TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/SPIN-FIELD-TEST.txt Original file SHA-256: d4d4a30b409b43aa8ea88913cdd52d6213139d800bc65f234c5d624cb58052a2 Inclusive original lines: 201-362 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 | This uses the unprojected factorized ambient state. An actual Gauss or 000202 | admissibility projection can change it and its allowed source legs. 000203 | Consequently the displayed R104 value is not asserted for an unevaluated 000204 | physical projected source. That caller must use its actual P and the 000205 | whole projected trace, as in TRANSFER-EXTRACTION(9). The word test does 000206 | not silently replace this state with the ambient one. 000207 | 000208 | There is a sharper PHYSICALLY PROJECTED subcase already admitted by 000209 | R104:147-179. Suppose the actual supplied gauge action is 000210 | U_R(g) tensor Gamma(r(g)), with det r(g)=1 and no extra implementer phase. 000211 | Both0 andF are then individually fixed. For D=span{0,F}, its actual Gauss 000212 | projector restricts on D tensor H_R to I_D tensor P_R, where P_R is the 000213 | SAME rest group-average projection. More generally the following 000214 | calculation needs just this restriction, [P,H]=0, [P_R,H_R]=0 and 000215 | H=H_R+2(I-P_phi), rather than a guessed gauge dictionary. 000216 | 000217 | On a finite regulator, for finite beta>0 and P_R!=0, the EXACT numerator is 000218 | 000219 | Tr[P exp(-beta H)Omega_16] 000220 | =(b exp(i phi)/2) Tr_R[P_R exp(-beta H_R)] !=0. 000221 | 000222 | Proof: Omega has support entirely in D, the identity term has zero mirror 000223 | trace, and Tr_D(P_phi Omega)=exp(i phi)/2. The rest trace is strictly 000224 | positive for nonzero P_R because exp(-beta H_R) is strictly positive. 000225 | Dividing by Z_P=Tr[P exp(-beta H)]>0 preserves nonvanishing. Other mirror 000226 | sectors can change this denominator but cannot cancel this numerator. 000227 | Thus the same one-word rotation contradiction also holds for THIS actual 000228 | Gauss subclass, whenever its retained D block is nonempty. If P_R=0, 000229 | the caller removes that entire determinant block and the witness is0. 000230 | The explicit conditional R104 gauge case is not an identified v1013 000231 | embedding; R104 itself says that embedding was not printed there. 000232 | No projected zero-temperature claim is inferred without the original 000233 | charge-resolved ground-energy ordering: the finite-beta ratio can tend 000234 | to0 when a different physical sector dominates the ground limit. 000235 | 000236 | 3C. THE EXISTING32-MODE CONSTRUCTION: SCALAR COMPLETION, NOT A16-CAT TRACE 000237 | 000238 | Retain BOTH local Weyl polarizations. The one-particle representation is 000239 | S tensor V=C^2 tensor C^16, dimension32, and 000240 | 000241 | det(g_spin tensor rho(g_int))=(det g_spin)^16(det rho(g_int))^2=1 000242 | 000243 | for spatial SU(2) and the given internal Spin(10) representation. Hence 000244 | its full32-field top determinant is a rotation/internal scalar. This 000245 | discharges that particular finite kinematical neutrality test; it does 000246 | not choose a parent, source state, 4D regulator or an IR realization. 000247 | 000248 | This size is already covered by v1027's ORIGINAL algebra: its proof is 000249 | for every even n>=4, explicitly including n=32. The native matrix 000250 | regression uses n=4, so no executed2^32-matrix validation is claimed. 000251 | The exact even unitary and all-field CAR result are algebraic for n32. 000252 | 000253 | With the native order putting the16 plus modes before the16 minus modes, 000254 | Lambda(V_+ + V_-) has its graded tensor decomposition F_+ tensor F_-. 000255 | All vacuum/full states used here are even. The scalar32-cat is 000256 | 000257 | Phi_32=(|0_+0_->+exp(i phi)|F_+F_->)/sqrt2. 000258 | 000259 | Taking the ORDINARY partial trace over the opposite spin sector gives 000260 | 000261 | Tr_- |Phi_32>=0. This is rank2, whereas the 000264 | native pure16-cat projector has rank1. In particular the reduced source 000265 | has q_red(Omega_16)=0, in agreement with3B; it does not retain the 000266 | exp(i phi)/2 coherence of the pure16-cat. No phase convention changes 000267 | orthogonality of the two traced states. 000268 | 000269 | For completeness, the same existing flat-parent thermal formula at n32 000270 | gives the exact reduced density 000271 | 000272 | rho_red=[2^16 a I_{F_+}+(b/2)(P_0+P_F)]/(2^32 a+b), 000273 | 000274 | where I_{F_+} is the identity on the65536-dimensional sixteen-mode Fock 000275 | space, not a one-particle16-dimensional identity. Thus the same 000276 | Omega_16 coherence is0 at every beta. This is only the 000277 | partial trace of that GIVEN32-mode source, not a newly selected action. 000278 | It explains why adding the missing physical spin slot can restore the 000279 | top-determinant symmetry while a plain trace cannot produce the old 000280 | scalar16-cat source. It supplies no 4D dynamics theorem. 000281 | 000282 | 4. EXACT SCOPE AND CONSEQUENCE FOR THE SOURCE MAP 000283 | 000284 | This closes the following literal shortcut: 000285 | 000286 | internal W between the two 16-dimensional Spin(10) packets 000287 | -> identify c_i with one fixed local Weyl polarization psi_{+i} 000288 | -> import a scalar-I_R determinant parent unchanged as a local 000289 | rotational scalar on the original 4D field legs. 000290 | 000291 | The first arrow is now exact. The second identification loses the other 000292 | Weyl polarization and is not an SU(2)-intertwining local component map. 000293 | The third term then has spin8 by the proof above. This is a stronger 000294 | test than matching only the internal representation. 000295 | 000296 | Allowed alternatives are not evaluated or excluded here: native UV modes 000297 | with physical spin emerging only in the IR; momentum-dependent helicity 000298 | or nonlocal mode embeddings; other internal invariant tensors; derivatives 000299 | or orbital angular momentum; a rest operator of spin8 coupled to a scalar; 000300 | dynamical bosonic fields; and a branch whose actual fixed background 000301 | reduces the spatial rotation group. Each changes one of the literal 000302 | same-local-Weyl/scalar-coefficient assumptions. No new one is supplied by 000303 | this report. A fixed seam direction must be included in the actual 000304 | source covariance, rather than compared with a symmetry it does not have. 000305 | 000306 | The correct next source evaluation is therefore the original full slab 000307 | insertion with its true spin/orbital/rest legs. If that source is locally 000308 | rotation covariant and the proposed coefficient is truly I_R, its pure 000309 | epsilon16 same-Weyl term must vanish in the scalar generator. If its 000310 | actual legs differ, the transfer extraction theorem applies to those 000311 | actual legs; native DET16 is not imposed as a universal answer. 000312 | 000313 | This is an elementary group/Grassmann theorem applied to an additional 000314 | explicit source identification, not new representation theory. Targeted 000315 | Atlas queries were checked first; no exact existing spin8 DET16 finding 000316 | was returned. That is a search boundary, not a corpus-wide novelty claim. 000317 | 000318 | 5. SOURCE RECORD 000319 | 000320 | paper-source:f4b8af4fa6ae03d1dfd4b2c2:197-292, 000321 | documents/archive--tfpt-45--source_extracts--02_carrier_source/sections/ 000322 | 002-subsection-internal-carrier-stabilizer-and-one-family-pa.tex 000323 | SHA256 ad8c8fced742cdbff008a9fb8b581d1aa75d92c3665ab9de21b4a821c74b42ef. 000324 | 000325 | paper-source:04c82396f7a983096b580bbd:24-29, 000326 | documents/archive--tfpt-45--source_extracts--technical_companion_source/ 000327 | sections/008-section-closed-theoretical-outputs-and-empirical-compari.tex 000328 | SHA256 a593716c6adbb9ec75c19ada46ddd926ec02eaa456da1600ab031f9056b17bbc. 000329 | 000330 | paper-source:b619fde355dbd98d93f59189:177-199, 000331 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000332 | 001-subsection-hadronic-admissibility-as-singlet-selection.tex 000333 | SHA256 f377e790ec73b8f5c56e1270b1bcf163022bf86e039af707dbda9fb8017a50df. 000334 | 000335 | Native internal field/action and its finite scope: 000336 | source:45419c072b6f371501f0:1-145,204-315, 000337 | verification/v1002_spin10_mirror_projector.py 000338 | SHA256 84f7cd7987887ee071d7e05872285ab274ae329f6bcfa693be69bb52add19a4c. 000339 | Original flat coefficient2 is reused from R104/J and the actual v1013 000340 | table, registered sources in FIELD-PROJECTOR-MAP.txt and J's 000341 | DET16-ORIGINAL-MAP.txt; it is not derived from S_adm here. 000342 | 000343 | [OMITTED: non-scientific conversation or execution metadata] 000344 | [OMITTED: non-scientific conversation or execution metadata] 000345 | [OMITTED: non-scientific conversation or execution metadata] 000346 | [OMITTED: non-scientific conversation or execution metadata] 000347 | [OMITTED: non-scientific conversation or execution metadata] 000348 | [OMITTED: non-scientific conversation or execution metadata] 000349 | [OMITTED: non-scientific conversation or execution metadata] 000350 | 000351 | Exact R104 ambient source response reused, not rerun: 000352 | research-source:52f35ed085f7e4e10c1bc207:1-200 and 000353 | research-source:e1f6b2dddcade38e6bd3aad1:201-217, 000354 | DET16-WHOLE-SOURCE-REDUCTION.txt 000355 | SHA256 fc43fcfef453bb278b8f5c28123078331a36d1e4350672b45c47a5d99c992897. 000356 | 000357 | Primary Weyl rotation law: David Tong, Quantum Field Theory, section4.4, 000358 | equations(4.365)-(4.368), both two-component Weyl chiralities restrict to 000359 | the fundamental rotation representation. Author's full text read: 000360 | https://davidtong.org/teaching/quantum-field-theory/qfthtml/S4 000361 | The symmetry, spin8 and no-singlet result above are proved explicitly, 000362 | rather than attributed to a lookup or numerical approximation.