TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/FULL-FINITE-ENERGY-AND-SECTOR-MENU.txt
Original file SHA-256: 331ad3b50a90ccd3e9ab16905e44b0a2ee04e84c92459739d0ac22f337d602b9
Inclusive original lines: 124-230
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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000124 | representation, or use a different sector vacuum and its energy subtraction.
000125 | The theorem does not exclude that move. It identifies precisely where it
000126 | ceases to be a sector generated inside the unchanged source Hilbert limit.
000127 | 
000128 | There is a separate all-operator obstruction if the actual folded Cartan
000129 | operators N_k are retained. They have integer excitation spectra, hence
000130 | exp(2pi i N_k)=I. Assume the ACTUAL integrated charge covariance
000131 | exp(it N_k) A exp(-it N_k)=exp(it q_k) A. For unbounded A this equation
000132 | is required on an invariant common domain. At t=2pi it gives
000133 | A=exp(2pi i q_k)A. Therefore a nonzero field with any half-integral q_k
000134 | cannot act in that unchanged representation. A merely formal commutator
000135 | is not sufficient to establish the integrated covariance premise.
000136 | This covers all words and their closed field realizations having that
000137 | covariance, not just bilinears. It is the already recorded Cartan-periodicity
000138 | obstruction from quotient-locality-e8-20261014, not a newly discovered one.
000139 | Intersector maps with shifted target charge representation evade this
000140 | premise; their microscopic origin is exactly what must then be constructed.
000141 | 
000142 | 5. All four uniform scalar Z4 twist classes in the eight-channel TARGET
000143 | 
000144 | This section is CONDITIONAL on the separately supplied rank8 D8 current
000145 | source. It does not introduce eight copies into the one-copy calculation.
000146 | For q=(r/4)^8+m, m in Z8, and a D8 root +/-e_i+/-e_j, locality requires
000147 | q dot root to be integral. Same-sign roots give r/2+integer. Therefore
000148 |   r0: integral class, D8-local;
000149 |   r1: quarter class, NOT D8-local;
000150 |   r2: half class, D8-local;
000151 |   r3: three-quarter class, NOT D8-local.
000152 | The order-two class is the only nontrivial uniform Z4 candidate compatible
000153 | with retaining ALL D8 currents. This is relative monodromy, not a proof of
000154 | existence, self-locality, positivity or physical microscopic selection.
000155 | 
000156 | At r1, q^2=2 gives exactly64 weights: m has one +1 and seven0 (8 choices),
000157 | or three -1 and five0 (56 choices). All have odd sum m. Every one has
000158 | q dot(e1+e2)=1/2+m1+m2 and monodromy -1 with that EXISTING D8 current.
000159 | A fermion-parity projection or cocycle sign cannot change this full
000160 | monodromy while retaining these charges and the full D8 current algebra.
000161 | Removing currents or changing a physical gauge/line structure is a changed
000162 | premise, not this proposed simple repair.
000163 | 
000164 | At r2 the256 minimum charges (+/-1/2)^8 have h1. They split into two128
000165 | chiralities. Inside each chirality pairings are integral; between opposite
000166 | chiralities they are half-integral. Thus one may adjoin ONE chirality to
000167 | D8 as the known bosonic E8 extension. The individual old CAR charge e_i
000168 | has half-integral pairing with it, so the unprojected CAR field algebra
000169 | and the bosonic E8 field algebra are not one ordinary local algebra.
000170 | The actual local endpoints, chirality selection/Gauss law and common time
000171 | have not been produced by this lattice statement.
000172 | 
000173 | The source-time check must also distinguish the two chiralities. With the
000174 | ACTUAL fixed fold shift delta_k=(k+alpha)/8-1/2 and K=q^2/2+delta.q,
000175 | the even-minus spinor class has one additional K0 state
000176 | q=(1/2)^4(-1/2)^4, in both quarters. The odd-minus class instead has
000177 | minimum K=1/32, uniquely. Thus the earlier second-ground-state obstruction
000178 | applies to the specified EVEN spinor extension and cannot be declared a
000179 | no-go for the opposite chirality. Both remain outside the unchanged integer
000180 | Cartan representation by the previous paragraph. Neither an odd-sector
000181 | projection nor its changed marking is selected by this energy comparison.
000182 | Here |delta_k|<1/2 makes +/-1/2 the coordinatewise minima on half-integers.
000183 | Moving outside that cube costs at least1-max|delta_k|=17/32, more than
000184 | the odd-chirality flip cost1/32. Nonnegative oscillator grades cannot lower
000185 | these values. The minima therefore hold on the full half-integral cosets,
000186 | not only on the enumerated256 conformal-weight-one charges.
000187 | 
000188 | 6. Primitive flow and both boundaries remain actual operator constraints
000189 | 
000190 | The existing source winding result is reused, not rediscovered:
000191 |   T1(p)=(p7+1,p0,...,p6),
000192 |   T4(p)=Pp+(1,1,1,1,0,0,0,0), P:k<->k+4.
000193 | Thus T4 is not the pure spinor translation p->p+(1/2)^8. The neutral
000194 | probe beta=e0-e4 transforms as beta.T4(p)=1-beta.p, whereas spinor
000195 | translation leaves beta.p unchanged. Equality of total charge4 and a
000196 | time increment cannot repair this operator mismatch. The original proof
000197 | derives an affine factorization ONLY on the already extended lattice;
000198 | its factors separately leave the original integer-charge lattice.
000199 | See seam-joint-ward-state-20261014/PROOF.txt sections16-17 and
000200 | joint-native-algebra-selection-20261016/
000201 | ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt section2.
000202 | 
000203 | Original unit winding has an unbounded integer carry. It is never divided
000204 | by8 or16. On the full microscopic cylinder the opposite edge compensates
000205 | the charge transport. Neither keeping only one edge nor adjoining a
000206 | four-state flux register is a consequence of the matrix identity above.
000207 | 
000208 | 7. Decision and the first genuinely additional structure
000209 | 
000210 | The unchanged representative's same-sea/time finite-energy carrier is now
000211 | exhausted. Its original global winding is not the missing spinor operator.
000212 | Within a separately supplied D8 source, the quarter and three-quarter
000213 | mechanisms fail relative locality; the half-twist is the only remaining
000214 | uniform scalar class, but needs actual intersector field generators.
000215 | These conclusions settle the specified mechanisms, not all original P1.
000216 | 
000217 | The primitive boundary declaration itself takes (A+,H+,D+,J,Gamma,B_Sigma)
000218 | as data, and labels the almost-commutative internal factorization a choice:
000219 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000220 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex65-91.
000221 | It is not permission to identify the one-copy QWZ coefficient bundle with
000222 | the separately reconstructed V16 or to append a flux register as derived.
000223 | 
000224 | The minimal missing physical TYPE is a marked LOCAL rank8 coefficient/net
000225 | realization together with an originally allowed parity-defect endpoint
000226 | extension, its chirality/Gauss projection and the SAME state/time map. If
000227 | rank8 is supplied independently, only the latter extension remains. A full
000228 | spinor-current extension theorem is not obtained here. No known E8 vertex
000229 | is substituted for its microscopic realization. The general P1 existence
000230 | question remains open; no claim of a complete TFPT solution is made.