) equivalent to (p,Jhat v)). 000086 | 000087 | A positive finite meridian lifts to an endpoint Delta12 p, so its parallel 000088 | transport on the fibre is exactly Jhat. This realizes the existing 000089 | peripheral all-J monodromy without turning it into a base rotation. 000090 | 000091 | The three roles are now explicit: 000092 | 000093 | [OMITTED: non-scientific conversation or execution metadata] 000094 | [OMITTED: non-scientific conversation or execution metadata] 000095 | [OMITTED: non-scientific conversation or execution metadata] 000096 | 000097 | Deck and base commute. A fibre automorphism descends through (2.1) precisely 000098 | when it commutes with Jhat. 000099 | 000100 | 3. Complete cyclic field-lift classification 000101 | 000102 | Write every candidate over c as 000103 | 000104 | chat_a=S_c D_a, a in A. 000105 | 000106 | Bits below refer, least significant bit first, to the dual of the predecessor's 000107 | ordered integral E8 basis. In doubled standard coordinates its column matrix 000108 | is 000109 | 000110 | B = [ 1 2 -2 0 0 0 0 0 ] 000111 | [-1 2 2 -2 0 0 0 0 ] 000112 | [-1 0 0 2 -2 0 0 0 ] 000113 | [-1 0 0 0 2 -2 0 0 ] 000114 | [-1 0 0 0 0 2 -2 0 ] 000115 | [-1 0 0 0 0 0 2 -2 ] 000116 | [-1 0 0 0 0 0 0 2 ] 000117 | [ 1 0 0 0 0 0 0 0 ]. 000118 | 000119 | The ninth-power equation is 000120 | 000121 | chat_a^9=Jhat 000122 | iff N9(a)=d, (3.1) 000123 | 000124 | where N9 is the exact character norm induced by the nine successive c actions. 000125 | In the ordered dual basis its eight columns, encoded as byte masks, and its 000126 | affine defect are 000127 | 000128 | columns(N9)=(1,2,93,128,119,1,215,128), 000129 | d=169, 000130 | rank_F2(N9)=6. 000131 | 000132 | Hence (3.1) has exactly 2^(8-6)=4 solutions. The complete list is 000133 | 000134 | a mask a as an F2 tuple b mask b as an F2 tuple 000135 | 31 (1,1,1,1,1,0,0,0) 63 (1,1,1,1,1,1,0,0) 000136 | 62 (0,1,1,1,1,1,0,0) 30 (0,1,1,1,1,0,0,0) 000137 | 151 (1,1,1,0,1,0,0,1) 183 (1,1,1,0,1,1,0,1) 000138 | 182 (0,1,1,0,1,1,0,1) 150 (0,1,1,0,1,0,0,1). 000139 | 000140 | Here b is defined by the exact fourth power 000141 | 000142 | sigmahat_a=chat_a^4=S_sigma D_b. (3.2) 000143 | 000144 | For every row the following identities hold on all lattice fields: 000145 | 000146 | chat_a^12=1, 000147 | sigmahat_a^3=1, 000148 | [Jhat,sigmahat_a]=1, 000149 | chat_a=Jhat sigmahat_a, 000150 | chat_a^9=Jhat, 000151 | chat_a^4=sigmahat_a. (3.3) 000152 | 000153 | Equality was reduced to the eight lattice generators only after retaining the 000154 | unchanged epsilon and the common lattice action. The oscillator action then 000155 | extends (3.3) to the complete VOA. The certificate also evaluates the fourth, 000156 | ninth and twelfth powers directly on all240 roots for every representative. 000157 | 000158 | The character gauges preserving the fixed Jhat are 000159 | 000160 | A^J={0,1,32,33,86,87,118,119,136,137,168,169,222,223,254,255}. 000161 | 000162 | Conjugation by D_t sends 000163 | 000164 | a -> a+t+c^*t. 000165 | 000166 | [OMITTED: non-scientific conversation or execution metadata] 000167 | 000168 | A^c={0,1,254,255}. 000169 | 000170 | The four solutions have one orbit of size16/4=4. Therefore the fixed-Jhat 000171 | cyclic compiler lift is unique up to character gauge. A convenient explicit 000172 | certificate representative is 000173 | 000174 | [OMITTED: non-scientific conversation or execution metadata] 000175 | [OMITTED: non-scientific conversation or execution metadata] 000176 | 000177 | The choice of mask31 in (3.4) is only a representative convention. The 000178 | single gauge orbit is the invariant result. 000179 | 000180 | There is no solution with b=0. Thus the two separately 000181 | generator-normalized sections S_J and S_sigma do not themselves extend to the 000182 | required normalized cyclic clock. The character D_63 in (3.4), or one of its 000183 | three gauge-equivalent alternatives, is forced by the original lattice 000184 | cocycle. It is not a new coupling. 000185 | 000186 | 4. The exact joint action 000187 | 000188 | For any of the four representatives define the marked action 000189 | 000190 | [OMITTED: non-scientific conversation or execution metadata]