TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SEAM-STATE-EVENT-POLAR-SELECTION.txt
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Inclusive original lines: 162-215
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000162 | 5. Why this is not yet a physically selected transport or 3+2 field split
000163 | 
000164 | In the actual GNS event space the coefficient inner product IS G. The map
000165 | D from that metric space into its source span is already an isometry.
000166 | Writing U uses a standard Euclidean coordinate realization of that space.
000167 | It becomes extra physical content only if an independent Euclidean label
000168 | record/observable or actual positive normalization procedure is identified.
000169 | 
000170 | The raw slot cut Q_W is not selfadjoint for G:
000171 | 
000172 |     ||[G,Q_W]||_HS^2=12.
000173 | 
000174 | Naive transport D Q_W D^-1 is idempotent but has antihermiticity defect
000175 | ||A-A^*||_HS^2=2. Requiring only a Hermitian physical projector onto the
000176 | raw weak span gives a different operator:
000177 | 
000178 |     ||supp K_W-P2_original||_HS^2=(28-8sqrt6)/25>0.
000179 | 
000180 | Nor does direct spectral calculus on a raw return select the target:
000181 | 
000182 |     ||[K_C,P2]||_HS^2=(3144-384sqrt6)/625>0,
000183 |     ||[K_W,P2]||_HS^2=(1044+216sqrt6)/625>0.
000184 | 
000185 | Even the signed raw contrast K_C-K_W has three positive and two negative
000186 | eigenvalues, namely(3,1,1,-1,-2), but its negative spectral projector has
000187 | distance squared(118-48sqrt6)/125>0 from original P2. This is a check of
000188 | the actual given contrast, not adoption of a new sign selection rule.
000189 | Same counts and clock compatibility do not fix the original orientation.
000190 | 
000191 | There is a concrete FULL-STATE control. Grant the normalized original mark
000192 | |q>=sqrt(6/5) F_n5|Omega>, without claiming its physical preparation. The
000193 | unprojected leading REGULAR product Delta_j(z)|q> at z^0 is the complete
000194 | weight8 polynomial sqrt(6/5) f_n5 f_dj. All its monomials are retained.
000195 | Its response Gram is exactly
000196 | 
000197 |     G_q=(43/5)I5+(73/5)J5.
000198 | 
000199 | The SAME vacuum-normalized fields then have Gram
000200 | 
000201 |     G^(-1/2) G_q G^(-1/2), spec=(68/5,(43/5)^4),
000202 | 
000203 | not I. These are source amplitudes, not probabilities. This does not say
000204 | that a physical instrument must have orthonormal outcome vectors; it shows
000205 | precisely why vacuum orthonormality is not a state-independent operator
000206 | normalization/instrument identity. We have not tested one finite component
000207 | as a surrogate for all locally smeared field action: it is an exact
000208 | component disproving equality of the corresponding full formal identities.
000209 | 
000210 | Native J acts as i on u and hence fixes ALL quartic fields and their
000211 | conformal translation descendants. It adds no polar direction to this
000212 | response. Physical geometric deck/reflection, OS positivity and this
000213 | native J are not automatically the same operator. Original boundary
000214 | action/seed data have not provided the missing source-derived identification
000215 | of a positive relative event kernel as the physically used observable.