TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/JOINT-LIFT-NATIVE-CLOCK-CONNECTION.txt
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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 |   <h,(epsilon_S/2,-sigma_j/2)>
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 |