TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_unification_20261027/analytic/C3-CAUCHY-HARDY-ARROW.txt Original file SHA-256: 3c8c2b55ca3a58469aaaa61be3e2d74b2075cc33d132fef799790fad22c1bdc2 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 | THE CUBIC CAUCHY RANGE: HARDY CLOSURE AND THE REMAINING OPERATOR DOMAIN 000002 | ============================================================================ 000003 | 27 October 2026. Source audit and exact local branch calculation. 000004 | No Atlas or paper source was changed. 000005 | 000006 | 000007 | QUESTION AND VERDICT 000008 | ==================== 000009 | 000010 | On 000011 | 000012 | C: eta^2=1-w^6, 000013 | 000014 | the original marked function 000015 | 000016 | z=eta/(1-w^3), w^3=(z^2-1)/(z^2+1) (1) 000017 | 000018 | defines a three sheeted map pi:C->P1_z. The chosen spin line S_0 satisfies 000019 | S_0^2=K_C and is the unique c invariant even theta characteristic. Does the 000020 | actual primitive boundary operator have Cauchy range 000021 | 000022 | H2(pi^(-1)D,S_0)=H2(D) direct-sum H2(D) direct-sum H2(D), (2) 000023 | 000024 | with the old seam shift represented by multiplication by z? 000025 | 000026 | The smallest decisive test has two exact outcomes. 000027 | 000028 | 1. CONDITIONAL ANALYTIC THEOREM. If the supplied operator is the standard 000029 | chiral spin Dirac on the normalization of pi^(-1)D, with its induced 000030 | half form norm, then its closed L2 Cauchy range is canonically (2). No 000031 | sheet metric, U(3) mixing, or boundary profile is selected. The cubic 000032 | branch exponent is exactly -1/3 and is L2 integrable. 000033 | 000034 | 2. SOURCE AND DOMAIN OBSTRUCTION. The smooth ambient spinor domain is not 000035 | the direct sum of three unrelated H1 circle domains. At each of the four 000036 | branch values it retains a six half edge matching condition. That 000037 | condition disappears only after L2 closure. The original TFPT boundary 000038 | source supplies an abstract self-adjoint B_Sigma on a smooth product 000039 | collar, but gives neither its formula nor a domain map to this cubic 000040 | boundary. The older D_Gamma and U_Gamma supply the circle generator and 000041 | winding, but not this missing map. Therefore the actual operator arrow 000042 | is not yet proved by the original source. 000043 | 000044 | This is a type level decision, not a new model. Standard Calderon theory can 000045 | transport the result after a marked collared spin intertwiner is given. It 000046 | cannot construct that intertwiner or replace the missing B_Sigma by a newly 000047 | chosen Dolbeault operator. 000048 | 000049 | 000050 | 1. THE OPEN PULLBACK IS THREE DISKS 000051 | =================================== 000052 | 000053 | Let D={|z|<1}. The only ramification values of pi are 000054 | 000055 | z_j in {1,-1,i,-i}. (3) 000056 | 000057 | They lie on the boundary. Hence pi is unramified over D. Formula (1) has 000058 | three holomorphic cubic roots over D, so 000059 | 000060 | pi^(-1)D = D_0 disjoint-union D_1 disjoint-union D_2, (4) 000061 | 000062 | and pi restricts to a biholomorphism D_a->D on every component. The centers 000063 | are q_a=-omega^a above z=0. The source symmetries act by 000064 | 000065 | c:(w,eta,z)->(omega w,eta,z), 000066 | Q:(w,eta,z)->(1/w,-i eta/w^3,i z). (5) 000067 | 000068 | Thus c cyclically permutes the three disks, while Q covers the quarter turn 000069 | z->i z and sends q_a to q_(-a). These are inherited marked actions, not a 000070 | choice of a three channel frame. 000071 | 000072 | On D_a choose the holomorphic half form e with 000073 | 000074 | e^2=dz. (6) 000075 | 000076 | It is unique up to sign. For psi=f(z)e the induced spin norm is 000077 | 000078 | ||psi||^2_boundary = integral_|z|=1 |f(z)|^2 |dz|. (7) 000079 | 000080 | Therefore the standard chiral spin kernel on the normalized disks has 000081 | orthogonal boundary range H2(D)^3 and orthogonal projector 000082 | 000083 | P_norm=P_H direct-sum P_H direct-sum P_H. (8) 000084 | 000085 | The scalar presentation H2(D)^3 is defined up to the three harmless frame 000086 | signs. The norm, closed subspace, and orthogonal projector in (7) and (8) are 000087 | intrinsic. No frame sign creates a metric fit or changes the range. 000088 | 000089 | 000090 | 2. THE EXACT CUBIC BRANCH EXPONENT 000091 | ================================== 000092 | 000093 | At a branch point over z_j=+/-1 use u=w. At a branch point over z_j=+/-i use 000094 | u=1/w. In either case (1) gives, after choosing the appropriate value z_j, 000095 | 000096 | z=z_j(1+u^3+(1/2)u^6+O(u^9)), 000097 | dz=3z_j u^2(1+O(u^3))du. (9) 000098 | 000099 | Write the boundary as z=z_j exp(i s). The six incident germs are 000100 | 000101 | u_(a,+)(s)=omega^a exp(i pi/6)s^(1/3)(1+O(s)), s>0, 000102 | u_(a,-)(s)=omega^a exp(-i pi/6)|s|^(1/3)(1+O(s)), s<0. (10) 000103 | 000104 | Let an ambient smooth spinor germ be 000105 | 000106 | psi=F(u)(du)^(1/2), 000107 | F(u)=F_0+F_1u+F_2u^2+O(u^3). (11) 000108 | 000109 | In the base frame e=(dz)^(1/2), the coefficient is 000110 | 000111 | f(z)=F(u)(du/dz)^(1/2) 000112 | =(3z_j)^(-1/2)u^(-1)(F_0+F_1u+F_2u^2+O(u^3)). (12) 000113 | 000114 | Consequently 000115 | 000116 | f_(a,+/-)(s)=kappa_(j,a,+/-)|s|^(-1/3) 000117 | [F_0+F_1u_(a,+/-)(s)+F_2u_(a,+/-)(s)^2+O(s)]. (13) 000118 | 000119 | Its squared norm has leading behavior 000120 | 000121 | |f(s)|^2 ds = O(|s|^(-2/3))ds, (14) 000122 | 000123 | which is integrable. The four branch values have measure zero. Hence the 000124 | half form trace extends to L2 on all three normalized boundary circles. 000125 | 000126 | This is the cubic analogue of the already proved square root exponent 000127 | -1/4 in TRACE-METRIC-AND-DOMAIN.txt. The change from -1/4 to -1/3 is forced 000128 | by dz approximately u^2du and is not a fitted weight. 000129 | 000130 | 000131 | 3. WHY THE CLOSED RANGE IS H2^3 000132 | ================================ 000133 | 000134 | The smooth ambient core retains equation (13). In particular all six 000135 | half-edge germs at one branch value arise from the same coefficients F_k. 000136 | Its closure in a specified Sobolev or graph norm retains exactly the traces 000137 | controlled by that norm. Ordinary intrinsic H1 on the six edges retains the 000138 | common vertex value F_0; it does not retain the complete Taylor series. 000139 | Higher Sobolev orders retain the corresponding finite jet. None of these 000140 | finite conditions reduces the closed L2 analytic range. 000141 | 000142 | There is a direct density proof at every fixed finite regularity. For m>=1, 000143 | multiplication by 000144 | 000145 | a_m(z)=(1-z^4)^m (15) 000146 | 000147 | forces as many boundary jets to vanish as required once m is large enough. 000148 | Therefore triples a_m p_a, with p_a polynomials, satisfy the matching needed 000149 | for any one fixed Sobolev order after m is chosen for that order. The 000150 | polynomial 1-z^4 has no zero in D and only boundary zeros, hence it is outer. 000151 | Outer cyclicity implies 000152 | 000153 | closure{(1-z^4)^m p:p polynomial}=H2(D). (16) 000154 | 000155 | For a literal C-infinity core one may use one explicit flat outer factor. 000156 | Choose 00 in D, this branch is holomorphic and zero-free. Near every 000161 | fourth root z_j of one, log|G| is of order -|s|^(-rho), which is integrable. 000162 | Thus G is outer. It and every derivative tend to zero at each z_j from inside 000163 | the disk. The intrinsic spin coefficients u G(z(u))p_a(z(u)) are therefore 000164 | flat at every vertex. They admit compatible smooth extensions through the 000165 | outside wedges by the ordinary smooth extension and partition construction; 000166 | holomorphy is required only on the inside disks. Polynomial multiples of G 000167 | are dense in H2 by outer cyclicity. 000168 | 000169 | Applying either finite-order construction, or (17) for the smooth core, 000170 | independently on the three disks proves that the L2 closure of the compatible 000171 | analytic core is all of H2(D)^3. This proves (8) even though a chosen strong 000172 | domain remains coupled. 000173 | 000174 | The result is precisely about the closed Cauchy range. It does not identify 000175 | an oblique Plemelj kernel, a finite-temperature covariance, or an interacting 000176 | state with P_norm. 000177 | 000178 | 000179 | 4. THE H1 DOMAIN DOES NOT SPLIT ON THE SINGULAR BOUNDARY 000180 | ======================================================== 000181 | 000182 | There are two different boundary objects. 000183 | 000184 | * On the normalization, the boundary is three disjoint smooth circles. 000185 | Its standard first-order domain is H1(S1)^3. This gives (8) directly. 000186 | 000187 | * In the ambient curve, the same boundary image has four vertices, each 000188 | with six incident half edges. Its smooth core obeys the common Taylor 000189 | relations (13). The common vertex value survives in its intrinsic H1 000190 | closure. Higher jets survive only in correspondingly stronger norms. 000191 | 000192 | The naked constant coefficient circle generator does not preserve the 000193 | natural traced ambient smooth core. Indeed an allowed leading coefficient 000194 | f(s) approximately s^(-1/3) is sent by -i partial_s to a term of order 000195 | s^(-4/3), which is not L2. The intrinsic smooth operator in the u coordinate, 000196 | when expressed through s approximately u^3, necessarily acquires a 000197 | degenerate variable coefficient and the spin connection term. Thus the 000198 | following three claims cannot be combined without another domain theorem: 000199 | 000200 | (i) the constant generator -i partial_theta,