TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SOURCE-INDUCED-SEAM-TRANSPORT.txt Original file SHA-256: 1474792f0edfc9f0fc34e354b73cfc8c0a8591d0ccc78312bcf8e57f68058e05 Inclusive original lines: 191-380 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. 000191 | 000192 | This is the requested common global frame. It also shows that the induced 000193 | flat system is reducible. In the character basis e_plus=(1,1) and 000194 | e_minus=(1,-1), P becomes Z=diag(1,-1), and the two global rank one 000195 | characters have peripheral values 000196 | 000197 | (+1,-1,+1,-1), (-1,+1,-1,+1). (6d) 000198 | 000199 | The decomposition does not yield a Q invariant line. Conjugation by Q sends 000200 | 000201 | lambda -> -A+i lambda modulo 2 Lambda, 000202 | 000203 | which cycles 0,A,A+B,B. The possible discrepancy by 2 Lambda is invisible 000204 | to chi_alpha. In the character basis one may take the raw Q intertwiner to 000205 | be X=[[0,1],[1,0]], so 000206 | 000207 | X M_j X^(-1)=M_(j+1), X Z=-Z X. (6e) 000208 | 000209 | Thus Q exchanges the two characters. With the holomorphic Spin Jacobian, 000210 | 000211 | Q_hat=zeta_8 X, h_hat=i Z, 000212 | 000213 | one obtains Q_hat^4=-I, h_hat^2=-I and the same central anticommutator as in 000214 | (7d) below. 000215 | 000216 | The half form Jacobian around a branch supplies a scalar c_j with c_j^2=-1. 000217 | For oriented peripheral generators its lifted matrices have the form 000218 | 000219 | M_tilde_j=c_j M_j. (6f) 000220 | 000221 | The punctured sphere product relation requires product_j c_j=1. Q covariance 000222 | requires c_(j+1)=c_j. Hence there are exactly two coherent Q equivariant 000223 | choices, c_j=+i for every j or c_j=-i for every j. For the latter, the local 000224 | diag(-i,+i) convention, the common character frame gives 000225 | 000226 | (M_tilde_0,M_tilde_1,M_tilde_2,M_tilde_3) 000227 | =(diag(-i,+i),diag(+i,-i),diag(-i,+i),diag(+i,-i)), 000228 | product_j M_tilde_j=I. (6g) 000229 | 000230 | The choice c=+i negates all four matrices and matches h_hat=i Z. Both choices 000231 | satisfy the product relation and Q covariance. Therefore the local half 000232 | density factor diag(-i,+i) does not by itself choose a global lift. The 000233 | isomorphism class of alpha fixes (6c) and (6d); choosing between (6g) and its 000234 | negative is the remaining deck Spin linearization or orientation convention. 000235 | The explicitly normalized source operator D fixes such a convention on its 000236 | stated twisted section carrier, up to the unavoidable overall sign. It does 000237 | not turn either character line into a Q invariant line. 000238 | 000239 | GLOBAL-SPIN-KERNEL.txt performs the further normalized Szego calculation. 000240 | It shows that the actual alpha kernel fixes the unordered pair of these two 000241 | choices and diagonalizes globally, while Q exchanges their labels. Thus no 000242 | extra rank two kernel is needed, although no single Q invariant line results. 000243 | 000244 | 2. The two radial push offs and Q 000245 | ================================= 000246 | 000247 | Along x=r exp(i theta), the quantity x^4-1 traces, during one quarter turn in 000248 | theta, a circle centered at -1. 000249 | 000250 | * For r<1 that circle does not enclose zero. A continuous square root 000251 | returns to itself after one quarter. Q preserves each radial component. 000252 | 000253 | * For r>1 it encloses zero once. Its continuous square root changes sign 000254 | after one quarter. The algebraic Q keeps y fixed, so it exchanges the two 000255 | continuously labelled radial components. 000256 | 000257 | Let S be the quarter shift theta -> theta+pi/2 and let q be the fixed Spin 000258 | lift scalar, q^2=i. Thus q=plus or minus zeta_8 and q^4=-1. In the continuous 000259 | component frames, 000260 | 000261 | Q_in=q S, Q_out=q P S, 000262 | Q_out=P Q_in. (7) 000263 | 000264 | Both lifts have fourth power -I and eighth power I. Formula (7) is the exact 000265 | relation between the two Q equivariant radial shards. Here P is the local 000266 | sheet permutation. It records the existing branch deck on component labels, 000267 | but is not by itself the full global deck Spin pullback on alpha. 000268 | 000269 | This distinction is forced globally on the actual square torus. With 000270 | 000271 | T=(1+i)/2, h z=-z+T, Q z=i z, 000272 | 000273 | one has 000274 | 000275 | hQz=-iz+T, Qhz=-iz+iT, hQz-Qhz=1. (7a) 000276 | 000277 | The given flat alpha character obeys chi_alpha(1)=chi_alpha(i)=-1. Therefore 000278 | the raw pullbacks on alpha satisfy the exact central relation 000279 | 000280 | U_h U_Q=-U_Q U_h. (7b) 000281 | 000282 | Adding the genuine holomorphic Spin derivative factors does not remove it. 000283 | For the standard source normalization 000284 | 000285 | h_hat=i U_h, Q_hat=zeta_8 U_Q, 000286 | h_hat^2=-I, Q_hat^4=-I, (7c) 000287 | 000288 | and hence 000289 | 000290 | h_hat Q_hat=-Q_hat h_hat. (7d) 000291 | 000292 | Consequently Q exchanges the plus i and minus i eigenspaces of h_hat. A 000293 | single deck quarter eigensector cannot be selected as a globally Q invariant 000294 | circle. This central sign comes from the already fixed alpha character and 000295 | the actual affine h,Q actions; it is not an auxiliary Spin or sheet factor. 000296 | 000297 | In the common quarter interval frame used in (1), define piecewise constant 000298 | 000299 | G_j=P^j on (theta_j,theta_(j+1)). (8) 000300 | 000301 | It removes the four P jumps because P^2=I and P^4=I. Hence it unitarily maps 000302 | the two unmarked radial operators to each other. Under the quarter clock it 000303 | changes by P. Therefore it is not an equivalence preserving the same Q mark. 000304 | The two resolutions are spectrally equal and marked differently. 000305 | 000306 | 3. Which operator is literally the pullback of -i partial_theta? 000307 | ================================================================ 000308 | 000309 | For r different from 1, p restricts on each radial component with degree one. 000310 | In a flat parallel alpha frame, the literal pullback is 000311 | 000312 | B_r=-i d/dtheta (9) 000313 | 000314 | on two copies of H1(S1) with periodic coefficients. Equivalently, in the 000315 | common cut frame it acts on piecewise H1 C2 valued functions obeying 000316 | 000317 | Psi(theta_j+)=T_j Psi(theta_j-), 000318 | 000319 | with all T_j=I on one resolution and all T_j=P on the other. These are 000320 | self adjoint domains because the T_j are unitary. 000321 | 000322 | The all mode proof is immediate. On each open quarter an eigenfunction is 000323 | exp(i lambda theta)v. Going around all four quarters gives 000324 | 000325 | exp(2 pi i lambda) (product_j T_j)v=v. 000326 | 000327 | Both products equal I. Hence 000328 | 000329 | lambda in Z, with multiplicity 2, (10) 000330 | 000331 | for either radial side. The gauge (8) gives the explicit unitary equivalence. 000332 | 000333 | If coefficients are written in the moving tangent frame, they are 000334 | antiperiodic because the total tangent Spin holonomy is -1. The same pulled 000335 | back connection is then 000336 | 000337 | B_pull,mov=-i d/dtheta+1/2 (11) 000338 | 000339 | up to the orientation convention. It still has the integer spectrum (10). 000340 | The naive operator -i d/dtheta on that antiperiodic domain is a different 000341 | intrinsic bounding circle Dirac operator and has spectrum Z+1/2. 000342 | 000343 | Neither reading gives the old quarter spectrum Z+1/4. The order 8 Q lift in 000344 | (7) is a quarter rotation symmetry. It is not parallel transport through a 000345 | full circle and cannot be substituted for a boundary condition. A quarter 000346 | twist must still come from the original phi=pi/2 domain declaration or from 000347 | an actual source intertwiner. 000348 | 000349 | 3.1 Failure at the ramified seam itself 000350 | --------------------------------------- 000351 | 000352 | At r=1, use x=a_j(1+z^2). Since dx/dtheta=i x, 000353 | 000354 | dz/dtheta=i(1+z^2)/(2z), 000355 | 000356 | and therefore 000357 | 000358 | p^*(-i partial_theta)=(1+z^2)/(2z) partial_z. (12) 000359 | 000360 | The coefficient has a pole. For a smooth local alpha section 000361 | 000362 | f(z)=f_0+f_1 z+f_2 z^2+..., 000363 | 000364 | the leading result is f_1/(2z). In the base dtheta measure its square is not 000365 | locally integrable. Regularity forces the odd first jet to vanish and, for a 000366 | full invariant domain, removes the deck odd local branch. The regular flat 000367 | alpha connection has local connection form zero and cannot cancel (12). 000368 | 000369 | The intrinsic Dirac operator using the smooth z coordinate is regular, but 000370 | it is a different operator. It is not the pullback of the written base 000371 | -i partial_theta. This is the exact operator obstruction at the branch. 000372 | 000373 | 4. Why both resolutions do not make one graph vertex rule 000374 | ========================================================== 000375 | 000376 | The full pullback of the marked equator has four valence 4 ramification 000377 | vertices. At one vertex the first order Green boundary form has signature 000378 | (2,2) on the four half edge traces. A self adjoint local graph domain must 000379 | choose a two dimensional Lagrangian subspace. 000380 |