TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PROJECTED-JOINT-SOURCE-GENERATOR-20261103.txt
Original file SHA-256: bac12849d64dff9abe77e0f02c6d47bd33359501ccaa0f5730fa291558c90796
Inclusive original lines: 99-148
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.

000099 | 
000100 | Let P=P_adm be the original whole physical selector on the SAME common
000101 | space, and A_t[B]=P Q_t[B] P. No fiber decomposition P=P_b is assumed.
000102 | No replacement by Gamma of a one-particle projector or fermion parity.
000103 | In particular a gauge/structural selector may mix boson boundary data.
000104 | 
000105 | Exact algebra applied to (3) gives
000106 |  A_t[1] c_i^dagger-sum_j c_j^dagger A_t[(R_t)_ji]
000107 |   =P Q_t[1][P,c_i^dagger]
000108 |        +sum_j[P,c_j^dagger]Q_t[(R_t)_ji]P.                   (8)
000109 | Proof: expand both commutators. The middle terms are
000110 |  -P Q_t[1]c_i^dagger P+sum_j P c_j^dagger Q_t[(R_t)_ji]P=0
000111 | by(3), leaving exactly the left side. No [P,Q_t[B]]=0 is used.
000112 | For distinct actual incoming/outgoing spaces retain P_in and P_out,
000113 | and their corresponding field operators, on their respective sides.
000114 | 
000115 | For ordered creation words C_I^dagger of length k the same exact identity
000116 | uses det(R_t)_(J,I) as its original scalar marks and commutators
000117 | [P,C_I^dagger],[P,C_J^dagger]. The formula follows from the already given
000118 | whole exterior source and the same two-line expansion. Adjoint and mixed
000119 | words retain the actual adjunction, ordering and field domains.
000120 | There is NO extra P between each charged letter of an original word.
000121 | Neutral composite words can contain charged intermediate propagation.
000122 | 
000123 | The original local net requires BOTH A and A^dagger to preserve Ran(P).
000124 | For bounded A this is equivalent to [A,P]=0. For these actual allowed
000125 | field combinations the corresponding selector contacts vanish.
000126 | This is not a requirement that every charged CAR coordinate be a physical
000127 | observable. No universal ambient-CAR prerequisite is introduced.
000128 | 
000129 | For the original color center let
000130 |  P_r=(1/3)sum_(k=0)^2 omega^(-kr) Z_c^k,
000131 |  Z_c c_q^dagger Z_c^-1=omega^q c_q^dagger.
000132 | Then EXACTLY
000133 |  P_r c_q^dagger=c_q^dagger P_(r-q),
000134 |  [P_0,c_q^dagger]=c_q^dagger(P_(-q)-P_0),
000135 |  P_0 c_q^dagger P_0=0 for q!=0.                             (9)
000136 | For the full original P=P_rest P_0 retain also
000137 |  [P,c]=P_rest[P_0,c]+[P_rest,c]P_0.
000138 | No field action of the gap, seam or determinant selector is invented.
000139 | The color example computes an ORIGINAL selector contact, not a new model.
000140 | 
000141 | 4. PHYSICAL RESTRICTION REQUIRES THE ACTUAL TIME SEWING
000142 | 
000143 | If the SAME original P reduces H, then the physical generator is the
000144 | restriction H|Ran(P), with (6) on that range. If it does not reduce H,
000145 | P exp(-tH)P is not automatically a closed physical semigroup. At a
000146 | compatible finite cutoff its exact sewing defect is
000147 |  P exp(-2tH)P-(P exp(-tH)P)^2
000148 |    =P exp(-tH)(I-P)exp(-tH)P >=0,