TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/uniqueness/THREE-CABLED-HARDY-SEAM-REVIEW.txt Original file SHA-256: ef86594c233412857f8a7d405ef7eea7557de8b1ba836f0a0a31169a8225cab7 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 | INDEPENDENT REVIEW OF THE THREE-CABLED HARDY SEAM 000002 | 2026-10-27 000003 | 000004 | VERDICT 000005 | 000006 | The proposed Hardy construction supplies the previously missing mathematical 000007 | one-copy line. On the existing source curve, the original coordinate z is a 000008 | degree-three sewing function with one simple zero on each of the three sheets 000009 | over the disk. Multiplication by z has a rank-three cokernel. Comparing its 000010 | spin defect line with the corresponding three-cabled scalar reference defect 000011 | gives exactly 000012 | 000013 | N_q(S_0)^(-1) 000014 | 000015 | once, with the correct c, Q, alpha and beta actions. The scalar i in 000016 | Q M_z Q^(-1)=i M_z does not add a determinant-line phase: multiplication of a 000017 | Fredholm operator by a nonzero scalar leaves its kernel and cokernel 000018 | unchanged. The induced action on the quotient is the relevant one. 000019 | 000020 | This closes the mathematical sewing origin of the one-copy norm factor on 000021 | the specified Hardy half of the source curve. It does not yet prove that the 000022 | physical P1 seam, state, reflection and Calderon domain are this Hardy 000023 | realization. It also does not construct N64 anew; it supplies the missing 000024 | one-dimensional line that the already accepted theta module tensors with. 000025 | 000026 | 000027 | 1. THE ORIGINAL FUNCTION HAS THE REQUIRED DIVISOR AND DEGREE 000028 | 000029 | On 000030 | 000031 | C: eta^2=1-w^6 000032 | 000033 | define 000034 | 000035 | z=eta/(1-w^3). 000036 | 000037 | Then 000038 | 000039 | z^2=(1+w^3)/(1-w^3). (1) 000040 | 000041 | At q_a, where w^3=-1, eta is a local coordinate and the denominator is 000042 | nonzero. Thus z has a simple zero at every q_a. At p_a, where w^3=1, the 000043 | same identity shows a simple pole. Therefore 000044 | 000045 | div(z)=D_q-D_p. (2) 000046 | 000047 | There are three poles, so z:C->P1 has degree three. Solving (1) gives 000048 | 000049 | w^3=(z^2-1)/(z^2+1). 000050 | 000051 | The only branch values are z=+1,-1,+i,-i, all on the unit circle. Hence over 000052 | the open disk D={|z|<1}, the map is an unramified three-sheeted cover. Since 000053 | D is simply connected, its inverse image is the disjoint union of three 000054 | disks, and each restriction maps biholomorphically to D. The boundary branch 000055 | points are measure-zero endpoints for the Hardy traces and do not alter the 000056 | interior shift index. 000057 | 000058 | 000059 | 2. THE HARDY SHIFT HAS THE FULL q EVALUATION QUOTIENT 000060 | 000061 | Let 000062 | 000063 | H_S=H^2(C_D,S_0), 000064 | 000065 | where C_D is the three-component inverse image of D. Point evaluation at the 000066 | interior zeros q_a is bounded on this Hardy space. On each disk, 000067 | multiplication by z is the ordinary unilateral shift, so M_z is injective, 000068 | has closed range, and has a one-dimensional cokernel. On all three sheets, 000069 | 000070 | index(M_z|H_S)=-3, 000071 | H_S/zH_S isomorphic direct_sum_a S_0|q_a. (3) 000072 | 000073 | The quotient isomorphism is evaluation at the three simple zeros. Therefore 000074 | the Fredholm defect line, in the convention det ker tensor det coker^(-1), is 000075 | 000076 | delta_S=det(H_S/zH_S)^(-1)=B_q^(-1). (4) 000077 | 000078 | Apply the same three-cabled shift to the trivial line: 000079 | 000080 | H_O=H^2(C_D,O_C), 000081 | delta_O=B_O^(-1), 000082 | B_O=det H^0(D_q,O_Dq). (5) 000083 | 000084 | Taking the spin defect relative to this scalar reference gives 000085 | 000086 | delta_S tensor delta_O^(-1) 000087 | =B_q^(-1) tensor B_O 000088 | =N_q(S_0)^(-1). (6) 000089 | 000090 | This identity is familywise, rather than confined to the trivial coefficient 000091 | line. Let L be any degree-zero line and use the same multiplier and the same 000092 | three Hardy components for S_0 tensor L and for L. Their cokernels are 000093 | 000094 | direct_sum_a (S_0|q_a tensor L|q_a), 000095 | direct_sum_a L|q_a. 000096 | 000097 | With one common ordering of the q points, 000098 | 000099 | det direct_sum_a(S_0|q_a tensor L|q_a) 000100 | =Nm_Dq(S_0) tensor det direct_sum_a L|q_a. 000101 | 000102 | Consequently 000103 | 000104 | delta(S_0 tensor L,z) tensor delta(L,z)^(-1) 000105 | =Nm_Dq(S_0)^(-1) (7) 000106 | 000107 | for every L. All coefficient fibers and all sheet-permutation signs cancel. 000108 | The equality is equivariant over the complete L family, including the actual 000109 | q sign line and its c orbit. It does not require L itself to be a c fixed 000110 | point and it does not require a Poincare rigidification merely to cancel the 000111 | coefficient line. The spin-relative Poincare construction remains the 000112 | stronger standalone trivialization of the individual norm family. 000113 | 000114 | Equations (6)-(7) are the decisive improvement over an arbitrary evaluation 000115 | restriction. The same original degree-three sewing function produces both 000116 | operators, and the ratio removes the ordering determinant of the three-point 000117 | scheme. It also explains why the old base shift has index -1 while its 000118 | actual three-sheet pullback has index -3: these are the base operator and its 000119 | three-cabled lift, not two operators asserted to be invertibly identical. 000120 | 000121 | 000122 | 2A. HIGHER WINDING AND WHY INDEX ONE SELECTS ONE COPY 000123 | 000124 | For n>=1, multiplication by z^n has index -3n and quotient supported on the 000125 | length-n divisor nD_q. At each q_a its jet filtration has graded pieces 000126 | 000127 | E|q_a tensor (T_q_a^*)^j, j=0,...,n-1. 000128 | 000129 | Apply this once with E=S_0 tensor L and once with E=L. The L fibers, the 000130 | cotangent jet factors and their permutation signs cancel degree by degree. 000131 | Every level retains one S_0|q_a. Hence 000132 | 000133 | delta(S_0 tensor L,z^n) tensor delta(L,z^n)^(-1) 000134 | =Nm_Dq(S_0)^(-n). (8) 000135 | 000136 | This proves the exponent, including all jets; it is not a phase inference. 000137 | On the base circle the same multiplier z^n has index -n. Therefore an 000138 | already specified oriented primitive seam with index -1, equivalently one 000139 | positive winding, selects n=1 inside this cable functor. The finite quarter 000140 | characters alone only see n modulo 4 or 8 and could not make that selection. 000141 | This conclusion uses the original integer index as part of the sewing data; 000142 | it does not identify an unrelated physical D_rel or D_ref with these Hardy 000143 | operators. 000144 | 000145 | 000146 | 3. THE Q SCALAR DOES NOT CREATE AN EXTRA LINE PHASE 000147 | 000148 | The source automorphisms act on z by 000149 | 000150 | c z=z, 000151 | Q z=i z. (9) 000152 | 000153 | Thus c commutes with M_z, while in the stated pullback convention 000154 | 000155 | Q M_z Q^(-1)=i M_z. (10) 000156 | 000157 | Equation (10) maps the Fredholm operator to its nonzero scalar multiple. The 000158 | canonical identification of their determinant lines is the identity on 000159 | kernel and cokernel, since 000160 | 000161 | ker(iM_z)=ker(M_z), 000162 | coker(iM_z)=coker(M_z). (11) 000163 | 000164 | Equivalently, to write a strictly commuting chain square one may multiply 000165 | the domain action by i. Because the kernel is zero, this domain scalar does 000166 | not enter the Fredholm defect line. The induced Q action on the cokernel is 000167 | the quotient action on H/zH and agrees with the action obtained from the 000168 | three evaluations. No additional factor i^(index) belongs to the Fredholm 000169 | line. Such a factor would concern a separately normalized determinant 000170 | section or a regularized determinant, neither of which is used in (4)-(8). 000171 | 000172 | The same chain scalar occurs for the spin and scalar three-cabled shifts and 000173 | would cancel in their relative ratio in any case. 000174 | 000175 | 000176 | 4. THE FULL FINITE LINE CHARACTER IS CORRECT 000177 | 000178 | At each q point, beta acts by +i on the S_0 fiber. It acts trivially on the 000179 | corresponding scalar reference fiber. Hence 000180 | 000181 | beta|B_q=i^3=-i, 000182 | beta|B_O=1, 000183 | beta|B_q^(-1) tensor B_O=+i. (12) 000184 | 000185 | This is exactly the beta character of N_q^(-1). At the Q level, both B_q 000186 | and B_O contain the sign of the transposition of q_1 and q_2. Their ratio 000187 | cancels that common permutation sign and retains precisely the spin norm 000188 | character. Thus the alpha square and the alpha sign convention agree with 000189 | the already proved marked-divisor line, rather than merely matching beta. 000190 | 000191 | The cubic c cycles the three points. Its permutation is even, and the 000192 | normalized spin lift has product one on N_q. Since c also commutes with 000193 | M_z, equation (6) carries the accepted trivial gamma character. Therefore 000194 | the relative Hardy defect supplies all line characters used in the 000195 | spin-rigidified theta-module theorem. 000196 | 000197 | The one-dimensional c-invariant summand of the quotient must not be confused 000198 | with this relative line. Its undualized fiber has beta=+i, whereas its 000199 | directed Fredholm defect is the inverse fiber and has beta=-i. The correct 000200 | +i character arises from the inverse determinant of all three q fibers,