TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/GLOBAL-SPIN-KERNEL.txt Original file SHA-256: 2f8a165804067b7522b2689239e2bcef8b58d925e3db2d09b2d8e50caabdecc8 Inclusive original lines: 1-180 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 | THE GLOBAL RANK TWO SPIN KERNEL ON THE ELLIPTIC DOUBLE 000002 | ====================================================== 000003 | 26 October 2026. Exact algebraic continuation and full kernel diagonalization. 000004 | Atlas audit frame: revision c76c7d969cabaa790d397657. No Atlas write. 000005 | 000006 | Question and verdict 000007 | ==================== 000008 | 000009 | On the already fixed curve and spin line 000010 | 000011 | E: y^2=x^4-1, Q(x,y)=(ix,y), alpha nontrivial and Q fixed, 000012 | 000013 | does the unique Q invariant branch partition give only the previously proved 000014 | local quarter principal part, or the complete global pushed forward Szego 000015 | kernel? 000016 | 000017 | It gives the complete kernel. Put 000018 | 000019 | A(x)=sqrt(x^2-1), B(x)=sqrt(x^2+1), y=A B, 000020 | g(x)=sqrt(A(x)/B(x)), g(x)^4=(x^2-1)/(x^2+1). (1) 000021 | 000022 | Then the actual normalized alpha Szego kernel is 000023 | 000024 | S_alpha(p,q)=sqrt(dx dz)/(2(x-z)) 000025 | [g(x)/g(z)+g(z)/g(x)]. (2) 000026 | 000027 | The square roots in (1) are sections on E, with their sheet signs retained. 000028 | In the two sheet pushforward, one constant unitary J diagonalizes the full 000029 | matrix kernel, not only its Cauchy principal part: 000030 | 000031 | J^* K_sheet(x,z) J 000032 | =sqrt(dx dz)/(x-z) 000033 | diag(g(x)/g(z),g(z)/g(x)). (3) 000034 | 000035 | There is no omitted elliptic correction in (3). The two diagonal entries are 000036 | the global quarter local systems selected as an unordered pair by the actual 000037 | alpha spin structure. Q exchanges them. Thus the original elliptic data give 000038 | a concrete global holomorphic source kernel without a fitted profile or a 000039 | free vertex matrix. 000040 | 000041 | This does not turn the four marked punctures into the old smooth quarter 000042 | circle. Nor does a holomorphic bilinear kernel alone prove positivity of an 000043 | equal time CAR covariance. Those two operator statements remain separately 000044 | typed. 000045 | 000046 | 1. The partition is the actual alpha spin structure 000047 | ==================================================== 000048 | 000049 | Let R_a denote the ramification point over a and let infinity_+, infinity_- 000050 | be the two points over infinity. The divisor 000051 | 000052 | D_A=R_1+R_(-1)-infinity_+-infinity_- (4) 000053 | 000054 | satisfies 000055 | 000056 | 2 D_A=div(x^2-1). (5) 000057 | 000058 | It is nonprincipal. Indeed, if it were principal, a function with at most 000059 | simple poles at the two infinities would vanish at both R_1 and R_(-1). 000060 | But 000061 | 000062 | H0(E,O(infinity_++infinity_-))=span(1,x), 000063 | 000064 | and no nonzero affine function of x vanishes at x=1 and x=-1. 000065 | 000066 | Likewise define 000067 | 000068 | D_B=R_i+R_(-i)-infinity_+-infinity_-. 000069 | 000070 | Since D_A+D_B=div(y), one has D_B~-D_A~D_A. The quarter map Q swaps the two 000071 | parts {1,-1} and {i,-i}, so the unordered partition 000072 | 000073 | {1,-1} | {i,-i} (6) 000074 | 000075 | is Q invariant. The other two nontrivial partitions are exchanged by Q. 000076 | Thus O(D_A) is the unique Q fixed nontrivial two torsion spin line. It is the 000077 | same alpha already fixed by the original elliptic spin audit. Since alpha is 000078 | nontrivial of degree zero, 000079 | 000080 | H0(E,alpha)=0. (7) 000081 | 000082 | No new spin characteristic is selected in (6). 000083 | 000084 | 2. Direct genus one Szego proof 000085 | =============================== 000086 | 000087 | For p=(x,y_x) and q=(z,y_z), define 000088 | 000089 | S_part(p,q)= 000090 | [A(x)B(z)+B(x)A(z)]/[2(x-z)] 000091 | (dx/y_x)^(1/2) (dz/y_z)^(1/2). (8) 000092 | 000093 | The signs of A and B are chosen coherently so that A B=y on E. Formula (8) 000094 | has the defining properties of the alpha Szego kernel: 000095 | 000096 | * It is a half form in each variable with spin multipliers D_A. 000097 | 000098 | * At p=q, its only pole has unit residue because both ratios tend to one. 000099 | 000100 | * At q=h p, the apparent x=z pole cancels between the two numerator terms. 000101 | 000102 | * It has no other poles. 000103 | 000104 | If S_part and the normalized Szego kernel differed, their difference for 000105 | fixed q would be a holomorphic section of alpha. Equation (7) makes that 000106 | difference zero. Hence (8) is the actual NS--NS elliptic Szego kernel. 000107 | 000108 | Now use y=A B. The two terms in (8) simplify exactly as 000109 | 000110 | A(x)B(z)/sqrt(y_x y_z)=g(x)/g(z), 000111 | B(x)A(z)/sqrt(y_x y_z)=g(z)/g(x). (9) 000112 | 000113 | Equations (8) and (9) prove (2). This is a direct genus one argument from 000114 | pole, multipliers and H0(E,alpha)=0. It does not import a genus two theorem. 000115 | 000116 | For comparison, the same algebraic shape for a hyperelliptic even spin 000117 | partition is displayed in E. D'Hoker, M. Hidding and O. Schlotterer, 000118 | "Cyclic products of Szego kernels and spin structure sums", JHEP 05 (2023) 000119 | 073, arXiv:2211.09069, equation (2.26): 000120 | 000121 | https://arxiv.org/abs/2211.09069 000122 | 000123 | That reference treats genus two. It is a primary literature check of the 000124 | partition formula, not the proof of the genus one identity used here. 000125 | 000126 | 3. Exact two sheet matrix and constant diagonalization 000127 | ======================================================= 000128 | 000129 | Set 000130 | 000131 | a=g(x)/g(z), b=g(z)/g(x)=a^(-1). 000132 | 000133 | Use the analytic square root of the Jacobian on the second sheet, so the 000134 | second half density differs by i. In the common ordered sheet frame the 000135 | complete pushforward of (8) is 000136 | 000137 | K_sheet(x,z)=sqrt(dx dz)/[2(x-z)] 000138 | [[a+b, i(a-b)], 000139 | [-i(a-b), a+b ]]. (10) 000140 | 000141 | The diagonal entries reconstruct the same sheet kernel (2). The off diagonal 000142 | numerator a-b vanishes when x=z, which is the explicit cancellation of the 000143 | false pole at the conjugate point. 000144 | 000145 | Let 000146 | 000147 | J=1/sqrt(2) [[i,-i],[1,1]]. (11) 000148 | 000149 | Then J is unitary and direct multiplication gives 000150 | 000151 | J^* K_sheet J=sqrt(dx dz)/(x-z) diag(a,b). (12) 000152 | 000153 | This proves (3). In particular, the regular part left unspecified by the 000154 | earlier local calculation is not an additional matrix mixing term. The full 000155 | global kernel splits into the two character kernels 000156 | 000157 | K_+(x,z)=sqrt(dx dz)/(x-z) g(x)/g(z), 000158 | K_-(x,z)=sqrt(dx dz)/(x-z) g(z)/g(x). (13) 000159 | 000160 | Their average reconstructs the cover kernel because the inverse character 000161 | transform has the usual factor 1/2. 000162 | 000163 | 4. Peripheral monodromy and Q transport 000164 | ======================================== 000165 | 000166 | The common factor 000167 | 000168 | t(x)=(x^4-1)^(-1/4)=y^(-1/2) 000169 | 000170 | turns A t and B t into g and g^(-1). Therefore no independent fourth root 000171 | has been fitted. Around x=1 and x=-1, a positive meridian gives 000172 | 000173 | (g,g^(-1)) -> (i g,-i g^(-1)). (14) 000174 | 000175 | Around x=i and x=-i it gives the inverse pair 000176 | 000177 | (g,g^(-1)) -> (-i g,+i g^(-1)). (15) 000178 | 000179 | In the cyclic branch order 1,i,-1,-i, the four peripheral matrices are 000180 |