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: 181-353 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. 000181 | =(1/(2pi)) integral_0^(2pi) 000182 | |tan theta|^(1/2) exp(-2i theta) dtheta 000183 | =-1/sqrt(2), (17) 000184 | 000185 | while 1 lies in H2 and exp(-2i theta) lies in H2_-. The imaginary part 000186 | vanishes by symmetry. The real integral follows from 000187 | 000188 | integral_0^(pi/2) tan(theta)^(1/2) cos(2theta)dtheta 000189 | =-pi sqrt(2)/4. 000190 | 000191 | Therefore 000192 | 000193 | C_+^* != C_+. 000194 | 000195 | Replacing w by w^(-1) changes the sign in (17), so C_- is not selfadjoint 000196 | either. Consequently neither C_+ nor C_- satisfies 0<=C<=I in the actual 000197 | standard trace Hilbert space (9). 000198 | 000199 | One can make C_+ orthogonal by declaring the target weight |g|^(-2), and C_- 000200 | by declaring |g|^2. But after (7)-(9) the actual induced metric is the identity 000201 | in both character channels. The diagonal replacement 000202 | 000203 | diag(|g|^(-2),|g|^2) (18) 000204 | 000205 | is therefore an additional nonunitary metric change. It is not supplied by 000206 | the flat NS--NS Spin metric and cannot be used to identify the state. 000207 | 000208 | This is also the conceptual distinction between the three kernels: 000209 | 000210 | * The algebraic Szego half form gives the bilinear two point function. 000211 | 000212 | * Its boundary Plemelj operator gives the oblique Cauchy idempotents (1). 000213 | 000214 | * The orthogonal Szego or Calderon projection uses the Hermitian boundary 000215 | metric and projects orthogonally onto the closed analytic trace range. 000216 | 000217 | Only the third object is automatically a positive one particle covariance. 000218 | 000219 | 4. Why the Hardy range nevertheless returns canonically 000220 | ======================================================== 000221 | 000222 | The function g is analytic and nonzero in |x|<1. Its boundary logarithm is 000223 | integrable and it has no inner factor. Both g and g^(-1) lie in H2 because 000224 | their four boundary powers are only +/-1/4. Hence both are outer. 000225 | 000226 | Outer cyclicity gives the domain-safe statements 000227 | 000228 | closure{g p:p a polynomial}=H2, 000229 | closure{g^(-1) p:p a polynomial}=H2. (19) 000230 | 000231 | Every displayed polynomial vector is fixed by the corresponding idempotent: 000232 | C_+(g p)=g p and C_-(g^(-1)p)=g^(-1)p. Conversely every vector in either 000233 | range is the L2 boundary value of an analytic Smirnov function and hence, 000234 | because that boundary value is in L2, belongs to H2. Since bounded 000235 | idempotents have closed ranges, (19) therefore sharpens to 000236 | 000237 | Ran(C_+)=Ran(C_-)=H2. (20) 000238 | 000239 | For r<1 the statement is even simpler: g and g^(-1) are bounded analytic and 000240 | boundedly invertible on the closed radial disk, so equality in (20) holds 000241 | before taking a closure. At r=1, outer cyclicity supplies the exact limit. 000242 | 000243 | Therefore the orthogonal projection onto the complete two channel inside 000244 | trace range, using the actual metric (9), is exactly (2). No U(2), profile or 000245 | new metric is selected. For r>1 the same argument in the exterior chart gives 000246 | 000247 | P_orth,out=P_- direct-sum P_-. (21) 000248 | 000249 | This is the precise positive recovery of the Hardy object. It is a statement 000250 | about the closed L2 analytic RANGE. It does not retain the pointwise branch 000251 | matching (13), and it does not say that the oblique correlator (1) equals the 000252 | orthogonal projection (2). 000253 | 000254 | 5. Comparison with the proved elliptic Gaussian state 000255 | ====================================================== 000256 | 000257 | The existing positive theorem uses the dividing Real form rho(z)=-bar(z), the 000258 | smooth half annulus 0