TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/REFERENCE.txt Original file SHA-256: 9f99952b6a75c28d65de9de9d45d2c409ac6135d2b82f39ba4f4bee35620e328 Inclusive original lines: 89-156 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. 000089 | Jacobians, gauge/Berezin conventions, changing domains and source frames 000090 | are part of these original transported data, not silently set to zero. 000091 | For an even background parameter and a fixed source chart, differentiation 000092 | of THIS integrand yields 000093 | partial_s log Z_rel,s[J] 000094 | =-_(s,J)-r_s. (2) 000095 | 000096 | Put q_s[J]=Z_adm,s[J]/Z_adm,s[0] and ell_s=log Z_rel,s[0]. Then 000097 | partial_s log q_s[J] 000098 | =-_(s,J)+_(s,0), 000099 | partial_s ell_s=-_(s,0)-r_s. (3) 000100 | Thus a full original normalized field source and its original action score 000101 | determine the vacuum variation, not just the field responses. A polynomial 000102 | K_s can be inserted by the corresponding full source derivatives; a 000103 | nonpolynomial spectral K_s requires its actual defined insertion/limit. 000104 | The local-force theorem controls its stated force words, not every 000105 | unbounded action insertion without checking that insertion. 000106 | 000107 | With the actual base value ell_0, the unique nonzero-branch solution is 000108 | Z_rel,1[J]=exp(ell_0) 000109 | exp[-int_0^1 (_(s,0)+r_s)ds] q_1[J]. (4) 000110 | Equation(4) is anchored thermodynamic integration on the original datum. 000111 | It reconstructs the lost vacuum scalar together with q; no free c(b) 000112 | has been added. At zeros use Z itself, and carry the original eta/kernel 000113 | branch rather than pretending one logarithm exists globally. 000114 | 000115 | The EXPLICIT identity-reference anchor is legitimate when the numerator 000116 | at b_0 is exactly the same full integral as the denominator reference: 000117 | same field domain, measure, regulator, boundary condition, orientation and 000118 | counterterms. Then Z_adm,b0[0]=Z_ref,b0[0], so ell_0=0. 000119 | For the fermion determinant quotient this follows immediately from equal 000120 | operators on the same determinant/Pfaffian contract. It is not a newly 000121 | chosen vacuum. If the denominator denotes only a sector, its identity 000122 | anchor normalizes that sector, not all remaining bosonic integrals. 000123 | The equation S_rel(b0)=0 alone does not establish equality of two integral 000124 | contracts. The original given Z_ref is the anchor object to bind, not an 000125 | arbitrary point H=0 or a dimensionless scale renamed as a normalization. 000126 | 000127 | At finite fermion cutoff, the particularly concrete version is 000128 | d log[det M_b/det M_ref,b] 000129 | =Tr(M_b^-1 dM_b)-Tr(M_ref,b^-1 dM_ref,b). (5) 000130 | Its integral from the identical reference operator gives the exact 000131 | determinant ratio1 at the base. A Pfaffian uses one half of the trace and 000132 | its actual orientation, so96 doubled chart slots must not be counted as 000133 | 96 independent determinant species. The separate Berezin/KMS source audit 000134 | owns which M and domain actually implement the original96 reader. 000135 | 000136 | 3. How the ORIGINAL spectral reference and eta enter the score 000137 | 000138 | For the current displayed spectral action set 000139 | X_s=(D_rel,s+A_Sigma,s)/chi_s, Y_s=D_ref,s/chi_s. 000140 | Whenever its supplied trace/domain calculus permits differentiation, 000141 | partial_s S_spec 000142 | =Tr[(partial_s f_s)(X_s)+f_s'(X_s)partial_s X_s] 000143 | -Tr[(partial_s f_s)(Y_s)+f_s'(Y_s)partial_s Y_s] 000144 | +(i pi/2)partial_s Delta_eta_s. (6) 000145 | Here partial_s X includes the actual chi derivative, connection/domain 000146 | transport and field frame; similarly for Y. The trace simplification in(6) 000147 | uses the ordinary cyclic trace actually displayed, not an arbitrary 000148 | weighted trace. Along a changing APS domain its additional boundary 000149 | transport is not represented merely by a bounded dD. 000150 | 000151 | The numerator's reference subtraction in(6) and the denominator score r_s 000152 | in(2) are separately named original data. They must not be merged or 000153 | double-counted on the assumption that their common word 'reference' 000154 | makes them the same functional. Eta is retained as a phase and can have 000155 | spectral-flow/kernel contributions on the original branch. It is not 000156 | replaced by a generic Quillen number or a nonzero-total-eta requirement.