TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/FIELD-PROJECTOR-MAP.txt
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000150 | R104, not values asserted for an uncomputed original smear. W alone does
000151 | not establish them. The whole-source equality must retain the SAME source
000152 | catalog: physical reducing observables and charged external legs are
000153 | different callers, as the next section proves.
000154 | 
000155 | 4. ACTUAL ADMISSIBLE NET: A STRONGER AND SMALLER REDUCTION TEST
000156 | 
000157 | Boundary001:315-345 expressly says that P A P is not the definition of a
000158 | local net. It defines
000159 | 
000160 |  A_adm(O)={ A|RanP : A in A_M(O), A RanP subset RanP,
000161 |                                 A^* RanP subset RanP }.
000162 | 
000163 | For bounded A these two invariances are equivalent to [A,P]=0. Assume an
000164 | identified ambient factor H=Lambda(C^16) tensor H_R and all the SAME bare
000165 | c_i tensor I, c_i^* tensor I belong to this reducing algebra. Then P
000166 | commutes with them all. Products of these CAR letters generate every
000167 | matrix unit |I><J| on Lambda(C^16): use the vacuum projector
000168 | product_i(1-n_i) between ordered creators and annihilators. Hence P commutes
000169 | with M_65536 tensor I. Its commutant is I tensor B(H_R), so
000170 | 
000171 |                          P=I tensor P_R.
000172 | 
000173 | No state, Hamiltonian or spectral approximation enters this argument.
000174 | 
000175 | An equivalent one-mode insertion test is obtained by Q=I-P:
000176 | 
000177 |  {Pc_iP,Pc_j^*P}=delta_ij P
000178 |                  -Pc_i Q c_j^*P-Pc_j^* Q c_iP.
000179 | 
000180 | For i=j the subtracted leakage is positive. Retaining all sixteen exact
000181 | CAR relations for these compressed SAME bare letters forces both leakage
000182 | terms to vanish and therefore the same reducing result. For a DET vacuum
000183 | line P_phi, P_phi c_i P_phi=0 by parity; its compressed anticommutator is0,
000184 | not P_phi. This is not a flaw in a rank-one vacuum STATE on the full CAR
000185 | algebra. It is the obstruction to treating that state projector as the
000186 | physical admissibility Hilbert subspace while retaining unchanged bare
000187 | CAR letters as operators within that subspace.
000188 | 
000189 | This is a source-map check, not a charged-field no-go. Gauge dressings can
000190 | act nontrivially on H_R; charged fields may be intertwiners between sectors;
000191 | and an original enlarged graded source may retain such legs even when
000192 | the physical observable net is restricted. Those possibilities require
000193 | their actual original insertions and do not satisfy the tensor-I premise.
000194 | 
000195 | Conversely, suppose P reduces the actual H, reflection and every source
000196 | letter, and the physical source is the trace/state on RanP. Then its full
000197 | ordered kernel depends only on PH P and the restrictions of the letters.
000198 | Replacing the unused Q block by any self-adjoint direct-sum completion
000199 | leaves EVERY such word, including its unnormalized partition trace,
000200 | unchanged. The same holds for the separately given reference block.
000201 | Thus equality of this actual PHYSICAL source does not require equality of
000202 | an inaccessible ambient DET kernel. If charged external legs leave RanP
000203 | and return, the intermediate Q block can be visible and this conclusion
000204 | does not apply. This is the precise caller distinction to settle, rather
000205 | than imposing full bare16 CAR on A_adm by default.
000206 | 
000207 | 5. ORIGINAL PROJECTORS AND THE TWO DIFFERENT TOP VERTICES
000208 | 
000209 | QFT002:14-43 specifies independent seam/color/determinant/time factors,
000210 | 
000211 |  P_sing=(I+Z_c+Z_c^2)/3,
000212 |  P_Theta=(I+Q_Theta)/2, Q_Theta^*=Q_Theta, Q_Theta^2=I.
000213 | 
000214 | P_sing is exactly color-CENTER neutrality. It is not Spin(10) Haar
000215 | projection and is not even a full SU(3) singlet projector merely by name.
000216 | On the natural second-quantized action of the ORIGINAL carrier family,
000217 | the one-particle center multiplicities are(4,6,6). The whole Fock center
000218 | sector dimensions are(21856,21840,21840), checked by exact residue counts.
000219 | Both physical empty/full states have center character1; P_sing is the
000220 | identity on their two-dimensional span and does not choose their coherence.
000221 | This is the natural CAR lift after identifying the factor, not a claim
000222 | that the unspecified abstract H_c in QFT002 already has that whole rank.
000223 | 
000224 | The original formal Q_Theta relations do not by themselves give its
000225 | matrix on this empty/full span. If an actual source map supplied
000226 | Q_Theta|0>=e^(i theta)|F>, Q_Theta|F>=e^(-i theta)|0>, its +projection on
000227 | that span would indeed be P_phi. That is a POSITIVE sufficient identity,
000228 | not an identity established by the word 'determinant'. Outside this span
000229 | other retained sectors would still have to be treated by the actual
000230 | seam/gap data. The compact bosonic determinant classes in Carrier007
000231 | are detE_2 and detE_3 over the normal S^2, with classes(1,0), and a Higgs
000232 | doublet. They are not, by a change of notation, the top line Lambda^16 S+.
000233 | 
000234 | Similarly the actual primitive top-form Yukawa in QFT003:1-145 is
000235 | 
000236 |  (E_3 tensor E_2) tensor Lambda^2 E_3 tensor E_2
000237 |        -> Lambda^3 E_3 tensor Lambda^2 E_2 -> C.
000238 | 
000239 | It has TWO fermionic legs and one Higgs leg, each internally expanded in
000240 | the five-dimensional carrier. The DET Omega has sixteen physical
000241 | fermionic letters, one for every component of S+. They are different
000242 | operators. Integrating original fields can induce higher vertices or
000243 | determinants, so this degree distinction does not exclude a full transfer
000244 | identity obtained by the original integration. It excludes the literal
000245 | shortcut 'both are top determinants, therefore they are the same vertex'.