TFPT scientific source excerpt Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/NATIVE-EVENT-CHANNELS.txt Original file SHA-256: f8c12b61871d9cc39777ea2752ce452d75f393b77546e839ad47152cbe51c1a0 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 EVENT CHANNEL INTERSECTION 000002 | ================================= 000003 | 000004 | Verdict 000005 | ------- 000006 | 000007 | The forgotten-outcome channel Phi_p of the six-effect covariant instrument is 000008 | an exact positive mixture of conjugations by the actual native G31 event words 000009 | if and only if 000010 | 000011 | 1/10 <= p <= 1/2. 000012 | 000013 | The square-root Lueders channel p=1 is therefore not a mixture of the original 000014 | native event unitaries. The nonempty interval is a genuine positive overlap 000015 | with the old event menu, but the original sources still do not select a weight 000016 | distribution or a value of p. 000017 | 000018 | 000019 | 1. Why central S6 weights are complete 000020 | -------------------------------------- 000021 | 000022 | On the original quartic H5, scalar phases and the P64 kernel disappear under 000023 | unitary conjugation. The actual native G31 event action therefore factors 000024 | through its existing S6 quotient. The checker reconstructs all 720 H5 000025 | matrices as products of fifteen selected representatives among the original 000026 | sixty native reflections. Every quotient element retains its word in the 000027 | original reflection indices. 000028 | 000029 | Suppose a random-unitary channel is fully G31 covariant. Conjugating its 000030 | probability distribution by any group element gives the same channel. 000031 | Averaging those distributions preserves positivity and the channel while 000032 | making the distribution constant on conjugacy classes. It is therefore both 000033 | necessary and sufficient to study nonnegative total class weights w_C. 000034 | 000035 | 000036 | 2. Actual End(H5) representation and class characters 000037 | ------------------------------------------------------ 000038 | 000039 | The conjugation representation decomposes as 000040 | 000041 | End(H5) = 1 + 5 + 9 + 10. 000042 | 000043 | This is checked on the actual native 5 by 5 word matrices. Central character 000044 | projectors reconstructed from all 720 words are exact orthogonal idempotents 000045 | of ranks 1,5,9,10 and sum to I25. 000046 | 000047 | For a cycle type C let m1 and m2 be its numbers of one-cycles and two-cycles. 000048 | Writing chi=chi5=m1-1 and chi2=chi5(g^2)=m1+2m2-1 gives 000049 | 000050 | chi9 = (chi^2+chi2)/2 - chi - 1, 000051 | chi10 = (chi^2-chi2)/2. 000052 | 000053 | The eleven exact class rows are: 000054 | 000055 | cycles size chi5 chi9 chi10 000056 | (6) 120 -1 0 1 000057 | (5,1) 144 0 -1 0 000058 | (4,2) 90 -1 1 0 000059 | (4,1,1) 90 1 -1 0 000060 | (3,3) 40 -1 0 1 000061 | (3,2,1) 120 0 0 -1 000062 | (3,1,1,1) 40 2 0 1 000063 | (2,2,2) 15 -1 3 -2 000064 | (2,2,1,1) 45 1 1 -2 000065 | (2,1,1,1,1) 15 3 3 2 000066 | (1,1,1,1,1,1) 1 5 9 10 000067 | 000068 | For every class average A_C and every one of the four actual isotypic 000069 | projectors Q_d, the checker verifies directly 000070 | 000071 | A_C Q_d = chi_d(C)/d Q_d. 000072 | 000073 | Thus the subsequent character calculation is an exact compression of the 000074 | actual native operator calculation, not a replacement representation. 000075 | 000076 | 000077 | 3. Exact channel intersection 000078 | ----------------------------- 000079 | 000080 | The classified six-effect channel has eigenvalues 000081 | 000082 | 1 on the trivial line, 000083 | p-1/5 on the five, 000084 | 0 on the nine and ten. 000085 | 000086 | A central native random-unitary mixture matches it precisely when 000087 | 000088 | sum_C w_C = 1, 000089 | sum_C w_C chi9(C) = 0, 000090 | sum_C w_C chi10(C) = 0, 000091 | w_C >= 0, 000092 | 000093 | and 000094 | 000095 | lambda5 = sum_C w_C chi5(C)/5 = p-1/5. 000096 | 000097 | The three equality constraints have rank three. Exact enumeration of every 000098 | basic feasible support of at most three classes gives 29 vertices. Each 000099 | vertex was also expanded to its actual 25 by 25 native channel and compared 000100 | entry by entry with Phi_p. 000101 | 000102 | 000103 | 4. Short analytic proof of the complete interval 000104 | ------------------------------------------------- 000105 | 000106 | Let expectation refer to the class distribution and put chi=chi5. The two 000107 | zero-eigenvalue equations imply 000108 | 000109 | E[chi^2] = 1 + E[chi]. 000110 | 000111 | The character chi is integral on every S6 class. Hence the following two 000112 | polynomials are pointwise nonnegative: 000113 | 000114 | chi(chi+1) >= 0, 000115 | (chi-1)(chi-2) >= 0. 000116 | 000117 | Averaging and inserting the moment identity gives 000118 | 000119 | 1 + 2 E[chi] >= 0, 000120 | 3 - 2 E[chi] >= 0. 000121 | 000122 | Therefore 000123 | 000124 | -1/2 <= E[chi] <= 3/2. 000125 | 000126 | Since p=(1+E[chi])/5, every feasible mixture obeys 000127 | 000128 | 1/10 <= p <= 1/2. 000129 | 000130 | Both bounds are attained by actual native word distributions below. Their 000131 | convex combinations attain every intermediate p, so this necessary interval 000132 | is also sufficient. 000133 | 000134 | 000135 | 5. Explicit endpoint mixtures 000136 | ----------------------------- 000137 | 000138 | At the lower endpoint p=1/10, take total class weights 000139 | 000140 | w_(6) = 1/2, 000141 | w_(3,2,1) = 1/2. 000142 | 000143 | Both classes have size 120, so each of their 240 actual native words receives 000144 | probability 1/240. 000145 | 000146 | At the upper endpoint p=1/2, take 000147 | 000148 | w_(4,1,1) = 1/4, 000149 | w_(3,1,1,1) = 1/2, 000150 | w_(2,2,1,1) = 1/4. 000151 | 000152 | The class sizes are 90,40,45. The corresponding per-word probabilities are 000153 | 1/360, 1/80 and 1/180. 000154 | 000155 | For any p in the full interval set 000156 | 000157 | t = (10p-1)/4 000158 | 000159 | and mix the lower and upper endpoint distributions with weights 1-t and t. 000160 | This gives an explicit native event law for every mathematically reachable 000161 | Phi_p. The JSON certificate records every contributing S6 permutation and 000162 | one word in original native reflection indices for all 415 endpoint members. 000163 | 000164 | 000165 | 6. Complete vertex census 000166 | ------------------------- 000167 | 000168 | The notation a[C] means total weight a on cycle class C. The 29 exact basic 000169 | feasible vertices are: 000170 | 000171 | 1. p=1/10: 1/2[3,3] + 1/2[3,2,1] 000172 | 2. p=1/10: 1/2[5,1] + 1/2[4,2] 000173 | 3. p=1/10: 1/2[6] + 1/2[3,2,1] 000174 | 4. p=1/10: 1/2[5,1] + 1/3[3,3] + 1/6[2,2,2] 000175 | 5. p=1/10: 1/3[6] + 1/2[5,1] + 1/6[2,2,2] 000176 | 6. p=3/20: 1/4[5,1] + 1/2[3,3] + 1/4[2,2,1,1] 000177 | 7. p=3/20: 1/2[6] + 1/4[5,1] + 1/4[2,2,1,1] 000178 | 8. p=1/5: 1/2[4,2] + 1/2[4,1,1] 000179 | 9. p=1/5: 1/4[4,1,1] + 1/2[3,3] + 1/4[2,2,1,1] 000180 | 10. p=1/5: 1/2[4,1,1] + 1/3[3,3] + 1/6[2,2,2] 000181 | 11. p=1/5: 4/5[5,1] + 1/6[2,2,2] + 1/30[1,1,1,1,1,1] 000182 | 12. p=1/5: 1/2[6] + 1/4[4,1,1] + 1/4[2,2,1,1] 000183 | 13. p=1/5: 1/3[6] + 1/2[4,1,1] + 1/6[2,2,2] 000184 | 14. p=1/4: 3/4[5,1] + 1/8[2,2,2] + 1/8[2,1,1,1,1] 000185 | 15. p=1/4: 9/20[5,1] + 1/2[3,2,1] + 1/20[1,1,1,1,1,1] 000186 | 16. p=3/10: 7/10[5,1] + 1/4[2,2,1,1] + 1/20[1,1,1,1,1,1] 000187 | 17. p=3/10: 1/2[5,1] + 1/3[3,1,1,1] + 1/6[2,2,2] 000188 | 18. p=3/10: 1/2[5,1] + 1/3[3,2,1] + 1/6[2,1,1,1,1] 000189 | 19. p=1/3: 2/3[5,1] + 1/6[2,2,1,1] + 1/6[2,1,1,1,1] 000190 | 20. p=17/50: 9/20[4,1,1] + 1/2[3,2,1] + 1/20[1,1,1,1,1,1] 000191 | 21. p=9/25: 4/5[4,1,1] + 1/6[2,2,2] + 1/30[1,1,1,1,1,1] 000192 | 22. p=2/5: 1/2[3,2,1] + 1/2[3,1,1,1] 000193 | 23. p=2/5: 3/4[4,1,1] + 1/8[2,2,2] + 1/8[2,1,1,1,1] 000194 | 24. p=2/5: 1/2[4,1,1] + 1/3[3,1,1,1] + 1/6[2,2,2] 000195 | 25. p=2/5: 1/2[4,1,1] + 1/3[3,2,1] + 1/6[2,1,1,1,1] 000196 | 26. p=11/25: 7/10[4,1,1] + 1/4[2,2,1,1] + 1/20[1,1,1,1,1,1] 000197 | 27. p=9/20: 1/4[5,1] + 1/2[3,1,1,1] + 1/4[2,2,1,1] 000198 | 28. p=7/15: 2/3[4,1,1] + 1/6[2,2,1,1] + 1/6[2,1,1,1,1] 000199 | 29. p=1/2: 1/4[4,1,1] + 1/2[3,1,1,1] + 1/4[2,2,1,1] 000200 |