TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/hodge_complement_20261028/E8-CLIFFORD-RESPONSE.txt
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000001 | ACTUAL E8 CURRENT RESPONSE, FULL CLIFFORD CHANNEL AND RETAINED CARRY
000002 | 28 October 2026
000003 | 
000004 | The actual source is loaded through
000005 | tfpt_explorer.current_block_geometry._actual_current_source, which loads
000006 | verification/v498_celestial_wp5b_singular_vector.py. No replacement E8
000007 | bracket or fitted coupling table is used.
000008 | 
000009 | Keep the existing first-five/last-three D5+D3 frame. Within E8, the 60
000010 | mixed integer roots form g2=(10,6). The 128 half-roots split into the
000011 | two64-dimensional modules g1=(16,4), g3=(16bar,4bar). The zero mode
000012 | of a root current acts on the weight-one root states by the actual
000013 | Chevalley bracket. Denote its restriction g1->g3 by B_beta.
000014 | 
000015 | For each of the60 mixed roots the matrix B_beta has16 nonzero columns,
000016 | each coefficient being exactly+1 or-1. Every input and output is incident
000017 | to15 of these actions. Consequently
000018 | 
000019 |   sum_beta B_beta^dagger B_beta = 15 I64,
000020 |   sum_beta B_beta B_beta^dagger = 15 I64.
000021 | 
000022 | Thus the mathematically specified inclusive current response
000023 | 
000024 |   E(rho) = (1/15) sum_beta B_beta rho B_beta^dagger
000025 | 
000026 | is completely positive, trace preserving and unital. The invariant source
000027 | metric fixes the equal orthonormal weights within this irreducible sector.
000028 | This fixes the mathematical response once that complete current sector is
000029 | specified; it does not prove P1 selects this sector as an event source.
000030 | 
000031 | The checker compares the actual current action to the exterior CAR action
000032 | on all112 D8 roots and all128 half-root states. Its3584 signed equations
000033 | have rank231 on240 weight/root phase variables. The resulting diagonal
000034 | weight-space phases identify actual root actions, not just spectra. All
000035 | eight Cartan charges match because the source roots and Fock occupation
000036 | weights are explicitly matched. Commutators transport the Cartan action.
000037 | The independent review confirms the full Spin16 representation bridge.
000038 | 
000039 | This phase transport is a joint tensor chart. The raw Chevalley bitbasis
000040 | does not admit a product phase correction into independent carrier and
000041 | family factors. An untransported raw-coordinate family trace would not
000042 | be justified. The fixed D5+A3 representation determines the tensor factors;
000043 | the actual source data must be transported before taking a marginal.
000044 | 
000045 | In that transported chart the channel factors exactly:
000046 | 
000047 |   E = Q_carrier tensor Q_family,
000048 | 
000049 |   Q_carrier(rho)=(1/5) sum_i=1..5
000050 |        [a_i rho a_i^dagger + a_i^dagger rho a_i],
000051 | 
000052 |   Q_family(sigma)=(1/3) sum_j=1..3
000053 |        [b_j sigma b_j^dagger + b_j^dagger sigma b_j].
000054 | 
000055 | The operators alternate the two chiral spinor modules. In real Majorana
000056 | coordinates the same normalization is
000057 | 
000058 |   B_Ap=(1/2) Gamma_A tensor Gamma_p,
000059 |   sum_A Gamma_A^dagger Gamma_A=10I16,
000060 |   sum_p Gamma_p^dagger Gamma_p=6I4,
000061 |   K_Ap=Gamma_A tensor Gamma_p/sqrt60.
000062 | 
000063 | Using the six complex epsilon basis matrices on4 instead gives
000064 | sum E_p^dagger E_p=3I4 and normalization sqrt30. These are different
000065 | orthonormal/Kraus-coordinate conventions for the same normalized response,
000066 | not a normalization disagreement.
000067 | 
000068 | For any joint input, including correlated and entangled inputs,
000069 | 
000070 |   Tr_family_out E(rho_CF)
000071 |       = Q_carrier(Tr_family_in rho_CF).
000072 | 
000073 | This follows from the full factorization and Q_family^dagger(I)=I.
000074 | It requires no chosen family state.
000075 | 
000076 | On an incoming two-word in Lambda2 C5, exactly three creation operations
000077 | lead to Lambda3 C5 and exactly two annihilation operations lead to
000078 | Lambda1 C5. Their total weights are3/5 and2/5. The volume-form
000079 | identification Lambda3 C5=Lambda2(C5)* turns the creation block into
000080 | the subnormalized Petersen population A_Pet/5. The other2/5 is a real
000081 | five-state output, not a stay operation on the original ten labels.
000082 | 
000083 | On the complete32-state Fock parent, the population graph is the five-cube
000084 | Q5. Its antipodal quotient is the Clebsch16 graph. The existing v810 already
000085 | establishes the Clebsch and Petersen geometry. The present result reuses
000086 | that old graph and supplies the actual signed current-response path to it.
000087 | 
000088 | There are three different statements to retain:
000089 | 
000090 | 1. Full E8 D5+A3 inclusive response -> full Clifford channel: exact.
000091 | 2. Pure Petersen A/3 from that channel: requires selection and normalization
000092 |    of its creation branch, followed by the dual target identification.
000093 | 3. Physical P1 event/time rule implementing this inclusive response:
000094 |    not established by the bracket identity.
000095 | 
000096 | The source currents used above are bosonic weight-one VOA currents with
000097 | internal spinor transformation types. The sixteen-dimensional label module
000098 | is not thereby a spacetime fermion field. The CAR operators realize the
000099 | internal Clifford action. No physical Hamiltonian, local apparatus, time
000100 | rate, additional bath or reset rule has been inferred.
000101 | 
000102 | Checks: check_e8_clifford_response.py; source phases and exact outputs in
000103 | e8-clifford-response-results.json. Normal and python -OO stdout agree.
000104 | Independent review: E8-CLIFFORD-INDEPENDENT-REVIEW.txt.