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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000153 | 5. COMPLETE LOCAL TWO-POINT KERNEL AND ALL MODE RESPONSES
000154 | 
000155 | Let m_i=f_i(b) and, for r=0,...,4, define the Hermitian source tensors
000156 |   (D_r)_ij=sum_(|nu|=r) conjugate(partial^nu f_i(b))
000157 |                               partial^nu f_j(b)/nu!.
000158 | Here D0=conjugate(m)m^T includes the disconnected mean term. The Riesz
000159 | vector of the one-point functional is conjugate(m), not m; the six
000160 | comparison rays are real in the literal N-chart, so either ray test agrees.
000161 | 
000162 | Wick's theorem for the SAME translated Gaussian current state gives exactly
000163 |   <F_i^(d)#(z) F_j^(d)(w)>
000164 |     =sum_(r=0..4) (D_r)_ij /
000165 |                    [(z-w)^(2r) z^(4-r) w^(4-r)].
000166 | The adjoint uses conjugate(b)/z and b/w respectively. This is the entire
000167 | separated-point kernel, not one leading OPE pole. All Taylor terms and
000168 | smooth local-profile integrals of this two-point distribution are thereby
000169 | specified, with the inherited radial/Wightman-i0 boundary-value prescription,
000170 | not an arbitrarily chosen contact-term extension. A particular physical
000171 | profile/instrument is not selected.
000172 | 
000173 | Radial expansion gives the complete oscillator-grade k response Gram:
000174 |   B0=D0,
000175 |   Bk=sum_(r=1..min(k,4)) binomial(k+r-1,2r-1) D_r, k>=1.
000176 | No cutoff or intermediate code projection occurs. The coefficient matrix
000177 | from D1..4 to B1..4 is triangular with determinant1:
000178 |   [[1,0,0,0],[2,1,0,0],[3,4,1,0],[4,10,6,1]].
000179 | Hence these four mode Grams determine all four connected Wick tensors;
000180 | this is not an added tensor-product or dynamical composition assumption.
000181 | 
000182 | Actual code/erasure normalization forces
000183 |   D4=I5, D3=||b||^2 I5=85/256 I5.
000184 | The D3 identity follows from the actual20 derivative states' orthonormality;
000185 | it is checked with the literal original polynomials. Thus the second-order
000186 | state perturbation is scalar. D2 begins at fourth order in b, D1 at sixth,
000187 | D0 at eighth. State orientation can be invisible to a lower-order probe
000188 | even though d has no native stabilizer. There is no contradiction.
000189 | 
000190 | For the fixed original grade4 input mode,
000191 |   B4=I+6||b||^2 I+10D2+4D1
000192 |      =(383/128)I+10D2+4D1.
000193 | An independent sparse Fock calculation of [z^0]F_i^(d)(z)Omega reproduces
000194 | EVERY entry of this matrix. It retains oscillator partitions4,31,22,211,
000195 | 1111, with20,28,28,28,35 monomials in the five unnormalized outputs.
000196 | Nothing is compressed back to the mode1-quartic subspace.
000197 |