TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/SAME-SOURCE-AFFINE-DEFECT.txt
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Inclusive original lines: 150-330
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000150 |   eta_R(n)=(-1)^(n7 sum_(j<7) n_j).
000151 | 
000152 | It obeys
000153 | 
000154 |   eta_R(n+m) eps_Z(n,m)
000155 |    = eta_R(n) eta_R(m) eps_Z(Rn,Rm).
000156 | 
000157 | To extend the lift to X one must set
000158 | 
000159 |   R_hat X R_hat^(-1) = kappa X.
000160 | 
000161 | Since eta_R(1)=-1 and X^2=-E_1, covariance of X^2 forces
000162 | 
000163 |   kappa^2=-1,
000164 |   kappa in {+i,-i}.
000165 | 
000166 | These are the two mu4 representatives of the cyclic lift on the Ramond
000167 | sectors.  They will be shown below to differ only by an intertwiner-basis
000168 | gauge.  They do not constitute an additional physical choice.  This basis
000169 | gauge must still be distinguished from the two genuinely different
000170 | operators obtained by choosing the geometric Cartan lift or the standard
000171 | g-defect lift, and from any finite compiler Arf word.
000172 | 
000173 | For either kappa define
000174 | 
000175 |   eta_*(n+c0 s)=eta_R(n) kappa^c0.
000176 | 
000177 | Then
000178 | 
000179 |   eta_*(p+q) eps_*(p,q)
000180 |    = eta_*(p) eta_*(q) eps_*(Rp,Rq)
000181 | 
000182 | for all p,q in D8*.  Hence R_hat is a unitary cocycle-compatible lift on the
000183 | whole four-sector charge algebra.
000184 | 
000185 | Define the diagonal basis change
000186 | 
000187 |   U_tau |n+c0 s> = (-1)^(c0 Q(n)) |n+c0 s>.
000188 | 
000189 | It is pointwise the identity on both already supplied local subtheories:
000190 | 
000191 |   * c0=0 on the whole CAR NS theory 0+v;
000192 |   * c0=1 and Q(n) even on the E8 spinor sector s, so it is also the identity
000193 |     on the whole E8 vacuum theory 0+s.
000194 | 
000195 | Since T1 changes Q(n) by one,
000196 | 
000197 |   U_tau W_kappa U_tau^* = W_(-kappa).
000198 | 
000199 | Thus kappa=+i and kappa=-i are isomorphic descriptions of the same extension
000200 | up to the normalization of the remaining v-to-c intertwiner.  In particular,
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