TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE48-PAULI-PH-FACE-BLOCK-REVIEW.txt Original file SHA-256: ebd4b08713fd0de18f58c41d4a330b7a122544558f8469a114c2d063fe9ee92f Inclusive original lines: 1-143 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 | 000004 | VERDICT 000005 | 000006 | The graph-face calculation is correct for the already given Phi=0 NS source. 000007 | It canonically defines a nonzero finite particle-hole contribution with the 000008 | claimed family trace and the exact phase -pi/3. Two corrections are required. 000009 | 000010 | [OMITTED: non-scientific conversation or execution metadata] 000011 | [OMITTED: non-scientific conversation or execution metadata] 000012 | [OMITTED: non-scientific conversation or execution metadata] 000013 | [OMITTED: non-scientific conversation or execution metadata] 000014 | [OMITTED: non-scientific conversation or execution metadata] 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 | 000019 | 1. ACTUAL TWO-FACE BLOCK 000020 | 000021 | Write, on the h=1/2 normal mode, 000022 | 000023 | V=(sqrt(a),sqrt(1-a))^t, 000024 | V_perp=(-sqrt(1-a),sqrt(a))^t, 000025 | a=(1+exp(-ell_NS))^-1. 000026 | 000027 | These are isometries with orthogonal ranges G and 1-G. Retaining the actual 000028 | J=Swap tensor j and the same two-face insertion used by the established Hardy 000029 | reader gives the exact face factors 000030 | 000031 | V* Swap V =2sqrt(a(1-a))=sech(ell_NS/2), 000032 | V_perp* Swap V =2a-1=tanh(ell_NS/2), 000033 | V* Swap V_perp =2a-1=tanh(ell_NS/2), 000034 | V_perp* Swap V_perp =-2sqrt(a(1-a)). 000035 | 000036 | Thus E^h=V_perp E0 is a canonical finite subspace of the actual unoccupied 000037 | graph range, and both particle-hole directions have the same positive graph 000038 | factor. Relative to the old particle-particle response, 000039 | 000040 | M_ph=sinh(ell_NS/2) M_pp. 000041 | 000042 | Since M_pp=(2/pi)sech(ell_NS/2) m tensor P_gamma,+, this becomes 000043 | 000044 | M_ph=M_hp=(2/pi)tanh(ell_NS/2) m tensor P_gamma,+. (1) 000045 | 000046 | Nothing in this derivation replaces j by a naked two-dimensional Swap. The 000047 | old spatial R_x, F_R, spin, normal reduction and P_gamma,+ factor remain in m 000048 | and in the already evaluated periods. Only the graph-face vector changed. 000049 | 000050 | 2. PAULI STATEMENT 000051 | 000052 | Let Pi=EE* and compress K=J* M_mu to Pi K Pi. Pi lies in the filled range of 000053 | the given pure graph projector. On the completely filled finite defect wedge, 000054 | 000055 | dGamma(Pi K Pi) Omega=Tr(Pi K Pi) Omega. 000056 | 000057 | For each of the six matrices F or N the family trace vanishes, so the actual 000058 | compressed particle-particle operator annihilates this vector. This is the 000059 | correct narrow Pauli statement. It does not make the whole original K_mu 000060 | invisible: the particle-hole blocks (1) are nonzero, and contributions outside 000061 | Pi and Pi_h have not been removed from the original field. 000062 | 000063 | 3. FINITE CONNECTED WICK BLOCK AND THE MISSING CARRIER TRACE 000064 | 000065 | For the finite Pi/Pi_h part of the standard ordered bilinear covariance, 000066 | 000067 | Tr[G K_mu (1-G) K_nu], 000068 | 000069 | equation (1) gives, per retained P_gamma,+ carrier component, 000070 | 000071 | C_family(mu,nu)=(2/pi)^2 tanh^2(ell_NS/2) 000072 | Tr_family(m_mu m_nu). (2) 000073 | 000074 | The actual v252 grading is 96-dimensional. P_gamma,+ has rank48 on all three 000075 | families, hence rank16 on the 32-dimensional carrier attached to one family. 000076 | With the ordinary unnormalized one-particle trace, the full finite block is 000077 | 000078 | C_full(mu,nu)=16 C_family(mu,nu). (3) 000079 | 000080 | A stated selfdual second-quantization convention may add its global one-half 000081 | normalization. It does not turn the carrier trace into one. Therefore (2) is 000082 | not the whole 96-dimensional trace unless a normalized carrier trace or one 000083 | fixed carrier component is explicitly declared. 000084 | 000085 | 4. EXACT FAMILY PAIRING AND PHASE 000086 | 000087 | For 000088 | 000089 | m_F=I02 E02+I21 E21, 000090 | m_N=-i(I02 E12+I21 E20), 000091 | 000092 | direct multiplication gives 000093 | 000094 | Tr(m_F^2)=Tr(m_N^2)=0, 000095 | Tr(m_F m_N)=Tr(m_N m_F)=-2i I02 I21. 000096 | 000097 | With I02=C L, I21=-C J and positive real L,J, 000098 | 000099 | -2i I02 I21=2i C^2 L J, 000100 | arg(2i C^2)=-pi/3. 000101 | 000102 | The carrier factor in (3) is positive real, so it does not change this phase. 000103 | The common-frame trace is also invariant under a simultaneous family basis 000104 | change. Hence -pi/3 is an exact phase of this directed finite complex channel 000105 | component, not a numerical fit. 000106 | 000107 | The original finite source is selfdual. A physical selfadjoint field includes 000108 | the ordinary adjoint and the actual J_F image, with P_gamma,- on the conjugate 000109 | slot. Their complete correlation is a sum of directed components and is real 000110 | for a selfadjoint observable. The single complex phase above must therefore 000111 | not be promoted by itself to a CP observable, a CKM phase or the phase of the 000112 | complete KO-real field. 000113 | 000114 | 5. SOURCE SCOPE 000115 | 000116 | Pi_h=E^h E^h* is canonically defined once the original E0 and pure graph G 000117 | are fixed. Selecting the Pi/Pi_h summand of the Wick trace is nevertheless a 000118 | finite compression of the whole source correlation. The checked calculation 000119 | does not prove that the original Z, sewing operation or second source 000120 | variation singles out precisely this summand, nor that it is the renormalized 000121 | regular part of the uncompressed local kernel. No new state, time or cutoff is 000122 | needed to define the summand, but its use as the complete original response 000123 | remains a separate source statement. 000124 | 000125 | REPRODUCIBILITY 000126 | 000127 | [OMITTED: non-scientific conversation or execution metadata] 000128 | [OMITTED: non-scientific conversation or execution metadata] 000129 | [OMITTED: non-scientific conversation or execution metadata] 000130 | [OMITTED: non-scientific conversation or execution metadata] 000131 | 000132 | Normal Python and python -OO produced byte-identical stdout and JSON. The 000133 | checker imports the actual v252 grading, verifies the 96/48/16 dimensions, 000134 | all four face blocks and the four family traces exactly. 000135 | 000136 | SOURCE ANCHORS 000137 | 000138 | INDUCED-HARDY-DEFECT-BMU-FINITE-READER.txt:45-71,87-108; 000139 | HARDY-BMU-FIRST-NONZERO-ENTRY.txt:21-30,77-90; 000140 | HARDY-BMU-FULL-MU01-READER.txt:104-118; 000141 | HARDY-BMU-KO-AND-FOUR-SECTOR-COMPLETION.txt:13-79,81-143; 000142 | GEOMETRIC-R-SPIN-COLLAR-REVIEW.txt:114-136; 000143 | verification/v252_full_finite_triple.py:72-78,119-132.