TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/dynamic_reconstruction_20261027/MCKAY-RECURSION-READING.txt Original file SHA-256: e48ae0a0b8d1d13e91baa28b948e47ee454e175fa5fca810deb55e1b69ba48ab Inclusive original lines: 1-51 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 | MCKAY RECURSION: A SOURCE BASED READING OF THE EXISTING NETWORK STEP 000002 | 27 October 2026 000003 | 000004 | Scope. This is a synthesis of the already proved v219/v299 McKay data and 000005 | the actual v395 network update. It adds no P1 selection theorem and does 000006 | not identify a representation index with a physical particle or spacetime. 000007 | 000008 | The original v219 constructs the binary icosahedral group from 120 unit 000009 | quaternions and obtains the affine E8 adjacency A from tensoring with its 000010 | natural two dimensional representation V. The irreducible dimensions are 000011 | the existing marks d=(1,2,3,4,5,6,4,2,3), in the v395 ordering. 000012 | They satisfy A d=2d and sum d_i^2=120. 000013 | 000014 | The actual recursive operator is T=(A+2I)/4. With D=diag(d), its canonical 000015 | positive eigenvector transform is 000016 | 000017 | P=D^-1 T D, 000018 | P_ij=(A_ij+2 delta_ij)d_j/(4d_i). 000019 | 000020 | This is entrywise nonnegative and each row sums to one. Its stationary law 000021 | is pi_i=d_i^2/120, and pi_i P_ij=pi_j P_ji follows from A_ij=A_ji. 000022 | Thus the mark vector and the Plancherel equilibrium are two normalizations 000023 | of the same existing recursion, not two independently chosen states. 000024 | 000025 | There is also an exact representation meaning. Tensor an irreducible i 000026 | with R=V direct-sum 1 direct-sum 1, and select summand j with probability 000027 | proportional to its multiplicity times d_j. R has dimension four and 000028 | multiplicity matrix A+2I, so the resulting tensor-product Markov chain is 000029 | precisely P. Its lazy weight was already fixed by v395; the representation 000030 | description does not derive that choice anew from P1. 000031 | 000032 | Primary literature check: Benkart, Diaconis, Liebeck and Tiep, Tensor Product 000033 | Markov Chains, arXiv:1810.00409, equation (1.1) and Theorem 1.1, give the 000034 | general multiplicity-times-dimension transition rule and its Plancherel 000035 | stationary distribution. The specialization above is a direct derivation 000036 | for the TFPT operator, using the existing v219/v299 source data. 000037 | https://arxiv.org/abs/1810.00409 000038 | 000039 | The exact rational calculation is saved as mckay-doob-exact.json. All rows, 000040 | positivity, detailed balance and normalization were checked without numeric 000041 | tolerances. Analytically those checks follow from the two displayed formulas. 000042 | 000043 | Meaning for the requested route. The hypergraph fixed point is already tied 000044 | to tensor fusion of representations and to the binary double cover. It need 000045 | not be interpreted as an arbitrary classical crystal. However the Markov 000046 | chain describes representation populations. It does not specify coherent 000047 | intertwiners, their phases, a physical field state, an electromagnetic 000048 | interaction or the physical clock. Promoting it to those objects would 000049 | repeat the source identification problem. This reading should be reused as 000050 | a finite structural connection, not as a new independent derivation of E8 000051 | or a complete quantum dynamics.