TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/REGISTER-GRADE-LIMITS-GAUSS-AND-DIRAC.txt
Original file SHA-256: 94dc3fe2f4b612b2a6f081db8339165f6b7beadfb764f4e3ae143b922d9aa8dd
Inclusive original lines: 236-344
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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000236 | does not select the chiral measure, native clock dictionary or E8 fields.
000237 | 
000238 | The Gauss comparison does not establish that this half flux even has the
000239 | required Spin(16) monodromy in the existing ONE-COPY fold. That first
000240 | identification is tested next, before any continuum profile optimization.
000241 | 
000242 | 10. A decisive monodromy distinction in the actual one-copy eightfold
000243 | 
000244 | Use the already specified fold eta_j(x)=psi((x+2pi j)/8)/sqrt8. In the arc
000245 | basis a circuit of the BASE x-circle cycles its eight sheets. If the
000246 | original COVER fermion holonomy is h=exp(2pi i alpha), its transition is
000247 |   C(h)e_j=e_(j+1) for j<7, C(h)e_7=h e_0,
000248 |   C(h)^8=h I.
000249 | In the diagonal Fourier frame its eigenvalues are exactly
000250 | exp(2pi i(k+alpha)/8), as in the existing source dictionary. The v1033
000251 | forward-seam convention reverses alpha's sign consistently; the half-flux
000252 | comparison below is unchanged.
000253 | Original fold/transition source: seam-joint-ward-state-20261014/PROOF.txt
000254 | section16, especially its D_alpha and bundle-endomorphism formulas.
000255 | 
000256 | The primitive REGISTER half flux changes the COVER holonomy h to -h, hence
000257 | the actual folded transition becomes C(-h), with eighth power -h I.
000258 | In contrast the uniform D8 spinor twist has OPE monodromy -1 with EACH of
000259 | the eight complex vector fermions, so on this same fixed base dictionary
000260 | it would give -C(h), with eighth power +h I. Therefore
000261 |   C(-h)^8=-h I != h I=(-C(h))^8.                     (6)
000262 | No complex charge-preserving fiber conjugacy, or periodic gauge change of
000263 | the GIVEN connection, identifies them. A nonperiodic change together with
000264 | a new connection/domain/field assignment is a different source dictionary
000265 | and must be specified; abstract bundle isomorphism alone is insufficient.
000266 | Smoothing a defect or the Gauss construction (3)-(5) does not change (6).
000267 | 
000268 | This distinguishes ONE original complex source reorganized into eight
000269 | sheets from EIGHT independent complex copies. For eight independent
000270 | copies a common half holonomy does give -I on all eight; that conditional
000271 | target calculation remains correct and is not substituted into (6).
000272 | 
000273 | Mandatory opposite-quarter countercontrol: for h=+/-i the two REALIFIED
000274 | SO(16) holonomies can be conjugate. Let S=diag((-1)^j), K be component
000275 | complex conjugation on the realification, O=realify(S) K. Then det O=+1,
000276 |   O realify(C(-h)) O^-1=realify(-C(h)),
000277 | but O reverses the complex structure and hence the original U(1) charge:
000278 |   O J_complex O^-1=-J_complex.
000279 | Thus (6) is a FIXED-CHARGE/CONNECTION obstruction, not a universal SO(16)
000280 | no-go. Calling charge conjugation an unchanged Cartan intertwiner would
000281 | repeat the earlier identification error. Physical charge reversal with
000282 | all state/time/clock data transported is a different explicitly typed map;
000283 | real conjugacy alone constructs no spinor field.
000284 | 
000285 | 11. What kind of sector transition the fixed fold actually needs
000286 | 
000287 | For even covering degree8, a Z2 change of the BASE spin lift pulls back to
000288 | trivial COVER holonomy: (-1)^8=1. Thus the desired uniform spinor twist is
000289 | a change of descent/lift on the base that is invisible to h, not the
000290 | primitive cover half-flux h->-h. There are two Z2 base holonomies with the
000291 | same pullback. Globally S C(h) S^-1=-C(h) confirms this invisibility.
000292 | 
000293 | This identifies the needed TYPE of datum, not its physical generation.
000294 | A global change of deck lift, or composing a deck action with total parity,
000295 | does not itself construct a localized charged intertwiner. Its actual
000296 | currents, charge, state, original source time, cocycles and support must
000297 | remain part of the map. Nor is the source's eightfold rotation already
000298 | identified with the primitive P1 deck/clock; original source section16
000299 | explicitly leaves the P1-core/source-circle map open.
000300 | 
000301 | No identification with the previously excluded internal Z8^f anomaly lift
000302 | is made: internal symmetry extension, spatial-cover descent, physical time
000303 | and geometric deck action are distinct objects. Equal group orders are
000304 | not an intertwiner. No new discrete Arf sign alone closes this problem.
000305 | 
000306 | The next original question is therefore more selective than 'smooth the
000307 | same half string': does P1 supply a localized transition of the relevant
000308 | DESCENT LIFT, with the correct unchanged-charge Ward action and chiral
000309 | measure/line, rather than a different global Wilson holonomy? Without that
000310 | map, the existing cover half-flux cannot be identified with the target
000311 | spinor by the fixed eightfold dictionary, even after fixing its locality.
000312 | 
000313 | 12. A physical projection inside the old carrier is not this Gauss map
000314 | 
000315 | There is also a complete finite-selector result under an explicit premise:
000316 | RETAIN the entire ORIGINAL even matter algebra and original time as
000317 | physical, and ask for an A/A*-admissible orthogonal projection P within
000318 | the old register-Fock Hilbert space. This is a tested comparison class,
000319 | not a new requirement imposed on all possible P1 physical subnets.
000320 | 
000321 | The even finite CAR algebra acts as full matrices on each parity block.
000322 | Schur's lemma therefore forces
000323 |   P=P_reg,+ tensor Pi_+ + P_reg,- tensor Pi_-.
000324 | For r!=s the time condition gives in each parity block
000325 |   (P_reg,eps)_(rs) Pi_eps(H_s-H_r)=0.
000326 | The actual QWZ seam hopping makes H_s-H_r nonzero on BOTH parity blocks:
000327 | a nonzero one-particle transfer gives an odd-sector witness, and adding
000328 | an untouched spectator gives an even-sector witness. Hence all register
000329 | offdiagonals vanish. P is diagonal in r and in particular [P,Z]=0.
000330 | If all F_A are also retained, selected r and r+2 blocks must agree; keeping
000331 | D_A requires all four r blocks agree. None makes Z inadmissible while
000332 | retaining those full original matter/time operations.
000333 | 
000334 | Thus merely relabeling the old Padm as a Gauss projector cannot implement
000335 | (3)-(5) under this premise. The actual completion changes the kinematic
000336 | local generators and their descent: the Wilson variable is global and
000337 | local electric links are present before restriction. If some old even
000338 | observables are instead removed, the premise is changed and the surviving
000339 | base currents/ALG-EXH require their actual derivation. This is not a demand
000340 | for code closure or an exclusion of all changed physical subnets.
000341 | 
000342 | check_register_words_and_fold.py verifies (1),(2),(6), the REAL quarter
000343 | countercontrol, actual width8 offdiagonal selector witnesses, and replays
000344 | every original Dirac assertion with explicit checks even under -OO.