TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/operator_audit.md
Original file SHA-256: f1849345563b8d81cd3861a5a5eaa96f1355ff7d5d55bdbc00e73ab90b7b64e5
Inclusive original lines: 87-125
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000087 | \qquad E_{\rm phys}(\rho)=W E(\rho)W^*.
000088 | \]
000089 | 
000090 | This is an actual erasure/Schmidt factorization, rather than a `20=5*4` dimension match.
000091 | 
000092 | On `V20`, `P64` acts trivially on `V5` and irreducibly on `bar V4`; therefore its commutant is `M5 tensor I4`.  Intersecting with the irreducible standard `S6` action on the multiplicity space leaves only scalars.  Thus `V20` is irreducible.  Together with the faithful irreducible `V4` and the irreducible quotient module `V5`, Burnside's theorem gives
000093 | 
000094 | \[
000095 | \mathbb C[\rho_4\oplus\rho_5\oplus\rho_{20}](G_{31})
000096 | =M_4\oplus M_5\oplus M_{20},\qquad
000097 | \rho(G_{31})'=\mathbb C^3.
000098 | \]
000099 | 
000100 | The kernels are `ker rho4=1`, `ker rho5=P64`, `ker rho20=1`.  Central `J=iI4` acts as
000101 | 
000102 | \[
000103 | (iI_4,\ I_5,\ -iI_{20}).
000104 | \]
000105 | 
000106 | Hence the three block projectors are already polynomials in the actual central operator.  This classifies a hidden block degree of freedom; it does not choose a dissipative rate or menu of Pauli-gated operations.
000107 | 
000108 | ## 3. Joint normalizer: proof and exact scope
000109 | 
000110 | The group calculation gives a normal 2-subgroup `P64` and quotient `S6`.  Since `O2(S6)=1`, every normal 2-subgroup of `G31` maps trivially to the quotient and lies in `P64`.  Therefore
000111 | 
000112 | \[
000113 | O_2(G_{31})=P_{64}.
000114 | \]
000115 | 
000116 | `O2` is characteristic, so every automorphism or unitary normalizer of `G31` preserves `P64`.  The unitary normalizer of the two-qubit Pauli group is, by definition, the two-qubit Clifford group.  Projectively it has order `11520`; the native `G31/mu4` has the same order and is already contained in it.  Consequently
000117 | 
000118 | \[
000119 | N_{U(4)}(G_{31})=U(1)G_{31}.
000120 | \]
000121 | 
000122 | For the joint representation, the dimensions `4,5,20` are distinct, so a normalizer preserves the three irreducible blocks.  Its faithful `V4` block induces an inner automorphism, by the previous equality.  Removing the corresponding common `rho(g)` leaves an operator commuting with `G31` on each irreducible block, hence a phase by Schur's lemma.  Therefore
000123 | 
000124 | \[
000125 | N_{U(4)\times U(5)\times U(20)}(\rho(G_{31}))