TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SOURCE-ERASURE-WARD-AND-CODE-CLOSURE.md
Original file SHA-256: b365eebb2d5419fdf6306d2a41d8ae40bb538b5b195e01ccec91636c1fc8cd38
Inclusive original lines: 70-109
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000070 | ## 3. Analytic full-return leakage witness
000071 | 
000072 | Use the already present doubled source root (0,0,0,0,0,0,-2,-2).
000073 | The original O and R_neut transport it to
000074 | 
000075 |     beta_native=sqrt2 e5, v_native=4sqrt2(e6+e7),
000076 |     b2=-i, c3=4-4i, all other b,c zero.
000077 | 
000078 | Write P0=sum_j x_j^4 and f0=P0/sqrt96. Define
000079 | 
000080 |     h=x0^4/sqrt24,
000081 |     q=(2/sqrt3)(h-f0/2)=(x0^4-P0/4)/sqrt18.
000082 | 
000083 | The original Fock product gives ||h||=||q||=1, <h,f0>=1/2,
000084 | and q is orthogonal to ALL5 code vectors. Because b,c have no0 component,
000085 | the FULL original vertex action gives S_beta h=0. Thus
000086 | 
000087 |     <h,L_beta f0>=<S_beta h,S_beta f0>=0.
000088 | 
000089 | The independently reproduced earlier K00=188/3, however, implies
000090 | 
000091 |     <h,Pi L_beta f0>=K00/2=94/3.
000092 | 
000093 | Consequently the exact full-operation matrix element is
000094 | 
000095 |     <q,L_beta f0>=-K00/sqrt3=-188/(3sqrt3) !=0,               (2)
000096 |     ||(I-Pi)L_beta f0||^2 >= 35344/27.                       (3)
000097 | 
000098 | This proves failure of code closure without constructing a large full matrix.
000099 | All norms are fixed original source/Fock norms; the bound is not a probability.
000100 | The sparse check also evaluates <S_beta q_raw,S_beta P0>=-1504 with
000101 | ||q_raw||^2=18 and ||P0||^2=96, reproducing (2) separately.
000102 | 
000103 | For this witness (1) is zero for a0,a1; it is iW_beta,2 for a2 and
000104 | (4+4i)E_beta,1 for a3. In particular there is no missing a0 Ward coefficient
000105 | that can repair the old decoder comparison. The adjoint also commutes with
000106 | a0: beta and v have no0 component. Therefore a0 L_beta f0=0, whereas the
000107 | compressed logical decoder has its nonzero188/3 diagonal entry. Compression
000108 | is a valid mathematical response operation; physical permission to impose it
000109 | has not been derived.