TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/GROUND-STATE-SOURCE-REDUCTION.txt
Original file SHA-256: 908420944efb265dbb57f5fc53e2331a416170f76a626f75e14f3445299046cb
Inclusive original lines: 201-266
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000201 | 5. Corrected primitive-source premise
000202 | -------------------------------------
000203 | 
000204 | The previous source audit separated
000205 | 
000206 |   S1: the collar's induced one-particle generator intertwines K;
000207 |   S2: omega on CAR is the quasifree state with P=1_(K<0).
000208 | 
000209 | Under the present hypotheses S2 is not independent. The sharper dependency is
000210 | 
000211 |   full graded local CAR embedding, including a(f) and a(f)^*
000212 |        +
000213 |   original reconstructed time restricts to alpha_t(a(f))=a(e^(itK)f)
000214 |        +
000215 |   K self-adjoint and gapped at zero
000216 |        +
000217 |   H>=0 and H Omega=0 on the same reconstructed representation
000218 |        =>
000219 |   omega|CAR is uniquely quasifree with P=1_(K<0).          (15)
000220 | 
000221 | If S1 is meant literally as D_coll|CAR=K, it must also include the proof that
000222 | the seed time generator on this cyclic subsystem is the second quantization
000223 | of that same K. Equality of unbounded symbols without the time intertwiner is
000224 | not enough. Identifying the resulting P_-=1_(K<0) with the geometric Calderon
000225 | projector P_C is a further collar-orientation dictionary unless the chosen
000226 | product-collar theorem already makes that equality explicit.
000227 | 
000228 | The types must be kept separate: K acts on the one-particle boundary space;
000229 | the reconstructed positive H acts on the many-body cyclic GNS/OS space.
000230 | The operator comparison is K_seed iota_1=iota_1 K on a one-particle core,
000231 | followed by its CAR-induced many-body time relation. It is not an equation
000232 | identifying a one-particle Dirac operator with a many-body Hamiltonian.
000233 | 
000234 | The smallest decisive source test is therefore:
000235 | 
000236 |   (i) embed one odd smeared generator a(f), its adjoint and the local CAR
000237 |       relations into the actual operational seed;
000238 |   (ii) verify time covariance with the fixed K on a generating one-particle
000239 |        core;
000240 |   (iii) verify that the image belongs to the same OS quotient carrying
000241 |         H_adm>=0 and Omega.
000242 | 
000243 | Passing (i)--(iii) removes the independent state premise automatically by the
000244 | proof above. Failure already at (i) leaves only the even-algebra counterexample
000245 | and cannot select the filled sea.
000246 | 
000247 | 6. Scope
000248 | --------
000249 | 
000250 | PROVED here:
000251 | 
000252 |   * the full-CAR ground-state reduction (7)--(11);
000253 |   * the sign dictated by anti-linearity of a(f);
000254 |   * the one-mode even-algebra counterexample;
000255 |   * the archived OS theorem supplies H_adm>=0 on the reconstructed theory
000256 |     under its explicit hypotheses.
000257 | 
000258 | STILL TO BE ESTABLISHED FOR THE TFPT/QWZ BRIDGE:
000259 | 
000260 |   * the full graded local CAR embedding into the operational seed;
000261 |   * equality of its induced time with the Bogoliubov dynamics of the same K;
000262 |   * its descent to the same OS vacuum representation and collar domain.
000263 | 
000264 | No claim is made that scalar P1 derives this embedding. The advance is the
000265 | removal of a redundant state input after the actual operator and time bridge
000266 | has been proved.