TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/session_coverage/supplements/CUT-GLUE-DIHEDRAL-COVER-BRIDGE.txt Original file SHA-256: d9fbd78c38af022e171d70fe35d5257cac0d79282bb297bfb129631b8c1078a9 Inclusive original lines: 201-344 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 | branch values mu4 on P1. Reflection swaps the two quarter lines and inverts 000202 | the rotation. This is the whole proved geometric content of the bridge. 000203 | 000204 | The compact normal S2 remains the separate normal index surface. Its 000205 | degree one line selects the weak doublet, but no source passage makes it the 000206 | base of C2, and none is needed for the representation result above. 000207 | 000208 | Smallest decisive certificate 000209 | ============================== 000210 | 000211 | research-20261026-fundamental-rethink/geometry/ 000212 | check_dihedral_cover_bridge.py 000213 | 000214 | Normal and python -OO runs have byte-identical output. The exact checks are: 000215 | 000216 | * A^2=-I, A^4=I and RAR=A^{-1}; 000217 | * Q2*=diag(i,-i) on eta,bar(eta), with conjugation inversion; 000218 | * Q2 preserves the intersection form, while kappa reverses it; 000219 | * both actions preserve the positive Hodge metric, with the linear and 000220 | anti-linear reflection actions kept separate; 000221 | * ord(R12)=12 and kappa R12 kappa=R12^{-1}; 000222 | * R12^4=Delta12^4, R12^9=Q12 and descent to C2; 000223 | * genus(C2)=1 with four finite simple branch points. 000224 | 000225 | Cable and knot terminology 000226 | =========================== 000227 | 000228 | No cable construction is needed for either bridge. A literal audit of the 000229 | original paper corpus found zero technical occurrences of cable, cabling, 000230 | knot, torus-knot, mapping-torus, orientation-double and Kabel. German 000231 | "Knoten" occurrences denote graph vertices or spectral zeros. The original 000232 | vocabulary is "oriented double cover", "cut geometry", "gluing" and "spin 000233 | lift". The maps used here are the existing cyclic branched covers and their 000234 | induced cohomology actions. 000235 | Calling them a cable would add a knot or solid torus structure that has not 000236 | been supplied by the seam sources. The useful cabling intuition is only the 000237 | elementary one that repeated gluing records powers of one monodromy. The 000238 | proof uses the displayed cover maps and operator relations instead. 000239 | 000240 | Original and pinned sources 000241 | =========================== 000242 | 000243 | 0. Original cut, double and Calderon reflection: 000244 | 000245 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000246 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex:58-145 000247 | Atlas source paper-source:8027564739689dd02e02f11c 000248 | SHA256 7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79 000249 | 000250 | The older four-corner statement is: 000251 | 000252 | documents/archive--paper-latex--tfpt-theory-fullv25/sections/ 000253 | 009-subsection-spin-lifted-deficit-factor.tex:1-20 000254 | Atlas sources paper-source:307a26903989dae663920511 and 000255 | paper-source:f00f8d45157e4dcca78bcd5f 000256 | SHA256 5c60e2ee85d7d3f3ab76c95c5ea48260583feb76c72ec9b88b3a6249ee4d3b76 000257 | 000258 | The correction of the archived Calderon equals deck claim is authoritative 000259 | for this report: 000260 | 000261 | experiments/theory-contracts/seam-full-collar-reconstruction-20261015/ 000262 | PROOF.txt:14-61,63-123,162-177 000263 | Atlas contract seam-full-collar-reconstruction-20261015 000264 | Atlas finding full-two-face-collar-recovers-signed-operator 000265 | SHA256 9f4effed34316798c6a35e1ac0ec2832cd3720180f476cb8747bb32359b56e6a 000266 | 000267 | 1. Original family surface and its actual D4 action: 000268 | 000269 | documents/archive--tfpt-45--source_extracts--02_carrier_source/ 000270 | sections/006-subsection-family-monodromy-and-occupancy.tex:74-125 000271 | SHA256 179b2d568e897e059c2d1ef804bb9b06ee053483d366fc9d24189f1df44253b2 000272 | 000273 | This defines the compact seam quotient from the deck involution and its spin 000274 | lift, derives four quarter corners, and gives r(z)=iz, s(z)=1/z and 000275 | P1 minus mu4. 000276 | 000277 | 2. Original normal surface, kept separate: 000278 | 000279 | documents/archive--tfpt-45--source_extracts--02_carrier_source/ 000280 | sections/007-subsection-bosonic-relative-index-and-higgs-selection.tex:96-180 000281 | SHA256 cf801a467465db8d72e4eb3f8e32eea2e23393bf42232a225116c34ed093f761 000282 | 000283 | This compactifies the oriented normal slice to S2 and uses L2=O(1). It gives 000284 | no map from that S2 to the flavor surface. 000285 | 000286 | 3. Existing C2/C12 maps and marked clock: 000287 | 000288 | experiments/theory-contracts/e-series-compiler-closure-20261023/ 000289 | COVERS.txt:6-33, 50-62, 91-115, 142-159 000290 | SHA256 cdd04f5f38819b60ddd8488eb1c3c316c6849829387205e78ec5e4bf767f80bf 000291 | 000292 | 4. Existing NS orientation cover and full Green result: 000293 | 000294 | research-20261026-p1-reflection/NS-COVER-FULL-DIRAC-RP.txt:14-31,107-160 000295 | SHA256 a18c7589ac87b0ea875af7a2a46790989c80c836369b9ee001b0417dda6ab154 000296 | 000297 | 5. Earlier source reviewed context for the same nonsplit extension class: 000298 | 000299 | documents/tfpt_research_contracts/sections/004-source.tex:229-284 000300 | SHA256 ea6eabe0b7bec647c69ac92510bb3206ef7939c3974dd11c94a2847c073eecb9 000301 | 000302 | This already relates the central D4 involution, the free deck and the 000303 | nonsplit NS implementation. The new step here is the explicit C2 000304 | cohomology intertwiner and its compatible descent from the fixed C12 clock. 000305 | 000306 | Terminology boundary: the original uses "spin lift", not the later specific 000307 | Pin terminology. Also its symbol C_2 in older family-cycle passages is a 000308 | family cycle, not the later named elliptic cover C2:y^2=x^4-1. 000309 | 000310 | 6. Primary geometry read for the Pin domain and monodromy: 000311 | 000312 | Anik Chakraborty, Pin Equivariance and Local Dirac Boundary Problems on 000313 | the Flat Moebius Band, arXiv:2610.03745v1, sections 3, 6 and 8. 000314 | https://arxiv.org/abs/2610.03745 000315 | 000316 | The A3 ALE route was checked as context, not used as a premise for this 000317 | proof: 000318 | 000319 | Roland Bittleston, Simon Heuveline and David Skinner, 000320 | The Celestial Chiral Algebra of Self-Dual Gravity on Eguchi-Hanson Space, 000321 | arXiv:2305.09451. 000322 | https://arxiv.org/abs/2305.09451 000323 | 000324 | Status 000325 | ====== 000326 | 000327 | PROVED: the exact real D4 representation intertwiner between the NS Pin pair 000328 | and H1(C2,R), including the mu4 branch marking. 000329 | 000330 | PROVED: reflection inversion of the fixed C12 marked clock and its descent 000331 | to C2. 000332 | 000333 | EXCLUDED IN GENERAL: geometric deck reflection equals 2P_C-I. The archived 000334 | formula is retained only as a corrected source claim. The three reflection, 000335 | deck and polarization slots remain distinct. 000336 | 000337 | NOT REQUIRED: a global Cartan choice or a map from the normal S2 to flavor 000338 | P1 for these representation statements. 000339 | 000340 | OPEN: an operator map identifying the full NS Green covariance, time, 000341 | support and field algebra with the original P1 boundary kernel. The present 000342 | bridge supplies one exact polarized monodromy and reflection carrier. Its 000343 | selection as the original physical carrier, and the state and field map, 000344 | remain open.