TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/MIXED-ORDERED-RETURN.md
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000095 | ## 4. A directly checked original channel and its erasure continuation
000096 | 
000097 | For source doubled root r=(0,0,0,0,0,0,-2,-2), the actual map gives
000098 | 
000099 |     beta_native=sqrt2 e5,
000100 |     v_native=4sqrt2(e6+e7),
000101 |     b2=-i, c3=4-4i (all other components zero).
000102 | 
000103 | Its five-return matrix is
000104 | 
000105 |     [ 188/3, -40sqrt3 i/3, 0, 0, 0 ]
000106 |     [ 40sqrt3 i/3, 80,    0, 0, 0 ]
000107 |     [ 0,             0, 48, 0, 0 ]
000108 |     [ 0,             0,  0,48, 0 ]
000109 |     [ 0,             0,  0, 0,32 ].
000110 | 
000111 | The actual native erasure twenty is Sym^3(H+), NOT the differently charged
000112 | E8 F20 current block. Its original isometry has Fock columns
000113 | 
000114 |     g_(i,a)=partial_a f_i, i=0..4, a=0..3.
000115 | 
000116 | All20 columns are checked orthonormal. The factor is fixed: annihilating
000117 | one oscillator in degree4 equals sqrt4 times tensor contraction, exactly
000118 | the existing W20 decoder's factor2. Thus this is the original intertwiner,
000119 | not a freely selected twenty-space of the same dimension.
000120 | 
000121 | Apply the SAME source operator o(W_beta) and its adjoint to these original
000122 | trimer columns. The computed return K20 has
000123 | 
000124 |     rank K5=5,  rank K20=13,
000125 |     operator-Schmidt rank K20 (5 x 4 factors)=4,
000126 |     ||K20-K5 tensor I4||_HS^2=124096/3.
000127 | 
000128 | An analytic witness needs no large matrix: g_(0,0)=x0^3/sqrt6 is killed
000129 | because D_b and D_c both vanish on it. Therefore K20 g_(0,0)=0, whereas
000130 | 
000131 |     <g_(0,0), (K5 tensor I4) g_(0,0)>=188/3.
000132 | 
000133 | This tests the direct decoder continuation of THIS return response. Even
000134 | the stronger allowance of an arbitrary register unitary cannot repair its
000135 | complete branchwise erasure action with given invertible K5. If
000136 | 
000137 |     T(rho tensor I4)T*=K5 rho K5* tensor I4  for every rho,
000138 | 
000139 | put S=(K5^(-1) tensor I4)T. Taking rho=I makes S unitary; the full matrix
000140 | algebra then makes S=I5 tensor V4. Hence T=K5 tensor V4 has rank20, in
000141 | contradiction to actual K20 rank13. Operator-Schmidt rank4 also rules out
000142 | any factorization with a register matrix, independent of an overall scale.
000143 | No unnormalized amplitude above is a detector probability.
000144 | 
000145 | The statement excludes this ordered response branch as a common full-erasure
000146 | lift. It does NOT classify arbitrary sums of histories, nonselective
000147 | instruments, other source terms, or all combined root/quadratic evolution.
000148 | Nor is K_beta itself the unitary quarter R5; constructing a desired
000149 | function of it would require an additional source-selected rule.