TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/NATIVE-SPIN2-ANNULUS-PULLBACK-REVIEW.txt Original file SHA-256: 2bdb013daad58a696abf36d62a6d8c9d4e7f314cc167013502e38a3f3298f6ed Inclusive original lines: 1-164 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 | THE NATIVE SPIN(2) ANNULUS LIFT AND THE ACTUAL S0 PHASE 000002 | Independent constructive operator/half-form review, 2026-10-30. 000003 | No new metric, cover, source, J class or change to pinned reports. 000004 | 000005 | VERDICT 000006 | 000007 | [OMITTED: non-scientific conversation or execution metadata] 000008 | [OMITTED: non-scientific conversation or execution metadata] 000009 | 000010 | Theta_spin=P_r Rx tensor(i_old Z) tensor I96. 000011 | 000012 | It is the genuine Spin(2) pi-rotation lift, with square -I. It transports 000013 | the whole D_coll to its natural p image with Phi_p=Phi_q composed with 000014 | inversion, without finite conjugation. It is NEW-linear, preserves the 000015 | oriented C_faces, and reverses the written circle parameter relative to 000016 | the target h_p. The existing source is transported; minus-time is not 000017 | an exclusion and no new time choice is required. 000018 | 000019 | The key geometric phase comparison is also exact. Reusing the ALREADY 000020 | PROVED canonical ramified base spin pullback, the actual S0 lift becomes 000021 | +/-i/z in the pulled sqrt(dz) frame, hence +/-i in the logarithmic NS 000022 | frame. Its local nonvanishing fiber phase at the cubic ramification 000023 | points is instead +/-exp(i pi/6), precisely because of the ramification 000024 | half-divisor. Thus the normal Spin(2) phase and the original curve spin 000025 | phase are connected by an explicit map, not equated from their powers. 000026 | 000027 | 1. SPIN(2) FROM THE ACTUAL PRINCIPAL SYMBOL 000028 | 000029 | Use D=N partial_r+S A+Z Phi, A=-i_old partial_x+a, 000030 | N=[[0,-1],[1,0]], Z=diag(1,-1), S=NZ. 000031 | Its two oriented Clifford matrices are N and -i_old S. Both square to 000032 | -I and anticommute. Their pi-rotation lift is exactly 000033 | 000034 | exp((pi/2) N(-i_old S))=N(-i_old S)=i_old Z. 000035 | 000036 | There are the usual two opposite spin lifts. No finite K_F or gamma_F 000037 | factor is required by this principal symbol. With P_r reversing r and 000038 | Rx reversing x, put A_p=-Rx A_q Rx^-1 and Phi_p=Phi_q composed with 000039 | inversion. Then exactly 000040 | 000041 | Theta_spin D_q Theta_spin^-1=D_p, 000042 | Theta_spin B_q Theta_spin^-1=-B_p, 000043 | Theta_spin^2=-I in the constant spin chart. 000044 | 000045 | The plus Phi sign follows because iZ commutes with Z and acts trivially 000046 | on the finite coefficient. It applies to the entire actual96 Phi, 000047 | including the Yukawa and Majorana blocks; no reality assumption or 000048 | new MR formula is used. General variable-frame formulas include the 000049 | actual pullback connection and its derivative terms as usual. 000050 | 000051 | 2. THE WHOLE F5 STRUCTURES, CHARGED CHART AND TIME 000052 | 000053 | For Gamma=Z tensor gamma_F, Ical=i_old Gamma and C0=i_old N, 000054 | Theta_spin commutes with Gamma/Ical and reverses C0. The face swap 000055 | reverses the sign in C_faces=C0 directsum(-C0), so the boundary lift 000056 | preserves C_faces. The full graph obeys 000057 | 000058 | Theta_faces G(B_q) Theta_faces^-1=G(B_p). 000059 | 000060 | Both crossing and diagonal blocks are included. The map is NEW-linear. 000061 | In the actual charged chart E v=(P_+v,i_old P_-v), its pullback is 000062 | 000063 | T_spin=i_old gamma_F Rx, 000064 | Theta_spin E=E T_spin, 000065 | T_spin h_q T_spin^-1=-h_p. 000066 | 000067 | Consequently the already given group transports by 000068 | 000069 | Theta_spin exp(t Ical B_q) Theta_spin^-1=exp(-t Ical B_p), 000070 | T_spin exp(i_old t h_q) T_spin^-1=exp(-i_old t h_p). 000071 | 000072 | This is a valid explicit source/time transport with reversed parameter 000073 | relative to the displayed target generator. It is different from the 000074 | plus-time K_F transport. The original functorial Seed/inner-fluctuation 000075 | rule supplies the target whole datum; no independent Phi_p is needed. 000076 | 000077 | The actual scalar clutch u=exp(ix) is retained in the constant smooth 000078 | chart because the two geometric inversions cancel. In a global frame 000079 | the precise formula is, as before, 000080 | 000081 | u_p(x)=L(ell,x)u_q(-x)^-1 L(0,x)^-1. 000082 | 000083 | The actual spin factors and domain are transported with this formula; 000084 | an arbitrary unchanged-domain assertion is not substituted for it. 000085 | 000086 | 3. REUSE THE EXISTING CANONICAL RAMIFIED PULLBACK 000087 | 000088 | The OLD CANONICAL-BRANCHED-SPIN-PULLBACK.txt and finding 000089 | source-spin-is-canonical-ramified-base-spin-pullback already prove 000090 | 000091 | pi:C->P1_z, w^3=(z^2-1)/(z^2+1), eta=z(1-w^3), 000092 | S0=pi*O_P1(-1) tensor O_C(R_pi/2), 000093 | e_z=sqrt(3) w/eta Phi, e_z^2=pi*dz. 000094 | 000095 | Here Phi^2=u dw, u=(w^3+1)/eta. The four ramification indices are three; 000096 | the half-ramification divisor consists of the two w=0 points and the 000097 | two points at infinity. This existing result is reused, not claimed as 000098 | a new spin selection. 000099 | 000100 | The actual positive hexagonal operator is 000101 | R(w,eta)=(rho w,eta), R*z=1/z. Its selected spin lift has 000102 | 000103 | Rhat Phi=lambda Phi/u, lambda^2=rho, 000104 | lambda=+/-exp(i pi/6). 000105 | 000106 | Direct substitution into the OLD exact frame now gives the new comparison 000107 | 000108 | Rhat e_z=(rho lambda/u)e_z=(rho lambda/z)e_z=(+/-i/z)e_z. 000109 | 000110 | The identity 1/u=1/z uses eta^2=(1-w^3)(1+w^3). The factor rho lambda 000111 | is exactly +/-i. This is the square-root derivative of z->1/z: 000112 | (+/-i/z)^2=-1/z^2. It is the actual old S0 lift in a pulled base frame. 000113 | 000114 | 4. NS FRAME AND THE RAMIFICATION FIBER CHECK 000115 | 000116 | On a consistent logarithmic seam chart write e_NS=e_z/sqrt(z), so that 000117 | e_NS^2=pi*(dz/z). Using sqrt(1/z)=1/sqrt(z) on the same local branch, 000118 | 000119 | Rhat e_NS=+/-i e_NS. 000120 | 000121 | The transition sign of sqrt(z) is the original NS spin transition; it 000122 | cannot be dropped globally by declaring a periodic scalar frame. The 000123 | displayed phase is precisely the positive or negative chiral Spin(2) 000124 | phase of iZ. 000125 | 000126 | At an R-fixed point w=0, z=z0=+1 or -1, the map has cubic ramification 000127 | z-z0=constant*w^3+... . The frame e_z vanishes simply there. The actual 000128 | nonvanishing spin frame is e_z/w, and hence its local fiber phase is 000129 | 000130 | (+/-i)/(rho z0). 000131 | 000132 | At z0=+1 this is +/-exp(i pi/6); at z0=-1 it is the opposite sign. 000133 | Equivalently the half-ramification frame contributes rho^-1 to the base 000134 | spin phase i. The two phases square to rho and retain Rhat^6=-I. 000135 | This explicitly avoids reading the coefficient of a vanishing meromorphic 000136 | frame as its fiber eigenvalue. It agrees with the original Phi/Chi lift. 000137 | 000138 | 5. DOMAIN AND COEFFICIENT SCOPE 000139 | 000140 | The displayed matrix D_coll identities are for the actual smooth product 000141 | collar and its transported connections/H1 clutch. The global spin-bundle 000142 | identity and ramification phase comparison are exact on C. A branched 000143 | coordinate pullback does not identify the unweighted H1 domain in w with 000144 | the smooth base domain at the four vertices; the actual Sobolev/trace 000145 | weights and corner prescription must be pulled along the cubic map. 000146 | 000147 | The existing G and signed W dictionaries can be used on their actual 000148 | given C3 pullback/coefficient subspaces. This calculation does not equate 000149 | their rank-three system with an unrestricted full pushforward, nor all 000150 | six spin sections with new elementary physical fields. No metric or 000151 | fermion count is inferred from their dimensions. 000152 | 000153 | A small exact Clifford check confirmed iZ=N(-iS), square -I, whole D 000154 | covariance, B to minus B_p, NEW linearity, the charged-chart identity and 000155 | h to minus h_p. These equations follow analytically from the displayed 000156 | original matrices and oddness of Phi, so they apply to all96 coefficients. 000157 | No second suite or new physical-source construction was run. 000158 | 000159 | Source consequence: the plain base Spin(2) lift is an actual omitted 000160 | alternative from the original data, and its half-form phase now matches 000161 | the actual R/S0 lift through the known ramified pullback. This is positive 000162 | progress on the global source comparison. It does not certify fixed marks, 000163 | singular corner domains, an interacting source selection or a physical CP 000164 | reading merely by the local symbol and phase equations.