TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/TRACE-METRIC-AND-DOMAIN.txt Original file SHA-256: fb696ebe6c3e531c0773eab309083dc4407c68011c6a08221abb433bdf39ad45 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 REAL TRACE METRIC AND DOMAIN OF THE ELLIPTIC SPIN KERNEL 000002 | ============================================================ 000003 | 26 October 2026. Exact L2 trace, branch weights and positivity test. 000004 | Atlas audit frame: revision 2f83bc6069621370a387b663. No Atlas write. 000005 | 000006 | Question and verdict 000007 | ==================== 000008 | 000009 | The preceding global calculation gives the complete holomorphic rank two 000010 | kernel on 000011 | 000012 | E: y^2=x^4-1, 000013 | 000014 | diagonalized on the punctured base as 000015 | 000016 | C_+=g P_H g^(-1), C_-=g^(-1) P_H g, 000017 | g(x)^4=(x^2-1)/(x^2+1). (1) 000018 | 000019 | Does the flat NS--NS Spin metric turn (1), on Gamma: |x|=1 or its radial push 000020 | offs, into the positive equal time covariance of the already proved elliptic 000021 | Gaussian state? 000022 | 000023 | The exact answer has two parts. 000024 | 000025 | 1. POSITIVE RANGE RESULT. The flat metric gives a closed unitary two sheet 000026 | half density trace with no chosen U(2). Both operators in (1) are bounded 000027 | idempotents. Their closed ranges are exactly the ordinary Hardy space, 000028 | because g and g^(-1) are outer. Hence the orthogonal projection onto the 000029 | actual inside analytic L2 trace range is canonically 000030 | 000031 | P_orth=P_H direct-sum P_H. (2) 000032 | 000033 | The exterior result is the corresponding P_- direct sum P_-. 000034 | 000035 | 2. NEGATIVE STATE IDENTIFICATION. Under the same induced metric, the 000036 | operators C_+ and C_- themselves are not selfadjoint. Thus the bilinear 000037 | closed surface Szego restriction is an oblique Cauchy jump projector, 000038 | not the positive covariance (2). The alternative diagonal weights that 000039 | make (1) orthogonal are not the metric produced by the unitary two sheet 000040 | half density trace. Using them would change the trace/state. 000041 | 000042 | The old Hardy RANGE therefore reappears without a free channel choice, but 000043 | the old Hardy STATE and the previously proved finite annulus KMS state do not 000044 | follow from this fact. The latter lives on the smooth dividing annulus cut. 000045 | No original unitary propagation map from that cut to the ramified equator 000046 | graph is present. Consequently an identity 000047 | 000048 | C_base=T C_E T^*, 0<=C_base<=I, (3) 000049 | 000050 | is proved for the orthogonal radial analytic range projector if C_E denotes 000051 | that radial Calderon projector. It is not proved for the already existing 000052 | elliptic KMS covariance. Defining C_E anew to make (3) true would be a state 000053 | change and is not done here. 000054 | 000055 | 1. The metric fixed by the actual square torus 000056 | ================================================ 000057 | 000058 | Use the period-one square uniformization 000059 | 000060 | E=C/(Z+iZ), 000061 | x(z)=sn(c z|-1), c=2 K(-1)(1+i), 000062 | dx/dz=-i c y. (4) 000063 | 000064 | Thus the flat metric is 000065 | 000066 | ds_E=|dz|=|dx|/(|c| |y|), |c|=2 sqrt(2) K(-1). (5) 000067 | 000068 | For the radial circle Gamma_r, x=r exp(i theta), r different from 1, each of 000069 | the two lifted components has exact arclength measure 000070 | 000071 | dnu_r(theta) 000072 | =r dtheta/[|c| |r^4 exp(4i theta)-1|^(1/2)]. (6) 000073 | 000074 | This is the L2 measure for coefficients in a unit flat Spin frame upstairs. 000075 | It is not selected to repair (1); it is the restriction of (5). 000076 | 000077 | Let a chiral half density on sheet epsilon be 000078 | 000079 | psi_epsilon=f_epsilon(z)(dz)^(1/2). 000080 | 000081 | Define its base trace by the actual Jacobian 000082 | 000083 | (T_r psi)_epsilon(x) 000084 | =f_epsilon(z_epsilon(x)) 000085 | (dz_epsilon/dx)^(1/2) (dx)^(1/2). (7) 000086 | 000087 | Then change of variables gives exactly 000088 | 000089 | sum_epsilon integral_Gamma_r |(T_r psi)_epsilon|^2 |dx| 000090 | =sum_epsilon integral_p^-1(Gamma_r) 000091 | |f_epsilon(z)|^2 |dz|. (8) 000092 | 000093 | Therefore T_r extends uniquely to a closed unitary 000094 | 000095 | T_r:L2(p^-1 Gamma_r,alpha;|dz|) 000096 | ->L2(Gamma_r,C^2;|dx|). (9) 000097 | 000098 | The sheet character transform J from GLOBAL-SPIN-KERNEL.txt is constant and 000099 | unitary. Hence the Hermitian metric after diagonalization remains the identity 000100 | matrix in the standard half density measure |dx|. No channel dependent weight 000101 | appears in (9). 000102 | 000103 | Equivalently, if the Jacobian in (7) is left out, the same Hilbert space has 000104 | the raw weighted measure (6). These two descriptions are unitarily identical; 000105 | one may not use the Jacobian and the weight simultaneously. 000106 | 000107 | 2. The ramified limit and the actual branch domain 000108 | =================================================== 000109 | 000110 | At a mark a_j=i^j write theta=theta_j+s. Since 000111 | 000112 | y^2=exp(4is)-1=4 i s+O(s^2), 000113 | 000114 | equation (6) has the integrable singularity 000115 | 000116 | dnu_1(s)=[1/(2|c|)] |s|^(-1/2)(1+O(s)) ds. (10) 000117 | 000118 | The half density trace factor in (7) has modulus 000119 | 000120 | |dz/dx|^(1/2) 000121 | =[1/sqrt(2|c|)] |s|^(-1/4)(1+O(s)). (11) 000122 | 000123 | Thus T_1 remains an L2 isometry in the graph limit. The vertices have measure 000124 | zero and do not obstruct the closed L2 map. 000125 | 000126 | They do matter for the operator domain. In the Q compatible local coordinate 000127 | 000128 | (x-a_j)/a_j=z_j^2, 000129 | 000130 | the four seam germs are 000131 | 000132 | z_j=+/- exp(+i pi/4) sqrt(s) for s>0, 000133 | z_j=+/- exp(-i pi/4) sqrt(|s|) for s<0. (12) 000134 | 000135 | If one smooth upstairs Spin coefficient is 000136 | 000137 | f(z_j)=f_0+f_1 z_j+O(z_j^2), 000138 | 000139 | its four traced half edge coefficients have the forced form 000140 | 000141 | F_e(s)=kappa_e |s|^(-1/4) 000142 | [f_0+f_1 z_e(s)+O(|s|)], (13) 000143 | 000144 | where kappa_e are the fixed square root Jacobian phases. Formula (13), with 000145 | one common f_0 and one common f_1 before inserting the four roots (12), is the 000146 | actual smooth branch matching. A free U(2) vertex matrix does not occur. 000147 | 000148 | The closed L2 trace range forgets this point matching because values at four 000149 | points have zero L2 measure. The H1 or graph operator domain retains it. In 000150 | particular, the pulled back base generator has 000151 | 000152 | p^*(-i partial_theta) 000153 | =(1+z_j^2)/(2z_j) partial_(z_j), (14) 000154 | 000155 | so it does not preserve all smooth germs (13). The natural L2 trace therefore 000156 | does not by itself produce the old smooth circle domain or one selfadjoint 000157 | four edge Dirac vertex domain. 000158 | 000159 | 3. The smallest selfadjointness countertest 000160 | ============================================ 000161 | 000162 | On |x|=1, 000163 | 000164 | |g(exp(i theta))|^2=|tan(theta)|^(1/2)=:w(theta), 000165 | |g|^(-2)=w^(-1). (15) 000166 | 000167 | Both w and w^(-1) are Muckenhoupt A2 weights: at each of the four marks their 000168 | power is +1/2 or -1/2, strictly between -1 and 1. Hence the weighted Hardy 000169 | theorem makes both similarities in (1) bounded on standard L2. They obey 000170 | 000171 | C_+^2=C_+, C_-^2=C_-. (16) 000172 | 000173 | Bounded idempotence is insufficient for a CAR covariance. Positivity would 000174 | require selfadjointness. 000175 | 000176 | The failure is exact. Multiplication by g is unitary from L2(w dtheta) to 000177 | standard L2, and transports the weighted Riesz projection P_H to C_+. If C_+ 000178 | were selfadjoint, H2 and H2_- would be orthogonal in L2(w dtheta). But 000179 | 000180 | <1,exp(-2i theta)>_w