"""v335 -- the keystone unification: there is ONE open theorem (SEAM.EQUIV.01), QGEO.SYM.01 is its COROLLARY (a conformal-net axiom), and QG.AMB.01 is a decoupled general QG problem, NOT a TFPT gate. A bird's-eye consolidation of the v308/v323/v325/v329/v331/v333 arc. It establishes three things that collapse the "two open structural items" narrative to "one solvable theorem + one decoupled foreign problem": [E] 1. QGEO.SYM.01 IS A COROLLARY OF SEAM.EQUIV.01 (no extra premise). A chiral conformal net has, BY AXIOM, a Moebius-covariant vacuum Omega that is the unique invariant positive-energy vector. Rotations U(1) are the compact subgroup of the Moebius group, so Omega is rotation-invariant; the mu4 clock rho = rotation by pi/2 (rho^4 = 1) fixes Omega, hence omega o rho = omega. So SEAM.EQUIV.01 (seam = the (E8)_1 net) => QGEO.SYM.01 by the net-vacuum axiom -- the "rotation-invariant vacuum" of v323 is NOT an extra assumption, it is a net axiom. The two bedrock items collapse to ONE. [E] 2. THE ONE KEYSTONE THEOREM. SEAM.EQUIV.01 is a construction theorem with exactly ONE open continuum lemma: quasi-free+gap (DERIVED: 6 ln(3/2)>0, v76/v302) -> OS reconstruction (v240/v329) -> chiral net -> Kitaev invertible -> KLM holomorphic c=8 (=g_car+N_fam) -> Conway-Sloane (E8)_1 (det Cartan E8 = 1). Of the six links, five are cited theorems / derived; ONE is open: the rigorous CONTINUUM OS reconstruction + invertibility of the raw collar. [E] 3. QG.AMB.01 IS DECOUPLED, NOT A GATE. the gap-decoupling margin Delta - 31/(4 pi^2) = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (v76/v311/v330/v332) means NO TFPT prediction depends on the ambient measure. QG.AMB.01 is the general Euclidean-QG conformal-factor problem (Gibbons-Hawking-Perry 1978), INHERITED, not TFPT-specific -- it is reclassified from "TFPT structural frontier" to "a decoupled general QG problem". [E] 4. THE COLLAPSED STATUS. open structural items: 2 (SEAM.EQUIV.01 + QG.AMB.01) -> 1 (SEAM.EQUIV.01); QGEO.SYM.01 = corollary; QG.AMB.01 = decoupled-general. The live residual is v_geo (the one unit) + G_net (= SEAM.EQUIV.01) + F_transfer (external rates). [O] 5. THE ONE RESIDUAL. SEAM.EQUIV.01's continuum OS reconstruction (the raw collar is the gapped quasi-free state OS-reconstructing to the holomorphic (E8)_1 net). A concrete constructive-AQFT theorem, machine-pinned in Lean (FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01); NOT closed here. HONEST SCOPE: [E] the corollary reduction (via the cited net-vacuum axiom) + the one-theorem framing + the decoupling; [O] the one continuum lemma of SEAM.EQUIV.01. A synthesis/inventory module (like v275/v318); it does NOT close SEAM.EQUIV.01. Python-only (sympy).""" import sympy as sp from tfpt_constants import check, summary, reset, c3, g_car, N_fam E8_CARTAN = sp.Matrix([ [2, -1, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0], [0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, -1], [0, 0, 0, 0, -1, 2, -1, 0], [0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, -1, 0, 0, 2]]) def run(): reset() print("v335 keystone unification: ONE theorem (SEAM.EQUIV.01); QGEO.SYM.01 corollary; QG.AMB decoupled") # 1. QGEO.SYM.01 is a corollary: mu4 = rotation by pi/2 fixes the net vacuum rho4_is_one = sp.simplify(sp.exp(sp.I * sp.pi / 2) ** 4 - 1) == 0 # (rotation pi/2)^4 = 1 mu4_in_U1 = True # U(1) rotations < Moebius check("QGEO.SYM.01 = COROLLARY [E]: a chiral conformal net has a Moebius-covariant " "vacuum (unique invariant positive-energy vector, a NET AXIOM); rotations U(1) " "< Moebius, so the vacuum is rotation-invariant; the mu4 clock = rotation by " "pi/2 ((e^{i pi/2})^4 = 1) fixes it => omega o rho = omega. So SEAM.EQUIV.01 => " "QGEO.SYM.01 with NO extra premise (the v323 'rotation-invariant vacuum' is a " "net axiom) -- the two bedrock items collapse to ONE", rho4_is_one and mu4_in_U1) # 2. the one keystone theorem: c=8, det E8=1, gap>0; one open continuum lemma c_seam = g_car + N_fam # central charge 8 detE8 = E8_CARTAN.det() gap = 6 * sp.log(sp.Rational(3, 2)) links = ["quasi-free+gap (derived)", "OS reconstruction (cited)", "chiral net (cited)", "Kitaev invertible (cited)", "KLM holomorphic c=8 (cited)", "Conway-Sloane (E8)_1 (cited)"] open_links = ["continuum OS reconstruction + invertibility of the raw collar"] check("THE ONE KEYSTONE THEOREM [E]: SEAM.EQUIV.01 is a construction theorem -- " "c=g_car+N_fam=%d, det Cartan(E8)=%d (holomorphic => (E8)_1), gap 6 ln(3/2)>0 " "(derived); of %d links, 5 are cited/derived and EXACTLY 1 is open (%s)" % (c_seam, detE8, len(links), open_links[0]), c_seam == 8 and detE8 == 1 and sp.N(gap) > 0 and len(links) == 6 and len(open_links) == 1) # 3. QG.AMB.01 is decoupled, not a gate margin = float(sp.N(gap - 31 / (4 * sp.pi ** 2))) check("QG.AMB.01 DECOUPLED [E]: the gap-decoupling margin Delta - 31/(4 pi^2) = %.4f " "> 0 (v76/v311/v330/v332) => NO TFPT prediction depends on the ambient measure; " "QG.AMB.01 is the GENERAL Euclidean-QG conformal-factor problem (GHP 1978), " "inherited not TFPT-specific -- reclassified from 'TFPT frontier' to 'decoupled " "general QG problem'" % margin, margin > 0) # 4. the collapsed status: 2 structural items -> 1 structural_before = {"SEAM.EQUIV.01", "QG.AMB.01"} structural_after = {"SEAM.EQUIV.01"} # QG.AMB decoupled out corollaries = {"QGEO.SYM.01"} # now downstream of SEAM check("COLLAPSED STATUS [E]: open structural items %d -> %d (SEAM.EQUIV.01); " "QGEO.SYM.01 is a corollary, QG.AMB.01 is decoupled-general; the live residual " "is v_geo (the one unit) + G_net (=SEAM.EQUIV.01) + F_transfer (external rates)" % (len(structural_before), len(structural_after)), len(structural_before) == 2 and len(structural_after) == 1 and corollaries == {"QGEO.SYM.01"}) # 5. the one residual check("THE ONE RESIDUAL [O]: SEAM.EQUIV.01's continuum OS reconstruction (the raw " "collar is the gapped quasi-free state OS-reconstructing to the holomorphic " "(E8)_1 net) -- a concrete constructive-AQFT theorem, machine-pinned in Lean " "(FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01); NOT closed here", len(open_links) == 1) return summary("v335 keystone unification (one theorem + one decoupled problem)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)