TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SPIN-TRACE-LOCAL-KERNEL.txt
Original file SHA-256: 591051be956f8bb3c464d039dca502ba66b1274b8faf84c73c411be10c84d0f7
Inclusive original lines: 191-274
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000191 |     regular-singular domain (2)-(4) on the four incident germs.
000192 | 
000193 | Consequently a punctured local gauge identifies (9) with a quarter kernel,
000194 | but it becomes singular as X^(+/-1/4) at the branch mark. A smooth bounded
000195 | circle conjugacy across that mark does not exist. Replacing the graph by one
000196 | normalized component, a radial push-off or a chosen vertex channel would be
000197 | an additional source selection unless it is supplied by the original trace
000198 | map.
000199 | 
000200 | 4. Quarter clock and loop typing
000201 | ================================
000202 | 
000203 | The geometric clock Q:x->ix permutes the four branch marks and transports
000204 | their local kernels. It does not turn a small branch meridian into the old
000205 | longitudinal circle generator. Therefore the following statements have
000206 | different scopes:
000207 | 
000208 |   proved: every marked cusp carries the natural spin quarter pair;
000209 |   proved: Q transports the four cusp kernels and their regular-singular
000210 |           domains;
000211 |   open:   one original Q-compatible trace map identifies a selected global
000212 |           graph channel with the old smooth quarter-circle domain.
000213 | 
000214 | This resolves the apparent coefficient mismatch. The raw p_*alpha
000215 | monodromy (+1,-1) is correct for coefficient functions. The ramified
000216 | spin-frame trace has the central-lift monodromy (-i,+i). Neither result
000217 | replaces the other, and the latter does not erase the global V4/E8 graph
000218 | obstruction or prove that the same ordinary connection is preserved.
000219 | 
000220 | 5. Consequence for the holomorphic source kernel
000221 | ================================================
000222 | 
000223 | The rank-two kernel (8) is obtained by an exact pushforward of the already
000224 | given elliptic Szego half-form, including its square-root Jacobian. Therefore
000225 | the holomorphic two-point kernel, and algebraically the determinant Wick
000226 | expressions built from that kernel, transport to the quasipermutation,
000227 | ramified spin-trace chart on the punctured base with the cover-induced domain
000228 | (4). The central lift and its phase are inherited from the original cover
000229 | spinor rather than selected independently.
000230 | 
000231 | This local half-form calculation does not by itself prove a positive
000232 | equal-time CAR covariance on a base Hilbert space. Such a claim additionally
000233 | requires a specified real slice and one-particle inner product, a closed trace
000234 | operator T across all branch vertices, and the operator statements
000235 | 
000236 |   C_base=T C_E T^*,       C_base=C_base^*,       0<=C_base<=I,
000237 | 
000238 | with the relevant isometry or coisometry property of T. None of those
000239 | Hilbert-space statements follows merely from the bilinear holomorphic formula
000240 | (7a). In particular, a square-root Jacobian gives the correct tensorial
000241 | transformation law but is not on its own a proof of unitarity or positivity.
000242 | 
000243 | The strongest positive statement proved here is therefore narrower:
000244 | 
000245 |   the same elliptic Szego half-form has the old quarter Cauchy principal part
000246 |   at every marked branch point, including its central spin phase.
000247 | 
000248 | It still does not supply a single periodic old-circle field. The first
000249 | remaining source equation is a Q-compatible global trace or vertex-domain
000250 | intertwiner from the actual ramified equator graph to the old smooth carrier.
000251 | Without that equation and the Hilbert-space conditions above, local quarter
000252 | conjugacy cannot be promoted to equality of CAR state, time or winding
000253 | domains.
000254 | 
000255 | Sources reused
000256 | ==============
000257 | 
000258 | For the standard twisted-fermion category in which square-root Jacobians and
000259 | quasipermutation monodromy are kept together, compare M. Bershtein,
000260 | P. Gavrylenko and A. Marshakov, "Twist-field representations of W-algebras,
000261 | exact conformal blocks and character identities", JHEP 08 (2018) 108,
000262 | arXiv:1705.00957, especially Sections 3 and 6. Equations (6)-(9) above are a
000263 | direct local calculation and do not depend on importing an additional model
000264 | from that paper.
000265 | 
000266 |   research-20261026-elliptic-source-continuation/
000267 |   ELLIPTIC-GREEN-RP.txt
000268 |   UPDATE-REAL-SLOTS-AND-SOURCE.txt
000269 |   ELLIPTIC-PUSHFORWARD-ANCHOR-TEST.txt
000270 | 
000271 |   experiments/theory-contracts/original-admissible-source-closure-20261025/
000272 |   ORIGINAL-BASES-AND-ELLIPTIC-SPIN.txt
000273 | 
000274 | No Atlas record or canonical project file was changed.