TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/GEOMETRIC-SOURCE-JET.txt Original file SHA-256: aa409fd5dbce2d3b32639aeb45be5666f0627172a73c933e1c07121a99b59058 Inclusive original lines: 131-261 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. 000131 | compatible collar lift, the family preserves all displayed P1 admissibility 000132 | properties. Self-adjointness and ellipticity are preserved by unitary 000133 | conjugation; positive principal symbol follows from phi_t'>0; 000134 | reflection regularity and positivity are preserved because reflection and 000135 | state are transported; seam spectral flow, eta data, clustering, relative 000136 | action finiteness and the declared reference are unchanged as unitary 000137 | invariants. In particular 000138 | 000139 | S_prim[D_rel,t,D_ref,t,chi_t,eta_t]=S_prim[D_rel,D_ref,chi,eta] 000140 | 000141 | when every defining datum is transported. This is an isospectral family of 000142 | presentations. It does not by itself select a physical pulse or new branch. 000143 | 000144 | Bare P1 does not establish the premise that the supplied quotient fold net, 000145 | its quarter state and its physical time are the displayed A_+, omega and 000146 | tau_t. That precise source-to-P1 embedding remains the condition on the 000147 | preceding paragraph. 000148 | 000149 | 3. Passive kernel transport and active preparation are different diagrams 000150 | ------------------------------------------------------------------------ 000151 | 000152 | Let V_t=Gamma(U_t). Passive covariant transport sends 000153 | 000154 | (source data, omega, O) 000155 | | Ad(V_t) on every datum 000156 | v 000157 | (transported source data, omega_t, O_t), 000158 | 000159 | where omega_t=omega o Ad(V_t*), O_t=V_t O V_t*. Consequently 000160 | 000161 | omega_t(O_t)=omega(O). (9) 000162 | 000163 | Equation (9) is the isospectral kernel/action statement. Calling it a new 000164 | prepared response would count a coordinate change as an intervention. 000165 | 000166 | An active preparation at a fixed detector instead keeps the source chart and 000167 | O fixed and changes only the vector/state: 000168 | 000169 | Omega_d -> Omega_d(t)=V_t Omega_d, 000170 | omega_t^active(O)=omega(V_t* O V_t). (10) 000171 | 000172 | Equations (9) and (10) are related by covariance, but they answer different 000173 | questions. The first transports the detector as well and is response 000174 | invariant. The second compares the transformed state with a fixed detector 000175 | and can have a nonzero octic response. 000176 | 000177 | 4. The first source jet is stress; the octic response is eighth order 000178 | --------------------------------------------------------------------- 000179 | 000180 | The fold relation 000181 | 000182 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_n0 000183 | 000184 | maps the cover vector v(theta)=g(8theta)/8 to the supplied shifted fold 000185 | stress 000186 | 000187 | A_d(g)=sum_n g_(-n) L_n^(d), 000188 | L_n^(d)=L_n+d.h_n+||d||^2/2 delta_n0. 000189 | 000190 | Thus (2), equivalently (5), is the one-particle source jet and A_d(g) is its 000191 | Fock stress implementer, up to the fixed projective scalar. Let c_-1 denote the 000192 | coefficient of L_-1^(d) in A_d(g). With the displayed Fourier convention 000193 | c_-1=g_1=conjugate(g_-1) for real g. 000194 | 000195 | The normalized original octic ray is 000196 | 000197 | F8hat=I8/(96 sqrt(140)) in Im M. 000198 | 000199 | The unshifted Sugawara part L_n has polarization degree zero, while the two 000200 | linear pieces of d.h_n have degree plus or minus one. F8hat has pure positive 000201 | degree eight. It follows exactly that 000202 | 000203 | =0, k<8. (11) 000204 | 000205 | At k=8 every factor must be the positive linear shift. Positive modes 000206 | annihilate Omega_d, and total oscillator weight eight forces all eight modes 000207 | to be -1. Therefore 000208 | 000209 | 000210 | =8! conjugate(F8hat(b)) c_-1^8, (12) 000211 | 000212 | where b=U_+^T O^T d is the actual native M/O transport. Both original 000213 | quarter sectors have F8hat(b) nonzero. Consequently 000214 | 000215 | 000216 | =t^8 conjugate(F8hat(b)) c_-1^8+o(t^8). (13) 000217 | 000218 | The projective scalar phase cannot change the first nonzero F8 coefficient. 000219 | A real smooth bump in any proper fold interval can be chosen with c_-1 nonzero, 000220 | and the existing affine energy bounds define its smeared stress unitary. 000221 | 000222 | 5. Correct origin criterion 000223 | ---------------------------- 000224 | 000225 | The earlier test 000226 | 000227 | != 0 000228 | 000229 | is a valid test for a DIRECT LINEAR F8 coupling. It is not a necessary test 000230 | for origin from a geometric one-parameter source family. In the actual 000231 | transported family the first jet is the linear stress generator and its F8 000232 | matrix element vanishes by (11), while the eighth ordered product is nonzero 000233 | by (12). The correct criterion is the first nonzero jet of the supplied 000234 | family: 000235 | 000236 | P_(C F8hat) P_(Im M) P_8 d^k/dt^k|_0 Omega_d(t) != 0 000237 | 000238 | for some source-derived k. Here the minimal k is exactly eight when 000239 | c_-1 != 0, the local profile class established in section4. 000240 | 000241 | This proves availability, not a law selecting a particular g or t. Choosing 000242 | an experimental profile and pulse strength is an intervention specification, 000243 | not a missing fundamental constant. The remaining fundamental obligation is 000244 | the embedding of the already given CAR/fold net, quarter state, reflection 000245 | and time into the P1 operational seed. No identification of the root-current 000246 | OPE coefficient with dot B or dot D is needed for this geometric route, and 000247 | none is inferred. 000248 | 000249 | Source locations 000250 | ---------------- 000251 | 000252 | experiments/theory-contracts/joint-native-algebra-selection-20261016/ 000253 | SCALAR-FIRST-ORDER-SOURCE-FIBER-AND-SEAM-WARD.txt:31-65,172-197 000254 | ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt:191-221 000255 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt:109-151 000256 | FULL-SOURCE-HULL-AND-QUARTIC-OPE.txt:81-91 000257 | experiments/theory-contracts/v16-operator-pullback-20261014/ 000258 | FOLD-AUDIT.md:12-65,67-86 000259 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000260 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex:65-80,302-343 000261 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex:103-143