TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/dynamic_reconstruction_20261027/PETERSEN-POLAR-DECK-INTERTWINER.txt Original file SHA-256: 0d2f283a3c6568bd77cc24f081c61dc706a7faef209138335336a1fbbd8007cd 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 | PETERSEN POLAR DECK INTERTWINER 000002 | ================================ 000003 | 000004 | 27 October 2026 000005 | 000006 | Question 000007 | -------- 000008 | 000009 | Does the signed Petersen radial transfer from v808 admit a natural marked 000010 | intertwiner with the actual lazy family operator from v486/v487 when the shell 000011 | charges and the existing deck involution are carried along? 000012 | 000013 | Answer 000014 | ------ 000015 | 000016 | Yes for the positive transfer, and no for the literal v487 deck marking. 000017 | 000018 | The reversible Petersen operator has a canonical polar decomposition 000019 | 000020 | P = J_P |P|. 000021 | 000022 | There is an explicit isometry W with 000023 | 000024 | W |P| W^{-1} = M_lazy. 000025 | 000026 | It maps the stationary shell mark to the family vacuum, the hypercharge mode 000027 | to the 2/3 even family mode, and the third v808 mode to the 1/3 odd family 000028 | mode. But under this same forced spectral mapping, 000029 | 000030 | W J_P W^{-1} = K = Z Delta, 000031 | 000032 | where Delta is the v487 deck swap and 000033 | 000034 | Z = diag(1,-1,-1) 000035 | 000036 | is the canonical reflection across the Perron line. Consequently no static 000037 | or time dependent isometric intertwiner can carry both the positive transfer 000038 | and the literal deck marking Delta. The obstruction already occurs on the 000039 | one dimensional 2/3 eigenspace. 000040 | 000041 | There are two exact surviving constructions: 000042 | 000043 | 1. a signed static correspondence 000044 | 000045 | W P W^{-1} = K M_lazy; 000046 | 000047 | 2. a period two correspondence to the positive lazy walk, 000048 | 000049 | W_n = K^n W, 000050 | W_{n+1} P W_n^{-1} = M_lazy. 000051 | 000052 | The period two frame transports the hypercharge sign rather than discarding 000053 | it. Equivalently, the canonical bipartite double cover turns P into the 000054 | positive lazy walk by a fixed six dimensional isometry. Its deck exchange is 000055 | transported to a K twisted exchange, not to the plain v487 deck. 000056 | 000057 | This is stronger than the old equality after six powers. The two transfers 000058 | are exactly equivalent at every even time. Their odd steps differ by K. 000059 | The transported shell multiplication operator is not conserved: its exact 000060 | one step defect has rank two. Also, Z has an older source candidate: v409 000061 | sets the same matrix as the seam sheet involution. What remains absent is a 000062 | source intertwiner proving that this involution and the v487 deck use the same 000063 | marked basis. 000064 | 000065 | 1. Atlas and source boundary 000066 | ---------------------------- 000067 | 000068 | The active Atlas at revision 4d210a8bebd5c91c7ee7ad75 records 000069 | `CARRIER.PETERSEN.RADIAL.01` as exact only for the finite Petersen geometry and 000070 | the sixth power spectrum. It explicitly says that the corpus freezes the 000071 | spectrum rather than a common basis and contains no deployed ten dimensional 000072 | operator or shells to cusp intertwiner. No reviewed finding already solves 000073 | the signed one step problem. 000074 | 000075 | The concrete sources used here are the existing finite machinery: 000076 | 000077 | * `verification/v808_petersen_sixthroot.py`, lines 478 to 583, for the 000078 | charge pure shells, the quotient and its three marked eigenvectors; 000079 | * `verification/v487_transfer_clock_rungs.py`, lines 73 to 119, for the 000080 | actual lazy operator and the even/odd deck assignment; 000081 | * `verification/v409_coxeter_prime2_lemma.py`, lines 10 to 40 and 61 to 76, 000082 | for the named sheet involution `diag(1,-1,-1)` and its prime two parity 000083 | scope; 000084 | * `verification/v97_sheet_conjugation_bridge.py`, lines 48 to 53 and 61 to 000085 | 110, for the distinct generation coordinate basis, cusp eigenvectors and 000086 | the still proposed lattice grading dictionary. 000087 | 000088 | No field quantization or new state is introduced. 000089 | 000090 | 2. Reversible radial Hilbert space 000091 | ---------------------------------- 000092 | 000093 | The shell sizes are 1, 3 and 6. The radial Markov operator is 000094 | 000095 | [ 0 1 0 ] 000096 | P = [1/3 0 2/3 ], 000097 | [ 0 1/3 2/3 ] 000098 | 000099 | with stationary measure 000100 | 000101 | pi = (1,3,6)/10. 000102 | 000103 | Detailed balance holds entry by entry: 000104 | 000105 | pi_i P_ij = pi_j P_ji. 000106 | 000107 | Thus P is selfadjoint on H_pi = R^3 with scalar product 000108 | 000109 | _pi = sum_i pi_i f_i g_i. 000110 | 000111 | The three source marked eigenfunctions from v808 are 000112 | 000113 | f0 = (1,1,1), eigenvalue 1, 000114 | q = (6,-4,1), eigenvalue -2/3, 000115 | r = (3,1,-1), eigenvalue 1/3. 000116 | 000117 | The second vector is literally the shell hypercharge observable evaluated on 000118 | the charge pure classes e^c, u^c and Q. Their squared H_pi norms are 1, 9 and 000119 | 9/5 respectively. Hence 000120 | 000121 | b0 = f0, 000122 | bY = q/3, 000123 | br = sqrt(5) r/3 000124 | 000125 | is an orthonormal marked basis. 000126 | 000127 | Conjugating by D^(1/2), D = diag(pi), gives the ordinary symmetric matrix 000128 | 000129 | [ 0 1/sqrt(3) 0 ] 000130 | P_sym = [1/sqrt(3) 0 sqrt(2)/3 ], 000131 | [ 0 sqrt(2)/3 2/3 ] 000132 | 000133 | with the same marked spectrum. 000134 | 000135 | 3. Canonical polar decomposition 000136 | -------------------------------- 000137 | 000138 | Since P is invertible and selfadjoint on H_pi, its polar factors are unique. 000139 | Interpolation on the three simple eigenvalues gives coordinate free formulas 000140 | 000141 | |P| = (4/5)P^2 - (1/15)P + (4/15)I, 000142 | J_P = -(6/5)P^2 + (8/5)P + (3/5)I. 000143 | 000144 | In the original shell coordinates these are 000145 | 000146 | [ 8/15 -1/15 8/15] 000147 | |P| = [-1/45 32/45 14/45], 000148 | [ 4/45 7/45 34/45] 000149 | 000150 | and 000151 | 000152 | [ 1/5 8/5 -4/5] 000153 | J_P = [ 8/15 -1/15 8/15]. 000154 | [-2/15 4/15 13/15] 000155 | 000156 | Direct multiplication gives 000157 | 000158 | J_P^2 = I, J_P |P| = |P| J_P, P = J_P |P|. 000159 | 000160 | Both factors are selfadjoint for the pi weighted scalar product. Their action 000161 | on the marked basis is 000162 | 000163 | b0 bY br 000164 | |P| 1 2/3 1/3, 000165 | J_P 1 -1 1. 000166 | 000167 | The negative sign is therefore attached canonically to the hypercharge mode. 000168 | 000169 | 4. Concrete intertwiner to M_lazy 000170 | --------------------------------- 000171 | 000172 | Write the target family basis as 000173 | 000174 | e0 = (1,0,0), 000175 | e+ = (0,1,1)/sqrt(2), 000176 | e- = (0,1,-1)/sqrt(2). 000177 | 000178 | Then 000179 | 000180 | M_lazy e0 = e0, 000181 | M_lazy e+ = (2/3)e+, 000182 | M_lazy e- = (1/3)e-. 000183 | 000184 | Define the isometry W by 000185 | 000186 | W b0 = e0, W bY = e+, W br = e-. 000187 | 000188 | This definition transports all three v808 markings and fixes W up to the 000189 | irrelevant simultaneous sign conventions of the one dimensional marked 000190 | lines. It proves immediately that 000191 | 000192 | W |P| W^{-1} = M_lazy. 000193 | 000194 | For a literal matrix, first send H_pi to the standard Euclidean shell frame by 000195 | D^(1/2). In that frame the orthogonal matrix U of W is 000196 | 000197 | [ 1/sqrt(10), sqrt(30)/10, sqrt(15)/5 ] 000198 | [ 1/2+sqrt(5)/5, -2sqrt(15)/15+sqrt(3)/6, -sqrt(6)/6+sqrt(30)/30 ] 000199 | [-1/2+sqrt(5)/5, -2sqrt(15)/15-sqrt(3)/6, sqrt(30)/30+sqrt(6)/6 ]. 000200 |