Embedding the specified finite dynamics in continuous time

DYN.MARKOV.EMBED.01

Original ledger classification: Contract [E] (finite kernel embedding, sympy-exact; v971) + [O] (field level)

Original claim text

THE MARKOV EMBEDDING CONTRACT (registered 2026-08-27; 'three meanings of dynamics' split, step (a)->(b)). FINITE KERNEL HALF [E] (v971, sympy-exact): for the v221 seam transfer kernel T (spectrum exactly {1, 64/729, 1/729}) the principal logarithm Q = log T is an EXACT classical Markov generator -- e^Q = T by exact spectral calculus; row sums symbolically zero; off-diagonals exactly {ln(9/8), 2 ln 3, 2 ln 3} = {0.117783, 2.197225, 2.197225} > 0 (the reviewer's independently verified numbers), diagonals -ln(81/8), -ln(81/8), -4 ln 3; uniformization certificate Q + (4 ln 3) I >= 0 entrywise (diagonals ln 8, ln 8, 0, exact) => e^{tQ} positivity-preserving doubly stochastic for ALL t >= 0, T embeddable, every root T^{1/n} stochastic; spec(Q) = -spec(H_mod) = {0, -6 ln(3/2), -6 ln 3} (gap = Delta = 6 ln(3/2)); detailed balance w.r.t. the uniform state. NEGATIVE CONTROL: second eigenvalue -1/10 gives det = -1/200 < 0 -- provably NOT embeddable (det e^Q = e^{tr Q} > 0 always); embeddability is not generic, the RP/OS spectral positivity (CONTRACT.F.01 axiom 3) is the enabling fact. WHAT STAYS [O]: the same embedding statement at FIELD level (reconstructed net / quasi-local algebra, uniform in system size). FIREWALL: e^{tQ} is irreversible statistical dynamics (H-theorem), NOT unitary evolution -- step (b)->(c) is the separate contract DYN.UNITARY.DILATION.01; the shared F_transfer relaxation shape is a universal CONTRACTION CLASS, not one physical clock (v723/v724/v777). Python-only per GATE.WOLFRAM.02 (matrix-log/spectral identities; the exact rationals 64/729, 1/729 already mirrored).

Original classification detail

[E] the finite kernel half: Q = log T is an exact Markov generator (row sums 0, off-diagonals ln(9/8), 2ln3, 2ln3 > 0, uniformization positivity for all t >= 0); [O] the field-level embedding

Dependencies as recorded

v221 (kernel); v957 (T = e^{-H_mod}); CONTRACT.F.01 (axiom 3 positivity); v723/v724/v777 (external clocks)