TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt
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000064 | IMPORTANT LOCALITY LIMIT: this is an original STATE/current intertwiner.
000065 | It does NOT identify the original folded quartic field with a point-local
000066 | source vertex Y_cover(P_i,z) of conformal weight32. Its true pullback still
000067 | involves eight original preimage arcs and the existing point splitting.
000068 | The actual field-product prescription is given separately in section4.
000069 | 
000070 | 3. Complete ordered current response, including all profile tails
000071 | 
000072 | Define the EXACT finite normal kernel
000073 |  K_ij(u,v)=<P_i(p), exp(sum_(m>0)u_m p_m)
000074 |                          P_j(p_m+m v_m)>_Hall.
000075 | Its weighted u and v degrees are at most32, by input energy, not a chosen
000076 | perturbative cutoff. An explicit finite formula in the saved coefficients is
000077 |  K_ij=sum_(lambda,mu)conj(c_(lambda,i))c_(mu,j)
000078 |           sum_(nu subset lambda,mu)
000079 |           s_(lambda/nu)(p_m=m u_m) s_(mu/nu)(p_m=m v_m).
000080 | Thus all coefficients, including mixed-mode correlations, are specified.
000081 | It is unnecessary to expand the very large polynomial to define it exactly.
000082 | 
000083 | For a smooth real original profile phi, phi_-m=conj(phi_m), put
000084 |  J(phi)=phi0 Q+sum_(m>0)[phi_m J_-m+conj(phi_m)J_m].
000085 | On charge0 the actual original Weyl response is
000086 |  <P_i,exp(iJ(phi))P_j>
000087 |    =exp(-1/2 sum_(m>0)m|phi_m|^2) K_ij(i phi,i conj(phi)).
000088 | The Gaussian retains EVERY Fourier tail, including m>32. Only the state
000089 | polynomial terminates. In particular a properly supported smooth local
000090 | profile is not replaced with a finite Fourier polynomial.
000091 | 
000092 | For any finite ordered list of such source operations, original CCR gives
000093 |  W(phi1)...W(phis)=exp(-1/2 sum_(i<j)[J(phii),J(phij)]) W(sum_i phii),
000094 |  [J(phi),J(psi)]=sum_(m>0)m[conj(phi_m)psi_m-phi_m conj(psi_m)] I.
000095 | All geometric-free time insertions are retained by the original evolution
000096 |  U(t)J_-m U(t)^*=exp(-itm/8)J_-m, U(t)=exp(-itK_alpha).
000097 | No independent time calibration or phase rule is supplied. This formula
000098 | computes all finite smooth density-Weyl histories on the actual five input
000099 | states, preserving their actual Gram and all intermediate source modes.
000100 | 
000101 | The complete first-mode slice is additionally saved as33 exact Gram
000102 | coefficients: K_ij(u1,v1)=sum_(k=0..32)(u1 v1)^k
000103 |  <J1^k P_i,J1^k P_j>/(k!)^2. This is a diagnostic slice, NOT a selected
000104 | physical profile. Arbitrary exponentiated geometric diffeomorphisms or