TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SOURCE-RETURN-MEMORY.txt
Original file SHA-256: c22c2144e4986a0f9c2f591fdc1667d3ad8233d4ff39e433c22031cc366dbd11
Inclusive original lines: 29-113
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.

000029 | ---------------------------------------------------------------
000030 | Let X_(n,A)=h_A(-n) be the original orthonormal complex Heisenberg creation
000031 | coordinates, with n>=1 and A in the8 J-positive/opposite directions. Their
000032 | Fock squared monomial norms w are product n^multiplicity times multiplicity
000033 | factorials.
000034 | 
000035 | For real beta and v perpendicular to beta, write b_a=u_a.beta,c_a=u_a.v.
000036 | The lattice-vertex formula on a neutral polynomial f is
000037 | 
000038 |   Y(e^beta,z) f(X) = E_-(beta,z) f(X_(n,A)-r_A z^-n) e^beta,
000039 |   E_-(beta,z)=exp(sum_(m>=1) beta(-m) z^m/m),
000040 |   r=(b0,b1,b2,b3,bar b0,bar b1,bar b2,bar b3).
000041 | 
000042 | Normal ordering Y(W_beta,z)=:v(z)Y(e^beta,z): then gives
000043 | 
000044 |   S f = [z^-2] E_-(beta,z) {
000045 |       sum_(m>=1) v(-m) z^(m-1) f(X-r z^-n)
000046 |     + sum_(n>=1) n z^(-n-1) (D_(v,n) f)(X-r z^-n)
000047 |   } e^beta.                                                     (1)
000048 | 
000049 | The derivative D_(v,n) uses c on the positive and bar c on the opposite
000050 | mode-n coordinates. The factor n follows from the original Heisenberg
000051 | commutator; it is not an inserted coupling. The neutral charge zero mode
000052 | vanishes and beta.v=0. This is the same source action as before, now on the
000053 | entire neutral grade4 PBW block rather than just positive-mode1 quartics.
000054 | 
000055 | Primary formula reference: Tuite and Zuevsky, arXiv:1106.6149v2,
000056 | Eq.(41), Heisenberg creation/annihilation exponentials; Prop.2.6/Eq.(33),
000057 | normal ordered descendants: https://arxiv.org/pdf/1106.6149 . These general
000058 | identities do not identify TFPT's physical source or select its history.
000059 | 
000060 | At total conformal grade4, a neutral input has726 PBW basis vectors.
000061 | Charge beta has conformal ground weight1, so its grade4 output has192
000062 | oscillator-grade3 PBW vectors. Formula(1) therefore gives the COMPLETE
000063 | 192 by726 single-beta matrix, with1998 nonzero entries in this chart.
000064 | These are finite exact graded blocks, not a numerical cutoff or particle count.
000065 | The output retains all192 rows and every allowed mode partition.
000066 | 
000067 | The adjoint is determined by those original Fock squared norms w:
000068 | 
000069 |   (S^dagger)_(m,n)=bar(S_(n,m)) w(n)/w(m).                        (2)
000070 | 
000071 | The complete map reproduces all5 previous code-forward polynomials, their
000072 | Gram K, and the normalized exterior amplitude -188/(3sqrt3) independently
000073 | of the earlier annihilated-h leakage argument. No intermediate P was applied.
000074 | All21 previously pinned original source files retain their recorded hashes.
000075 | 
000076 | Scope of the finite block is essential. A single-beta L preserves final
000077 | lattice charge0. The diagonal root-input sum also preserves charge0.
000078 | A COHERENT sum of different S_beta has cross terms with nonzero final
000079 | charge gamma-beta and needs the corresponding other grade4 charge sectors.
000080 | The present full-operator numbers are for the SINGLE-beta pair only.
000081 | 
000082 | 3. Exact feedback before any reduced dynamical assumption
000083 | --------------------------------------------------------
000084 | P is the orthogonal original code projector, Q=I-P within the neutral
000085 | grade4 block. They only SPLIT a vector; neither is executed as a measurement.
000086 | In the original normalized code basis define
000087 | 
000088 |   K=P L P,  B=Q L P,  D=Q L Q.
000089 | 
000090 | Because L is selfadjoint, the complete block matrix is [[K,B^dagger],[B,D]].
000091 | For an input psi in the code set x_n=P L^n psi, y_n=Q L^n psi. Then
000092 | 
000093 |   x_(n+1)=K x_n+B^dagger y_n,
000094 |   y_(n+1)=B x_n+D y_n,             y_0=0.                       (3)
000095 | 
000096 | Elimination, with no approximation or intervention, gives
000097 | 
000098 |   x_(n+1)=K x_n+sum_(j=0)^(n-1) B^dagger D^(n-1-j)B x_j.      (4)
000099 | 
000100 | The kernel is fixed by the existing source and the existing code readout.
000101 | It is the elementary linear block form of exact projection/elimination
000102 | identities, related to the discrete Mori-Zwanzig formalism. See Lin, Tian,
000103 | Perez and Livescu, arXiv:2205.05135, Sec.2.7/Eq.(5), for the general memory
000104 | decomposition: https://arxiv.org/pdf/2205.05135 . No statistical projection,
000105 | training, noise model or physical time from that paper is imported here.
000106 | 
000107 | In particular the next ordered response is
000108 | 
000109 |   H=P L^2 P=K^2+Delta,   Delta=B^dagger B.                      (5)
000110 | 
000111 | Unlike K^2, H includes the actual exit and re-entry through original
000112 | source states. This is a FOUR-W-field-insertion answer, not a proposed
000113 | physical four-step protocol.