TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt
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Inclusive original lines: 191-247
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000191 | 5. Original geometric modes are not all folded canonical stress histories
000192 | 
000193 | The original one-channel fermion stress with its retained twist has
000194 | 
000195 |  Lcal_m^alpha=sum_r (r+m/2+alpha):a_r^* a_(r+m):
000196 |                      +(alpha-1/2)^2/2 delta_m0.
000197 | 
000198 | The constant is the original twisted conformal ground weight, while the
000199 | excitation generator is Lcal_0^alpha-(alpha-1/2)^2/2=8K_alpha.
000200 | This stress is also fixed by the same original density Sugawara identity;
000201 | it is not an independently chosen E8 improvement or clock.
000202 | 
000203 | The original cover/fold relation is
000204 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_n0.
000205 | This is the ACTUAL twisted c8 geometric action. In the fixed current
000206 | dictionary its stress is Y_d(T)=T+d.h/z+||d||^2/(2z^2), not silently the
000207 | unshifted T on the old primary/vacuum seeds. Even for the folded geometric
000208 | subalgebra, the previous all-mode theorem cannot be extended to a different
000209 | operator/input pair merely by the same symbol 'stress'. Additionally, the
000210 | folded geometric c8 action samples only original indices divisible by8.
000211 | Original Lcal_+-1 changes fold energy by1/8 and produces charged cyclic D8
000212 | outputs. It is outside the earlier unshifted canonical c8-only class. The
000213 | earlier theorem is retained in its precise original source/input premises.
000214 | 
000215 | The full original CAR calculation verifies on every actual input vector
000216 | 
000217 |  [Lcal_1^alpha,J_-1]=(Q+alpha-1/2),
000218 |  [Lcal_1^alpha,Lcal_-1^alpha]=2Lcal_0^alpha.
000219 | 
000220 | The second RHS equals(64+(alpha-1/2)^2) on these neutral source energy32
000221 | inputs; omitting the twisted ground term fails this test.
000222 | 
000223 | Let G_L be the actual Lcal_1 response Gram. The two B4-selfadjoint response
000224 | operators B4^-1 G_- and B4^-1 G_L have common commutant dimension1, checked
000225 | over Gaussian rationals. Hence their generated complex *-algebra, with the
000226 | actual B4 adjoint, is M5, and they have no common reducing rank2/3 split.
000227 | This excludes a COMMON SPECTRAL-SECTOR identification for this response pair,
000228 | not a physical axiom requiring every mark to commute with all responses.
000229 | Nonlinear event readouts and other full source histories remain available.
000230 | 
000231 | In the saved COMPLETE current family, mode30 has rank2 and mode29 rank3.
000232 | This is not a derived Coxeter30 dictionary: J30 lowers original energy by30,
000233 | while a Coxeter clock is a specified unitary finite-order action. Moreover
000234 | the exact mode30 nullspace is none of the60 original polar rank3 kernels,
000235 | in BOTH quarters. The positive generalized overlap
000236 | tr((B4^-1 G29)(B4^-1 G30)) is nonzero, so these adjacent rank3/rank2
000237 | responses are not ORTHOGONALLY complementary in the actual B4 metric.
000238 | Their sum has rank5: they DO form an oblique algebraic direct sum. A positive
000239 | relative normalization can consequently build a rank2 projector for this
000240 | SUPPLIED mode pair, as in the previously proved general relative-response
000241 | theorem. This control does not exclude that construction or establish its
000242 | literal match to an original polar flag. No original rule selecting this
000243 | particular pair29/30 is supplied. They end in charge0 energies3 and2,
000244 | whose full source dimensions are p(3)=3 and p(2)=2. Dimension counts alone
000245 | do not identify them with the native carrier split. No mode30 selection
000246 | rule or eigenvalue grouping is adopted.
000247 |