"""Reproduce the exact proof schematic and its stated finite checks.""" from pathlib import Path SVG='\n Actual singular hypertrace, Jones ideal and bounded entropy cost\n The bounded left cuts use only bounded approximating operators and control trace tails. They show a singular compatible hypertrace kills all coefficients of the actual Jones corner ideal. Physical M remains faithful in the quotient. The exact bounded functional balance gives a nonnegative logarithmic entropy production equivalent to canonical cost annihilation, with quantified quotient-order and commutator certificates. Their general amenability input remains unproved.\n \n The singular branch has an actual operator boundary\n e = e_R^M; A = <N,e> inside B = <M,e>; psi is compatible, M-central and centrally singular.\n \n Bounded cuts control a singular-state limit. (SC.3)-(SC.7)\n r_m a e = h_m e, h_m in N bounded; tau(1-r_m) <= 4^(-m)/12.\n psi(e) = 0 implies psi(a e a*) <= ||a||^2 tau(1-r_m), hence zero.\n Only the physical trace takes a monotone limit; no affiliated-vector assumption or state normality.\n \n \n \n The actual finite basis extends this to all B coefficients. (SC.5)-(SC.8)\n b_0 = sum a_i c_i, c_i in A; psi(b_0 e b_0*) = 0.\n J_e = norm-closed span B e B; psi(J_e) = 0. Physical M stays faithful in B/J_e.\n e t_hat = t e does not identify the two quotient images once e is killed.\n \n One exact bounded logarithmic test controls the canonical cost. (SC.9)-(SC.11)\n theta = ell/w in [1/H,H]; H = (q/p)^2; j = (theta-1) log(theta) >= (theta-1)^2/H.\n b_log = Q(j); nu(b_log) = gamma(log theta) - alpha(log theta) >= 0.\n b <= d R_* sqrt(H b_log); b_log <= [log(H)/(d r)] b. Zero-cost tests are equivalent.\n r = p/q; R_* = q/p; p = (1-sqrt(1-4/d))/2; all logarithms are bounded.\n \n \n \n \n \n Positive quotient-order certificate\n a <= y + k + epsilon 1, y in M_+, k in J_e+.\n psi(a) <= tau(y) + epsilon. (SC.12)-(SC.13)\n Use a = b_hat or the entropy lift b_log_hat.\n \n Norm defect certificate\n c_log = lift(P(log theta) - log theta).\n nu(b_log) <= dist(c_log,L). (SC.14)-(SC.15)\n L uses actual commutators, E_A deficits and J_e.\n The new certificates are not yet supplied by general relative amenability.\n Exact original residual: a suitable state must annihilate b = d^2 sqrt(1-4/d) Q(|E_D0(f-E_C(f))|).\n Schematic proof maps; no trace areas encoded. Full common-support BF, partition and generation remain assigned.\n' Path(__file__).with_suffix(".svg").write_bytes(SVG.encode("utf-8"))