"""Reproduce the exact compatible-state transfer and ergodic-cost schematic.""" from pathlib import Path SVG='\n Automatic basis control and the exact remaining ergodic cost\n The actual canonical transfer gives T equals E_A I divided by d on the smaller center, physically Q P(w times t). Cesaro weights automatically have vanishing total basis energy and preserve compatibility and the physical trace. Their state functionals return in norm to the actual B. The normal trace in the state GNS completion is used only for the averaging proof, without replacing the original P. The minimum cost among boundedly dominated compatible states equals the lower spectral endpoint of the conditional ergodic cost bar a. General amenability has not been shown to make that endpoint zero for one suitable state.\n \n Automatic commutator control; the original cost remains\n a = b_hat; b = d^2 sqrt(1-4/d) Q(|E_D0(f-E_C(f))|). Full joint centers retained. (LC.1)\n \n Use the actual basis transfer and the original larger expectation. (LC.3)-(LC.4)\n I(h) = sum a_i h a_i*; T = E_A I/d on Z(A); physically T(t) = Q(P(w t)).\n sum psi([h,a_i]*[h,a_i]) = 2d psi(h(h-T(h))); psi T = psi on Z(A).\n The transfer is basis-independent. No new P_alpha or original P(ell)=1 is inserted.\n \n \n \n Every positive smaller-center seed can be averaged. (LC.5)-(LC.6)\n h_n = (1/n) sum_(k=0)^(n-1) T^k(h); c = psi(h) > 0; A2 = sum ||a_i||^2 <= d ceil(d).\n sum W_(n,i) <= 4d psi(h^2)/n; uniform conjugation error <= 2 sqrt(d A2 psi(h^2)/n)/c.\n Compatibility and restriction to M equal tau are exact for every n.\n W_(n,i) = psi([h_n,a_i]*[h_n,a_i]); the averaged positive weights need not be projections.\n \n The state completion supplies a proved return map to actual B. (LC.7)-(LC.10)\n In state L2: h_n --> Pi(h) in the Markov fixed algebra F; bar a = Pi(a).\n ||psi(z dot)||_(B*) <= ||z||_(2,psi): normalized weights converge in actual B* norm.\n Limit state psi_H is compatible, M-central, has physical trace tau and retains J_e annihilation.\n Exact limiting cost: psi_(Pi h)(a) = alpha(h bar a)/c; the original a is never replaced.\n \n One-state spectral target, with its exact limitation. (LC.11)-(LC.12)\n inf {sigma(a): sigma compatible and sigma <= C psi for some finite C} = min sp_F(bar a).\n For one psi, a zero lower endpoint gives one actual zero-cost state. Kernel mass gives it directly.\n At a true cost minimizer psi_0: bar a_(psi_0) = psi_0(a) 1. Its scalar need not be proved zero yet.\n A positive endpoint for one psi excludes only its bounded domination class, not every suitable state.\n General amenability has not yet forced min sp_F(bar a) = 0 for one available state.\n Full BF, exact full partition, common-stage alignment and generation remain assigned.\n Schematic proof maps; no encoded trace areas. Complete constructions and all constants: LC.1-LC.12.\n' Path(__file__).with_suffix(".svg").write_bytes(SVG.encode("utf-8"))