TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt
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000230 | 7. THE EXISTING ACTUAL SOURCE WINDING: A FALSE RANK2 SHORTCUT REMOVED
000231 | 
000232 | We also evaluate the SPECIFIED original source intervention, not a newly
000233 | fitted field history. Seam-joint-ward-state-20261014 sections16,17,21 derive
000234 | unit winding on the retained chiral source and its fourth power W4:
000235 |   W4: p -> P p+a, a=(1,1,1,1,0,0,0,0), P:k<->k+4.
000236 | It permutes every nonzero current oscillator by P. Its actual Klein/cocycle
000237 | factorization on even D8 and the given E8 extension is already proved there;
000238 | it is not re-derived from raw P1 or called a local instrument here.
000239 | 
000240 | Using SAME existing M and O, its native real oscillator action is
000241 |   G=O^T M P M^T O.
000242 | On the positive native directions this acts EXACTLY as
000243 |   u0 -> i conjugate(u0), u1 -> -u1,
000244 |   u2 -> u2,             u3 -> -i conjugate(u3).
000245 | It neither commutes nor anticommutes with J. The positive-to-positive block
000246 | is diag(0,-1,1,0), and the opposite-polarization block diag(i,0,0,-i).
000247 | Thus it is not silently a native four-register U4 clock. Both polarizations
000248 | and all exterior source components are retained in the following test.
000249 | 
000250 | If one compares only the unshifted homogeneous quartic frames, their
000251 | oscillator overlap is
000252 |   A0=diag(1/2,0,0,1/2,0), rank2.
000253 | This tempting2+3 count is NOT the full original state response. It is not
000254 | declared to be the original polar mark or a physical sector decomposition.
000255 | 
000256 | Let R_b(f)=[z^0]Y_d(F,z)Omega be the COMPLETE grade4 response from section5.
000257 | For the source's actual outgoing charge a use the literal SAME FIELDS on
000258 | that state, hence b_out=U_+^T O^T(d+M a). Define the cross-response matrix
000259 |   C_ji=< R_(b_out)(f_j), Gamma(G) R_b(f_i) >.
000260 | This is a matrix element between complete source states, with the outgoing
000261 | charge a kept. It is not an executed projection or a requirement that the
000262 | five-dimensional observer slice be closed.
000263 | 
000264 | For BOTH retained quarter phases C has rank5, not rank2. All source modes
000265 | and opposite-polarization components remain in Gamma(G) R_b. Its full Gram
000266 | equals the incoming Gram exactly, as it must for the actual unitary W4.
000267 | No exterior amplitude is thrown away. The lower-degree terms fixed by the
000268 | source's normal-ordering shift remove the homogeneous-chart rank deficiency.
000269 | 
000270 | The physical source time increments are respectively7 and9 (fold units7/8,
000271 | 9/8), agreeing with the ORIGINAL Ward law
000272 |   [K_alpha,W4]=W4(Q/2+3/4+alpha/2), Q=0 on the inputs.
000273 | The outgoing actual charge is M a=(1,0,1,1,0,1,0,0), and W4^2 carries p to
000274 | p+(1)^8 in the source chart. No closed finite clock is asserted. A chart
000275 | overlap rank is not a probability, a physical admissibility projector or
000276 | a no-go for every other charged process.
000277 | 
000278 | This adds a concrete source/operator check to the neutral state result:
000279 | the original winding is now evaluated on the full state-dependent native
000280 | field response, including charge carry, time and polarization. A static
000281 | 2+3 count obtained after suppressing those state terms is not preserved.
000282 |