TFPT scientific source excerpt
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000243 | \paragraph{T3/T4: canonical DET excitations and the signed wall.}
000244 | \texttt{CHIRAL4D.MIRROR.SIGNED.CAR.01}.
000245 | \veri{v1027\_\allowbreak signed\_\allowbreak det\_\allowbreak car\_\allowbreak wall.py}
000246 | checks the following algebra and its finite countermodels.
000247 | The old Hubbard operators $b_i=|\Omega\rangle\langle e_i|$ satisfy
000248 | $\{b_i,b_j^*\}=\delta_{ij}P_\Omega+|e_j\rangle\langle e_i|$, not CAR,
000249 | and $b_i^*b_j^*=0$. For even $n\ge4$ (in particular $n=32$), write
000250 | $F=|{\rm full}\rangle\langle0|$ and define
000251 | \[
000252 |  U=I+(2^{-1/2}-1)(P_0+P_F)+(F-F^*)/\sqrt2,\qquad a_i=Uc_iU^*.
000253 | \]
000254 | This is a two-dimensional rotation and identity elsewhere. It is even;
000255 | under the inherited full-determinant singlet premise it is onsite gauge
000256 | invariant. It maps $|0\rangle$ to $\Omega=(|0\rangle+|{\rm full}\rangle)/\sqrt2$,
000257 | so the $a_i$ satisfy CAR on the entire $2^n$-dimensional Fock space and
000258 | $a_i^*\Omega=e_i$. Products of even onsite rotations preserve cross-site CAR.
000259 | The actual new parent is additive:
000260 | \[
000261 |  N_a=UNU^*=N+\tfrac n2(P_0-P_F-F-F^*),\qquad
000262 |  Q\le N_a\le nQ,\quad Q=I-P_\Omega.
000263 | \]
000264 | Thus $\Delta N_a$ has the same unique ground state and gap $\Delta$ as
000265 | $\Delta Q$, but is not the old flat parent. The high-degree local rotation
000266 | is not asserted to be a linear Bogoliubov map or an efficient Fock simulator.
000267 | 
000268 | Let $A=A^*$ be a fixed c-number, gauge-covariant quasilocal one-particle
000269 | operator, $\|A\|\le L$. Define
000270 | \[
000271 |  H=p^*Ap+\Delta a^*a+\lambda(a+gAp)^*(a+gAp),\quad
000272 |  M=\Delta+\lambda,\quad\beta=\lambda g^2,\quad\eta=\lambda g,
000273 |  \quad c=\lambda g^2\Delta/M.
000274 | \]
000275 | If $cL<1$ and $B=L+\beta L^2<M$, the spectral block
000276 | \[
000277 |  h(s)=\begin{pmatrix}s+\beta s^2&\eta s\\\eta s&M\end{pmatrix},\quad
000278 |  \det h(s)=Ms(1+cs)
000279 | \]
000280 | has precisely the original kernel, no extra zero, high band $\mu_+\ge M$,
000281 | and $|\mu_-|\le B$. Expansion gives
000282 | $\mu_-(s)=s+cs^2-\eta^2s^3/M^2+O(s^4)$, hence $z=1$ when $A$ is linear.
000283 | For $(\Delta,\lambda,g,L)=(3,1,1/2,2)$, $M=4$, $B=3$, $cL=3/8$.
000284 | Filling the negative band and subtracting its energy gives a nonnegative
000285 | full-Fock Hamiltonian for each fixed classical background;
000286 | $E_0\ge-Lr|\Lambda|$ with $r$ physical modes per site.
000287 | The bare mirror operator nevertheless has low-band weight of order $s^2$.
000288 | There is no hard bare-mirror response gap, nor an isolated entire
000289 | many-body mirror continuum. Operator-valued quantum-link diagonalization
000290 | does not automatically lift to a canonical Fock transformation (the
000291 | explicit SWAP mutant fails CAR).
000292 | 
000293 | For a free illustration only, $\gamma_i=\tau_1\otimes\sigma_i$,
000294 | $\gamma_5=\tau_3\otimes I$, $w=\sum_i(1-\cos k_i)-1$,
000295 | $R^2=1+2\sum_{i<j}(1-\cos k_i)(1-\cos k_j)\in[1,25]$ and
000296 | $D=I+(w+i\gamma\cdot\sin k)/R$ give $A=\gamma_5D$,
000297 | $A^2=2+2w/R$ and $\|A\|\le2$, with one Dirac node at zero.
000298 | The inverse-square-root series has norm ratio $12/13$ and exponential
000299 | length $2/\log(13/12)$. This four-component regulator contains opposite
000300 | Weyl sectors: it is not the demanded two-component chiral SM construction.
000301 | The common-parent dynamical gauge, measure, anomaly/index and continuum
000302 | obligations remain open.