TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/JOINT-LIFT-NATIVE-CLOCK-CONNECTION.txt Original file SHA-256: edd2cdaea4c4debd337b4850d48da95de68e9367d780b9e3f0a96755588a45c8 Inclusive original lines: 37-145 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. 000037 | 2. Reuse the existing global carrier and connection 000038 | 000039 | The September22 contract source-joint-spin-family-lift proves the full 000040 | natural family for a GIVEN unitary rank5 E, L=det E: 000041 | 000042 | K=(Spin(10) x SU4)/diagonal Z4, 000043 | iota_m(g)=[z^-1 Lambda-even(g), diag(z^(2m_j+1))], 000044 | z^2=det g, sum m_j=-2, 000045 | W_m=Lambda-even(E) tensor direct-sum_j L^m_j. 000046 | 000047 | Changing z to -z multiplies both factors by -1, which is in the quotient 000048 | kernel. Thus no global square root of L is selected or needed. 000049 | These are all natural homomorphisms with the fixed ordinary U5 Spin 000050 | projection, up to family conjugacy. This is a REUSED result, not new. 000051 | 000052 | The existing source-boundary-lift-selection contract further proves that 000053 | a GIVEN unitary connection A_E induces, on W_m, 000054 | 000055 | A_Wm=rho_Lambda-even(A_E) tensor I4 + I16 tensor M_m a, 000056 | M_m=diag(m_j), a=tr A_E. 000057 | 000058 | Spin and family half-determinant connections cancel in this expression. 000059 | That formula neither chooses A_E nor forbids an independently supplied 000060 | family connection. The latter would have its own compatibility condition. 000061 | 000062 | 3. Keep the correct chiral weights in the actual E8 roots 000063 | 000064 | Use the original 112 roots +-e_i+-e_j and128 half roots with an even 000065 | TOTAL number of minus signs. Let sigma_j be (+++),(+--),(-+-),(--+). 000066 | For S subset{1,...,5} with |S| even, set epsilon_i=+1 on S and -1 off S. 000067 | The Spin weight epsilon/2 has ODD minus parity, since5-|S| is odd. 000068 | The paired family4 therefore has weights -sigma_j/2, not +sigma_j/2. 000069 | 000070 | For a primitive supplied toral clutch with sum k_i=1, put 000071 | 000072 | beta_j=m_j+1/2, sum beta_j=0, 000073 | v=(beta0+beta1,beta0+beta2,beta0+beta3), 000074 | h(k,m)=(k,-v). 000075 | 000076 | Then sigma_j dot v/2=beta_j and the actual E8 half root satisfies 000077 | 000078 | 000079 | =sum_(i in S) k_i -1/2+beta_j 000080 | =sum_(i in S) k_i+m_j. 000081 | 000082 | These are literally the weights of W_m, not just matching multiplicities. 000083 | The mixed(10,6) weights are +-k_i+m_j+m_l+1. The checker independently 000084 | compares all248 root/Cartan weights against all five actual branches. 000085 | Replacing only -v by +v is not the same signed W_m identification. 000086 | The adjoint character alone cannot determine this orientation. 000087 | 000088 | 4. The explicit ternary clock is color center times a family2+2 matrix 000089 | 000090 | Write omega=exp(2pi i/3), g_col=diag(omega I3,I2) in SU5. On W_m, 000091 | 000092 | sigma0=Lambda-even(g_col) tensor R, 000093 | R=diag(omega,omega^2,omega^2,omega). 000094 | 000095 | This identity follows on EVERY individual half root from Section3. 000096 | For S, let n_col be its number of color indices. Pairing with b0 gives 000097 | =n_col+f_j mod3, 000098 | (f0,f1,f2,f3)=(1,2,2,1). 000099 | The Spin path sign is compensated by the common(-1,-I4) kernel. 000100 | Because det g_col=1, the L^m factors acquire no additional phase here. 000101 | The same expression is a group element of K on all its E8 branches. 000102 | 000103 | The primitive kappa is central in K: it fixes all D5 and A3 adjoint 000104 | roots, has square the D8 spinor sign and exact order4 on the E8 glue. 000105 | In the signed W_m convention of Section3 its phase is -i. Reversing all 000106 | conventions also reverses that orientation. No orientation is inferred 000107 | from the unordered adjoint spectrum alone. 000108 | 000109 | 5. Globality and connection preservation, not just conjugacy 000110 | 000111 | Every transition of the given split carrier lies in U3 x U2. Its image 000112 | under iota_m commutes with sigma0 because: 000113 | 000114 | g_col is central in U3 x U2; 000115 | R commutes with every diagonal determinant factor diag(z^(2m_j+1)). 000116 | 000117 | Kappa is central in K, hence also commutes with every transition. 000118 | Consequently kappa, sigma0 and c0 define globally well-defined vertical 000119 | bundle automorphisms on the associated full E8 field bundle. On overlaps 000120 | their definitions agree. No square-root trivialization is hidden. 000121 | 000122 | For EVERY GIVEN split connection A_E=A_E3 direct-sum A_E2, its induced 000123 | A_Wm from Section2 commutes with these constant automorphisms. Thus 000124 | 000125 | nabla_Wm(c0 s)=c0 nabla_Wm(s). 000126 | 000127 | The same holds on every associated E8 branch because the Lie algebra 000128 | image iota_m(u3+u2) is centralized. Checking the actual root generators 000129 | e_i-e_j within the color and weak blocks proves this directly. 000130 | 000131 | [OMITTED: non-scientific conversation or execution metadata] 000132 | [OMITTED: non-scientific conversation or execution metadata] 000133 | [OMITTED: non-scientific conversation or execution metadata] 000134 | [OMITTED: non-scientific conversation or execution metadata] 000135 | [OMITTED: non-scientific conversation or execution metadata] 000136 | [OMITTED: non-scientific conversation or execution metadata] 000137 | [OMITTED: non-scientific conversation or execution metadata] 000138 | [OMITTED: non-scientific conversation or execution metadata] 000139 | [OMITTED: non-scientific conversation or execution metadata] 000140 | 000141 | For a source Dirac operator c_ext nabla +Phi with an external Clifford 000142 | factor, full operator commutation also requires [Phi,c0]=0 and a clock 000143 | invariant closed domain. A connection identity alone does not supply 000144 | these data, a CAR representation, a state or physical time. If the 000145 | original clock acts on the base as well, its base lift must be compared;