TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/source/ORIGINAL-BOUNDARY-NORM-ORIGIN.txt Original file SHA-256: 2a92d7c75aeed3fb09d85de0f9cf8be3c9fb8757b6686cf0c6724cd7b4a977a6 Inclusive original lines: 1-200 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 | ORIGINAL BOUNDARY NORM ORIGIN 000002 | The exact three-cabled seam determinant 000003 | 2026-10-27 000004 | 000005 | QUESTION 000006 | 000007 | The accepted finite source construction has 000008 | 000009 | H_Theta = H^0(Jac(C),O(2 Theta)), 000010 | W_q = H_Theta tensor N_q^(-1), 000011 | 000012 | where N_q is the permutation-free norm of the source-selected spin line S_0 000013 | over D_q=q_0+q_1+q_2. Why should the original boundary operation contribute 000014 | exactly one inverse N_q, rather than a fitted line power? 000015 | 000016 | RESULT 000017 | 000018 | There is an exact answer at the relative Fredholm-line level. It uses the 000019 | original seam multiplier itself. 000020 | 000021 | On the already reconstructed genus-two source 000022 | 000023 | C: eta^2=1-w^6 000024 | 000025 | one has the original covering coordinate 000026 | 000027 | z=eta/(1-w^3), z^2=(1+w^3)/(1-w^3), (1) 000028 | 000029 | and therefore 000030 | 000031 | div(z)=D_q-D_p, c*z=z, Q*z=i z. (2) 000032 | 000033 | The old fullv21 seam gluing is exactly multiplication by 000034 | U_Gamma(theta)=exp(i theta), hence by z. Over the open unit z-disc the 000035 | three-sheeted map C->P1_z is unramified and splits into three discs, centered 000036 | at q_0,q_1,q_2. Pulling the original shift to the selected spin line S_0 000037 | gives an injective three-cabled Hardy shift whose cokernel is 000038 | 000039 | K_q = direct_sum_a S_0|q_a. (3) 000040 | 000041 | Its Fredholm determinant line is the inverse evaluation determinant 000042 | 000043 | Det_Fred(z:S_0)^{} = det(K_q)^(-1)=B_q^(-1). (4) 000044 | 000045 | More generally, for every degree-zero family line L, compare the spin 000046 | cable S_0 tensor L with the otherwise identical reference cable L. If B(E) 000047 | denotes the ordered evaluation determinant of E at D_q, their relative line is 000048 | 000049 | B(S_0 tensor L)^(-1) tensor B(L) 000050 | = Nm_Dq(S_0)^(-1) 000051 | = N_q^(-1). (5) 000052 | 000053 | All L fibers and all permutation signs cancel between the two determinant 000054 | lines. For L=O_C this is B_q^(-1) tensor B_O. Equation (5) is exactly one 000055 | inverse norm, fiberwise over the full Picard family; it does not require a 000056 | c-fixed L. 000057 | The exponent is the integer winding. For multiplication by z^n the same 000058 | relative construction gives N_q^(-n). The original scalar seam spectral 000059 | flow is one, so it selects n=1 and hence exactly one inverse norm. This is 000060 | stronger than the finite quarter-clock test, which sees only a residue of n. 000061 | It does not turn the three-dimensional cokernel into three independent 000062 | physical insertions. 000063 | 000064 | This closes the algebraic sewing question. One physical-source premise 000065 | remains: the actual relative boundary field must be the S_0 cable in (3), 000066 | and the declared physical reference D_ref must induce the same degree-zero L 000067 | cable without the S_0 twist, as used in (5). The original texts supply 000068 | that seam multiplier and a relative operator split, but they do not yet prove 000069 | this common-L spin-twist identification. Thus (5) is an original-operation 000070 | reconstruction with a sharply stated field/reference premise, not yet an unconditional 000071 | identification of the full P1 operator. 000072 | 000073 | 1. THE ORIGINAL MAP AND ITS DIVISOR 000074 | 000075 | The algebra in (1) is immediate from eta^2=(1-w^3)(1+w^3). At 000076 | q_a=-omega^a, eta is a local parameter and 1-w^3 is nonzero, so z has a 000077 | simple zero. At p_a=omega^a, eta is again a local parameter and 000078 | 1-w^3 has a double zero, so z has a simple pole. Hence 000079 | 000080 | div(z)=q_0+q_1+q_2-p_0-p_1-p_2. 000081 | 000082 | The two source automorphisms act by 000083 | 000084 | c(w,eta)=(omega w,eta), 000085 | Q(w,eta)=(1/w,-i eta/w^3). 000086 | 000087 | Substitution gives c*z=z and Q*z=i z exactly. Thus z is simultaneously 000088 | 000089 | * the cyclic quotient coordinate for c; 000090 | * the original quarter-clock coordinate for Q; 000091 | * the old seam multiplier exp(i theta) on |z|=1. 000092 | 000093 | The branch values of w^3=(z^2-1)/(z^2+1) are z=1,-1,i,-i. They lie on the 000094 | unit circle. The open unit disc is therefore unramified. Since it is simply 000095 | connected, its inverse image is the disjoint union of three discs, one about 000096 | each q_a. 000097 | 000098 | This is the missing geometric reason that the old one-circle shift and the 000099 | three q evaluations belong to the same map. It does not yet say which 000100 | coefficient bundle the physical field carries. 000101 | 000102 | 2. HARDY DEFECT AND THE ONE INVERSE LINE 000103 | 000104 | Let H_q be the direct sum of the three holomorphic Hardy spaces of S_0 on 000105 | the lifted discs. Multiplication by z is an isometry after the standard 000106 | boundary normalization and satisfies 000107 | 000108 | H_q / z H_q ~= direct_sum_a S_0|q_a (6) 000109 | 000110 | by evaluation at the three simple zeros. No boundary Sobolev trace theorem 000111 | is needed for (6): on each disc it is the elementary exact sequence 000112 | 0->zH^2->H^2->fiber_at_0->0. 000113 | 000114 | 2.1 THE HARDY NORM IS FIXED BY THE HALF-FORM 000115 | 000116 | This Hardy realization does not require a fitted measure. On each open lifted 000117 | disc, dz has no zero and the theta characteristic S_0 has a holomorphic 000118 | half-form e with e^2=dz. For psi=f(z)e, and likewise after tensoring with a 000119 | unitary flat frame of L, 000120 | 000121 | integral_boundary |psi|^2 000122 | = integral_|z|=1 |f(z)|^2 |dz|. 000123 | 000124 | Thus the source spin structure supplies the ordinary Hardy norm, while the 000125 | unitary coefficient metric is the same in the relative and reference cables. 000126 | Changing e by its other square root changes no norm or determinant line. 000127 | 000128 | At each of the four branch points on |z|=1, a local coordinate u has 000129 | z-z_0=a u^3. Hence dz has a zero of order two and e has a zero of order one. 000130 | Those boundary points have measure zero in the Hardy norm and create no 000131 | additional interior evaluation defect or H^1 point term. This proves the 000132 | natural half-form Hardy realization used in (6). It does not identify that 000133 | Hardy polarization with the physical Calderon polarization of D_rel and 000134 | D_ref; that remains the operator premise stated in section 7. 000135 | 000136 | The shift has no kernel and three-dimensional cokernel, hence 000137 | 000138 | index(z|H_q)=-3. 000139 | 000140 | With the determinant-line convention 000141 | 000142 | Det_Fred(T)=det ker(T) tensor det coker(T)^(-1), 000143 | 000144 | (6) gives B_q^(-1), exactly as in (4). This is one line. Its inverse power 000145 | is fixed by the cokernel orientation in the Fredholm determinant functor. 000146 | 000147 | For any degree-zero line L, apply the same shift to S_0 tensor L and to 000148 | L. Their cokernel determinants obey 000149 | 000150 | B(S_0 tensor L)=Nm_Dq(S_0) tensor B(L). 000151 | 000152 | Consequently the relative Fredholm line is 000153 | 000154 | Det_rel(L) 000155 | = Det_Fred(z:S_0 tensor L) tensor Det_Fred(z:L)^(-1) 000156 | = B(S_0 tensor L)^(-1) tensor B(L) 000157 | = Nm_Dq(S_0)^(-1)=N_q^(-1). (7) 000158 | 000159 | Both ordered determinants acquire the same sign when the sheets are 000160 | permuted, so the ratio is the permutation-free norm automatically. All 000161 | three L fibers cancel. Thus (7) holds over the full Picard family, not only 000162 | at a c-fixed fiber. The spin-relative Poincare normalization from the 000163 | accepted construction then makes the remaining N_q line constant over that 000164 | family. 000165 | 000166 | 2.2 INTEGER WINDING FIXES THE NORM POWER 000167 | 000168 | For n>=1, multiplication by z^n has cokernel equal to the length-n jets at 000169 | each q point: 000170 | 000171 | H_q(E)/z^n H_q(E) ~= H^0(nD_q,E|nD_q). 000172 | 000173 | Its natural filtration has graded pieces 000174 | 000175 | gr_j ~= direct_sum_a E|q_a tensor (T_q_a^* C)^j, 000176 | j=0,...,n-1. (8) 000177 | 000178 | Take E=S_0 tensor L and compare with E_ref=L. In the ratio of the two 000179 | Fredholm determinant lines, every L fiber, every cotangent/jet factor and 000180 | every sheet-permutation sign in (8) cancels. Each value of j leaves one 000181 | copy of Nm_Dq(S_0)^(-1). Therefore 000182 | 000183 | Det_rel(z^n; S_0 tensor L,L)=N_q(S_0)^(-n), 000184 | index(z^n on each cable)=-3n. (9) 000185 | 000186 | Downstairs the scalar shift by z^n has spectral flow n and index -n. The 000187 | fullv21 source chooses the primitive class SF=1, so n=1. Thus the original 000188 | integer winding fixes the literal power in (7). The earlier finite 000189 | relations could only constrain n modulo 4, or modulo 8 after retaining the 000190 | Spin-lift sign; the Fredholm winding removes that residual ambiguity. 000191 | 000192 | The executable audit is 000193 | 000194 | check_three_cabled_seam.py 000195 | 000196 | with checked output three-cabled-seam.json. It verifies (1), both symmetry 000197 | actions, and, for n=1,...,8, the explicit rectangular matrix of z^n on three 000198 | truncated polynomial sheets. It computes rank, kernel, the 3n-dimensional 000199 | jet quotient, its c character, and the Q and beta determinant characters 000200 | zeta_8^(-3n) and i^n. This finite calculation checks the algebraic exact