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Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/MOBIUS-QUARTER-OPERATOR-DICTIONARY.txt
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000171 | 3. Hilbert spaces, domains, and unitary longitudinal decomposition
000172 | -------------------------------------------------------------------
000173 | 
000174 | Use normalized measures dtheta/(2 pi) and dx/(2 pi). Let
000175 | 
000176 |   H_cov = {longitudinal L2 sections satisfying (5)},
000177 |   Dom(B_cov) = H1 sections satisfying (5).              (8)
000178 | 
000179 | Let K=L2(S1_x) with periodic Sobolev domain H1_per, and define
000180 | 
000181 |   D=-i partial_x,
000182 |   B_a=D+a,                    a in {1/4,3/4}.            (9)
000183 | 
000184 | Define maps W_0,W_1:K -> H_cov by
000185 | 
000186 |   (W_0 g)(theta)=exp(i theta/2)  g(2 theta),
000187 |   (W_1 g)(theta)=exp(3 i theta/2)g(2 theta).             (10)
000188 | 
000189 | They obey the NS boundary condition because both phases change sign after
000190 | theta -> theta+2 pi. Their Fourier images are respectively
000191 | 
000192 |   W_0 exp(i m x) = exp(i(2m+1/2)theta),
000193 |   W_1 exp(i m x) = exp(i(2m+3/2)theta).                 (11)
000194 | 
000195 | These are the even-n and odd-n parts of the complete NS basis
000196 | exp(i(n+1/2)theta). They are orthogonal and their union is complete.
000197 | 
000198 | The norm is preserved without an extra square root:
000199 | 
000200 |   integral_0^2pi |g(2theta)|^2 dtheta/(2pi)
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.