TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/MARKED-Q-DIVISOR-CENTRAL-TWIST-INDEPENDENT-REVIEW.txt Original file SHA-256: f2b4f084ace7506fd9e5ea90f293082d6ad7149fbba1ac8d830159aeb80eda8f Inclusive original lines: 1-130 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 | INDEPENDENT REVIEW OF THE MARKED q DIVISOR CENTRAL TWIST 000002 | 2026-10-26 000003 | 000004 | Reviewed source 000005 | 000006 | MARKED-Q-DIVISOR-CENTRAL-TWIST.txt 000007 | SHA-256 da1f75831a46a0c0662f931486b77e6e2085126df452b009d1a90351d413d904 000008 | 000009 | VERDICT 000010 | 000011 | ACCEPTED. The finite evaluation line has the stated central phase, and the 000012 | source-selected spin normalization removes the moving two-torsion factor over 000013 | the full Jacobian family. The construction is functorial under c and Q. It 000014 | still does not prove that the physical boundary or conformal-block functor 000015 | inserts one inverse copy of this line; the report states that remaining bridge 000016 | precisely. 000017 | 000018 | 1. FIXED-FIBER CHARACTER 000019 | 000020 | For D_q=q_0+q_1+q_2 the permutation-free norm line is 000021 | 000022 | N_q = tensor_a S_0|q_a. 000023 | 000024 | The selected beta phase is +i at q_0. If gamma is the normalized order-three 000025 | lift of c, gamma beta gamma^(-1)=zeta beta implies zeta^3=1, while beta^2=-I 000026 | implies zeta^2=1. Thus zeta=1 and beta has phase +i on all three q fibers. 000027 | Consequently 000028 | 000029 | beta|N_q = i^3 = -i, 000030 | beta|N_q^(-1) = +i. 000031 | 000032 | The canonical even action on H^0(O(2Theta)) is +I. Hence the fixed-fiber 000033 | candidate has beta=alpha^2=+i I and alpha^4=-I. These square relations do not 000034 | depend on the remaining sign alpha -> -alpha. The cubic cycle has positive 000035 | permutation sign, and gamma^3=1 makes its normalized action on N_q trivial. 000036 | 000037 | At p_0 the local derivative has phase -i. The same argument gives beta=-i 000038 | on every p fiber, beta=+i on N_p, and beta=-i on N_p^(-1). The relation 000039 | sQs=Q^(-1) therefore exchanges conjugate q and p blocks. This is compatible 000040 | with a paired eight-dimensional system and does not identify that pair with 000041 | one native four-dimensional module. 000042 | 000043 | 2. SPIN-RELATIVE POINCARE NORMALIZATION 000044 | 000045 | Let P be any Poincare family on C x A and put 000046 | 000047 | R=, 000048 | P_spin=P tensor pr_A^*(R^(-1)). 000049 | 000050 | Because deg S_0=1, Deligne-pairing bilinearity gives a canonical 000051 | trivialization 000052 | 000053 | =O_A. 000054 | 000055 | This condition removes the usual freedom to tensor a universal family by a 000056 | line from A. Indeed, such a tensor changes the pairing by that base line to 000057 | the first power. It therefore defines the Poincare family relative to the 000058 | already selected spin line, without choosing a new point or a flat 000059 | character. 000060 | 000061 | The divisor identity used next is exact. From 000062 | 000063 | S_0=O(D_q-K_C), S_0^2=K_C 000064 | 000065 | one obtains O(D_q)=S_0^3. Hence 000066 | 000067 | Norm_Dq(P_spin) 000068 | = 000069 | =^3 000070 | =O_A. 000071 | 000072 | An isomorphism O(D_q)->S_0^3 is unique only up to a nonzero scalar. That 000073 | scalar acts in this pairing with exponent deg(P_spin|C)=0, so the displayed 000074 | trivialization is independent of it. The same proof applies to D_p because 000075 | O(D_p)=O(D_q). 000076 | 000077 | At the origin this normalization does not change N_q(S_0) or its beta phase. 000078 | Over the family it gives 000079 | 000080 | mathcal N_q(S_0 tensor P_spin)=N_q(S_0) tensor O_A. 000081 | 000082 | It also leaves the ordinary theta determinant unchanged: a line pulled back 000083 | from A enters determinant of cohomology with exponent 000084 | chi(S_0 tensor L)=0. Thus the norm changes with the required divisor length 000085 | three while the theta line has exponent zero. 000086 | 000087 | 3. FUNCTORIALITY AND THE EARLIER P_DELTA TERM 000088 | 000089 | For a Poincare bundle normalized at q_0 one instead finds 000090 | 000091 | Norm_Dq(P)=P_delta_q, 000092 | delta_q=D_q-3q_0=q_1-q_2. 000093 | 000094 | Delta_q is a nonzero element of V_q and c moves it through that plane. The 000095 | resulting failure of c invariance is real for the q_0-normalized family, but 000096 | it is not intrinsic to the source family. It disappears under the 000097 | spin-relative normalization above. 000098 | 000099 | The c and Q actions preserve the classes of S_0 and D_q. Pulling P_spin back 000100 | by either action gives another universal family with the same trivialized 000101 | pairing with S_0. Uniqueness of this normalized universal family supplies 000102 | the comparison isomorphism. Any scalar ambiguity in the chosen lift of S_0 000103 | acts on the pairing with exponent deg(P_spin|C)=0 and disappears. An 000104 | automorphism of P_spin preserving its pairing trivialization is itself the 000105 | identity because deg S_0=1. These facts make the comparisons satisfy the 000106 | group cocycles, rather than giving only equality of Picard classes. 000107 | 000108 | This also sharpens the rigidification distinction. A scalar change of a 000109 | q_0 rigidification cannot alter P_delta_q. The successful construction is a 000110 | different canonical normalization supplied by S_0 itself. It is not a 000111 | scalar repair of the point-normalized bundle. 000112 | 000113 | 4. DETERMINANT EXPONENT AND SCIENTIFIC SCOPE 000114 | 000115 | For the evaluation exact sequence, imposing vanishing first changes the 000116 | determinant line D to D tensor N_q^(-1). Taking the inverse square that 000117 | produces the ordinary two-theta line gives 000118 | 000119 | (D tensor N_q^(-1))^(-2)=D^(-2) tensor N_q^2. 000120 | 000121 | Its beta phase is (-i)^2=-1, not +i. The full-family one-copy candidate 000122 | 000123 | H^0(O(2Theta)) tensor N_q(S_0)^(-1) 000124 | 000125 | has the desired +i phase, but determinant exactness alone does not select 000126 | that one-copy insertion. The report correctly leaves as open the original 000127 | field, boundary, sewing, Calderon, or conformal-block rule that would attach 000128 | this inverse norm line to the four-dimensional theta module. 000129 | 000130 | No material mathematical correction is required.