TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SOURCE-INDUCED-SEAM-TRANSPORT.txt Original file SHA-256: 1474792f0edfc9f0fc34e354b73cfc8c0a8591d0ccc78312bcf8e57f68058e05 Inclusive original lines: 1-190 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 | SOURCE INDUCED SEAM TRANSPORT ON THE ELLIPTIC DOUBLE 000002 | ==================================================== 000003 | 26 October 2026. Exact local branch, global push off and domain calculation. 000004 | Atlas audit frame: revision c76c7d969cabaa790d397657. 000005 | 000006 | Question and answer 000007 | =================== 000008 | 000009 | On the existing curve 000010 | 000011 | E: y^2=x^4-1, p(x,y)=x, 000012 | h(x,y)=(x,-y), Q(x,y)=(i x,y), 000013 | 000014 | take the already fixed Q invariant nontrivial spin line alpha with flat 000015 | character chi_alpha(A)=chi_alpha(B)=-1. Does its flat connection turn the 000016 | ramified pullback of the original seam into one canonical circle operator and 000017 | one canonical vertex rule? 000018 | 000019 | It gives more than a bare monodromy list, but less than that conclusion. 000020 | 000021 | PROVED: 000022 | 000023 | 1. The two one sided resolutions have no free U(2) matrix. In a common 000024 | Q compatible sheet frame their local flat transports are exactly 000025 | 000026 | T_out=I_2, T_in=P, 000027 | P=[[0,1],[1,0]]. (1) 000028 | 000029 | Reversing all orientation conventions exchanges the two names. The 000030 | invariant statement is T_out^(-1)T_in=P. 000031 | 000032 | 2. In a separately normalized local frame, a full positive or negative loop 000033 | around any one branch point has flat alpha monodromy P, because 000034 | P^(-1)=P. Its moving tangent Spin phase is respectively +i or -i. These 000035 | are distinct pieces of data: 000036 | 000037 | M_flat,local^ccw=M_flat,local^cw=P, 000038 | M_tangent^ccw=+i P, M_tangent^cw=-i P. (2) 000039 | 000040 | In one common global frame the four flat peripheral matrices are instead 000041 | (P,-P,P,-P). Their product is I. The signs are the actual alpha 000042 | translation character and cannot be discarded when comparing vertices. 000043 | 000044 | 3. The two radial push offs are both exact self adjoint realizations of the 000045 | pulled back scalar operator. Unmarked, they are unitarily equivalent and 000046 | have two copies of the integer spectrum. As Q and deck marked objects 000047 | they differ by P. The outer quarter action is the inner quarter action 000048 | times the deck exchange. 000049 | 000050 | On the full flat alpha line, however, the genuine deck and quarter Spin 000051 | pullbacks anticommute. Thus Q exchanges the two deck eigensectors. No 000052 | single deck eigenline is a global Q preserved circle carrier. 000053 | 000054 | 4. At the ramification point itself the vector field lifting 000055 | -i partial_theta has a 1/z pole. The regular flat alpha connection does 000056 | not cancel it. Therefore neither radial realization extends as a regular 000057 | first order operator on all smooth alpha spinors of the ramified graph. 000058 | 000059 | 5. Keeping both push offs gives the canonical pair of smooth resolutions. 000060 | It does not give a single self adjoint four edge vertex domain. Smooth 000061 | ambient continuity is only a one dimensional trace condition, whereas a 000062 | self adjoint first order graph vertex requires a two dimensional 000063 | Lagrangian trace space. 000064 | 000065 | Thus the existing spin and complex choices canonically fix the two radial 000066 | resolutions and leave one exact Z2 relative cocycle between them. They do not 000067 | select a self adjoint vertex condition from the full mathematical family, 000068 | identify the original matching operator U_Gamma, or produce the old quarter 000069 | twisted Hardy domain. No free vertex matrix is inserted here. 000070 | 000071 | 1. Local branch calculation 000072 | ============================ 000073 | 000074 | Write the four marks as 000075 | 000076 | a_j=i^j, theta_j=j pi/2, j=0,1,2,3, 000077 | 000078 | and use the same local coordinate at every mark, 000079 | 000080 | w_j=(x-a_j)/a_j=z_j^2. (3) 000081 | 000082 | For x=r exp(i(theta_j+s)), 000083 | 000084 | w_j=r exp(i s)-1=(r-1)+i s+O(s^2). (4) 000085 | 000086 | This normalization is Q equivariant because 000087 | 000088 | w_(j+1)(Qx)=w_j(x). 000089 | 000090 | The sheet deck is z_j -> -z_j. Let the incoming and outgoing root frames be 000091 | 000092 | z_- = exp(-i pi/4)sqrt(|s|), 000093 | z_+ = exp(+i pi/4)sqrt(|s|). 000094 | 000095 | For r>1, the path in (4) passes to the right of w=0 and its argument changes 000096 | by +pi. Analytic continuation sends z_- to z_+, hence T_out=I. For r<1 it 000097 | passes to the left and the continuously chosen argument changes by -pi. It 000098 | sends z_- to -z_+, hence T_in=P. This proves (1) at all four marks at once. 000099 | 000100 | The labels in (1) depend on the displayed endpoint roots. Changing a sheet 000101 | frame conjugates both matrices and may exchange which one is written as I. 000102 | Their ratio P is invariant. It is the monodromy of a full small base loop. 000103 | 000104 | 1.1 Flat alpha transport 000105 | ------------------------ 000106 | 000107 | The lift of a small branch disk is an ordinary z disk. Alpha is flat and 000108 | locally trivial there, so its parallel transport contributes scalar 1 on 000109 | each lifted path. The full base loop is not a closed path on one sheet. It 000110 | joins z to -z and therefore acts in that local frame on the rank 2 pushforward 000111 | by P. Clockwise and counterclockwise give P and P^(-1), which are equal. This 000112 | is a branchwise normalized statement. Transporting all four local frames to 000113 | one base fiber introduces the alpha character signs computed in Section 1.3. 000114 | 000115 | There is no hidden sign from alpha itself. The closed radial components for 000116 | r<1 contract to the two points over x=0, and those for r>1 contract to the 000117 | two points over x=infinity. Consequently alpha has holonomy +1 around every 000118 | one of these four component loops. 000119 | 000120 | 1.2 Moving tangent Spin phase 000121 | ------------------------------ 000122 | 000123 | A positive full base loop lifts to a half turn z -> -z. Its tangent rotates 000124 | by +pi and the Spin lift contributes exp(+i pi/2)=+i. A negative loop gives 000125 | -i. This proves the second line of (2). For the two half detours in (1), the 000126 | lifted tangent rotates by plus or minus pi/2, so a standard endpoint tangent 000127 | frame gives 000128 | 000129 | H_out=zeta_8 I, H_in=zeta_8^(-1) P, 000130 | zeta_8=exp(i pi/4). (5) 000131 | 000132 | Their fourth products are both -I. This is the bounding Spin sign of a 000133 | positively oriented contractible component. Their relative transport is 000134 | 000135 | H_out^(-1)H_in=-i P. (6) 000136 | 000137 | Equations (1), (5) and (6) must not be merged. Equation (1) is flat alpha 000138 | parallel transport. Equation (5) expresses the same paths in a moving tangent 000139 | Spin frame. The individual eighth root phases change with endpoint tangent 000140 | frames; the full loop statement (2) and the total sign -I are invariant. 000141 | 000142 | There are two further common traps: 000143 | 000144 | * The complex coordinate factor (2z)^(-1/2) changes phase around the branch 000145 | and is not a single valued gauge of p_*alpha. 000146 | 000147 | * The metric half density |2z|^(-1/2) is positive and contributes no complex 000148 | phase at all. 000149 | 000150 | The ramified base orbifold has local order 2, while a pi tangent rotation has 000151 | Spin lifts plus or minus i. It therefore does not define an ordinary Spin 000152 | quotient downstairs. The honest object is the upstairs alpha line together 000153 | with its central Spin extension. 000154 | 000155 | 1.3 The global flat pushforward and all four peripherals 000156 | -------------------------------------------------------- 000157 | 000158 | Let Lambda=Z A plus Z B with A=1 and B=i, and write t_lambda for translation 000159 | by lambda. In the affine torus coordinate used above, 000160 | 000161 | h z=-z+T, T=(1+i)/2, 000162 | 000163 | and the four branch reflections can be represented by 000164 | 000165 | r_lambda=t_lambda h, lambda in {0,A,A+B,B}. (6a) 000166 | 000167 | Use the Q cyclic order 000168 | 000169 | lambda_0=0, lambda_1=A, lambda_2=A+B, lambda_3=B. 000170 | 000171 | After naming the first fixed point x=1, this is the branch order 000172 | 1,i,-1,-i. A cyclic relabeling changes none of the relations below. 000173 | 000174 | Since r_a r_b=t_(a-b), the global peripheral relation is 000175 | 000176 | r_0 r_A r_(A+B) r_B 000177 | =t_(-A+A+B-B)=1. (6b) 000178 | 000179 | Now induce the actual flat alpha character 000180 | 000181 | chi_alpha(A)=chi_alpha(B)=-1 000182 | 000183 | to the two sheets. In the common sheet basis, 000184 | 000185 | rho(t_lambda)=chi_alpha(lambda) I, rho(h)=P. 000186 | 000187 | Therefore the four peripheral matrices are exactly 000188 | 000189 | (M_0,M_1,M_2,M_3)=(P,-P,P,-P), 000190 | M_j^2=I, M_0 M_1 M_2 M_3=I. (6c)