TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/MOBIUS-QUARTER-OPERATOR-DICTIONARY.txt Original file SHA-256: 0b5f5cb8a4e47bedc3c6febb00666893da1797335644d0fb058072379589bd59 Inclusive original lines: 201-392 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 | = integral_0^2pi |g(x)|^2 dx/(2pi), (12) 000202 | 000203 | because x=2theta traverses the base circle twice and the factor from 000204 | dx=2dtheta cancels that multiplicity. Hence 000205 | 000206 | W=W_0 direct-sum W_1 000207 | 000208 | is unitary from K direct-sum K onto H_cov, including the stated H1 domains. 000209 | Tensoring the two summands with the corresponding K_j coefficient spaces in 000210 | (F1) gives the full Fourier and Sobolev decomposition just described. 000211 | 000212 | Differentiating (10) gives 000213 | 000214 | B_cov W_0 = 2 W_0 B_1/4, 000215 | B_cov W_1 = 2 W_1 B_3/4, (13) 000216 | 000217 | which proves (1) as an unbounded-operator identity with domains. 000218 | 000219 | The equivalent quasiperiodic-domain presentation uses 000220 | 000221 | H_a={h:h(x+2pi)=exp(2pi i a)h(x)}, 000222 | Q_a=-i partial_x on H1_a. (14) 000223 | 000224 | Multiplication G_a g=exp(i a x)g is unitary K -> H_a and 000225 | 000226 | Q_a G_a = G_a B_a. (15) 000227 | 000228 | Thus the two summands in (1) are exactly the alpha=1/4 and alpha=3/4 000229 | domains, with boundary phases +i and -i. No alpha was fitted or added. 000230 | 000231 | 4. Half-deck action 000232 | ------------------- 000233 | 000234 | Equations (10) and periodicity of g give 000235 | 000236 | T_pi W_0=+i W_0, 000237 | T_pi W_1=-i W_1. (16) 000238 | 000239 | Therefore T_pi^2=-I on both sectors, exactly the A4 order-four lift. The two 000240 | quarter domains are its complete reduced longitudinal character 000241 | decomposition. In the full geometry these are the cj characters of cJ in 000242 | (F2). The signs +i and -i belong to one selected Pin-minus lift; they are not 000243 | two competing spin structures. 000244 | 000245 | 5. Winding-one action 000246 | --------------------- 000247 | 000248 | Let 000249 | 000250 | (u_cov f)(theta)=exp(i theta)f(theta), 000251 | (u_base g)(x)=exp(i x)g(x). (17) 000252 | 000253 | Both preserve the corresponding Sobolev domains. Direct substitution in 000254 | (10) gives 000255 | 000256 | u_cov W_0 g = W_1 g, 000257 | u_cov W_1 g = W_0(u_base g). (18) 000258 | 000259 | In the ordered coefficient basis (1/4,3/4), this is exactly (2). Squaring 000260 | gives (3). 000261 | 000262 | In the quasiperiodic gauge (14), the off-diagonal arrows carry multiplication 000263 | by exp(i x/2). Formula (2) is the same operation after both sectors are 000264 | written in their common periodic coefficient basis. Omitting this gauge 000265 | distinction would incorrectly replace a domain-changing map by an ordinary 000266 | periodic multiplier. 000267 | 000268 | The scalar multiplier u_cov is defined on the orientable-double seam. Under 000269 | the one-step Moebius relation it changes c to -c while leaving J fixed. Thus 000270 | it exchanges the two reduced longitudinal quarter subspaces in (18), but it 000271 | does not preserve one fixed descended Pin lift by itself. A full descended 000272 | endomorphism with this longitudinal factor would also need a specified 000273 | operator anticommuting with J. No such extra operator is needed for the 000274 | present A3 comparison because the original seam multiplier is stated after 000275 | cutting the orientable double. This distinction keeps the full monodromy cJ 000276 | separate from its reduced longitudinal coefficient action. 000277 | 000278 | 6. Spectral flow and the non-double-counting rule 000279 | ------------------------------------------------- 000280 | 000281 | The NS cover spectrum is 000282 | 000283 | spec(B_cov)={n+1/2:n in Z}. (19) 000284 | 000285 | The path implementing u_cov is B_cov(t)=B_cov+t, 000286 | 0<=t<=1. Only n=-1 crosses zero, at t=1/2, so 000287 | 000288 | SF_cov(u_cov)=1. (20) 000289 | 000290 | This is the half-shifted correction of the A3 family. It retains the original 000291 | winding-one spectral flow exactly. 000292 | 000293 | The square implements B_cov(t)=B_cov+2t. The modes n=-1 and n=-2 cross, so 000294 | 000295 | SF_cov(u_cov^2)=2. (21) 000296 | 000297 | By (3), the same index is the direct sum of the two base-quarter paths 000298 | B_a(t)=B_a+t: 000299 | 000300 | SF_1/4(u_base)+SF_3/4(u_base)=1+1=2. (22) 000301 | 000302 | Equations (20)--(22) impose the required bookkeeping rule: 000303 | 000304 | * u_cov is a full-cover winding-one operator and has SF=1; 000305 | * it exchanges the quarter sectors, so it is not a within-sector winding; 000306 | * u_base has SF=1 on each individual quarter domain; 000307 | * the sum of those two indices describes u_cov squared, not u_cov. 000308 | 000309 | Therefore one must not identify the full-cover SF=1 of u_cov with two 000310 | simultaneous within-quarter SF=1 contributions. That would count the cover 000311 | degree twice. 000312 | 000313 | 7. Comparison with the original per-sheet factor d=2 000314 | ----------------------------------------------------- 000315 | 000316 | The later original source states 000317 | 000318 | J=d J_1, 000319 | J_1=SF, 000320 | d=2, 000321 | SF=1, (23) 000322 | 000323 | under the premise of identical sheet contributions. It also proves the 000324 | separate integral identity for a deck-invariant source, 000325 | 000326 | integral_(M_tilde) J = d integral_M J. (24) 000327 | 000328 | Source: 000329 | 000330 | documents/archive--paper-latex--tfpt-theory-fullv27/sections/ 000331 | 003-subsection-the-cubic-fixed-point-equation-cfe.tex:286-326 000332 | SHA256 8ae0867de9ddbd50e3684aa9b9b023d58791252c3fa146da7fc7b8ddf9cc1dd4 000333 | 000334 | The operator dictionary proves the exact candidate equality 000335 | 000336 | d SF(u_cov)=2 SF(u_cov)=2 000337 | =SF(u_cov^2) 000338 | =SF(u_base on 1/4)+SF(u_base on 3/4). (25) 000339 | 000340 | But the equal integers in (25) have different native meanings. The original 000341 | A3 spectral flow is introduced on the already doubled seam Gamma: 000342 | 000343 | documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000344 | 006-section-numerical-constants-high-precision.tex:344-368 000345 | SHA256 034f97c898e1c10caef9480e42f746ef1cfb58b770924e766f3dd5eceb526db8 000346 | 000347 | That passage does not restrict its U_Gamma to one position sheet. Conversely, 000348 | the d=2 lemma calls SF the per-sheet normalization but does not define a 000349 | per-sheet boundary operator or the Fourier transform (10). 000350 | 000351 | There is a precise route to equality. A deck-invariant local source which is 000352 | the same on both position sheets has a two-dimensional sheet trace. Changing 000353 | from the position-sheet basis to the T_pi character basis is unitary, so its 000354 | trace becomes the sum of the 1/4 and 3/4 blocks. If its seam insertion is 000355 | u_cov^2, equation (22) is exactly the original d=2 sum. 000356 | 000357 | What is not yet in the source is the premise in the last sentence: the 000358 | original response functional is not identified with u_cov^2 or with 000359 | diag(u_base,u_base). Since u_cov itself is off diagonal in (2), its SF=1 000360 | cannot simply be called SF=1 on each sheet or each quarter. 000361 | 000362 | The smallest remaining index-map obligation is therefore: 000363 | 000364 | exhibit the original deck-invariant response on the two-sheet seam fiber 000365 | and prove that, under W, its Fredholm or spectral-flow class is 000366 | 000367 | [u_cov^2]=[u_base] direct-sum [u_base], (26) 000368 | 000369 | or give the different explicit per-sheet operator whose two copies produce 000370 | (23). No new geometry or free alpha is needed. 000371 | 000372 | 8. Scope 000373 | -------- 000374 | 000375 | This result derives the two quarter domains and their exact winding action 000376 | from the local A4 NS cover. It retains the direct full-cover source 000377 | (B_1/2,u_cov), for which 0<1/2<1 and the rank-one spectral-flow construction 000378 | needs no quarter choice. It does not identify either quarter character with 000379 | the internal E8/P2 quarter time, construct the marked D8/CAR local-net 000380 | intertwiner, or supply the global spin-surface quadratic refinement and 000381 | diagonal Arf gauging. 000382 | 000383 | Exact checker: 000384 | 000385 | experiments/theory-contracts/canonical-seam-fermionization-20261025/ 000386 | check_mobius_quarter_operator_dictionary.py 000387 | 000388 | It verifies the reduced longitudinal identities (1)--(4), the deck 000389 | characters, coefficient-norm preservation, and all four spectral-flow counts 000390 | using exact rational arithmetic. The full transverse reflection and Sobolev 000391 | statements are covered by the cited Fourier theorem and are not replaced by 000392 | this finite-label checker.