TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-observation-arity-20261025/NATIVE-OBSERVATION-ARITY.txt Original file SHA-256: 3beb984ed44fdc1c9aab40626fc19357aaa5a0a15dcaa57259f6d71e487e5b39 Inclusive original lines: 1-200 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. 000001 | NATIVE OBSERVATION ARITY 000002 | ======================== 000003 | 000004 | Result 000005 | ------ 000006 | 000007 | PROVED for the finite native G31 representation reconstructed by the checker: 000008 | the first tensor power carrying a nonconstant POVM with the actual six Ovoid 000009 | label action is n=3. At n=1 and n=2 every covariant six-effect POVM is the 000010 | uniform POVM. On the full 64 dimensional V4 tensor3, every such POVM belongs 000011 | to one exact one-parameter family. Its informative centered part is forced 000012 | to lie on the original twenty-dimensional Sym^3(V4) carrier. 000013 | 000014 | The result covers arbitrary linear Hermitian effects, followed by positivity 000015 | and normalization. It is not restricted to projective or rank-one guesses. 000016 | 000017 | Actual input, with no fitted representation 000018 | -------------------------------------------- 000019 | 000020 | The checker imports these tracked constructors: 000021 | 000022 | * tfpt_explorer/quartic_selection.py: the original N64=mu4.P2 matrices; 000023 | * tfpt_explorer/process.py: the original five quartic word supports; 000024 | * atlas-defect-synthesis-20261024/doily_quartic_bridge.py: the actual sixty 000025 | Gaussian root reflections, their E8/(1-J)E8 Ovoid permutations, the native 000026 | quartic H5 matrices, the six Ovoid rays, and the all-sixty intertwiner; 000027 | * atlas-defect-synthesis-20261024/native_erasure_readout.py: the original 000028 | gauge-first isometry W:C4 tensor H5 -> C64; 000029 | * compiler-correlated-event-clock-20260919/source_channel.py: provenance for 000030 | the Bell10 and outer-S6 comparison, reconstructed here from the matrices. 000031 | 000032 | No new four-dimensional representation or relabelling is solved for. 000033 | 000034 | Why covariance becomes an intertwiner question 000035 | ----------------------------------------------- 000036 | 000037 | Let F_a, a=0,...,5, be effects with sum F_a=I and with the actual Ovoid 000038 | permutation pi_g: 000039 | 000040 | rho_n(g) F_a rho_n(g)* = F_{pi_g(a)}. 000041 | 000042 | Write X_a=F_a-I/6. Then sum X_a=0. The six labels therefore contribute 000043 | their centered five-dimensional permutation representation. A nonconstant 000044 | POVM exists only if this target five occurs in End(V4 tensor n). Conversely, 000045 | at n=3 the checker supplies positive effects explicitly. This reduces the 000046 | minimality statement to a finite exact Hom computation. 000047 | 000048 | Arity one 000049 | --------- 000050 | 000051 | The Ovoid labels are fixed by the normal Pauli kernel N64. Hence every F_a 000052 | must commute with N64. The sixteen phase-free Pauli matrices span End(V4), 000053 | and their exact simultaneous commutant has dimension one. Thus every F_a is 000054 | scalar. The transitive S6 action makes the six scalars equal, and normalization 000055 | gives 000056 | 000057 | F_a = I4/6. 000058 | 000059 | Arity two: the decisive outer-action mismatch 000060 | --------------------------------------------- 000061 | 000062 | On V4 tensor V4, the sixteen vectors vec(P)/2, for the sixteen phase-free 000063 | Paulis P, form an orthonormal Bell basis. The checker diagonalizes every 000064 | P tensor P in that basis and finds sixteen distinct joint characters. The 000065 | full N64-invariant operator algebra is therefore exactly the span of the 000066 | sixteen Bell projectors. Its dimension is 16. 000067 | 000068 | Transpose parity splits these projectors canonically: 000069 | 000070 | 16 = 10 symmetric + 6 antisymmetric. 000071 | 000072 | Every one of the sixty actual root reflections preserves this split. Its 000073 | cycle types are 000074 | 000075 | actual Ovoid6 labels : (2,1,1,1,1), 000076 | symmetric Bell10 labels : (2,2,2,1,1,1,1), 000077 | antisymmetric Bell6 labels : (2,2,2). 000078 | 000079 | Thus the antisymmetric six is precisely the outer six-point S6 action. It is 000080 | not the actual Ovoid six. The fifteen target transpositions together with 000081 | their Bell actions generate 720 paired permutations. Exact character inner 000082 | products over those 720 elements give 000083 | 000084 | Hom(target5, Bell10) = 0, 000085 | Hom(target5, Bell6) = 0. 000086 | 000087 | The Bell10 centered character has norm one and dimension nine. The Bell6 000088 | centered character has norm one and dimension five, but its inner product 000089 | with the target five is zero. Equal dimension five therefore does not supply 000090 | the needed dictionary. The complete N64 commutant contains no target five, 000091 | so 000092 | 000093 | F_a = I16/6 000094 | 000095 | for every covariant six-effect POVM at n=2. 000096 | 000097 | Mixed-orientation control 000098 | ------------------------- 000099 | 000100 | The checker does not infer the result for V4 tensor conjugate(V4) from the 000101 | ordinary tensor square. It separately reconstructs the similarity action 000102 | 000103 | vec(P) -> vec(R P R*) 000104 | 000105 | on all sixteen Pauli Bell lines. It separately diagonalizes every 000106 | P tensor conjugate(P) and again finds sixteen distinct joint characters, so 000107 | the mixed N64 commutant is exactly the span of these sixteen projectors. The 000108 | identity line is fixed, and the nonzero 000109 | fifteen-label character decomposes exactly as 000110 | 000111 | 15 = 1 + 9 + outer5. 000112 | 000113 | The paired root permutations again generate all 720 elements. The inner 000114 | product of this permutation character with the actual Ovoid target five is 000115 | zero. Thus this alternative orientation also supplies only the uniform 000116 | six-effect Ovoid POVM. This is a separate exact check and does not identify 000117 | the two tensor orientations. 000118 | 000119 | Arity three: the existing original carrier 000120 | ------------------------------------------- 000121 | 000122 | From the original quartic word supports the checker reconstructs the same 000123 | 64 by 20 isometry W used by the erasure contract. It proves exactly 000124 | 000125 | W*W = I20, 000126 | WW* = Pi20 = the three-slot symmetrizer, 000127 | image(W) = Sym^3(V4), 000128 | R4 tensor3 W = W [conjugate(R4) tensor C5] 000129 | 000130 | for all sixty native root reflections. For all sixteen phase-free Pauli 000131 | kernel matrices it also proves 000132 | 000133 | P tensor3 W = W [conjugate(P) tensor I5]. 000134 | 000135 | Consequently the N64 commutant on this carrier is 000136 | 000137 | I4 tensor End(H5). 000138 | 000139 | The actual quartic H5 is the actual Ovoid standard five by the independently 000140 | reconstructed all-sixty bridge. Over the generated S6, if chi is its 000141 | character, the checker verifies the exact character decomposition 000142 | 000143 | End(H5) = 1 + 5 + 9 + 10 000144 | 000145 | with pairwise orthonormal characters. Equivalently, 000146 | 000147 | = 1. 000148 | 000149 | The target five occurs exactly once on the carrier. The known centered Ovoid 000150 | effects E_a-I5/6 are a nonzero copy of it. Therefore every covariant 000151 | six-effect POVM restricted to Sym^3(V4) has the form 000152 | 000153 | A_a(t) = I4 tensor [t E_a + (1-t) I5/6] 000154 | 000155 | for one real t. This is a classification, not only an example. 000156 | 000157 | Full G31 character count 000158 | ------------------------ 000159 | 000160 | The carrier calculation alone would still allow another target five on the 000161 | 44 dimensional complementary sector. The checker closes that possibility 000162 | by an independent full representation character count. 000163 | 000164 | It chooses one of the actual native four-dimensional root matrices above each 000165 | of the fifteen Ovoid transpositions. A breadth-first word construction gives 000166 | one matrix G_pi for every one of the 720 permutations pi in S6. The checker 000167 | then verifies all multiplication data needed for a complete transversal: 000168 | 000169 | * all sixty original root lifts lie in the expected Pauli coset; 000170 | * the fifteen chosen lifts normalize the projective Pauli16; 000171 | * every one of the 720 times 15 right-generator products lands in the expected 000172 | next coset; 000173 | * the matrices P G_pi give exactly 11,520 distinct projective elements. 000174 | 000175 | The four central phases of G31 do not change |tr|. Hence the full G31 000176 | multiplicity equals the projective average 000177 | 000178 | m_n = (1/11520) sum_(pi in S6, P in Pauli16) 000179 | chi5(pi) |tr(P G_pi)|^(2n). 000180 | 000181 | The exact unnormalized sums for n=1,2,3 are 000182 | 000183 | 0, 0, 11520, 000184 | 000185 | so 000186 | 000187 | m_1=0, m_2=0, m_3=1. 000188 | 000189 | Thus the target five occurs exactly once in the entire End(V4 tensor3), not 000190 | once per chosen sector. The already constructed centered family is nonzero, 000191 | so it spans that unique full Hom space. Every full covariant POVM is therefore 000192 | 000193 | F_a(t) = I64/6 + t W[I4 tensor (E_a-I5/6)]W* 000194 | 000195 | or, equivalently, 000196 | 000197 | F_a(t) = W[I4 tensor (t E_a+(1-t)I5/6)]W* 000198 | + (I64-Pi20)/6. 000199 | 000200 | The uniform action on the 44 dimensional complement and the support of the