TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/E0-COMPENSATOR-K-CLOCK-REVIEW.txt Original file SHA-256: 3baf383b118e2c67ff0abdfacc674a0cded634e42e080b79b29f74b925c98185 Inclusive original lines: 1-173 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 | E0 COMPENSATOR: GENUINE K/E8 LIFT AND EXACT CLOCK TRANSPORT 000002 | Independent bounded analytic review, 2026-10-29. 000003 | No new model or gauge calibration; no Atlas or old-source mutation. 000004 | 000005 | RESULT 000006 | W_E=V_omega L_S has an explicit parallel extension to the existing 000007 | common K bundle on C and hence to every associated E8 branch and its 000008 | Lie products. The extension is not a central E8/G31 phase. It preserves 000009 | the adapted central kappa and the adapted sigma0,c0, but changes the 000010 | marked flavor-quarter coefficient. It also fails to descend as a 000011 | parallel endomorphism of the irreducible original z-plane local system. 000012 | Thus it is a valid whole-bundle equivalence with transported marks; 000013 | it is not automatically a fixed-clock symmetry of the original source. 000014 | 000015 | 1. ACTUAL SLOT DICTIONARY, NOT DIMENSION MATCHING 000016 | 000017 | The algebraic G columns carry characters (h0,h2,h1), hence line order 000018 | (L0,L2,L1). This follows from the ACTUAL deck action 000019 | G->G diag(epsilon0,epsilon2,epsilon1), not from the number3. 000020 | The particle compensator in that horizontal frame is 000021 | 000022 | W_G=omega S, S=diag(1,-1,1). 000023 | 000024 | Therefore in the actual old q line order (L0,L1,L2) it is 000025 | diag(omega,omega,-omega), and its conjugate acts on antiparticles. 000026 | 000027 | Extend by the flat-trivial fourth line O. In q four-slot order the 000028 | required SU4 element is 000029 | 000030 | F_q=diag(omega,omega,-omega,-1), det F_q=1. 000031 | 000032 | It restricts to exactly W_G after the explicit triplet permutation. 000033 | The extra fourth eigenvalue -1 is forced by det1 once this action on 000034 | the triplet and the identity Spin10 component have been specified. 000035 | 000036 | The old printed F comparison orders the triplet as (L1,L0,-L2), followed 000037 | by O. Its diagonal F is still (omega,omega,-omega,-1). The proposal 000038 | diag(-1,omega,-omega,omega) is also correct, but in the DIFFERENT order 000039 | (O,L0,L2,L1). This moves the old last slot to first and its old first 000040 | slot to last. It must not be silently called the unchanged old F order. 000041 | Reordering summands as a dictionary is legitimate; it does not assert 000042 | an isomorphism of the nontrivial line L1 with O inside a fixed bundle. 000043 | 000044 | 2. EXPLICIT GLOBAL K ELEMENT AND ALL E8 PRODUCTS 000045 | 000046 | Reuse the already given q-compatible joint assignment 000047 | 000048 | K=(Spin10 x SU4)/diagonal Z4, 000049 | m_q=(0,-1,0,-1), 000050 | W=Lambda_even E tensor [L0 + D^-1 L1 + L2 + D^-1], D=det E. 000051 | 000052 | Its common transition functions are 000053 | k_ij=iota_m(g_ij)[1,diag(l0,l1,l2,1)], with l0l1l2=1, and 000054 | iota_m has diagonal family factor z^(2m_j+1), z^2=det g. 000055 | 000056 | The constant element f=[1,F_q] commutes with every one of these 000057 | transition functions: the Spin factor is identity and all family 000058 | factors are diagonal. It also commutes with the diagonal induced 000059 | determinant/flat connection. Thus it is an actual GLOBAL PARALLEL 000060 | bundle automorphism on C for the given split connection. The common 000061 | (-1,-I4) root ambiguity and diagonal Z4 quotient remain unchanged. 000062 | No trivialization of L_a or square root of D has been added. 000063 | 000064 | On the full E8 branching its action is precisely 000065 | 000066 | (45,1): identity, 000067 | (1,15): Ad(F_q), 000068 | (16,4): I16 tensor F_q, 000069 | (bar16,bar4): Ibar16 tensor bar(F_q), 000070 | (10,6): I10 tensor Lambda^2(F_q). 000071 | 000072 | All Lie brackets and invariant forms are preserved because these are 000073 | the restrictions of the same K element. The action on the marked 000074 | three (16,4) lines is the given particle compensator and the dual 000075 | restriction is its conjugate. This establishes the associated-bundle 000076 | statement under the EXISTING joint slot assignment. It does not derive 000077 | a microscopic CAR identification of the v252 fields with E8 adjoint 000078 | components or turn K into the inner gauge group of its displayed A_F. 000079 | 000080 | F_q has order6 and unequal family eigenvalues. It is not a scalar SU4 000081 | center element and gives no guessed G31 central action. 000082 | 000083 | 3. WHICH NATIVE CLOCKS STAY FIXED 000084 | 000085 | The original joint source proves kappa central in K, with phase -i 000086 | on its oriented W branch. Consequently f kappa f^-1=kappa. 000087 | The adapted sigma0 has Spin color-center factor and, in q order, 000088 | 000089 | R_q=diag(omega^2,omega,omega^2,omega). 000090 | 000091 | F_q commutes with both factors, so it fixes sigma0 and the adapted 000092 | native-class c0=kappa sigma0. The determinant endomorphism m_q is fixed 000093 | as well. These are exact statements for the common connection and 000094 | clock sections already specified in C3-QUARTER-EQUIVARIANT-JOINT-LIFT. 000095 | An untransported originally marked clock in some other chart is not 000096 | automatically the same section; the old joint-source scope is retained. 000097 | 000098 | 4. THE QUARTER IS A DIFFERENT MARK AND IS CHANGED 000099 | 000100 | In actual G order the right quarter matrix is 000101 | 000102 | K_Q=[[0,1,0],[-1,0,0],[0,0,1]], 000103 | S K_Q S=K_Q^-1, [S,K_Q] not zero. 000104 | 000105 | The scalar omega cancels in this conjugation. In the old q order K_Q 000106 | is A_q^-1, with A_q=[[0,0,-1],[0,1,0],[1,0,0]]. The inverse is the 000107 | already declared pullback/right-solution convention, not a changed 000108 | character ordering. In either orientation F_q conjugates this 000109 | coefficient matrix to its inverse. The fourth line is fixed by Q. 000110 | 000111 | IMPORTANT BASE DISTINCTION: f is vertical. Thus f Qlift f^-1 still 000112 | covers the SAME base map Q, not the base map Q^-1. Only its coefficient 000113 | matrix becomes K_Q^-1. The literal full inverse Qlift^-1 covers Q^-1 000114 | and cannot be equated to this conjugate without another base map. 000115 | In (O,L0,L2,L1) order the local coefficient difference is the parallel 000116 | vertical sign 000117 | 000118 | D_Q=diag(1,-1,-1,1), 000119 | coefficient(Q_new)=D_Q coefficient(Q_old)=coefficient(Q_old)^-1. 000120 | 000121 | Conjugation preserves the order4 relations, but changes the specified 000122 | lift over Q. At a fixed basepoint this is already a decisive matrix 000123 | counterexample to preserving the original quarter mark. It does not 000124 | invert kappa and must not be described as doing so. 000125 | 000126 | 5. COVER AUTOMORPHISM VERSUS DESCENT TO THE ORIGINAL z SOURCE 000127 | 000128 | On C the flat monodromy is diagonal V4, so F_q is parallel. On the 000129 | original z source the actual monodromy also contains the cubic Pc. 000130 | For Pc=[[0,0,1],[1,0,0],[0,1,0]], 000131 | 000132 | S Pc e0=-e1, Pc S e0=+e1. 000133 | 000134 | Hence [S,Pc] is nonzero. The compensator fails the necessary commutant 000135 | test for a parallel endomorphism downstairs. Equivalently the concrete 000136 | differential expression G W_G G^-1 is invariant under deck V4 but not 000137 | under the full cubic deck transformation. It is single valued on C, 000138 | not on the original punctured z sphere. Irreducibility gives the same 000139 | conclusion by Schur's lemma; the explicit commutator is the smaller test. 000140 | 000141 | The earlier m/R assignments also distinguish fixed C3 descent from a 000142 | transported marking orbit. No new simultaneous fixed C3 descent follows 000143 | from this diagonal compensator. 000144 | 000145 | 6. CONCRETE CONTINUATION AND SCOPE 000146 | 000147 | The K/E8 existence question is positively settled on the already given 000148 | cover and joint assignment. For real finite constructor inputs the old 000149 | identity W_E Phi^E W_E*=Phi may therefore be combined with this genuine 000150 | whole-bundle lift, PROVIDED its specified marks are transported as above. 000151 | For generic complex inputs the coefficientwise conjugation remains, 000152 | as in the existing whole-finite-source review. 000153 | 000154 | If the original action menu requires the same marked quarter, W_E 000155 | fails that requirement already by the 3x3 calculation. The next concrete 000156 | original test is whether the actual source permits this transported 000157 | quarter lift and the lack of parallel base descent, or instead fixes 000158 | the original lift. That source/domain/mark comparison is required; 000159 | an arbitrary relabeling of the positive quarter is not a proof of it. 000160 | No physical gauge permission, state choice, P1 identification or universal 000161 | obstruction to other legitimate source equivalences is inferred here. 000162 | 000163 | Sources read directly: 000164 | E0-WHOLE-FINITE-SOURCE-TYPE-REVIEW.txt, sections1,5; 000165 | ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt, sections2,5; 000166 | C3-COVER-JOINT-CONNECTION-REVIEW.txt; 000167 | C3-QUARTER-EQUIVARIANT-JOINT-LIFT.txt; 000168 | source-joint-spin-family-lift-20260922/PROOF.md:38-78; 000169 | original-admissible-source-closure-20261025/ 000170 | JOINT-LIFT-NATIVE-CLOCK-CONNECTION.txt:88-146. 000171 | Atlas lookup reused finding 000172 | actual-q-quarter-extends-common-k-connection-and-real-coefficient-lifts; 000173 | the source proofs, not the Atlas label, support the statements above.