TFPT scientific source excerpt Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-CONNECTION-AND-WHOLE-CURRENT-SOURCE.txt Original file SHA-256: 4ef9447060d2139e6ff1d4de9961c1f463945549712da7cb8274766e00441514 Inclusive original lines: 93-186 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. 000093 | Use the actual regulated minimal historical parent, not a new action: 000094 | L_E=A S^2-S.Q+L_rest, Q=r+c0 J5, r=iB da, 000095 | A=3/(4 kappa_phys^2), c0=3/4. 000096 | All other original fields and their mixed correlations remain integrated. 000097 | With an external S source ell the exact finite Gaussian gives 000098 | Z[ell,eta]=C_A int Drest exp(-S_rest+eta.fields) 000099 | exp[(ell+Q)^2/(4A)], 000100 | C_A=product_cells(pi/A_cell)^(number_of_real_components/2), 000101 | with the original cell/metric factors, same reference and Berezin orientation. 000102 | This reuses the established Gaussian integration. The continuation here 000103 | retains ALL external fermion coefficients and extracts the whole connected 000104 | S/current response; it is not a new Gaussian identity or a one-loop ansatz. 000105 | 000106 | Fermions first gives the SAME polynomial Berezin Gaussian (Pfaffian with 000107 | the actual real pairing, or complex determinant in a supplied complex 000108 | pairing), with every external source leg. Integrating S first creates 000109 | the fixed J5^2 contact. At a finite regulator their equality is coefficientwise 000110 | Gaussian integration of a finite Grassmann polynomial. No inverse fermion 000111 | matrix is required, so zero Pfaffians do not invalidate the identity. 000112 | The independent control keeps all six general antisymmetric mass entries, 000113 | all24 general current entries and all eight nonzero external eta sectors 000114 | in the four-variable exterior-algebra identity. This is universal algebra 000115 | verification, not a four-mode physical substitute for original full96. 000116 | 000117 | Differentiating W=log Z, where the actual source is nonzero, gives 000118 | _ell=(ell+_ell)/(2A), 000119 | G_SS,c(x,y)=delta(x-y)/(2A)+G_QQ,c(x,y)/(4A^2). 000120 | The second term contains the full phase/phase, phase/current, current/phase 000121 | and current/current responses in the SAME interacting parent state. 000122 | None is replaced by a bosonic saddle or a fixed fermion background. 000123 | 000124 | For constant original couplings and co-closed transverse source tests, 000125 | the exact-gradient r has zero overlap, giving at nonzero momentum 000126 | G_STST,c=(2 kappa_phys^2/3) P_T 000127 | +(kappa_phys^4/4) G_J5T J5T,c. 000128 | Subtract the fixed local contact and distinguish kinematic factors of P_T 000129 | from propagating singularities. Any propagating transverse torsion response 000130 | in this minimal parent is carried by its full axial-current correlations. 000131 | It can include composite/bound-state modes. An independent elementary 000132 | torsion mass or new pole cannot be read from a truncated heat coefficient. 000133 | A physical spectral-density interpretation additionally uses the actual 000134 | valid common time/reflection reconstruction; this note does not prove it. 000135 | 000136 | 5. DECISIVE WHOLE-SOURCE TEST AND THE REMAINING PRIMITIVE IDENTITY 000137 | The same parent obeys the exact source equation 000138 | 2A partial_ell Z=(ell+r+c0 Jhat5)Z, 000139 | where Jhat5 inserts the ACTUAL current with its original fermion pairing. 000140 | Its zero-source normalization, current contact and the mixed derivative 000141 | identities above provide a concrete test of a proposed source map. 000142 | They are necessary tests, not sufficient reconstruction of all source words. 000143 | 000144 | If a further original S-dependent factor R[S] is inside the same integral, 000145 | integration by parts adds int(partial_S R) exp(...) to this equation. 000146 | For R=exp(-U), this is the insertion -partial_S U. Moving integration 000147 | domains and selectors carry their actual additional terms. One cannot 000148 | factor R out, delete its force, or complete the old Gaussian as if absent. 000149 | In particular the bound S_minus reference and a moving full Calderon 000150 | domain require their actual variation. No new counterterm is introduced. 000151 | 000152 | There is also an explicit second-order identity, not just an unspecified 000153 | correction. Let I[F] denote the same original regulated integral with a 000154 | remaining S-dependent factor F, and Z=I[F] at ell=0. Then 000155 | 2A Z_x=I[Q_x F+F_x], 000156 | 4A^2 Z_xy=2A delta_xy Z 000157 | +I[Q_x Q_y F+Q_x F_y+F_x Q_y+F_xy]. 000158 | Here subscripts on Z are external ell derivatives; on F they are actual 000159 | S derivatives, including its original reference/selector/measure carry. 000160 | The connected answer is Z_xy/Z-Z_x Z_y/Z^2 where Z is nonzero. 000161 | This form needs no division by F, so isolated determinant/Pfaffian zeros 000162 | do not create an artificial inverse. Differentiability, common regulator, 000163 | convergence and actual integration-boundary terms remain necessary. 000164 | When F is nonzero one may equivalently put Y=delta logF/dS and obtain 000165 | G_SS,c=delta/(2A)+[G_(Q+Y)(Q+Y),c+]/(4A^2). 000166 | The second derivative is a required contact, not merely a covariance of 000167 | the extra force. This includes mixed reference/current responses. 000168 | If the same fermions are integrated into F, remove c0 J5 from Q in that 000169 | representation. F_S and F_SS then reproduce the conditional current and 000170 | its contact, and the total-covariance identity reconstructs the same 000171 | whole response. Keeping both would count the same fermions twice. 000172 | 000173 | The geometric i*c0 spin vertex alone does NOT identify the historical 000174 | -c0 S.J5 fermion caller: transport its actual Euclidean pairing/current, 000175 | real structure, chiral restriction, reference and the same integration 000176 | operation. Equal c0 values are not a complete fermionic source map. 000177 | The operator frame in section3 supplies an actual local connection map; 000178 | it does not select that variable in the primitive field integral. 000179 | 000180 | [OMITTED: non-scientific conversation or execution metadata] 000181 | [OMITTED: non-scientific conversation or execution metadata] 000182 | [OMITTED: non-scientific conversation or execution metadata] 000183 | [OMITTED: non-scientific conversation or execution metadata] 000184 | [OMITTED: non-scientific conversation or execution metadata] 000185 | [OMITTED: non-scientific conversation or execution metadata] 000186 | [OMITTED: non-scientific conversation or execution metadata]