TFPT scientific source excerpt Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/NATIVE-ERASURE-READOUT.txt Original file SHA-256: 5be50c2142b498e91350c011c904a13ce823d7a0eef1ff490d89469187e72357 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 ERASURE READOUT 000002 | ======================= 000003 | 000004 | Verdict 000005 | ------- 000006 | 000007 | PASS, exact. 000008 | 000009 | The six doily effects define a concrete additional POVM on the original 000010 | twenty-dimensional erasure output. This does not identify that POVM with the 000011 | original erasure/recovery instrument. No new dynamics and no auxiliary 000012 | Hilbert space are introduced. 000013 | 000014 | 000015 | 1. Original decoder, reconstructed rather than assumed 000016 | -------------------------------------------------------- 000017 | 000018 | Let S_i, i=0,...,4, be the five original quartic word supports from 000019 | tfpt_explorer/process.py:242-258. Their sizes are 000020 | 000021 | |S_i| = (4,12,12,12,24). 000022 | 000023 | After erasing tensor slot x0, define the gauge-first isometry 000024 | 000025 | W : C4_gauge tensor H5 -> C4 tensor C4 tensor C4 000026 | 000027 | by 000028 | 000029 | W[(x1,x2,x3),(a,i)] = 2/sqrt(|S_i|) 000030 | 000031 | when (a,x1,x2,x3) belongs to S_i, and zero otherwise. In integer row and 000032 | column indices this is 000033 | 000034 | row = 16*x1 + 4*x2 + x3, 000035 | column = 5*a + i. 000036 | 000037 | The checker reconstructs all 64 nonzero entries. Exact algebra gives 000038 | 000039 | W* W = I20, 000040 | Pi20 = W W*, 000041 | rank(Pi20) = 20, 000042 | Pi20^2 = Pi20. 000043 | 000044 | Thus the actual decoder on its physical carrier is U_dec=W*. The earlier 000045 | original checker uses columns 4*i+a, hence H5 tensor C4 order. The present 000046 | columns 5*a+i are the requested C4 tensor H5 order and differ only by the 000047 | explicit factor-swap permutation. The complete sparse W and U_dec matrices 000048 | are stored in native_erasure_readout_results.json. 000049 | 000050 | This exactly reuses the original constructor in 000051 | experiments/theory-contracts/joint-native-algebra-selection-20261016/ 000052 | operator_checker.py:85-94. Its original isometry and selected intertwiner 000053 | checks are at lines 169-178. 000054 | 000055 | 000056 | 2. Six effects on H5 and H20 000057 | ---------------------------- 000058 | 000059 | In the unnormalised original word basis the invariant metric and six rays are 000060 | 000061 | G = diag(1,3,3,3,6), 000062 | q_r^T G q_r = 10, 000063 | E5_native,r = q_r q_r^T G / 12. 000064 | 000065 | The normalized original quartic code basis is obtained with 000066 | 000067 | D = diag(1,sqrt(3),sqrt(3),sqrt(3),sqrt(6)). 000068 | 000069 | Consequently the orthonormal H5 effects are 000070 | 000071 | E5_r = D E5_native,r D^-1 000072 | = (D q_r)(D q_r)^T / 12. 000073 | 000074 | They obey 000075 | 000076 | sum_r E5_r = I5, 000077 | rank(E5_r) = 1, 000078 | E5_r^2 = (5/6) E5_r. 000079 | 000080 | The requested twenty-dimensional effects are the explicit matrices 000081 | 000082 | E20_r = I4 tensor E5_r. 000083 | 000084 | Therefore 000085 | 000086 | sum_r E20_r = I20, 000087 | rank(E20_r) = 4. 000088 | 000089 | On the actual three-register physical carrier they are 000090 | 000091 | Ephys_r = W E20_r W*. 000092 | 000093 | All six 20 by 20 matrices and all six sparse 64 by 64 physical matrices are 000094 | stored in the result JSON. Their physical sum is Pi20, which is the identity 000095 | on H20 inside the ambient C64. 000096 | 000097 | 000098 | 3. Exact probabilities on the original erasure output 000099 | ------------------------------------------------------ 000100 | 000101 | For an arbitrary symbolic 5 by 5 matrix rho5, define the original erasure 000102 | state in the physical three-register space by 000103 | 000104 | rho_out = W (I4/4 tensor rho5) W*. 000105 | 000106 | Using W*W=I20 gives, for every r, 000107 | 000108 | Tr(rho_out Ephys_r) 000109 | = Tr[(I4/4 tensor rho5)(I4 tensor E5_r)] 000110 | = Tr(rho5 E5_r). 000111 | 000112 | The checker verifies this as a polynomial identity in all 25 symbolic entries 000113 | of rho5 for each of the six outcomes. Positivity or unit trace is not needed 000114 | for the algebraic check, so the result covers every density matrix by 000115 | specialisation. 000116 | 000117 | 000118 | 4. Actual native action on all twenty output directions 000119 | ------------------------------------------------------- 000120 | 000121 | For each of the 60 original Gaussian reflections, let R4 be its exact native 000122 | four-dimensional matrix and C5 its action on the normalized original quartic 000123 | code. The checker verifies the full 64 by 20 identity 000124 | 000125 | R4^tensor3 W = W [conjugate(R4) tensor C5]. 000126 | 000127 | Thus the gauge-first twenty-dimensional action is the actual matrix 000128 | 000129 | rho20(g) = conjugate(R4(g)) tensor C5(g). 000130 | 000131 | It is not inferred from matching dimensions. It is checked entry by entry 000132 | against the explicit W for all 60 reflections. The six effects obey 000133 | 000134 | rho20(g) E20_r rho20(g)* = E20_{g.r} 000135 | 000136 | for all 60 times 6 cases. The result JSON records every induced permutation, 000137 | both exact verdicts, and a hash of each concrete 20 by 20 action matrix. 000138 | 000139 | The C5 matrices are independently reconstructed from the fourth tensor power 000140 | of the same R4 in doily_quartic_bridge.py:49-102 and 438-510. Their invariant 000141 | metric and six-ray covariance are proved there at lines 638-693. 000142 | 000143 | 000144 | 5. The 120-word stabilizer average 000145 | ---------------------------------- 000146 | 000147 | The 60 native reflections induce the 15 transpositions of the six ovoids, 000148 | each four times. Selecting the first native representative for each distinct 000149 | transposition gives 15 actual reflection generators. Exact breadth-first 000150 | closure produces all 720 elements of S6 together with a word in those native 000151 | reflection indices. 000152 | 000153 | For each ray r, its stabilizer has exactly 120 elements. Averaging their 000154 | original H5 matrices in the native word basis gives 000155 | 000156 | (1/120) sum_{g in Stab_S6(r)} C5_native(g) 000157 | = q_r q_r^T G / 10 000158 | = (6/5) E5_native,r. 000159 | 000160 | After the fixed D conversion this is precisely 000161 | 000162 | P_r = (6/5) E5_r. 000163 | 000164 | So P_r is the invariant projector onto the one-dimensional fixed space of the 000165 | actual S5 stabilizer. This is the representation-theoretic normalization of 000166 | the doily effect, not merely a rank or trace coincidence. The JSON contains 000167 | all six lists of 120 group words. Each word is a list of original native 000168 | reflection indices; the certificates found with the 15 transposition 000169 | generators have maximum length four. 000170 | 000171 | 000172 | 6. Meaning of the connection 000173 | ---------------------------- 000174 | 000175 | The previous instrument comparison remains unchanged: the original recovered 000176 | erasure branch effects are scalar on H5, whereas these six effects are 000177 | rank-one and noncommuting. The positive statement proved here is narrower and 000178 | concrete. The existing decoder transports the new logical POVM to an actual 000179 | observable on the already present twenty-dimensional output register: 000180 | 000181 | E5_r -> I4 tensor E5_r -> W(I4 tensor E5_r)W*. 000182 | 000183 | The native E8/G31 conjugation words act on this observable through the already 000184 | verified original decoder intertwiner. This supplies an observable readout 000185 | on the existing erasure consumer without claiming a new preparation, 000186 | Hamiltonian, recovery rule, or equality with the original erasure instrument. 000187 | 000188 | 000189 | Machine certificate 000190 | ------------------- 000191 | 000192 | Run from the repository root: 000193 | 000194 | python3 -B experiments/theory-contracts/atlas-defect-synthesis-20261024/native_erasure_readout.py 000195 | 000196 | Observed result: 000197 | 000198 | PASS: W=64x20 exact; six H20 effects rank 4 and sum I20; 000199 | 60 native actions covariant; six S5 stabilizers have 120 words. 000200 |