TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/CLUSTERING-ADMISSIBLE-SECTOR.txt
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000065 | this the color-center singlet projector.  It is a center-neutrality projector,
000066 | not the Haar projector onto a full connected SU(3) singlet.
000067 | 
000068 | 
000069 | 2. THE SMALLEST DECISIVE ACTION TEST
000070 | 
000071 | Use the actual ordered carrier frame E=C^3 direct-sum C^2.  The scaled
000072 | hypercharge mark is
000073 | 
000074 |     X=6Y=(-2,-2,-2,3,3,0,0,0).
000075 | 
000076 | For q=M p_*=e1+e4,
000077 | 
000078 |     X.q = 1,              Y(q)=1/6,
000079 |     Q_source.q = 0.
000080 | 
000081 | The SU(3) center exponent in this frame is the coweight pairing
000082 | 
000083 |     t(q) = q1+q2-2q3 mod 3.
000084 | 
000085 | It equals one on e1+e4.  Thus Z_c acts by omega.  Reversing the convention for
000086 | the fundamental would replace omega by omega^2 and leaves the projector result
000087 | unchanged.  The vacuum has t=0 and survives.
000088 | 
000089 | On the two-dimensional full ground space the exact actions are therefore
000090 | 
000091 |                   Omega       |p_*>
000092 |     H8              0            0
000093 |     Z_c             1          omega
000094 |     P_sing          1            0.
000095 | 
000096 | This also explains the observable statement without treating P_sing as a
000097 | global compression.  For a center-neutral field A, so that [A,Z_c]=0,
000098 | 
000099 |     <p_*|A|Omega>
000100 |       = <p_*|Z_c^{-1} A Z_c|Omega>
000101 |       = omega^{-1}<p_*|A|Omega>
000102 |       = 0.
000103 | 
000104 | The same vanishing follows directly from the original local-net definition for
000105 | any A for which A and A* preserve the center-neutral subspace, because A Omega
000106 | then lies in a subspace orthogonal to p_*.  The colored root current may remain
000107 | an intersector field in the graded field net.  It is not an operator of the
000108 | restricted center-neutral local observable net because it sends the vacuum out
000109 | of the admissible subspace.
000110 | 
000111 | 
000112 | 3. COMPLETE ZERO AND GAP CLASSIFICATION
000113 | 
000114 | For a lattice charge p and oscillator level N,
000115 | 
000116 |     K_alpha(p,N) = N + ||p||^2/2 + delta_alpha.p.
000117 | 
000118 | Because |delta_(alpha,j)|<1/2, every nonzero integer coordinate contributes
000119 | 
000120 |     n^2/2 + delta_j n = n(n/2+delta_j) > 0.