TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/FOUR-STARTING-POINTS-STATE-RELATIVE-EQUIVALENCE.txt Original file SHA-256: 53dd97e3d9fc5b9d62a3f772fbe5d578b258c84bff82afac68668422c90d2ac8 Inclusive original lines: 69-157 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. 000069 | 3. THE FOUR FIELDS AND THEIR COMPLETE NEUTRAL RESPONSE 000070 | 000071 | Let f_i be the actual normalized quartics with norms(96,8,8,8,1), and N 000072 | the previously INTRINSICALLY reconstructed six-ray matrix, in the existing 000073 | literal comparison chart. NN*=I5, N*N=I6-J6/6. It is not fitted here. 000074 | Use unit ray w_q=sqrt(6/5)N[:,q], and F_w=sum_i(w_i)F_i. 000075 | 000076 | The actual fold fixes delta_k=(k+alpha)/8-1/2, d=M delta and 000077 | b=U_+^T O^T d, 000078 | alpha1/4: b=((7-9i)/32,(1-i)/4,-i/4,(3-3i)/32), 000079 | alpha3/4: b=((5-11i)/32,(1-i)/4,-i/4,(1-i)/32). 000080 | This b is from the SAME covariance subtraction, not a q-dependent state. 000081 | 000082 | For the fixed NS reference F_i^d(z)=:f_i(h_+(z)+b/z): and 000083 | (D_r)_ij=sum_|nu|=r conjugate(partial^nu f_i(b))partial^nu f_j(b)/nu!. 000084 | The FULL separated two-point kernel is 000085 | K_ij(z,w)=sum_r=0..4 (D_r)_ij /[(z-w)^(2r)z^(4-r)w^(4-r)]. 000086 | D0 is the disconnected mean; the connected function is r1..4. 000087 | Define c_r(q)=w_q*D_r w_q. The kernel basis is independent: multiply by 000088 | (zw)^4 to get the powers of t=zw/(z-w)^2. Thus equality of full connected 000089 | functions forces equality of EACH c1..4, not merely a sampled profile. 000090 | 000091 | For every q2..5 in both quarters: 000092 | c4=1, c3=85/256, c2=4335/131072. 000093 | But the exact c1 numerators, with COMMON denominator1006632960, are: 000094 | 000095 | q 2 3 4 5 000096 | alpha1/4 1200580 1300621 982285 1446340 000097 | alpha3/4 1308100 1521805 1089805 1338820 000098 | 000099 | All six unordered comparisons are different in each row. c1 is positive, 000100 | invariant under a phase of F_w and under jointly transported unitary charts, 000101 | and unchanged by the correctly joint b->-b/conjugation convention change. 000102 | The first difference along THESE six-ray directions is at sixth order in 000103 | b (D1); D2's fourth-order ray diagonal happens to be equal. D2 is not 000104 | asserted scalar as a whole matrix. Distinct mean magnitudes D0 corroborate 000105 | but are not needed. No largest-c1 or nearest-q rule is adopted. 000106 | In fact, for each fixed quarter and any two of the four normalized rays, 000107 | K_q_connected(z,w)-K_q'_connected(z,w) 000108 | =(c1(q)-c1(q'))/[(z-w)^2 z^3 w^3]. 000109 | This nonzero separated-point distribution is an entire-kernel witness, 000110 | not a difference obtained by tuning a profile. Appropriate local smears 000111 | can distinguish unequal distributions; no physical instrument is thereby 000112 | postulated or selected. 000113 | 000114 | If a state-preserving *-isomorphism A sends F_wq to exp(i phi)F_wq', it 000115 | preserves _connected. The table contradicts this for 000116 | q!=q'. This argument does NOT depend on Stab_G31(d)=1; it also obstructs 000117 | an arbitrary larger equivalence that respects this actual marked-field 000118 | comparison class. It is not a classification of all P1 gauges. 000119 | 000120 | A stronger test does not need the distinguished ray to be preserved: 000121 | lambda_q=tr(P(q,{0,1})D1). 000122 | These four EXACT real algebraic numbers are ALSO pairwise distinct in 000123 | both quarters (all values are retained in results). The central leading 000124 | OPE coefficient D4=I fixes the quartic-space metric. A state-preserving 000125 | linear *-equivalence carrying the whole actual quartic space and Pq to 000126 | Pq' acts unitarily in that metric, must transport D1, and hence must 000127 | preserve this trace. Therefore even forgetting q as an extra label does 000128 | not identify these four planes within that comparison class. This is a 000129 | source-response invariant on a supplied plane, not an executed projection 000130 | or a choice of its physical detector. A transformation abandoning the 000131 | specified quartic field space is not covered by this stronger test. 000132 | 000133 | For the complete rooted five-field frames let J_q={0,...,5}\{q}, with 000134 | the original sorted labels retained, D_q[:,j]=n_j-n_q and G=I5+J5. 000135 | U_q=D_q G^-1/2, U_q[:,j]=n_j-(sqrt6-1)n_q/5, U_q*U_q=I. 000136 | Their raw kernels are D_q*D_r D_q; orthonormal kernels U_q*D_r U_q. 000137 | Both retain all cross-correlations. The two columns j0,1 reproduce the 000138 | actual P(q,{0,1}); they are not selected as a detector in this calculation. 000139 | 000140 | Let g_ia=partial_a f_i be the actual erasure20. Its original vacuum Gram 000141 | is I20, explicitly checked; it is not a charged E8 twenty of equal size. 000142 | For the 25 fields p=(f_i,g_ia), degrees k=(4^5,3^20), compute 000143 | J_r[A,B]=sum_|nu|=r conjugate(partial^nu p_A(b))partial^nu p_B(b)/nu!. 000144 | Every mixed separated kernel is exactly 000145 | sum_r J_r[A,B]/[(z-w)^(2r)z^(k_A-r)w^(k_B-r)]. 000146 | The new results store ALL five 25x25 coefficient matrices for EACH of the 000147 | four frames and BOTH quarters, via 000148 | L_q=diag(U_q,U_q tensor I4), J_(q),r=L_q*J_r L_q. 000149 | Indices are (i,a). No independently normalized erasure, reset, executed 000150 | projection or missing cross-block is introduced. These 25 fields are not 000151 | claimed to be the entire E8 algebra; their arbitrary products are covered 000152 | by the next generating formula. 000153 | Here L_q is a PASSIVE quartic-frame change with the derivative/Cartan basis 000154 | held fixed. An ACTIVE native G transport must additionally carry the 000155 | contragredient bar4 erasure action, as supplied by the full Y_d transport 000156 | below. L_q alone is not a physical transversal operation. 000157 |