Tier 2 Specification: Abiotic, Continuum & Tectonic Physics¶

Document ID: tier_2_abiotic_continuum_physics.md
Mediating Spine ($\mathcal{T}$): foundational-manifold-2026-10-20-r7.json and foundational-manifold-companion-2026-10-20-r10.md

0. Tier Architecture & Four-Part Functional Homology¶

Tier 2 maps the laws of abiotic matter, continuum fluid and wave mechanics, radiative-thermal boundary physics, and geomorphic and structural mechanics onto Spine $\mathcal{T}$. All four apertures are empirical apertures declared in Branch II of r7, the bounded non-equilibrium steady state (aperture_generation_guidance.md §2.3). Each declares, in its §1, a domain, physical metric, cone speed, free energy with datum, transport tensor, interior sink, drive, and Branch II evidence, and instantiates all nine coordinates $O_0 \to O_\Omega$ on that domain. Each declared domain is of one kind: a conserved, stored quantity diffusing through a passive medium, with supply crossing the boundary. The wider phenomena named in the aperture titles enter the mappings through the declared parameters, as boundary conditions, through the hysteresis record $\mathcal{H}$, or as illustrative context, and each §1 states which. Each aperture is formulated at the level of general continuum and statistical physics, so that it applies across any terrestrial, aquatic, cryospheric, atmospheric, or built environment without presupposing a specific habitat or site inventory.

Slot & Functional Role	Aperture Designation	Declared Domain and Diffusing Quantity	Titled Phenomena Outside the Domain	Vertical Homology Across Tiers
Slot 2.1: Structural Substrate	Condensed-Matter & Phase Transition Physics ($\mathcal{E}_{\mathrm{condensed}}$)	Supersaturated solute diffusing through a condensed host toward smooth interior growth zones.	Intermolecular networks, cohesion, nucleation, latent heat: through the material parameters, $\mathcal{H}$, or context.	$1.1 \leftrightarrow \mathbf{2.1} \leftrightarrow 3.1 \leftrightarrow 4.1 \leftrightarrow 5.1$
Slot 2.2: Dynamic Flow & Waves	Continuum Fluid & Wave Mechanics ($\mathcal{E}_{\mathrm{continuum}}$)	Pore-water head diffusing through a deformable porous medium with specific storage.	Inertial flow, vorticity, boundary layers, waves: through $\mathfrak{K}$ and $c_*$, as boundary data, or context.	$1.2 \leftrightarrow \mathbf{2.2} \leftrightarrow 3.2 \leftrightarrow 4.2 \leftrightarrow 5.2$
Slot 2.3: Boundary Gating	Atmospheric Radiative & Thermal Boundary Physics ($\mathcal{E}_{\mathrm{radiative}}$)	Thermal energy carried by radiative diffusion through an optically thick layer in local thermodynamic equilibrium.	Scattering, Fresnel partitioning, refraction, ray optics: through $\kappa_R$ and $c_*$, as boundary data, or context.	$1.3 \leftrightarrow \mathbf{2.3} \leftrightarrow 3.3 \leftrightarrow 4.3 \leftrightarrow 5.3$
Slot 2.4: Coalescence & Hysteresis	Geomorphic & Structural Mechanics ($\mathcal{E}_{\mathrm{tectonic}}$)	Pore-water head diffusing through a confined, elastically swelling stratum, with distributed leakage.	Erosion, deposition, bedload, consolidation under load, structural stress: as boundary conditions, through $\mathcal{H}$, or context.	$1.4 \leftrightarrow \mathbf{2.4} \leftrightarrow 3.4 \leftrightarrow 4.4 \leftrightarrow 5.4$

0.1 Conventions Common to the Four Tier 2 Apertures¶

The following conventions apply to every aperture of this tier. Each aperture's §1 and coordinate blocks state its specific values and cite these items where they apply.

Branch and windows. Branch II is declared before any test, with the estimators of item 6 and §1, and no Tier 2 domain is re-evaluated in Branch I on the same data (closure (v)). Statements that hold in Branch I only (r10 Decision 6) are never applied. Every observation window is a relaxation window: it begins with a declared step increase of the supply-boundary data from a prior steady state, after which the data are held, and it follows the domain as it settles toward the new steady state. Under the linear storage law of each declared domain, the comparison principle for parabolic equations then gives $\partial_\tau\mathcal{F} \ge 0$ at every point, so $\mathcal{F}$, and every response it drives such as the skeleton strain of 2.2 and 2.4, is one-signed within the window. The argument is exact on the rigid domains. On the deforming domains the equation is non-autonomous at order $\varepsilon_v$ ($\theta \neq 0$, $g$ depends on $\tau$), so there the conclusion holds only to $O(\varepsilon_v)$, and their precondition 3 checks $\theta \ge 0$ directly from data. The rate of change of the flux, $\partial_\tau J_W$, is not one-signed (item 8).
Metric. The physical metric is the pull-back of the Euclidean metric to the material points of the declared body under its deformation map. It is flat at every $\tau$, so $G_{\mu\nu} \equiv 0$ and $R_{\mu\nu} \equiv 0$ (aperture_generation_guidance.md §2.3 item 2). No Tier 2 aperture declares a non-Euclidean material metric in this epoch, so the curved configurations of the r10 §5.11 and §5.13 witnesses are realized by none of them.
Rigid domains (2.1, 2.3). In a Cartesian chart $g_{\mu\nu} = \delta_{\mu\nu}$, so $\Omega \equiv 1$, $\hat{h} \equiv 0$, $\theta \equiv 0$, and $\sigma^\mu_{\;\nu} \equiv 0$. The domain is declared rigid within a pre-declared tolerance on its measured volumetric strain; windows in which the measured volume decreases beyond that tolerance are excluded, so that $\theta \ge 0$ is not violated. With $\sigma \equiv 0$ the $O_3$ shear law holds trivially, and these apertures support no shear-relaxation test.
Deforming domains (2.2, 2.4). The solid skeleton swells elastically, with small strain, as pore pressure rises during the window, so $\theta \ge 0$ and $\theta \to 0$. Windows that compact the skeleton ($\theta < 0$) are excluded.
Stress sector. With $G_{\mu\nu} \equiv 0$, $T_{\mu\nu}^{\mathrm{mat}} \equiv 0$; its trace gives $\nabla_\alpha\xi^\alpha = 0$ for $n = 3$, and then $\nabla_{(\mu}\xi_{\nu)} = 0$, so $\xi$ is a Killing field. A passive medium has no global control field, so Tier 2 declares $\eta^\lambda \equiv 0$, hence $\xi_\nu \equiv 0$ and $R = \kappa_G\nabla_\alpha\xi^\alpha = 0$, and the $O_4$ equivalence $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$ holds identically. The $\tau$-compatibility law then reduces to $\mathbb{L}_{\mathrm{Ein}}[\sigma]_{\mu\nu} = \frac{n-2}{n}(\nabla_\mu\nabla_\nu\theta - g_{\mu\nu}\nabla^2\theta)$, and its trace, the traced elliptic equation, to $\frac{n-1}{n}\nabla^2\theta = \nabla^\alpha\nabla^\beta\sigma_{\alpha\beta}$. These are the Saint-Venant compatibility conditions of the strain rate, which every deformation of an embedded body satisfies because its metric stays flat. The load-bearing stresses of the physical medium (Cauchy stress, effective stress, overburden) are not $T_{\mu\nu}^{\mathrm{mat}}$; they enter as boundary conditions and through $\mathcal{H}$.
Cone speed and trajectories. Each aperture declares $c_*$ as a characteristic signal speed of its medium. Rays of that signal are null for the declared cone and, like all null rays, are not trajectories of $O_2$ Eq. [3]. Every material speed in $\mathcal{M}$ (diffusive, seepage, and deformation speeds) is $\ll c_*$, so $u^2 \ll 1$. No Tier 2 aperture maps a force-free timelike tracer or declares a drive: Eq. [3] is instantiated with $a^\mu_{\mathrm{d}} = 0$, and Tier 2 makes no trajectory claim (the driven trajectory equation is untested in this epoch, empirical_validation_protocol.md §3.3).
Currents. In Branch II, $J_E^\mu = J_W^\mu/p_0$ is fixed. The expansion current $J_{\mathrm{geom}}$ enters only through its divergence $\theta$ and its boundary flux $dV_{\mathcal{M}}/d\tau$ (r7 $O_0$ branch scope), and no supply current is ever mapped to it. In general $\nabla_\mu J_E^\mu = -p_0^{-1}(\partial_\tau\mathcal{F} + \mathcal{F}\theta + \mathcal{R}_{\mathcal{F}}) \neq \theta$.
Measured dissipation. The Branch II evidence requires $\lim\langle\Theta_E\rangle > 0$ to be estimated from measured dissipation, never from $\mathfrak{K}\nabla\mathcal{F}$. Integrating $-J_E^\mu\nabla_\mu\mathcal{F}$ by parts and using the $O_4$ balance law gives the identity $$p_0\dot{W}_{\mathrm{diss}} = \oint_{\partial\mathcal{M}}\mathcal{F}\,(-J_W^\mu)\,d\Sigma_\mu - \int_{\mathcal{M}}\mathcal{F}\,\mathcal{R}_{\mathcal{F}}\sqrt{\vert g\vert}\,d^nx - \frac{1}{2}\frac{d}{d\tau}\int_{\mathcal{M}}\mathcal{F}^2\sqrt{\vert g\vert}\,d^nx - \frac{1}{2}\int_{\mathcal{M}}\mathcal{F}^2\theta\sqrt{\vert g\vert}\,d^nx$$ Every Tier 2 datum is $\mathcal{F}_{\mathrm{ref}} = 0$ on the outflow boundary, so the boundary term reduces to the supply face. Each term is evaluated from measured $\mathcal{F}$, measured supply flux, and measured or independently declared consumption; none uses the declared $\mathfrak{K}$. The supply-face record used here comes from instruments and a surface different from those of the $O_1$ flux-prediction test, so that the evidence is independent of that test. Because the identity uses the balance law, this estimate is conditional on it; each §1 also names the directly measured physical dissipation whose sign it must share. Only signs and $s_0$-independent rates of $\Theta_E$ are tested.
Gate sector ($O_4$). $\rho_{\mathrm{total}}(\tau)$ is a declared positive trace-class state on $L^2(\mathcal{M}, d\mathrm{vol}_g)$, and $Q$, $P$, and the $O_6$ marginals $Q_i$ and joints $Q_{ij}$ are densities of this declared state sector. No Tier 2 aperture identifies them with $\mathcal{F}$ or with any measured field in this epoch, and none makes an empirical claim about the gate (empirical_validation_protocol.md §3.3). What each domain's physics supplies is, first, the spectral setting: the Neumann eigenmodes of $-\nabla^2_g$ on $\mathcal{M}$, whose eigenvalues are fixed on rigid domains, drift by a fraction of the order of the strain on deforming ones, and converge as $g \to G_\infty$. Second, it supplies a physical low-pass behaviour of $\mathcal{F}$: under the balance law, fine spatial structure in $\mathcal{F}$ relaxes faster than coarse structure. That behaviour is mapped via Spine $\mathcal{T}$ to the gate and is not itself the gate.
Cover, overlaps, and coupling ($O_5$, $O_6$).
The cover $\{\mathcal{U}_i\}$ consists of $\tau$-independent material subregions of $\mathcal{M}$ (layers or zones) with Lipschitz boundaries and overlaps $\mathcal{U}_{ij}$ that are convex at $\tau_0$. Each $g^{(i)}$ is the restriction of the one physical metric, so $g^{(i)} = g^{(j)}$ on every overlap, the geometric mean returns the common metric ($g \# g = g$), and the gluing of $O_5$ Eq. [3] holds identically.
The anisotropy of a medium lives in $\mathfrak{K}$, never in $g$. Effective-medium averages of conductivities, including geometric-mean averages, are statements about $\mathfrak{K}$ and are not mapped to $O_5$ Eq. [1].
For each pair, the partner fields are the rate of change of the free-energy flux read in the two orientations, $\eta_{i\to j}^\lambda = \partial_\tau J_W^\lambda$ and $\eta_{j\to i}^\lambda = -\partial_\tau J_W^\lambda$, defined on all of $\mathcal{M}$, so $\eta_{[i\to j]}^\lambda = \partial_\tau J_W^\lambda$.
Where the supply boundary holds a value of $\mathcal{F}$ (2.1, 2.2, 2.4), this field is not one-signed. After the step increase, the flux near the supply boundary decays from an initial overshoot, so there $\partial_\tau J_W$ points toward the supply; farther downstream the flux rises toward its new steady value, so there $\partial_\tau J_W$ points away from the supply. Between the two lies the zero set of $\partial_\tau J_W$. It moves during the window and, in the late single-mode regime, settles on a fixed surface (for a column with fixed values at both ends, its mid-plane). On that set $\eta_{[i\to j]} = 0$ and $J_{[i\to j]} = 0$, so a construction that included it would itself decide the non-degeneracy hypothesis there. Where the supply is a prescribed flux (2.3), $\partial_\tau J_W$ is one-signed and vanishes only on the supply face (see 2.3 $O_6$).
Each aperture therefore declares overlaps $\bar{\mathcal{U}}_{ij}$ that the zero set, predicted before the window from the independently estimated parameters, does not reach during the window. The zero crossing travels during the early window (in the semi-infinite early-time regime roughly as $\sqrt{2D(\tau - \tau_0)}$, possibly overshooting its late-time position), so this choice rests on the predicted solution, not on the late-time zero set alone. Before the transient arrives, $\partial_\tau J_W$ on a downstream overlap is non-zero but exponentially small. So the check is made against a pre-declared detection threshold: each overlap's window starts once the predicted $\vert\partial_\tau J_W\vert$ exceeds that threshold on $\bar{\mathcal{U}}_{ij}$, and the data must then show $\vert\partial_\tau J_W\vert$ above it there (a declared, checkable precondition). On such an overlap $\eta_{[i\to j]}$ keeps one direction throughout the window, which each aperture states.
The precision potentials $\Phi^{(i)}$ are declared before data collection as quadratic potentials with constant Hessians. Their trace-free parts $\hat{\Pi}^{(i)}$ cannot be positive definite, because their eigenvalues sum to zero, so $J_{[i\to j]} = \frac{1}{2}(\hat{\Pi}^{(i)} + \hat{\Pi}^{(j)})\eta_{[i\to j]}$ carries no sign relation to $\eta_{[i\to j]}$. Tier 2 declares $\frac{1}{2}(\hat{\Pi}^{(i)} + \hat{\Pi}^{(j)})$ non-singular, so that $J_{[i\to j]} \neq 0$ wherever $\eta_{[i\to j]} \neq 0$. What is claimed for $J_{[i\to j]}$ is its oddness under $i \leftrightarrow j$ and its non-vanishing on the declared overlaps, not a direction. Tier 2 also declares the two trace-free Hessians distinct, $\hat{\Pi}^{(i)} \neq \hat{\Pi}^{(j)}$. With these antisymmetric partner fields, $J_{i\to j} + J_{j\to i} = (\hat{\Pi}^{(j)} - \hat{\Pi}^{(i)})\eta_{[i\to j]}$, so equal Hessians would remove the gradient term of r7 $O_6$ Eq. [3] and leave $M^{(ij)}$ equal to $\gamma_M$ times the trace-free part of $J_{[i\to j]} \otimes J_{[i\to j]}$, which is non-zero wherever $J_{[i\to j]} \neq 0$: the hypothesis would then hold by construction. With distinct Hessians the gradient term depends on $\nabla\eta_{[i\to j]}$ and can cancel the quadratic term (r10 Decision 4). With these declarations the construction neither forces nor forbids $M^{(ij)} \neq 0$, which remains a standing hypothesis.
On a connected diffusive domain the transient reaches every overlap for $\tau > \tau_0$, so every edge of $\mathcal{K}_0$ lies in $\mathcal{K}$ and $\phi_{\mathrm{perc}} = 1 > \phi_c$; empirically, an edge is counted when the transient flux across its overlap exceeds a pre-declared detection threshold. The percolation condition therefore carries no separate empirical content on these domains.
$M^{(ij)} \neq 0$ and $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$ are standing hypotheses of r7 $O_6$, not consequences of diffusion. In Branch II the transient decays as the domain settles, so each coupling-tensor boundary flux decays and is integrable on $[\tau_0, \infty)$; no sustained coupling is asserted.
Every Tier 2 domain is declared simply connected, so $H^1_{\mathrm{dR}}(\mathcal{M}) = 0$ and $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$. A domain with a non-contractible loop, such as an annulus around an impermeable or opaque core, lies outside the declarations. It would carry the harmonic Neumann field of its loop, a property of its topology that is not identified with any transport current.
Gauge sector and hysteresis ($O_7$, $O_0$). Tier 2 declares the trivial gauge sector: $A_\mu \equiv 0$ and a uniform unit section $\hat{u}$. Then $F_{\mu\nu}^{\mathrm{YM}} \equiv 0$, $[F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] = 0$ identically, and $\hat{\mathcal{E}}_\mu = (\partial_\mu\chi_E)\hat{u}$, which tends to $0$ in $C^1$ when $\sigma \to 0$ and $\theta \to 0$ in $C^2$ with $\liminf\langle\Theta_E\rangle > 0$. The gauge limit therefore carries no content beyond the shear and expansion limits, and $O_7$ gauge transport is untested in this epoch. The hysteresis record obeys the r7 law $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ from a declared initial record $\mathcal{H}_0$, chosen convex because $O_7$ Eq. [2] requires $\nabla_\alpha\nabla_\beta\mathcal{H} \succ 0$ at every $\tau$. $\mathcal{H}$ then evolves through a state sector that is declared but not specified, so the declarations do not establish convexity at later $\tau$. Tier 2 therefore states it as a declared condition, with a sufficient form: the largest eigenvalue magnitude of $\nabla_\alpha\nabla_\beta(\mathcal{H} - \mathcal{H}_0)$ (with indices raised by $g$) stays below the smallest eigenvalue of $\nabla_\alpha\nabla_\beta\mathcal{H}_0$ on $\mathcal{M}$ for all $\tau \ge \tau_0$. Both Hessians are covariant Hessians of $g(\tau)$, evaluated at each $\tau$, which matters on the deforming domains 2.2 and 2.4. Then $\nabla_\alpha\nabla_\beta\mathcal{H} \succ 0$ persists. The alternative, $\nabla q \equiv 0$, would keep $\mathcal{H}$ static and convex, but would make the $O_\Omega$ generator condition hold trivially, so it is not adopted. Physical memory variables of each medium are mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and are not identified with it. $\hat{\rho}_g \to \frac{1}{n}\delta^\mu_\nu$ follows from $\hat{h} \to \hat{h}_\infty$, and the alignment $D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0$ then requires the Hessian of the record to become isotropic: an asymptotic condition that each aperture states and does not test.
Finite windows and coverage. No finite window exhibits a $\tau \to \infty$ limit. Each aperture's empirical claims are the within-window tests of aperture_generation_guidance.md §5.5 that its domain supports, and each $O_\Omega$ block lists them. In this epoch Apertures 2.1 and 2.3 are exercised by no test, and Apertures 2.2 and 2.4 only by the flux-prediction family (empirical_validation_protocol.md §3.3); everything else is stated as untested, not as confirmed.

Aperture 2.1: Condensed-Matter & Phase Transition Physics ($\mathcal{E}_{\mathrm{condensed}}$)¶

1. Epistemological & Physical Scope¶

Aperture 2.1 resolves how microscopic intermolecular interactions (hydrogen-bond and dipolar coordination networks, lattice bonding, crystal nucleation) generate macroscopic thermodynamic extensivity, cohesion, and transitions between fluid, amorphous, and crystalline states, and how a supersaturated phase relaxes by the diffusion-limited growth of a new phase. Lexicon is strictly restricted to statistical mechanics, condensed-matter physics, and chemical thermodynamics.

Declaration (aperture_generation_guidance.md §2.3; Branch II).

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A condensed sample in which a dissolved component diffuses through a single host phase: a single-crystal solid solution, or a gel or quiescent solution column in which convection is suppressed. Smooth interior growth zones, fixed within the window, hold a dispersed second phase that grows by consuming the solute. $\partial\mathcal{M}$ consists of a supply face in contact with a reservoir held at a declared concentration $c_{\mathrm{s}} > c_{\mathrm{sat}}$, an outflow face held at saturation, and no-flux lateral faces. All supply crosses the supply face; there are no interior sources.
Window. A relaxation window (§0.1 item 1), opened by a declared step increase of $c_{\mathrm{s}}$. It ends before the volume fraction of the second phase changes by more than a declared tolerance, so that $\mathcal{M}$ and its growth zones are fixed.
Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart.
Cone speed. $c_*$ is the longitudinal sound speed of the host. Diffusive solute speeds (of order $D/\ell$) and growth-front speeds are $\ll c_*$.
Free energy $\mathcal{F}$ and datum. $\mathcal{F} = \epsilon_{\mathrm{s}}(c - c_{\mathrm{sat}})$, with $c$ the solute concentration (mol m⁻³) and $\epsilon_{\mathrm{s}} > 0$ a declared constant supersaturation yield (J mol⁻¹, the free energy per mole of excess solute at a declared reference supersaturation). The datum is saturation, $\mathcal{F}_{\mathrm{ref}} = 0$. This is the declared linear form of aperture_generation_guidance.md §2.3 item 3: the free energy of a supersaturated solution is not linear in $c - c_{\mathrm{sat}}$, so the declaration is used only where $(c - c_{\mathrm{sat}})/c_{\mathrm{sat}}$ stays within a declared range. The relaxation time $\ell^2/D$ is finite and resolvable.
Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D^{\alpha\beta}$, the solute diffusivity tensor, parametric along the host's crystal axes: one principal value for a cubic host, a gel, or a solution (the default $D_{\mathcal{F}}g^{\alpha\beta}$), two for a uniaxial host, three for an orthorhombic host. Because $\mathcal{F}$ is linear in $c$, $J_W = \epsilon_{\mathrm{s}}\,j$, with $j$ the molar solute flux; $\epsilon_{\mathrm{s}}$ is the declared conversion factor.
Sink $\mathcal{R}_{\mathcal{F}}$. $\mathcal{R}_{\mathcal{F}} = k_{\mathrm{g}}(\mathbf{x})(\mathcal{F} - \mathcal{F}_{\mathrm{ref}}) \ge 0$: growth kinetics first order in supersaturation, with $k_{\mathrm{g}}$ a smooth, non-negative profile supported on the growth zones and fixed by a finite number of declared parameters (amplitude, centre, and width for each zone). A sharp growth front is represented by such a smooth zone.
Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 4).
Branch II evidence (closure (v)). The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau$ come from dilatometric or interferometric measurement of the sample volume, which must stay within the rigid tolerance. $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 6): the supply reservoir's depletion record and $\mathcal{F}$ at the supply face give the boundary term, and growth rates measured in the growth zones, weighted by measured local $\mathcal{F}$, give the consumption term. The physical dissipation of diffusion, the entropy produced as solute flows down its chemical-potential gradient, must share its sign.
Admissible domains (declared, checkable preconditions):
supersaturation throughout $\mathcal{M}$ ($c \ge c_{\mathrm{sat}}$, so $\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$), with no nucleation of new growth zones and no dissolution during the window;
$\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$, so no stagnant region lies behind a growth zone;
purely diffusive transport: no convection in a gel or solution, and no fast grain-boundary paths in the host;
isothermal conditions within a declared tolerance, so that $c_{\mathrm{sat}}$ is constant;
volumetric strain of the sample within the rigid tolerance.
Titled phenomena outside the domain.
Nucleation lies before the window or outside $\mathcal{M}$, and enters through the initial growth-zone configuration and $\mathcal{H}_0$.
Phase transitions inside $\mathcal{M}$ enter only through the declared growth zones and $\mathcal{R}_{\mathcal{F}}$. The latent heat they release is conducted away (precondition 4) and enters as context. Melting and solidification fronts lie on $\partial\mathcal{M}$ or outside the window.
Intermolecular networks, cohesion, and surface tension set the material parameters ($D^{\alpha\beta}$, $c_{\mathrm{sat}}$, $k_{\mathrm{g}}$), and the host's crystal symmetry sets the principal axes of $\mathfrak{K}$.

2. Coordinate-by-Coordinate Mapping ($O_0 \to O_\Omega$)¶

$O_0$ — Exogenous Flux & Emergent Foliation Time¶

Spine Equations: $\theta \equiv -N K \overset{N=1,\,\beta^\mu=0}{=} \frac{1}{2}g^{\mu\nu}\frac{\partial g_{\mu\nu}}{\partial\tau} = \frac{1}{\sqrt{\vert g\vert}}\frac{\partial\sqrt{\vert g\vert}}{\partial\tau} = \nabla_\mu J_{\mathrm{geom}}^\mu \ge 0$; $\Theta_E \equiv \frac{\dot{s}_{\mathrm{prod}}}{s_0} = -\frac{\beta_{\mathrm{th}}}{s_0}J_E^\mu\nabla_\mu\mathcal{F}$; $\Theta_{\mathrm{tot}} \equiv \sqrt{\theta^2 + \Theta_E^2}$, $\tau_\Theta(\tau) \equiv \int_{\tau_0}^{\tau}\langle\Theta_{\mathrm{tot}}\rangle_{\mathcal{M}}\,d\tau'$; $\frac{dV_{\mathcal{M}}}{d\tau} = \int_{\mathcal{M}}\theta\sqrt{\vert g\vert}\,d^nx = \oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$, with $\lim_{\tau\to\infty}V_{\mathcal{M}}(\tau) = V_\infty < \infty$ (Branch II); $\frac{\partial\mathcal{H}}{\partial\tau} \equiv J_E^\mu\nabla_\mu q$, $q(\mathbf{x}, \tau) \equiv -\ln(v_0Q(\mathbf{x}, \tau))$.
Physical Instantiation: The sample is rigid, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$ at every $\tau$. The expansion current enters only through its vanishing divergence and boundary flux, and the solute supply is never mapped to it (§0.1 item 5). Openness is carried by throughput: diffusion of supersaturated solute from the supply face toward the growth zones and the outflow face produces entropy at every point, so $\Theta_E > 0$ (by $O_1$) and $\Theta_{\mathrm{tot}} = \Theta_E$. $\tau_\Theta$ counts entropy production per $s_0$, while $\tau$ remains laboratory proper time. Along the throughput current $J_E^\mu = J_W^\mu/p_0$ the record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ (§0.1 item 9). The sample's physical memory of its growth history (the size distribution and arrangement of the second phase, growth banding) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it.

$O_1$ — Non-Equilibrium Thermodynamic Openness¶

Spine Equations: $\dot{W}_{\mathrm{ext}} = p_0\dot{V}_{\mathcal{M}} = p_0\oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$; $\dot{W}_{\mathrm{diss}} \equiv \int_{\mathcal{M}}(-J_E^\mu\nabla_\mu\mathcal{F})\sqrt{\vert g\vert}\,d^nx > 0$; $\dot{W}_{\mathrm{tot}} \equiv \dot{W}_{\mathrm{ext}} + \dot{W}_{\mathrm{diss}} > 0$; $\nabla_\mu\mathcal{F} \neq 0$; $J_E^\mu \equiv \frac{J_W^\mu}{p_0} = -\frac{1}{p_0}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$ (Branch II, fixed); $J_E^\mu\nabla_\mu\mathcal{F} < 0$; $\dot{s}_{\mathrm{prod}} = -\beta_{\mathrm{th}}J_E^\mu\nabla_\mu\mathcal{F} > 0$.
Physical Instantiation: The maintained concentration difference between the supply reservoir and the growth zones and outflow face holds the sample far from equilibrium. With $J_E^\mu = -p_0^{-1}D^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} = -p_0^{-1}\nabla_\mu\mathcal{F}\,D^{\mu\nu}\nabla_\nu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 2), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. The physical solute flux runs down the supersaturation gradient, from the supply face toward the growth zones and the outflow face, in the direction of $J_E^\mu$. On the rigid sample $\dot{W}_{\mathrm{ext}} = 0$, so $\dot{W}_{\mathrm{tot}} = \dot{W}_{\mathrm{diss}} > 0$. The empirical content of $O_1$ is the flux-prediction test (aperture_generation_guidance.md §2.3, §5.5 item 6). The declared $D^{\alpha\beta}$ has its parameters estimated on the same sample per closure (ii), by a different modality (for example an isotope-tracer diffusion measurement) or in a declared perturbation regime run before the window. It must predict the flux $J_W^{\mathrm{meas}} = \epsilon_{\mathrm{s}}\,j^{\mathrm{meas}}$ measured across a declared surface $S$ (for example the outflow face, from the solute uptake of the reservoir in contact with it), with $\nabla\mathcal{F}$ from in-situ concentration profiling that uses neither $J_W^{\mathrm{meas}}$ nor the declared $D^{\alpha\beta}$ or $k_{\mathrm{g}}$.

$O_2$ — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

Spine Equations: $g_{\mu\nu}(\tau, \mathbf{x}) = \Omega^2(\tau, \mathbf{x})\hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}(\tau, \mathbf{x}))^\alpha_{\;\nu}$ with $\det\hat{g} = 1$, $\mathrm{Tr}(\hat{h}) = 0 \implies \det g = \Omega^{2n}$, $\frac{\partial\ln\Omega}{\partial\tau} = \frac{1}{n}\theta$; $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$; $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$, $R = \kappa_G\nabla_\alpha\xi^\alpha$; $\frac{2(n-1)}{n}\nabla^2\theta + \frac{2R}{n}\theta = -\kappa_G\frac{\partial(\nabla_\alpha\xi^\alpha)}{\partial\tau} + 2\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} - 2\sigma^{\alpha\beta}R_{\alpha\beta}$; $\left.\left(\frac{dv^\mu}{d\tau} + \Gamma^\mu_{\rho\epsilon}v^\rho v^\epsilon + g^{\mu\lambda}\left(\frac{\partial g_{\lambda\nu}}{\partial\tau}\right)v^\nu - \frac{1}{2}\left(\frac{\partial g_{\rho\epsilon}}{\partial\tau}v^\rho v^\epsilon\right)v^\mu\right)\right\vert_{N=1,\,\beta^\mu=0} = f^\mu_{\mathrm{drive}}$, $f^\mu_{\mathrm{drive}} = (1-u^2)\left(a^\mu_{\mathrm{d}} - c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu\right)$, $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2 < 1$, with the default $a^\mu_{\mathrm{d}} = 0$.
Physical Instantiation: On the rigid sample the unimodular Weyl split is trivial: $\Omega \equiv 1$, $\hat{h} \equiv 0$, $\det g = 1$, and $\partial_\tau\ln\Omega = \theta/3 = 0$. The split isolates the volume element only algebraically; $\theta$ remains coupled to $\xi_\alpha = \hat{\Pi}_{\alpha\lambda}\eta^\lambda$ and to $\sigma_{\alpha\beta}$ through the traced elliptic $\tau$-compatibility equation (r10 Decision 2), whose two sides vanish identically on this domain ($G_{\mu\nu} \equiv 0$, $\xi \equiv 0$, $\sigma \equiv 0$; §0.1 items 2–3). The host's crystal lattice does not enter $g$: its symmetry enters through the principal axes of $D^{\alpha\beta}$, and lattice strain around the second phase is context. $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$ is the Hessian of a declared precision potential and enters only the coupling sector ($O_6$). Eq. [3] is instantiated with the default drive, and on the static flat metric its solutions are straight lines traversed at constant speed. No tracer is mapped to it: solute transport is diffusive and is described by the $O_4$ balance law, and growth fronts advance by interface kinetics.

$O_3$ — Shear Damping & Hysteresis Consolidation¶

Spine Equations: $\left.\frac{\partial g_{\mu\nu}}{\partial\tau}\right\vert_{N=1,\,\beta^\mu=0} = \frac{2}{n}\theta g_{\mu\nu} + 2\sigma_{\mu\nu}$; $\nabla_\tau\sigma^\mu_{\;\nu} \equiv \partial_\tau\sigma^\mu_{\;\nu} = -\gamma_\sigma\sigma^\mu_{\;\nu}$ ($\gamma_\sigma = \mathrm{const} > 0$, $\sigma^\mu_{\;\mu} = 0$) $\implies \lim_{\tau\to\infty}\Vert\sigma^\mu_{\;\nu}\Vert_{C^2(\mathcal{M})} = 0$; $\hat{h}(\tau, \mathbf{x}) \xrightarrow{C^2(\mathcal{M})} \hat{h}_\infty(\mathbf{x})$, $(\hat{g}_\infty(\mathbf{x}))_{\mu\nu} \equiv \hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}_\infty(\mathbf{x}))^\alpha_{\;\nu}$ ($\det\hat{g}_\infty = \det\hat{g} = 1$); $V_{\mathrm{local}}(\tau) \equiv \int_{\mathcal{U}_i}\mathcal{H}(\mathbf{x}, \tau)\sqrt{\vert g\vert}\,d^nx$.
Physical Instantiation: On the rigid sample $\sigma^\mu_{\;\nu} \equiv 0$, so the mixed-form shear law holds trivially for any $\gamma_\sigma > 0$, $\hat{h} = \hat{h}_\infty = 0$, and $\hat{g}_\infty = \hat{g} = \delta$. This aperture supports no shear-relaxation test (§0.1 item 2). The relaxation of $\mathcal{F}$ toward its new steady profile is a property of the $O_4$ balance law and is not mapped to $\sigma$. Shear relaxation of the host itself (for example Maxwell relaxation of a glass-forming melt) would require a deforming domain and lies outside this declaration. $V_{\mathrm{local}}$ is the hysteresis content of a subregion $\mathcal{U}_i$; the growth record of the second phase within $\mathcal{U}_i$ is mapped to it via Spine $\mathcal{T}$.

$O_4$ — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

Spine Equations: $\frac{\partial\mathcal{F}}{\partial\tau} + \mathcal{F}\theta = -\nabla_\alpha J_W^\alpha - \mathcal{R}_{\mathcal{F}}$, $J_W^\alpha \equiv -\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}$, $\mathfrak{K}^{\alpha\beta} = \mathfrak{K}^{\beta\alpha} \succ 0$, $\mathcal{R}_{\mathcal{F}} \ge 0$; $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}} \equiv \mathbf{1}_{(-\nabla^2 \le \Lambda_{\mathrm{gate}}^2)}$; $Q(\mathbf{x}, \tau) = \langle\mathbf{x}\vert\rho_{\mathrm{local}}(\tau)\vert\mathbf{x}\rangle > 0$; $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P) > 0$; $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$.
Physical Instantiation: With $\theta \equiv 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha(D^{\alpha\beta}\nabla_\beta\mathcal{F}) - k_{\mathrm{g}}(\mathcal{F} - \mathcal{F}_{\mathrm{ref}})$: the linearized reaction–diffusion law of diffusion-limited growth from a supersaturated phase, first order in supersaturation. It follows its boundary data with the finite relaxation time $\ell^2/D$, so the field is stored, not quasi-static. Every departure from the default ($D_{\mathcal{F}}g^{\alpha\beta}$, $\mathcal{R}_{\mathcal{F}} = 0$) satisfies the binding closure. The anisotropic tensor and the growth law are named laws with finite pre-declared parameters, estimated independently and tested against pre-declared tolerances, and each is rejected if the default fits at least as well (closure (iv)). The gate sector follows §0.1 item 7: the Neumann eigenmodes of $-\nabla^2$ on the rigid sample are fixed, so the gate does not drift, and the faster decay of fine spatial structure in $\mathcal{F}$ is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The stress-conservation equivalence holds identically (§0.1 item 3); elastic stresses in the lattice and around inclusions are not $T_{\mu\nu}^{\mathrm{mat}}$ and enter as context.

$O_5$ — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

Spine Equations: $g_{\mu\nu}^{(ij)} \equiv (g^{(i)} \# g^{(j)})_{\mu\nu}$; $\mathcal{U}_{ij}$ geodesically convex for $g^{(ij)}(\tau) \; \forall \tau \ge \tau_0$; $\phi_{\mathrm{perc}}(\mathcal{K}) = \vert\mathcal{K}\vert/\vert\mathcal{K}_0\vert > \phi_c$; $\Delta_{\mathrm{Hodge}}\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} = 0$; $\partial_\tau\sqrt{\vert g^{(i)}\vert} \ge 0$; $g_{\mu\nu}^{(i)} \xrightarrow{C^2(\mathcal{U}_i)} G_{\infty,\mu\nu}^{(i)} \succ 0$ (standing hypothesis with $H_0 = 0$).
Physical Instantiation: The cover consists of $\tau$-independent subregions of the sample, for example each growth zone with its surroundings and the host regions between them, with convex overlaps $\mathcal{U}_{ij}$. Each $g^{(i)}$ is the restriction of the rigid metric, so the gluing holds identically, each overlap stays convex for every $\tau$, $\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$, and $g^{(i)} = G_\infty^{(i)}$ at every $\tau$ ($H_0 = 0$; §0.1 item 8). Every overlap is crossed by the transient that follows the declared step, so $\phi_{\mathrm{perc}} = 1 > \phi_c$ (§0.1 item 8). Fast-diffusion networks in a polycrystalline host would enter through $\mathfrak{K}$ and are excluded by precondition 3. The declared sample is simply connected, so $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$ (§0.1 item 8).

$O_6$ — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

Spine Equations: $(J_{[i\to j]})_\nu = -(J_{[j\to i]})_\nu$; $M^{(ij)}(\mathbf{x}) \neq 0 \; \forall \mathbf{x} \in \bar{\mathcal{U}}_{ij}$ (hypothesis) $\wedge \; \mathrm{Tr}\,M^{(ij)} = 0 \implies m_+ > 0 > m_-$; $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) = k_{\mathcal{I}}\oint_{\partial\mathcal{U}_{ij}} S_{\mu\nu}^{(ij)}\hat{n}^\mu\hat{n}^\nu\,dA > 0$ (hypothesis, $\langle i,j\rangle \in \mathcal{K}$), with this boundary flux integrable on $[\tau_0, \infty)$ (Branch II).
Physical Instantiation: Take a pair of adjacent subregions, for example a growth zone and the host region between it and the supply face. The partner fields are the rate of change of the solute free-energy flux in the two orientations (§0.1 item 8), so $\eta_{[i\to j]} = \partial_\tau J_W$. On a declared overlap that the predicted zero set of $\partial_\tau J_W$ does not reach during the window (§0.1 item 8), $\eta_{[i\to j]}$ keeps one direction: toward the supply face where the flux is decaying from its overshoot, and away from it where the flux is rising. The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is its image under the declared non-singular precision map and is non-zero there. Under the non-degeneracy hypothesis $M^{(ij)} \neq 0$, with $M^{(ij)}$ built per r7 $O_6$ Eq. [3] from these fields and the declared precision potentials, $\mathrm{Tr}\,M^{(ij)} = 0$ gives $m_+ > 0 > m_-$, with $m_+ \neq -m_-$ in general. Under the flux-positivity hypothesis the coupling transmits correlation between the subsystem marginals of the declared state sector ($\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$). Both are standing hypotheses, not consequences of diffusion. In Branch II the transient decays as the sample approaches its new steady profile, so the coupling flux decays and is integrable on $[\tau_0, \infty)$. With $\sigma \equiv 0$ the overlaps keep their declared convex shape; the r10 §5.12 constraint $a_\infty = 2s_\sigma/\gamma_\sigma < a_c(E)$, a design constraint of that witness, has no counterpart here.

$O_7$ — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

Spine Equations: $D_\mu = \partial_\mu - i g_{\mathrm{YM}}\,\mathrm{ad}_{A_\mu}$; $\Vert\hat{\mathcal{E}}_\mu\Vert_{C^1(\mathcal{M})} \to 0 \implies [F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] \to 0$; $\chi_E \equiv 1 + \frac{n\,\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu}}{\theta^2 + \langle\Theta_E\rangle_{\mathcal{M}}^2} \to 1$, $\hat{\chi} \equiv \chi_E\hat{u}$ (Branch II); $D_{\mathrm{KL}}(\hat{\rho}_g(\tau) \parallel \hat{\rho}_{\mathcal{H}}(\tau)) \to 0$.
Physical Instantiation: The gauge sector is the trivial declaration of §0.1 item 9, so $[F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] = 0$ identically. With $\sigma \equiv 0$, $\chi_E \equiv 1$ and $\hat{\mathcal{E}}_\mu \equiv 0$ at every $\tau$, while $\chi$ itself is undefined because $\theta = 0$. $\hat{\rho}_g = \frac{1}{n}\delta^\mu_\nu$ at every $\tau$, so the alignment $D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0$ holds where the Hessian of the declared convex record becomes isotropic, an asymptotic condition stated and not tested. The $O_7$ Eq. [3] flux identity holds with $J_{\mathrm{geom}}$ and $u_{\mathrm{exp}}$ both divergence-free, $J_E$ is bounded in $C^1$ under the held supply, and $\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0$. This aperture claims no orientational ordering, grain alignment, or annealing within the window.

$O_\Omega$ — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Spine Equations: Branch II ($H_0 = 0$): $\theta \xrightarrow[\tau\to\infty]{C^2(\mathcal{M})} 0$ with $\int_{\tau_0}^{\infty}\langle\theta\rangle_{\mathcal{M}}\,d\tau < \infty$ and $\langle\Theta_E\rangle_{\mathcal{M}} \to \Theta_\infty > 0$; $\sigma^\mu_{\;\nu} \to 0$; $\partial_\tau g_{\mu\nu} \to 0$; $V_{\mathcal{M}} \to V_\infty$ and $\mathcal{I}(\tau) \to s_0V_\infty$; $\left.\hat{\mathbb{L}}_{\mathrm{stat}}(\mathbf{x})\Psi[g_{\mu\nu}, \mathcal{H}]\right\vert_{\mathcal{H}=\mathcal{H}_{\mathrm{sol}}[g]} = (J_E^\lambda\nabla_\lambda q)\frac{\delta\Psi}{\delta\mathcal{H}(\mathbf{x})} \xrightarrow[\tau\to\infty]{} 0 \iff J_E^\lambda\nabla_\lambda q \to 0 \iff \frac{\partial\mathcal{H}}{\partial\tau} \to 0$ (with $\vert\delta\Psi/\delta\mathcal{H}\vert$ bounded and bounded away from $0$ uniformly as $\tau \to \infty$).
Physical Instantiation: Under maintained supply the sample approaches a steady supersaturation profile: solute enters across the supply face and is consumed by growth and outflow. This is a bounded non-equilibrium steady state. $\theta \equiv 0$ (so $\int\langle\theta\rangle\,d\tau = 0$ and $V_{\mathcal{M}} = V_\infty$), $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while supply is maintained, $\sigma \equiv 0$, $\partial_\tau g \equiv 0$, and the transient coupling decays, so $\mathcal{I}$ converges to $s_0V_\infty$. The generator condition requires $J_E^\lambda\nabla_\lambda q \to 0$, that is, the gated density of the declared state sector must become uniform along the steady throughput current; this is an asymptotic hypothesis of the declared branch, which no finite window tests. The state is neither static equilibrium, which would require $c = c_{\mathrm{sat}}$ throughout and no growth, nor a periodic return. The limit belongs to the declared idealization of fixed growth zones under maintained supply; the physical sample leaves that idealization when the second-phase fraction exceeds its tolerance, which bounds the usable window. Within windows the supported tests (aperture_generation_guidance.md §5.5) are the non-degeneracy check (4), the signs of $\Theta_E$, $\dot{s}_{\mathrm{prod}}$, and $\dot{W}_{\mathrm{tot}}$ (5), the flux prediction (6), and the Branch II evidence (7); tests (1)–(3) hold trivially on a rigid domain with $\sigma \equiv 0$.

Aperture 2.2: Continuum Fluid & Wave Mechanics ($\mathcal{E}_{\mathrm{continuum}}$)¶

1. Epistemological & Physical Scope¶

Aperture 2.2 maps the macroscopic mechanics of deformable fluid continua (viscous flow and boundary layers, vorticity transport, dispersive surface and internal waves, and diffraction around obstacles) onto Spine $\mathcal{T}$. Its empirical domain lies in the regime where these mechanics reduce to storage and diffusion: slow viscous flow through a deformable porous medium. Lexicon is strictly restricted to continuum mechanics, hydrodynamics, poromechanics, and linear and nonlinear wave theory.

Declaration (aperture_generation_guidance.md §2.3; Branch II).

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A saturated, deformable porous medium with specific storage $S_s > 0$, occupying a laterally confined column of length $L$ along the axis $x^3$. $\mathcal{M}$ is the material region of the solid skeleton (material chart). $\partial\mathcal{M}$ consists of an inlet boundary (supply) at a declared head $h_{\mathrm{in}}$, an outlet boundary at the head $h_{\mathrm{ref}}$, and impermeable lateral walls.
Window. A relaxation window (§0.1 item 1), opened by a declared step increase of $h_{\mathrm{in}}$. The head then rises monotonically at every point, so pore pressure rises, effective stress falls, and the skeleton swells within its elastic, recoverable range.
Metric. Flat, with small-strain deformation (§0.1 item 2). Under lateral confinement the strain is uniaxial, so in the material chart $g_{\mu\nu} = \mathrm{diag}(1, 1, (1+e)^2)$, with $e(x^3, \tau)$ the axial strain. Then $\theta = \partial_\tau\ln(1+e) \ge 0$ and $\sigma^\mu_{\;\nu} = \theta\,\mathrm{diag}(-\tfrac{1}{3}, -\tfrac{1}{3}, \tfrac{2}{3})$.
Cone speed. $c_*$ is the compressional wave speed of the saturated medium. Seepage velocities and skeleton speeds are $\ll c_*$.
Free energy $\mathcal{F}$ and datum. $\mathcal{F} = \rho g(h - h_{\mathrm{ref}})$, the head energy density (J m⁻³), with datum the outlet head ($\mathcal{F}_{\mathrm{ref}} = 0$). Equivalently, $\mathcal{F}$ is the stored water volume per unit volume, $S_s(h - h_{\mathrm{ref}})$, times the declared conversion factor $\rho g/S_s$. Storage makes the relaxation time $L^2/D$ finite; the incompressible creeping limit $S_s \to 0$ is quasi-static and is excluded.
Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = K_h^{\alpha\beta}/S_s$, the hydraulic diffusivity tensor, with parametric anisotropy (declared principal values and axes); isotropic $K_h$ is the default. With $q$ the Darcy flux, $J_W = (\rho g/S_s)\,q$.
Sink $\mathcal{R}_{\mathcal{F}}$. The default $0$, or first-order leakage $k_L(\mathcal{F} - \mathcal{F}_{\mathrm{ref}})$ to a drainage system at the outlet head, distributed through $\mathcal{M}$, with $k_L$ a declared rate constant. Leakage through a bounding wall is a boundary condition, not a sink.
Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 4).
Branch II evidence (closure (v)). The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau$ (which equals $\ln(V_{\mathcal{M}}(\tau_{\mathrm{obs}})/V_{\mathcal{M}}(\tau_0))$) come from the column's axial-strain record, bounded by the final strain. $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 6). Its leading term is the hydraulic power delivered at the inlet, $(\rho g/S_s)\,\rho g(h_{\mathrm{in}} - h_{\mathrm{ref}})\,Q_{\mathrm{in}}$, with $Q_{\mathrm{in}}$ from inlet flow metering and $h_{\mathrm{in}}$ from an inlet transducer, not from the outlet volumetry and piezometers of the $O_1$ test. The physical viscous dissipation of the through-flow, $\rho g K_h\vert\nabla h\vert^2$ per unit volume for isotropic $K_h$, must share its sign.
Admissible domains (declared, checkable preconditions):
Darcy regime: pore Reynolds number $\ll 1$, so inertia and recirculation are negligible;
$h \ge h_{\mathrm{ref}}$ throughout (so $\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$) and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$;
elastic, recoverable skeleton deformation with volumetric strain $\varepsilon_v \ll 1$ and $\theta \ge 0$ throughout the window, verified from the strain record. The linear storage equation and the r7 balance law then differ by terms of relative order $\varepsilon_v$ (the $\mathcal{F}\theta$ term among them), and the pre-declared flux tolerance must exceed them. Windows that compact the skeleton are excluded;
no change of $K_h$ or $S_s$ within the window (no migration of fines or clogging beyond tolerance);
laterally uniform response, so that the strain is uniaxial.
Titled phenomena outside the domain.
Inertial flow governed by the full Navier–Stokes equations (boundary layers, vorticity transport, turbulence) lies outside the Darcy regime; pore-scale viscous flow is the microscale origin of $K_h$ and enters through $\mathfrak{K}$.
Waves enter in two ways: the compressional wave speed sets $c_*$, and pressure waves arriving at the inlet are boundary data. Wave propagation through the column obeys a wave equation, not the storage–diffusion balance law, and lies outside the domain; wave rays are null or dispersive and are not trajectories of $O_2$ Eq. [3].
Diffraction around obstacles and free-surface waves are illustrative context.

2. Coordinate-by-Coordinate Mapping ($O_0 \to O_\Omega$)¶

$O_0$ — Exogenous Flux & Emergent Foliation Time¶

Spine Equations: $\theta \equiv -N K \overset{N=1,\,\beta^\mu=0}{=} \frac{1}{2}g^{\mu\nu}\frac{\partial g_{\mu\nu}}{\partial\tau} = \frac{1}{\sqrt{\vert g\vert}}\frac{\partial\sqrt{\vert g\vert}}{\partial\tau} = \nabla_\mu J_{\mathrm{geom}}^\mu \ge 0$; $\Theta_E \equiv \frac{\dot{s}_{\mathrm{prod}}}{s_0} = -\frac{\beta_{\mathrm{th}}}{s_0}J_E^\mu\nabla_\mu\mathcal{F}$; $\Theta_{\mathrm{tot}} \equiv \sqrt{\theta^2 + \Theta_E^2}$, $\tau_\Theta(\tau) \equiv \int_{\tau_0}^{\tau}\langle\Theta_{\mathrm{tot}}\rangle_{\mathcal{M}}\,d\tau'$; $\frac{dV_{\mathcal{M}}}{d\tau} = \int_{\mathcal{M}}\theta\sqrt{\vert g\vert}\,d^nx = \oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$, with $\lim_{\tau\to\infty}V_{\mathcal{M}}(\tau) = V_\infty < \infty$ (Branch II); $\frac{\partial\mathcal{H}}{\partial\tau} \equiv J_E^\mu\nabla_\mu q$, $q(\mathbf{x}, \tau) \equiv -\ln(v_0Q(\mathbf{x}, \tau))$.
Physical Instantiation: During the window the skeleton swells as pore pressure rises, so $\theta = \partial_\tau\ln(1+e) \ge 0$. The volume growth $dV_{\mathcal{M}}/d\tau = \int\theta\sqrt{\vert g\vert}\,d^3x \ge 0$ equals the boundary flux of an expansion current $J_{\mathrm{geom}}$ of divergence $\theta$, of which only that divergence and boundary flux are used (§0.1 item 5); the inflow of water across the inlet is throughput and is never mapped to $J_{\mathrm{geom}}$. As the head approaches its new steady profile, $\theta \to 0$ and $V_{\mathcal{M}}$ approaches a finite $V_\infty$. Openness is carried by throughput: viscous dissipation of the through-flow at every point gives $\Theta_E > 0$ (by $O_1$). During the transient $\tau_\Theta$ is driven by both $\theta$ and $\Theta_E$, and by $\Theta_E$ alone as $\theta \to 0$. The record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput current. Lasting changes of the porous fabric from earlier flow events (rearranged grains, altered pore structure) are mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and are not identified with it; within the window such changes are excluded (precondition 4).

$O_1$ — Non-Equilibrium Thermodynamic Openness¶

Spine Equations: $\dot{W}_{\mathrm{ext}} = p_0\dot{V}_{\mathcal{M}} = p_0\oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$; $\dot{W}_{\mathrm{diss}} \equiv \int_{\mathcal{M}}(-J_E^\mu\nabla_\mu\mathcal{F})\sqrt{\vert g\vert}\,d^nx > 0$; $\dot{W}_{\mathrm{tot}} \equiv \dot{W}_{\mathrm{ext}} + \dot{W}_{\mathrm{diss}} > 0$; $\nabla_\mu\mathcal{F} \neq 0$; $J_E^\mu \equiv \frac{J_W^\mu}{p_0} = -\frac{1}{p_0}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$ (Branch II, fixed); $J_E^\mu\nabla_\mu\mathcal{F} < 0$; $\dot{s}_{\mathrm{prod}} = -\beta_{\mathrm{th}}J_E^\mu\nabla_\mu\mathcal{F} > 0$.
Physical Instantiation: The maintained head difference between inlet and outlet holds the medium far from equilibrium. With $J_E^\mu = -p_0^{-1}(K_h^{\mu\nu}/S_s)\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$. The Darcy flux runs down the head gradient from inlet to outlet, in the direction of $J_E^\mu$, so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. The expansion work $\dot{W}_{\mathrm{ext}} = p_0\dot{V}_{\mathcal{M}} \ge 0$ is positive during swelling and tends to $0$, so $\dot{W}_{\mathrm{tot}} > 0$ is carried asymptotically by $\dot{W}_{\mathrm{diss}}$. Measured hydraulic power tests the signs of $\Theta_E$, $\dot{s}_{\mathrm{prod}}$, and $\dot{W}_{\mathrm{tot}}$ and is never identified with $\dot{W}_{\mathrm{ext}}$. The empirical content of $O_1$ is the flux-prediction test (aperture_generation_guidance.md §2.3, §5.5 item 6). The declared $K_h^{\alpha\beta}/S_s$ is estimated on the same specimen per closure (ii), for example from a pressure-pulse decay run as a declared perturbation regime before the window, with $S_s$ from the specimen's measured compressibility. It must predict $J_W^{\mathrm{meas}} = (\rho g/S_s)\,q^{\mathrm{meas}}$ across a declared section $S$, with $q^{\mathrm{meas}}$ from outlet volumetry and $\nabla\mathcal{F}$ from piezometers that use neither $q^{\mathrm{meas}}$ nor $\mathfrak{K}$.

$O_2$ — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

Spine Equations: $g_{\mu\nu}(\tau, \mathbf{x}) = \Omega^2(\tau, \mathbf{x})\hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}(\tau, \mathbf{x}))^\alpha_{\;\nu}$ with $\det\hat{g} = 1$, $\mathrm{Tr}(\hat{h}) = 0 \implies \det g = \Omega^{2n}$, $\frac{\partial\ln\Omega}{\partial\tau} = \frac{1}{n}\theta$; $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$; $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$, $R = \kappa_G\nabla_\alpha\xi^\alpha$; $\frac{2(n-1)}{n}\nabla^2\theta + \frac{2R}{n}\theta = -\kappa_G\frac{\partial(\nabla_\alpha\xi^\alpha)}{\partial\tau} + 2\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} - 2\sigma^{\alpha\beta}R_{\alpha\beta}$; $\left.\left(\frac{dv^\mu}{d\tau} + \Gamma^\mu_{\rho\epsilon}v^\rho v^\epsilon + g^{\mu\lambda}\left(\frac{\partial g_{\lambda\nu}}{\partial\tau}\right)v^\nu - \frac{1}{2}\left(\frac{\partial g_{\rho\epsilon}}{\partial\tau}v^\rho v^\epsilon\right)v^\mu\right)\right\vert_{N=1,\,\beta^\mu=0} = f^\mu_{\mathrm{drive}}$, $f^\mu_{\mathrm{drive}} = (1-u^2)\left(a^\mu_{\mathrm{d}} - c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu\right)$, $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2 < 1$, with the default $a^\mu_{\mathrm{d}} = 0$.
Physical Instantiation: The physical metric is the flat metric of the swelling skeleton, $g_{\mu\nu} = \mathrm{diag}(1, 1, (1+e)^2)$ in the material chart. Its Weyl split has $\Omega = (1+e)^{1/3}$ (so $\det g = \Omega^6$) and $\hat{h}$ equal to the trace-free part of the logarithmic axial stretch, and $\partial_\tau\ln\Omega = \theta/3$. The split isolates the volume element only algebraically; $\theta$ remains coupled to $\xi_\alpha = \hat{\Pi}_{\alpha\lambda}\eta^\lambda$ and $\sigma_{\alpha\beta}$ through the traced elliptic $\tau$-compatibility equation (r10 Decision 2). On this domain ($G_{\mu\nu} \equiv 0$, $\xi \equiv 0$; §0.1 item 3) it reduces to $\frac{2}{3}\nabla^2\theta = \nabla^\alpha\nabla^\beta\sigma_{\alpha\beta}$, which the uniaxial strain rate satisfies identically because the metric stays flat. The anisotropy of the pore fabric is carried by $\mathfrak{K}$, not by $g$, and walls and grains enter as boundary data and through $K_h$. $\Pi_{\mu\nu}$ is the Hessian of a declared precision potential and enters only the coupling sector ($O_6$). Eq. [3] is instantiated with the default drive and maps no tracer. Fluid parcels in Darcy flow are driven by pressure gradients against drag, which is forced motion; no drag-law drive is declared in this epoch, so their paths are not trajectories of Eq. [3]. Wave rays are excluded (§0.1 item 4).

$O_3$ — Shear Damping & Hysteresis Consolidation¶

Spine Equations: $\left.\frac{\partial g_{\mu\nu}}{\partial\tau}\right\vert_{N=1,\,\beta^\mu=0} = \frac{2}{n}\theta g_{\mu\nu} + 2\sigma_{\mu\nu}$; $\nabla_\tau\sigma^\mu_{\;\nu} \equiv \partial_\tau\sigma^\mu_{\;\nu} = -\gamma_\sigma\sigma^\mu_{\;\nu}$ ($\gamma_\sigma = \mathrm{const} > 0$, $\sigma^\mu_{\;\mu} = 0$) $\implies \lim_{\tau\to\infty}\Vert\sigma^\mu_{\;\nu}\Vert_{C^2(\mathcal{M})} = 0$; $\hat{h}(\tau, \mathbf{x}) \xrightarrow{C^2(\mathcal{M})} \hat{h}_\infty(\mathbf{x})$, $(\hat{g}_\infty(\mathbf{x}))_{\mu\nu} \equiv \hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}_\infty(\mathbf{x}))^\alpha_{\;\nu}$ ($\det\hat{g}_\infty = \det\hat{g} = 1$); $V_{\mathrm{local}}(\tau) \equiv \int_{\mathcal{U}_i}\mathcal{H}(\mathbf{x}, \tau)\sqrt{\vert g\vert}\,d^nx$.
Physical Instantiation: Under lateral confinement the trace-free strain rate is $\sigma^\mu_{\;\nu} = \theta\,\mathrm{diag}(-\tfrac{1}{3}, -\tfrac{1}{3}, \tfrac{2}{3})$, so all its components share one time dependence. After the step, the head transient is a sum of decaying modes of the column. Once the higher modes have decayed below a declared tolerance, the slowest mode dominates and $\sigma$ decays exponentially at a single constant rate. The aperture declares that rate as $\gamma_\sigma = D_{33}\mu_1 + k_L$, with $D_{33} = K_{33}/S_s$ and $\mu_1 = \pi^2/L^2$ the lowest eigenvalue of $-\partial_3^2$ on the column with fixed heads at inlet and outlet, all estimated independently before the window. Because the head rises monotonically, $\sigma$ keeps one sign and does not oscillate. The shear rate decays while the accumulated deformation increases monotonically to its final value, so $\hat{h} \to \hat{h}_\infty$ (the final trace-free logarithmic stretch) and $(\hat{g}_\infty)_{\mu\nu} = \delta_{\mu\alpha}(\exp\hat{h}_\infty)^\alpha_{\;\nu}$, with unit determinant. The late part of the window therefore supports the shear-rate test of aperture_generation_guidance.md §5.5 item 1, with $\gamma_\sigma$ predicted in advance, to within a declared residual from the higher modes; this aperture is not among the systems of the current shear-relaxation test (empirical_validation_protocol.md §3.3). $V_{\mathrm{local}}$ is the hysteresis content of a layer; elastic recovery leaves no plastic record within the window.

$O_4$ — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

Spine Equations: $\frac{\partial\mathcal{F}}{\partial\tau} + \mathcal{F}\theta = -\nabla_\alpha J_W^\alpha - \mathcal{R}_{\mathcal{F}}$, $J_W^\alpha \equiv -\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}$, $\mathfrak{K}^{\alpha\beta} = \mathfrak{K}^{\beta\alpha} \succ 0$, $\mathcal{R}_{\mathcal{F}} \ge 0$; $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}} \equiv \mathbf{1}_{(-\nabla^2 \le \Lambda_{\mathrm{gate}}^2)}$; $Q(\mathbf{x}, \tau) = \langle\mathbf{x}\vert\rho_{\mathrm{local}}(\tau)\vert\mathbf{x}\rangle > 0$; $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P) > 0$; $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$.
Physical Instantiation: On the declared domain the balance law reads $\partial_\tau\mathcal{F} + \mathcal{F}\theta = \nabla_\alpha((K_h^{\alpha\beta}/S_s)\nabla_\beta\mathcal{F}) - \mathcal{R}_{\mathcal{F}}$: the groundwater storage equation in energy-density form, with the $\mathcal{F}\theta$ term of relative order $\varepsilon_v$ (precondition 3). Specific storage gives the finite relaxation time. Departures from the default (anisotropic $K_h$, leakage) satisfy the binding closure, and each is rejected if the default fits at least as well (closure (iv)). The gate sector follows §0.1 item 7: the Neumann spectrum of $-\nabla^2_g$ changes with $g$ by a fraction of the order of the strain across the window, which is negligible, and it converges as $g \to G_\infty$. The faster decay of fine spatial structure in the head field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The stress-conservation equivalence holds identically (§0.1 item 3). The effective stress of the skeleton is not $T_{\mu\nu}^{\mathrm{mat}}$; it enters through the boundary conditions and $\mathcal{H}$.

$O_5$ — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

Spine Equations: $g_{\mu\nu}^{(ij)} \equiv (g^{(i)} \# g^{(j)})_{\mu\nu}$; $\mathcal{U}_{ij}$ geodesically convex for $g^{(ij)}(\tau) \; \forall \tau \ge \tau_0$; $\phi_{\mathrm{perc}}(\mathcal{K}) = \vert\mathcal{K}\vert/\vert\mathcal{K}_0\vert > \phi_c$; $\Delta_{\mathrm{Hodge}}\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} = 0$; $\partial_\tau\sqrt{\vert g^{(i)}\vert} \ge 0$; $g_{\mu\nu}^{(i)} \xrightarrow{C^2(\mathcal{U}_i)} G_{\infty,\mu\nu}^{(i)} \succ 0$ (standing hypothesis with $H_0 = 0$).
Physical Instantiation: The cover consists of $\tau$-independent material layers of the column, with slab-shaped overlaps $\mathcal{U}_{ij}$ perpendicular to its axis. Under laterally uniform uniaxial strain, planes perpendicular to the axis stay planar, so each overlap stays convex for $g(\tau)$ at every $\tau$. Each $g^{(i)}$ is the restriction of $g$, so the gluing holds identically (§0.1 item 8). $\partial_\tau\sqrt{\vert g^{(i)}\vert} = \theta\sqrt{\vert g^{(i)}\vert} \ge 0$, and $g^{(i)} \to G_\infty^{(i)}$ as the strain approaches its final value ($H_0 = 0$). Every overlap is crossed by the transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The connectivity of the pore space is a property of the medium that enters through $K_h$: a medium below its pore-percolation threshold has no continuum $K_h$ and is not an admissible domain. That connectivity is mapped via Spine $\mathcal{T}$ to the condition $\phi_{\mathrm{perc}} > \phi_c$ and is not identified with it. The declared column is simply connected, so $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$ (§0.1 item 8). An annular column around an impermeable core, an illustration outside this declaration, would carry the harmonic Neumann field of its loop. Under isotropic $K_h$ the Darcy flux is the gradient of a single-valued head and has no circulation around that loop, so that field would not be identified with it.

$O_6$ — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

Spine Equations: $(J_{[i\to j]})_\nu = -(J_{[j\to i]})_\nu$; $M^{(ij)}(\mathbf{x}) \neq 0 \; \forall \mathbf{x} \in \bar{\mathcal{U}}_{ij}$ (hypothesis) $\wedge \; \mathrm{Tr}\,M^{(ij)} = 0 \implies m_+ > 0 > m_-$; $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) = k_{\mathcal{I}}\oint_{\partial\mathcal{U}_{ij}} S_{\mu\nu}^{(ij)}\hat{n}^\mu\hat{n}^\nu\,dA > 0$ (hypothesis, $\langle i,j\rangle \in \mathcal{K}$), with this boundary flux integrable on $[\tau_0, \infty)$ (Branch II).
Physical Instantiation: Take two layers of differing permeability that share a transition zone $\mathcal{U}_{ij}$. The partner fields are the rate of change of the head free-energy flux in the two orientations (§0.1 item 8), so $\eta_{[i\to j]} = \partial_\tau J_W$: a spatially resolved flux field, not a temporal lag between scalar signals. With fixed heads at both ends, the late-time zero set of $\partial_\tau J_W$ is the column's mid-plane. Upstream of it $\partial_\tau J_W$ points toward the inlet, and downstream of it away from the inlet. The declared overlap is one that the predicted zero set does not reach during the window (§0.1 item 8), for example a transition zone in the downstream half that the predicted zero crossing, which travels downstream during the early window, never reaches. There the flux rises throughout the window and $\eta_{[i\to j]}$ points away from the inlet. The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is its image under the declared non-singular precision map and is non-zero there. Under the non-degeneracy hypothesis $M^{(ij)} \neq 0$, built per r7 $O_6$ Eq. [3] from these fields and the declared precision potentials, $\mathrm{Tr}\,M^{(ij)} = 0$ gives $m_+ > 0 > m_-$, with $m_+ \neq -m_-$ in general. The eigendirections are directions of the operator $M^{(ij)}$; in the laterally confined column the transverse ones carry no flow. Under the flux-positivity hypothesis the coupling transmits correlation between the subsystem marginals of the declared state sector ($\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$). Both are standing hypotheses. In Branch II the transient decays at the rate $\gamma_\sigma$ of $O_3$, so the coupling flux is integrable on $[\tau_0, \infty)$ and no sustained coupling is asserted. The planar overlaps remain convex at every $\tau$, so the r10 §5.12 constraint $a_\infty = 2s_\sigma/\gamma_\sigma < a_c(E)$, a design constraint of a curved witness overlap, does not apply. Any convexity threshold for curved overlaps in a deforming porous medium would be an aperture-level empirical conjecture mapped to that witness criterion, and none is asserted here.

$O_7$ — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

Spine Equations: $D_\mu = \partial_\mu - i g_{\mathrm{YM}}\,\mathrm{ad}_{A_\mu}$; $\Vert\hat{\mathcal{E}}_\mu\Vert_{C^1(\mathcal{M})} \to 0 \implies [F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] \to 0$; $\chi_E \equiv 1 + \frac{n\,\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu}}{\theta^2 + \langle\Theta_E\rangle_{\mathcal{M}}^2} \to 1$, $\hat{\chi} \equiv \chi_E\hat{u}$ (Branch II); $D_{\mathrm{KL}}(\hat{\rho}_g(\tau) \parallel \hat{\rho}_{\mathcal{H}}(\tau)) \to 0$.
Physical Instantiation: The gauge sector is the trivial declaration of §0.1 item 9, so $[F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] = 0$ identically and $\hat{\mathcal{E}}_\mu \to 0$ as $\sigma \to 0$. Here $\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu} = \frac{2}{3}\theta^2$, so $\chi_E - 1 = 2\theta^2/(\theta^2 + \langle\Theta_E\rangle^2) \to 0$ as $\theta \to 0$ with $\langle\Theta_E\rangle \to \Theta_\infty > 0$. Once $\theta^2 \ll \langle\Theta_E\rangle^2$, late in the window, $\chi_E - 1$ decays at $2\gamma_\sigma$. Because $\langle\Theta_E\rangle$ depends on $s_0$, the onset of that regime is not $s_0$-independent, so the aperture's $s_0$-independent observable is the decay rate of $\sigma$ itself ($O_3$). $\chi$ is undefined where $\theta = 0$ and is not used. $\hat{\rho}_g = e^{-\hat{h}_\infty}e^{\hat{h}}/\mathrm{Tr}(e^{-\hat{h}_\infty}e^{\hat{h}}) \to \frac{1}{n}\delta^\mu_\nu$ as $\hat{h} \to \hat{h}_\infty$, a residual that decays at $\gamma_\sigma$ and can be computed from the strain record. Alignment with $\hat{\rho}_{\mathcal{H}}$ requires the Hessian of the record to become isotropic, an asymptotic condition stated and not tested. In $O_7$ Eq. [3], $u_{\mathrm{exp}}$ and $J_{\mathrm{geom}}$ share the divergence $\theta$, so the boundary-flux identity holds; $J_E$ is bounded in $C^1$ and $\hat{h} \to \hat{h}_\infty$, so $\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \to 0$. No alignment of the flow with the porous fabric is claimed beyond what $\mathfrak{K}$ declares.

$O_\Omega$ — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Spine Equations: Branch II ($H_0 = 0$): $\theta \xrightarrow[\tau\to\infty]{C^2(\mathcal{M})} 0$ with $\int_{\tau_0}^{\infty}\langle\theta\rangle_{\mathcal{M}}\,d\tau < \infty$ and $\langle\Theta_E\rangle_{\mathcal{M}} \to \Theta_\infty > 0$; $\sigma^\mu_{\;\nu} \to 0$; $\partial_\tau g_{\mu\nu} \to 0$; $V_{\mathcal{M}} \to V_\infty$ and $\mathcal{I}(\tau) \to s_0V_\infty$; $\left.\hat{\mathbb{L}}_{\mathrm{stat}}(\mathbf{x})\Psi[g_{\mu\nu}, \mathcal{H}]\right\vert_{\mathcal{H}=\mathcal{H}_{\mathrm{sol}}[g]} = (J_E^\lambda\nabla_\lambda q)\frac{\delta\Psi}{\delta\mathcal{H}(\mathbf{x})} \xrightarrow[\tau\to\infty]{} 0 \iff J_E^\lambda\nabla_\lambda q \to 0 \iff \frac{\partial\mathcal{H}}{\partial\tau} \to 0$ (with $\vert\delta\Psi/\delta\mathcal{H}\vert$ bounded and bounded away from $0$ uniformly as $\tau \to \infty$).
Physical Instantiation: Under a maintained head difference the column approaches a steady head profile with continuing through-flow and dissipation: a bounded non-equilibrium steady state. Swelling ceases ($\theta \to 0$), $\int\langle\theta\rangle\,d\tau$ is bounded by the final volumetric strain, $V_{\mathcal{M}} \to V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$, $\sigma \to 0$, and $\partial_\tau g \to 0$. The transient coupling decays, so $\mathcal{I} \to s_0V_\infty$. The generator condition requires $J_E^\lambda\nabla_\lambda q \to 0$, an asymptotic hypothesis on the declared state sector that no finite window tests. The state is neither static equilibrium, which would require a uniform head and no flow, nor a periodic return. Within windows the supported tests (aperture_generation_guidance.md §5.5) are the shear-rate test (1); decaying $\partial_\tau g_{\mu\nu}$ residuals at the rate $\gamma_\sigma$ (3); the non-degeneracy check (4); the signs of $\Theta_E$, $\dot{s}_{\mathrm{prod}}$, and $\dot{W}_{\mathrm{tot}}$ (5); the flux prediction (6); and the Branch II evidence (7). The $\chi_E$ rate test (2) is not among them: its late-window regime has an $s_0$-dependent onset (see $O_7$), so $\chi_E$ may be reported only as a quantity that is not $s_0$-independent.

Aperture 2.3: Atmospheric Radiative & Thermal Boundary Physics ($\mathcal{E}_{\mathrm{radiative}}$)¶

1. Epistemological & Physical Scope¶

Aperture 2.3 maps the physics of electromagnetic radiation in matter and of atmospheric and other radiative-thermal boundary layers (absorption and emission, Rayleigh and Mie scattering, Fresnel partitioning at interfaces, refraction, and the coupling of radiation to sensible heat) onto Spine $\mathcal{T}$. Its empirical domain lies in the regime where radiative transfer becomes a storage–diffusion process: an optically thick layer in local thermodynamic equilibrium (LTE). Lexicon is strictly restricted to radiative transfer, atmospheric physics, non-equilibrium thermodynamics, and geometrical and physical optics.

Declaration (aperture_generation_guidance.md §2.3; Branch II).

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. An optically thick layer in LTE through which radiation diffuses: a translucent condensed or particulate medium, or an optically thick, convectively stable gas layer. The layer's upper face, the irradiated boundary (supply), lies several extinction lengths below the physical surface on which radiation is incident, at a depth where the incident radiation has been absorbed. The absorbed flux therefore enters $\mathcal{M}$ across this face, and the absorption layer above it lies outside $\mathcal{M}$. The lower boundary is held at the temperature $T_{\mathrm{ref}}$, and the lateral faces are adiabatic. The absorbed flux is laterally uniform across the supply face, so the declared problem is one-dimensional in $x^3$.
Window. A relaxation window (§0.1 item 1), opened by a declared step increase of the absorbed flux. The temperature then rises monotonically at every point, so any residual thermal expansion is non-negative.
Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, with thermal expansion within the rigid tolerance.
Cone speed. $c_* = c/n_r$, the speed of radiation in the medium, with $n_r$ its refractive index. Radiation rays are null for this cone; all material speeds are $\ll c_*$.
Free energy $\mathcal{F}$ and datum. $\mathcal{F} = \rho c_{\mathrm{eff}}(T - T_{\mathrm{ref}})$, the effective thermal energy density, linearized about the datum $T_{\mathrm{ref}}$ of the lower boundary ($\mathcal{F}_{\mathrm{ref}} = 0$). Storage is dominated by the heat capacity of the matter: the radiation's own heat capacity per unit volume, $4aT^3$, is negligible against $\rho c_{\mathrm{eff}}$. The relaxation time $\ell^2/D_R$ is finite and resolvable.
Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D_R(x^3)\,g^{\alpha\beta}$, with $D_R = 16\sigma_{\mathrm{SB}}T_{\mathrm{ref}}^3/(3\kappa_R\rho c_{\mathrm{eff}})$, the effective radiative diffusivity. $\kappa_R(x^3)$ is the Rosseland mean extinction coefficient (m⁻¹), stratified with a declared parametric profile, and $T^3$ is evaluated at the declared reference; constant $\kappa_R$ is the default. Then $J_W = -D_R\nabla\mathcal{F} = -k_R\nabla T$, with $k_R = 16\sigma_{\mathrm{SB}}T_{\mathrm{ref}}^3/(3\kappa_R)$: the radiative heat flux (W m⁻²), so the only conversion factor is $\rho c_{\mathrm{eff}}$.
Sink $\mathcal{R}_{\mathcal{F}}$. Default, $0$: in LTE local emission balances local absorption, so there is no net interior loss.
Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 4).
Branch II evidence (closure (v)). The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau$ come from dilatometric or thickness measurement of the layer, which must stay within the rigid tolerance. $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 6), using a heat-flux and temperature record at the supply face, separate from the interior section and sensors of the $O_1$ test. The physical entropy production of heat flowing down a temperature gradient, $k_R\vert\nabla T\vert^2/T^2$ per unit volume, must share its sign.
Admissible domains (declared, checkable preconditions):
optical depth $\gg 1$ across $\mathcal{M}$ and on the scale of its temperature gradients, so the diffusion approximation holds. The error it makes within about one extinction length of each boundary falls within the declared tolerance, which requires the extinction length to be small against the layer thickness;
LTE, with no phase change and no chemical heat release in $\mathcal{M}$ (no interior source or sink);
no convection or other advection;
$\vert T - T_{\mathrm{ref}}\vert \ll T_{\mathrm{ref}}$, so that the $T^3$ linearization holds within tolerance, and radiative transport dominates conduction by the matter, so that $D_R$ is the effective diffusivity within tolerance;
$T \ge T_{\mathrm{ref}}$ throughout (so $\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$) and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$;
thermal expansion within the rigid tolerance, and $\kappa_R$ constant in time (no sintering, recrystallization, or compositional change within the window).
laterally uniform absorbed flux, checked by radiometry across the supply face to within a declared tolerance.
Titled phenomena outside the domain.
Rayleigh and Mie scattering and spectral absorption enter only through $\kappa_R$, the Rosseland mean of the medium's extinction spectrum.
Fresnel partitioning, refraction, and the absorption of the incident beam at and above the supply face enter as boundary data, through the absorbed flux.
Geometrical-optics rays, caustics, and refracted beams are null and are not trajectories of $O_2$ Eq. [3].
Optically thin atmospheric layers, convective boundary layers, and turbulent exchange lie outside the diffusion regime and enter as boundary conditions or context. Latent-heat exchange at a two-medium boundary lies on or outside $\partial\mathcal{M}$.

2. Coordinate-by-Coordinate Mapping ($O_0 \to O_\Omega$)¶

$O_0$ — Exogenous Flux & Emergent Foliation Time¶

Spine Equations: $\theta \equiv -N K \overset{N=1,\,\beta^\mu=0}{=} \frac{1}{2}g^{\mu\nu}\frac{\partial g_{\mu\nu}}{\partial\tau} = \frac{1}{\sqrt{\vert g\vert}}\frac{\partial\sqrt{\vert g\vert}}{\partial\tau} = \nabla_\mu J_{\mathrm{geom}}^\mu \ge 0$; $\Theta_E \equiv \frac{\dot{s}_{\mathrm{prod}}}{s_0} = -\frac{\beta_{\mathrm{th}}}{s_0}J_E^\mu\nabla_\mu\mathcal{F}$; $\Theta_{\mathrm{tot}} \equiv \sqrt{\theta^2 + \Theta_E^2}$, $\tau_\Theta(\tau) \equiv \int_{\tau_0}^{\tau}\langle\Theta_{\mathrm{tot}}\rangle_{\mathcal{M}}\,d\tau'$; $\frac{dV_{\mathcal{M}}}{d\tau} = \int_{\mathcal{M}}\theta\sqrt{\vert g\vert}\,d^nx = \oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$, with $\lim_{\tau\to\infty}V_{\mathcal{M}}(\tau) = V_\infty < \infty$ (Branch II); $\frac{\partial\mathcal{H}}{\partial\tau} \equiv J_E^\mu\nabla_\mu q$, $q(\mathbf{x}, \tau) \equiv -\ln(v_0Q(\mathbf{x}, \tau))$.
Physical Instantiation: The layer is rigid, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$; the absorbed supply is throughput and is never mapped to $J_{\mathrm{geom}}$ (§0.1 item 5). Radiative diffusion of energy from the supply face to the lower boundary produces entropy at every point, so $\Theta_E > 0$ (by $O_1$), $\Theta_{\mathrm{tot}} = \Theta_E$, and $\tau_\Theta$ counts entropy production per $s_0$. The record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput current. Lasting changes of the medium under its irradiation history, such as sintering or recrystallization of a particulate layer, are mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and are not identified with it; within the window they are excluded (precondition 6). The temperature profile itself is $\mathcal{F}$, not the record.

$O_1$ — Non-Equilibrium Thermodynamic Openness¶

Spine Equations: $\dot{W}_{\mathrm{ext}} = p_0\dot{V}_{\mathcal{M}} = p_0\oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$; $\dot{W}_{\mathrm{diss}} \equiv \int_{\mathcal{M}}(-J_E^\mu\nabla_\mu\mathcal{F})\sqrt{\vert g\vert}\,d^nx > 0$; $\dot{W}_{\mathrm{tot}} \equiv \dot{W}_{\mathrm{ext}} + \dot{W}_{\mathrm{diss}} > 0$; $\nabla_\mu\mathcal{F} \neq 0$; $J_E^\mu \equiv \frac{J_W^\mu}{p_0} = -\frac{1}{p_0}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$ (Branch II, fixed); $J_E^\mu\nabla_\mu\mathcal{F} < 0$; $\dot{s}_{\mathrm{prod}} = -\beta_{\mathrm{th}}J_E^\mu\nabla_\mu\mathcal{F} > 0$.
Physical Instantiation: The layer sits between the absorbed supply and the colder lower boundary and is held far from equilibrium by the energy flux between them. With $J_E^\mu = -p_0^{-1}D_R\nabla^\mu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} = -p_0^{-1}D_R\vert\nabla\mathcal{F}\vert^2 < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$. The radiative heat flux runs down the temperature gradient from the supply face to the lower boundary, in the direction of $J_E^\mu$, so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. On the rigid layer $\dot{W}_{\mathrm{ext}} = 0$ and $\dot{W}_{\mathrm{tot}} = \dot{W}_{\mathrm{diss}} > 0$. The empirical content of $O_1$ is the flux-prediction test (aperture_generation_guidance.md §2.3, §5.5 item 6). The parameters of $\kappa_R(x^3)$ are estimated on the specimen itself by a different modality, for example spectral transmission measurements averaged with the Rosseland weight, and $c_{\mathrm{eff}}$ by calorimetry. The declared law must then predict the heat flux $J_W^{\mathrm{meas}}$ measured across a declared interior section $S$, with $\nabla T$ from embedded thermometry that uses neither $J_W^{\mathrm{meas}}$ nor $\kappa_R$. No current test exercises this aperture (empirical_validation_protocol.md §3.3).

$O_2$ — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

Spine Equations: $g_{\mu\nu}(\tau, \mathbf{x}) = \Omega^2(\tau, \mathbf{x})\hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}(\tau, \mathbf{x}))^\alpha_{\;\nu}$ with $\det\hat{g} = 1$, $\mathrm{Tr}(\hat{h}) = 0 \implies \det g = \Omega^{2n}$, $\frac{\partial\ln\Omega}{\partial\tau} = \frac{1}{n}\theta$; $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$; $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$, $R = \kappa_G\nabla_\alpha\xi^\alpha$; $\frac{2(n-1)}{n}\nabla^2\theta + \frac{2R}{n}\theta = -\kappa_G\frac{\partial(\nabla_\alpha\xi^\alpha)}{\partial\tau} + 2\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} - 2\sigma^{\alpha\beta}R_{\alpha\beta}$; $\left.\left(\frac{dv^\mu}{d\tau} + \Gamma^\mu_{\rho\epsilon}v^\rho v^\epsilon + g^{\mu\lambda}\left(\frac{\partial g_{\lambda\nu}}{\partial\tau}\right)v^\nu - \frac{1}{2}\left(\frac{\partial g_{\rho\epsilon}}{\partial\tau}v^\rho v^\epsilon\right)v^\mu\right)\right\vert_{N=1,\,\beta^\mu=0} = f^\mu_{\mathrm{drive}}$, $f^\mu_{\mathrm{drive}} = (1-u^2)\left(a^\mu_{\mathrm{d}} - c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu\right)$, $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2 < 1$, with the default $a^\mu_{\mathrm{d}} = 0$.
Physical Instantiation: On the rigid layer the unimodular Weyl split is trivial ($\Omega \equiv 1$, $\hat{h} \equiv 0$, $\partial_\tau\ln\Omega = 0$). The split isolates the volume element only algebraically; $\theta$ remains coupled to $\xi_\alpha = \hat{\Pi}_{\alpha\lambda}\eta^\lambda$ and $\sigma_{\alpha\beta}$ through the traced elliptic $\tau$-compatibility equation (r10 Decision 2), whose two sides vanish identically here (§0.1 items 2–3). The refractive-index field and any birefringence do not enter $g$. The optical (Fermat) metric of ray optics is a different object from the physical metric and is not mapped to $g_{\mu\nu}$; the refractive index enters only through $c_*$, $\kappa_R$, and the boundary data. Dielectric interfaces, where Snell's law and Fresnel partitioning apply, lie on or outside $\partial\mathcal{M}$. $\Pi_{\mu\nu}$ is the Hessian of a declared precision potential and enters only the coupling sector ($O_6$). Eq. [3] is instantiated with the default drive and maps no tracer. Radiation is null for the declared cone, dispersive wavepackets follow Hamiltonian ray equations ($\dot{x}^\mu = \partial\omega/\partial k_\mu$), and the diffusing energy is described by the $O_4$ balance law; none of these is a timelike trajectory of Eq. [3].

$O_3$ — Shear Damping & Hysteresis Consolidation¶

Spine Equations: $\left.\frac{\partial g_{\mu\nu}}{\partial\tau}\right\vert_{N=1,\,\beta^\mu=0} = \frac{2}{n}\theta g_{\mu\nu} + 2\sigma_{\mu\nu}$; $\nabla_\tau\sigma^\mu_{\;\nu} \equiv \partial_\tau\sigma^\mu_{\;\nu} = -\gamma_\sigma\sigma^\mu_{\;\nu}$ ($\gamma_\sigma = \mathrm{const} > 0$, $\sigma^\mu_{\;\mu} = 0$) $\implies \lim_{\tau\to\infty}\Vert\sigma^\mu_{\;\nu}\Vert_{C^2(\mathcal{M})} = 0$; $\hat{h}(\tau, \mathbf{x}) \xrightarrow{C^2(\mathcal{M})} \hat{h}_\infty(\mathbf{x})$, $(\hat{g}_\infty(\mathbf{x}))_{\mu\nu} \equiv \hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}_\infty(\mathbf{x}))^\alpha_{\;\nu}$ ($\det\hat{g}_\infty = \det\hat{g} = 1$); $V_{\mathrm{local}}(\tau) \equiv \int_{\mathcal{U}_i}\mathcal{H}(\mathbf{x}, \tau)\sqrt{\vert g\vert}\,d^nx$.
Physical Instantiation: On the rigid layer $\sigma^\mu_{\;\nu} \equiv 0$, so the shear law holds trivially, $\hat{h} = \hat{h}_\infty = 0$, and $\hat{g}_\infty = \hat{g} = \delta$; this aperture supports no shear-relaxation test (§0.1 item 2). The relaxation of the temperature profile toward its new steady state, which proceeds through modes decaying at rates set by $D_R$ and the layer thickness, is a property of the $O_4$ balance law and is not mapped to $\sigma$. Thermal-stress deformation of the layer lies outside the rigid declaration. $V_{\mathrm{local}}$ is the hysteresis content of a layer $\mathcal{U}_i$.

$O_4$ — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

Spine Equations: $\frac{\partial\mathcal{F}}{\partial\tau} + \mathcal{F}\theta = -\nabla_\alpha J_W^\alpha - \mathcal{R}_{\mathcal{F}}$, $J_W^\alpha \equiv -\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}$, $\mathfrak{K}^{\alpha\beta} = \mathfrak{K}^{\beta\alpha} \succ 0$, $\mathcal{R}_{\mathcal{F}} \ge 0$; $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}} \equiv \mathbf{1}_{(-\nabla^2 \le \Lambda_{\mathrm{gate}}^2)}$; $Q(\mathbf{x}, \tau) = \langle\mathbf{x}\vert\rho_{\mathrm{local}}(\tau)\vert\mathbf{x}\rangle > 0$; $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P) > 0$; $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$.
Physical Instantiation: With $\theta \equiv 0$ and $\mathcal{R}_{\mathcal{F}} = 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha(D_R(x^3)\nabla^\alpha\mathcal{F})$: linearized radiative diffusion in the Rosseland approximation, with heat storage in the matter. The stratified $\kappa_R(x^3)$ departs from the default constant diffusivity and satisfies the binding closure; it is rejected if the constant default fits at least as well (closure (iv)). The gate sector follows §0.1 item 7: the Neumann eigenmodes of $-\nabla^2$ on the rigid layer are fixed, so the gate does not drift, and the faster decay of fine spatial structure in the temperature field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The gate is a spatial filter and is distinct from any spectral (wavelength) filtering of the incident beam above the supply face. The stress-conservation equivalence holds identically (§0.1 item 3).

$O_5$ — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

Spine Equations: $g_{\mu\nu}^{(ij)} \equiv (g^{(i)} \# g^{(j)})_{\mu\nu}$; $\mathcal{U}_{ij}$ geodesically convex for $g^{(ij)}(\tau) \; \forall \tau \ge \tau_0$; $\phi_{\mathrm{perc}}(\mathcal{K}) = \vert\mathcal{K}\vert/\vert\mathcal{K}_0\vert > \phi_c$; $\Delta_{\mathrm{Hodge}}\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} = 0$; $\partial_\tau\sqrt{\vert g^{(i)}\vert} \ge 0$; $g_{\mu\nu}^{(i)} \xrightarrow{C^2(\mathcal{U}_i)} G_{\infty,\mu\nu}^{(i)} \succ 0$ (standing hypothesis with $H_0 = 0$).
Physical Instantiation: The cover consists of $\tau$-independent layers of differing $\kappa_R$, with slab-shaped overlaps $\mathcal{U}_{ij}$ that stay convex on the rigid metric. Each $g^{(i)}$ is the restriction of $g$, so the gluing holds identically, $\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$, and $g^{(i)} = G_\infty^{(i)}$ at every $\tau$ ($H_0 = 0$; §0.1 item 8). Every overlap is crossed by the transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. In a particulate medium the connectivity of radiative pathways between particles enters through $\kappa_R$; it is mapped via Spine $\mathcal{T}$ to the percolation condition and is not identified with it. The declared layer is simply connected, so $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$ (§0.1 item 8).

$O_6$ — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

Spine Equations: $(J_{[i\to j]})_\nu = -(J_{[j\to i]})_\nu$; $M^{(ij)}(\mathbf{x}) \neq 0 \; \forall \mathbf{x} \in \bar{\mathcal{U}}_{ij}$ (hypothesis) $\wedge \; \mathrm{Tr}\,M^{(ij)} = 0 \implies m_+ > 0 > m_-$; $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) = k_{\mathcal{I}}\oint_{\partial\mathcal{U}_{ij}} S_{\mu\nu}^{(ij)}\hat{n}^\mu\hat{n}^\nu\,dA > 0$ (hypothesis, $\langle i,j\rangle \in \mathcal{K}$), with this boundary flux integrable on $[\tau_0, \infty)$ (Branch II).
Physical Instantiation: Take two layers of differing $\kappa_R$ that share a transition zone $\mathcal{U}_{ij}$. The partner fields are the rate of change of the radiative heat flux in the two orientations (§0.1 item 8): a spatially resolved flux field across the zone, not a temporal lag between scalar signals. Because the supply is a prescribed absorbed flux, after the step the flux at the supply face is held constant, so $\partial_\tau J_W = 0$ there. The lower boundary is held at fixed temperature, so the flux has zero gradient there. With a laterally uniform supply (precondition 7) the flux $J = -D_R\partial_3\mathcal{F}$ obeys the autonomous parabolic law $\partial_\tau J = D_R\partial_3^2 J$. The prior uniform flux is a subsolution for the raised supply flux, so comparison gives $J(\cdot, \tau_0 + h) \ge J(\cdot, \tau_0)$, and time-translation invariance then gives $J(\cdot, \tau + h) \ge J(\cdot, \tau)$, that is, $\partial_\tau J_W \ge 0$. The strong maximum principle, with Hopf's lemma at the lower face, makes it positive everywhere except on the supply face. Its zero set is therefore the supply face itself, and on every declared overlap away from that face $\eta_{[i\to j]} = \partial_\tau J_W$ points away from the supply face throughout the window (§0.1 item 8). The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is its image under the declared non-singular precision map and is non-zero there. Under the non-degeneracy hypothesis $M^{(ij)} \neq 0$, built per r7 $O_6$ Eq. [3], $\mathrm{Tr}\,M^{(ij)} = 0$ gives $m_+ > 0 > m_-$, with $m_+ \neq -m_-$ in general. The eigendirections are directions of the operator $M^{(ij)}$, not of separate fluxes; in the laterally adiabatic layer the transverse ones carry no flux. Under the flux-positivity hypothesis the coupling transmits correlation between the subsystem marginals of the declared state sector ($\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$). Both are standing hypotheses. In Branch II the transient decays as the layer settles, so the coupling flux is integrable on $[\tau_0, \infty)$. With $\sigma \equiv 0$ the overlaps keep their declared convex shape, and the r10 §5.12 constraint has no counterpart here. Sensible and latent heat exchange across a boundary between two different media lies on or outside $\partial\mathcal{M}$ (precondition 2).

$O_7$ — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

Spine Equations: $D_\mu = \partial_\mu - i g_{\mathrm{YM}}\,\mathrm{ad}_{A_\mu}$; $\Vert\hat{\mathcal{E}}_\mu\Vert_{C^1(\mathcal{M})} \to 0 \implies [F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] \to 0$; $\chi_E \equiv 1 + \frac{n\,\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu}}{\theta^2 + \langle\Theta_E\rangle_{\mathcal{M}}^2} \to 1$, $\hat{\chi} \equiv \chi_E\hat{u}$ (Branch II); $D_{\mathrm{KL}}(\hat{\rho}_g(\tau) \parallel \hat{\rho}_{\mathcal{H}}(\tau)) \to 0$.
Physical Instantiation: The gauge sector is the trivial declaration of §0.1 item 9, so $[F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] = 0$ identically. With $\sigma \equiv 0$, $\chi_E \equiv 1$ and $\hat{\mathcal{E}}_\mu \equiv 0$ at every $\tau$, and $\chi$ is undefined ($\theta = 0$). $\hat{\rho}_g = \frac{1}{n}\delta^\mu_\nu$ at every $\tau$, and alignment with $\hat{\rho}_{\mathcal{H}}$ holds where the Hessian of the declared convex record becomes isotropic, an asymptotic condition stated and not tested. The $O_7$ Eq. [3] identities hold with divergence-free $J_{\mathrm{geom}}$ and $u_{\mathrm{exp}}$, bounded $J_E$, and $\hat{h} \equiv \hat{h}_\infty$. No alignment of the energy distribution with a stored profile is claimed beyond these declarations.

$O_\Omega$ — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Spine Equations: Branch II ($H_0 = 0$): $\theta \xrightarrow[\tau\to\infty]{C^2(\mathcal{M})} 0$ with $\int_{\tau_0}^{\infty}\langle\theta\rangle_{\mathcal{M}}\,d\tau < \infty$ and $\langle\Theta_E\rangle_{\mathcal{M}} \to \Theta_\infty > 0$; $\sigma^\mu_{\;\nu} \to 0$; $\partial_\tau g_{\mu\nu} \to 0$; $V_{\mathcal{M}} \to V_\infty$ and $\mathcal{I}(\tau) \to s_0V_\infty$; $\left.\hat{\mathbb{L}}_{\mathrm{stat}}(\mathbf{x})\Psi[g_{\mu\nu}, \mathcal{H}]\right\vert_{\mathcal{H}=\mathcal{H}_{\mathrm{sol}}[g]} = (J_E^\lambda\nabla_\lambda q)\frac{\delta\Psi}{\delta\mathcal{H}(\mathbf{x})} \xrightarrow[\tau\to\infty]{} 0 \iff J_E^\lambda\nabla_\lambda q \to 0 \iff \frac{\partial\mathcal{H}}{\partial\tau} \to 0$ (with $\vert\delta\Psi/\delta\mathcal{H}\vert$ bounded and bounded away from $0$ uniformly as $\tau \to \infty$).
Physical Instantiation: Under maintained irradiation the layer approaches a steady temperature profile that carries a steady energy flux from the supply face to the lower boundary: a bounded non-equilibrium steady state. $\theta \equiv 0$, $V_{\mathcal{M}} = V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while the absorbed flux is maintained, $\sigma \equiv 0$, $\partial_\tau g \equiv 0$, and the transient coupling decays, so $\mathcal{I} \to s_0V_\infty$. The generator condition requires $J_E^\lambda\nabla_\lambda q \to 0$, an asymptotic hypothesis on the declared state sector that no finite window tests. The state is neither thermodynamic equilibrium at a uniform temperature, which would carry no flux, nor a periodic return. Within windows the supported tests (aperture_generation_guidance.md §5.5) are the non-degeneracy check (4), the signs of $\Theta_E$, $\dot{s}_{\mathrm{prod}}$, and $\dot{W}_{\mathrm{tot}}$ (5), the flux prediction (6), and the Branch II evidence (7); tests (1)–(3) hold trivially on the rigid layer.

Aperture 2.4: Geomorphic & Structural Mechanics ($\mathcal{E}_{\mathrm{tectonic}}$)¶

1. Epistemological & Physical Scope¶

Aperture 2.4 maps the mechanics of solid-earth substrates and load-bearing bodies (landform evolution, sediment transport and bedform memory, consolidation and swelling of saturated soils and rocks, and the stress fields of structures) onto Spine $\mathcal{T}$. Its empirical domain is the saturated stratum of a landform or load-bearing earth body, where pore-pressure diffusion couples groundwater flow to the deformation of the solid skeleton. Lexicon is strictly restricted to geomorphology, sedimentology, hydrogeology, geotechnical engineering, and structural mechanics.

Declaration (aperture_generation_guidance.md §2.3; Branch II).

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A confined aquifer stratum within a landform or load-bearing earth body: a saturated, deformable porous stratum bounded above and below by confining layers, which lie outside $\mathcal{M}$ and are treated as no-flow boundaries. $\mathcal{M}$ is the material region of the stratum's skeleton (material chart). $\partial\mathcal{M}$ consists of a recharge boundary (supply) at a declared head, a drain boundary at the drain level $h_{\mathrm{ref}}$, and the no-flow confining surfaces.
Window. A relaxation window (§0.1 item 1), opened by a declared increase of the recharge-boundary head from a prior steady state, after which the head is held. Pore pressure then rises monotonically at every point and the stratum swells elastically, which is the recoverable rebound counterpart of consolidation. Windows that compact the stratum (consolidation under added load, or a declining head) give $\theta < 0$ and are excluded.
Metric. Flat, with small-strain deformation (§0.1 item 2): the pull-back of the Euclidean metric under the measured displacement field of the skeleton. For a laterally extensive stratum the strain is approximately uniaxial and normal to the bedding, and $\theta \ge 0$.
Cone speed. $c_*$ is the compressional wave speed of the saturated stratum. Seepage velocities and deformation speeds are $\ll c_*$.
Free energy $\mathcal{F}$ and datum. $\mathcal{F} = \rho g(h - h_{\mathrm{ref}})$, the head energy density, with datum the drain level ($\mathcal{F}_{\mathrm{ref}} = 0$): the stored water volume per unit volume, $S_s(h - h_{\mathrm{ref}})$, times the declared conversion factor $\rho g/S_s$. Storage makes the relaxation time finite and resolvable.
Transport $\mathfrak{K}$. Layered hydraulic diffusivity, $\mathfrak{K}^{\alpha\beta} = \mathrm{diag}(D_H, D_H, D_V)$ in a chart aligned with the bedding, with $D_H = K_H/S_s$ and $D_V = K_V/S_s$ (two declared parameters, plus the declared bedding orientation). With $q$ the Darcy flux, $J_W = (\rho g/S_s)\,q$.
Sink $\mathcal{R}_{\mathcal{F}}$. $\mathcal{R}_{\mathcal{F}} = k_L(\mathcal{F} - \mathcal{F}_{\mathrm{ref}}) \ge 0$: distributed leakage through interbedded semi-confining layers to a drainage system at the drain level, with $k_L$ a declared rate constant. In this quasi-three-dimensional idealization $k_L$ is of order $K'/(b'bS_s)$, for semi-confining layers of conductivity $K'$ and thickness $b'$ in a stratum of thickness $b$. Leakage across a bounding confining layer is a boundary condition, not a sink.
Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 4).
Branch II evidence (closure (v)). The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau = \ln(V_{\mathcal{M}}(\tau_{\mathrm{obs}})/V_{\mathcal{M}}(\tau_0))$ come from geodetic surface uplift and borehole extensometers, bounded by the final uplift. $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 6). The boundary term uses the recharge-boundary record (recharge rate and head), and the leakage term uses the declared, independently estimated $k_L$ with $\mathcal{F}$ from a set of piezometers not used for the $O_1$ gradient. The physical viscous dissipation of the groundwater flow must share its sign.
Admissible domains (declared, checkable preconditions):
Darcy regime;
$h \ge h_{\mathrm{ref}}$ throughout (so $\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$) and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$;
elastic, recoverable swelling with volumetric strain $\varepsilon_v \ll 1$ and $\theta \ge 0$ throughout the window, verified from the deformation record. The linear storage equation and the r7 balance law then differ by terms of relative order $\varepsilon_v$, which the pre-declared flux tolerance must exceed;
no erosion, deposition, or change of load on $\mathcal{M}$ within the window; these act on $\partial\mathcal{M}$ or between windows;
$K_H$, $K_V$, $S_s$, and $k_L$ constant within the window.
Titled phenomena outside the domain.
Landform evolution acts at the ground surface, outside $\mathcal{M}$, and between windows: erosion, deposition, and the divergence-form Exner sediment balance. It enters through changed loads and boundary heads and through $\mathcal{H}$.
Bedforms, granular bedload, and saltation are forced transport outside the domain; no drive is declared for them in this epoch.
Consolidation under loading ($\theta < 0$) and visco-plastic creep lie outside the declared windows.
Structural stress fields (overburden, effective stress, the stresses of load-bearing spans) and their principal-stress lines, arches, and catenaries are equilibrium objects. They enter through the boundary conditions and $\mathcal{H}$, not through $T_{\mu\nu}^{\mathrm{mat}}$ or $O_2$ Eq. [3].
Granular force-chain networks in the skeleton set its compressibility, and hence $S_s$ and $\mathfrak{K}$, and enter otherwise as context. Fault and contact-zone mechanics lie outside the domain.

2. Coordinate-by-Coordinate Mapping ($O_0 \to O_\Omega$)¶

$O_0$ — Exogenous Flux & Emergent Foliation Time¶

Spine Equations: $\theta \equiv -N K \overset{N=1,\,\beta^\mu=0}{=} \frac{1}{2}g^{\mu\nu}\frac{\partial g_{\mu\nu}}{\partial\tau} = \frac{1}{\sqrt{\vert g\vert}}\frac{\partial\sqrt{\vert g\vert}}{\partial\tau} = \nabla_\mu J_{\mathrm{geom}}^\mu \ge 0$; $\Theta_E \equiv \frac{\dot{s}_{\mathrm{prod}}}{s_0} = -\frac{\beta_{\mathrm{th}}}{s_0}J_E^\mu\nabla_\mu\mathcal{F}$; $\Theta_{\mathrm{tot}} \equiv \sqrt{\theta^2 + \Theta_E^2}$, $\tau_\Theta(\tau) \equiv \int_{\tau_0}^{\tau}\langle\Theta_{\mathrm{tot}}\rangle_{\mathcal{M}}\,d\tau'$; $\frac{dV_{\mathcal{M}}}{d\tau} = \int_{\mathcal{M}}\theta\sqrt{\vert g\vert}\,d^nx = \oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$, with $\lim_{\tau\to\infty}V_{\mathcal{M}}(\tau) = V_\infty < \infty$ (Branch II); $\frac{\partial\mathcal{H}}{\partial\tau} \equiv J_E^\mu\nabla_\mu q$, $q(\mathbf{x}, \tau) \equiv -\ln(v_0Q(\mathbf{x}, \tau))$.
Physical Instantiation: During the window the stratum swells as pore pressure rises, so $\theta \ge 0$, and the surface rises. The volume growth equals the boundary flux of an expansion current of divergence $\theta$, of which only that divergence and boundary flux are used (§0.1 item 5); recharge is throughput and is never mapped to $J_{\mathrm{geom}}$. As the head approaches its new steady field, $\theta \to 0$ and $V_{\mathcal{M}} \to V_\infty$. Openness is carried by throughput: groundwater flow from recharge to drain and leakage dissipates energy at every point, so $\Theta_E > 0$ (by $O_1$), and $\tau_\Theta$ is driven by $\theta$ and $\Theta_E$ during the transient and by $\Theta_E$ alone as $\theta \to 0$. The record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput current. The stratum's consolidation and stress history (its preconsolidation state, fabric, and stratigraphic record) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$, and enters each window through the declared $\mathcal{H}_0$. The sediment mass balance at the surface is the Exner equation; its memory content is mapped via Spine $\mathcal{T}$ to the hysteresis law, which is not itself the Exner equation.

$O_1$ — Non-Equilibrium Thermodynamic Openness¶

Spine Equations: $\dot{W}_{\mathrm{ext}} = p_0\dot{V}_{\mathcal{M}} = p_0\oint_{\partial\mathcal{M}}J_{\mathrm{geom}}^\mu\,d\Sigma_\mu \ge 0$; $\dot{W}_{\mathrm{diss}} \equiv \int_{\mathcal{M}}(-J_E^\mu\nabla_\mu\mathcal{F})\sqrt{\vert g\vert}\,d^nx > 0$; $\dot{W}_{\mathrm{tot}} \equiv \dot{W}_{\mathrm{ext}} + \dot{W}_{\mathrm{diss}} > 0$; $\nabla_\mu\mathcal{F} \neq 0$; $J_E^\mu \equiv \frac{J_W^\mu}{p_0} = -\frac{1}{p_0}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$ (Branch II, fixed); $J_E^\mu\nabla_\mu\mathcal{F} < 0$; $\dot{s}_{\mathrm{prod}} = -\beta_{\mathrm{th}}J_E^\mu\nabla_\mu\mathcal{F} > 0$.
Physical Instantiation: The maintained head difference between the recharge and drain boundaries holds the stratum far from equilibrium. With $J_E^\mu = -p_0^{-1}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$ for the layered $\mathfrak{K}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$. Groundwater flows down the head gradient from recharge to drain, in the direction of $J_E^\mu$, so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. $\dot{W}_{\mathrm{ext}} = p_0\dot{V}_{\mathcal{M}} \ge 0$ is positive during uplift and tends to $0$, so $\dot{W}_{\mathrm{tot}} > 0$ is carried asymptotically by $\dot{W}_{\mathrm{diss}}$. The empirical content of $O_1$ is the flux-prediction test (aperture_generation_guidance.md §2.3, §5.5 item 6). $D_H$ and $D_V$ are estimated per closure (ii) from a declared hydraulic perturbation test on the same stratum before the window, with the skeleton part of $S_s$ from the stratum's measured strain response to a known head change and the water-compressibility part from tabulated values. The declared law must predict $J_W^{\mathrm{meas}} = (\rho g/S_s)\,q^{\mathrm{meas}}$ across a declared section $S$, for example from measured drain discharge, with $\nabla\mathcal{F}$ from piezometers that use neither $q^{\mathrm{meas}}$ nor $\mathfrak{K}$.

$O_2$ — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

Spine Equations: $g_{\mu\nu}(\tau, \mathbf{x}) = \Omega^2(\tau, \mathbf{x})\hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}(\tau, \mathbf{x}))^\alpha_{\;\nu}$ with $\det\hat{g} = 1$, $\mathrm{Tr}(\hat{h}) = 0 \implies \det g = \Omega^{2n}$, $\frac{\partial\ln\Omega}{\partial\tau} = \frac{1}{n}\theta$; $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$; $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$, $R = \kappa_G\nabla_\alpha\xi^\alpha$; $\frac{2(n-1)}{n}\nabla^2\theta + \frac{2R}{n}\theta = -\kappa_G\frac{\partial(\nabla_\alpha\xi^\alpha)}{\partial\tau} + 2\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} - 2\sigma^{\alpha\beta}R_{\alpha\beta}$; $\left.\left(\frac{dv^\mu}{d\tau} + \Gamma^\mu_{\rho\epsilon}v^\rho v^\epsilon + g^{\mu\lambda}\left(\frac{\partial g_{\lambda\nu}}{\partial\tau}\right)v^\nu - \frac{1}{2}\left(\frac{\partial g_{\rho\epsilon}}{\partial\tau}v^\rho v^\epsilon\right)v^\mu\right)\right\vert_{N=1,\,\beta^\mu=0} = f^\mu_{\mathrm{drive}}$, $f^\mu_{\mathrm{drive}} = (1-u^2)\left(a^\mu_{\mathrm{d}} - c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu\right)$, $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2 < 1$, with the default $a^\mu_{\mathrm{d}} = 0$.
Physical Instantiation: The physical metric is the flat pull-back metric of the swelling skeleton. Its Weyl split puts the local volume ratio in $\Omega^n = \sqrt{\det g}$ and the trace-free part of the logarithmic strain in $\hat{h}$, with $\partial_\tau\ln\Omega = \theta/3$. The split isolates the volume element only algebraically; $\theta$ remains coupled to $\xi_\alpha = \hat{\Pi}_{\alpha\lambda}\eta^\lambda$ and $\sigma_{\alpha\beta}$ through the traced elliptic $\tau$-compatibility equation (r10 Decision 2). On this domain ($G_{\mu\nu} \equiv 0$, $\xi \equiv 0$; §0.1 item 3) it reduces to $\frac{2}{3}\nabla^2\theta = \nabla^\alpha\nabla^\beta\sigma_{\alpha\beta}$, the traced Saint-Venant compatibility, which the skeleton's strain rate satisfies because its metric stays flat. Bedrock strata and load-bearing spans do not curve the physical metric: their stresses balance applied load in the Cauchy sense, which is not $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$. $\Pi_{\mu\nu}$ is the Hessian of a declared precision potential and enters only the coupling sector ($O_6$). Eq. [3] is instantiated with the default drive and maps no tracer. Principal-stress lines are integral curves of eigenvector fields of the stress, and funicular arches and catenaries solve equilibrium problems, so neither is a trajectory of Eq. [3]. Bedload is forced and frictional, and saltating grains are accelerated by gravity and drag; no drive is declared for them in this epoch.

$O_3$ — Shear Damping & Hysteresis Consolidation¶

Spine Equations: $\left.\frac{\partial g_{\mu\nu}}{\partial\tau}\right\vert_{N=1,\,\beta^\mu=0} = \frac{2}{n}\theta g_{\mu\nu} + 2\sigma_{\mu\nu}$; $\nabla_\tau\sigma^\mu_{\;\nu} \equiv \partial_\tau\sigma^\mu_{\;\nu} = -\gamma_\sigma\sigma^\mu_{\;\nu}$ ($\gamma_\sigma = \mathrm{const} > 0$, $\sigma^\mu_{\;\mu} = 0$) $\implies \lim_{\tau\to\infty}\Vert\sigma^\mu_{\;\nu}\Vert_{C^2(\mathcal{M})} = 0$; $\hat{h}(\tau, \mathbf{x}) \xrightarrow{C^2(\mathcal{M})} \hat{h}_\infty(\mathbf{x})$, $(\hat{g}_\infty(\mathbf{x}))_{\mu\nu} \equiv \hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}_\infty(\mathbf{x}))^\alpha_{\;\nu}$ ($\det\hat{g}_\infty = \det\hat{g} = 1$); $V_{\mathrm{local}}(\tau) \equiv \int_{\mathcal{U}_i}\mathcal{H}(\mathbf{x}, \tau)\sqrt{\vert g\vert}\,d^nx$.
Physical Instantiation: The trace-free part of the skeleton's strain rate is $\sigma^\mu_{\;\nu}$; for approximately uniaxial strain normal to the bedding it is proportional to $\theta$, with $\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu} \approx \frac{2}{3}\theta^2$. After the declared increase, the head transient is a sum of decaying modes of the stratum. Once the higher modes have decayed below a declared tolerance, the slowest mode dominates and $\sigma$ decays exponentially at a single constant rate. The aperture declares that rate as $\gamma_\sigma = \lambda_1$, the lowest eigenvalue of $-\nabla_\alpha(\mathfrak{K}^{\alpha\beta}\nabla_\beta\,\cdot\,) + k_L$ on the stratum, with fixed heads on the recharge and drain boundaries and no flow across the confining surfaces, computed from the independently estimated parameters. This is the swelling counterpart of Terzaghi consolidation. The head rises monotonically, so $\sigma$ keeps one sign. The shear rate decays while the accumulated strain increases monotonically to its final value, so $\hat{h} \to \hat{h}_\infty$ and $(\hat{g}_\infty)_{\mu\nu} = \hat{g}_{\mu\alpha}(\exp\hat{h}_\infty)^\alpha_{\;\nu}$, with unit determinant. The late part of the window supports the shear-rate test of aperture_generation_guidance.md §5.5 item 1, with $\gamma_\sigma$ predicted in advance, to within a declared residual from the higher modes; this aperture is not among the systems of the current shear-relaxation test. Elastic swelling leaves no plastic record in the window, so $V_{\mathrm{local}}$, the hysteresis content of a stratum, carries the earlier plastic history entered through $\mathcal{H}_0$.

$O_4$ — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

Spine Equations: $\frac{\partial\mathcal{F}}{\partial\tau} + \mathcal{F}\theta = -\nabla_\alpha J_W^\alpha - \mathcal{R}_{\mathcal{F}}$, $J_W^\alpha \equiv -\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}$, $\mathfrak{K}^{\alpha\beta} = \mathfrak{K}^{\beta\alpha} \succ 0$, $\mathcal{R}_{\mathcal{F}} \ge 0$; $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}} \equiv \mathbf{1}_{(-\nabla^2 \le \Lambda_{\mathrm{gate}}^2)}$; $Q(\mathbf{x}, \tau) = \langle\mathbf{x}\vert\rho_{\mathrm{local}}(\tau)\vert\mathbf{x}\rangle > 0$; $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P) > 0$; $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$.
Physical Instantiation: On the declared domain the balance law reads $\partial_\tau\mathcal{F} + \mathcal{F}\theta = \nabla_\alpha(\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}) - k_L(\mathcal{F} - \mathcal{F}_{\mathrm{ref}})$: the leaky confined-aquifer storage equation in energy-density form, with the $\mathcal{F}\theta$ term of relative order $\varepsilon_v$ (precondition 3). The layered $\mathfrak{K}$ and the leakage law depart from the default and satisfy the binding closure, and each is rejected if the default fits at least as well (closure (iv)). The gate sector follows §0.1 item 7: the Neumann spectrum of $-\nabla^2_g$ drifts by a fraction of the order of the strain, which is negligible within the window, and it converges as $g \to G_\infty$. The faster decay of fine spatial structure in the head field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. Seismic and vibrational filtering by the landform involves elastic waves, which lie outside the storage–diffusion domain, and is not mapped to the gate. The stress-conservation equivalence holds identically (§0.1 item 3). Effective stress and overburden are not $T_{\mu\nu}^{\mathrm{mat}}$, and vertically stratified stiffness and confinement profiles do not make $G_{\mu\nu} \not\equiv 0$ on this flat domain; they enter through the boundary conditions, $S_s$, and $\mathcal{H}$.

$O_5$ — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

Spine Equations: $g_{\mu\nu}^{(ij)} \equiv (g^{(i)} \# g^{(j)})_{\mu\nu}$; $\mathcal{U}_{ij}$ geodesically convex for $g^{(ij)}(\tau) \; \forall \tau \ge \tau_0$; $\phi_{\mathrm{perc}}(\mathcal{K}) = \vert\mathcal{K}\vert/\vert\mathcal{K}_0\vert > \phi_c$; $\Delta_{\mathrm{Hodge}}\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} = 0$; $\partial_\tau\sqrt{\vert g^{(i)}\vert} \ge 0$; $g_{\mu\nu}^{(i)} \xrightarrow{C^2(\mathcal{U}_i)} G_{\infty,\mu\nu}^{(i)} \succ 0$ (standing hypothesis with $H_0 = 0$).
Physical Instantiation: The cover consists of $\tau$-independent material strata, with overlaps $\mathcal{U}_{ij}$ across bedding contacts or an unconformity, convex at $\tau_0$ in $g$. Under the small, approximately uniaxial strain they stay convex, and the aperture checks this on the measured metric. Each $g^{(i)}$ is the restriction of $g$, so the gluing holds identically (§0.1 item 8). $\partial_\tau\sqrt{\vert g^{(i)}\vert} = \theta\sqrt{\vert g^{(i)}\vert} \ge 0$, and $g^{(i)} \to G_\infty^{(i)}$ as the strain approaches its final value ($H_0 = 0$). Every overlap is crossed by the transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The rigidity percolation of force chains in a granular skeleton is a microstructural property that sets the skeleton's compressibility; it is mapped via Spine $\mathcal{T}$ to the percolation condition only as context. The declared stratum is simply connected, so $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$ (§0.1 item 8); self-equilibrated stress states of the body are structural stresses outside $T_{\mu\nu}^{\mathrm{mat}}$.

$O_6$ — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

Spine Equations: $(J_{[i\to j]})_\nu = -(J_{[j\to i]})_\nu$; $M^{(ij)}(\mathbf{x}) \neq 0 \; \forall \mathbf{x} \in \bar{\mathcal{U}}_{ij}$ (hypothesis) $\wedge \; \mathrm{Tr}\,M^{(ij)} = 0 \implies m_+ > 0 > m_-$; $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) = k_{\mathcal{I}}\oint_{\partial\mathcal{U}_{ij}} S_{\mu\nu}^{(ij)}\hat{n}^\mu\hat{n}^\nu\,dA > 0$ (hypothesis, $\langle i,j\rangle \in \mathcal{K}$), with this boundary flux integrable on $[\tau_0, \infty)$ (Branch II).
Physical Instantiation: Take two strata that share a bedding contact or unconformity $\mathcal{U}_{ij}$. The partner fields are the rate of change of the groundwater free-energy flux in the two orientations (§0.1 item 8): a spatially resolved flux field transmitted from one stratum into the other, not a temporal lag between recharge and response. With fixed heads on the recharge and drain boundaries, the late-time zero set of $\partial_\tau J_W$ is an interior surface between them: nearer the recharge boundary $\partial_\tau J_W$ points toward it, and nearer the drain away from it. On a declared overlap that the predicted zero set of $\partial_\tau J_W$ does not reach during the window (§0.1 item 8), $\eta_{[i\to j]}$ keeps one direction, which the declaration states from the predicted solution. The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is its image under the declared non-singular precision map and is non-zero there. Under the non-degeneracy hypothesis $M^{(ij)} \neq 0$, built per r7 $O_6$ Eq. [3] from these fields and the declared precision potentials, $\mathrm{Tr}\,M^{(ij)} = 0$ gives $m_+ > 0 > m_-$, with $m_+ \neq -m_-$ in general. Under the flux-positivity hypothesis the coupling transmits correlation between the subsystem marginals of the declared state sector ($\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$). Both are standing hypotheses. In Branch II the transient decays at the rate $\gamma_\sigma$ of $O_3$, so the coupling flux is integrable on $[\tau_0, \infty)$ and no sustained coupling is asserted. For a curved contact, any threshold of accumulated strain beyond which the overlap would lose convexity is an aperture-level empirical conjecture mapped to the r10 §5.12 witness criterion $a_\infty = 2s_\sigma/\gamma_\sigma < a_c(E)$, which is a design constraint of that witness. No such threshold is asserted, and convexity is checked on the measured metric.

$O_7$ — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

Spine Equations: $D_\mu = \partial_\mu - i g_{\mathrm{YM}}\,\mathrm{ad}_{A_\mu}$; $\Vert\hat{\mathcal{E}}_\mu\Vert_{C^1(\mathcal{M})} \to 0 \implies [F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] \to 0$; $\chi_E \equiv 1 + \frac{n\,\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu}}{\theta^2 + \langle\Theta_E\rangle_{\mathcal{M}}^2} \to 1$, $\hat{\chi} \equiv \chi_E\hat{u}$ (Branch II); $D_{\mathrm{KL}}(\hat{\rho}_g(\tau) \parallel \hat{\rho}_{\mathcal{H}}(\tau)) \to 0$.
Physical Instantiation: The gauge sector is the trivial declaration of §0.1 item 9, so $[F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] = 0$ identically and $\hat{\mathcal{E}}_\mu \to 0$ as $\sigma \to 0$. With $\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu} \approx \frac{2}{3}\theta^2$, $\chi_E - 1 \approx 2\theta^2/(\theta^2 + \langle\Theta_E\rangle^2) \to 0$ as swelling ceases. As in Aperture 2.2, its late-window decay rate $2\gamma_\sigma$ applies only once $\theta^2 \ll \langle\Theta_E\rangle^2$, whose onset depends on $s_0$, so the $s_0$-independent observable is the decay rate of $\sigma$ ($O_3$). $\chi$ is not used. $\hat{\rho}_g \to \frac{1}{n}\delta^\mu_\nu$ as $\hat{h} \to \hat{h}_\infty$, a residual computable from the deformation record. Alignment with $\hat{\rho}_{\mathcal{H}}$ requires the Hessian of the record to become isotropic, an asymptotic condition stated and not tested. The $O_7$ Eq. [3] identities hold as in Aperture 2.2. The aperture claims no smoothing of stress concentrations or alignment of load paths with the consolidated morphology; those belong to the structural stress field, which lies outside $T_{\mu\nu}^{\mathrm{mat}}$.

$O_\Omega$ — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Spine Equations: Branch II ($H_0 = 0$): $\theta \xrightarrow[\tau\to\infty]{C^2(\mathcal{M})} 0$ with $\int_{\tau_0}^{\infty}\langle\theta\rangle_{\mathcal{M}}\,d\tau < \infty$ and $\langle\Theta_E\rangle_{\mathcal{M}} \to \Theta_\infty > 0$; $\sigma^\mu_{\;\nu} \to 0$; $\partial_\tau g_{\mu\nu} \to 0$; $V_{\mathcal{M}} \to V_\infty$ and $\mathcal{I}(\tau) \to s_0V_\infty$; $\left.\hat{\mathbb{L}}_{\mathrm{stat}}(\mathbf{x})\Psi[g_{\mu\nu}, \mathcal{H}]\right\vert_{\mathcal{H}=\mathcal{H}_{\mathrm{sol}}[g]} = (J_E^\lambda\nabla_\lambda q)\frac{\delta\Psi}{\delta\mathcal{H}(\mathbf{x})} \xrightarrow[\tau\to\infty]{} 0 \iff J_E^\lambda\nabla_\lambda q \to 0 \iff \frac{\partial\mathcal{H}}{\partial\tau} \to 0$ (with $\vert\delta\Psi/\delta\mathcal{H}\vert$ bounded and bounded away from $0$ uniformly as $\tau \to \infty$).
Physical Instantiation: Under maintained recharge the stratum approaches a steady head field with continuing recharge, leakage, and drain discharge: a bounded non-equilibrium steady state. Uplift ceases ($\theta \to 0$), $\int\langle\theta\rangle\,d\tau$ is bounded by the final volumetric strain, $V_{\mathcal{M}} \to V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$, $\sigma \to 0$, and $\partial_\tau g \to 0$. The transient coupling decays, so $\mathcal{I} \to s_0V_\infty$. The generator condition requires $J_E^\lambda\nabla_\lambda q \to 0$, an asymptotic hypothesis on the declared state sector that no finite window tests. The state is neither static equilibrium, which would require a uniform head at the drain level and no flow, nor a periodic return. The limit belongs to the window's held boundary data. A physical landform keeps evolving by erosion and deposition at its surface between windows, which changes the boundary data and opens a new window; this is never read as a return. Within windows the supported tests (aperture_generation_guidance.md §5.5) are the same as for Aperture 2.2: (1) and (3) with the predicted $\gamma_\sigma$, and (4)–(7). The $\chi_E$ rate test (2) is not among them, because its regime has an $s_0$-dependent onset.

Tier 2 Specification: Abiotic, Continuum & Tectonic Physics¶

0. Tier Architecture & Four-Part Functional Homology¶

0.1 Conventions Common to the Four Tier 2 Apertures¶

Aperture 2.1: Condensed-Matter & Phase Transition Physics (\(\mathcal{E}_{\mathrm{condensed}}\))¶

1. Epistemological & Physical Scope¶

2. Coordinate-by-Coordinate Mapping (\(O_0 \to O_\Omega\))¶

\(O_0\) — Exogenous Flux & Emergent Foliation Time¶

\(O_1\) — Non-Equilibrium Thermodynamic Openness¶

\(O_2\) — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

\(O_3\) — Shear Damping & Hysteresis Consolidation¶

\(O_4\) — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

\(O_5\) — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

\(O_6\) — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

\(O_7\) — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

\(O_\Omega\) — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Aperture 2.2: Continuum Fluid & Wave Mechanics (\(\mathcal{E}_{\mathrm{continuum}}\))¶

1. Epistemological & Physical Scope¶

2. Coordinate-by-Coordinate Mapping (\(O_0 \to O_\Omega\))¶

\(O_0\) — Exogenous Flux & Emergent Foliation Time¶

\(O_1\) — Non-Equilibrium Thermodynamic Openness¶

\(O_2\) — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

\(O_3\) — Shear Damping & Hysteresis Consolidation¶

\(O_4\) — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

\(O_5\) — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

\(O_6\) — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

\(O_7\) — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

\(O_\Omega\) — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Aperture 2.3: Atmospheric Radiative & Thermal Boundary Physics (\(\mathcal{E}_{\mathrm{radiative}}\))¶

1. Epistemological & Physical Scope¶

2. Coordinate-by-Coordinate Mapping (\(O_0 \to O_\Omega\))¶

\(O_0\) — Exogenous Flux & Emergent Foliation Time¶

\(O_1\) — Non-Equilibrium Thermodynamic Openness¶

\(O_2\) — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

\(O_3\) — Shear Damping & Hysteresis Consolidation¶

\(O_4\) — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

\(O_5\) — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

\(O_6\) — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

\(O_7\) — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

\(O_\Omega\) — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶

Aperture 2.4: Geomorphic & Structural Mechanics (\(\mathcal{E}_{\mathrm{tectonic}}\))¶

1. Epistemological & Physical Scope¶

2. Coordinate-by-Coordinate Mapping (\(O_0 \to O_\Omega\))¶

\(O_0\) — Exogenous Flux & Emergent Foliation Time¶

\(O_1\) — Non-Equilibrium Thermodynamic Openness¶

\(O_2\) — Unimodular Weyl Factorization, Constraint Precision & Trajectories¶

\(O_3\) — Shear Damping & Hysteresis Consolidation¶

\(O_4\) — Free-Energy Balance, Spectral Low-Pass Gating & Boundary Stress Conservation¶

\(O_5\) — Riemannian Geometric Mean Gluing, Percolation & Harmonic Circulation¶

\(O_6\) — Asymmetric Phase-Lead Coupling & Trace-Free Deformation Operator¶

\(O_7\) — Principal-Bundle Gauge Coherence, Impedance Decay & Hysteresis Alignment¶

\(O_\Omega\) — Topological Asymptote, Branch II: Bounded Non-Equilibrium Steady State¶