TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CANONICAL-BRANCHED-SPIN-PULLBACK.txt Original file SHA-256: c61a15753a64da286cef464067e221860cda9495f6efa61082bd39398606d5c3 Inclusive original lines: 1-87 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 ORIGINAL SOURCE SPIN IS THE CANONICAL RAMIFIED BASE SPIN PULLBACK 000002 | Independent exact divisor and square-map proof, 29 October 2026. 000003 | No new spin choice, source, model, test run or Atlas mutation. 000004 | 000005 | 1. Original map and the two divisor representatives 000006 | Use the ACTUAL original cubic map 000007 | pi:C -> P1_z, w^3=(z^2-1)/(z^2+1), eta=z(1-w^3), eta^2=1-w^6. 000008 | Let K=I_++I_- represent K_C, q_a=(-omega^a,0), p_a=(omega^a,0), 000009 | Dq=sum q_a=pi^-1(0), Dp=sum p_a=pi^-1(infinity). 000010 | There are four ramification points r_j, above z=1,i,-1,-i, each of index3: 000011 | the two w=0 points and I_+,I_-. Hence 000012 | R_pi=2 sum r_j, Rhalf=sum r_j, Rhalf~2K. 000013 | Indeed div(w)=(the two w=0 points)-K. Also 000014 | div(w^3+1)=2Dq-3K, so 2Dq~3K. 000015 | 000016 | The sphere has its UNIQUE spin line O_P1(-1). Since all ramification 000017 | indices are odd, its canonical ramified spin pullback is 000018 | S_can=pi*O_P1(-1) tensor O_C(Rhalf). 000019 | Riemann-Hurwitz gives S_can^2=K_C. Representing the base spin by O(-[0]), 000020 | S_can=O(Rhalf-Dq)~O(2K-Dq)~O(Dq-K)=S0. 000021 | This is the ALREADY source-selected S0, not a further theta choice. 000022 | 000023 | 2. Explicit square-compatible map, not only equality in Picard 000024 | Retain the old common meromorphic spin frame Phi0, with 000025 | Phi0^2=mu0=(w^3+1)dw/eta, div(Phi0)=Dq-K. 000026 | Direct differentiation of the actual z gives 000027 | dz=3w^2/[eta(1-w^3)] dw, 000028 | pi*(dz/z^2)=3w^2/[(w^3+1)eta] dw. 000029 | For f=w/(w^3+1), 000030 | div(f)=Rhalf-Dq-(Dq-K), 000031 | pi*(dz/z^2)=3 f^2 mu0. 000032 | Therefore the meromorphic canonical pullback frame is EXACTLY 000033 | e_can=sqrt(3) f Phi0, e_can^2=pi*(dz/z^2). 000034 | Its divisor is Rhalf-Dq, as required. Multiplying by z gives the equally 000035 | useful frame pulled from sqrt(dz): 000036 | sqrt(dz)=z e_can=(sqrt(3)w/eta)Phi0, (1) 000037 | mu0/dz=eta^2/(3w^2), div(sqrt(dz))=Rhalf-Dp. 000038 | The positive real sqrt(3) is the displayed derivative normalization; 000039 | the remaining common spin sign is the ordinary sign of a spin frame. 000040 | No fitted quarter or relative source scalar occurs in this identity. 000041 | 000042 | 3. The original c,Q and Real lifts, with the actual spin phase 000043 | Use the already fixed order-three spin representative and marked quarter: 000044 | c(w,eta)=(omega w,eta), c*Phi0=omega^2 Phi0, 000045 | Q(z,w,eta)=(iz,1/w,-i eta/w^3), Q*Phi0=(xi/w)Phi0, 000046 | xi=exp(-i pi/4). 000047 | Then c*f=omega f and Q*f=w f. Consequently 000048 | c*e_can=e_can, Q*e_can=xi e_can, 000049 | c*sqrt(dz)=sqrt(dz), Q*sqrt(dz)=i xi sqrt(dz)=bar(xi)sqrt(dz). 000050 | This is precisely the square-root derivative of z -> iz. It retains the 000051 | old S0 lift and does not infer a fiber value from a vanishing meromorphic 000052 | frame. In particular e_can has poles at q_a; its meromorphic coefficient 000053 | xi is not its nonvanishing local fiber eigenvalue there. 000054 | 000055 | Take tau0(w,eta)=(bar(w),bar(eta)) with the common Real meromorphic 000056 | frame Phi0. All rational factors in (1) are real, so its anti-pullback 000057 | fixes e_can and sqrt(dz). For E0=Q tau0, E0*z=i bar(z), and the 000058 | anti-pullback on sections is CONTRAVARIANTLY T0 Q. It follows that 000059 | J_E e_can=bar(xi)e_can, J_E sqrt(dz)=xi sqrt(dz), J_E^2=I. 000060 | This is a spin-bundle Real identity, not identification with CAR C0 or OS. 000061 | 000062 | 4. Precisely what has been gained 000063 | The old base map and the old selected S0 now have an explicit canonical 000064 | spin pullback, its ramification correction, square map and common c/Q/E0 000065 | transport. This closes the spin-line and local half-form part of the 000066 | previously missing cabling comparison. S0 must no longer be listed as an 000067 | independent spin input on this original cubic source. 000068 | The actual tangential metric, connection, trace Sobolev domain at the four 000069 | ramification vertices, full coefficient system and Gamma_+/Gamma_- face 000070 | assignment must still be transported before identifying the physical B_Sigma. 000071 | No ordinary smooth H1 domain is inferred merely from a Picard identity. 000072 | 000073 | [OMITTED: non-scientific conversation or execution metadata] 000074 | [OMITTED: non-scientific conversation or execution metadata] 000075 | [OMITTED: non-scientific conversation or execution metadata] 000076 | [OMITTED: non-scientific conversation or execution metadata] 000077 | [OMITTED: non-scientific conversation or execution metadata] 000078 | [OMITTED: non-scientific conversation or execution metadata] 000079 | [OMITTED: non-scientific conversation or execution metadata] 000080 | [OMITTED: non-scientific conversation or execution metadata] 000081 | [OMITTED: non-scientific conversation or execution metadata] 000082 | [OMITTED: non-scientific conversation or execution metadata] 000083 | [OMITTED: non-scientific conversation or execution metadata] 000084 | [OMITTED: non-scientific conversation or execution metadata] 000085 | [OMITTED: non-scientific conversation or execution metadata] 000086 | [OMITTED: non-scientific conversation or execution metadata] 000087 | [OMITTED: non-scientific conversation or execution metadata]