TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-FUCHS-COEFFICIENT-READER-REVIEW.txt Original file SHA-256: e6e3afc9a14e29f25ba045a11d768c2322d4b0efe94d40afa5f176a5ca89512c Inclusive original lines: 1-158 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 FUCHS COEFFICIENT READER REVIEW 000002 | Independent narrow source review, 2026-10-30. No Atlas or original source changed. 000003 | 000004 | QUESTION 000005 | 000006 | The stored geometric Spin(2) whole-source arrow leaves a sharp final reader 000007 | question. The geometric rho line has c-character rho^2=omega. Total deck 000008 | descent therefore retains it only with a coefficient leg of inverse character 000009 | omega^2. Does the actual original v252/P2 Yukawa and Pi_triangle consumer 000010 | already contract precisely that coefficient channel? 000011 | 000012 | VERDICT 000013 | 000014 | No such contraction is present in the checked original reader. The exact 000015 | Fuchs matrix does contain the required omega^2 coefficient eigenspace, but it 000016 | contains all three Pc eigenspaces. Its deck matrix Pc is a cyclic permutation 000017 | with spectrum {1,omega,omega^2} and trace zero. It is not the native scalar 000018 | family centre Z_F=omega^2 I3. The original open hard holonomies use the full 000019 | transport matrix, and the closed phase consumer is exactly 000020 | 000021 | Pi_triangle=Hol_12 Hol_23 Hol_31, 000022 | delta_CKM=arg tr Pi_triangle. 000023 | 000024 | Neither formula inserts the rank-one omega^2 coefficient projector. The 000025 | positive K_f^str kernel also does not provide it. Thus the geometric rho line 000026 | and the original CKM reader are not yet connected by an actual source 000027 | operator. This is a scoped statement about the displayed original reader, 000028 | not an exclusion of a relative phase in a larger open-leg evaluation. 000029 | 000030 | 1. ACTUAL ORIGINAL CONSUMER 000031 | 000032 | The primary source formula is 000033 | 000034 | (Y_f)_ij=, 000035 | 000036 | and, after diagonal reduction, 000037 | 000038 | (Y_f)_ij=lambda_Y^(L_i^L+L_j^R) 000039 | . 000040 | 000041 | The phase consumer then closes three full holonomies and takes their ordinary 000042 | trace. These are Hom(F,F) operators. No geometric coefficient line, inverse 000043 | character evaluation, or one-sided coefficient projector occurs in the 000044 | displayed operator formula. 000045 | 000046 | The structural kernel is 000047 | 000048 | K_f^str=(P_str D_f^dagger D_f P_str)^(-1/2), 000049 | 000050 | with D_f=y I-delta U6. It is a positive, orientation-even function of the 000051 | hexagonal transport. It cannot by itself supply the oriented one-dimensional 000052 | Pc character required below. 000053 | 000054 | 2. EXACT FUCHS COEFFICIENT ACTION 000055 | 000056 | The actual algebraic fundamental matrix satisfies 000057 | 000058 | G(c y)=G(y) Pc, 000059 | Pc=[[0,0,1],[1,0,0],[0,1,0]], 000060 | Pc^3=I. 000061 | 000062 | Thus Pc is a right action on the coefficient columns. For ordinary open 000063 | transport on a common lifted sheet, 000064 | 000065 | H_ij=G(y_i)G(y_j)^(-1), 000066 | 000067 | the simultaneous right action cancels: 000068 | 000069 | (G_i Pc)(G_j Pc)^(-1)=G_i G_j^(-1). 000070 | 000071 | For a closed path carrying deck c, the normalized monodromy is instead 000072 | 000073 | M_c=G_base Pc G_base^(-1). 000074 | 000075 | It has the same character as Pc. In particular 000076 | 000077 | spec(Pc)={1,omega,omega^2}, tr(Pc)=tr(M_c)=0. 000078 | 000079 | The exact projector onto the needed inverse-character line is 000080 | 000081 | P_(omega^2)=(I+omega Pc+omega^2 Pc^2)/3, 000082 | Pc P_(omega^2)=omega^2 P_(omega^2), 000083 | 000084 | and has rank one. This projector is absent from the original Y_f and 000085 | Pi_triangle formulas. Even under the strongest direct identification of the 000086 | canonical family basis with the three coefficient slots, the displayed e_i 000087 | are not individual Pc eigenvectors. Their use therefore does not silently 000088 | perform this projection. 000089 | 000090 | 3. SMALLEST DECISIVE COUNTEREXAMPLE 000091 | 000092 | The tempting identification is 000093 | 000094 | Pc ?= Z_F=omega^2 I3. 000095 | 000096 | It fails before any dynamics or state is considered: 000097 | 000098 | tr(Pc)=0, tr(Z_F)=3 omega^2, 000099 | minpoly(Pc)=x^3-1, minpoly(Z_F)=x-omega^2. 000100 | 000101 | Hence Pc and Z_F are not conjugate. Replacing one by the other changes the 000102 | actual closed consumer. Moreover, in the Hom(F,F) Yukawa action the scalar 000103 | centre cancels by conjugation, 000104 | 000105 | Z_F H Z_F^dagger=H. 000106 | 000107 | It becomes observable only if the source places it on a single open leg before 000108 | the final evaluation. No such one-sided insertion occurs in the checked 000109 | Pi_triangle definition. 000110 | 000111 | This also separates two labels that must not be merged. The q/p labels of the 000112 | geometric Kummer cover are source-sheet labels. The quark/lepton split in the 000113 | positive P2 module is a physical carrier grading. The established target 000114 | operator Z_F S_col has the desired rho/rho^4 eigenvalues, but that fact does not 000115 | define an intertwiner from the Fuchs coefficient line to the P2 family leg. 000116 | 000117 | 4. FIRST MISSING TYPED ARROW 000118 | 000119 | The smallest remaining source statement is an open-leg coefficient reader, 000120 | for example an intertwiner J_coeff with 000121 | 000122 | J_coeff : im P_(omega^2)(Pc) -> H_family, 000123 | J_coeff Pc=Z_F J_coeff 000124 | 000125 | up to the inverse convention chosen for the geometric leg. It must occur 000126 | before the Hom(F,F) closure and trace. The existing geometric whole-source 000127 | map proves that this is the correct inverse character; it does not provide 000128 | J_coeff. The checked original v252/P2/Yukawa sources likewise do not display 000129 | it. 000130 | 000131 | This is the first missing operator, not a request to introduce a new rho line, 000132 | a second p source, a freely chosen state, or another lift. A future positive 000133 | identification must point to an original insertion or evaluation leg that 000134 | satisfies this intertwining equation in the actual source basis. 000135 | 000136 | SOURCES CHECKED 000137 | 000138 | Original source: 000139 | documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/ 000140 | 003-subsection-discrete-phase-operator-and-transport-grammar.tex:3-13, 000141 | 80-84,102-117 000142 | 004-source.tex:185-197,234-239 000143 | 005-source.tex:252-281 000144 | 000145 | Exact current contracts: 000146 | aps_source_review/ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt 000147 | SHA256 3a57907902631b58e0abee61e9a424c3ac03ff7a8fa074cc4d27e3379b406cf3 000148 | aps_source_review/GEOMETRIC-R-SPIN-COLLAR-REVIEW.txt 000149 | SHA256 f41d9aa53c27d8e167ec82e7692f9367d073b287a157869a2a39a30114dfe164 000150 | aps_source_review/ORIGINAL-U6-TRIALITY-SECTOR-SELECTION-REVIEW.txt 000151 | SHA256 958c55a73b64fc2a35eec07d5242a497a18cefcf28d08acedb5f955b6ef7df4a 000152 | aps_source_review/HEXAGONAL-OPEN-SOURCE-PHASE-REVIEW.txt 000153 | SHA256 34e897a08c91ad4110f56baaceac82745640a17b675e5a70a1619a105f5f8b8c 000154 | aps_source_review/SPIN-ZERO-MODE-C12-CLOCK-TYPE-REVIEW.txt 000155 | SHA256 5d5f6bfb960f20d004bf5442920e09826fe9f963b5ac888f3f45c3a4a568edbf 000156 | 000157 | The Atlas was read at revision 3ee3dd60e91640c3fa79a567. Nothing was written 000158 | to it.