TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-joint-conjugation-20261015/PROOF.txt Original file SHA-256: bbea15f5ebb1dff698f1ecdfd8a18e724089c8fe9880dea684a5f9fd45cb1c8a Inclusive original lines: 80-187 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. 000080 | 3. ONE FIELD LIFT, NOT TWO INDEPENDENT CHARGE REVERSALS 000081 | 000082 | For the specified chiral source use f_(n,alpha)=exp(i(n+alpha)theta), 000083 | D_alpha=-i partial_theta, and alpha=1/4 or 3/4. Its actual domain is 000084 | f(theta+2pi)=exp(2pi i alpha)f(theta). Set beta=1-alpha. 000085 | Coefficient conjugation sends n to -n-1 and D_alpha to -D_beta. 000086 | With P_alpha=1_(D_alpha<0), it sends P_alpha to 1-P_beta. 000087 | 000088 | The single polarized CAR intertwiner is 000089 | U a_n U^-1 = a'*(-n-1), U Omega_alpha=Omega_beta. 000090 | The ordered finite-excitation-word construction in the previous contract proves 000091 | it unitary between the two sea representations, with all CAR signs. It preserves 000092 | the positive excitation Hamiltonian H_exc=:dGamma(D):_P and reverses the relative 000093 | particle number. The energy of every excitation is |n+alpha|. No zero mode or 000094 | second vacuum is created. Neither domain is replaced by periodic boundary data. 000095 | 000096 | For an additional GIVEN internal antiunitary j in such a CAR realization, the 000097 | general normal-ordering law is 000098 | U Q_alpha(q) U^-1 = -Q_beta(j q j^-1). 000099 | It follows by a a*=1-a* a, with the constant cancelled by the transformed sea. 000100 | Consequently j I j^-1=I reverses relative number, whereas jYj^-1=-Y' preserves 000101 | the second-quantized charge Q(Y). A particle of internal charge y becomes a hole 000102 | of a target mode with charge -y, so its physical charge is still y. This is a 000103 | contragredient identification, not a second independent physical charge reversal. 000104 | The positive time identity uses :D:, not :|D|: (the latter gives wrong hole signs). 000105 | 000106 | This general formula does NOT authorize adjoining 16 complex spinor species to 000107 | the original source. The actual internal coefficient module must be established. 000108 | 000109 | 4. THE EXACT FOLD DETERMINES THE COMMON QUARTER ACTION 000110 | 000111 | Write an original mode n=8l+j, j=0,...,7. Then 000112 | -n-1=8(-l-1)+(7-j). 000113 | Thus the conjugation reverses the folded branch index: R e_j=e_(7-j). 000114 | In the recorded folded boson/current convention 000115 | delta_(alpha,j)=(j+alpha)/8-1/2, 000116 | we have delta_beta=-R delta_alpha, exactly for every alpha. 000117 | On source normal-ordered Cartan charges p, the action is p -> -R p. 000118 | This follows from the field lift; it is not a guessed reflection of root labels. 000119 | At the actual source level the map is F_beta U F_alpha^-1, using the existing 000120 | fold unitaries and their actual covariances, not an eight-copy product vacuum. 000121 | 000122 | Use the already recorded rational E8 coordinate dictionary M. With d_alpha=M 000123 | delta_alpha, the induced coordinate action is 000124 | O=-M R M^T, 000125 | O(x1,...,x8)=(x4,-x5,-x6,x1,-x2,-x3,x8,x7), 000126 | O d_alpha=d_beta. 000127 | O is an involutive signed permutation with four sign changes; it preserves D8 000128 | and its even half-integral E8 coset. It therefore preserves all 240 E8 roots. 000129 | It sends the actual source cocharacter q'=2(e3+e6) to -q'. 000130 | For the fixed original-carrier hypercharge coordinate 000131 | y=(-1/3,-1/3,-1/3,1/2,1/2,0,0,0), 000132 | O y=(1/2,-1/2,0,-1/3,1/3,1/3,0,0). 000133 | Hence it is not the fixed-carrier antiparticle operation y -> -y. 000134 | The previously discussed root e1+e4 is fixed by O; its behavior cannot stand in 000135 | for the full carrier or the full source. 000136 | 000137 | Conditional on the ALREADY STIPULATED E8 level-one lattice theory, O has a 000138 | current/vertex lift. This is stronger than matching energies: let b_i be any 000139 | integer E8 basis L and epsilon(x,z)=(-1)^c(x,z) a standard ordered cocycle gauge. 000140 | The isometry makes A_ij=c(O b_i,O b_j)-c(b_i,b_j) symmetric modulo 2. Then 000141 | eta(x)=(-1)^[sum_i A_ii x_i(x_i-1)/2 + sum_(i=eta(x)|Ox>, with O on every oscillator, preserves vertex products 000144 | and adjoints; other cocycle gauges require their cochain rephasing. V_O is unitary, 000145 | fixes the vacuum, and preserves L0. This standard cocycle construction does 000146 | not choose a canonical phase of the lift or assert V_O^2=1 without that choice. 000147 | The current law is V_O J_(r,n) V_O^-1=eta(L^-1 r) J_(Or,n), and 000148 | V_O K_alpha V_O^-1=K_beta, K_alpha=L0+d_alpha.H0. 000149 | No new E8 extension of the original odd CAR algebra is proved by constructing 000150 | this lift inside the given E8 theory. In particular the previously found E8 000151 | ground doublet is not equated to the unique original-source sea vacuum. 000152 | 000153 | 5. THE MARKING IS TRANSPORTED, AND CANNOT BE REPAIRED AT FIXED TIME 000154 | 000155 | Let P5 project onto coordinates {1,2,3,4,5} and P3 onto {6,7,8}, as in the 000156 | existing dictionary. O transports them to P5' on {1,2,4,5,6} and P3' on {3,7,8}. 000157 | Exactly 24/40 D5 roots stay in the old D5, and 16 leave it. Of the D3 roots, 000158 | 4/12 stay and 8 leave. This says which embedding moves, not that E8 is lost. 000159 | 000160 | There is NO orthogonal correction T simultaneously satisfying 000161 | T d_beta=d_beta, T P5' T^-1=P5. 000162 | Indeed those equalities require ||P5' d_beta||^2=||P5 d_beta||^2. But 000163 | beta=1/4: fixed 129/256, transported 137/256; 000164 | beta=3/4: fixed 137/256, transported 129/256. 000165 | The family norms are respectively 41/256 versus 33/256, and vice versa. 000166 | This proves the obstruction for the whole orthogonal class, hence every Weyl 000167 | correction, without a GPU search. It is scoped to the fixed folded d and marked 000168 | D5+A3. Nonlinear/dynamical dictionaries or a different original source are 000169 | not ruled out. Old module symmetries that preserved an unshifted norm do not 000170 | automatically preserve this quarter-dependent physical time. 000171 | 000172 | There IS a parameter-free covariant completion from a GIVEN initial marking: 000173 | P5_beta=O P5_alpha O^-1, P3_beta=O P3_alpha O^-1, 000174 | y_beta=O y_alpha, k_beta(p)=k_alpha(O^-1 p). 000175 | Here k_alpha is the original given Z4 character. Its neutral-root second moment 000176 | M0=16P5+8P3 transports to O M0 O^-1. Therefore the same previously proved 000177 | formula P5=(M0-8I)/8 reconstructs precisely the transported projector; no new 000178 | independent projector or fitted phase is required. The 3+2 color/weak subspaces 000179 | transport in the same way. O^2=I makes the lattice/marking transport coherent. 000180 | On operators, H_y -> H_(Oy), so the transported state and observable retain 000181 | the same expectation. Retaining the old y instead would describe a different 000182 | comparison. A second-quarter coordinate choice can make charge labels appear 000183 | as simple negation, but does not turn the result into a symmetry of fixed M. 000184 | 000185 | This reduces the joint problem to the origin of ONE marked source realization, 000186 | rather than independent choices of two markings, two vacua and two times. 000187 | It is a derived relation between supplied data, not a P1 selection theorem.