[
  {
    "O_0 (The Foundational Manifold)": {
      "macroscopic_vector": "The foundational geometric tensor and normal extrinsic expansion field (\\theta \\equiv -N K) on the foliated spacetime (\\mathcal{M} \\times \\mathbb{R}_\\tau, \\; ds^2 = -N^2 d\\tau^2 + g_{\\mu\\nu}(dx^\\mu + \\beta^\\mu d\\tau)(dx^\\nu + \\beta^\\nu d\\tau)) over a compact spatial manifold (\\mathcal{M}, smooth boundary \\partial\\mathcal{M}, \\dim \\mathcal{M} = n \\ge 3) that structurally spatializes continuous non-equilibrium thermodynamic flow directly into the physical geometry of the state space.",
      "relational_edges": {
        "integrates": "(\\text{Branch I: } \\mathbf{J}_E = \\mathbf{J}_{\\mathrm{geom}} \\equiv \\mathbf{J}_{\\mathrm{ambient}}\\vert_{\\mathcal{M}}; \\text{ Branch II: } \\mathbf{J}_E = \\mathbf{J}_W/p_0 \\text{ (fixed), and the volume element is carried by } \\mathbf{J}_{\\mathrm{geom}}) The unmediated mathematical reality of continuous exogenous flux, restricted to the spatial manifold, directly into the spatial coordinates and metric volume element of the macroscopic manifold.",
        "conditions": "(v^\\mu = dx^\\mu / d\\tau, \\; N = 1, \\; \\beta^\\mu = 0) The continuous wave and phase flow in synchronous normal gauge, where localized test trajectories navigate the expanding spatial hypersurfaces (\\Sigma_\\tau).",
        "actualizes": "(\\theta \\equiv -N K = \\frac{1}{\\sqrt{\\vert g \\vert}} \\partial_\\tau \\sqrt{\\vert g \\vert} = \\nabla_\\mu J_{\\mathrm{geom}}^\\mu > 0 \\text{ in Branch I}) The unbroken spatial continuity and strictly positive normal extrinsic volume expansion of the macroscopic manifold, which, together with the asymptotic rate \\theta \\to n H_0 > 0 of O_\\Omega, drives the unbounded spatial volume limit; in synchronous gauge the foliation time \\tau equals the proper time of normal observers."
      },
      "transductive_prose": "This coordinate anchors the continuous wave on a compact n-dimensional spatial manifold (n \\ge 3) carried by a Lorentzian foliation (ds^2 = -N^2 d\\tau^2 + g_{\\mu\\nu}(dx^\\mu + \\beta^\\mu d\\tau)(dx^\\nu + \\beta^\\nu d\\tau)) in synchronous normal gauge (N = 1, \\; \\beta^\\mu = 0). It is the unmediated thermodynamic baseline where ambient stochastic flux (\\mathbf{J}_E) directly impacts the topological field. Rather than statically containing this force as a passive Newtonian arena, the manifold absorbs the exogenous flux into its normal extrinsic expansion scalar (\\theta \\equiv -N K = \\nabla_\\mu J_{\\mathrm{geom}}^\\mu, > 0 \\text{ in Branch I}), precluding any collapse into static equilibrium. On each constant-\\tau hypersurface (\\Sigma_\\tau) the connection is Levi-Civita (\\nabla_\\alpha g_{\\mu\\nu} = 0); across the foliation, the rate of volumetric dilation equals the net boundary flux (dV_{\\mathcal{M}}/d\\tau = \\oint_{\\partial\\mathcal{M}} J_{\\mathrm{geom}}^\\mu d\\Sigma_\\mu), and \\tau is the proper time of normal observers.",
      "mathematical_equations": [
        "x^\\mu \\in \\mathcal{M} \\; (\\mathcal{M} \\text{ compact, connected}, \\; \\partial\\mathcal{M} \\text{ smooth}, \\; \\dim \\mathcal{M} = n \\ge 3), \\quad g_{\\mu\\nu} \\in C^3(\\mathcal{M}), \\; \\partial_\\tau g_{\\mu\\nu} \\in C^2(\\mathcal{M}), \\quad ds^2 = -N^2 d\\tau^2 + g_{\\mu\\nu} (dx^\\mu + \\beta^\\mu d\\tau)(dx^\\nu + \\beta^\\nu d\\tau), \\quad N = 1, \\quad \\beta^\\mu = 0, \\quad v^\\mu = \\frac{dx^\\mu}{d\\tau}",
        "\\theta \\equiv -N K \\overset{N=1, \\, \\beta^\\mu=0}{=} \\frac{1}{2} g^{\\mu\\nu} \\frac{\\partial g_{\\mu\\nu}}{\\partial \\tau} = \\frac{1}{\\sqrt{\\vert g \\vert}} \\frac{\\partial \\sqrt{\\vert g \\vert}}{\\partial \\tau} = \\nabla_\\mu J_{\\mathrm{geom}}^\\mu \\ge 0 \\; (\\theta > 0 \\text{ in Branch I}), \\quad \\Theta_E \\equiv \\frac{\\dot{s}_{\\mathrm{prod}}}{s_0} \\; (s_0 \\text{: the } O_\\Omega \\text{ Eq. } [0] \\text{ constant, fixed by that limit}) = -\\frac{\\beta_{\\mathrm{th}}}{s_0} J_E^\\mu \\nabla_\\mu \\mathcal{F} \\; (> 0 \\text{ by } O_1 \\text{ Eq. } [1]), \\quad \\Theta_{\\mathrm{tot}} \\equiv \\sqrt{\\theta^2 + \\Theta_E^2} \\; (> 0 \\text{ by } O_1 \\text{ Eq. } [1]), \\quad \\tau_\\Theta(\\tau) \\equiv \\int_{\\tau_0}^{\\tau} \\langle \\Theta_{\\mathrm{tot}} \\rangle_{\\mathcal{M}} \\, d\\tau' \\text{ strictly increasing by } O_1 \\text{ Eq. } [1] \\text{ (dimensionless; when } \\theta = 0 \\text{ it is the count of entropy production per } s_0 \\text{; } \\tau \\text{ remains the proper time of normal observers, and } \\tau_\\Theta \\text{ is a monotone reparametrization that orders events when } \\theta = 0), \\quad \\langle f \\rangle_{\\mathcal{M}} \\equiv \\frac{\\int_{\\mathcal{M}} f \\sqrt{\\vert g \\vert} \\, d^n x}{\\int_{\\mathcal{M}} \\sqrt{\\vert g \\vert} \\, d^n x} \\quad (\\text{Branch I: } \\tau_\\Theta \\text{ is driven by both } \\theta > 0 \\text{ and } \\Theta_E; \\text{ Branch II: by } \\Theta_E \\text{ alone when } \\theta = 0)",
        "\\frac{\\partial \\mathcal{H}}{\\partial \\tau} \\equiv J_E^\\mu \\nabla_\\mu q, \\quad \\mathcal{H}(\\tau_0, \\mathbf{x}) = \\mathcal{H}_0(\\mathbf{x}), \\quad q(\\mathbf{x}, \\tau) \\equiv -\\ln\\left(v_0 \\, Q(\\mathbf{x}, \\tau)\\right) \\; (Q > 0), \\quad \\int_{\\mathcal{M}} Q \\sqrt{\\vert g \\vert} \\, d^n x = 1 \\; (v_0 > 0 \\text{ reference volume})",
        "\\left. \\nabla_{\\alpha} g_{\\mu\\nu} \\right\\vert_{\\Sigma_\\tau} = 0, \\quad \\frac{d V_{\\mathcal{M}}}{d\\tau} = \\int_{\\mathcal{M}} \\theta \\sqrt{\\vert g \\vert} \\, d^n x = \\oint_{\\partial\\mathcal{M}} J_{\\mathrm{geom}}^\\mu \\, d\\Sigma_\\mu \\ge 0 \\; (\\text{Branch I: } J_{\\mathrm{geom}}^\\mu \\equiv J_{\\mathrm{ambient}}^\\mu\\vert_{\\mathcal{M}}, \\text{ as in r6}), \\quad d\\Sigma_\\mu \\equiv \\hat{n}_\\mu dA, \\; \\hat{n}_\\mu \\hat{n}^\\mu = 1 \\; (\\hat{n}_\\mu \\text{ outward unit normal})",
        "\\text{Branch I } (H_0 > 0): \\; \\lim_{\\tau \\to \\infty} V_{\\mathcal{M}}(\\tau) = V_{\\mathrm{field}} = \\infty \\quad (\\text{consequence of } \\theta \\to n H_0 > 0 \\text{ in } O_\\Omega); \\qquad \\text{Branch II } (H_0 = 0, \\; \\int_{\\tau_0}^{\\infty} \\langle \\theta \\rangle_{\\mathcal{M}} \\, d\\tau < \\infty): \\; \\lim_{\\tau \\to \\infty} V_{\\mathcal{M}}(\\tau) = V_\\infty < \\infty"
      ],
      "r7_branch_scope": "r7 (Decision 6, r10): expansion (\\theta = \\nabla_\\mu J_{\\mathrm{geom}}^\\mu \\ge 0) is separated from non-equilibrium throughput (\\Theta_E = \\dot{s}_{\\mathrm{prod}}/s_0 > 0). Statements below that \\theta > 0, that the boundary flux grows without bound, or that volume becomes infinite hold in Branch I (H_0 > 0) and recover r6 there; in Branch II (H_0 = 0) the manifold is bounded and non-equilibrium openness is carried by \\Theta_E > 0, with events ordered by \\tau_\\Theta (\\tau remains proper time). Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I. The divergence equation fixes J_{\\mathrm{geom}} only up to a divergence-free field; in Branch I this is resolved pointwise by J_{\\mathrm{geom}} \\equiv J_{\\mathrm{ambient}}\\vert_{\\mathcal{M}} (O_0 Eq. [3]), and in Branch II every use of J_{\\mathrm{geom}} depends only on its divergence or boundary flux (in Branch I, J_E \\equiv J_{\\mathrm{geom}} also enters pointwise, as in r6)."
    }
  },
  {
    "O_1 (Thermodynamic Invariants)": {
      "macroscopic_vector": "The foundational physical geometry and non-equilibrium conservation law of the continuous wave, actualizing the irreducible flux divergence that drives continuous topological expansion.",
      "relational_edges": {
        "integrates": "(\\text{Branch I: } \\oint_{\\partial\\mathcal{M}} J_{\\mathrm{geom}}^\\mu \\, d\\Sigma_\\mu \\to \\infty; \\text{ Branch II: } \\dot{W}_{\\mathrm{diss}} > 0 \\text{ with bounded volume}) The unbounded net boundary flux of the exogenous matrix directly into the foundational kinetic capacity of the active state.",
        "conditions": "(\\nabla_\\mu \\mathcal{F} \\neq 0, \\; J_E^\\mu \\nabla_\\mu \\mathcal{F} < 0) The permanent phase-state of dynamic non-equilibrium, spatializing the non-vanishing scalar free-energy gradient as continuous boundary control exertion (O_4).",
        "actualizes": "(\\dot{W}_{\\mathrm{ext}} = p_0 \\oint_{\\partial\\mathcal{M}} J_{\\mathrm{geom}}^\\mu \\, d\\Sigma_\\mu \\ge 0, \\; > 0 \\text{ in Branch I}; \\; \\dot{W}_{\\mathrm{tot}} > 0 \\text{ in both branches}) The net outward flux asymmetry and strictly positive mechanical expansion power across the boundary that lock the continuous wave into an open, non-equilibrium structural trajectory."
      },
      "transductive_prose": "The physical law of perpetual non-equilibrium. Because the net boundary flux of the exogenous matrix grows without bound, an open dynamical system can never rest at detailed balance. It must perpetually execute dissipative mechanical expansion work (\\dot{W}_{\\mathrm{ext}} > 0, parameterized by reference pressure p_0 = (\\beta_{\\mathrm{th}} v_0)^{-1} with inverse temperature \\beta_{\\mathrm{th}} > 0 and elementary reference volume v_0 > 0) across its boundary to prevent structural dissolution. This invariant enforces a permanent spatial gradient of the scalar free-energy density (\\nabla_\\mu \\mathcal{F} \\neq 0) down which the exogenous current flows (J_E^\\mu \\nabla_\\mu \\mathcal{F} < 0), ensuring strictly positive entropy production (\\dot{s}_{\\mathrm{prod}} > 0) and continuous outward state-space expansion.",
      "mathematical_equations": [
        "\\dot{W}_{\\mathrm{ext}} = p_0 \\dot{V}_{\\mathcal{M}} = p_0 \\oint_{\\partial\\mathcal{M}} J_{\\mathrm{geom}}^\\mu \\, d\\Sigma_\\mu = p_0 \\int_{\\mathcal{M}} \\theta \\sqrt{\\vert g \\vert} \\, d^n x \\ge 0 \\; (> 0 \\text{ in Branch I}), \\quad p_0 \\equiv (\\beta_{\\mathrm{th}} v_0)^{-1} \\; (\\beta_{\\mathrm{th}} > 0, \\, v_0 > 0), \\quad \\dot{W}_{\\mathrm{diss}} \\equiv \\int_{\\mathcal{M}} \\left( -J_E^\\mu \\nabla_\\mu \\mathcal{F} \\right) \\sqrt{\\vert g \\vert} \\, d^n x = \\frac{s_0}{\\beta_{\\mathrm{th}}} \\int_{\\mathcal{M}} \\Theta_E \\sqrt{\\vert g \\vert} \\, d^n x > 0, \\quad \\dot{W}_{\\mathrm{tot}} \\equiv \\dot{W}_{\\mathrm{ext}} + \\dot{W}_{\\mathrm{diss}} > 0",
        "\\mathcal{F} \\in C^2(\\mathcal{M}), \\quad \\nabla_\\mu \\mathcal{F} \\neq 0, \\quad J_E^\\mu \\nabla_\\mu \\mathcal{F} < 0, \\quad \\dot{s}_{\\mathrm{prod}} = -\\beta_{\\mathrm{th}} J_E^\\mu \\nabla_\\mu \\mathcal{F} > 0, \\quad J_E^\\mu \\equiv J_{\\mathrm{geom}}^\\mu \\; (\\text{Branch I; recovers r6}), \\quad J_E^\\mu \\equiv \\frac{J_W^\\mu}{p_0} = -\\frac{1}{p_0} \\mathfrak{K}^{\\mu\\nu} \\nabla_\\nu \\mathcal{F} \\; (\\text{Branch II, fixed; then } J_E^\\mu \\nabla_\\mu \\mathcal{F} = -p_0^{-1} \\nabla_\\mu \\mathcal{F} \\, \\mathfrak{K}^{\\mu\\nu} \\nabla_\\nu \\mathcal{F} < 0 \\text{ holds identically wherever } \\nabla_\\mu \\mathcal{F} \\neq 0; \\text{ no other Branch II } J_E^\\mu \\text{ is admitted}; \\text{ the empirical content of this coordinate in Branch II is that a declared parametric } \\mathfrak{K}^{\\mu\\nu}, \\text{ with parameters estimated per the binding constitutive closure of } O_4, \\text{ predicts the independently measured flux } J_W^{\\mathrm{meas}} \\text{ within the pre-declared tolerance})"
      ],
      "r7_branch_scope": "r7: \\dot{W}_{\\mathrm{ext}} = p_0\\dot{V}_{\\mathcal{M}} is expansion work (> 0 in Branch I, \\ge 0 in Branch II); \\dot{W}_{\\mathrm{diss}} > 0 and \\dot{W}_{\\mathrm{tot}} > 0 hold in both branches. Prose below about unbounded flux describes Branch I. Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I. The switch between the Branch I and Branch II forms of J_E^\\mu (and of \\chi versus \\chi_E in O_7) is governed by clause (v) of the O_4 constitutive closure: the branch is declared before the test and cannot be changed on the same data."
    }
  },
  {
    "O_2 (Extrinsic Geometry)": {
      "macroscopic_vector": "The non-linear spatial actualization of the continuous wave, mapping accumulated thermodynamic work, ADM extrinsic curvature (K_{\\mu\\nu}), and environmental precision weighting (\\Pi_{\\mu\\nu}) into the constraint geometry that governs local trajectories.",
      "relational_edges": {
        "integrates": "(\\Pi_{\\mu\\nu} = \\nabla_\\mu \\nabla_\\nu \\Phi, \\; K_{\\mu\\nu} = -\\frac{1}{2 N}(\\partial_\\tau g_{\\mu\\nu} - \\mathcal{L}_\\beta g_{\\mu\\nu})) The variance of O_1 directly into load-bearing structural curvature, pairing the covariant Hessian of the environmental potential (\\Phi) with the ADM extrinsic curvature tensor of each spatial leaf (\\Sigma_\\tau).",
        "conditions": "(G_{\\mu\\nu} = \\kappa_G T_{\\mu\\nu}^{\\mathrm{mat}}, \\; \\kappa_G \\partial_\\tau T_{\\mu\\nu}^{\\mathrm{mat}} = -\\frac{n-2}{n} \\left( \\nabla_\\mu \\nabla_\\nu \\theta - g_{\\mu\\nu} \\nabla^2 \\theta \\right) + \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu}, \\; g_{\\mu\\nu} = \\Omega^2 \\hat{g}_{\\mu\\alpha}(\\exp \\hat{h})^\\alpha_{\\;\\nu}) The tensor manifold (n \\ge 3) obeying the per-slice Einstein constraint, whose \\tau-derivative (a conformal term plus the linearized Einstein operator on trace-free perturbations) binds localized boundary exertion (O_4) into unified macroscopic coherence (O_7); its trace couples the expansion scalar \\theta to the boundary-stress field through an elliptic equation.",
        "actualizes": "(\\frac{d v^\\mu}{d\\tau} + \\Gamma^\\mu_{\\rho\\epsilon} v^\\rho v^\\epsilon + g^{\\mu\\lambda}(\\partial_\\tau g_{\\lambda\\nu}) v^\\nu - \\frac{1}{2}(\\partial_\\tau g_{\\rho\\epsilon} v^\\rho v^\\epsilon) v^\\mu = f^\\mu_{\\mathrm{drive}}, \\; f^\\mu_{\\mathrm{drive}} = 0 \\text{ under the default drive}) The synchronous-gauge trajectory equation on the evolving spatial metric (the exact geodesic equation under the default drive; with a declared drive, its cone-preserving driven form), steering localized test trajectories along timelike paths that are proper-time extremal under the default drive (g_{\\mu\\nu} v^\\mu v^\\nu < 1) with both conformal drag and coordinate-time reparametrization terms."
      },
      "transductive_prose": "The condensation of kinetic energy into formal, macroscopic constraint geometry. As the continuous wave transduces variance across the foliated manifold (n \\ge 3), the covariant precision Hessian (\\Pi_{\\mu\\nu} = \\nabla_\\mu \\nabla_\\nu \\Phi) and the ADM extrinsic curvature tensor (K_{\\mu\\nu}) define the embedding and stiffness of each spatial slice (\\Sigma_\\tau). The per-slice Einstein constraint (G_{\\mu\\nu} = \\kappa_G T_{\\mu\\nu}^{\\mathrm{mat}}) holds on every slice, so its \\tau-derivative is an identity: the evolution of T_{\\mu\\nu}^{\\mathrm{mat}} is fixed by a conformal term in \\theta and the linearized Einstein operator on trace-free perturbations (\\mathbb{L}_{\\mathrm{Ein}}) acting on the shear, and its trace (using R = \\kappa_G \\nabla_\\alpha \\xi^\\alpha) is an elliptic equation tying the expansion scalar \\theta to the boundary-stress field \\xi_\\alpha = \\hat{\\Pi}_{\\alpha\\lambda}\\eta^\\lambda. The spatial metric undergoes a non-perturbative unimodular Weyl factorization that isolates the volume element (\\det g = \\Omega^{2n}) from anisotropic shear; being a change of variables, it separates volume from shear but does not decouple \\theta from precision, which remains coupled through the elliptic equation above. The factors are a unit-determinant reference metric (\\hat{g}_{\\mu\\nu}, \\det \\hat{g} = 1 in a fixed reference chart), a conformal factor (\\Omega, a scalar density of weight 1/n), and the exponential map of the trace-free shear endomorphism (\\hat{h}^\\alpha_{\\;\\nu} \\equiv \\hat{g}^{\\alpha\\beta} h_{\\beta\\nu}, with h_{\\mu\\nu} symmetric), while test trajectories obey the exact timelike geodesic equation of the Lorentzian foliation under the default drive (and its cone-preserving driven form under a declared drive) in synchronous normal gauge (N = 1, \\; \\beta^\\mu = 0).",
      "mathematical_equations": [
        "G_{\\mu\\nu} = R_{\\mu\\nu} - \\frac{1}{2} R g_{\\mu\\nu} = \\kappa_G T_{\\mu\\nu}^{\\mathrm{mat}}, \\quad \\kappa_G \\frac{\\partial T_{\\mu\\nu}^{\\mathrm{mat}}}{\\partial \\tau} = -\\frac{n-2}{n} \\left( \\nabla_\\mu \\nabla_\\nu \\theta - g_{\\mu\\nu} \\nabla^2 \\theta \\right) + \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu}, \\quad \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu} \\equiv \\nabla^\\alpha \\nabla_\\mu \\sigma_{\\nu\\alpha} + \\nabla^\\alpha \\nabla_\\nu \\sigma_{\\mu\\alpha} - \\nabla^2 \\sigma_{\\mu\\nu} - g_{\\mu\\nu} \\left( \\nabla^\\alpha \\nabla^\\beta \\sigma_{\\alpha\\beta} - \\sigma^{\\alpha\\beta} R_{\\alpha\\beta} \\right) - R \\, \\sigma_{\\mu\\nu} \\quad (n \\ge 3, \\; \\kappa_G = \\mathrm{const}), \\quad R = \\kappa_G \\nabla_\\alpha \\xi^\\alpha, \\quad \\frac{2(n-1)}{n} \\nabla^2 \\theta + \\frac{2R}{n} \\theta = -\\kappa_G \\frac{\\partial (\\nabla_\\alpha \\xi^\\alpha)}{\\partial \\tau} + 2 \\nabla^\\alpha \\nabla^\\beta \\sigma_{\\alpha\\beta} - 2 \\sigma^{\\alpha\\beta} R_{\\alpha\\beta}",
        "\\Phi \\in C^4(\\mathcal{M}), \\quad \\Pi_{\\mu\\nu} = \\nabla_\\mu \\nabla_\\nu \\Phi = \\frac{\\partial^2 \\Phi}{\\partial x^\\mu \\partial x^\\nu} - \\Gamma^\\lambda_{\\mu\\nu} \\frac{\\partial \\Phi}{\\partial x^\\lambda}, \\quad K_{\\mu\\nu} = -\\frac{1}{2 N} \\left( \\frac{\\partial g_{\\mu\\nu}}{\\partial \\tau} - \\mathcal{L}_\\beta g_{\\mu\\nu} \\right)",
        "g_{\\mu\\nu}(\\tau, \\mathbf{x}) = \\Omega^2(\\tau, \\mathbf{x}) \\, \\hat{g}_{\\mu\\alpha}(\\mathbf{x}) \\left( \\exp \\hat{h}(\\tau, \\mathbf{x}) \\right)^\\alpha_{\\;\\nu}, \\quad \\Omega(\\tau, \\mathbf{x}) > 0, \\quad h_{\\mu\\nu} = h_{\\nu\\mu}, \\quad \\hat{g}_{\\mu\\nu} \\in C^3(\\mathcal{M}), \\quad \\det \\hat{g}_{\\mu\\nu} = 1 \\; (\\text{fixed reference chart}; \\; \\Omega \\text{ a scalar density of weight } 1/n), \\quad \\mathrm{Tr}(\\hat{h}) \\equiv \\hat{g}^{\\mu\\nu} h_{\\mu\\nu} = 0 \\implies \\det g_{\\mu\\nu} = \\Omega^{2n}, \\quad \\frac{\\partial \\ln \\Omega}{\\partial \\tau} = \\frac{1}{n} \\theta = \\frac{1}{n} \\left( \\nabla_\\mu J_{\\mathrm{geom}}^\\mu \\right)",
        "\\left. \\left( \\frac{d v^\\mu}{d\\tau} + \\Gamma^\\mu_{\\rho\\epsilon} v^\\rho v^\\epsilon + g^{\\mu\\lambda} \\left( \\frac{\\partial g_{\\lambda\\nu}}{\\partial \\tau} \\right) v^\\nu - \\frac{1}{2} \\left( \\frac{\\partial g_{\\rho\\epsilon}}{\\partial \\tau} v^\\rho v^\\epsilon \\right) v^\\mu \\right) \\right\\vert_{N=1, \\, \\beta^\\mu=0} = f^\\mu_{\\mathrm{drive}}, \\quad f^\\mu_{\\mathrm{drive}} \\equiv \\left( 1 - u^2 \\right) \\left( a^\\mu_{\\mathrm{d}} - \\left( g_{\\alpha\\beta} a^\\alpha_{\\mathrm{d}} v^\\beta \\right) v^\\mu \\right), \\; u^2 \\equiv g_{\\mu\\nu} v^\\mu v^\\nu \\quad (a^\\mu_{\\mathrm{d}}(\\mathbf{x}, v, \\tau) \\text{ a declared drive acceleration, jointly continuous in } (\\mathbf{x}, v, \\tau) \\text{ and locally Lipschitz in } (\\mathbf{x}, v) \\text{ locally uniformly in } \\tau, \\text{ with } g_{\\mu\\nu}, \\; \\partial_\\tau g_{\\mu\\nu}, \\; \\Gamma^\\mu_{\\rho\\epsilon}, \\text{ and } g^{\\mu\\lambda} \\partial_\\tau g_{\\lambda\\nu} \\text{ locally Lipschitz in } \\mathbf{x} \\text{ locally uniformly in } \\tau \\; (\\text{e.g. } \\partial_{\\mathbf{x}} \\Gamma \\text{ and } \\partial_{\\mathbf{x}} \\partial_\\tau g \\text{ jointly continuous on } \\mathcal{M} \\times \\mathbb{R}_\\tau), \\text{ so the whole right side is locally Lipschitz in } (\\mathbf{x}, v) \\text{ and solutions are unique}; \\; a^\\mu_{\\mathrm{d}} \\equiv 0 \\text{ by default, recovering the r6 geodesic equation}; \\; a^\\mu_{\\mathrm{d}} \\neq 0 \\text{ only under the binding constitutive closure of } O_4, \\text{ r7\\_constitutive\\_closure}), \\quad \\frac{d u^2}{d\\tau} = \\left( 1 - u^2 \\right) \\left( -\\partial_\\tau g_{\\rho\\epsilon} v^\\rho v^\\epsilon + 2 \\left( 1 - u^2 \\right) g_{\\mu\\nu} a^\\mu_{\\mathrm{d}} v^\\nu \\right) \\implies u^2 = 1 \\text{ is invariant and } u^2 < 1 \\text{ is preserved} \\quad  (\\text{Lorentzian}, \\; g_{\\tau\\tau} = -1, \\; \\text{timelike: } g_{\\mu\\nu} v^\\mu v^\\nu < 1)"
      ],
      "r7_branch_scope": "r7: the trajectory equation carries a declared drive f^\\mu_{\\mathrm{drive}}, zero by default (recovering r6's exact geodesic equation). Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I."
    }
  },
  {
    "O_3 (Intrinsic Topology)": {
      "macroscopic_vector": "The internal Riemannian geometry of localized structural hysteresis, encoding accumulated interaction history strictly as load-bearing topological density and volumetric capacity.",
      "relational_edges": {
        "integrates": "(f_\\mu^{\\mathrm{top}} = -\\nabla_\\mu \\Phi(\\mathbf{x})) The continuous localized resolution of thermodynamic work directly into the geometric curvature of the active metric tensor (g_{\\mu\\nu}) and internal hysteresis density (\\mathcal{H}).",
        "conditions": "(\\partial_\\tau g_{\\mu\\nu} = -2 N K_{\\mu\\nu} = \\frac{2}{n} \\theta g_{\\mu\\nu} + 2\\sigma_{\\mu\\nu}, \\; \\nabla_\\tau \\sigma^\\mu_{\\;\\nu} \\equiv \\partial_\\tau \\sigma^\\mu_{\\;\\nu} = -\\gamma_\\sigma \\sigma^\\mu_{\\;\\nu}) The normal ADM evolution of the macroscopic metric across the Markov blanket (\\partial\\mathcal{U}_i), where incoming flux divergence drives isotropic dilation while internal dissipation exponentially damps C^2-regular trace-free shear (O_4).",
        "actualizes": "(V_{\\mathrm{local}}(\\tau) \\equiv \\int_{\\mathcal{U}_i} \\mathcal{H}(\\mathbf{x}, \\tau) \\sqrt{\\vert g \\vert} \\, d^n x, \\; (\\hat{g}_\\infty)_{\\mu\\nu} = \\hat{g}_{\\mu\\alpha}(\\exp \\hat{h}_\\infty)^\\alpha_{\\;\\nu}) The physical retention of transduced variance, where shear relaxation anchors the asymptotic unimodular reference metric (\\hat{g}_\\infty) and expands internal state-space capacity."
      },
      "transductive_prose": "The dense, internal geometry of accumulated adaptation. When external thermodynamic pressure bears down on a localized domain (\\mathcal{U}_i), this interior topology absorbs and redistributes the load. It is the physical volume of localized structural hysteresis (\\mathcal{H})—the multi-dimensional internal curvature forged by resolving past shear. In the synchronous normal gauge (N = 1, \\; \\beta^\\mu = 0), normal ADM evolution of the macroscopic metric (g_{\\mu\\nu} \\equiv g_{\\mu\\nu}^{\\mathrm{macro}}, O_5) channels exogenous flux divergence into the extrinsic expansion trace (\\theta \\equiv -N K = \\nabla_\\lambda J_{\\mathrm{geom}}^\\lambda \\ge 0) while exponentially damping the C^2-regular mixed trace-free shear endomorphism (\\nabla_\\tau \\sigma^\\mu_{\\;\\nu} \\equiv \\partial_\\tau \\sigma^\\mu_{\\;\\nu} = -\\gamma_\\sigma \\sigma^\\mu_{\\;\\nu} with constant \\gamma_\\sigma > 0; the identification \\nabla_\\tau = \\partial_\\tau is exact for the mixed form because [\\mathcal{D}, \\sigma] = 0, and it preserves \\sigma^\\mu_{\\;\\mu} = 0), so that \\hat{h}(\\tau, \\mathbf{x}) converges in C^2(\\mathcal{M}) to an asymptotic profile \\hat{h}_\\infty(\\mathbf{x}) absorbed into the unit-determinant reference metric \\hat{g}_\\infty(\\mathbf{x}). The local subsystem metrics (g_{\\mu\\nu}^{(i)} on \\mathcal{U}_i) from which g_{\\mu\\nu} is glued are specified in O_5. Throughout O_3 and O_4, \\mathcal{U}_i denotes the fixed domain of a representative subsystem i \\in \\{1, \\dots, N_{\\mathrm{nodes}}\\} of that cover, while the control field \\eta^\\lambda and the stress tensor T_{\\mu\\nu}^{\\mathrm{mat}} are defined on all of \\mathcal{M}.",
      "mathematical_equations": [
        "f_\\mu^{\\mathrm{top}} = -\\nabla_\\mu \\Phi(\\mathbf{x}), \\quad V_{\\mathrm{local}}(\\tau) \\equiv \\int_{\\mathcal{U}_i} \\mathcal{H}(\\mathbf{x}, \\tau) \\sqrt{\\vert g \\vert} \\, d^n x",
        "\\left. \\frac{\\partial g_{\\mu\\nu}}{\\partial \\tau} \\right\\vert_{N=1, \\, \\beta^\\mu=0} = -2 K_{\\mu\\nu} = \\frac{2}{n} \\theta g_{\\mu\\nu} + 2\\sigma_{\\mu\\nu}, \\quad \\theta \\equiv -N K = \\nabla_\\lambda J_{\\mathrm{geom}}^\\lambda \\ge 0 \\; (> 0 \\text{ in Branch I})",
        "\\sigma^\\mu_{\\;\\nu}(\\tau_0, \\mathbf{x}) \\in C^2(\\mathcal{M}), \\quad \\sigma^\\mu_{\\;\\nu} \\equiv g^{\\mu\\alpha} \\sigma_{\\alpha\\nu}, \\quad \\sigma^\\mu_{\\;\\mu} = 0, \\quad \\nabla_\\tau \\sigma^\\mu_{\\;\\nu} \\equiv \\frac{\\partial \\sigma^\\mu_{\\;\\nu}}{\\partial \\tau} = -\\gamma_\\sigma \\sigma^\\mu_{\\;\\nu} \\; (\\gamma_\\sigma = \\mathrm{const} > 0) \\implies \\lim_{\\tau \\to \\infty} \\Vert \\sigma^\\mu_{\\;\\nu} \\Vert_{C^2(\\mathcal{M})} = 0",
        "\\hat{h}(\\tau, \\mathbf{x}) \\xrightarrow[\\tau \\to \\infty]{C^2(\\mathcal{M})} \\hat{h}_\\infty(\\mathbf{x}), \\quad \\left(\\hat{g}_\\infty(\\mathbf{x})\\right)_{\\mu\\nu} \\equiv \\hat{g}_{\\mu\\alpha}(\\mathbf{x}) \\left( \\exp \\hat{h}_\\infty(\\mathbf{x}) \\right)^\\alpha_{\\;\\nu}, \\quad \\det \\hat{g}_\\infty = \\det \\hat{g} = 1"
      ],
      "r7_branch_scope": "r7: \\theta = \\nabla_\\lambda J_{\\mathrm{geom}}^\\lambda \\ge 0; the shear law is unchanged in both branches. Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I."
    }
  },
  {
    "O_4 (Autopoiesis)": {
      "macroscopic_vector": "The continuous kinetic generation of the selectively gated Markov blanket (\\partial\\mathcal{U}_i), regulating boundary flux (\\mathbf{J}_W) via the boundary control vector field (\\eta^\\lambda) to transduce exogenous variance into internal metric dilation while minimizing variational free energy.",
      "relational_edges": {
        "integrates": "(\\frac{\\partial \\mathcal{H}}{\\partial \\tau} \\equiv J_E^\\mu \\nabla_\\mu q, \\; \\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}} \\equiv \\mathbf{1}_{(-\\nabla^2 \\le \\Lambda_{\\mathrm{gate}}^2)}, \\; \\rho_{\\mathrm{local}}(\\tau) \\equiv \\frac{\\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}} \\rho_{\\mathrm{total}}(\\tau) \\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}}}{\\mathrm{Tr}(\\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}} \\rho_{\\mathrm{total}}(\\tau))}, \\; Q(\\mathbf{x}, \\tau) \\equiv \\langle \\mathbf{x} \\vert \\rho_{\\mathrm{local}}(\\tau) \\vert \\mathbf{x} \\rangle) Spectrally low-pass-gated exogenous stochasticity (\\mathbf{J}_E), filtered by the projector \\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}} onto Neumann eigenmodes of -\\nabla^2 with eigenvalue at most \\Lambda_{\\mathrm{gate}}^2 on L^2(\\mathcal{M}, d\\mathrm{vol}_g) (with \\langle \\mathbf{x} \\vert \\mathbf{x}' \\rangle = \\delta(\\mathbf{x} - \\mathbf{x}')/\\sqrt{\\vert g \\vert}, \\rho_{\\mathrm{total}}(\\tau) a positive trace-class operator on L^2(\\mathcal{M}, d\\mathrm{vol}_g), and \\mathrm{Tr}(\\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}} \\rho_{\\mathrm{total}}) > 0), directly into the load-bearing structural hysteresis (\\mathcal{H}) of the localized domain (O_3).",
        "conditions": "(\\dot{W}_{\\mathrm{var}} = \\frac{1}{t_0 \\beta_{\\mathrm{th}}} D_{\\mathrm{KL}}(Q \\parallel P) > 0, \\; \\partial_\\tau \\mathcal{F} + \\mathcal{F} \\theta = -\\nabla_\\alpha J_W^\\alpha - \\mathcal{R}_{\\mathcal{F}}, \\; J_W^\\alpha = -\\mathfrak{K}^{\\alpha\\beta}\\nabla_\\beta\\mathcal{F}) The continuous active inference and local free-energy balance of the Markov blanket across relaxation timescale t_0 > 0, maintaining operational distinction from the normalized ambient reference prior (P, with Q \\ll P).",
        "actualizes": "(\\lim_{\\tau \\to \\infty} \\frac{V_Q(\\tau)}{V_P(\\tau)} = 1) The volumetric matching of the entropy (typical-set) volume of the internal generative density (V_Q) to that of the environmental density (V_P), coupling localized expansion to continuous flux transduction."
      },
      "transductive_prose": "The continuous thermodynamic and cybernetic engine of the Markov blanket (\\partial\\mathcal{U}_i). The localized subsystem actively regulates its boundary via the control vector field (\\eta^\\lambda \\in \\Gamma(T\\mathcal{M})), dynamically tuning its spectral gate (\\hat{\\mathcal{P}}_{\\Lambda_{\\mathrm{gate}}}) to minimize expected free energy. Uninformative high-frequency perturbations are attenuated at the perimeter, while informative variance is admitted to drive internal volumetric expansion governed by the local balance law (\\partial_\\tau \\mathcal{F} + \\mathcal{F}\\theta = -\\nabla_\\alpha J_W^\\alpha) and the Reynolds transport identity on the fixed (\\tau-independent) coordinate domain \\mathcal{U}_i with constant diffusivity D_{\\mathcal{F}}. The Einstein-type (trace-adjusted) symmetric stress-energy tensor (T_{\\mu\\nu}^{\\mathrm{mat}}, with trace (1 - n/2)\\nabla_\\alpha \\xi^\\alpha) is exactly conserved if and only if the control-weighted field satisfies the elliptic vector condition (\\nabla^2 \\xi_\\nu + R_\\nu^{\\;\\alpha} \\xi_\\alpha = 0), as required by the contracted Bianchi identity; the \\tau-compatibility law for T_{\\mu\\nu}^{\\mathrm{mat}} then follows identically from the Einstein constraint holding on every slice.",
      "mathematical_equations": [
        "Q(\\mathbf{x}, \\tau) = \\langle \\mathbf{x} \\vert \\rho_{\\mathrm{local}}(\\tau) \\vert \\mathbf{x} \\rangle > 0, \\quad \\langle \\mathbf{x} \\vert \\mathbf{x}' \\rangle = \\frac{\\delta(\\mathbf{x} - \\mathbf{x}')}{\\sqrt{\\vert g \\vert}} \\implies \\int_{\\mathcal{M}} Q \\sqrt{\\vert g \\vert} \\, d^n x = \\mathrm{Tr}\\,\\rho_{\\mathrm{local}} = 1, \\quad \\dot{W}_{\\mathrm{var}} = \\frac{1}{t_0 \\beta_{\\mathrm{th}}} D_{\\mathrm{KL}}(Q \\parallel P) > 0 \\; (t_0 > 0), \\quad \\int_{\\mathcal{M}} P \\sqrt{\\vert g \\vert} \\, d^n x = 1, \\quad Q \\ll P, \\quad \\frac{\\partial \\mathcal{H}}{\\partial \\tau} \\equiv J_E^\\mu \\nabla_\\mu q, \\quad V_Q(\\tau) \\equiv v_0 \\exp\\left( -\\int_{\\mathcal{M}} Q \\ln(v_0 Q) \\sqrt{\\vert g \\vert} \\, d^n x \\right), \\quad V_P(\\tau) \\equiv v_0 \\exp\\left( -\\int_{\\mathcal{M}} P \\ln(v_0 P) \\sqrt{\\vert g \\vert} \\, d^n x \\right)",
        "\\mathcal{U}_i \\text{ is } \\tau\\text{-independent}, \\; \\partial\\mathcal{U}_i \\text{ Lipschitz}, \\quad \\mathcal{F} \\in C^2(\\mathcal{M}), \\quad \\frac{\\partial \\mathcal{F}}{\\partial \\tau} + \\mathcal{F} \\theta = -\\nabla_\\alpha J_W^\\alpha - \\mathcal{R}_{\\mathcal{F}}, \\quad J_W^\\alpha \\equiv -\\mathfrak{K}^{\\alpha\\beta} \\nabla_\\beta \\mathcal{F}, \\quad \\mathfrak{K}^{\\alpha\\beta} = \\mathfrak{K}^{\\beta\\alpha} \\succ 0, \\; \\mathfrak{K}^{\\alpha\\beta} \\in C^1, \\; \\mathcal{R}_{\\mathcal{F}} \\ge 0, \\; \\mathcal{R}_{\\mathcal{F}} \\in C^0 \\; (\\text{default } \\mathfrak{K}^{\\alpha\\beta} = D_{\\mathcal{F}} g^{\\alpha\\beta}, \\; D_{\\mathcal{F}} = \\mathrm{const} > 0, \\; \\mathcal{R}_{\\mathcal{F}} = 0, \\text{ recovering r6}; \\text{ departures only under the binding constitutive closure, r7\\_constitutive\\_closure}) \\implies \\frac{d \\mathcal{F}_{\\mathcal{U}_i}}{d\\tau} = \\frac{d}{d\\tau} \\int_{\\mathcal{U}_i} \\mathcal{F} \\sqrt{\\vert g \\vert} \\, d^n x = \\int_{\\mathcal{U}_i} \\left( \\frac{\\partial \\mathcal{F}}{\\partial \\tau} + \\mathcal{F} \\theta \\right) \\sqrt{\\vert g \\vert} \\, d^n x = -\\oint_{\\partial\\mathcal{U}_i} J_W^\\alpha \\, d\\Sigma_\\alpha - \\int_{\\mathcal{U}_i} \\mathcal{R}_{\\mathcal{F}} \\sqrt{\\vert g \\vert} \\, d^n x = \\int_{\\mathcal{U}_i} \\left( \\nabla_\\alpha \\left( \\mathfrak{K}^{\\alpha\\beta} \\nabla_\\beta \\mathcal{F} \\right) - \\mathcal{R}_{\\mathcal{F}} \\right) \\sqrt{\\vert g \\vert} \\, d^n x",
        "\\eta^\\lambda \\in \\Gamma(T\\mathcal{M}) \\cap C^2(\\mathcal{M}), \\quad \\hat{\\Pi}_{\\nu\\lambda} \\equiv \\Pi_{\\nu\\lambda} - \\frac{1}{n} g_{\\nu\\lambda} \\left( g^{\\alpha\\beta} \\Pi_{\\alpha\\beta} \\right), \\quad \\xi_\\nu \\equiv \\hat{\\Pi}_{\\nu\\lambda} \\eta^\\lambda, \\quad T_{\\mu\\nu}^{\\mathrm{mat}} = \\nabla_{(\\mu} \\xi_{\\nu)} - \\frac{1}{2} g_{\\mu\\nu} \\nabla_\\alpha \\xi^\\alpha, \\quad g^{\\mu\\nu} T_{\\mu\\nu}^{\\mathrm{mat}} = \\left(1 - \\frac{n}{2}\\right) \\nabla_\\alpha \\xi^\\alpha, \\quad \\nabla^\\mu T_{\\mu\\nu}^{\\mathrm{mat}} = \\frac{1}{2} \\left( \\nabla^2 \\xi_\\nu + R_\\nu^{\\;\\alpha} \\xi_\\alpha \\right), \\quad \\nabla^2 \\xi_\\nu + R_\\nu^{\\;\\alpha} \\xi_\\alpha = 0 \\iff \\nabla^\\mu T_{\\mu\\nu}^{\\mathrm{mat}} = 0",
        "R_{\\mu\\nu} - \\frac{1}{2} R g_{\\mu\\nu} = \\kappa_G T_{\\mu\\nu}^{\\mathrm{mat}}, \\quad \\kappa_G \\frac{\\partial T_{\\mu\\nu}^{\\mathrm{mat}}}{\\partial \\tau} = -\\frac{n-2}{n} \\left( \\nabla_\\mu \\nabla_\\nu \\theta - g_{\\mu\\nu} \\nabla^2 \\theta \\right) + \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu}, \\quad \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu} \\equiv \\nabla^\\alpha \\nabla_\\mu \\sigma_{\\nu\\alpha} + \\nabla^\\alpha \\nabla_\\nu \\sigma_{\\mu\\alpha} - \\nabla^2 \\sigma_{\\mu\\nu} - g_{\\mu\\nu} \\left( \\nabla^\\alpha \\nabla^\\beta \\sigma_{\\alpha\\beta} - \\sigma^{\\alpha\\beta} R_{\\alpha\\beta} \\right) - R \\, \\sigma_{\\mu\\nu}, \\quad \\frac{d V_{\\mathcal{M}}}{d \\tau} = \\oint_{\\partial\\mathcal{M}} J_{\\mathrm{geom}}^\\mu \\hat{n}_\\mu \\, dA \\ge 0 \\; (> 0 \\text{ in Branch I})"
      ],
      "r7_branch_scope": "r7: the balance law admits a declared transport tensor \\mathfrak{K}^{\\alpha\\beta} \\succ 0 and interior sink \\mathcal{R}_{\\mathcal{F}} \\ge 0 (defaults recover r6); prose below about constant diffusivity describes the default. Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I. Notation: \\mathfrak{K}^{\\alpha\\beta} is the transport tensor; in the spine, K, K_{\\mu\\nu}, and K^\\mu_{\\;\\nu} denote extrinsic curvature (the companion's §5.9 uses K for Gaussian curvature).",
      "r7_constitutive_closure": "Constitutive closure (binding, Decision 6). The defaults are a^\\mu_{\\mathrm{d}} = 0 (so f^\\mu_{\\mathrm{drive}} = 0), \\mathfrak{K}^{\\alpha\\beta} = D_{\\mathcal{F}} g^{\\alpha\\beta}, and \\mathcal{R}_{\\mathcal{F}} = 0; in Branch II, J_E^\\mu = J_W^\\mu/p_0 is fixed, not a default, and admits no alternative. Any departure must satisfy all of: (i) its functional form is a named constitutive law fixed before the test; (ii) it has a finite number of parameters, declared in advance (tabulated or non-parametric fields are not admissible), and those parameters are estimated from data that are independent of the test, not merely disjoint in time: either a different measurement modality from the one used to test the coordinate (for example, \\mathfrak{K} from an imposed-gradient conductivity measurement when J_W^{\\mathrm{meas}} is calorimetric), or a declared perturbation or non-stationary regime of the system distinct from the test window; (iii) every acceptance tolerance and comparison criterion is fixed before the test; (iv) the default model is fitted to the same test-window data and its residual reported alongside, and if the default fits at least as well as the declared law under the pre-declared criterion, the departure is rejected and the coordinate is evaluated under the default. The test-window fit of the default is used only for the comparison of (iv); whenever a coordinate is evaluated under the default, the default's parameters (for example D_{\\mathcal{F}}) are estimated per (ii), from data independent of the test. A term fitted to the data it closes does not count as satisfying any coordinate in which the term appears. In Branch II, with the fixed J_E^\\mu = J_W^\\mu/p_0, the O_1 sign condition holds identically, so the empirical content of O_1 is that the declared parametric \\mathfrak{K}^{\\alpha\\beta}, with parameters estimated per (ii), predicts the flux: the independently measured free-energy flux J_W^{\\mathrm{meas}} agrees with -\\mathfrak{K}^{\\alpha\\beta}\\nabla_\\beta\\mathcal{F} within the pre-declared tolerance. Here J_W^{\\mathrm{meas}} is the free-energy flux measured directly, within the test window, across a declared surface S (a boundary component of \\mathcal{M} or a declared interior section), by a method that does not use the declared \\mathfrak{K}^{\\alpha\\beta} (for example calorimetric, volumetric, or tracer flux), and reported with its uncertainty. The gradient \\nabla\\mathcal{F} (or \\mathcal{F}) used in the comparison must also be obtained within the test window by a method that uses neither J_W^{\\mathrm{meas}} nor the declared \\mathfrak{K}^{\\alpha\\beta} or \\mathcal{R}_{\\mathcal{F}}. The compared quantity is pre-declared: either the total flux \\int_S J_W^{\\mathrm{meas}} \\cdot d\\Sigma versus \\int_S (-\\mathfrak{K}\\nabla\\mathcal{F}) \\cdot d\\Sigma, or the normal flux density pointwise on S. Every evaluation, under the defaults or a declared departure, also requires: (v) the branch (I or II) is declared before the test, together with a pre-declared estimator and acceptance criterion for the asymptotic regime (for Branch I, an estimate of H_0 > 0; for Branch II, the trend of \\langle\\theta\\rangle, a bound on \\int\\langle\\theta\\rangle\\,d\\tau, and an estimate of \\lim\\langle\\Theta_E\\rangle > 0), supported by evidence obtained independently of the O_1 test; a configuration declared in one branch may not be re-evaluated in the other on the same test data."
    }
  },
  {
    "O_5 (Topological Coalescence)": {
      "macroscopic_vector": "The continuous volumetric intersection and simplicial percolation of expanding Markov blankets, fusing localized subsystem metrics into the single contiguous, positive-definite macroscopic metric of the manifold.",
      "relational_edges": {
        "integrates": "(\\rho_{\\mathrm{top}}^{(ij)} = \\int_{\\mathcal{U}_{ij}} g_{\\mu\\nu}^{(i)} (g^{(j)})^{\\mu\\nu} \\sqrt{\\vert g^{(ij)} \\vert} \\, d^n x) The localized metric dilations of autopoietic subsystems (O_4) across geodesically convex overlaps (\\mathcal{U}_{ij} \\equiv \\mathcal{U}_i \\cap \\mathcal{U}_j) directly into the unified volumetric manifold.",
        "conditions": "(\\Delta_{\\mathrm{Hodge}} \\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}} = (d d^\\dagger + d^\\dagger d)\\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}} = 0) The unbroken physical medium spatializing divergence-free and curl-free harmonic lateral tension transfer, with Hodge operators of the glued metric g_{\\mu\\nu} and absolute boundary conditions, across non-trivial cohomology cycles (H^1_{\\mathrm{dR}}(\\mathcal{M}) \\neq 0), supporting continuous thermodynamic exchange (O_6).",
        "actualizes": "(g_{\\mu\\nu} \\equiv g_{\\mu\\nu}^{\\mathrm{macro}} = \\sum_i \\psi_i g_{\\mu\\nu}^{(i)} + \\sum_{\\langle i,j \\rangle \\in \\mathcal{K}_0} \\psi_{ij} g_{\\mu\\nu}^{(ij)} \\text{ on } \\mathcal{M} = \\bigcup_i \\mathcal{U}_i) The global positive-definite metric synthesis built by Riemannian geometric-mean interpolation (g^{(i)} \\# g^{(j)}) over all geometric overlaps (\\mathcal{K}_0), with the fraction of actively coupled overlaps above the critical percolation threshold (\\phi_{\\mathrm{perc}}(\\mathcal{K}) = \\vert \\mathcal{K} \\vert / \\vert \\mathcal{K}_0 \\vert > \\phi_c); this glued metric is the single macroscopic metric g_{\\mu\\nu} used in O_0–O_4 and O_6–O_\\Omega."
      },
      "transductive_prose": "The dimensional aperture where isolated geometries coalesce into a shared simplicial complex. The manifold is covered by a fixed (\\tau-independent) finite open cover of subsystem domains (\\mathcal{M} = \\bigcup_i \\mathcal{U}_i) with non-empty pairwise overlaps (\\mathcal{U}_{ij} \\equiv \\mathcal{U}_i \\cap \\mathcal{U}_j \\neq \\emptyset for \\langle i,j \\rangle \\in \\mathcal{K}_0), each carrying its own local subsystem metric (g_{\\mu\\nu}^{(i)}). 'As autopoietic boundaries dilate, they intersect' refers to the growth of metric volume on these fixed domains (\\partial_\\tau \\sqrt{\\vert g^{(i)} \\vert} > 0) across the shared overlaps, not to moving coordinate boundaries; edge occupation (\\phi_{\\mathrm{perc}}(\\mathcal{K}) = \\vert \\mathcal{K} \\vert / \\vert \\mathcal{K}_0 \\vert) is the fraction of geometric overlaps (\\mathcal{K}_0) whose relational coupling is active (\\mathcal{K}: \\eta_{[i \\to j]}^\\lambda \\not\\equiv 0, O_6), and the metric is glued over all of \\mathcal{K}_0. Each overlap carries the positive-definite Riemannian geometric-mean metric (g_{\\mu\\nu}^{(ij)} = (g^{(i)} \\# g^{(j)})_{\\mu\\nu}) and is geodesically convex with respect to it for all \\tau \\ge \\tau_0. Local metric tensors are parallel-transported to a fiducial basepoint (\\mathbf{x}_0 \\in \\mathcal{U}_{ij}) via Synge's bitensor propagator (I_{\\mu'}^{\\;\\mu}) along the Levi-Civita connection (\\nabla^{(ij)}) of g_{\\mu\\nu}^{(ij)}, while a \\tau-independent non-negative partition of unity subordinate to the cover glues local and interpolant metrics into the single strictly positive-definite macroscopic metric (g_{\\mu\\nu} \\equiv g_{\\mu\\nu}^{\\mathrm{macro}}). As standing hypotheses, every subsystem metric is asymptotically homothetic with a common rate H_0: its rescaled form converges in C^2 to a positive-definite limit (e^{-2 H_0 \\tau} g_{\\mu\\nu}^{(i)} \\to G_{\\infty,\\mu\\nu}^{(i)} \\succ 0) and its mixed rate tends to H_0 \\delta^\\mu_\\nu. The glued metric inherits both properties, including e^{-2 H_0 \\tau} g_{\\mu\\nu} \\to G_{\\infty,\\mu\\nu} \\succ 0 in C^2(\\mathcal{M}), because the geometric mean is homogeneous of degree one and C^\\infty on the cone of positive-definite pairs and the gluing is linear; consequently the Christoffel symbols of g_{\\mu\\nu} stay bounded in C^1 as \\tau \\to \\infty. Tangential Hodge harmonic currents of g_{\\mu\\nu} satisfying full absolute boundary conditions circulate around non-trivial 1-cycles to redistribute lateral kinetic load without dissipative loss.",
      "mathematical_equations": [
        "\\mathcal{M} = \\bigcup_{i=1}^{N_{\\mathrm{nodes}}} \\mathcal{U}_i \\; (\\mathcal{M} \\text{ connected}, \\; N_{\\mathrm{nodes}} \\ge 2, \\; \\mathcal{U}_i \\text{ open}, \\; \\tau\\text{-independent}), \\quad \\partial\\mathcal{U}_i \\text{ Lipschitz}, \\quad \\mathcal{K}_0 \\equiv \\{ \\langle i,j \\rangle : i \\neq j, \\; \\mathcal{U}_{ij} \\equiv \\mathcal{U}_i \\cap \\mathcal{U}_j \\neq \\emptyset \\}, \\quad \\mathcal{K} \\equiv \\{ \\langle i,j \\rangle \\in \\mathcal{K}_0 : \\eta_{[i \\to j]}^\\lambda \\not\\equiv 0 \\}, \\quad \\phi_{\\mathrm{perc}}(\\mathcal{K}) \\equiv \\frac{\\vert \\mathcal{K} \\vert}{\\vert \\mathcal{K}_0 \\vert}, \\quad g_{\\mu\\nu}^{(i)}(\\tau, \\mathbf{x}) \\succ 0 \\text{ on } \\mathcal{U}_i, \\; g_{\\mu\\nu}^{(i)} \\in C^3, \\quad \\partial_\\tau \\sqrt{\\vert g^{(i)} \\vert} \\ge 0 \\; (> 0 \\text{ in Branch I})",
        "g_{\\mu\\nu}^{(ij)} \\equiv \\left( g^{(i)} \\# g^{(j)} \\right)_{\\mu\\nu} = g_{\\mu\\alpha}^{(i)} \\left[ \\left(g^{(i)}\\right)^{-1} g^{(j)} \\right]^{1/2\\,\\alpha}_{\\quad\\;\\nu}, \\quad \\mathcal{U}_{ij} \\text{ geodesically convex for } g^{(ij)}(\\tau) \\; \\forall \\tau \\ge \\tau_0, \\quad \\rho_{\\mathrm{top}}^{(ij)} = \\int_{\\mathcal{U}_{ij}} g_{\\mu\\nu}^{(i)} (g^{(j)})^{\\mu\\nu} \\sqrt{\\vert g^{(ij)} \\vert} \\, d^n x",
        "\\left(\\mathbf{C}_{\\mathrm{top}}^{(ij)}(\\mathbf{x}_0)\\right)_{\\mu'\\nu'\\alpha'\\beta'} = \\int_{\\mathcal{U}_{ij}} I_{\\mu'}^{\\;\\mu}(\\mathbf{x}_0, \\mathbf{x}) I_{\\nu'}^{\\;\\nu}(\\mathbf{x}_0, \\mathbf{x}) I_{\\alpha'}^{\\;\\alpha}(\\mathbf{x}_0, \\mathbf{x}) I_{\\beta'}^{\\;\\beta}(\\mathbf{x}_0, \\mathbf{x}) \\, g_{\\mu\\nu}^{(i)}(\\mathbf{x}) g_{\\alpha\\beta}^{(j)}(\\mathbf{x}) \\sqrt{\\vert g^{(ij)}(\\mathbf{x}) \\vert} \\, d^n x",
        "g_{\\mu\\nu}(\\tau, \\mathbf{x}) \\equiv g_{\\mu\\nu}^{\\mathrm{macro}}(\\tau, \\mathbf{x}) = \\sum_{i=1}^{N_{\\mathrm{nodes}}} \\psi_i(\\mathbf{x}) g_{\\mu\\nu}^{(i)}(\\tau, \\mathbf{x}) + \\sum_{\\langle i,j \\rangle \\in \\mathcal{K}_0} \\psi_{ij}(\\mathbf{x}) g_{\\mu\\nu}^{(ij)}(\\tau, \\mathbf{x}), \\quad \\mathbf{x} \\in \\mathcal{M}, \\quad \\psi_i, \\psi_{ij} \\in C^\\infty(\\mathcal{M}), \\; \\psi_i, \\psi_{ij} \\ge 0, \\quad \\mathrm{supp}(\\psi_i) \\subset \\mathcal{U}_i, \\quad \\mathrm{supp}(\\psi_{ij}) \\subset \\mathcal{U}_{ij}, \\quad \\sum_i \\psi_i + \\sum_{\\langle i,j \\rangle \\in \\mathcal{K}_0} \\psi_{ij} = 1",
        "e^{-2 H_0 \\tau} g_{\\mu\\nu}^{(i)} \\xrightarrow{C^2(\\mathcal{U}_i)} G_{\\infty,\\mu\\nu}^{(i)} \\succ 0, \\quad \\frac{1}{2} \\left(g^{(i)}\\right)^{\\mu\\alpha} \\partial_\\tau g_{\\alpha\\nu}^{(i)} \\xrightarrow{C^2(\\mathcal{U}_i)} H_0 \\delta^\\mu_\\nu \\quad (\\text{standing hypotheses, common } H_0 \\ge 0 \\; \\forall i; \\; H_0 > 0 \\text{ in Branch I}, \\; H_0 = 0 \\text{ in Branch II}) \\implies e^{-2 H_0 \\tau} g_{\\mu\\nu} \\xrightarrow{C^2(\\mathcal{M})} G_{\\infty,\\mu\\nu} \\equiv \\sum_{i=1}^{N_{\\mathrm{nodes}}} \\psi_i G_{\\infty,\\mu\\nu}^{(i)} + \\sum_{\\langle i,j \\rangle \\in \\mathcal{K}_0} \\psi_{ij} \\left( G_\\infty^{(i)} \\# G_\\infty^{(j)} \\right)_{\\mu\\nu} \\succ 0 \\; \\wedge \\; \\frac{1}{2} g^{\\mu\\alpha} \\partial_\\tau g_{\\alpha\\nu} \\xrightarrow{C^2(\\mathcal{M})} H_0 \\delta^\\mu_\\nu \\quad (\\text{by both hypotheses, the linearity of the gluing, the homogeneity } (cA) \\# (cB) = c \\, (A \\# B), \\text{ the } C^\\infty \\text{ smoothness of } \\# \\text{ on the cone of positive-definite pairs, and compactness of } \\mathrm{supp}\\,\\psi_i, \\mathrm{supp}\\,\\psi_{ij})",
        "\\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}}(\\tau) \\in \\Omega^1(\\mathcal{M}) \\cap C^2(\\bar{\\mathcal{M}}) \\; (\\text{defined slice-wise on } \\Sigma_\\tau, \\text{ with the Hodge operators of } g_{\\mu\\nu}(\\tau)), \\quad \\Delta_{\\mathrm{Hodge}} \\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}} = (d d^\\dagger + d^\\dagger d)\\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}} = 0 \\iff \\left( d\\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}} = 0 \\; \\wedge \\; d^\\dagger \\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}} = 0 \\right) \\; (d^\\dagger \\text{ of } g_{\\mu\\nu}), \\quad \\left. \\hat{n}_\\mu J_{\\mathcal{M}}^{\\mathrm{harm}\\,\\mu} \\right\\vert_{\\partial\\mathcal{M}} = 0, \\quad \\left. \\hat{n}_\\mu \\left(d\\mathbf{J}_{\\mathcal{M}}^{\\mathrm{harm}}\\right)^{\\mu\\nu} \\right\\vert_{\\partial\\mathcal{M}} = 0, \\quad \\partial\\mathcal{U}_{ij} \\text{ Lipschitz}, \\quad \\phi_{\\mathrm{perc}}(\\mathcal{K}) > \\phi_c \\in (0, 1)"
      ],
      "r7_branch_scope": "r7: the homothety hypotheses hold with common H_0 \\ge 0 (Branch I: H_0 > 0; Branch II: H_0 = 0); 'dilate' in the prose below describes Branch I. Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I."
    }
  },
  {
    "O_6 (The Relational Edge)": {
      "macroscopic_vector": "The continuous spatial manifold where coupled Markov blankets achieve topological entanglement, actualizing the magnitude of non-reciprocal phase-lead asymmetry and exterior curvature flux to sustain mutual information growth.",
      "relational_edges": {
        "integrates": "(\\frac{d}{d\\tau} \\mathcal{I}(Q_i ; Q_j) > 0, \\; \\mathcal{I}(Q_i ; Q_j) \\equiv D_{\\mathrm{KL}}(Q_{ij} \\parallel Q_i \\otimes Q_j)) The localized thermodynamic flux of autopoiesis (O_4) and the continuous spatial boundaries of topological coalescence (O_5) directly into the shared, coordinate-fixed overlap boundary (\\partial\\mathcal{U}_{ij}), whose metric geometry evolves in \\tau.",
        "conditions": "((J_{i \\to j})_\\nu \\equiv \\hat{\\Pi}_{\\nu\\lambda}^{(j)} \\eta_{i \\to j}^\\lambda, \\; (J_{[i \\to j]})_\\nu \\equiv \\frac{1}{2}(\\hat{\\Pi}_{\\nu\\lambda}^{(i)} + \\hat{\\Pi}_{\\nu\\lambda}^{(j)}) \\eta_{[i \\to j]}^\\lambda, \\; \\eta_{[i \\to j]}^\\lambda \\not\\equiv 0 \\text{ for } \\langle i,j \\rangle \\in \\mathcal{K}) The active, asymmetric coupling 1-forms entering the trace-free coupling operator; M^{(ij)}(\\mathbf{x}) \\neq 0 \\; \\forall \\mathbf{x} \\in \\bar{\\mathcal{U}}_{ij} is an independent hypothesis, under which the mixed endomorphism (M^{(ij)})^\\mu_{\\;\\nu} has stretching and folding eigenvalues of both signs summing to zero (\\mathrm{Tr}\\,M^{(ij)} = 0, so m_+ > 0 > m_-), superposed on the strictly positive extrinsic deformation spectrum (\\lambda_{\\min}(\\mathcal{D}) > 0).",
        "actualizes": "(V_{i \\cup j}(\\tau) / V_{\\mathrm{macro}}(\\tau) \\to c_{ij}, \\; 0 < c_{ij} \\le 1, \\; g^{\\mu\\nu} S_{\\mu\\nu}^{(ij)} = 0) The localized structural coupling and doubly symmetric, strictly trace-free coupling tensor (S_{\\mu\\nu}^{(ij)}) that bind the overarching manifold into an unbroken macroscopic continuum (O_7) while preserving boundary sovereignty; because the rescaled glued metric converges (e^{-2 H_0 \\tau} g_{\\mu\\nu} \\to G_{\\infty,\\mu\\nu} \\succ 0 in C^2(\\mathcal{M}), O_5), each pairwise volume keeps a fixed positive asymptotic share c_{ij} of the total."
      },
      "transductive_prose": "The selectively permeable membrane of structural coupling. It is the exact transductive surface where sovereign active states exchange mutual information. Each subsystem carries its own normalized generative density (Q_i on \\mathcal{U}_i, an independent object that need not coincide with the restriction of the global Q of O_0), and each coupled pair carries a normalized joint density (Q_{ij} on \\mathcal{U}_i \\times \\mathcal{U}_j, with marginals Q_i and Q_j), so mutual information is well-posed (\\mathcal{I}(Q_i ; Q_j) = D_{\\mathrm{KL}}(Q_{ij} \\parallel Q_i \\otimes Q_j) \\ge 0). The antisymmetric part of the coupling vector (\\eta_{[i \\to j]}^\\lambda) defines the precision-symmetrized phase-lead 1-form (J_{[i \\to j]} = -J_{[j \\to i]}), whose sign records which partner leads. Only the magnitude of this asymmetry enters the dynamics: the 1-form appears in the trace-free coupling operator (M_{\\mu\\nu}^{(ij)}) through its symmetric outer product, so M^{(ij)}, S^{(ij)}, and d\\mathcal{I}/d\\tau are invariant under exchanging leader and follower, and the asymmetry acts even when \\hat{\\Pi}^{(i)} = \\hat{\\Pi}^{(j)}. Taking M^{(ij)}(\\mathbf{x}) \\neq 0 \\; \\forall \\mathbf{x} \\in \\bar{\\mathcal{U}}_{ij} as a hypothesis, the zero trace of the mixed endomorphism (M^{(ij)})^\\mu_{\\;\\nu} guarantees stretching and folding eigenvalues of both signs summing to zero (\\mathrm{Tr}\\,M^{(ij)} = 0, so m_+ > 0 > m_-), riding atop the strictly positive extrinsic deformation spectrum (\\mathcal{D}^\\mu_{\\;\\nu} \\equiv -N K^\\mu_{\\;\\nu}, hypothesis \\theta/n > \\vert\\lambda_{\\min}(\\sigma)\\vert \\iff \\lambda_{\\min}(\\mathcal{D}) > 0). Combining M_{\\mu\\nu}^{(ij)} with the n-dimensionally trace-free exterior-curvature bracket (F_{\\mu\\alpha}^{(i \\to j)} = (d J_{i \\to j})_{\\mu\\alpha}, weighted by the coupling constant \\ell_F^2 > 0) yields a doubly symmetric, strictly trace-free coupling tensor (S_{\\mu\\nu}^{(ij)}); its boundary flux, weighted by the positive constant k_{\\mathcal{I}}, is hypothesized to sustain positive mutual information growth (d\\mathcal{I}/d\\tau > 0) even as the inter-subsystem marginal divergence on the overlap approaches zero (D_{\\mathrm{KL}}(Q_i^{\\vert ij} \\parallel Q_j^{\\vert ij}) \\to 0).",
      "mathematical_equations": [
        "\\int_{\\mathcal{U}_i \\times \\mathcal{U}_j} Q_{ij} \\, d\\mathrm{vol}_g(\\mathbf{x}_i) \\, d\\mathrm{vol}_g(\\mathbf{x}_j) = \\int_{\\mathcal{U}_i} Q_i \\, d\\mathrm{vol}_g = \\int_{\\mathcal{U}_j} Q_j \\, d\\mathrm{vol}_g = 1, \\quad \\int_{\\mathcal{U}_j} Q_{ij} \\, d\\mathrm{vol}_g(\\mathbf{x}_j) = Q_i(\\mathbf{x}_i), \\quad \\int_{\\mathcal{U}_i} Q_{ij} \\, d\\mathrm{vol}_g(\\mathbf{x}_i) = Q_j(\\mathbf{x}_j), \\quad Q_{ij} \\ll Q_i \\otimes Q_j, \\quad \\mathcal{I}(Q_i ; Q_j) \\equiv D_{\\mathrm{KL}}(Q_{ij} \\parallel Q_i \\otimes Q_j) = \\int_{\\mathcal{U}_i \\times \\mathcal{U}_j} Q_{ij} \\ln \\frac{Q_{ij}}{Q_i \\otimes Q_j} \\, d\\mathrm{vol}_g(\\mathbf{x}_i) \\, d\\mathrm{vol}_g(\\mathbf{x}_j) \\ge 0, \\quad \\frac{d}{d\\tau} \\mathcal{I}(Q_i ; Q_j) = k_{\\mathcal{I}} \\oint_{\\partial\\mathcal{U}_{ij}} S_{\\mu\\nu}^{(ij)} \\hat{n}^\\mu \\hat{n}^\\nu \\, dA \\; (k_{\\mathcal{I}} = \\mathrm{const} > 0, \\text{ in nats per unit time per unit of } \\textstyle\\oint S \\, dA), \\quad \\frac{d}{d\\tau} \\mathcal{I}(Q_i ; Q_j) > 0 \\; (\\text{hypothesis}, \\; \\langle i,j \\rangle \\in \\mathcal{K})",
        "\\int_{\\mathcal{U}_{ij}} Q_i \\, d\\mathrm{vol}_g > 0, \\quad \\int_{\\mathcal{U}_{ij}} Q_j \\, d\\mathrm{vol}_g > 0, \\quad Q_i^{\\vert ij} \\equiv \\frac{Q_i \\, \\mathbf{1}_{\\mathcal{U}_{ij}}}{\\int_{\\mathcal{U}_{ij}} Q_i \\, d\\mathrm{vol}_g}, \\quad Q_i^{\\vert ij} \\ll Q_j^{\\vert ij}, \\quad D_{\\mathrm{KL}}(Q_i^{\\vert ij} \\parallel Q_j^{\\vert ij}) = \\int_{\\mathcal{U}_{ij}} Q_i^{\\vert ij} \\ln \\frac{Q_i^{\\vert ij}}{Q_j^{\\vert ij}} \\, d\\mathrm{vol}_g \\ge 0 \\quad (\\text{inter-subsystem marginal divergence on } \\mathcal{U}_{ij})",
        "(J_{i \\to j})_\\nu \\equiv \\hat{\\Pi}_{\\nu\\lambda}^{(j)} \\eta_{i \\to j}^\\lambda, \\quad \\hat{\\Pi}_{\\nu\\lambda}^{(j)} \\equiv \\Pi_{\\nu\\lambda}^{(j)} - \\frac{1}{n} g_{\\nu\\lambda} \\, g^{\\alpha\\beta} \\Pi_{\\alpha\\beta}^{(j)}, \\quad \\Pi_{\\nu\\lambda}^{(j)} \\equiv \\nabla_\\nu \\nabla_\\lambda \\Phi^{(j)} \\; (\\Phi^{(j)} \\in C^4(\\mathcal{M}), \\; \\eta_{i \\to j}^\\lambda \\in C^2(\\mathcal{M}) \\text{ defined on all of } \\mathcal{M}, \\text{ so } S_{\\mu\\nu}^{(ij)} \\text{ is defined on } \\partial\\mathcal{U}_{ij}), \\quad \\eta_{[i \\to j]}^\\lambda = \\frac{1}{2}\\left(\\eta_{i \\to j}^\\lambda - \\eta_{j \\to i}^\\lambda\\right) \\not\\equiv 0 \\text{ on } \\mathcal{U}_{ij}, \\quad (J_{[i \\to j]})_\\nu \\equiv \\frac{1}{2} \\left( \\hat{\\Pi}_{\\nu\\lambda}^{(i)} + \\hat{\\Pi}_{\\nu\\lambda}^{(j)} \\right) \\eta_{[i \\to j]}^\\lambda = -(J_{[j \\to i]})_\\nu \\; (\\text{precision-symmetrized phase-lead 1-form; in general} \\neq \\tfrac{1}{2}(J_{i \\to j} - J_{j \\to i})_\\nu, \\text{ with equality when } \\hat{\\Pi}^{(i)} = \\hat{\\Pi}^{(j)}), \\quad F_{\\mu\\alpha}^{(i \\to j)} \\equiv \\nabla_\\mu (J_{i \\to j})_\\alpha - \\nabla_\\alpha (J_{i \\to j})_\\mu",
        "M_{\\mu\\nu}^{(ij)} \\equiv \\nabla_{(\\mu} \\left( J_{i \\to j} + J_{j \\to i} \\right)_{\\nu)} - \\frac{1}{n} g_{\\mu\\nu} \\nabla^\\alpha \\left( J_{i \\to j} + J_{j \\to i} \\right)_\\alpha + \\gamma_M \\left( (J_{[i \\to j]})_\\mu (J_{[i \\to j]})_\\nu - \\frac{1}{n} g_{\\mu\\nu} J_{[i \\to j]}^\\alpha (J_{[i \\to j]})_\\alpha \\right) \\; (\\gamma_M > 0), \\quad (M^{(ij)})^\\mu_{\\;\\nu} \\equiv g^{\\mu\\alpha} M_{\\alpha\\nu}^{(ij)}, \\quad M^{(ij)}(\\mathbf{x}) \\neq 0 \\; \\forall \\mathbf{x} \\in \\bar{\\mathcal{U}}_{ij} \\; (\\text{hypothesis}) \\implies m_+ \\equiv \\lambda_{\\max}\\left((M^{(ij)})^\\mu_{\\;\\nu}\\right) > 0 > \\lambda_{\\min}\\left((M^{(ij)})^\\mu_{\\;\\nu}\\right) \\equiv m_-, \\quad S_{\\mu\\nu}^{(ij)} = M_{\\mu\\nu}^{(ij)} + \\frac{\\ell_F^2}{2} \\left( F_{\\mu\\alpha}^{(i \\to j)} F_{\\nu}^{(j \\to i)\\alpha} + F_{\\mu\\alpha}^{(j \\to i)} F_{\\nu}^{(i \\to j)\\alpha} - \\frac{2}{n} g_{\\mu\\nu} F_{\\alpha\\beta}^{(i \\to j)} F^{(j \\to i)\\alpha\\beta} \\right) \\; (\\ell_F^2 > 0), \\quad g^{\\mu\\nu} S_{\\mu\\nu}^{(ij)} = 0",
        "V_{i \\cup j}(\\tau) = \\int_{\\mathcal{U}_i \\cup \\mathcal{U}_j} \\sqrt{\\vert g \\vert} \\, d^n x, \\quad V_{\\mathrm{macro}}(\\tau) \\equiv \\int_{\\mathcal{M}} \\sqrt{\\vert g \\vert} \\, d^n x = V_{\\mathcal{M}}(\\tau), \\quad \\lim_{\\tau \\to \\infty} \\frac{V_{i \\cup j}(\\tau)}{V_{\\mathcal{M}}(\\tau)} = c_{ij} \\equiv \\frac{\\int_{\\mathcal{U}_i \\cup \\mathcal{U}_j} \\sqrt{\\vert G_\\infty \\vert} \\, d^n x}{\\int_{\\mathcal{M}} \\sqrt{\\vert G_\\infty \\vert} \\, d^n x} \\; (0 < c_{ij} \\le 1) \\; (\\text{by } e^{-2 H_0 \\tau} g_{\\mu\\nu} \\xrightarrow{C^2(\\mathcal{M})} G_{\\infty,\\mu\\nu} \\succ 0, \\; O_5 \\text{ Eq. } 4, \\; H_0 \\ge 0), \\quad \\mathcal{D}^\\mu_{\\;\\nu} \\equiv -N K^\\mu_{\\;\\nu} \\overset{N=1, \\, \\beta^\\mu=0}{=} \\frac{1}{2} g^{\\mu\\alpha} \\frac{\\partial g_{\\alpha\\nu}}{\\partial \\tau} = \\frac{1}{n} \\theta \\delta^\\mu_\\nu + \\sigma^\\mu_{\\;\\nu}, \\quad \\frac{\\theta}{n} > \\vert\\lambda_{\\min}(\\sigma)\\vert \\iff \\lambda_{\\min}(\\mathcal{D}) > 0 \\; (\\text{standing hypothesis in Branch I, as in r6; not asserted in Branch II, where } \\mathcal{D} \\to 0), \\quad \\mathrm{Tr}(\\mathcal{D}) = \\sum_{a=1}^{n} \\lambda_a(\\mathcal{D}) = \\theta = \\nabla_\\mu J_{\\mathrm{geom}}^\\mu \\ge 0 \\; (> 0 \\text{ in Branch I})"
      ],
      "r7_branch_scope": "r7: the deformation-spectrum hypothesis \\lambda_{\\min}(\\mathcal{D}) > 0 is a Branch I hypothesis; c_{ij} is defined for H_0 \\ge 0. In Branch II, \\mathcal{I}(\\tau) \\to s_0 V_\\infty < \\infty (O_\\Omega Eq. [0]) while d\\mathcal{I}(Q_i;Q_j)/d\\tau > 0 for each pair in \\mathcal{K} (hypothesis), so each coupling-tensor boundary flux \\oint S^{(ij)}_{\\mu\\nu}\\hat{n}^\\mu\\hat{n}^\\nu\\,dA must be integrable on [\\tau_0, \\infty); the O_6 Eq. [3] hypothesis M^{(ij)} \\neq 0 at every finite \\tau remains compatible with this, since the boundary flux need only be integrable on [\\tau_0, \\infty) (it tends to 0 under the additional hypothesis that it is uniformly continuous in \\tau). Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I."
    }
  },
  {
    "O_7 (Macroscopic Coherence)": {
      "macroscopic_vector": "The continuous spatial volume where resonating localized boundaries structurally phase-lock via a dynamic principal-bundle gauge connection, establishing a singular, scale-free topological manifold.",
      "relational_edges": {
        "integrates": "(\\mathcal{A}_{\\mathcal{M}} = \\int_{\\mathcal{M}} (\\eta^\\mu \\nabla_\\mu q) \\sqrt{\\vert g \\vert} \\, d^n x) The localized geometric densities and global surprisal gradients of distinct autopoietic boundaries (O_4, O_5, O_6) actualizing directly as an unbroken, overarching thermodynamic field.",
        "conditions": "(D_\\mu \\hat{\\mathcal{F}}_{\\mathrm{macro}} = \\sum_{i=1}^{N_{\\mathrm{nodes}}} D_\\mu \\hat{\\mathcal{F}}_i, \\; \\Vert \\hat{\\mathcal{E}}_\\mu \\Vert_{C^1(\\mathcal{M})} \\to 0 \\implies [F_{\\mu\\nu}^{\\mathrm{YM}}, \\hat{\\chi}] \\to 0) The state of scale-free autopoiesis on a principal G_{\\mathrm{gauge}}-bundle (\\mathcal{P}_{\\mathrm{gauge}} \\to \\mathcal{M}, G_{\\mathrm{gauge}} compact), where a bounded adjoint gauge connection (A_\\mu) regulates spatial variations in the spectral anisotropy section (\\hat{\\chi}) and drives it to C^1-asymptotic covariant constancy across coupled geometries.",
        "actualizes": "(D_{\\mathrm{KL}}(Q_i^{\\vert ij} \\parallel Q_j^{\\vert ij}) \\to 0, \\; D_{\\mathrm{KL}}(\\hat{\\rho}_g(\\tau) \\parallel \\hat{\\rho}_{\\mathcal{H}}(\\tau)) \\to 0) The continuous phase-state where the structural expansion of localized coordinates seamlessly constitutes the conformal dilation of the entire macrostate."
      },
      "transductive_prose": "The phase-state of macroscopic geometric resonance. Localized boundaries maintain their distinct internal loci across N_{\\mathrm{nodes}} coupled subsystems but synchronize on a principal G_{\\mathrm{gauge}}-bundle (\\mathcal{P}_{\\mathrm{gauge}} \\to \\mathcal{M}, compact G_{\\mathrm{gauge}} with Hermitian Lie algebra generators \\hat{T}_a^\\dagger = \\hat{T}_a) via an adjoint covariant derivative (D_\\mu = \\partial_\\mu - i g_{\\mathrm{YM}}\\,\\mathrm{ad}_{A_\\mu} with constant g_{\\mathrm{YM}} \\neq 0, curvature F_{\\mu\\nu}^{\\mathrm{YM}} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu - i g_{\\mathrm{YM}}[A_\\mu, A_\\nu], and A_\\mu \\in C^1 uniformly bounded in \\tau). The connection acts on Lie-algebra-valued free-energy sections (\\hat{\\mathcal{F}}_i \\equiv \\psi_i \\mathcal{F} \\, \\hat{e}_i: the partition-weighted scalar free energy of O_1 and O_4 along a unit-norm direction \\Vert \\hat{e}_i \\Vert_{\\mathfrak{g}} = 1, smooth and supported in \\mathcal{U}_i; the sum \\hat{\\mathcal{F}}_{\\mathrm{macro}} = \\sum_i \\hat{\\mathcal{F}}_i runs over the single-node weights \\psi_i only, so the overlap weights \\psi_{ij} carry no free-energy section) and on the spectral anisotropy section (\\hat{\\chi} \\equiv \\chi \\, \\hat{u}, with a C^2 unit-norm section \\Vert \\hat{u} \\Vert_{\\mathfrak{g}} = 1, so \\Vert\\hat{\\chi}\\Vert_{\\mathfrak{g}} = \\chi). The anisotropy scalar is the C^2 index \\chi \\equiv n\\,\\mathrm{Tr}(\\mathcal{D}^2)/(\\mathrm{Tr}\\,\\mathcal{D})^2 = 1 + n\\,\\sigma^\\mu_{\\;\\nu}\\sigma^\\nu_{\\;\\mu}/\\theta^2 \\ge 1, which has no eigenvalue-crossing kinks and tends to 1 because shear decays (O_3) while \\theta \\to n H_0 > 0 (O_\\Omega). Transient spatial impedance gradients (D_\\mu \\hat{\\chi} = \\hat{\\mathcal{E}}_\\mu) decay in C^1(\\mathcal{M}) norm as \\tau \\to \\infty, and with A_\\mu bounded and g_{\\mathrm{YM}} \\neq 0 this forces D_{[\\mu} \\hat{\\mathcal{E}}_{\\nu]} \\to 0 and hence [F_{\\mu\\nu}^{\\mathrm{YM}}, \\hat{\\chi}] \\to 0. The global surprisal action (\\mathcal{A}_{\\mathcal{M}}) pairs the global control field \\eta^\\mu (O_4) with the surprisal gradient of the global density Q (O_0). Raising an index via the absorbed asymptotic unit-determinant metric (\\hat{g}_\\infty^{\\mu\\alpha}) constructs coordinate-invariant, positive-definite type-(1,1) density endomorphisms (\\hat{\\rho}_g \\to \\frac{1}{n}\\delta^\\mu_\\nu, \\; \\hat{\\rho}_{\\mathcal{H}}) over the strongly convex hysteresis basin (\\nabla_\\alpha \\nabla_\\beta \\mathcal{H} \\succ 0) whose relative entropy vanishes at phase-lock.",
      "mathematical_equations": [
        "\\mathcal{A}_{\\mathcal{M}} \\equiv \\int_{\\mathcal{M}} \\left( \\eta^\\mu \\nabla_\\mu q \\right) \\sqrt{\\vert g \\vert} \\, d^n x \\quad (\\text{global control field } \\eta^\\mu \\text{ of } O_4, \\text{ global surprisal } q = -\\ln(v_0 Q) \\text{ of } O_0)",
        "D_{\\mathrm{KL}}(Q_i^{\\vert ij} \\parallel Q_j^{\\vert ij}) \\to 0, \\quad D_{\\mathrm{KL}}\\left( \\hat{\\rho}_g(\\tau) \\parallel \\hat{\\rho}_{\\mathcal{H}}(\\tau) \\right) = \\mathrm{Tr}\\left[ \\hat{\\rho}_g \\left( \\ln \\hat{\\rho}_g - \\ln \\hat{\\rho}_{\\mathcal{H}} \\right) \\right] \\to 0",
        "(\\hat{\\rho}_g)^\\mu_{\\;\\nu} \\equiv \\frac{\\hat{g}_\\infty^{\\mu\\alpha} g_{\\alpha\\nu}}{\\hat{g}_\\infty^{\\alpha\\beta} g_{\\alpha\\beta}} = \\frac{\\left(e^{-\\hat{h}_\\infty} e^{\\hat{h}}\\right)^\\mu_{\\;\\nu}}{\\mathrm{Tr}\\left(e^{-\\hat{h}_\\infty} e^{\\hat{h}}\\right)} \\to \\frac{1}{n} \\delta^\\mu_\\nu, \\quad (\\hat{\\rho}_{\\mathcal{H}})^\\mu_{\\;\\nu} \\equiv \\frac{\\hat{g}_\\infty^{\\mu\\alpha} \\nabla_\\alpha \\nabla_\\nu \\mathcal{H}}{\\hat{g}_\\infty^{\\alpha\\beta} \\nabla_\\alpha \\nabla_\\beta \\mathcal{H}}, \\quad \\nabla_\\alpha \\nabla_\\beta \\mathcal{H} \\succ 0, \\quad \\left. \\nabla_{\\alpha} g_{\\mu\\nu} \\right\\vert_{\\Sigma_\\tau} = 0",
        "u_{\\mathrm{exp}}^\\mu \\in \\Gamma(T\\mathcal{M}), \\quad \\nabla_\\mu u_{\\mathrm{exp}}^\\mu = \\theta = \\frac{1}{2} g^{\\mu\\nu} \\partial_\\tau g_{\\mu\\nu} \\implies \\oint_{\\partial\\mathcal{M}} \\left( J_{\\mathrm{geom}}^\\mu - u_{\\mathrm{exp}}^\\mu \\right) d\\Sigma_\\mu = \\int_{\\mathcal{M}} \\left( \\nabla_\\mu J_{\\mathrm{geom}}^\\mu - \\theta \\right) \\sqrt{\\vert g \\vert} \\, d^n x = 0, \\quad \\sup_{\\tau \\ge \\tau_0} \\Vert J_E^\\mu \\Vert_{C^1(\\mathcal{M})} < \\infty \\; (\\text{coordinate components}) \\; \\wedge \\; \\hat{h} \\xrightarrow{C^2(\\mathcal{M})} \\hat{h}_\\infty \\; (O_3) \\implies \\lim_{\\tau \\to \\infty} \\Vert \\mathcal{L}_{\\mathbf{J}_E} \\left( \\hat{h} - \\hat{h}_\\infty \\right) \\Vert_{C^0(\\mathcal{M})} = 0",
        "G_{\\mathrm{gauge}} \\text{ compact}, \\quad \\hat{T}_a^\\dagger = \\hat{T}_a, \\quad g_{\\mathrm{YM}} = \\mathrm{const} \\neq 0, \\quad F_{\\mu\\nu}^{\\mathrm{YM}} \\equiv \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu - i g_{\\mathrm{YM}} [A_\\mu, A_\\nu], \\quad A_\\mu \\in C^1(\\mathcal{M}), \\quad \\sup_{\\tau \\ge \\tau_0} \\Vert A_\\mu \\Vert_{C^0(\\mathcal{M})} < \\infty, \\quad \\hat{\\mathcal{F}}_i \\equiv \\psi_i \\, \\mathcal{F} \\, \\hat{e}_i, \\; \\hat{e}_i \\in \\Gamma(\\mathrm{Ad}(\\mathcal{P}_{\\mathrm{gauge}})\\vert_{\\mathcal{U}_i}), \\; \\Vert \\hat{e}_i \\Vert_{\\mathfrak{g}} = 1, \\quad \\hat{\\mathcal{F}}_{\\mathrm{macro}} \\equiv \\sum_{i=1}^{N_{\\mathrm{nodes}}} \\hat{\\mathcal{F}}_i, \\quad D_\\mu \\hat{\\mathcal{F}}_{\\mathrm{macro}} = \\sum_{i=1}^{N_{\\mathrm{nodes}}} \\left( \\partial_\\mu \\hat{\\mathcal{F}}_i - i g_{\\mathrm{YM}} [A_\\mu, \\hat{\\mathcal{F}}_i] \\right)",
        "\\chi(\\tau, \\mathbf{x}) \\equiv \\frac{n \\, \\mathrm{Tr}(\\mathcal{D}^2)}{\\left(\\mathrm{Tr}\\,\\mathcal{D}\\right)^2} = 1 + \\frac{n \\, \\sigma^\\mu_{\\;\\nu} \\sigma^\\nu_{\\;\\mu}}{\\theta^2} \\ge 1 \\; (\\text{defined wherever } \\theta \\neq 0; \\text{ undefined where } \\theta = 0), \\quad \\hat{\\chi} \\equiv \\chi \\, \\hat{u} \\; (\\text{Branch I}), \\quad \\hat{u} \\in \\Gamma(\\mathrm{Ad}(\\mathcal{P}_{\\mathrm{gauge}})) \\cap C^2(\\mathcal{M}), \\; \\Vert \\hat{u} \\Vert_{\\mathfrak{g}} = 1 \\implies \\Vert \\hat{\\chi} \\Vert_{\\mathfrak{g}} = \\chi, \\quad D_\\mu \\hat{\\chi} = \\hat{\\mathcal{E}}_\\mu(\\tau, \\mathbf{x}), \\quad \\lim_{\\tau \\to \\infty} \\Vert \\hat{\\mathcal{E}}_\\mu \\Vert_{C^1(\\mathcal{M})} = 0 \\implies \\lim_{\\tau \\to \\infty} [F_{\\mu\\nu}^{\\mathrm{YM}}, \\hat{\\chi}] = 0, \\quad \\left( \\sigma^\\mu_{\\;\\nu} \\to 0 \\; (O_3) \\; \\wedge \\; \\theta \\to n H_0 > 0 \\; (O_\\Omega, \\text{ Branch I}) \\right) \\implies \\lim_{\\tau \\to \\infty} \\chi(\\tau, \\mathbf{x}) = 1, \\quad \\chi_E(\\tau, \\mathbf{x}) \\equiv 1 + \\frac{n \\, \\sigma^\\mu_{\\;\\nu} \\sigma^\\nu_{\\;\\mu}}{\\theta^2 + \\langle \\Theta_E \\rangle_{\\mathcal{M}}^2} \\ge 1 \\; (\\text{defined in both branches}; \\; \\chi_E \\le \\chi \\text{ wherever } \\theta > 0; \\; \\chi_E \\text{ is not } n\\,\\mathrm{Tr}(\\mathcal{D}^2)/(\\mathrm{Tr}\\,\\mathcal{D})^2), \\quad \\left( \\sigma^\\mu_{\\;\\nu} \\to 0 \\; \\wedge \\; \\liminf_{\\tau \\to \\infty} \\langle \\Theta_E \\rangle_{\\mathcal{M}} > 0 \\right) \\implies \\lim_{\\tau \\to \\infty} \\chi_E(\\tau, \\mathbf{x}) = 1, \\quad \\hat{\\chi} \\equiv \\chi_E \\, \\hat{u} \\text{ in Branch II}"
      ],
      "r7_branch_scope": "r7: \\chi is retained unchanged (Branch I); \\chi_E is the throughput-normalized index used in Branch II, where \\hat{\\chi} \\equiv \\chi_E \\hat{u}. Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I."
    }
  },
  {
    "O_\\Omega (The Topological Asymptote)": {
      "macroscopic_vector": "The homothetic (conformally stationary) horizon of the continuous wave, where the representative \\Omega^{-2} g_{\\mu\\nu} of the conformal class [g_{\\mu\\nu}] converges in C^2(\\mathcal{M}) to the stationary unit-determinant reference metric (\\hat{g}_\\infty) while the physical volume expands without bound (Branch I; in Branch II the volume tends to a finite V_\\infty under sustained throughput) (O_0 \\equiv O_\\Omega).",
      "relational_edges": {
        "integrates": "(\\lim_{\\tau \\to \\infty} \\frac{\\mathcal{I}(\\tau)}{s_0 V_{\\mathcal{M}}(\\tau)} = 1, \\; V_{\\mathcal{M}}(\\tau) \\to V_{\\mathrm{field}} = \\infty \\text{ (Branch I) or } V_\\infty < \\infty \\text{ (Branch II)}) Unbroken foliation time (\\tau \\to \\infty) strictly as homothetic volumetric dilation and extensive mutual-information saturation (s_0 > 0, information per unit volume) across the overarching manifold (\\mathcal{M}).",
        "conditions": "(\\text{Branch I: } \\theta \\xrightarrow{C^2(\\mathcal{M})} n H_0 > 0 \\; [\\text{Branch II: } \\theta \\to 0, \\; \\langle\\Theta_E\\rangle \\to \\Theta_\\infty > 0, \\; \\dot{W}_{\\mathrm{tot}} > 0], \\; \\sigma^\\mu_{\\;\\nu} \\xrightarrow{C^2(\\mathcal{M})} 0, \\; \\lim_{\\tau \\to \\infty} D_{\\mathrm{KL}}(Q_i^{\\vert ij} \\parallel Q_j^{\\vert ij}) = 0, \\; \\lim_{\\tau \\to \\infty} D_{\\mathrm{KL}}(\\hat{\\rho}_g \\parallel \\hat{\\rho}_{\\mathcal{H}}) = 0 \\mid \\dot{W}_{\\mathrm{ext}} > 0 \\text{ (Branch I)}; \\; \\dot{W}_{\\mathrm{tot}} > 0 \\text{ (both branches)}) The perpetual non-equilibrium homothetic phase-state, where trace-free shear vanishes in C^2(\\mathcal{M}) and \\hat{h} \\to \\hat{h}_\\infty while C^2-uniform conformal expansion (H_0 > 0) sustains open mechanical boundary work and harmonic circulation redistributes lateral load.",
        "actualizes": "(\\partial_\\tau (\\Omega^{-2} g_{\\mu\\nu}) \\to 0, \\; \\hat{\\mathbb{L}}_{\\mathrm{stat}}(\\mathbf{x}) \\Psi[g_{\\mu\\nu}, \\mathcal{H}] = 0 \\text{ (Branch I)}; \\; \\hat{\\mathbb{L}}_{\\mathrm{stat}}(\\mathbf{x}) \\Psi \\to 0 \\text{ as } \\tau \\to \\infty \\text{ (Branch II)}) The conformally stationary closure of the tensor cascade, establishing that the expanding physical metric on \\Sigma_\\tau and the asymptotically stationary rescaled metric (\\tilde{g}_{\\mu\\nu} = \\Omega^{-2} g_{\\mu\\nu} \\to \\hat{g}_\\infty) are dual representations of a single homothetic invariant."
      },
      "transductive_prose": "The homothetic mathematical horizon of the continuous wave. As foliation time approaches infinity, the normal expansion scalar converges in C^2(\\mathcal{M}) to a strictly positive spatially uniform rate (Branch I: \\theta = \\nabla_\\lambda J_{\\mathrm{geom}}^\\lambda \\xrightarrow{C^2} n H_0 > 0; because the Christoffel symbols stay bounded in C^1 by O_5 Eq. 4, \\nabla_\\mu \\nabla_\\nu \\theta \\to 0) while trace-free shear decays to zero in C^2(\\mathcal{M}) (\\sigma^\\mu_{\\;\\nu} \\xrightarrow{C^2} 0, so, with the same Christoffel bound, \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu} \\to 0 and \\partial_\\tau T_{\\mu\\nu}^{\\mathrm{mat}} \\to 0) and \\hat{h}(\\tau, \\mathbf{x}) \\to \\hat{h}_\\infty(\\mathbf{x}). The physical metric expands homothetically in mixed form (\\frac{1}{2} g^{\\mu\\alpha} \\partial_\\tau g_{\\alpha\\nu} \\to H_0 \\delta^\\mu_\\nu; the lower-index components themselves grow like e^{2 H_0 \\tau}, precisely e^{-2 H_0 \\tau} g_{\\mu\\nu} \\to G_{\\infty,\\mu\\nu} \\succ 0 in C^2(\\mathcal{M}) by O_5 Eq. 4), driving extensive volume to infinity (V_{\\mathrm{field}} = \\infty) and sustaining positive conformal boundary work (\\dot{W}_{\\mathrm{ext}} > 0), while the conformally rescaled metric (\\tilde{g}_{\\mu\\nu} = \\Omega^{-2} g_{\\mu\\nu} \\to \\hat{g}_\\infty) becomes asymptotically time-independent (\\partial_\\tau \\tilde{g}_{\\mu\\nu} = 2 \\tilde{g}_{\\mu\\alpha} \\sigma^\\alpha_{\\;\\nu} \\to 0). On the non-degenerate stationary graph locus of an ultralocal solution (\\mathcal{H} = \\mathcal{H}_{\\mathrm{sol}}(g_{\\mu\\nu}(\\mathbf{x}), \\mathbf{x}), where \\delta\\Psi/\\delta g_{\\mu\\nu} + (\\partial\\mathcal{H}_{\\mathrm{sol}}/\\partial g_{\\mu\\nu})\\delta\\Psi/\\delta\\mathcal{H} = 0, \\; \\delta\\Psi/\\delta\\mathcal{H} \\neq 0, and (\\partial \\mathcal{H}_{\\mathrm{sol}}/\\partial g_{\\mu\\nu})_{\\mathrm{tf}} = 0), invariance of the smooth functional \\Psi[g_{\\mu\\nu}, \\mathcal{H}] on \\mathrm{Met}(\\mathcal{M}) \\times C^\\infty(\\mathcal{M}) under the pointwise first-order stationary flow generator (\\hat{\\mathbb{L}}_{\\mathrm{stat}}(\\mathbf{x}) \\Psi[g_{\\mu\\nu}, \\mathcal{H}] = 0 \\; \\forall \\mathbf{x} \\in \\Sigma_\\tau) closes the partial chain rule bidirectionally with a finite non-zero denominator (n H_0 > 0), returning O_\\Omega seamlessly to O_0.",
      "mathematical_equations": [
        "\\mathcal{I}(\\tau) \\equiv \\sum_{\\langle i,j \\rangle \\in \\mathcal{K}} \\mathcal{I}(Q_i ; Q_j), \\quad \\lim_{\\tau \\to \\infty} \\frac{\\mathcal{I}(\\tau)}{s_0 \\int_{\\Sigma_\\tau} \\sqrt{\\vert g \\vert} \\, d^n x} = 1 \\; (s_0 > 0), \\quad \\text{Branch I: } \\lim_{\\tau \\to \\infty} \\int_{\\Sigma_\\tau} \\sqrt{\\vert g \\vert} \\, d^n x = V_{\\mathrm{field}} = \\infty; \\quad \\text{Branch II: } \\lim_{\\tau \\to \\infty} \\int_{\\Sigma_\\tau} \\sqrt{\\vert g \\vert} \\, d^n x = V_\\infty < \\infty \\; (\\text{so } \\mathcal{I}(\\tau) \\to s_0 V_\\infty, \\text{ a finite capacity}), \\quad \\sqrt{\\vert \\,^{(n+1)}g \\, \\vert} \\, d^{n+1}x = N \\sqrt{\\vert g \\vert} \\, d\\tau \\, d^n x",
        "\\theta \\xrightarrow[\\tau \\to \\infty]{C^2(\\mathcal{M})} n H_0, \\; H_0 \\ge 0 \\; (\\text{Branch I: } H_0 > 0; \\; \\text{Branch II: } H_0 = 0 \\text{ with } \\int_{\\tau_0}^{\\infty} \\langle \\theta \\rangle_{\\mathcal{M}} \\, d\\tau < \\infty \\text{ and } \\langle \\Theta_E \\rangle_{\\mathcal{M}} \\to \\Theta_\\infty > 0), \\quad \\sigma^\\mu_{\\;\\nu} \\xrightarrow[\\tau \\to \\infty]{C^2(\\mathcal{M})} 0, \\quad \\lim_{\\tau \\to \\infty} \\left( \\frac{1}{2} g^{\\mu\\alpha} \\frac{\\partial g_{\\alpha\\nu}}{\\partial \\tau} - H_0 \\delta^\\mu_\\nu \\right) = 0 \\iff \\left( \\lim_{\\tau \\to \\infty} \\left( \\frac{1}{2} g^{\\mu\\nu} \\frac{\\partial g_{\\mu\\nu}}{\\partial \\tau} - n H_0 \\right) = 0 \\; \\wedge \\; \\lim_{\\tau \\to \\infty} \\sigma^\\mu_{\\;\\nu} = 0 \\right), \\quad \\sup_{\\tau \\ge \\tau_0} \\Vert \\Gamma^\\lambda_{\\mu\\nu} \\Vert_{C^1(\\mathcal{M})} < \\infty \\; (\\text{from } e^{-2 H_0 \\tau} g_{\\mu\\nu} \\xrightarrow{C^2(\\mathcal{M})} G_{\\infty,\\mu\\nu} \\succ 0, \\; O_5 \\text{ Eq. } 4, \\; H_0 \\ge 0) \\implies \\left( \\nabla_\\mu \\nabla_\\nu \\theta \\to 0, \\; \\mathbb{L}_{\\mathrm{Ein}}[\\sigma]_{\\mu\\nu} \\to 0, \\; \\frac{\\partial T_{\\mu\\nu}^{\\mathrm{mat}}}{\\partial \\tau} \\to 0 \\right) \\; (\\text{coordinate components})",
        "\\Psi \\in C^\\infty\\left(\\mathrm{Met}(\\mathcal{M}) \\times C^\\infty(\\mathcal{M})\\right) \\; (\\delta / \\delta g_{\\mu\\nu}(\\mathbf{x}) \\text{ and } \\delta / \\delta \\mathcal{H}(\\mathbf{x}) \\text{ taken with respect to the } g\\text{-independent coordinate measure } d^n x, \\text{ so both are densities of weight } 1), \\quad \\hat{\\mathbb{L}}_{\\mathrm{stat}}(\\mathbf{x}) \\equiv 2 H_0 g_{\\mu\\nu}(\\mathbf{x}) \\frac{\\delta}{\\delta g_{\\mu\\nu}(\\mathbf{x})} + \\left( J_E^\\lambda \\nabla_\\lambda q \\right) \\frac{\\delta}{\\delta \\mathcal{H}(\\mathbf{x})}, \\quad \\mathcal{H}_{\\mathrm{sol}}[g](\\mathbf{x}) = \\mathcal{H}_{\\mathrm{sol}}(g_{\\mu\\nu}(\\mathbf{x}), \\mathbf{x}) \\; (\\text{ultralocal}), \\quad \\frac{\\delta \\Psi}{\\delta g_{\\mu\\nu}(\\mathbf{x})} + \\frac{\\partial \\mathcal{H}_{\\mathrm{sol}}(\\mathbf{x})}{\\partial g_{\\mu\\nu}(\\mathbf{x})} \\frac{\\delta \\Psi}{\\delta \\mathcal{H}(\\mathbf{x})} = 0, \\quad \\frac{\\delta \\Psi}{\\delta \\mathcal{H}(\\mathbf{x})} \\neq 0, \\quad \\left( \\left. \\hat{\\mathbb{L}}_{\\mathrm{stat}}(\\mathbf{x}) \\Psi\\left[g_{\\mu\\nu}, \\mathcal{H}\\right] \\right\\vert_{\\mathcal{H}=\\mathcal{H}_{\\mathrm{sol}}[g]} = 0 \\; \\forall \\mathbf{x} \\in \\Sigma_\\tau \\; \\wedge \\; \\left(\\frac{\\partial \\mathcal{H}_{\\mathrm{sol}}}{\\partial g_{\\mu\\nu}}\\right)_{\\mathrm{tf}} = 0 \\right) \\iff \\frac{\\partial \\mathcal{H}_{\\mathrm{sol}}(\\mathbf{x})}{\\partial g_{\\mu\\nu}(\\mathbf{x})} = \\frac{1}{2} \\left( \\frac{J_E^\\lambda \\nabla_\\lambda q}{n H_0} \\right) g^{\\mu\\nu}(\\mathbf{x}) \\; (\\text{Branch I}, H_0 > 0), \\quad \\text{Branch II } (H_0 = 0; \\text{ the condition is asymptotic, and finite-}\\tau \\text{ surprisal transduction is permitted}): \\; \\left. \\hat{\\mathbb{L}}_{\\mathrm{stat}}(\\mathbf{x}) \\Psi\\left[g_{\\mu\\nu}, \\mathcal{H}\\right] \\right\\vert_{\\mathcal{H}=\\mathcal{H}_{\\mathrm{sol}}[g]} = \\left( J_E^\\lambda \\nabla_\\lambda q \\right) \\frac{\\delta \\Psi}{\\delta \\mathcal{H}(\\mathbf{x})} \\xrightarrow[\\tau \\to \\infty]{} 0 \\iff J_E^\\lambda \\nabla_\\lambda q \\xrightarrow[\\tau \\to \\infty]{} 0 \\iff \\frac{\\partial \\mathcal{H}}{\\partial \\tau} \\xrightarrow[\\tau \\to \\infty]{} 0 \\; (\\text{asymptotically stationary hysteresis, by } O_0 \\text{ Eq. } 2, \\text{ with } \\vert \\delta\\Psi/\\delta\\mathcal{H} \\vert \\text{ bounded and bounded away from } 0, \\text{ uniformly as } \\tau \\to \\infty)"
      ],
      "r7_branch_scope": "r7: O_\\Omega has two branches. Branch I (H_0 > 0) is the r6 homothetic horizon described in the prose below. Branch II (H_0 = 0) is a bounded non-equilibrium steady state: \\theta \\to 0 with \\int\\langle\\theta\\rangle d\\tau < \\infty, \\langle\\Theta_E\\rangle \\to \\Theta_\\infty > 0, \\sigma \\to 0, \\partial_\\tau g_{\\mu\\nu} \\to 0, finite volume V_\\infty and information capacity s_0 V_\\infty, and, with \\vert\\delta\\Psi/\\delta\\mathcal{H}\\vert bounded and bounded away from 0 uniformly as \\tau \\to \\infty, \\hat{\\mathbb{L}}_{\\mathrm{stat}}\\Psi \\to 0 \\iff J_E^\\lambda\\nabla_\\lambda q \\to 0 as \\tau \\to \\infty (finite-\\tau surprisal transduction is permitted). In Branch II the relational coupling flux of O_6 is integrable on [\\tau_0, \\infty), so \\mathcal{I} converges; its limit, with O_\\Omega Eq. [0], defines s_0 V_\\infty. Relational edges, transductive prose, and macroscopic vectors retained from r6 describe Branch I with the default constitutive terms, except where they state r7 formulas; r6 clauses retained alongside r7 formulas within the same edge also describe Branch I."
    }
  }
]