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: 191-279
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 | Thus Q exchanges K_+ and K_- and transports (14) to (15). The unordered
000192 | rank two kernel is Q invariant, while neither diagonal line is. The two
000193 | abstract coherent choices of local square root sign found from the product
000194 | relation amount here to swapping the two character labels. The normalized
000195 | kernel, its residue and the unordered partition fix the full rank two object;
000196 | they do not canonically name one of its two lines.
000197 | 
000198 | 5. Exact finite replay
000199 | ======================
000200 | 
000201 | An inline SymPy replay checked
000202 | 
000203 |   J^*J=I,
000204 |   J^* [[a+b,i(a-b)],[-i(a-b),a+b]] J /2=diag(a,a^(-1)),
000205 |   (P)(-P)(P)(-P)=I,
000206 |   product of the four matrices in (16)=I,
000207 |   every matrix in (16) has square -I,
000208 |   [(x^2-1)/(x^2+1)] at x->ix equals its reciprocal.
000209 | 
000210 | The pole check is also exact: as z approaches x, a and b approach one, so
000211 | (12) has identity residue. The conjugate sheet numerator a-b vanishes to
000212 | first order and removes the other apparent pole.
000213 | 
000214 | As an independent diagnostic, the square torus uniformization was evaluated
000215 | at 80 decimal digits using
000216 | 
000217 |   x(z)=sn(c z|-1),       c=2 K(-1)(1+i),
000218 |   y(z)=i cn(c z|-1) dn(c z|-1),
000219 |   A(z)=i sqrt(1-x(z)^2),       B(z)=y(z)/A(z).
000220 | 
000221 | At three generic complex point pairs, the algebraic value
000222 | 
000223 |   -i c [A(z)B(w)+B(z)A(w)]/[2(x(z)-x(w))]
000224 | 
000225 | agreed with the normalized NS--NS theta kernel
000226 | 
000227 |   pi theta_3(pi(z-w)) theta_1'(0)
000228 |        /[theta_3(0) theta_1(pi(z-w))]
000229 | 
000230 | to absolute errors 8.5e-81, 1.3e-80 and 9.5e-81. This is a diagnostic of
000231 | normalization and spin characteristic. The proof remains the pole, multiplier
000232 | and H0(alpha)=0 argument in Section 2.
000233 | 
000234 | 6. Scope and consequence
000235 | ========================
000236 | 
000237 | PROVED:
000238 | 
000239 |   [E] The Q invariant partition (6) is the already fixed alpha spin class.
000240 |   [E] Formula (2) is the actual normalized NS--NS elliptic Szego kernel.
000241 |   [E] The complete rank two spin trace diagonalizes as (12), with no
000242 |       remainder and no fitted connection.
000243 |   [E] Its four global peripheral matrices, product relation and Q exchange
000244 |       are (16)-(17).
000245 | 
000246 | This is a constructive global source connection in the holomorphic sense:
000247 | the original curve, spin class and normalized elliptic Green function already
000248 | supply the full global two point kernel on the four marked punctured carrier.
000249 | It strengthens the earlier local quarter result substantially.
000250 | 
000251 | OPEN:
000252 | 
000253 |   [O] A holomorphic Szego kernel is not by itself an ordered equal time CAR
000254 |       covariance. Positivity still requires the original real slice, trace
000255 |       map and one particle inner product.
000256 | 
000257 |   [O] The four punctured or ramified base carrier is not the old smooth
000258 |       circle. Equations (12)-(17) do not identify its graph domain with the
000259 |       original B_(1/4) direct-sum B_(3/4) circle domain.
000260 | 
000261 |   [O] Since Q exchanges K_+ and K_-, selecting one diagonal channel would be
000262 |       an extra symmetry breaking choice. The canonical source object is the
000263 |       full unordered rank two pair.
000264 | 
000265 | Source anchors
000266 | ==============
000267 | 
000268 | [OMITTED: non-scientific conversation or execution metadata]
000269 | [OMITTED: non-scientific conversation or execution metadata]
000270 | [OMITTED: non-scientific conversation or execution metadata]
000271 | [OMITTED: non-scientific conversation or execution metadata]
000272 | 
000273 |   research-20261026-elliptic-source-continuation/ELLIPTIC-GREEN-RP.txt:1-216
000274 |   SHA256 d8799f04851a9bd539b05b4d013b38eb00de9bec07dddde6438023aaf1facd80.
000275 | 
000276 |   research-20261026-spin-transport/SPIN-TRACE-LOCAL-KERNEL.txt:1-253
000277 |   Earlier local principal-part result; (10)-(13) provide the new exact global
000278 |   completion. That file was not changed here.
000279 |