TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/ORIGINAL-BASES-AND-ELLIPTIC-SPIN.txt
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Inclusive original lines: 37-120
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000037 | 
000038 |   E: y^2=x^4-1,   p(x,y)=x,
000039 |   delta(x,y)=(x,-y),   q(x,y)=(i x,y).
000040 | 
000041 | Only if p is supplied as the actual pullback of the original normal S2,
000042 | with its spin/twist data, does it act on the original normal index problem.
000043 | Its abstract source index test can nevertheless be solved completely.
000044 | 
000045 | For every elliptic spin line S, deg S=0. Hence Riemann-Roch gives
000046 | 
000047 |   ind D_(S tensor p*O(1)) = deg p*O(1) = 2.
000048 | 
000049 | A degree-two pullback does not preserve the ordinary complex index one.
000050 | More generally the ordinary spin Dirac index of the degreeq pullback is q.
000051 | This is not a reinterpretation of the old bosonic doublet object count.
000052 | Any physical source factor that uses that count needs its actual kernel
000053 | representation and source pairing, not this ordinary index substituted for it.
000054 | 
000055 | The natural ramification spin is trivial:
000056 |  p*O(-1) tensor O(R/2) = O_E,  O(R/2)=p*O(1).
000057 | Since p_*O_E=O plus O(-2), the two sections of p*O(1) lie in the same
000058 | algebraic deck sector. They do not form a one-plus-one deck splitting.
000059 | 
000060 | For a nontrivial elliptic two-torsion spin alpha, h0(alpha)=0 and
000061 | p_*alpha=O(-1) plus O(-1). After twisting by O(1) this gives one section
000062 | in each deck sector, for a chosen compatible algebraic delta linearization.
000063 | The quarter q fixes one of the three even spin classes and exchanges the
000064 | other two. The fixed class does not itself give a q-stable one-dimensional
000065 | kernel sector, as the explicit calculation below shows.
000066 | 
000067 | Primary background for the four elliptic spin classes and their parities:
000068 | M. F. Atiyah, Riemann surfaces and spin structures (1971), introduction,
000069 | https://www.numdam.org/item/10.24033/asens.1205.pdf
000070 | The following lift calculation is explicit on the original curve.
000071 | 
000072 | 3. The exact deck/quarter commutator in the invariant even spin class
000073 | --------------------------------------------------------------------
000074 | Let O=(0,i), T=(0,-i), and u=(y+i)/x^2. Then
000075 | 
000076 |   div u = 2T-2O,   delta* u=-1/u,   q* u=-u.
000077 | 
000078 | The unique q-fixed nontrivial spin class is alpha=O(T-O). For the twist
000079 | L=p*O(1)=O(O+T), the kernel line bundle is
000080 | 
000081 |   alpha tensor L=O(2T) isomorphic to O(2O),
000082 |   H0(alpha tensor L)=span(1,u).
000083 | 
000084 | One genuine deck spin lift acts by D f=u(f composed delta), so in that
000085 | basis
000086 | 
000087 |   D = [[0,-1],[1,0]],       D^2=-I.
000088 | 
000089 | Up to its required scalar phase, a quarter lift acts by
000090 | 
000091 |   Q = a diag(1,-1).
000092 | 
000093 | The spin square-root convention fixes a fourth-power sign, but the following
000094 | relation is independent of every scalar choice:
000095 | 
000096 |   Q D = -D Q,      Q D Q^-1 D^-1 = -I.
000097 | 
000098 | Thus q exchanges the two D eigenlines (eigenvalues +i and -i). No rank-one
000099 | kernel subspace is invariant under both operators. Multiplying either lift
000100 | by a scalar or changing basis cannot remove this commutator. The half-turn
000101 | Q^2 preserves each line. A changed spin class is an additional input.
000102 | 
000103 | This result is for the specified normal spin/twist factor only. Another
000104 | ACTUAL factor with the opposite projective cocycle could cancel the sign.
000105 | The original full Spin/family source needs its real tensor factors and
000106 | transport matrices to establish such a cancellation. No artificial two-state
000107 | factor is inserted, and no exclusion of the whole TFPT construction follows.
000108 | 
000109 | 4. Consequence for the current research path
000110 | --------------------------------------------
000111 | The old C2 curve is not automatically an index-preserving, quarter-compatible
000112 | repair of a globally chosen Cartan chart. Neither that repair nor a global
000113 | Cartan chart is needed for the internally invariant field W8 now explicitly
000114 | constructed in QUANTUM-OCTIC-COMPLETION.txt. The classical invariant quotient
000115 | and the quantum field are different objects; the latter is checked by actual
000116 | root zero modes, Virasoro modes, full Fock norm and the original CAR map.
000117 | 
000118 | The original task returns to the common source operator/reflection/collar
000119 | identification. The topological and spin statements above remain available
000120 | when an actual same-base cover or marked charged-field transport is supplied.