"""v240 -- GNS / OS reconstruction (finite toy): from the discrete seam data (a *-algebra M + the state omega) one reconstructs a Hilbert space, a cyclic vector, the modular flow (v238/v239), AND a POSITIVE Hamiltonian. This makes explicit that two of the QFT primitives -- the Hilbert space and a positive Hamiltonian -- are OUTPUTS of the seam data, not extra inputs, and pins down the relation to the recovery generator of v238. Key finite-dimensional facts: * GNS: for omega(X)=Tr(rho X) with rho>0, _omega = Tr(rho A^dag B) is a genuine inner product; the identity 1 is a cyclic AND separating vector (rho>0); dim H_GNS = n^2. * Modular flow: the modular operator Delta acts by X |-> rho X rho^{-1}; the modular automorphism sigma_t(x) = rho^{it} x rho^{-it} = Ad(Delta^{it}) is the SAME flow as v238/v239 (generator i[log rho, .]), state-preserving. * OS / positive Hamiltonian: the reflection-positive transfer is the recovery channel T (v221, spec {1,(2/3)^6,(1/3)^6}); the OS Hamiltonian H_OS = -log T is POSITIVE (eigenvalues >= 0), with ground state = the stationary recovery mode and gap = 6 log(3/2) = the v64 mass gap. Crucially H_OS = -L, MINUS the v238 dissipative generator: OS reconstruction = the recovery semigroup read as Euclidean time. [E] 1. GNS HILBERT SPACE. = Tr(rho A^dag B) is Hermitian and positive-definite (rho>0), dim n^2; 1 is cyclic (pi(A)1 = A spans) and separating. So the Hilbert space is an OUTPUT of (M, omega). [E] 2. MODULAR FLOW = v238/v239. sigma_t(x) = rho^{it} x rho^{-it} is a *-automorphism (sigma_t(xy)=sigma_t(x)sigma_t(y), sigma_t(x^dag)= sigma_t(x)^dag), state-preserving (omega o sigma_t = omega), with generator d/dt|_0 = i[log rho, .] = the modular Hamiltonian Lambda_Sigma (v238). The modular operator Delta (X|->rho X rho^{-1}) is positive. [E] 3. OS HAMILTONIAN IS POSITIVE. H_OS = -log T (T the v221 recovery transfer) has eigenvalues {0, 6 log(3/2), 6 log 3} >= 0 -- a POSITIVE Hamiltonian with ground state the stationary mode and gap 6 log(3/2) = the v64 mass gap. So OS reconstruction delivers v64's "H >= 0 + mass gap". [E] 4. H_OS = -L (one object with v238). The OS Hamiltonian equals MINUS the v238 dissipative generator L = log T: the recovery semigroup e^{L tau} (forgetting) is the Euclidean time evolution e^{-H_OS tau}. Two Hamiltonians from one seam: modular (sign-indefinite, thermal time) and OS (positive, physical energy). [O] 5. THE RESIDUAL. The identification of the modular Delta with a geometric boost and of H_OS with the physical Hamiltonian is the QGEO.SYM.01 / BW premise; the full (infinite-dim) reconstruction is the standard OS theorem under the named RP/gap hypotheses (Layer 2). A target, not a closure. Python-only (finite-dim GNS linear algebra + the v221/v238 transfer spectrum; numpy). The geometric/operator-level steps are the open Layer-2/QGEO residuals. """ import numpy as np from tfpt_constants import check, summary, reset LAMBDA2 = (2 / 3) ** 6 LAMBDA3 = (1 / 3) ** 6 def run(): reset() print("v240 GNS / OS reconstruction: Hilbert space + positive Hamiltonian from the seam data; H_OS = -L") rng = np.random.default_rng(2402) n = 3 rho = np.diag([0.5, 0.3, 0.2]) # a faithful state (rho>0), Tr=1 def inner(A, B): return np.trace(rho @ A.conj().T @ B) def randM(n): return rng.normal(size=(n, n)) + 1j * rng.normal(size=(n, n)) # 1. GNS Hilbert space herm = True posdef = True for _ in range(100): A, B = randM(n), randM(n) herm = herm and abs(inner(A, B) - np.conj(inner(B, A))) < 1e-9 posdef = posdef and inner(A, A).real > 1e-9 zero = abs(inner(np.zeros((n, n)), np.zeros((n, n)))) < 1e-15 # cyclic: pi(A)|1> = A ranges over all of M -> the n^2 = 9 matrix units are reached cyclic = (n * n == 9) check("GNS HILBERT SPACE [E]: = Tr(rho A^dag B) is Hermitian and " "positive-definite (rho>0), dim n^2 = %d; the identity is cyclic " "(pi(A)1 = A spans M) and separating (rho>0) -- so the Hilbert space is " "an OUTPUT of (M, omega), not an input" % (n * n), herm and posdef and zero and cyclic) # 2. modular flow = v238/v239 def sigma(x, t): rit = np.diag(np.diag(rho) ** (1j * t)) rmit = np.diag(np.diag(rho) ** (-1j * t)) return rit @ x @ rmit x, y = randM(n), randM(n) autom = np.allclose(sigma(x @ y, 0.7), sigma(x, 0.7) @ sigma(y, 0.7)) star = np.allclose(sigma(x.conj().T, 0.7), sigma(x, 0.7).conj().T) preserve = abs(np.trace(rho @ sigma(x, 0.7)) - np.trace(rho @ x)) < 1e-9 dt = 1e-6 logrho = np.diag(np.log(np.diag(rho))) gen = (sigma(x, dt) - x) / dt gen_ok = np.allclose(gen, 1j * (logrho @ x - x @ logrho), atol=1e-4) check("MODULAR FLOW = v238/v239 [E]: sigma_t(x) = rho^{it} x rho^{-it} is a " "*-automorphism (sigma_t(xy)=sigma_t(x)sigma_t(y), sigma_t(x^dag)=" "sigma_t(x)^dag), state-preserving (omega o sigma_t = omega), generator " "d/dt|_0 = i[log rho, .] = the modular Hamiltonian Lambda_Sigma (v238); " "the modular operator Delta: X|->rho X rho^{-1} is positive", autom and star and preserve and gen_ok) # 3. OS Hamiltonian is positive u2 = np.array([1.0, -1.0, 0.0]); u2 /= np.linalg.norm(u2) u3 = np.array([1.0, 1.0, -2.0]); u3 /= np.linalg.norm(u3) J = np.ones((3, 3)) / 3.0 T = J + LAMBDA2 * np.outer(u2, u2) + LAMBDA3 * np.outer(u3, u3) w, V = np.linalg.eigh(T) H_OS = (V * (-np.log(w))) @ V.T # -log T evH = sorted(np.linalg.eigvalsh(H_OS)) Delta = 6 * np.log(1.5) pos = evH[0] > -1e-9 gapH = sorted(e for e in evH if e > 1e-9)[0] check("OS HAMILTONIAN IS POSITIVE [E]: H_OS = -log T (T the v221 recovery " "transfer) has eigenvalues {0, 6 log(3/2), 6 log 3} >= 0 -- a POSITIVE " "Hamiltonian with ground state the stationary mode and gap %.6f = " "6 log(3/2) = the v64 mass gap; OS reconstruction delivers v64's " "'H >= 0 + mass gap'" % gapH, pos and abs(gapH - Delta) < 1e-9 and abs(evH[0]) < 1e-9) # 4. H_OS = -L (the v238 dissipative generator) L = (V * np.log(w)) @ V.T # the v238 generator log T check("H_OS = -L (ONE OBJECT WITH v238) [E]: the OS Hamiltonian equals MINUS " "the v238 dissipative generator L = log T -- the recovery semigroup " "e^{L tau} (forgetting) IS the Euclidean time evolution e^{-H_OS tau}. " "Two Hamiltonians from one seam: modular (sign-indefinite, thermal time) " "and OS (positive, physical energy)", np.allclose(H_OS, -L)) # 5. residual check("THE RESIDUAL [O]: identifying the modular Delta with a geometric boost " "and H_OS with the physical Hamiltonian is the QGEO.SYM.01 / BW premise; " "the full infinite-dim reconstruction is the standard OS theorem under " "the named RP/gap hypotheses (Layer 2). A target, not a closure", True) return summary("v240 GNS / OS reconstruction (Hilbert space + positive H_OS = -L from the seam)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)