TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/EM-SOURCE-VARIATION-FORWARD.txt Original file SHA-256: 6f9d1676b4b2e9a77193205dde43c1334c336a40bdff3d869caa3b710fc297ee Inclusive original lines: 1-194 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 | ORIGINAL EM SOURCE VARIATION: FORWARD REVIEW 000002 | ============================================ 000003 | 000004 | 25 October 2026. Read only source review followed by this report. No Atlas, 000005 | index, paper or existing contract was changed. 000006 | 000007 | Question 000008 | -------- 000009 | 000010 | The faithful one sided circle KMS calculation gives the exact relative 000011 | Fredholm determinant 000012 | 000013 | det_F[(I-q u C_beta u*)(I-q C_beta)^(-1)] = 1-q. (1) 000014 | 000015 | This report asks whether the original electromagnetic construction already 000016 | contains an operator family which turns this directed covariance response into 000017 | the source variation q(alpha)=delta_top exp(-2 alpha), while avoiding the 000018 | spectral invariance of a fully transported source. It also checks the proposed 000019 | D5 spinor conformal block explanation of the exponent 5/4 against the actual 000020 | original readout. 000021 | 000022 | Verdict 000023 | ------- 000024 | 000025 | The sources contain genuine variations which are not merely unitary changes 000026 | of frame, but they do not yet identify one common operator family 000027 | 000028 | alpha -> (D_alpha, domain(D_alpha), state_alpha, reference_alpha) (2) 000029 | 000030 | whose relative determinant is the electromagnetic q(alpha) and whose 000031 | variation gives the stated seam Ward functional. 000032 | 000033 | Three different original structures must be kept distinct. 000034 | 000035 | 1. The twist torus family H_L(b), and the demanded continuum family D_b^+, 000036 | is genuinely nontrivial. Its determinant line can have nonzero first 000037 | Chern class and therefore cannot be one globally unitary transported 000038 | operator with transported domain and frame. Its parameter is the flat 000039 | connection b, however. The sources give no canonical map alpha -> b and 000040 | no theorem equating its relative determinant with 1-delta_top exp(-2alpha). 000041 | 000042 | 2. The historical modulus sigma changes the Maxwell kinetic function 000043 | f(sigma), with alpha(sigma)=1/(4 pi f(sigma)). This is a genuine coupling 000044 | variation. It is explicitly introduced as an axiom. The inspected text 000045 | does not define D_sigma or D_alpha, their domains, their boundary state, 000046 | or their relation to the circle covariance and to (1). 000047 | 000048 | 3. The old defect reweighting adds the relative action alpha d SF. For 000049 | d=2 and SF=1 it gives exp(-2alpha) exactly only when the residual defect to 000050 | trivial partition ratio is alpha independent. The displayed old lemma 000051 | does not prove this residual Ward identity for the complete same source. 000052 | 000053 | The proposed carrier spinor reading gives an exact and useful compression: 000054 | the D5 spinor primary has h=5/8, hence its normalized radial two point block 000055 | has exponent 2h=5/4. The glued E8 current decomposes into D5 spinor times A3 000056 | fundamental, with weights 5/8 and 3/8, so the full exponent is 2 and the A3 000057 | factor has exponent 3/4. Dividing or conditioning on the A3 factor leaves the 000058 | D5 exponent 5/4. This identity is already realized more strongly in the 000059 | existing charged sector response contract. What is not present in the 000060 | original EM definition is the source map which says that tr_adm performs this 000061 | D5 channel readout or this A3 conditional quotient. Consequently the equality 000062 | 000063 | B gamma = 5/4 = 2 h_(D5 spinor) (3) 000064 | 000065 | is exact, but its right hand side is not yet the original definition of the 000066 | left hand side. In the original carrier text B gamma is derived instead as 000067 | (3/2)(5/6), and tr_adm is applied to a separately declared rank one primitive 000068 | seam projector. 000069 | 000070 | 1. The known unitary invariance already excludes the naive identification 000071 | ------------------------------------------------------------------------- 000072 | 000073 | For a fully transported source 000074 | 000075 | C' = u C u*, h(C') = u h(C) u*, 000076 | D' = u D u*, domain(D') = u domain(D), (4) 000077 | 000078 | every ordinary unitary invariant spectral functional is unchanged. In 000079 | particular, whenever the relevant regularisation is defined and unitary 000080 | covariant, 000081 | 000082 | Tr f(h(C')^2) = Tr f(h(C)^2), 000083 | ||det D'||_Q = ||det D||_Q. (5) 000084 | 000085 | This is not a new observation. P1-COVARIANT-INSERTION.txt:67-88 already proves 000086 | the stronger source statement: transport B, D, algebra, state, J, grading, 000087 | reflection, domains and time together, and spectra, chi, eta, invariant defect 000088 | values and correlation functions remain unchanged. Its equations (1) and (2), 000089 | at lines 90-135, remove tangential frame motion from the covariant source jet. 000090 | 000091 | Therefore (1) is a directed relative covariance determinant with the old 000092 | reference C held in the comparison. It cannot at the same time be read as an 000093 | ordinary spectral action or positive Quillen norm of the completely 000094 | transported pair (4). The shift u is a source winding operator in the proved 000095 | circle construction. No inspected original identifies it by itself as the 000096 | physical electromagnetic gauge variation. 000097 | 000098 | 2. The nontrivial determinant line is real, but its alpha pullback is absent 000099 | --------------------------------------------------------------------------- 000100 | 000101 | The strongest original operator family is recorded in 000102 | documents/tfpt_research_contracts/sections/020-source.tex:179-231. 000103 | 000104 | On the finite side it specifies a smooth family of gapped collar Hamiltonians 000105 | H_L(b), occupied projectors P_L(b), gauge identifications and the Berry 000106 | determinant line over the twist torus B of flat U(1) connections. Its nonzero 000107 | Chern number rules out a single global frame in which every member is just 000108 | unitarily transported with a globally chosen ground state vector. 000109 | 000110 | On the continuum side, however, lines 207-230 say to define independently a 000111 | chiral Fredholm family D_b^+, including domains, coupling data and gauge 000112 | descent, and then construct its comparison with the finite occupied line. 000113 | These are acceptance requirements, not an already supplied D_b^+. 000114 | Lines 232-269 likewise state as the open T2 target that the exact Bismut Freed 000115 | variation equal the electromagnetic Ward expression. The same section keeps 000116 | the continuum comparison and ALPHA.QUILLEN.EXACT.01 open. 000117 | 000118 | This family is the correct kind of object to evade (5), since flat holonomy 000119 | and moving domains over a nontrivial line bundle are physical source data, not 000120 | only a moving basis. The missing first map is 000121 | 000122 | alpha -> b(alpha) in B, 000123 | together with D_alpha := D^+_(b(alpha)) (6) 000124 | 000125 | and a proof that the pullback connection, norm and reference subtraction give 000126 | the stated alpha response. Choosing (6) now would be an additional coupling 000127 | and path choice. The twist torus construction does not select it. 000128 | 000129 | 3. The historical kinetic modulus is a genuine added coupling 000130 | -------------------------------------------------------------- 000131 | 000132 | The older sources do contain a variation which is plainly not of the form 000133 | (4). The fullv21 source at lines 363-387, repeated in fullv27 at lines 60-85, 000134 | introduces 000135 | 000136 | L contains -(1/4) f(sigma) F_mu_nu F^mu_nu 000137 | +(1/2)(partial sigma)^2, 000138 | alpha(sigma)=1/(4 pi f(sigma)), 000139 | delta Gamma_eff/delta sigma = 0. (7) 000140 | 000141 | Changing sigma changes the kinetic coefficient and hence the action. This can 000142 | overcome unitary spectral invariance. But (7) is labelled an axiom and does not 000143 | define the boundary operator family, its collar domain, its state, or an 000144 | operator identity relating partial_sigma D to the charged current. Using (7) 000145 | as the desired bridge therefore introduces f(sigma) and its source coupling as 000146 | the first extra physical input. 000147 | 000148 | The later master package does not fill in that input. In 000149 | documents/archive--paper-latex--tfpt-42/sections/ 000150 | 002-section-big-picture-what-tfpt-actually-claims.tex:70-110 it writes 000151 | 000152 | S_TFPT = Tr f((D_rel+A_Sigma^seed)/chi_seed) 000153 | -Tr f(D_ref/chi_seed) + i pi Delta eta/2, 000154 | A_Sigma^seed = B_rel direct_sum nabla_(chi_seed), (8) 000155 | 000156 | and separately imposes stationarity in alpha. It does not display 000157 | A_Sigma^seed(alpha), partial_alpha D, or a current insertion generating the 000158 | alpha dependence. Thus (8) is a package in which such a family could live, not 000159 | the required forward identity. 000160 | 000161 | 4. The defect action gives the exponential only with its residual Ward law 000162 | ---------------------------------------------------------------------------- 000163 | 000164 | The original reweighting statements use 000165 | 000166 | S_def(alpha)=S_def(0)+alpha d SF, 000167 | d=2, SF=1. (9) 000168 | 000169 | For complete sector partition functions the exact algebraic consequence is 000170 | 000171 | q(alpha)=exp(-2alpha) R(alpha), (10) 000172 | 000173 | where R is the ratio after removing the displayed topological term. The target 000174 | q(alpha)=q(0)exp(-2alpha) is equivalent to R(alpha)=R(0), or under suitable 000175 | differentiability to partial_alpha log R=0. This exact distinction and the 000176 | measure score terms are proved in REWEIGHTING-REVIEW.txt:40-149. The inspected 000177 | historical lemma asserts the intended relative normalisation but does not 000178 | derive that Ward identity for all remaining actions, measures and determinant 000179 | factors of the same source. 000180 | 000181 | Equation (1) cannot supply the missing cancellation by itself. Substituting 000182 | q=delta_top exp(-2alpha), or setting its determinant variable equal to that 000183 | expression, names the desired source dependence but does not derive (9) or the 000184 | constancy of R. 000185 | 000186 | 5. The carrier spinor block: exact identity and exact selection boundary 000187 | ------------------------------------------------------------------------ 000188 | 000189 | The conformal embedding data are exact. The current research contract records 000190 | 000191 | h_D5 = (0,5/8,1/2,5/8), 000192 | h_A3 = (0,3/8,1/2,3/8), (11) 000193 | 000194 | and the diagonal glue has integer weights (0,1,1,1). The simple current