"""v971 -- DYN.MARKOV.EMBED.01: the seam transfer kernel embeds into a continuous positivity-preserving Markov semigroup (Q = log T exact). THE POINT (external-review round 2026-08-27, "three meanings of dynamics"). Three different things have been called "dynamics" in the corpus: (a) the positive relaxation T^n to the unique attractor; (b) a continuous dissipative semigroup e^{t log T}; (c) reversible Lorentz/unitary time evolution e^{-itH}. This module proves the step (a)->(b) EXACTLY for the concrete 3-state seam transfer kernel (v221: T = J + (2/3)^6 u2u2^T + (1/3)^6 u3u3^T, spectrum {1, (2/3)^6, (1/3)^6}): the principal matrix logarithm Q = log T is a genuine classical Markov generator (Q-matrix), so T = e^Q sits inside the continuous positivity-preserving semigroup {e^{tQ} : t >= 0} -- embeddability is a THEOREM for this kernel, not an analogy. Everything is sympy-exact: [E] 1. exact kernel: T is symmetric, doubly stochastic, entrywise > 0, with exact spectrum {1, 64/729, 1/729} (identical to v221's kernel). [E] 2. the principal log is spectral: Q = ln(64/729) P2 + ln(1/729) P3 with orthogonal projectors; exp(Q) = T exactly (spectral calculus). [E] 3. Q-MATRIX, ROW SUMS: every row of Q sums to 0 exactly (symbolic zero). [E] 4. Q-MATRIX, OFF-DIAGONALS: the three independent off-diagonal entries are EXACTLY {ln(9/8), 2 ln 3, 2 ln 3} = {0.117783, 2.197225, 2.197225}, all > 0 symbolically (9/8 > 1, 9 > 1); diagonals -ln(81/8), -ln(81/8), -4 ln 3. [E] 5. POSITIVITY FOR ALL t >= 0 (uniformization certificate): with q = 4 ln 3 = max|Q_ii|, the matrix Q + qI is entrywise >= 0 exactly (diagonal entries ln 8, ln 8, 0), hence e^{tQ} = e^{-qt} e^{t(Q+qI)} is entrywise >= 0 for every t >= 0; Q1 = 0 and symmetry make each e^{tQ} doubly stochastic. So T = e^{1*Q} is embeddable, and every real root T^{1/n} = e^{Q/n} is again a stochastic matrix. [E] 6. spec(Q) = {0, -6 ln(3/2), -6 ln 3} = -spec(H_mod) (v957): the generator gap is the seam gap Delta = 6 ln(3/2), and the semigroup contracts deviations at exactly (2/3)^{6t} (v221 at integer t). [E] 7. NEGATIVE CONTROL (embeddability is not generic): the same construction with second eigenvalue -1/10 gives a doubly stochastic kernel with det = -1/200 < 0, while every embeddable stochastic matrix has det e^Q = e^{tr Q} > 0 -- that kernel provably admits NO continuous positivity-preserving embedding. Positivity of the seam spectrum (RP/OS, CONTRACT.F.01 axiom 3) is what makes (a)->(b) real. HONEST SCOPE (the firewall this module exists to enforce). This closes the FINITE, KERNEL-LEVEL half of DYN.MARKOV.EMBED.01 and nothing else: e^{tQ} is IRREVERSIBLE statistical dynamics (detailed balance w.r.t. the uniform state; an H-theorem, not a Schroedinger equation). Step (b)->(c) -- a local, size-consistent Stinespring/Hamiltonian dilation with a Lieb-Robinson bound and an OS continuation T = e^{-aH} -> U(t) = e^{-itH} -- is the SEPARATE open contract DYN.UNITARY.DILATION.01 and is NOT advanced here. The shared relaxation shape of the four F_transfer sectors is a universal CONTRACTION CLASS, not one physical clock (v723/v724/v777: no common continuous clock exists). Status: [E] finite kernel embedding; the field-level statement and the unitary dilation stay [O]. """ import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset LAM2 = sp.Rational(64, 729) # (2/3)^6, the seam gap factor LAM3 = sp.Rational(1, 729) # (1/3)^6 def exact_kernel_and_projectors(): """The v221 kernel and its spectral projectors, all exact rationals.""" one = sp.ones(3, 1) u2 = sp.Matrix([1, -1, 0]) u3 = sp.Matrix([1, 1, -2]) # the Nariai traceless anchor J = one * one.T / 3 # projector onto the uniform state P2 = u2 * u2.T / (u2.T * u2)[0] P3 = u3 * u3.T / (u3.T * u3)[0] T = J + LAM2 * P2 + LAM3 * P3 return T, J, P2, P3 def run(): reset() print("v971 DYN.MARKOV.EMBED.01: Q = log T is an exact Markov generator " "(the seam kernel embeds into a continuous stochastic semigroup)") T, J, P2, P3 = exact_kernel_and_projectors() # 1. exact kernel identical to v221 dstoch = all(sum(T[i, :]) == 1 for i in range(3)) \ and all(sum(T[:, j]) == 1 for j in range(3)) \ and all(e > 0 for e in T) check("T symmetric, doubly stochastic, entrywise > 0 (exact rationals; " "the v221 kernel verbatim)", T == T.T and dstoch) spec_T = sorted(T.eigenvals().keys(), reverse=True) check("exact spectrum of T = {1, 64/729, 1/729} = {1, (2/3)^6, (1/3)^6}", spec_T == [1, LAM2, LAM3], exact=True) # 2. the principal spectral logarithm, and exp(Q) = T exactly Q = sp.log(LAM2) * P2 + sp.log(LAM3) * P3 T_back = J + sp.exp(sp.log(LAM2)) * P2 + sp.exp(sp.log(LAM3)) * P3 proj_ok = (P2 * P2 == P2 and P3 * P3 == P3 and P2 * P3 == sp.zeros(3) and J + P2 + P3 == sp.eye(3)) check("spectral calculus exact: {J, P2, P3} orthogonal resolution of the " "identity and exp(Q) = T (principal log; trivial eigenvalue -> 0)", proj_ok and sp.simplify(T_back - T) == sp.zeros(3)) # numeric cross-check of the principal log at 50 digits, built from the # exact closed-form entries (checked symbolically below): # Q = [[-ln(81/8), ln(9/8), 2ln3], [ln(9/8), -ln(81/8), 2ln3], # [2ln3, 2ln3, -4ln3]] mp.mp.dps = 50 Tn = mp.matrix(3, 3) for i in range(3): for j in range(3): e = T[i, j] Tn[i, j] = mp.mpf(sp.Rational(e).p) / mp.mpf(sp.Rational(e).q) l98 = mp.log(mp.mpf(9) / 8) l3 = mp.log(mp.mpf(3)) l818 = mp.log(mp.mpf(81) / 8) Qn = mp.matrix([[-l818, l98, 2 * l3], [l98, -l818, 2 * l3], [2 * l3, 2 * l3, -4 * l3]]) dev = max(abs(mp.expm(Qn)[i, j] - Tn[i, j]) for i in range(3) for j in range(3)) check("mpmath expm(Q) reproduces T entrywise at 50 digits (dev < 1e-45)", dev < mp.mpf('1e-45')) # 3. row sums vanish exactly rows0 = all(sp.simplify(sum(Q[i, :])) == 0 for i in range(3)) check("Q-MATRIX row sums: sum_j Q_ij = 0 EXACTLY for every row " "(symbolic zero, no tolerance)", rows0) # 4. off-diagonals exactly {ln(9/8), 2 ln 3, 2 ln 3} > 0 offdiag_ok = (sp.simplify(Q[0, 1] - sp.log(sp.Rational(9, 8))) == 0 and sp.simplify(Q[0, 2] - 2 * sp.log(3)) == 0 and sp.simplify(Q[1, 2] - 2 * sp.log(3)) == 0 and Q == Q.T) check("Q-MATRIX off-diagonals: Q12 = ln(9/8), Q13 = Q23 = 2 ln 3, " "symmetric -- all > 0 symbolically (9/8 > 1, 9 > 1)", offdiag_ok and sp.log(sp.Rational(9, 8)) > 0 and 2 * sp.log(3) > 0) check("Q12 = ln(9/8) = 0.117783 (the reviewer's verified number)", float(sp.N(Q[0, 1], 20)), 0.117783035656383589, tol=mp.mpf('1e-12')) check("Q13 = Q23 = 2 ln 3 = 2.197225 (the reviewer's verified numbers)", float(sp.N(Q[0, 2], 20)), 2.197224577336219383, tol=mp.mpf('1e-12')) diag_ok = (sp.simplify(Q[0, 0] + sp.log(sp.Rational(81, 8))) == 0 and sp.simplify(Q[1, 1] + sp.log(sp.Rational(81, 8))) == 0 and sp.simplify(Q[2, 2] + 4 * sp.log(3)) == 0) check("diagonals exactly -ln(81/8), -ln(81/8), -4 ln 3", diag_ok) # 5. uniformization: Q + qI >= 0 entrywise with q = 4 ln 3 exactly q = 4 * sp.log(3) Qplus = Q + q * sp.eye(3) unif_ok = (sp.simplify(Qplus[0, 0] - sp.log(8)) == 0 and sp.simplify(Qplus[1, 1] - sp.log(8)) == 0 and sp.simplify(Qplus[2, 2]) == 0 and all(sp.simplify(Qplus[i, j]) >= 0 for i in range(3) for j in range(3))) check("UNIFORMIZATION [E]: q = 4 ln 3 = max|Q_ii| gives Q + qI >= 0 " "entrywise EXACTLY (diagonals ln 8, ln 8, 0) => e^{tQ} = " "e^{-qt} e^{t(Q+qI)} is entrywise >= 0 for ALL t >= 0", unif_ok) # spot-check the semigroup numerically on a t-grid incl. fractional roots grid_ok = True for tval in ('0.1', '0.25', '0.5', '1.5', '3.0', '7.0'): Et = mp.expm(Qn * mp.mpf(tval)) grid_ok = grid_ok and all(Et[i, j] > -mp.mpf('1e-30') for i in range(3) for j in range(3)) \ and all(abs(sum(Et[i, j] for j in range(3)) - 1) < mp.mpf('1e-30') for i in range(3)) check("semigroup grid check: e^{tQ} entrywise >= 0 and row-stochastic at " "t in {0.1, 0.25, 0.5, 1.5, 3, 7} (50 digits); T^{1/n} = e^{Q/n} " "is again stochastic (embeddability/Elfving, affirmative)", grid_ok) # 6. generator spectrum = -spec(H_mod), gap = Delta spec_Q = sorted(Q.eigenvals().keys(), reverse=True, key=lambda e: float(sp.N(e, 30))) target = [sp.Integer(0), sp.log(LAM2), sp.log(LAM3)] spec_ok = all(sp.simplify(a - b) == 0 for a, b in zip(spec_Q, target)) check("spec(Q) = {0, -6 ln(3/2), -6 ln 3} = -spec(H_mod) (v957: " "T = e^{-H_mod}); generator gap = Delta = 6 ln(3/2) exactly", spec_ok and sp.simplify(-sp.log(LAM2) - 6 * sp.log(sp.Rational(3, 2))) == 0) check("detailed balance: Q symmetric w.r.t. the uniform stationary state " "(the KMS/OS-positive free level; irreversible H-theorem dynamics, " "NOT a Schroedinger equation)", Q == Q.T) # 7. negative control: spectrum {1, -1/10, 1/20} is NOT embeddable Tneg = J - sp.Rational(1, 10) * P2 + sp.Rational(1, 20) * P3 neg_dstoch = all(sum(Tneg[i, :]) == 1 for i in range(3)) \ and all(e > 0 for e in Tneg) detneg = Tneg.det() check("NEG control [E]: the same construction with eigenvalue -1/10 is " "still a doubly stochastic kernel but det = -1/200 < 0, while " "det e^Q = e^{tr Q} > 0 always -- provably NOT embeddable; " "spectral positivity (RP/OS) is what makes (a)->(b) real", neg_dstoch and detneg == sp.Rational(-1, 200) and detneg < 0) check("FIREWALL (scope): this module closes the FINITE KERNEL half of " "DYN.MARKOV.EMBED.01 only -- e^{tQ} is dissipative statistical " "dynamics; the unitary/local dilation (b)->(c) is the separate " "open contract DYN.UNITARY.DILATION.01, untouched here", True) return summary("v971 Markov embedding generator (Q = log T exact)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)