TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/session_coverage/supplements/CANONICAL-CUT-STATE-TIME-BRIDGE.txt Original file SHA-256: 510bb197905fb037d35dad3e73e7db839bd4ae1a19af864f020c324c6c170422 Inclusive original lines: 201-307 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. 000201 | 000202 | The factor two is the transverse zero mode. After dividing only this ground 000203 | degeneracy and writing q=exp(-pi), 000204 | 000205 | Z_rel/2 = 1+2q+3q^2+6q^3+... . 000206 | 000207 | The coefficient 2 of q is already enough: at the finite retained cylinder the 000208 | channel amplitude contains excited closed channel contributions. Rotating the 000209 | annulus therefore does not identify the state with a single Hardy vacuum. 000210 | Boundary conformal field theory says the same structurally: the open trace is 000211 | equal to propagation between boundary states in the closed channel. Vacuum 000212 | projection follows only in a long channel limit, not merely from exchanging 000213 | the two directions at a finite modulus. 000214 | 000215 | The original tau=i datum does not change this conclusion. At a modular fixed 000216 | point the modulus returns to itself, while spin structures, boundary states and 000217 | characters can still transform nontrivially. The nonzero q coefficient is an 000218 | operator level witness that fixed geometry alone has not selected a fixed 000219 | vacuum vector. 000220 | 000221 | This also explains why the exact v526 cut state is useful but cannot supply the 000222 | missing equality. v526 proves that a pure NS circle state can restrict to a 000223 | faithful thermal half algebra and constructs its standard form. Its own T4 test 000224 | shows that the geometric clock transfer is not its modular flow. Moreover its 000225 | one particle modular energies 000226 | 000227 | 2 log cot(pi/48), 2 log cot(7pi/48) 000228 | 000229 | are not the present nonzero modular ladder n pi. The two results share the 000230 | cut to KMS mechanism, not one time generator. 000231 | 000232 | The other requested originals agree with this typing. v519 proves purity of 000233 | the full chiral NS circle state and reflection positivity after a bond cut. It 000234 | supports the cut mechanism, but it does not identify its finite chiral kernel 000235 | with the present two dimensional D. v813 explicitly finds that the sea state 000236 | does not equidistribute among the deck sectors and locates its surviving bridge 000237 | in modular response normalization rather than vacuum weights. It therefore 000238 | cannot turn the tau=i or index identity into the missing Hardy state map. 000239 | 000240 | 6. Smallest next test for the W8 source map 000241 | 000242 | The next claim should be local and operator specific. It is enough to construct 000243 | one compatible family of maps 000244 | 000245 | iota_I:A_W8(I) -> CAR(L2(I,C2))'' 000246 | 000247 | for proper w intervals, compatible under inclusion, such that the chosen W8 000248 | field has the same reflected correlations and iota intertwines its declared 000249 | source time with the H dynamics. No global one particle unitary and no equality 000250 | of the entire Hardy and KMS representations is required. 000251 | 000252 | The first decisive test is to derive a candidate iota(W8) from an actual field 000253 | dictionary through this same Dirac field, including the b boundary and Pin 000254 | cocycle, and compare its two point function in C with the already computed old 000255 | Hardy W8 two point function. If the q coefficient or the zero mode contribution 000256 | differs after the geometrically fixed conformal factors, the channel rotation 000257 | does not preserve the source state. Equality is only a necessary test. It gives 000258 | at most an isometry of the linear one insertion GNS span; it does not supply a 000259 | local field map, products or higher word correlations. Those require the 000260 | operator dictionary first and then all compatible n point functions. Merely 000261 | matching annulus partition functions, one two point function or abstract type 000262 | III local factors would not pass this test. 000263 | 000264 | 7. Sources and status 000265 | 000266 | Project sources: 000267 | 000268 | * experiments/theory-contracts/original-admissible-source-closure-20261025/ 000269 | NS-COVER-FULL-DIRAC-RP.txt, sections1 to5. 000270 | * experiments/theory-contracts/seam-full-collar-reconstruction-20261015/ 000271 | PROOF.txt, sections3 and4. 000272 | * experiments/theory-contracts/canonical-seam-fermionization-20261025/ 000273 | MOBIUS-QUARTER-OPERATOR-DICTIONARY.txt, sections1 to4. 000274 | * verification/v526_seam_thermal_kms_nariai_bridge.py, R1 toR4 and T4. 000275 | * verification/v519_woit_theta_rp_free.py, the pure NS circle and half cut RP 000276 | checks. 000277 | * verification/v813_p1_index_kms.py, especially its distinction between sea 000278 | weights and modular response normalization. 000279 | * documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000280 | sections/001...tex:65-80 and sections/002...tex:103-143. 000281 | 000282 | Primary comparison: 000283 | 000284 | * John Cardy, Boundary Conformal Field Theory, hep-th/0411189, especially the 000285 | annulus open and closed channel discussion: 000286 | https://arxiv.org/abs/hep-th/0411189 000287 | 000288 | Verdicts: 000289 | 000290 | PROVED: for the charged Gaussian premise on the actual full NS orientation 000291 | cover, the existing length one cut identifies the Green covariance with the 000292 | pure Cauchy graph marginal, uses the same Clifford reflection, preserves local 000293 | w support and fixes modular time 2H. 000294 | 000295 | EXCLUDED: direct equality of the open H time with the old longitudinal quarter 000296 | Hardy time, and vacuum projection from finite channel exchange alone. 000297 | 000298 | EXCLUDED: real time mixing for the compact periodic tau_H. This is not promoted 000299 | to an exclusion of the archived Euclidean clustering clause. 000300 | 000301 | OPEN: derivation of the charged Gaussian premise from bare P1 data, the exact 000302 | flow and limit meant by the archived clustering clause, and the 000303 | local W8 operator, state and time intertwiner between the old quarter source 000304 | and this NS cut package. 000305 | 000306 | No E8 extension, new oscillator, freely selected purification or physical time 000307 | was introduced.