TFPT scientific source excerpt Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/NEUTRAL-DEFECT-PAIRS-AND-VECTOR-WARDS.txt Original file SHA-256: e798405b02c979e0f9e77e223a097ea9ae269c3bd677a501f2becc8f00c0dda5 Inclusive original lines: 207-292 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. 000207 | 8. Smooth unitary endpoint separation at the SAME compact source time 000208 | 000209 | The already established original Weyl/time identity gives, on the given 000210 | chiral vacuum and original time K=H_cover/8, 000211 | Delta K(W(phi)Omega) = (1/8) sum_(m>0) m^2 |phi_m|^2. 000212 | See ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt:210-223; 000213 | the zero-charge vacuum specialization follows directly from the same CCR. 000214 | For the ACTUAL uniform base profile phi(theta)=f(8theta), phi_(8n)=f_n 000215 | and all other Fourier coefficients vanish. Parseval therefore gives 000216 | Delta K = 8 sum_(n>0) n^2 |f_n|^2 000217 | = (2/pi) integral_(base circle) |f'(u)|^2 du. (11) 000218 | There is no choice of eight independent source vacua or changed time here. 000219 | The quarter-dependent linear charge term is unchanged: these periodic 000220 | real phase profiles have zero winding and preserve total charges. 000221 | 000222 | Suppose one endpoint stays in a fixed interval, while the RETURN from 000223 | phase pi to0 is squeezed into a deleted base-cut interval of length eps. 000224 | Cauchy-Schwarz on that return gives, for ANY such smooth profile, 000225 | integral_return |f'|^2 >= pi^2/eps, 000226 | Delta K >= 2pi/eps -> infinity. (12) 000227 | This covers the entire stated smooth UNITARY phase separation class, 000228 | not a fitted profile. Piecewise linear H1 ramps of widths a,eps saturate 000229 | the separate ramp bound and have Delta K=2pi(1/a+1/eps); smooth H1 000230 | approximations retain the lower bound. With a=pi/8 and eps=pi/(8q), the 000231 | exact energies are16(1+q), not finite as the return is removed. 000232 | 000233 | Scalar multiplication cannot repair this normalized-state energy: 000234 | /||cW Omega||^2 equals the same Rayleigh quotient. 000235 | The result does NOT exclude general nonunitary sums/charged intertwiners, 000236 | new positive charged representations with their correctly derived energy 000237 | connection, or a different decompactified time/geometry. Those are not 000238 | the same smooth unitary vector-state limit on the fixed source circle. 000239 | No universal statement about arbitrary operator smearing is made. 000240 | 000241 | Local normal convergence away from the removed point can thus coexist 000242 | with divergence of the original GLOBAL energy. The usual punctured-line 000243 | sector construction is not automatically the requested full compact 000244 | source/time theorem. Conversely the divergence in(12) is not a no-go for 000245 | the standard locally normal spinor sectors. Their charged Hilbert-space 000246 | and energy/cocycle comparison is precisely the additional map at issue. 000247 | 000248 | 9. The charged current-side energy rule is already fixed conditionally 000249 | 000250 | Do not turn(12) into a new free-energy-input claim. In a GIVEN charge-lambda 000251 | module with the SAME current normalization and fixed source delta, 000252 | K_lambda(p,N)=N+|p+lambda|^2/2+delta.(p+lambda) 000253 | =K_0(p,N)+lambda.p+|lambda|^2/2+delta.lambda. (13) 000254 | This is the established Sugawara/spectral-flow comparison, not a tunable 000255 | coupling. For spinor lambda=(+/-1/2)^8 its conformal weight is1. The 000256 | corresponding mode-level spectral flow shifts H_i,0 by lambda_i and root 000257 | mode E_beta,n by lambda.beta; for D8 roots this is integral. The original 000258 | fixed-time minima of the two spinor cosets are already known and retained 000259 | from FULL-FINITE-ENERGY-AND-SECTOR-MENU.txt: the even spinor adds a K0 000260 | state; the odd spinor minimum is1/32. The exact global Phi with the odd 000261 | source module is likewise retained, not claimed as a new local inclusion. 000262 | 000263 | Consequently(12) does not exclude the finite positive charged generators 000264 | of the standard sector representations. It shows that their known energy 000265 | connection cannot simply be identified with normalized W_eps Omega at 000266 | fixed old global energy. The unknown is the ACTUAL microscopic charged 000267 | intertwiner and physical measure/support map implementing this connection, 000268 | not another arbitrary sector Hamiltonian or abstract E8 construction. 000269 | 000270 | 10. Remaining existence question and source boundaries 000271 | 000272 | Positive result: an existing neutral transporter product has exactly the 000273 | uniform sixteen-component parity-cut PAIR ACTION and no local-register 000274 | obstruction. This is an actual originally available operator class, not 000275 | a formally attached Ramond sector. The single cover-half-flux exclusion 000276 | does not dispose of it. 000277 | 000278 | Still unproved: separating one endpoint into a locally normal charged 000279 | field sector, its positive complete word limits and finite-energy/domain 000280 | bounds, the 128 chirality and all OPEs, FE-GEN and ALG-EXH on one physical 000281 | carrier, the original physical time and primitive P1/source-circle map, 000282 | and a justified single-edge sector or right absorber. The full-vector 000283 | Dirac Ward result requires the actual chiral measure/current-map origin; 000284 | it cannot be repaired by scalar endpoint normalization. 000285 | 000286 | The next mathematical step is a genuine nonunitary charged-intertwiner 000287 | limit, or a microscopic realization of the known charged energy/measure 000288 | connection, for this SPECIFIC neutral pair. Merely separating a smooth unitary phase while 000289 | discarding its return point is now excluded at finite original global 000290 | energy by(12). A single-chiral locally normal comparison cannot replace 000291 | the full source/state/time contract. No extra Arf bit, fitted current, 000292 | added projector or newly chosen interaction is claimed as derived.