TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/MIXED-ORDERED-RETURN.md
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000151 | ## 5. Positive root reweighting cannot repair this two-insertion class
000152 | 
000153 | The individual mismatch is not just an unfortunate channel choice. For
000154 | ANY original root beta and ANY REAL v perpendicular to beta, define
000155 | K5(beta,v), K20(beta,v) by this same normalized adjoint-return calculation.
000156 | Resolve v into its component parallel to Jbeta and its component orthogonal
000157 | to both beta and Jbeta. Exact trace forms give
000158 | 
000159 |     tr K5=(45/16)||v_parallel||^2+(203/48)||v_orthogonal||^2,
000160 |     tr K20=(23/3)||v_parallel||^2+(32/3)||v_orthogonal||^2,
000161 | 
000162 |     4 tr K5-tr K20
000163 |       =(43/12)||v_parallel||^2+(25/4)||v_orthogonal||^2 >0
000164 | 
000165 | whenever v is nonzero. This is for real transverse v; it is not a claimed
000166 | classification of arbitrary complex weight-two fields.
000167 | 
000168 | Proof: the actual native G31 acts transitively on all240 roots, preserves J
000169 | and the code, and acts unitarily on the entire Sym3 trimer. Therefore it
000170 | suffices to use beta=sqrt2 e0. The action is linear in the seven real
000171 | coordinates of v, so both traces are real quadratic forms. Computing ALL
000172 | entries for v=sqrt2 e_j, j=1..7, gives trace matrices
000173 | 
000174 |     C5=diag(45/8,203/24,203/24,203/24,203/24,203/24,203/24),
000175 |     C20=diag(46/3,64/3,64/3,64/3,64/3,64/3,64/3).
000176 | 
000177 | The exact difference is positive definite; each tested basis vector has
000178 | squared norm2. The imaginary antisymmetric Gram part is retained, not
000179 | claimed zero, and drops out for REAL v. Root transitivity is checked by
000180 | an exact240-element orbit under the original five native generators; code
000181 | invariance uses the existing native representation theorem. As a separate
000182 | cross-check, these formulas reproduce the traces of ALL240 complete code
000183 | return matrices. For the original aggregate they give
000184 | 
000185 |     tr K5_sum=18400, tr K20_sum=46848, gap=26752.
000186 | 
000187 | Now let W(a)=sum_beta a_beta W_beta, with arbitrary complex input profile
000188 | coefficients and the ORIGINAL real transverse v_beta. Orthogonal charge
000189 | sectors force Kd(a)=sum_beta |a_beta|^2 Kd,beta, d=5,20. Every nontrivial
000190 | profile has the same strictly positive trace gap. It therefore cannot
000191 | obey K20(a)=K5(a) tensor I4. Since every nonzero original K5,beta is positive
000192 | definite, so is K5(a); positivity plus the preceding commutant argument
000193 | also excludes complete unrenormalized branchwise erasure with an arbitrary
000194 | register unitary. No positive root weights can repair this response class.
000195 | 
000196 | The SAME original source normalization is essential. Inserting an extra
000197 | relative normalization on the two carriers would change the tested map.
000198 | The theorem covers these positive ADJOINT two-insertion profiles, not
000199 | arbitrary nonadjoint outgoing/return pairs, new complex transverse fields,
000200 | longer charge-balanced histories, or all physical channels.