TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE29-CHIRAL-WEIGHT-DICTIONARY.txt Original file SHA-256: 269d2272e43d610dc25a1f7c08143667331ba0184aa9e8ffaa0d4dc58f38f054 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 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | [OMITTED: non-scientific conversation or execution metadata] 000004 | [OMITTED: non-scientific conversation or execution metadata] 000005 | 000006 | VERDICT 000007 | The actual v252 positive internal grade is already proved to be the ORIGINAL 000008 | P2 even exterior module under the full global SM group. Reusing that result, 000009 | the opposite grade can be assigned explicitly to the odd exterior module, 000010 | with real signs. The resulting whole96 unitary intertwines the actual SM 000011 | action, hypercharge, gamma_F and J_F. Crucially, componentwise conjugation 000012 | becomes split coefficient conjugation, whereas J_F becomes compact spinor 000013 | charge conjugation. These two antiunitaries are different. 000014 | 000015 | [OMITTED: non-scientific conversation or execution metadata] 000016 | [OMITTED: non-scientific conversation or execution metadata] 000017 | [OMITTED: non-scientific conversation or execution metadata] 000018 | [OMITTED: non-scientific conversation or execution metadata] 000019 | 000020 | 1. PREVIOUS POSITIVE RESULT AND EXACT ORIGINAL INPUTS 000021 | 000022 | [OMITTED: non-scientific conversation or execution metadata] 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | [OMITTED: non-scientific conversation or execution metadata] 000025 | [OMITTED: non-scientific conversation or execution metadata] 000026 | [OMITTED: non-scientific conversation or execution metadata] 000027 | [OMITTED: non-scientific conversation or execution metadata] 000028 | [OMITTED: non-scientific conversation or execution metadata] 000029 | 000030 | Actual original verification/v252_full_finite_triple.py: 000031 | 63-83: labels and three generations; 86-116: actual full_D(Y,MR); 000032 | 119-121: gamma_F=+ on particle L/antiparticle R, - on the other slots; 000033 | 124-132: J_F=Kswap kappa, with kappa ordinary component conjugation; 000034 | 139-168: actual left bimodule representation pi. 000035 | Original SHA256: 000036 | 820254440e178430422e50605067775656236602996307f67c9e69124ba5d163. 000037 | The physical SM action compared is U(u)=pi(u) J_F pi(u) J_F^-1, 000038 | on the connected unimodular branch, not pi(u) alone. 000039 | 000040 | Original P2 source: 000041 | documents/archive--paper-latex--tfpt-v29/sections/ 000042 | 001-section-overview-and-claim-map.tex:166-197,200-223,236-282. 000043 | It gives Y5=diag(-1/3,-1/3,-1/3,1/2,1/2), the global group 000044 | S(U3 x U2), and its complete exterior-even decomposition. Its SHA256 is 000045 | 4869b6ba74ae2ecd5c316d99f97c63140f67090b52b79032c16962f1e7fab8e5. 000046 | 000047 | 2. REUSE THE POSITIVE-GRADE BASIS, WITH ITS ACTUAL SM ACTION 000048 | 000049 | Use ordered orthonormal basis (c1,c2,c3,w1,w2) of the carrier E. 000050 | Write vc=c1 c2 c3, vw=w1 w2 and 000051 | eta_c=(1/2) sum_jk epsilon_cjk c_j c_k, 000052 | where juxtaposition denotes exterior multiplication. In particular 000053 | eta_2=c3 c1, with its minus sign in increasing-index order. 000054 | At every fixed original generation the existing map W_+ is: 000055 | 000056 | actual positive slot exterior image hypercharge 000057 | nu_R^c 1 0 000058 | u_L,c, d_L,c c_c w1, c_c w2 1/6 000059 | u_R,c^c eta_c -2/3 000060 | e_R^c vw 1 000061 | d_R,c^c eta_c vw 1/3 000062 | nu_L, e_L vc w1, vc w2 -1/2 000063 | 000064 | For (x,z) in S(U3 x U2), the corresponding six complete actions are 000065 | 1; x tensor z; det(x) conjugate(x); det(z); 000066 | conjugate(x); det(x) z. 000067 | These are exactly the original real SM bimodule action and wedge actions. 000068 | Thus nonabelian generators, conjugate color, weak doublets and charges 000069 | all intertwine. They are not inferred from dimension or equal spectra. 000070 | 000071 | 3. EXPLICIT NEGATIVE GRADE AND THE TWO DIFFERENT CONJUGATIONS 000072 | 000073 | Let e_S be the increasing-index exterior basis of Lambda C5. Define the 000074 | antilinear Hodge complement by 000075 | star e_S=epsilon(S,S^c) e_(S^c), 000076 | where epsilon(S,S^c) is the sign of the concatenated permutation to 000077 | (1,2,3,4,5). Let P act on degree k by (-1)^(k(k-1)/2), and set 000078 | C_comp=P star. 000079 | The degree signs of P are (+,+,-,-,+,+). The operators obey 000080 | C_comp^2=1, C_comp (-1)^N=-(-1)^N C_comp. 000081 | Define the missing basis by the actual original J_F: 000082 | W_-(J_F v)=C_comp W_+(v), for every positive-grade v. 000083 | Both sides are antilinear in v, so this defines a LINEAR W_-. 000084 | Its explicit values are: 000085 | 000086 | actual negative slot exterior image hypercharge 000087 | nu_R vc vw 0 000088 | u_R,c -c_c vw 2/3 000089 | e_R -vc -1 000090 | u_L,c^c -eta_c w2 -1/6 000091 | d_L,c^c +eta_c w1 -1/6 000092 | d_R,c c_c -1/3 000093 | nu_L^c w2 1/2 000094 | e_L^c -w1 1/2 000095 | 000096 | These are sixteen distinct signed odd basis vectors. Put W=W_+ directsum 000097 | W_- at each of the three original generation labels. Then W is a real 000098 | signed unitary 96x96 matrix and exactly 000099 | W gamma_F W^-1=(-1)^N, 000100 | W J_F W^-1=C_comp tensor I_generation, 000101 | W kappa W^-1=kappa_split, 000102 | where kappa_split conjugates coefficients of these real exterior vectors. 000103 | The sign convention is explicitly specified. The original SM action on 000104 | the negative grade is the conjugate action, since J_F commutes with U(u). 000105 | The table therefore also intertwines the complete negative SM action. 000106 | 000107 | [OMITTED: non-scientific conversation or execution metadata] 000108 | [OMITTED: non-scientific conversation or execution metadata] 000109 | [OMITTED: non-scientific conversation or execution metadata] 000110 | [OMITTED: non-scientific conversation or execution metadata] 000111 | [OMITTED: non-scientific conversation or execution metadata] 000112 | [OMITTED: non-scientific conversation or execution metadata] 000113 | [OMITTED: non-scientific conversation or execution metadata] 000114 | [OMITTED: non-scientific conversation or execution metadata] 000115 | [OMITTED: non-scientific conversation or execution metadata] 000116 | [OMITTED: non-scientific conversation or execution metadata] 000117 | 000118 | 4. SIGNED E8 WEIGHTS AND GLOBAL DETERMINANT FACTORS 000119 | 000120 | The already established signed root dictionary is 000121 | original-admissible-source-closure-20261025/ 000122 | JOINT-LIFT-NATIVE-CLOCK-CONNECTION.txt:37-86. 000123 | For S even, epsilon_i=+ on S and - off S. Hence the D5 weight 000124 | epsilon_S/2 has ODD minus parity. Its actual family4 weight is 000125 | -sigma_j/2, sigma_j=(+++),(+--),(-+-),(--+). 000126 | The corresponding E8 half root (epsilon_S/2,-sigma_j/2) has even TOTAL 000127 | minus parity. Complementing S gives the conjugate half-spinor; the full 000128 | compact root conjugation also conjugates the family4. Hypercharge is 000129 | epsilon_S dot Y5/2=sum_(i in S)Y5_i because tr(Y5)=0. 000130 | This fixes the same charge and chirality orientation as the previous map. 000131 | 000132 | The displayed real signed table is a FIBRE dictionary in a specified 000133 | unitary carrier frame. It is not a new global trivialization. In the 000134 | original full U5/K lift let D=det E and retain every supplied power: 000135 | W_m=Lambda_even E tensor directsum_j D^m_j, 000136 | W_m^*=Lambda_odd E tensor directsum_j D^(-1-m_j). 000137 | The latter identity follows from the wedge pairing to det E. It is 000138 | essential whenever D is nontrivial. The Spin and family half-determinant 000139 | factors cancel exactly as in the September22 K construction; they are not 000140 | discarded by a local real matrix. A global source identification must 000141 | also carry the given connection, line cocycles and base-reflection map. 000142 | The S(U3 x U2) comparison above has det=1 and needs no extra D power. 000143 | 000144 | 5. ACTUAL ALL96 D_K RESTRICTION, WITH THE CORRECT FAMILY ORDER 000145 | 000146 | The differential frame fixes G order (L0,L2,L1), since its three columns 000147 | have V4 characters (epsilon0,epsilon2,epsilon1). See 000148 | ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt:81-84,116-138. 000149 | Thus the derived D_E,G is omega^2 S_G with S_G=diag(1,-1,1). 000150 | In the original q order (L0,L1,L2), it is 000151 | D_E,q=omega^2 diag(1,1,-1). 000152 | Reordering these lines must simultaneously reorder all given marks. 000153 | 000154 | In the G generation order, the proposed CHIRAL restriction is exactly: 000155 | particle nu_L,e_L,u_L,d_L: D_E,G; 000156 | particle nu_R,e_R,u_R,d_R: conjugate(D_E,G); 000157 | antiparticle nu_L^c,e_L^c,u_L^c,d_L^c: conjugate(D_E,G); 000158 | antiparticle nu_R^c,e_R^c,u_R^c,d_R^c: D_E,G. 000159 | Equivalently it is D_E on gamma_F=+ and its conjugate on gamma_F=-. 000160 | Under W these are the even/odd spinor family and conjugate-family actions. 000161 | This differs from the OLD PARTICLEWISE extension on both L and R particles. 000162 | 000163 | [OMITTED: non-scientific conversation or execution metadata] 000164 | [OMITTED: non-scientific conversation or execution metadata] 000165 | [OMITTED: non-scientific conversation or execution metadata] 000166 | [OMITTED: non-scientific conversation or execution metadata] 000167 | [OMITTED: non-scientific conversation or execution metadata] 000168 | [OMITTED: non-scientific conversation or execution metadata] 000169 | [OMITTED: non-scientific conversation or execution metadata] 000170 | [OMITTED: non-scientific conversation or execution metadata] 000171 | [OMITTED: non-scientific conversation or execution metadata] 000172 | [OMITTED: non-scientific conversation or execution metadata] 000173 | [OMITTED: non-scientific conversation or execution metadata] 000174 | [OMITTED: non-scientific conversation or execution metadata] 000175 | 000176 | 6. EXACT CHECK PERFORMED AND THE PRECISE GAIN 000177 | 000178 | [OMITTED: non-scientific conversation or execution metadata] 000179 | [OMITTED: non-scientific conversation or execution metadata] 000180 | [OMITTED: non-scientific conversation or execution metadata] 000181 | [OMITTED: non-scientific conversation or execution metadata] 000182 | [OMITTED: non-scientific conversation or execution metadata] 000183 | [OMITTED: non-scientific conversation or execution metadata] 000184 | [OMITTED: non-scientific conversation or execution metadata] 000185 | [OMITTED: non-scientific conversation or execution metadata] 000186 | [OMITTED: non-scientific conversation or execution metadata] 000187 | [OMITTED: non-scientific conversation or execution metadata] 000188 | [OMITTED: non-scientific conversation or execution metadata] 000189 | [OMITTED: non-scientific conversation or execution metadata] 000190 | [OMITTED: non-scientific conversation or execution metadata] 000191 | 000192 | This supplies the missing whole96 charge/grade/real-basis COMPARISON and 000193 | therefore the stated chiral restriction of a given K family action. The 000194 | unitary W induces the usual CAR star-isomorphism on this same internal 000195 | one-particle space and preserves every CAR product. Transporting the given 000196 | operator/state through that map introduces no new state or time choice. 000197 | 000198 | It does not alone certify all E8 Lie products of the physical source: 000199 | v252 implements the SM bimodule, not all E8 generators. The full E8 bundle 000200 | has a fourth family slot and the other K branches with their fixed bracket