TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/EXISTING-C3-SOURCE-DICTIONARY.txt Original file SHA-256: 9653148d8235dff822254d2b5be96da73f17c374741e1d6275d0e3e649826817 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 | EXISTING C3 SOURCE DICTIONARY 000002 | ============================= 000003 | 000004 | 26 October 2026 000005 | 000006 | Question 000007 | -------- 000008 | 000009 | Does an already existing order three operator identify, on the actual marked 000010 | source and without a new VOA or a fitted basis, all of the following data? 000011 | 000012 | 1. the Fermat cover deck on y^3=x^4-1; 000013 | 2. the native E8 compiler element sigma=c^4; 000014 | 3. the P2 puncture class with eigenvalues (1,omega,omega^2); 000015 | 4. the v117 matrices M0_EXACT,U_EXACT; 000016 | 5. the fixed original compiler Q. 000017 | 000018 | Verdict 000019 | ------- 000020 | 000021 | NO for the claimed direct identification. There are two exact positive 000022 | statements, but they meet only after forgetting decisive marks. 000023 | 000024 | * The native three slot family permutation and M0_EXACT are isomorphic as 000025 | bare C3 modules. Both have characteristic polynomial x^3-1. 000026 | 000027 | * The existing September dictionary gives a fully marked integral model of 000028 | the v117 pair and transports Q with it. 000029 | 000030 | The actual native permutation is not that marked integral model. The linear 000031 | system for an intertwiner which also fixes Q has only the zero solution. The 000032 | same is true if the v117 quarter operator is retained. Moreover the Fermat 000033 | deck does not even have the regular C3 character on its natural rank three 000034 | holomorphic carrier. The current C12 Phi12 bridge is rank four and has yet 000035 | another C3 character. 000036 | 000037 | There is also a direct defect in the displayed monodromy formula of the 000038 | current TFPT45 extract. Its fourth puncture monodromy is the identity, so it 000039 | does not have the stipulated local spectrum. That formula cannot supply the 000040 | missing source bridge. The exact v117 matrices do satisfy the four puncture 000041 | flatness relation, but their identification with the analytic ODE monodromy 000042 | remains the recorded numerical step and their identification with native E8 000043 | sigma remains absent. 000044 | 000045 | This is a bounded source dictionary result. It does not revoke P2, the exact 000046 | v117 W(A3) representation, the C12 cover theorem, or the native E8 compiler. 000047 | 000048 | 000049 | 1. Keep the three C3 carriers separate 000050 | -------------------------------------- 000051 | 000052 | Write omega=exp(2 pi i/3). 000053 | 000054 | 1.1 Fermat cover 000055 | 000056 | For C:y^3=x^4-1, v610 gives the holomorphic basis 000057 | 000058 | dx/y, dx/y^2, x dx/y^2. 000059 | 000060 | Under the deck y -> omega y its characters are 000061 | 000062 | omega^2, omega, omega. 000063 | 000064 | Thus the multiplicities on H^{1,0}(C) are 000065 | 000066 | 1 omega omega^2 000067 | Fermat H10 0 2 1. 000068 | 000069 | On full rational H1 they are (0,3,3), equivalently 000070 | 000071 | charpoly(D_omega on H1) = Phi_3(x)^3. 000072 | 000073 | Multiplication by a scalar cubic character only cyclically shifts the three 000074 | multiplicities: 000075 | 000076 | (0,2,1) -> (1,0,2) -> (2,1,0). 000077 | 000078 | It never gives (1,1,1). Hence neither a scalar phase nor duality turns the 000079 | natural Fermat deck into the P2 regular cusp class. 000080 | 000081 | The existing maximal E8/Fermat transport is different again. It is the 000082 | rank four Phi12 summand of the C12 action. Its order three matrix is 000083 | 000084 | D_C = [ -1 0 -1 0 ] 000085 | [ 0 -1 0 -1 ] 000086 | [ 1 0 0 0 ] 000087 | [ 0 1 0 0 ], 000088 | 000089 | with 000090 | 000091 | charpoly(D_C)=Phi_3(x)^2, 000092 | multiplicities=(0,2,2). 000093 | 000094 | The exact C12 theorem therefore supplies an unpolarized rank four clock 000095 | intertwiner. It does not supply the rank three P2 local system or its Q 000096 | operator. This agrees with COVERS.txt, which explicitly does not promote the 000097 | older Q/V dictionary. 000098 | 000099 | 1.2 Native E8 family cycle 000100 | 000101 | The native compiler uses sigma=c^4. In v634 it is the coordinate pair cycle 000102 | 000103 | (pair1,pair2,pair3,pair4) -> (pair2,pair3,pair1,pair4), 000104 | 000105 | and v689 identifies the same action on the quotient as the cycle of the three 000106 | family bits with the anchor fixed. On the three family slots its integral 000107 | matrix, in the convention of v417, is 000108 | 000109 | P_nat = [0 0 1] 000110 | [1 0 0] 000111 | [0 1 0]. 000112 | 000113 | It is the regular C3 representation: 000114 | 000115 | P_nat^3=I, 000116 | charpoly(P_nat)=x^3-1, 000117 | multiplicities=(1,1,1). 000118 | 000119 | This is an exact existing native operator. It is not the Fermat deck module 000120 | above. 000121 | 000122 | 1.3 P2 and v117 cusp class 000123 | 000124 | v117 uses 000125 | 000126 | M = [ 0 -(1+i)/2 (1-i)/2 ] 000127 | [ -(1+i)/2 -i/2 -1/2 ] 000128 | [ (1-i)/2 -1/2 i/2 ], 000129 | 000130 | U = diag(1,i,-i). 000131 | 000132 | It proves exactly 000133 | 000134 | M^3=I, 000135 | charpoly(M)=x^3-1, 000136 | product_{k=0}^3 U^k M U^{-k}=I, 000137 | is W(A3)=S4 of order 24. 000138 | 000139 | The pair is irreducible: its simultaneous complex commutant has dimension 000140 | one, equivalently the algebra generated by U and M is all M_3(C). 000141 | 000142 | Therefore P_nat and M agree as one generator after forgetting Q, U, the 000143 | puncture loops, and the source basis. This is only the statement that every 000144 | complex regular C3 representation is diagonalizable to 000145 | diag(1,omega,omega^2). It is not yet a source dictionary. 000146 | 000147 | 000148 | 2. The existing marked v117 dictionary 000149 | -------------------------------------- 000150 | 000151 | The September22 contract already transports the complete v117 pair. With 000152 | 000153 | Q = [3 1 0] Sigma = diag(1,-1,-1), 000154 | [3 2 0] 000155 | [3 2 1], 000156 | 000157 | T = [0 1 0], G=T Sigma, 000158 | [1 0 0] 000159 | [2 -2 1] 000160 | 000161 | and its exact S, it proves 000162 | 000163 | S(-G)S^{-1}=U, 000164 | S P_* S^{-1}=M, 000165 | 000166 | where 000167 | 000168 | P_* = [0 -2 1] 000169 | [1 0 0] 000170 | [2 1 0]. 000171 | 000172 | It also transports the unchanged Q: 000173 | 000174 | Q_c=SQS^{-1}, 000175 | 000176 | and consequently transports every noncommutative word in Q, Sigma, -G, -T, 000177 | and P_*. This is a genuine marked dictionary, not a spectral comparison. 000178 | 000179 | Its own stated boundary is decisive here: it does not identify the code/G31 000180 | three cycle with M. The new test below checks exactly that missing sentence. 000181 | 000182 | 000183 | 3. Smallest decisive Q equivariance test 000184 | ---------------------------------------- 000185 | 000186 | Let Z be an arbitrary complex 3 by 3 matrix. First forget Q and solve 000187 | 000188 | Z P_nat = P_* Z. (3.1) 000189 | 000190 | The coefficient matrix has rank 6, so the intertwiner space has dimension 3