TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/SAME-SOURCE-AFFINE-DEFECT.txt
Original file SHA-256: 514356e9ca01f8aa02399020a503f6cd782c879b00220274b7c4377cbbad7255
Inclusive original lines: 201-400
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000201 | they cannot be counted as an independent physical phase after intertwiner
000202 | bases are quotiented out.
000203 | 
000204 | For the marked spinor intertwiner X=E_s, conjugation by U_tau changes the
000205 | sign only on the v<->c channel.  Therefore every mixed word is unchanged if
000206 | W and X are transported together.  If X is instead held fixed as an extra
000207 | marking, then
000208 | 
000209 |   W_kappa X W_kappa^* X^* = -i kappa I.
000210 | 
000211 | This scalar distinguishes the two representatives only relative to that
000212 | additional fixed action of X.  It is not an invariant of the unmarked
000213 | four-sector extension.
000214 | 
000215 | 
000216 | 4. Explicit affine defect operator
000217 | ----------------------------------
000218 | 
000219 | Define the affine charge map
000220 | 
000221 |   T1(p)=Rp+e0.
000222 | 
000223 | On the finite-energy basis set
000224 | 
000225 |   W_ext [ a1(-m1)...ar(-mr) |n+c0 s> ]
000226 | 
000227 |     = eta_R(n) kappa^c0
000228 |       (Ra1)(-m1)...(Rar)(-mr)
000229 |       |Rn+e0+c0 s>.
000230 | 
000231 | The translation factor eps_*(e0,Rp) is identically one in this chosen ordered
000232 | section, so this is exactly E_e0 R_hat, not an abbreviation which drops a
000233 | Klein phase.
000234 | 
000235 | The map of charge labels is bijective, R is orthogonal on every oscillator
000236 | mode, and all displayed coefficients have absolute value one.  W_ext is
000237 | therefore an isometry of H_fin with dense inverse, and extends uniquely to a
000238 | unitary on H_*.
000239 | 
000240 | For c0=0 the formula is precisely the old ordered-CAR W1 action.  In
000241 | particular it retains the actual CAR phase eta_R(n); the extension has not
000242 | replaced W1 by a charge-only permutation.
000243 | 
000244 | The sector action is
000245 | 
000246 |   0 <-> v,
000247 |   s <-> c.
000248 | 
000249 | Consequently
000250 | 
000251 |   0+s  -> v+c,     E8 vacuum representation to g-twisted representation,
000252 |   0+v  -> 0+v,     fermionic NS sum preserved,
000253 |   s+c  -> s+c.     Ramond sum preserved.
000254 | 
000255 | Eight iterations give the actual integer carry, including its cocycle action:
000256 | 
000257 |   W_ext^8 = E_1.
000258 | 
000259 | On a charge state |p>, the right hand side acts as
000260 | 
000261 |   E_1 |p> = eps_*(1,p) |p+1>.
000262 | 
000263 | This is an infinite-lattice statement.  It is not a replacement by a finite
000264 | eight-state register.
000265 | 
000266 | 
000267 | 5. Exact charge and quarter-time Ward identities
000268 | ------------------------------------------------
000269 | 
000270 | Let
000271 | 
000272 |   delta_k=(k+alpha)/8-1/2,
000273 |   K_delta=N+p^2/2+delta.p,
000274 |   H=8 K_delta,
000275 | 
000276 | with alpha=1/4 or alpha=3/4.  A direct polynomial calculation gives, for
000277 | every real p and hence for every integer or half-integer charge,
000278 | 
000279 |   K_delta(Rp+e0)-K_delta(p)=(Q(p)+alpha)/8.
000280 | 
000281 | Therefore on H_fin,
000282 | 
000283 |   W_ext^* Q W_ext = Q+1,
000284 |   W_ext^* K_delta W_ext = K_delta+(Q+alpha)/8,
000285 |   W_ext^* H W_ext = H+Q+alpha.
000286 | 
000287 | The half-charge sectors require no new energy prescription.  As the smallest
000288 | quarter-state check, for
000289 | 
000290 |   p_*=(1,1,1,1,-1,-1,-1,-1)/2
000291 | 
000292 | one has
000293 | 
000294 |   K_delta(p_*)=0,
000295 |   K_delta(T1 p_*)=alpha/8,
000296 | 
000297 | exactly as for p=0 and T1p=e0.  Thus the extra E8 half-charge zero-energy line
000298 | is transported by the same Ward law.  This calculation does not decide which
000299 | zero-energy line the physical state selects.
000300 | 
000301 | 
000302 | 6. Vertex transport and the twist
000303 | ----------------------------------
000304 | 
000305 | For beta=m+d0 s define
000306 | 
000307 |   lambda_kappa(beta)=eta_R(m) kappa^d0,
000308 |   C_*(u,v)=eps_*(u,v)/eps_*(v,u).
000309 | 
000310 | On the common algebraic domain, conjugation of the lattice exponential field
000311 | is
000312 | 
000313 |   W_ext Y(e^beta,z) W_ext^(-1)
000314 | 
000315 |     = lambda_kappa(beta) C_*(e0,R beta)
000316 |       z^(-(R beta).e0) Y(e^(R beta),z),
000317 | 
000318 | with the Heisenberg oscillators transported by R.  If gamma=R beta, the
000319 | monomial is exactly z^(-gamma.e0).
000320 | 
000321 | For D8 roots the exponent is integral.  For E8 spinor roots it is
000322 | half-integral.  The latter branch is the required sector change from 0+s to
000323 | v+c.  It is the lattice realization of the Li Delta(-e0,z) twist.  Reading
000324 | this formula as an ordinary local automorphism of V_E8 would erase the
000325 | half-integral monodromy and is false.
000326 | 
000327 | The construction is therefore a global defect or representation
000328 | intertwiner.  It is not itself a point-local vertex insertion.  On a simply
000329 | connected punctured chart the chosen branch has the standard formal twisted
000330 | locality.  If a smeared field has independently been constructed, multiplying
000331 | its test function by the displayed analytic monomial preserves support.
000332 | Li's formal Delta theorem alone does not prove strong convergence on a common
000333 | operator core or produce a bounded localized conformal-net defect.
000334 | 
000335 | 
000336 | 7. Two different lifts of the same untwisted E8 character
000337 | ---------------------------------------------------------
000338 | 
000339 | There are two relevant involutions on the four sector labels.
000340 | 
000341 | The direct Cartan lift is
000342 | 
000343 |   G_Cartan(n+c0 s)=(-1)^c0.
000344 | 
000345 | In sector order (0,v,s,c) its signs are
000346 | 
000347 |   (+,+,-,-).
000348 | 
000349 | It equals exp(2 pi i e0.p).  It is +I on the entire integer NS charge space
000350 | 0+v and commutes with W_ext.  Therefore the original retained Cartan
000351 | four-word stays
000352 | 
000353 |   W1^* U_k W1 U_k^* = +I.
000354 | 
000355 | The standard spin-sector lift is
000356 | 
000357 |   G_std(n+c0 s)=(-1)^(Q(n)+c0).
000358 | 
000359 | Its signs are
000360 | 
000361 |   (+,-,-,+)
000362 | 
000363 | in sector order (0,v,s,c), equivalently (0+,s-,v-,c+) in the order often used
000364 | for the four D8 modules.  It anticommutes with W_ext, so its corresponding
000365 | four-word is -I.
000366 | 
000367 | This is the lift convention encoded by equation (5.3) of Smith, Lin,
000368 | Tachikawa and Zheng:
000369 | 
000370 |   Z[E8]^a   = chi_0-chi_s,
000371 |   Z[E8]_a^a = chi_c-chi_v.
000372 | 
000373 | The accompanying fermionization convention also fixes a spin invertible
000374 | theory and an Arf sign.  Those are extra defect-junction data; they are not
000375 | contained in the original NS winding operator.
000376 | 
000377 | Both lifts restrict to the same E8 g-character on the untwisted E8 vacuum
000378 | representation 0+s: sector 0 has sign + and sector s has sign -.  They differ
000379 | by the discriminant or Klein character
000380 | 
000381 |   chi_v=(+,-,+,-),
000382 | 
000383 | which is invisible on the choice of untwisted E8 character but changes the
000384 | lift on the twisted module v+c.  Hence the E8 character alone does not fix the
000385 | twisted-module phase.  The original Cartan word and the standard spin-sector
000386 | word are both mathematically valid, but they are different operators once W1
000387 | and the source Klein factors are fixed.
000388 | 
000389 | The following data must therefore remain typed separately:
000390 | 
000391 |   1. eta_R, the fixed ordered-CAR Klein lift;
000392 |   2. kappa=+i or -i, the two gauge-equivalent mu4 representatives of the
000393 |      cyclic R lift on X=E_s;
000394 |   3. chi_v, the Z2 choice of lift on the g-twisted E8 module;
000395 |   4. any finite compiler Gaussian or Arf word.
000396 | 
000397 | The kappa representatives are identified by U_tau, while no equality between
000398 | the remaining typed operators follows from the present construction.  In
000399 | particular the old Cartan +I word must not be replaced by the -I word of
000400 | G_std merely because both extend the same untwisted E8 character.