TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/GRADED-HS-REFLECTION-AND-TIME.txt
Original file SHA-256: 0d9727a9e167c64ef4a8413736c1555b90a7c5da5392eeea3a3455d0bb037986
Inclusive original lines: 201-265
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000201 |   U_t a_L(w) U_t^*=a_L(exp(i t h)w).
000202 | The annihilator formulas are antilinear in the test vector: in an h eigenmode
000203 | they carry exp(-i t lambda_j), exactly as they must.
000204 | 
000205 | On the geometric boundary space U_geo,1(t)=exp(t Ical B), applied on both
000206 | faces, becomes in the T_faces chart
000207 |   (v0,v1,bar(w0),bar(w1))
000208 |     ->(exp(i t h)v0,exp(i t h)v1,
000209 |                        overline(exp(i t h)w0),overline(exp(i t h)w1)).
000210 | The conjugate slot has generator -bar(h), since
000211 | overline(exp(i t h)w)=exp(-i t bar(h))bar(w).
000212 | Consequently the explicit WHOLE map (1) satisfies
000213 |   iota(c(U_geo,1(t)u))=U_t iota(c(u)) U_t^*
000214 | with the SAME t. The graph state is invariant under this group. Its cyclic
000215 | GNS implementation therefore transports to U_t under the graph-to-HS unitary.
000216 | This closes the time dictionary INSIDE THIS given graph source and field map.
000217 | No kappa_h or inverse time parameter is needed in the natural creation chart.
000218 | 
000219 | A previously used annihilator-coordinate chart instead produced an inverse
000220 | parameter. That was a property of that alternative field labelling, not an
000221 | obstruction to the creation map (1). It is superseded here as the proposed
000222 | whole-field/time identification. Equality to the original physical P1 time,
000223 | or to the old scalar NS/QWZ source, still requires its actual source map.
000224 | On matrix units its frequencies are energy differences, with both signs
000225 | for a nontrivial faithful thermal source. H>=0 on K does not make L_HS>=0
000226 | on the entire HS space. Since H is even, Q_Gamma commutes with this group.
000227 | One has J_R L_HS J_R=-L_HS and J_R U_t J_R=U_t; the second identity also
000228 | uses J_R's antilinearity. The modular operator is exp(-beta L_HS), with its
000229 | own unbounded functional-calculus domain. No equality of all these operators
000230 | with the classical geometric Dirac or with radial normal transfer is claimed.
000231 | Finite temperature therefore needs no invented ground-state vector for the
000232 | whole two-face Hilbert space to retain the given source and its reflection.
000233 | 
000234 | 7. Exact type boundary and original-source limitation
000235 | 
000236 | If one instead forces the section3 field rule to be an ordinary CAR
000237 | ANTILINEAR ANTI-HOMOMORPHISM, let n0=p0^*p0 and n1=p1^*p1 for the geometric
000238 | positive-grade slots in (1). The wrong reflection sends n0 to n1. In one
000239 | graph eigenmode their occupations are (1-c,c), their cross is sqrt(c(1-c)),
000240 | and <n0 n1>=0. The even element A=n0-(1-c)I then gives
000241 |   <Theta_wrong(A) A>=-c(1-c)<0.
000242 | This counterexample excludes that wrong word type only. It does not apply
000243 | to the explicitly order-preserving geometric/HS reflection proved above.
000244 | Grassmann source reversal and its reconstruction into operators must be
000245 | tracked before translating between the two descriptions.
000246 | 
000247 | The physical P1 insertion, reference/domain, charge and bosonic-measure
000248 | identifications remain as in charge_transport/ORIGINAL-SOURCE-AUDIT.txt.
000249 | The new exact content is that the original candidate product reflection
000250 | is now realized by a definite graded HS antiunitary on the same GIVEN source,
000251 | and its even positive form agrees with the independent whole-CAR proof.
000252 | The actual h_Phi, rather than an artificially scalar surrogate, is retained.
000253 | No interacting bosonic integration, physical P1 selection, universal vacuum
000254 | condition, or complete fundamental solution is claimed.
000255 | 
000256 | Sources reused directly:
000257 |   FINITE-COUPLED.txt:10-81;
000258 |   EVEN-REFLECTION-POSITIVITY.txt:15-142;
000259 |   original_insertions/PROOF.txt:180-188 in the parent contract;
000260 |   experiments/theory-contracts/native-arc-charge-source-interface-20261026/
000261 |     HALF-PERIOD-GNS-SEWING.txt,
000262 |     sections2-4, as registered by
000263 |     finding:existing-elliptic-half-period-has-canonical-graded-gns-intertwiner.
000264 | The last path is relative to the repository root. It is an existing general
000265 | source/result, not a second independently selected physical source.