TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/CAR-OCTIC-DICTIONARY.txt Original file SHA-256: c82fce3ad57904fb7628a1abaf2c6eacd94d230d73f2ac69d8da0fe9d49f22c2 Inclusive original lines: 1-109 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 | THE OCTIC IS AN OPERATOR OF THE ORIGINAL EVEN CAR SOURCE 000002 | ========================================================= 000003 | 000004 | Claim: MATH.NATIVE.OCTIC-CAR-COMPOSITE.01 000005 | Status: PROVED in the given quarter CAR source and its retained fold and 000006 | conformal point-splitting reference. No full E8 root or spinor extension 000007 | is needed for this operator or the stress response. 000008 | 000009 | 1. Preserve the original fermions and their correlations 000010 | -------------------------------------------------------- 000011 | The supplied unitary fold is eta_r(x)=psi((x+2pi r)/8)/sqrt8. Its actual 000012 | eigenchannel chart is chi=U_alpha* eta with 000013 | 000014 | U_alpha(r,k)=exp(2pi i(k+alpha)r/8)/sqrt8. 000015 | 000016 | For the negative-D sea, the covariance in this chart is 000017 | delta_jk C^-_(mu_k), mu_k=(k+alpha)/8. In the arc chart it remains 000018 | U_alpha diag(C^-_mu) U_alpha*. No independent arc seas are added. 000019 | 000020 | Write j_k(z)=:chi_k*(z)chi_k(z):_sea for the mean-zero currents, with 000021 | the charge ordering of the original source. Their current OPE is 000022 | 000023 | j_k(z)j_l(w) ~ delta_kl/(z-w)^2. 000024 | 000025 | The actual sea-to-conformal-reference counterterm is 000026 | delta_k=mu_k-1/2. It follows from the finite point-splitting difference 000027 | of the two specified kernels, not from choosing a response coefficient. 000028 | 000029 | 2. The original matrix map already supplies the target field 000030 | ------------------------------------------------------------- 000031 | Let M,O,U_+ be the same fixed original matrices as in the Ward checker. 000032 | Set A=U_+^T O^T M. Exactly, 000033 | 000034 | A A*=I4, A A^T=0, b=A delta. 000035 | 000036 | The checker reconstructs A and proves all three identities for both 000037 | quarter cases. The four positive currents are therefore 000038 | 000039 | H_+(z)=A j(z), 000040 | h_+^conf(z)=A(j(z)+delta/z)=H_+(z)+b/z. 000041 | 000042 | They have no mutual singular contractions and retain the positive 000043 | pairing with their adjoints. Substitution into the ALREADY IDENTIFIED 000044 | normalized native polynomial gives the literal source operator 000045 | 000046 | F8^(d)(z)=:I8(A(j(z)+delta/z))/(96 sqrt140):. (1) 000047 | 000048 | This field is a composite of fermion bilinears, has even CAR parity and 000049 | zero source charge, and acts within the given source Fock space. The 000050 | isometric/isotropic map A transports the factorial Fock norm; equation 000051 | (1) has exactly the unit-normalized octic state used in the Ward proof. 000052 | No root vertex is used in (1), nor is odd-CAR locality in a full E8 000053 | extension asserted. G31 remains the already supplied native dictionary; 000054 | its use to label this polynomial does not select the microscopic P1 net. 000055 | 000056 | The highest homogeneous oscillator vector |F8;d> has fold excitation 000057 | weight8 and ORIGINAL source excitation energy64. This is the inherited 000058 | factor8 of the actual source Hamiltonian, not a calibration to seconds. 000059 | 000060 | 3. Normal ordering, adjoints and localization 000061 | ---------------------------------------------- 000062 | The colon in (1) is the retained BOSONIC CURRENT normal ordering. In 000063 | fermions it means the original finite point-splitting limit, including 000064 | all subtraction and finite derivative counterterms. Replacing (1) by a 000065 | naively fully fermionically normal-ordered degree16 monomial is wrong. 000066 | The existing source proof already specifies the required limits for 000067 | arbitrary finite current polynomials; apply that same prescription. 000068 | 000069 | The adjoint uses A-bar and b-bar, with the original radial/i0 ordering. 000070 | Every separated product is computed by the SAME CAR covariance C_alpha 000071 | and its complement according to the actual creator/annihilator order. 000072 | At finite splitting it is an even CAR word with a finite Pfaffian. The 000073 | same limits therefore determine all its ordered correlation functions. 000074 | No independent state, new subtraction or second b shift is inserted. 000075 | 000076 | For a proper fold interval I, equation (1) is affiliated with the even 000077 | CAR algebra on the union F^(-1)(I) of eight original arcs. It need not be 000078 | an original point-local c1 field. The quotient c8 description is a 000079 | description of this actual multiarc operator; its identification with 000080 | P1 point locality is a separate origin statement. 000081 | 000082 | 4. Consequence for the source problem 000083 | -------------------------------------- 000084 | The positive eighth-order response in LOCAL-STRESS-HISTORY.txt is 000085 | therefore already an original even-CAR response. A full E8 root-field 000086 | completion is unnecessary for its existence, its shifted ground state, 000087 | the original physical source time and its ordered current correlations. 000088 | The bounded root-OPE check in NATIVE-FOLDED-OCTIC-ORIGIN.txt is a distinct 000089 | statement about that optional larger algebra and is not a premise of 000090 | this CAR mechanism. 000091 | 000092 | The remaining original P1 obligation is precise: identify this source's 000093 | reflection, actual collar operator/domain and reference sector with 000094 | Theta_P1, D_coll,P1 and D_ref,P1 by one compatible intertwiner. The 000095 | ordinary source state, operator response and time are already supplied. 000096 | Introducing another state, profile or coupling would not prove that 000097 | identification. P1-SOURCE-EMBEDDING-AUDIT.txt examines these data directly. 000098 | 000099 | Original sources under experiments/theory-contracts/ 000100 | ----------------------------------------------------- 000101 | joint-native-algebra-selection-20261016/ 000102 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt:51-120 000103 | ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt:137-157,188-214 000104 | ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt:109-156 000105 | v16-operator-pullback-20261014/FOLD-AUDIT.md:15-86 000106 | 000107 | All matrices in the executable checker are rebuilt from their original 000108 | source records. There is no Cartan-direction fit, new coupling constant, 000109 | or claim that all original P1 data have thereby been reconstructed.