TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SOURCE-ERASURE-WARD-AND-CODE-CLOSURE.md
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000111 | ## 4. The unchanged all-root sum also fails full code closure
000112 | 
000113 | This is NOT an arbitrary fitted linear combination. Use the same original
000114 | root-current INPUT basis and unrenormalized R_neut responses as before:
000115 | 
000116 |     L_sum = sum_beta S_beta^dagger S_beta.
000117 | 
000118 | Direct exact evaluation on the same q_raw and P0 gives
000119 | 
000120 |     sum_beta <S_beta q_raw,S_beta P0> = -50208,
000121 |     <q,L_sum f0> = -2092/sqrt3 !=0,                         (4)
000122 |     ||(I-Pi)L_sum f0||^2 >= 4376464/3.                      (5)
000123 | 
000124 | All4 ORIGINAL family omissions yield exactly the same (4). The computation
000125 | rebuilds each original A16 and A_R=4I-A16 before transport by O; it does not
000126 | rotate the code to fit a branch. Thus summing all original root responses or
000127 | changing among these4 supplied branches does not restore code closure.
000128 | 
000129 | As full operators, L_sum and S^dagger S for S=sum_beta a_beta S_beta
000130 | are different: the latter also has cross terms with nonzero final lattice
000131 | charge gamma-beta. For the NEUTRAL q and f0 used in (4), charge orthogonality
000132 | eliminates all beta!=gamma cross terms. The same witness therefore also
000133 | excludes the full coherent S^dagger S with |a_beta|=1, whatever the phases.
000134 | No arbitrary positive-weight code-closure theorem is claimed here. The
000135 | separate previously proved strict trace-gap theorem excludes every such
000136 | nontrivial positive profile as the fixed common erasure lift, with its stated
000137 | real-transverse/common-normalization premises.