TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/INTRINSIC-EVENT-GEOMETRY-AND-COMPOSITION.txt
Original file SHA-256: f5599348967407c285776b61e5fcecde42b88a50ef3aaf48b36bf85ddc380547
Inclusive original lines: 177-260
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.

000177 | 4. What 'intrinsic under basis changes' actually means
000178 | 
000179 | For unitary field-coordinate change S, H transforms under Sym² S, and
000180 | its projectors and Z transform covariantly. The six-ray projective
000181 | configuration, ranks, event actions and polynomial relations are the same
000182 | objects in different coordinates. Numeric matrix entries are not invariants.
000183 | 
000184 | For nonunitary coordinate changes the SOURCE/LABEL METRICS must also be
000185 | transported. A response T on label space with positive metric L_label,
000186 | and an L_label-selfadjoint label operator Q, gives
000187 | 
000188 |   R=T L_label^-1 T*, K(Q)=T Q L_label^-1 T*.
000189 | 
000190 | Under a=A a', take T'=T A, L'=A* L_label A, Q'=A^-1 Q A.
000191 | Then R'=R,K'=K. Resetting L' to I would change the geometry, not merely
000192 | the basis. The checker confirms this for a nonunitary shear in the actual
000193 | original event map. The same covariant-adjoint rule makes H a similarity-
000194 | covariant endomorphism under all invertible field basis changes.
000195 | 
000196 | For a genuinely transformed SOURCE response S T, the tensors instead
000197 | transform by congruence: R'=S R S*, K'=S K S*. Hermitian relative tensors
000198 | E=R^-1/2 K R^-1/2 then obey
000199 | 
000200 |   E'=V E V*, V=R'^-1/2 S R^1/2,
000201 | 
000202 | with V unitary on corresponding response supports. The nonsymmetric
000203 | relative operator R^-1 K is similarity-covariant. These are consistent
000204 | metric geometries; equality of coordinate matrices is neither expected
000205 | nor required. For noninvertible responses use supports and Moore-Penrose
000206 | inverses, and retain the label quotient by the response kernel.
000207 | 
000208 | Crucially, if T is injective and the original Euclidean label Q is an
000209 | orthogonal projector, W=T(T*T)^-1/2 is isometric and
000210 | 
000211 |   E(Q)=W Q W*.
000212 | 
000213 | It INHERITS the rank and spectrum of the supplied Q. Applying this with
000214 | a previously supplied rank2 cut cannot explain why that cut was rank2.
000215 | For redundant T, W is partial isometric and the label kernel/compatibility
000216 | matters; E need not be an idempotent projector.
000217 | 
000218 | 5. Reference state: central algebra versus complete response
000219 | 
000220 | For actual supported fields Delta_j(f) and a source/GNS vector in the
000221 | common domain with finite second moments, define
000222 | 
000223 |   T_omega a=sum a_j Delta_j(f)|Omega_omega>,
000224 |   H_omega=T_omega* T_omega.
000225 | 
000226 | This is the canonical POSITIVE RESPONSE FORM for the specified state,
000227 | fields and profile. For H0>0 a relative-state form is
000228 | 
000229 |   C_(omega|0)=H0^-1/2 H_omega H0^-1/2.
000230 | 
000231 | Its generalized eigenvalues and subspaces are coordinate intrinsic as
000232 | a PAIR of forms. They depend on omega and reference0; different GNS
000233 | Hilbert spaces have no automatically physical identification. A chosen
000234 | metric isometry cannot prove two physical preparations equivalent.
000235 | 
000236 | The original complete regular mode-product controls are explicit. With
000237 | normalized mark w_q, multiplication into that state has input-field Gram
000238 | 
000239 |   B_q=(43/5)I5+5 w_q w_q*, spec=(68/5,(43/5)^4).
000240 | 
000241 | For normalized original f0 and f4 states the corresponding Grams are
000242 | 
000243 |   B_f0=diag(19,8,8,8,5), B_f4=diag(5,9,9,9,16).
000244 | 
000245 | These are complete degree8 source polynomials, not compressed code
000246 | responses or probabilities. For the original event differences, H_q=D*B_qD.
000247 | Their central OPE Gram is still G=D*D=I+J. Thus central OPE invariance and
000248 | finite-state dependence are perfectly compatible; no single inverse root
000249 | normalizes every full response. Keeping one vacuum whitening is conformal
000250 | up to scale exactly if H_omega=c_omega H0. Commutation [H_omega,H0]=0 alone
000251 | does not suffice; the earlier Gq commutes withG but is not proportional.
000252 | 
000253 | These positive finite-energy vector/density states are mathematical
000254 | controls in the actual source representation. Original seed480-520 also
000255 | requires RP, clustering, finite relative action and specified time/collar
000256 | data. We have not shown every such control is a physically admissible P1
000257 | seed or that changing only omega preserves that seed. Morphisms preserving
000258 | ALL displayed seed data preserve omega too. This prevents confusing a
000259 | coordinate/reference relabeling with a different physical state.
000260 |