TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000199 | 8. Full known chiral quarter source: no finite-window promotion
000200 | 
000201 | For the existing DECLARED charged chiral CAR source with D_alpha=-i partial_theta
000202 | and boundary condition f(theta+2pi)=exp(2pi i alpha)f(theta), keep alpha=1/4.
000203 | Relative to its supplied filled negative sea, positive particle/hole energies are
000204 |   epsilon_(+,n)=n+alpha, epsilon_(-,n)=n+1-alpha, n=0,1,... .
000205 | This is the original source model, not a derivation of its physical identity
000206 | from P1/P2. The original signed graph transfer exp(-l D_alpha) is unbounded,
000207 | not a contraction. At every finite cutoff, change the negative modes to holes and
000208 | divide by the heat eigenvalue of the filled sea. This produces the relative
000209 | excitation transfer exp(-l|D_alpha|), with the sea scalar kept separate. The
000210 | infinite construction below is directly in the given sea-relative CAR
000211 | representation; no unitary particle-hole transform of infinitely many modes
000212 | in the empty-vacuum representation is claimed.
000213 | At cutoff n=0,...,N the divided sea factor is
000214 | exp(l (N+1)(N/2+1-alpha)); it diverges as the cutoff is removed. The Z_p below
000215 | are relative excitation partition sums and do not restore that absolute factor.
000216 | Fix the supplied positive collar length l and put q=exp(-l).
000217 | The one-particle excitation transfer has eigenvalues q^(n+alpha) and
000218 | q^(n+1-alpha). Its full multi-field lift is defined explicitly on the countable
000219 | occupation basis of finite subsets S of these modes by
000220 |   E(S)=sum_(sigma,n in S) epsilon_(sigma,n),
000221 |   H|S>=E(S)|S>,  Q_l|S>=exp(-l E(S))|S>.
000222 | The domain sum_S E(S)^2 |psi_S|^2<infinity makes H selfadjoint and positive.
000223 | Finite linear combinations are a core. Real time exp(-itH) is unitary and
000224 | strongly continuous; Q_l obeys the semigroup law on the entire Fock space.
000225 | Creator covariance and the determinant multi-field rule hold on the invariant
000226 | finite-particle core and extend for bounded creators/transfers.
000227 | 
000228 | There is explicit cutoff control, not just numerical stability. R=exp(-l h)
000229 | satisfies
000230 |   Tr R^p = (q^(p alpha)+q^(p(1-alpha)))/(1-q^p), p>0.
000231 | After retaining n=0,...,N in both sectors, the omitted trace equals
000232 |   q^(p(N+1)) (q^(p alpha)+q^(p(1-alpha)))/(1-q^p).
000233 | Hence Q_l is trace class, Z_p=Tr Q_l^p=det(I+R^p) is finite and
000234 |   0 <= log Z_p-log Z_(p,N) <= omitted Tr R^p.
000235 | With the truncated operator extended by the tail vacuum projector,
000236 |   ||Q_l-Q_(l,N)|| = q^(N+1+min(alpha,1-alpha)).
000237 | The normalized Choi vectors vec(Q_(l,N))/||Q_(l,N)||_HS converge in the COMMON
000238 | Hilbert-Schmidt/Choi space of the full source Fock space. Their squared distance
000239 | is at most the omitted Tr R^2. This follows from their overlap sqrt(Z_2,N/Z_2)
000240 | and 2(1-exp(-x/2))<=x. The equivalent graph-state interpretation uses a separately
000241 | declared filled INPUT-face reference in the doubled boundary CAR representation.
000242 | Raw finite Slater graph vectors embedded into the EMPTY doubled Fock space have
000243 | increasing particle number and do not converge there. No infinite particle-hole
000244 | unitary or normalizable maximally entangled identity vector is invoked.
000245 | The complete state and transfer therefore exist beyond every finite window in
000246 | this specified representation. Thermal state Q_l^2/Z_2 has the same beta=2l
000247 | relation; no trace-class claim is made for an arbitrary continuum.
000248 | 
000249 | The actual retained charged quarter transport acts as i on particle source
000250 | modes and its conjugate -i on hole source modes. Its lift is
000251 |   W|S>=i^(N_+(S)-N_-(S))|S>.
000252 | It commutes with this source H and retains the charged quarter rule on every
000253 | field word. Indeed W=exp(2pi i H) in this declared model; with real-time
000254 | U_t=exp(-itH), this is the positively oriented loop U_(-2pi).
000255 | This typed charged-source statement is not an identification of
000256 | the unrelated internal i^N rotation or of all 16 carrier components with the
000257 | same quarter eigenvalue. Nor does the single chiral construction identify all
000258 | components of the doubled graded collar with this same time/loop generator.
000259 | Overall determinant-line/partition phases still need
000260 | the original unnormalized source data.