TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/REWEIGHTING-REVIEW.txt Original file SHA-256: aa7479fc2d4bb9eb11fd071e81550906bd4105e43a19af759da7054efbc3d49a Inclusive original lines: 43-149 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. 000043 | Let X_s and X_0 be the defect and trivial configuration spaces. After putting 000044 | both sectors over fixed reference descriptions, suppose 000045 | 000046 | S_s[x;alpha]=S_s^0[x;alpha]+alpha d s, (3) 000047 | S_0[x;alpha]=S_0^0[x;alpha], (4) 000048 | 000049 | where d s is independent of x. Allow the residual measures to depend on 000050 | alpha. Define 000051 | 000052 | Z_s^0(alpha)=integral_(X_s) exp(-S_s^0[x;alpha]) 000053 | d mu_(s,alpha)(x), 000054 | Z_0^0(alpha)=integral_(X_0) exp(-S_0^0[x;alpha]) 000055 | d mu_(0,alpha)(x), 000056 | R(alpha)=Z_s^0(alpha)/Z_0^0(alpha). (5) 000057 | 000058 | Then direct factorization gives 000059 | 000060 | Z_s(alpha)=exp(-alpha d s) Z_s^0(alpha), 000061 | Z_0(alpha)=Z_0^0(alpha), 000062 | 000063 | and therefore 000064 | 000065 | q(alpha):=Z_s(alpha)/Z_0(alpha) 000066 | =exp(-alpha d s) R(alpha). (6) 000067 | 000068 | At alpha=0 one has q(0)=R(0). Consequently the archived form (1) is exact on 000069 | an interval precisely when 000070 | 000071 | R(alpha)=R(0) (7) 000072 | 000073 | throughout that interval. If R is positive and differentiable, (7) is 000074 | equivalent to the relative Ward identity 000075 | 000076 | partial_alpha log R(alpha)=0 (8) 000077 | 000078 | throughout the interval. At only one value of alpha, equation (8) proves 000079 | first-order stationarity there, not global constancy. 000080 | 000081 | Equivalently, 000082 | 000083 | partial_alpha log q(alpha) 000084 | =-d s+partial_alpha log R(alpha). (9) 000085 | 000086 | Thus -d s is the complete logarithmic slope exactly when the residual sector 000087 | ratio contributes no slope. 000088 | 000089 | If the supposed insertion is instead a field-dependent functional J[x], the 000090 | factorization in (6) generally fails already within the defect sector: 000091 | 000092 | Z_s(alpha)=Z_s^0(alpha) 000093 | _(s,alpha;0). (10) 000094 | 000095 | Its logarithm contains the cumulants of J. The old single exponential is 000096 | therefore exact only when the identified spectral-flow insertion is 000097 | sectorwise constant, or when an equivalent zero-fluctuation theorem is 000098 | provided. The intended identification J=d SF treats it as the constant 000099 | topological label and satisfies this requirement. 000100 | 000101 | 2. Fixed-measure Ward test 000102 | -------------------------- 000103 | 000104 | Suppose first that mu_s and mu_0 do not depend on alpha and differentiation 000105 | under both integrals is justified. Normalize expectations with the residual 000106 | weights exp(-S_s^0) and exp(-S_0^0). Then 000107 | 000108 | partial_alpha log Z_s^0 000109 | =-_s, 000110 | partial_alpha log Z_0^0 000111 | =-_0. (11) 000112 | 000113 | Hence 000114 | 000115 | partial_alpha log R 000116 | =-_s 000117 | +_0. (12) 000118 | 000119 | The old reweighting follows exactly if and only if these two residual response 000120 | expectations agree for every alpha in the claimed interval. A sufficient 000121 | special case is that the residual actions are alpha independent. Another is 000122 | a common alpha deformation whose normalized expectations agree in the two 000123 | sectors. Neither conclusion follows merely from additivity of alpha d s in 000124 | the defect action. 000125 | 000126 | 3. Alpha-dependent measures 000127 | ---------------------------- 000128 | 000129 | If the measures vary with alpha, first transport each sector to a fixed 000130 | reference space and write 000131 | 000132 | d mu_(r,alpha)=w_r(alpha,x) d mu_(r,*), r=s,0. 000133 | 000134 | Let 000135 | 000136 | sigma_r(alpha,x)=partial_alpha log w_r(alpha,x) (13) 000137 | 000138 | be the measure score. Under the usual differentiation and integrability 000139 | hypotheses, 000140 | 000141 | partial_alpha log R 000142 | =-_s+_s 000143 | +_0-_0. (14) 000144 | 000145 | Equation (14), rather than (12), is the complete Ward test. Moving domains, 000146 | gauge-fixing Jacobians and determinant normalizations must be included in the 000147 | scores after pullback. The archived exponential remains exact if the right 000148 | side of (14) vanishes. Measure compatibility alone means the ratio is 000149 | well-defined; cancellation still has to be stated or proved.