TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/NATIVE-FOLDED-OCTIC-ORIGIN.txt Original file SHA-256: c694cb1cee45e65145fdbf144925e45178a7d0ee29adb7142400a00a970e259b Inclusive original lines: 1-150 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 | NATIVE FOLDED OCTIC ORIGIN 000002 | ========================== 000003 | 000004 | Verdict 000005 | ------- 000006 | 000007 | PROVED in the supplied original CAR/fold current dictionary: the actual 000008 | original fold has a nonzero normalized-octic matrix element in both quarter 000009 | sectors. CAR-OCTIC-DICTIONARY.txt gives the field directly as an even CAR 000010 | current composite; a full E8 root extension is not needed for this result. 000011 | The earlier exclusion for the unshifted vacuum Omega and its pure 000012 | stress descendants does not apply to this state. The supplied fold changes 000013 | the conformal vector itself, 000014 | 000015 | Y_d(T,z)=T(z)+d.h(z)/z+||d||^2/(2z^2), 000016 | L_n^d=L_n+d.h(n)+||d||^2 delta_(n,0)/2. 000017 | 000018 | The resulting octic state is not primary for L^d. This is the precise new 000019 | premise that opens the matrix element. No fitted Cartan direction or chosen 000020 | root is used. 000021 | 000022 | 1. Original transport and normalization 000023 | --------------------------------------- 000024 | 000025 | The checker reconstructs exactly the already supplied objects 000026 | 000027 | O = T_target B_lattice^(-1)/sqrt(2), 000028 | U_+[:,k]=(e_(2k)-i e_(2k+1))/sqrt(2), 000029 | d_alpha=M delta_alpha, 000030 | delta_alpha,j=(j+alpha)/8-1/2, 000031 | b_alpha=U_+^T O^T d_alpha. 000032 | 000033 | Here M is read from the original fold result, O from the original E8 simple 000034 | root maps, and U_+ is the original J-positive polarization. The checker 000035 | verifies O^T O=I8, U_+^*U_+=I4 and U_+^T U_+=0 before evaluating the octic. 000036 | For alpha=1/4 and 3/4 it independently obtains 000037 | 000038 | ||d_alpha||^2=85/128, h_d=85/256, 000039 | ||b_alpha||^2=85/256. 000040 | 000041 | In the exact native positive Fock coordinates, 000042 | 000043 | I8=sum_a x_a^8+14 sum_(a=F8hat(a_+,-1) Omega_d. For a homogeneous degree-eight oscillator 000066 | polynomial P, 000067 | 000068 | 000069 | =8! conjugate(P(b)). (1) 000070 | 000071 | The conjugation in (1) is required because the retained Hilbert product is 000072 | linear in its second slot. It agrees with the original fold kernel, whose 000073 | one-point functional has Riesz vector conjugate(m), not m. 000074 | 000075 | For P=F8hat, equation (1) gives 000076 | 000077 | alpha=1/4: 000078 | A_1/4=3 sqrt(35)(5416993+18690336i)/2^31, 000079 | 000080 | alpha=3/4: 000081 | A_3/4=429 sqrt(35)(-23473+110880i)/2^31. 000082 | 000083 | Thus the actual folded source has a nonzero F8hat component in both 000084 | quarters. The norm of the full Virasoro descendant is fixed by the sl2 000085 | recursion and independently by all oscillator partitions in the checker, 000086 | 000087 | ||(L_-1^d)^8 Omega_d||^2=8!(2h_d)_8=8!(85/128)_8 000088 | =330344196035583751126875/562949953421312. 000089 | 000090 | The exact squared normalized projections are respectively 000091 | 000092 | 2611534296241/59248596692359104307200, 000093 | 23879517043811/781004229126551829504000. 000094 | 000095 | They are small and strictly positive. A dimension count plays no role. 000096 | 000097 | The state F8hat(a_+,-1)Omega_d is not an L^d primary. Already the coefficient of 000098 | x0^7 in the exact directional derivative D_b F8hat is 000099 | 000100 | b_0/(12 sqrt(140)) != 0. 000101 | 000102 | This explains exactly why the old unshifted-primary orthogonality argument 000103 | cannot be transported to the original fold. 000104 | 000105 | 3. Actual source mode and the full Möbius orbit 000106 | ---------------------------------------------- 000107 | 000108 | The existing original cover/fold relation is 000109 | 000110 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_(n,0). 000111 | 000112 | Consequently 000113 | 000114 | L_-1^d=Lhat_-1=Lcal_-8^alpha/8, 000115 | =8^8 A_alpha. (2) 000116 | 000117 | This is an actual original source mode relation. It is not a relabeling of 000118 | the unshifted c=8 vacuum stress. 000119 | 000120 | For the standard disk Möbius element with coordinate w, |w|<1, the positive 000121 | energy highest-weight representation gives the complete normalized orbit 000122 | 000123 | U(w)Omega_d=(1-|w|^2)^h_d exp(w L_-1^d)Omega_d. 000124 | 000125 | This is the full all-level vector. Its word coefficients and normalization 000126 | are fixed at every level: 000127 | 000128 | [U(w)Omega_d]_n=(1-|w|^2)^h_d w^n(L_-1^d)^n Omega_d/n!, 000129 | sum_(n>=0) |w|^(2n)(2h_d)_n/n!=(1-|w|^2)^(-2h_d). 000130 | 000131 | Thus no unlisted higher word is discarded. Weight orthogonality and (1) 000132 | select its level-eight matrix element without truncating the state: 000133 | 000134 | 000135 | =(1-|w|^2)^(85/256) w^8 conjugate(F8hat(b_alpha)). (3) 000136 | 000137 | Equation (3) supplies an original geometric source operation with a nonzero 000138 | octic amplitude for every w != 0. A passive simultaneous change of the 000139 | field dictionary and background is covariance. It does not prepare a fixed 000140 | ray. Equation (3) concerns the active Möbius action on the stipulated 000141 | shifted source state. LOCAL-STRESS-HISTORY.txt extends this to genuinely 000142 | local smooth profiles, with the first nonzero response in order eight. 000143 | GEOMETRIC-SOURCE-JET.txt derives the geometric source operator family. 000144 | The remaining P1 question is its actual net/state/reflection/time and 000145 | collar embedding. Profiles and strengths specify interventions; they do 000146 | not need to be selected as fundamental constants by P1. 000147 | 000148 | 4. Bounded root OPE result retained 000149 | ----------------------------------- 000150 |