TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/DIRECT-DEFECT-CLOSURE.txt Original file SHA-256: 84a79c74da8b18387a3d2d8a9928e4a437a608120b52a46c7ede6392cf1128ff Inclusive original lines: 1-200 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 | DIRECT CLOSURE OF THE OLD SEAM DEFECT 000002 | ===================================== 000003 | 25 October 2026 000004 | 000005 | Question and outcome 000006 | --------------------- 000007 | Can the supplied older circle determine the later primitive line directly, 000008 | instead of fitting a vector or requiring a heat to EM parameter map? 000009 | 000010 | YES on that circle: its polarization change is exactly one rank-one line. 000011 | With the already specified full CAR/time interface, that line is the excess 000012 | covariance of the original source winding. A directed relative Fredholm 000013 | determinant gives the exact single-line factor, without a heat limit. 000014 | The physical use of this determinant remains a source/response contract. 000015 | 000016 | The earlier SOURCE W1 implementer is REUSED, not claimed as a new result. 000017 | What is connected here is its actual mode action to the older fullv21 A3 000018 | operator and to the later minimal EM formula. Original papers are unchanged. 000019 | 000020 | 1. The original pair, with its explicit orientation repair 000021 | ----------------------------------------------------------- 000022 | fullv21, section001:230-251 supplies the twisted circle, i partial_theta, 000023 | u=exp(i theta), and displayed eigenvalues n+a. Those eigenvalues belong to 000024 | B=-i partial_theta, so fix B=-D_Gamma rather than mixing the printed signs. 000025 | For 0. (3) 000061 | 000062 | Pi acts on the ONE-PARTICLE space. The Fock density Pi_F=|Xi> Fock, V f=c(f)*Omega, 000066 | V*V=I_(H_+), V Pi V*=Pi_F, (4) 000067 | 000068 | with e^(itH)V=V e^(itB) on H_+. In particular V Dom(B|H_+) lies in Dom(H) 000069 | and HVf=VBf there. Thus the two rank-one traces can be transported when 000070 | this is the actual source map. 000071 | They are not literally identical operators on the same Hilbert space. 000072 | 000073 | The charged U need not be an observable preserving the vacuum sector. 000074 | For a gauge-invariant A, the full word U*A U is gauge invariant and maps 000075 | charge-zero back to charge-zero. Therefore no demand that U itself be a 000076 | neutral preparation follows from using (3). Its physical permission and 000077 | localization still have to be checked on the full word in the original net. 000078 | 000079 | 3. All equal-time CAR moments of this defect, without a new source model 000080 | ------------------------------------------------------------------------ 000081 | Both C and C+Pi are projections, so the known Wick rule applies. For any 000082 | finite normal-ordered word with k creators followed by k annihilators put 000083 | 000084 | M_ij=omega_C(c(f_i)*c(g_j)), 000085 | v_i w_j=omega_Xi(c(f_i)*c(g_j))-M_ij. 000086 | 000087 | The matrix change v w^T has rank at most one. Hence 000088 | 000089 | omega_Xi(c(f_1)*...c(f_k)*c(g_k)...c(g_1)) 000090 | =det(M+v w^T) 000091 | =det(M)+sum_(i,j)cof_ij(M) v_i w_j. (5) 000092 | 000093 | The cofactor formula also holds if M is singular; no inverse is needed. 000094 | Adjunction and CAR reduce other finite words to these. The entire finite 000095 | word response is fixed by the existing source and the line in (1). This 000096 | does not identify it with every field or interaction of S_adm. 000097 | 000098 | 4. The exact relative determinant 000099 | ---------------------------------- 000100 | C has infinite rank, so det_F(I-zC) alone is not defined. For z!=1 define 000101 | 000102 | A(z)=I-zC, A_u(z)=I-z(C+Pi). 000103 | 000104 | Because C Pi=0, 000105 | 000106 | A_u A^(-1)=I-zPi, 000107 | A A_u^(-1)=I+[z/(1-z)]Pi. (6) 000108 | 000109 | Both right sides are identity plus trace class. Their Fredholm determinants 000110 | are respectively 000111 | 000112 | D_rel(z)=1-z, D_rel(z)^(-1)=(1-z)^(-1). (7) 000113 | 000114 | This is an exact finite-rank RELATIVE change of infinite polarizations, 000115 | not a quotient of two separately defined ordinary Fredholm determinants. 000116 | The ordered pair fixes the sign of the logarithm. With the already supplied 000117 | carrier susceptibility b=5/4, the chosen inverse response is 000118 | 000119 | D_rel(z)^(-b)=(1-z)^(-5/4). (8) 000120 | 000121 | For |z|<1 choose the holomorphic logarithm with Log(1)=0; for real 000122 | 0<=z<1 this gives the positive branch in (8). 000123 | No heat time, ground-energy subtraction or isolated-mode approximation 000124 | enters (6)--(8). The minimal line of the later definition can be realized 000125 | canonically as Pi on this old representative. 000126 | 000127 | The original EM response must still be shown to be this ordered relative 000128 | response on its actual admissible image. A fermion occupancy partition has 000129 | Tr(z^N)=1+z, its supertrace is1-z, and a boson partition has1/(1-z). A 000130 | Berezin inverse-kernel deformation can give1-z with z as coupling instead. 000131 | These meanings cannot be substituted for one another merely by matching 000132 | the scalar factor. No claim of a new physical alpha prediction is made. 000133 | 000134 | 5. The older weight already has a topological contract 000135 | ------------------------------------------------------ 000136 | fullv21 uses a defect-sector action shift alpha*J and minimal per-sheet J1=1. 000137 | fullv27 section003:285-325 and fullv28 section004:17-59 explicitly identify 000138 | 000139 | J1=SF, J=d SF. 000140 | 000141 | With d=2 and primitive SF=1, their declared topological reweighting gives 000142 | 000143 | z=q(alpha_EM)=delta_top exp(-2alpha_EM). (9) 000144 | 000145 | Thus the DIRECT DEFECT route does not require q_heat=q_EM. The heat route 000146 | is a different comparison; the q4/q6 heat corrections remain correctly scoped 000147 | there. They do not automatically modify the minimal relative response (8). 000148 | 000149 | What the source sets is an index coupling, not a coupling to B energy, 000150 | particle number or net CAR charge. A neutral opposite pump would have 000151 | total charge0, excitation count2 and energy1 at the same twist. The original 000152 | adds two identical positively oriented SF contributions. A physical sheet 000153 | map must preserve the specified orientation rather than identify all these 000154 | quantities because they equal1 on a single forward transition. 000155 | 000156 | For a COMPLETE deformed functional an additional normalization issue is 000157 | explicit. Writing the two sectors as 000158 | 000159 | Z_s(alpha)=exp(-alpha*d*s) Z_s^0(alpha), 000160 | Z_0(alpha)=Z_0^0(alpha), 000161 | 000162 | gives the exact identity 000163 | 000164 | Z_s(alpha)/Z_0(alpha)=exp(-alpha*d*s) R(alpha), 000165 | R(alpha)=Z_s^0(alpha)/Z_0^0(alpha). (10) 000166 | 000167 | The original fixed-prefactor rule follows if R(alpha)=R(0). This holds for 000168 | the intended fixed-background, defect-only reweighting prescription. It is 000169 | not a consequence of SF alone when the rest of the full source also varies. 000170 | For a fixed measure, the decisive identity is 000171 | 000172 | partial_alpha log R 000173 | =-_s+_0=0. (11) 000174 | 000175 | An alpha-dependent measure adds its corresponding logarithmic score terms. 000176 | The elementary test Z_s=exp(-2alpha),Z_0=exp(-alpha) violates the unqualified 000177 | prefactor conclusion, but is not a counterexample to TFPT or to the intended 000178 | relative-action prescription. See REWEIGHTING-REVIEW.txt for its scope. 000179 | 000180 | 6. The smallest actual source test and its success criterion 000181 | ------------------------------------------------------------ 000182 | Reuse the original circle and the existing full CAR source. Produce an 000183 | isometry on the originally specified LOCAL/REDUCING image, not a demanded 000184 | global equivalence of all Hilbert spaces, such that 000185 | 000186 | T psi_n=f_n, 000187 | C_source f_n=1_(n<0) f_n, 000188 | u_source f_n=f_(n+1), 000189 | actual time, adjoints, reflection and local supports are intertwined. (12) 000190 | 000191 | Then Pi_source=T Pi T* is already fixed. If the minimal line is required 000192 | to lie in the physical admissible space, the test is 000193 | 000194 | P_adm Pi_source=Pi_source. (13) 000195 | 000196 | Commutation alone is insufficient: a line in ker(P_adm) commutes too. 000197 | If Pi_source acts in a larger field sector, compare the actual gauge-invariant 000198 | response word before applying the observable restriction. Do not require 000199 | every charged field to preserve the observable Hilbert space. 000200 |