Tier 3 Specification: Ecological, Metabolic & Phenological Networks¶

Document ID: tier_3_ecological_living_networks.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 3 maps vascular plant physiology, movement ecology, benthic and edaphic biogeochemistry, and chronobiology 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, storage coefficient, 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 stored quantity (xylem water, prey biomass, a dissolved oxidant, a signal molecule) spreading through a living or biologically structured medium, with supply crossing the boundary and consumption, where declared, by a named interior sink. 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 physiology, ecology, and biogeochemistry, so that it applies across terrestrial, freshwater, and marine ecosystems 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 3.1: Structural Substrate	Vascular Plant Physiology & Rhizosphere Coupling ($\mathcal{E}_{\mathrm{botanical}}$)	Water in the conducting xylem of the main stem, moving from the root collar to the crown base down its water-potential gradient.	Rhizosphere and mycorrhizal uptake, crown and leaf storage, stomatal regulation, phloem transport, growth, stem sway, reaction wood: as boundary data, through $\mathcal{H}$, or context.	$1.1 \leftrightarrow 2.1 \leftrightarrow \mathbf{3.1} \leftrightarrow 4.1 \leftrightarrow 5.1$
Slot 3.2: Dynamic Flow & Waves	Movement Ecology & Ecotone Foraging Dynamics ($\mathcal{E}_{\mathrm{trophic}}$)	Prey biomass dispersing into a three-dimensional foraging volume across an ecotone and consumed by predators, whose trajectories follow a declared taxis drive.	Foraging decisions, population cycles, prey reproduction, migration, learned routes: through the declared parameters, as boundary data, through $\mathcal{H}$, or context.	$1.2 \leftrightarrow 2.2 \leftrightarrow \mathbf{3.2} \leftrightarrow 4.2 \leftrightarrow 5.2$
Slot 3.3: Boundary Gating	Benthic & Edaphic Biogeochemical Metabolism ($\mathcal{E}_{\mathrm{metabolic}}$)	A dissolved oxidant diffusing from overlying water through the diffusive boundary layer into a sediment column, consumed by microbial respiration.	The sequence of other electron acceptors, bioturbation, humification, extracellular enzymes: through the declared parameters, the lower boundary, $\mathcal{H}$, or context.	$1.3 \leftrightarrow 2.3 \leftrightarrow \mathbf{3.3} \leftrightarrow 4.3 \leftrightarrow 5.3$
Slot 3.4: Coalescence & Hysteresis	Chronobiology & Phenological Synchrony ($\mathcal{E}_{\mathrm{chrono}}$)	A humoral or paracrine entraining signal diffusing into an avascular target tissue from cells outside it, and degraded there.	Pacemaker oscillators, entrainment and phase response, dormancy, vernalization, mast seeding, succession: as boundary data, through $\mathcal{H}$, or context.	$1.4 \leftrightarrow 2.4 \leftrightarrow \mathbf{3.4} \leftrightarrow 4.4 \leftrightarrow 5.4$

0.1 Conventions Common to the Four Tier 3 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. Where an item repeats a convention of tier_2_abiotic_continuum_physics.md §0.1, it is restated here so that this tier can be read on its own.

Branch and windows. Branch II is declared before any test, with the estimators of item 7 and §1, and no Tier 3 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 at a declared $\tau_0$ from a prior steady state, a declared transient starts at $\tau_0$, the external data are then held within declared tolerances, and the window follows the domain as it settles.
In 3.1, 3.3, and 3.4 the transient is a declared step of the boundary data in the direction that raises $\mathcal{F}$: a step decrease of evaporative demand at the leaves (3.1), and a step increase of the oxidant concentration of the overlying water (3.3) or of the signal concentration at the supply interface (3.4). Each declared balance law is a scalar parabolic equation whose sink is Lipschitz and non-decreasing in $\mathcal{F}$ (zero, first order, Michaelis–Menten), so the comparison principle holds. The prior steady state is a subsolution for the raised data, so $\mathcal{F}(\cdot, \tau_0 + h) \ge \mathcal{F}(\cdot, \tau_0)$; the held data make the problem invariant under time translation, so $\mathcal{F}(\cdot, \tau + h) \ge \mathcal{F}(\cdot, \tau)$, that is, $\partial_\tau\mathcal{F} \ge 0$ at every point. Where the step is in fact a ramp (non-decreasing boundary data that reach their held value over a finite rise time, as for secretion in 3.4), the same conclusion follows without time translation: $\partial_\tau\mathcal{F}$ obeys the linearized equation with zero initial data (the prior steady state) and non-negative boundary data, so it stays non-negative. In 3.1 the xylem also exchanges water with capacitive living tissue outside the $\mathcal{F}$ field; that coupled linear system is cooperative, and the same conclusion holds for it (3.1 §1). The argument is exact on rigid domains. Under the optional torsional perturbation of 3.1 the operator depends on $\tau$ through $g(\tau)$ at the order of the shear strain, so there the conclusion holds only to that order, and 3.1 checks it from its water-potential record.
In 3.2 the prey data are held at their prior steady values, so $\mathcal{F}$ is steady within the window, which aperture_generation_guidance.md §2.3 item 3 admits. The transient is the mixing of two predator groups, labelled at $\tau_0$ by the patch each animal occupies (item 9 and 3.2 $O_6$).
The rate of change of the flux, $\partial_\tau J_W$, is not one-signed in general (item 9).
Metric. The physical metric is the pull-back of the Euclidean metric to the material points of the declared body under its deformation map (for 3.2, to a fixed region of space). 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 3 aperture declares a non-Euclidean material metric in this epoch: the growth material metric that the master table admits for 3.1 is discussed in 3.1 §1 and not declared. The curved configurations of the r10 §5.11 and §5.13 witnesses are therefore realized by none of these apertures, and r7's curvature sector is exercised by no Tier 3 aperture.
Rigid domains (3.2, 3.3, 3.4). In a Cartesian chart $g_{\mu\nu} = \delta_{\mu\nu}$, so $\Omega \equiv 1$, $\hat{h} \equiv 0$, $\theta \equiv 0$, and $\sigma^\mu_{\;\nu} \equiv 0$. For 3.2 this holds by construction, because the declared domain is a fixed region of space. For 3.3 and 3.4 the body is declared rigid within a pre-declared tolerance on its measured volumetric strain, and 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.
Volume-rigid domain (3.1). The volumetric strain of the conducting xylem stays within a declared tolerance, so $\theta \equiv 0$ in the declared metric. Without the optional torsional perturbation, the shear strain also stays within tolerance and 3.1 is treated as rigid. With it, the metric is the pull-back under an isochoric twist: in a Cartesian material chart $g(\tau) = \mathbf{F}^{\mathsf{T}}\mathbf{F}$ with $\det\mathbf{F} = 1$, so $\Omega \equiv 1$, $\theta \equiv 0$, $\exp\hat{h} = g$, and $\sigma^\mu_{\;\nu} \neq 0$. On that deforming domain every covariant quantity (Laplacian, Hessians, convexity, harmonic fields) is evaluated with $g(\tau)$.
Storage. The r7 $O_4$ balance law carries unit storage: $\partial_\tau\mathcal{F}$ is the rate of change of the stored quantity per unit volume. A living medium stores its diffusing quantity $u$ with a coefficient $s$ (stored amount per unit volume per unit of $u$): the hydraulic capacitance of stem sapwood in 3.1, unity in 3.2, the sediment porosity in 3.3, and the extracellular volume fraction times a binding retardation factor in 3.4. With $s$ uniform, the physical balance $s\,\partial_\tau u = \nabla_\alpha(k^{\alpha\beta}\nabla_\beta u) - r$ becomes the r7 law for $\mathcal{F} = \epsilon u$ (with $\epsilon$ a declared constant yield), $\mathfrak{K} = k/s$, and $\mathcal{R}_{\mathcal{F}} = \epsilon r/s$, and the r7 flux is $J_W = (\epsilon/s)\,j$, with $j = -k\nabla u$ the physical flux. The factor $\epsilon/s$ is a declared conversion factor, estimated independently (closure (ii)). If $s$ varies in space, $s^{-1}\nabla_\alpha(k^{\alpha\beta}\nabla_\beta u) = \nabla_\alpha(s^{-1}k^{\alpha\beta}\nabla_\beta u) + s^{-2}k^{\alpha\beta}\nabla_\alpha s\,\nabla_\beta u$, and the last term is a drift term that the r7 balance law does not contain. Each aperture therefore declares $s$ uniform on $\mathcal{M}$ within a tolerance (a declared, checkable precondition), or, in 3.3, states where the declared law departs from the physical one and bounds that departure.
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. No Tier 3 medium has an identified global control field in this epoch, so Tier 3 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; with $\theta \equiv 0$ (all Tier 3 domains) they read $\mathbb{L}_{\mathrm{Ein}}[\sigma] = 0$ and $\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} = 0$. The mechanical stresses of living media (turgor, growth stresses in wood, tissue and sediment stresses) are not $T_{\mu\nu}^{\mathrm{mat}}$; they enter as boundary conditions, through the declared parameters, and through $\mathcal{H}$. The partner fields $\eta_{i\to j}$ of $O_6$ (item 9) are distinct objects from the global $\eta$ of $O_4$.
Cone speed and trajectories. Each aperture declares $c_*$ as a characteristic signal speed of its medium (a sound or pressure-wave speed). 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 and animal speed in $\mathcal{M}$ is $\ll c_*$, so $u^2 \ll 1$ and $f_{\mathrm{drive}} \approx a_{\mathrm{d}}$. The trajectory equation is displayed in physical units: both $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2$ and the projection term $c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu$ carry $c_*$, and with $c_* = 1$ the display reduces to the r7 form. Apertures 3.1, 3.3, and 3.4 instantiate Eq. [3] with $a^\mu_{\mathrm{d}} = 0$ and map no tracer. Aperture 3.2 declares a taxis drive for predator trajectories; 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 $$p_0\dot{W}_{\mathrm{diss}} = \oint_{\partial\mathcal{M}}\mathcal{F}\,(-J_W^\mu)\,d\Sigma_\mu - \int_{\mathcal{M}}\mathcal{F}\,\mathcal{R}_{\mathcal{F}}\sqrt{\vert g\vert}\,d^nx - \frac{1}{2}\frac{d}{d\tau}\int_{\mathcal{M}}\mathcal{F}^2\sqrt{\vert g\vert}\,d^nx - \frac{1}{2}\int_{\mathcal{M}}\mathcal{F}^2\theta\sqrt{\vert g\vert}\,d^nx$$
Where an outflow face is held at the datum $\mathcal{F}_{\mathrm{ref}} = 0$ (3.2, 3.3, 3.4), it contributes nothing and the boundary term reduces to the supply face. The outflow faces of 3.1 (crown base, cambial interface) are not at the datum, and their terms are evaluated from the measured outflow and the measured $\mathcal{F}$ there.
Each term is evaluated from measured $\mathcal{F}$, measured boundary fluxes, and measured or independently declared consumption; none uses the declared $\mathfrak{K}$. Each §1 names instruments and surfaces different from those of the $O_1$ flux-prediction test, so that the evidence is independent of that test; 3.3 states where this independence is weaker.
Because the identity uses the balance law, the estimate is conditional on it. Each §1 also names a directly measured physical dissipation whose sign the estimate must share. Every such dissipation is non-negative, by the second law (3.1, 3.3, 3.4) or because a living system's metabolic power is positive (3.2), so this comparison tests only the sign of the identity-based estimate, and it is stated as such. 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 3 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 shear strain under the torsional perturbation of 3.1, 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. Temporal smoothing of fluctuating supply by the medium is a property in time and is not the spatial gate.
Cover, overlaps, and coupling ($O_5$, $O_6$).
The cover $\{\mathcal{U}_i\}$ consists of $\tau$-independent material subregions of $\mathcal{M}$ (for 3.2, fixed subregions of space) with Lipschitz boundaries, and every overlap $\mathcal{U}_{ij}$ is convex at $\tau_0$. Where $\partial\mathcal{M}$ has concave parts (the core boundary of the conducting xylem in 3.1), each concave part lies in a single cover set, which need not be convex: r7 requires convex overlaps, not convex cover sets. 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 or diffusivities, including geometric-mean averages, are statements about $\mathfrak{K}$ and are not mapped to $O_5$ Eq. [1].
Partner fields. In 3.1, 3.3, and 3.4 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 $\eta_{[i\to j]}^\lambda = \partial_\tau J_W^\lambda$. In 3.2 they are the movement-flux fields of the two labelled predator groups (3.2 $O_6$), which are not antisymmetric by construction.
Zero sets. $\partial_\tau J_W$ is not one-signed in general. Where the supply boundary holds a value of $\mathcal{F}$ (3.3, 3.4), the flux near the supply decays after the step from an initial overshoot, so there $\partial_\tau J_W$ points toward the supply, while farther downstream the flux rises toward its new steady value, so there it points away from the supply. Between the two lies the zero set of $\partial_\tau J_W$. It travels during the early window, possibly overshooting its late position, and in the late single-mode regime it settles on a fixed surface. In a laterally uniform column, a precondition that 3.3 and 3.4 declare for this purpose, that surface is the level at which the slowest relaxation mode of the linearized problem peaks; without lateral uniformity it is located only by the predicted solution. In 3.1 the boundary data are of mixed kind (3.1 $O_6$), and the zero set is located only by the predicted solution. On the zero 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.
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. Before the transient arrives, the partner field 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 $\vert\eta_{[i\to j]}\vert$ exceeds that threshold on $\bar{\mathcal{U}}_{ij}$, and the data must then show $\vert\eta_{[i\to j]}\vert$ above it there (a declared, checkable precondition). On such an overlap $\eta_{[i\to j]}$ keeps one direction throughout the overlap's window, which each aperture states.
Precision potentials. The potentials $\Phi^{(i)}$ are declared before data collection as quadratic potentials (in a Cartesian chart, or in the Cartesian material chart for 3.1, with Hessians computed with $g(\tau)$ under the torsional perturbation). 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 3 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 3 also declares the two trace-free Hessians distinct, $\hat{\Pi}^{(i)} \neq \hat{\Pi}^{(j)}$. With antisymmetric partner fields, $J_{i\to j} + J_{j\to i} = (\hat{\Pi}^{(j)} - \hat{\Pi}^{(i)})\eta_{[i\to j]}$. Equal Hessians would therefore remove the gradient term of r7 $O_6$ Eq. [3] and leave $M^{(ij)}_{\mu\nu} = \gamma_M\big((J_{[i\to j]})_\mu(J_{[i\to j]})_\nu - \frac{1}{n}g_{\mu\nu}J_{[i\to j]}^\alpha(J_{[i\to j]})_\alpha\big)$, the trace-free part of a rank-one tensor, which is non-zero wherever $J_{[i\to j]} \neq 0$: the hypothesis would 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), so the construction neither forces nor forbids $M^{(ij)} \neq 0$, which remains a standing hypothesis. The labelled fields of 3.2 are not antisymmetric by construction, but their sum is small under its uniform-density precondition, so the same declaration is made there.
On a connected domain the declared transient reaches every overlap for $\tau > \tau_0$ (in 3.2, the labelled groups mix across every overlap), so every edge of $\mathcal{K}_0$ lies in $\mathcal{K}$ and $\phi_{\mathrm{perc}} = 1 > \phi_c$; empirically, an edge is counted when its partner field exceeds a pre-declared detection threshold on its overlap. 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 or of movement. In Branch II the transient decays as the domain settles (in 3.2, as the groups mix), so each coupling-tensor boundary flux decays and is integrable on $[\tau_0, \infty)$; no sustained coupling is asserted.
Topology. The domains of 3.2, 3.3, and 3.4 are declared simply connected, so $H^1_{\mathrm{dR}}(\mathcal{M}) = 0$ and $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$. The conducting xylem of 3.1 generally surrounds a non-conducting core (pith or heartwood), so its first Betti number $b_1(\mathcal{M})$ is at least one and the space of harmonic Neumann fields $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}}$ of r7 $O_5$ Eq. [5] is non-trivial: a property of the domain's topology, computed slice-wise with $g(\tau)$. Tier 3 identifies no transport current or measured field with it and makes no claim about it.
Gauge sector and hysteresis ($O_7$, $O_0$). Tier 3 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. In particular, no Tier 3 aperture makes a synchronization, alignment, or collective-coherence claim: such claims would belong to gauge transport (aperture_generation_guidance.md §5.5), not to $\chi_E$. 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 3 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 for 3.1 under the torsional perturbation. 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. Biological memory variables of each system (wood anatomy, learned routes, a sediment's chemical record, an entrained clock phase) 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 Aperture 3.1 is exercised by the shear-relaxation and flux-prediction families, 3.2 by the coupling family, 3.3 by the flux-prediction family, and 3.4 by no test (empirical_validation_protocol.md §3.3). Everything else is stated as untested, not as confirmed.

0.2 Coordinates That Hold Trivially or by Construction¶

On these declared domains several coordinates hold identically and carry no empirical content. They are listed here so that no block below is read as evidence for them.

Statement	Apertures	Why it holds
$\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, $V_{\mathcal{M}} = V_\infty$, $\int\langle\theta\rangle\,d\tau = 0$, $\dot{W}_{\mathrm{ext}} = 0$, and the $\theta$ parts of the Branch II evidence	All	3.2 by construction (a fixed region of space); 3.1, 3.3, and 3.4 by their rigid or volume-rigid declarations, so the evidence reduces to a tolerance check on measured volume.
$\sigma \equiv 0$, the $O_3$ shear law, $\hat{h} = \hat{h}_\infty$, and the tests (1)–(3) of `aperture_generation_guidance.md` §5.5	3.2, 3.3, 3.4; 3.1 without the torsional perturbation	Rigid declaration.
The $\partial_\tau g$ residual test (3); the $\chi_E$ rate test (2) where $\langle\Theta_E\rangle$ is constant	3.1 under the torsional perturbation	With $\theta \equiv 0$, $\partial_\tau g = 2\sigma$, so (3) is equivalent to the shear-rate test (1) unconditionally. (2) is equivalent to (1) only on sub-intervals of constant measured dissipation, because the decay rate of $\chi_E - 1$ is $2\gamma_\sigma + 2\,d\ln\langle\Theta_E\rangle/d\tau$ (3.1 $O_7$).
$J_E^\mu\nabla_\mu\mathcal{F} < 0$, $\dot{s}_{\mathrm{prod}} > 0$, $\dot{W}_{\mathrm{diss}} > 0$	All	Identities of the fixed Branch II current wherever $\nabla\mathcal{F} \neq 0$; the empirical content of $O_1$ is the flux prediction.
Sign agreement with a measured physical dissipation	All	The named dissipation is non-negative by the second law or by metabolism (item 7). In 3.2, $\dot{s}_{\mathrm{prod}}$ is the entropy production of the declared mapping, not a measured physical entropy production (3.2 §1).
Stress sector: $T^{\mathrm{mat}} \equiv 0$, the $O_4$ conservation equivalence, and the compatibility law (Saint-Venant form)	All	Flat metric and declared $\eta \equiv 0$ (items 2 and 4).
$O_2$ Eq. [3]	3.1, 3.3, 3.4	Default drive and no tracer mapped.
$O_5$ gluing, $g^{(i)} = G^{(i)}_\infty$, and overlap convexity	All (convexity is non-trivial only for 3.1 under the torsional perturbation)	Restricted metrics on a rigid or volume-rigid body (item 9).
$\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$	All	$\theta \equiv 0$ on every declared domain.
$\phi_{\mathrm{perc}} = 1 > \phi_c$	All	A connected domain whose transient reaches every overlap (item 9).
$\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$	3.2, 3.3, 3.4	Declared simple connectivity; for 3.1 the harmonic space is non-trivial and no claim is made about it.
$O_7$ Eq. [3] identities: divergence-free $J_{\mathrm{geom}}$ and $u_{\mathrm{exp}}$, and $\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0$	3.2, 3.3, 3.4; 3.1 without the torsional perturbation	$\theta \equiv 0$ and $\hat{h} \equiv \hat{h}_\infty$ on rigid domains.
$\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$	3.2, 3.3, 3.4	A zero datum for a non-negative concentration or density.
$\mathcal{I} \to s_0V_\infty$	All	The Branch II limit with which `r7` defines the finite capacity $s_0V_\infty$; it follows from the integrable coupling and is not tested.
$[F^{\mathrm{YM}}_{\mu\nu}, \hat{\chi}] = 0$; $\chi_E \equiv 1$ and $\hat{\mathcal{E}}_\mu \equiv 0$	All; the $\chi_E$ statements except 3.1 under the torsional perturbation	Trivial gauge sector and $\sigma \equiv 0$ (item 10).
The $O_4$ gate, the $O_\Omega$ generator condition, and the alignment $D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0$	All	Declared but not identified with any field; stated and not tested.

The empirical content of Tier 3 therefore sits in the $O_1$/$O_4$ flux predictions; the $O_3$ shear law of 3.1 under the torsional perturbation; the $O_6$ standing hypotheses on the declared overlaps; the sign of the measured-dissipation estimate; and the checkable preconditions of each domain.

Aperture 3.1: Vascular Plant Physiology & Rhizosphere Coupling ($\mathcal{E}_{\mathrm{botanical}}$)¶

1. Epistemological & Biological Scope¶

Aperture 3.1 resolves how vascular plants lift water from the soil, through the rhizosphere, roots, and conducting xylem, to the leaves, where it is transpired: water moves under the tension that evaporation generates (the cohesion–tension mechanism), is buffered by the water stored in tissues, is regulated at the leaves by stomata, and is coupled at the roots to the rhizosphere and its mycorrhizal symbionts. Lexicon is strictly restricted to plant physiology, plant hydraulics, dendrology, plant biomechanics, and rhizosphere and mycorrhizal ecology.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. The conducting xylem (sapwood) of the main stem, between the root collar and the crown base: the functional conduits and fibres of a single organ class, with the living ray and axial parenchyma embedded among them, treated at the tissue scale as a porous continuum. The domain is confined to one organ class because xylem capacitance per unit volume differs systematically between trunk, branches, petioles, and veins: a capacitance jump at an organ junction would make $J_W = q/C$ discontinuous where $q$ is continuous, which the unit-storage balance law reads as an interface source or sink (§0.1 item 3). $\partial\mathcal{M}$ consists of:
the root collar (supply face), across which water enters from the root system;
the crown-base cross-section (outflow), across which water leaves $\mathcal{M}$ for the crown: the branch, petiole, and vein xylem and the leaf mesophyll, from which it is transpired, all of which lie outside $\mathcal{M}$;
the cambial interface with the bark, across which water is exchanged with the bark's living tissue;
the boundary with the non-conducting core (pith or heartwood), which is impermeable.

All supply crosses the root collar; there are no interior sources. The soil, the rhizosphere, and the roots lie outside $\mathcal{M}$ and enter through the root-collar boundary condition: water enters at the rate set by the soil water potential, held during the window, and by declared root and rhizosphere conductances. For predictions (as in $O_6$) the crown (its xylem and leaf storage, with transpiration leaving it) and the bark are represented as capacitive compartments outside $\mathcal{M}$, with declared capacitances and conductances estimated independently. * Window. A relaxation window (§0.1 item 1), opened from a prior steady state by a declared step decrease of evaporative demand at the leaves (for example a step increase of air humidity around an enclosed shoot), after which evaporative demand, stomatal conductance, and soil water potential are held within declared tolerances. Water potential then rises at every point and tension relaxes, so no conduit embolizes, and water flows from the xylem into the living tissue, which it refills. Windows in which demand rises are excluded: there the xylem contracts as tension rises ($\theta < 0$), and living tissue discharges water into the xylem, which would be an interior source. The window is fixed: growth (new conduits, extension, radial increment) stays within a declared tolerance throughout it. * Coupled compartments. The stem xylem, the parenchyma embedded in it, and the bark and crown compartments form a linear system in which each compartment gains water from the others in proportion to their potential differences. Such a system is cooperative, so a step of the external data in the direction that raises potentials raises every potential monotonically: $\partial_\tau\mathcal{F} \ge 0$ throughout $\mathcal{M}$, and every compartment's potential stays below the xylem potential that feeds it, so its exchange with the xylem is one-signed, into the living tissue. * Optional torsional perturbation. A stem segment may be twisted and released at $\tau_0$ within such a window. The recovery of the twist then supplies a shear transient ($O_3$), while transpiration-driven flow continues to supply the throughput. * Metric. Flat and volume-rigid (§0.1 item 2): the volumetric strain of the conducting xylem stays within a declared tolerance (checked by point dendrometers anchored in the xylem), so $\theta \equiv 0$. Without the torsional perturbation the shear strain also stays within tolerance, and $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian material chart. With it, $g(\tau) = \mathbf{F}^{\mathsf{T}}\mathbf{F}$ is the pull-back under the twist, with $\det\mathbf{F} = 1$. For a segment twisted about its axis by an angle $\Phi_{\mathrm{tw}}(z, \tau)$, the deformation $(r, \varphi, z) \mapsto (r, \varphi + \Phi_{\mathrm{tw}}, z)$ is isochoric and maps each cylinder $r = \mathrm{const}$ and each cross-section onto itself. * Cone speed. $c_*$ is the speed of pressure waves in the sap-filled conduits, set by the compressibility of sap and the compliance of conduit walls. Sap velocities and twist-recovery speeds are $\ll c_*$. * Storage. The hydraulic capacitance $C$ of the stem sapwood: the water released per unit volume per unit decrease of water potential (m³ m⁻³ Pa⁻¹), from capillary and elastic storage in conduits and fibres and from the apoplast of the embedded parenchyma. It is declared uniform over $\mathcal{M}$ within a tolerance and linear over the window's range of potentials (§0.1 item 3). * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = \psi_{\mathrm{h}} - \psi_{\mathrm{ref}}$ (J m⁻³). Here $\psi_{\mathrm{h}}$ is the water potential of the xylem sap in hydraulic form: pressure plus gravitational potential $\rho_w g z$. The osmotic component of xylem sap is small, does not drive flow through the conduit network, and is declared constant within tolerance; psychrometric readings are corrected for it and for the gravitational term. The datum $\psi_{\mathrm{ref}}$ is the declared window minimum of leaf water potential, in the same hydraulic form. Potentials only rise during the window, so that minimum is the lowest leaf value of the prior steady state, measured at $\tau_0$ and fixed before the window opens. Water flows from the root collar toward the leaves, so the leaves hold the lowest potentials, and $\mathcal{F} \ge 0$ throughout $\mathcal{M}$. $\mathcal{F}$ is linear in the stored water: the water stored per unit volume above the datum is $C\mathcal{F}$, so $\mathcal{F}$ is that stored water times the declared conversion factor $1/C$. The relaxation time $\ell^2C/k$ is finite and resolvable. * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = k^{\alpha\beta}/C$, with $k^{\alpha\beta}$ the hydraulic conductivity tensor of the conducting xylem, parametric: axial and radial (and, if declared, tangential) conductivities along the stem's axes. The master table's "conductances" are these conductivities per unit area and gradient. Isotropic $k$ is the default. With $q$ the volumetric sap flux density (m s⁻¹), $J_W = q/C$; $1/C$ is the declared conversion factor. * Sink $\mathcal{R}_{\mathcal{F}}$. The default $0$, or a declared lateral loss to the living parenchyma embedded in $\mathcal{M}$, first order relative to the datum: $\mathcal{R}_{\mathcal{F}} = k_{\mathrm{p}}(\mathcal{F} - \mathcal{F}_{\mathrm{ref}}) \ge 0$, with $k_{\mathrm{p}}$ a declared rate constant. The physical exchange is driven by the difference between the xylem potential and the lagging potential of the parenchyma, so the first-order datum form is a declared idealization. It is admitted only under the binding closure, and it is rejected if the default fits at least as well (closure (iv)). Exchange with the bark across the cambial interface is a boundary flux, not a sink. * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). * Growth material metric (not declared in this epoch). The master table admits an optional incompatible-growth material metric for this aperture. It is not declared, for three reasons. 1. Within a fixed window growth is excluded, so a growth metric would be $\tau$-independent: $\theta = \sigma = 0$ in it. The imposed twist, an elastic or viscoelastic deformation, would not appear in it at all, so the shear-relaxation test of this aperture requires the physical metric. 2. r7 ties curvature to the control field. With $G_{\mu\nu} = \kappa_GT^{\mathrm{mat}}_{\mu\nu}$ and $R = \kappa_G\nabla_\alpha\xi^\alpha$ ($O_2$ Eq. [0]), $T^{\mathrm{mat}}_{\mu\nu} = \nabla_{(\mu}\xi_{\nu)} - \frac{1}{2}g_{\mu\nu}\nabla_\alpha\xi^\alpha$ gives $R_{\mu\nu} = \kappa_G\nabla_{(\mu}\xi_{\nu)}$, with $\xi_\nu = \hat{\Pi}_{\nu\lambda}\eta^\lambda$ ($O_4$ Eq. [2]). No plant-physiological field is identified as the global control field $\eta$. A $\xi$ constructed from a declared growth metric so as to satisfy this equation would satisfy the stress sector by construction. 3. The incompatibility of a growth metric (its curvature) is carried by residual growth strains, whose measurement by release cuts is destructive and so cannot be made on the intact plants that the tests require.

A future declaration would need a growth tensor whose incompatibility is estimated by an independent modality; an identified control field for which $R_{\mu\nu} = \kappa_G\nabla_{(\mu}\xi_{\nu)}$ is a tested hypothesis rather than a construction; and a statement, coordinate by coordinate, of which metric is used. Until then the stress sector of this aperture is that of §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 the xylem dendrometers, which must stay within the volume tolerance; for this volume-rigid domain the evidence is that tolerance check. * $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 7). The supply term uses the root-collar inflow, from a stem heat-balance gauge at the collar, and $\mathcal{F}$ there, from a collar psychrometer. The crown-base term uses the outflow into the crown, from a second stem heat-balance gauge at the crown base, with $\mathcal{F}$ there from a crown-base psychrometer. The cambial term uses bark dendrometry with $\mathcal{F}$ at the cambium. The storage term uses $\mathcal{F}$ along the flow path. None of these are the heat-pulse sensors or stem psychrometers of the $O_1$ section. * The physical dissipation is the viscous dissipation of sap flow, $q\cdot(-\nabla\psi_{\mathrm{h}})$ per unit volume. Under the declarations it equals the r7 dissipation density $-J_E^\mu\nabla_\mu\mathcal{F}$ times the positive constant $p_0C$, and its measured sign must agree with that of the estimate. * Admissible domains (declared, checkable preconditions): 1. intact conduit network: no new embolism and no refilling of embolized conduits within the window, checked by a declared method (for example acoustic emission for new embolism and a conductance or imaging check for refilling), so that $k$ and $C$ are constant; 2. $\psi_{\mathrm{h}} \ge \psi_{\mathrm{ref}}$ throughout (so $\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$) and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$; 3. volumetric strain of the conducting xylem within the volume tolerance. Without the torsional perturbation, shear strain is also within tolerance. With it, the twist is recoverable, with shear strain $\ll 1$, and $\partial_\tau\mathcal{F} \ge 0$ is checked from the psychrometer record (§0.1 item 1); 4. exchange with living tissue (embedded parenchyma, bark) one-signed into the tissue, checked by bark dendrometry: living tissue never discharges into the xylem within the window; 5. evaporative demand, stomatal conductance, and soil water potential held within tolerance after the step, checked by gas exchange and a soil or root-collar psychrometer; 6. capacitance and sap osmotic potential uniform over the stem sapwood within tolerance (§0.1 item 3), checked against pressure–volume measurements at two or more heights; a stem whose sapwood capacitance varies with height beyond tolerance is not admissible; 7. growth within tolerance throughout the window. * Titled phenomena outside the domain. * Rhizosphere coupling. Soil water potential and the conductances of roots and rhizosphere, including the extension of the absorbing surface by mycorrhizal hyphae, set the root-collar boundary condition. The soil, roots, and fungal networks lie outside $\mathcal{M}$; common mycorrhizal networks between plants are context. * Leaves and stomata. The crown lies outside $\mathcal{M}$: stomatal regulation and transpiration, with the storage of the crown's xylem and leaves, set the crown-base boundary data. * Phloem. Pressure-driven phloem transport in the bark lies outside $\mathcal{M}$ and enters only through the cambial exchange. * Growth and wood. Growth, wood formation, reaction wood, conduit dimensions, and the history of cavitation lie between windows; they enter through the declared parameters and $\mathcal{H}_0$. * Canopy mechanics. Stem and canopy sway is underdamped oscillation, which the mixed-form shear law excludes, so it lies outside the declared windows. Canopy architecture and spacing between crowns are 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: The conducting xylem is volume-rigid, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$ at every $\tau$; under the torsional perturbation the twist is isochoric, so this still holds. The expansion current enters only through its vanishing divergence and boundary flux. The water entering at the root collar and leaving at the crown base is throughput and is never mapped to it (§0.1 item 6). Openness is carried by throughput: sap flow down the water-potential gradient dissipates energy at every point, so $\Theta_E > 0$ (by $O_1$), $\Theta_{\mathrm{tot}} = \Theta_E$, and $\tau_\Theta$ 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 10). The plant's hydraulic memory (conduit dimensions and their vulnerability to embolism, the history of cavitation, reaction wood) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it. Within the window such changes are excluded (preconditions 1 and 7), and they enter through $\mathcal{H}_0$.

$O_1$ — 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 difference between the soil's water potential and the atmosphere's evaporative demand holds the shoot far from equilibrium. With $J_E^\mu = -p_0^{-1}(k^{\mu\nu}/C)\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 2), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. Sap flows down the gradient of $\psi_{\mathrm{h}}$, from the root collar toward the leaves and into the refilling tissue, in the direction of $J_E^\mu$. On the volume-rigid xylem $\dot{W}_{\mathrm{ext}} = 0$, 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), which exercises this aperture in the current epoch (empirical_validation_protocol.md §3.3). The parameters of $k^{\alpha\beta}$ are estimated on the intact test plant per closure (ii), from its water-potential response to a declared step change of transpiration run as a perturbation regime before the window, with transpiration from gas exchange. $C$, the stem-sapwood capacitance, comes from water-release (pressure–volume) curves on sapwood cores from the test stem itself, taken after the window at the heights of precondition 6 (or by a non-destructive method), never from another plant. The declared law must predict $J_W^{\mathrm{meas}} = q^{\mathrm{meas}}/C$ across a declared stem section $S$, with $q^{\mathrm{meas}}$ from heat-pulse sap-flux sensors, and with $\nabla\mathcal{F}$ from stem psychrometers at two heights, with the gravitational term added; neither $\nabla\mathcal{F}$ nor $q^{\mathrm{meas}}$ uses the other or the declared $k^{\alpha\beta}$.

$O_2$ — 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: Without the torsional perturbation the unimodular Weyl split is trivial: $\Omega \equiv 1$, $\hat{h} \equiv 0$, and $\partial_\tau\ln\Omega = \theta/3 = 0$. Under it, in the Cartesian material chart $\hat{g} = \delta$, $\Omega \equiv 1$ because the twist is isochoric ($\det g = 1$), and $\exp\hat{h} = g(\tau)$, so $\hat{h}$ is the logarithm of the right Cauchy–Green tensor of the twist, trace-free. 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 ($G_{\mu\nu} \equiv 0$, $\xi \equiv 0$, $\theta \equiv 0$; §0.1 items 2 and 4) it reduces to $\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} = 0$, and the full law to $\mathbb{L}_{\mathrm{Ein}}[\sigma] = 0$: the Saint-Venant compatibility of an isochoric strain rate, which the twist satisfies because its metric stays flat. Lignified walls and turgid tissue carry Cauchy stresses (bending stresses, growth stresses), which are not $T_{\mu\nu}^{\mathrm{mat}}$. The tendency of stems to grow toward uniform surface stress is an adaptive growth rule and is context, not $G_{\mu\nu} = \kappa_GT_{\mu\nu}^{\mathrm{mat}}$. Grain and fibre lines are integral curves of material fields, not trajectories. $\Pi_{\mu\nu}$ 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. Sap is driven by pressure gradients against viscous drag, which is forced motion; no drag-law drive is declared in this epoch. The paths of material points under the twist are prescribed by the deformation, not by 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: Without the torsional perturbation, $\sigma^\mu_{\;\nu}$ stays within tolerance of $0$, the shear law holds trivially for any $\gamma_\sigma > 0$, $\hat{h} = \hat{h}_\infty = 0$, and the aperture supports no shear-relaxation test. The hydraulic relaxation of $\mathcal{F}$ is a property of the $O_4$ balance law and is not mapped to $\sigma$. Under the perturbation, $\sigma^\mu_{\;\nu} = \frac{1}{2}g^{\mu\alpha}\partial_\tau g_{\alpha\nu}$ is the isochoric strain rate of the recovering segment, trace-free because $\theta \equiv 0$. If the twist recovers as a single overdamped exponential at a constant rate, every component of $\sigma$ decays at that rate, and the mixed-form shear law holds to first order in the twist strain; the second-order terms fall within a declared residual. That rate is $\gamma_\sigma$. It is not predicted from independent parameters: it is estimated per recovery trajectory, and the test is whether one constant rate fits within each trajectory, with the pre-declared precondition that $\sigma$ does not change sign during the window (the mixed-form law forbids oscillation, so underdamped sway is excluded). This is the aperture's contribution to the shear-relaxation family (empirical_validation_protocol.md §3.3). As the twist recovers, $\hat{h} \to \hat{h}_\infty$, the logarithmic strain of any residual twist (set), and $(\hat{g}_\infty)_{\mu\nu} = \delta_{\mu\alpha}(\exp\hat{h}_\infty)^\alpha_{\;\nu}$ has unit determinant. $V_{\mathrm{local}}$ is the hysteresis content of a cover set. Lasting responses of wood to loading, such as permanent set or reaction-wood formation, lie between windows and enter 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: With $\theta \equiv 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha((k^{\alpha\beta}/C)\nabla_\beta\mathcal{F}) - \mathcal{R}_{\mathcal{F}}$: the storage–conductance equation of plant water transport in water-potential form, the continuum version of capacitance–resistance models of the soil–plant–atmosphere continuum. Capacitance gives the finite relaxation time, so the field is stored, not quasi-static. Every departure from the default (organ-wise anisotropic $k$, lateral loss) satisfies the binding closure: each is a named law 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)). Under the torsional perturbation the Laplacian and $J_W$ are evaluated with $g(\tau)$. The gate sector follows §0.1 item 8: the Neumann spectrum of $-\nabla^2_g$ is fixed on the volume-rigid xylem and drifts by a fraction of the order of the shear strain under the perturbation, which is negligible within the window. The faster decay of fine spatial structure in the water potential is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The stress-conservation equivalence holds identically (§0.1 item 4); stresses in wood are not $T_{\mu\nu}^{\mathrm{mat}}$.

$O_5$ — 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 subregions of the conducting xylem. A single cover set, a thin shell, contains the concave part of $\partial\mathcal{M}$: the boundary with the non-conducting core. The other cover sets are intersections of $\mathcal{M}$ with strictly convex bodies that stay clear of those concave parts. Every overlap is then an intersection of strictly convex bodies with the convex side of a cylinder $r = \mathrm{const}$ or of a cross-section, so it is convex at $\tau_0$ (§0.1 item 9). The twist maps each cylinder $r = \mathrm{const}$ and each cross-section onto itself, and strictly convex parts stay convex for twist strains below a declared bound. Convexity for $g(\tau)$ is checked on the measured metric. Each $g^{(i)}$ is the restriction of $g$, so the gluing holds identically, $\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$, and $g^{(i)} \to G_\infty^{(i)}$ as the twist recovers ($H_0 = 0$). Every overlap is crossed by the hydraulic transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The connectivity of the conduit network through inter-conduit pits is a property of the tissue that enters through $k$; a network below its percolation threshold has no continuum conductivity 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 conducting xylem surrounds a non-conducting core, so $b_1(\mathcal{M}) \ge 1$. On an annular stem segment the harmonic Neumann field is the angular field around the core, computed with $g(\tau)$. No transport current is identified with it, and no claim is made about it (§0.1 item 9).

$O_6$ — 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 adjacent cover sets along the flow path, for example a lower and an upper stem region that share an overlap $\mathcal{U}_{ij}$, or an outer and an inner sapwood region of differing conductivity. The partner fields are the rate of change of the sap free-energy flux in the two orientations (§0.1 item 9), so $\eta_{[i\to j]} = \partial_\tau J_W$: a spatially resolved flux field, not a temporal lag between scalar signals. The boundary data are of mixed kind:
at the root collar the inflow declines as the collar potential rises;
at the crown base the outflow changes continuously: the step acts on transpiration inside the crown compartment, so at $\tau_0^+$ the crown-base outflow still equals the old transpiration rate, and it then declines toward the new rate as the crown's xylem and leaf tissue refill;
at the cambial interface water flows outward into the bark as it refills.

Whether $\partial_\tau J_W$ changes sign in the stem is not decided by these descriptions: it may be one-signed or not, and its zero set, if any, has no simple form. The sign structure comes from the prediction. The zero set is predicted before the window from the declared model, using the independently estimated $k$, $C$, and $k_{\mathrm{p}}$, the root and rhizosphere conductances, and the declared crown and bark compartments. The declared overlap is one that the predicted zero set does not reach during the overlap's window, with the detection threshold of §0.1 item 9. On it $\eta_{[i\to j]}$ keeps one direction, which the declaration states from the predicted solution; typically it points toward the root collar where the flux is declining. The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is its image under the declared non-singular precision map, with distinct Hessians, 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, not consequences of sap flow, and this aperture is not among the systems of the current coupling test. In Branch II the hydraulic transient decays as the shoot settles, so the coupling flux is integrable on $[\tau_0, \infty)$. Under the torsional perturbation the overlaps are convex for $g(\tau)$ ($O_5$). Any threshold of twist beyond which an 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.

$O_7$ — 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 10, so $[F_{\mu\nu}^{\mathrm{YM}}, \hat{\chi}] = 0$ identically. Without the torsional perturbation, $\chi_E \equiv 1$ and $\hat{\mathcal{E}}_\mu \equiv 0$. Under it, $\theta \equiv 0$ gives $\chi_E - 1 = 3\sigma^\mu_{\;\nu}\sigma^\nu_{\;\mu}/\langle\Theta_E\rangle^2$. The slope of $\ln(\chi_E - 1)$ is then $-2\gamma_\sigma - 2\,d\ln\langle\Theta_E\rangle/d\tau$, and the second term is a relative rate of measured dissipation, so the slope contains no $s_0$. On sub-intervals where measured dissipation is constant within a pre-declared relative tolerance, $\chi_E - 1$ decays at $2\gamma_\sigma$. Under the declarations this is equivalent to the shear-rate test of $O_3$ and adds no separate content. Its $s_0$-independence also rests on the declared $\theta \equiv 0$: whether the measured residual volumetric strain rate is negligible against $\langle\Theta_E\rangle$ depends on $s_0$. $\chi$ is undefined ($\theta = 0$) 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 the twist recovers, a residual computable from the twist 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 with divergence-free $J_{\mathrm{geom}}$ and $u_{\mathrm{exp}}$, $J_E$ bounded in $C^1$ under the held data, and $\hat{h} \to \hat{h}_\infty$, so $\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \to 0$. The aperture claims no alignment of flow with conduit orientation beyond what $\mathfrak{K}$ declares, and no synchrony of canopy motion.

$O_\Omega$ — 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 held evaporative demand and soil water potential, the shoot approaches a steady water-potential field that carries a steady transpiration stream from the root collar to the leaves: 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 transpiration is maintained. $\sigma \to 0$ as any imposed twist recovers, and $\partial_\tau g \to 0$. The transient coupling decays, so $\mathcal{I}$ converges 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 hydrostatic potential field with no transpiration, nor a periodic return: repeated cycles of evaporative demand form a sequence of windows, each settling toward its own held data, and are never mapped to $\theta$ or read as a return. The limit belongs to the declared idealization of a non-growing shoot under held data; the physical plant leaves that idealization as it grows, which bounds the usable window.
Within windows the supported tests (aperture_generation_guidance.md §5.5) are the shear-rate test (1) under the torsional perturbation, with $\gamma_\sigma$ estimated per trajectory; 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).
Under the perturbation the $\chi_E$ rate test (2) and the residual test (3) ($\partial_\tau g_{\mu\nu} = 2\sigma_{\mu\nu}$) are equivalent to (1) and are not counted separately. Without it, tests (1)–(3) hold trivially.

Aperture 3.2: Movement Ecology & Ecotone Foraging Dynamics ($\mathcal{E}_{\mathrm{trophic}}$)¶

1. Epistemological & Biological Scope¶

Aperture 3.2 maps the movement ecology of foraging animals (random search, area-restricted search, taxis toward resources, and movement across habitat edges and ecotones) and the trophic dynamics of predator and prey onto Spine $\mathcal{T}$. Its empirical domain is the redistribution and consumption of mobile prey in a three-dimensional foraging volume. There, random prey dispersal is a storage–diffusion process and predation is an interior sink, while the predators' trajectories follow a declared taxis drive. Lexicon is strictly restricted to movement ecology, behavioural ecology, trophic dynamics, and spatial population biology.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A three-dimensional foraging volume within one habitat (a layer of vegetation canopy or a body of the water column), fixed in space over one foraging window. $\partial\mathcal{M}$ consists of:
an ecotone boundary with an adjacent prey-source habitat (supply), across which prey disperse into $\mathcal{M}$;
an outflow boundary with a habitat into which prey that cross are lost to $\mathcal{M}$ without return, so that prey density there is held at the datum within tolerance;
closed faces with no prey flux.

No prey reproduce inside $\mathcal{M}$ and none enter except across the ecotone boundary, so all supply crosses $\partial\mathcal{M}$. Within $\mathcal{M}$, patches of differing structure (vegetation strata or plant types, or water layers) meet along internal transition zones. * Window. A foraging window (§0.1 item 1). It begins from a prior steady prey distribution, with prey supply at the ecotone, predator numbers, and habitat structure held within declared tolerances, so that $\mathcal{F}$ is steady within the window. At $\tau_0$ each tracked predator is labelled by the patch it occupies, and the window follows the mixing of the two labelled groups. The window is shorter than the prey's generation time and than the time over which predator numbers respond to prey. * Metric. Flat and rigid (§0.1 item 2): the domain is a fixed region of space, with $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, so $\theta \equiv 0$ and $\sigma \equiv 0$ by construction. Vegetation or water moving within the region is part of the medium and enters through $\mathfrak{K}$ and as context. * Cone speed. $c_*$ is the speed of sound in the ambient medium (air or water). Prey and predator speeds are $\ll c_*$. * Storage. Unity: $\mathcal{F}$ is itself the stored prey energy per unit volume, so no storage factor enters (§0.1 item 3). * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = \epsilon_{\mathrm{p}}b$ (J m⁻³), with $b$ the prey biomass density (g m⁻³) and $\epsilon_{\mathrm{p}}$ the declared energy content of prey biomass (J g⁻¹), from calorimetry of the prey population. The datum is zero ($\mathcal{F}_{\mathrm{ref}} = 0$), and $\mathcal{F} \ge 0$ holds trivially. $\mathcal{F}$ is linear in prey biomass. The relaxation time $\ell^2/D$ of prey dispersal across the volume is finite and resolvable. * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D_{\mathrm{p}}^{\alpha\beta}$, the prey dispersal tensor, parametric: for example one horizontal and one vertical dispersal coefficient in a canopy layer or water column. Isotropic dispersal is the default. With $j_b$ the prey biomass flux, $J_W = \epsilon_{\mathrm{p}}j_b$. * Sink $\mathcal{R}_{\mathcal{F}}$. Predation with a type II functional response: $$\mathcal{R}_{\mathcal{F}} = \frac{aP\,\mathcal{F}}{1 + (ah/\epsilon_{\mathrm{p}})\,\mathcal{F}}$$ Here $a$ is the attack rate (volume searched per predator per unit time), $h$ the handling time, and $P$ the predator density, uniform within the window at an independently estimated value. $\mathcal{R}_{\mathcal{F}}$ is continuous, non-negative, non-decreasing in $\mathcal{F}$, and zero at the datum. $a$ and $h$ are estimated from functional-response trials with the same predator individuals in a declared perturbation regime before the window (controlled prey densities), and $P$ from a census, per closure (ii). * Drive. Predator taxis toward prey gradients, a named law (linear taxis with velocity relaxation, the deterministic part of a biased Ornstein–Uhlenbeck movement model): $$a^\mu_{\mathrm{d}} = \tau_v^{-1}\left(\chi_{\mathrm{P}}\,g^{\mu\nu}\nabla_\nu b - v^\mu\right)$$ Predators relax their velocity, at the rate $\tau_v^{-1}$, toward a taxis velocity proportional to the prey-density gradient. The taxis sensitivity $\chi_{\mathrm{P}}$ (the measured sensitivity of the master table) and $\tau_v$ are estimated per closure (ii) from the same individuals' responses to an imposed prey gradient, in a declared perturbation regime before the window, never from the trajectories the law predicts. $b \in C^2$ on the steady prey field of the window, so $a^\mu_{\mathrm{d}}$ is jointly continuous and locally Lipschitz in $(\mathbf{x}, v)$, as r7 $O_2$ Eq. [3] requires. * Eq. [3] is deterministic, so the declared reading is that $f^\mu_{\mathrm{drive}}$ is the mean acceleration of predators conditional on position and velocity. The variability of search about that mean lies outside Eq. [3] and is not mapped. * The drive acts on individual trajectories. The uniform predator density declared for the sink is compatible with it only where taxis-driven redistribution of predators within the window stays below the uniformity tolerance (precondition 3). * Coupling-sector data. A predator search diffusivity $D_{\mathrm{P}}$, estimated from predator tracks in a declared period before the window, is used, with the drive, only to predict the labelled-flux fields of $O_6$ and their zero set. * Branch II evidence (closure (v)). * The trend of $\langle\theta\rangle$ and $\int\langle\theta\rangle\,d\tau$ are zero by construction for a fixed region of space; this part of the evidence holds trivially and is stated as such. * $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 7): * the supply term uses the prey flux across the ecotone boundary, from bidirectional interception traps or counts at that boundary, with $\mathcal{F}$ there from a census; * the outflow face, at the datum, contributes nothing; * the consumption term uses predation rates from observed kills or feeding records of tracked predators, weighted by the local $\mathcal{F}$; * the storage term is checked to be zero by repeated census.

None of these instruments or surfaces is that of the $O_1$ section.

$\mathcal{F}$ is a resource energy density, not a thermodynamic potential of the dispersing medium, so $\dot{s}_{\mathrm{prod}}$ is the entropy production of the declared mapping, not a measured physical entropy production. The physical dissipation named for the sign comparison is the metabolic power of the foraging community (by respirometry or isotope turnover), which is positive in any living system. The comparison therefore holds trivially and is stated as such.
Admissible domains (declared, checkable preconditions):
prey movement is unbiased random dispersal at the declared scale, with no drift from predator avoidance or resource tracking beyond tolerance, and prey numbers are large enough for a continuum density;
no prey reproduction, immigration other than across the ecotone boundary, or mortality other than predation within the window;
predator density uniform within tolerance and predator numbers constant throughout the window, checked by census. Sustained taxis on the steady prey field would drive predators toward a profile $P \propto \exp(\chi_{\mathrm{P}}b/D_{\mathrm{P}})$, so the relative non-uniformity it can produce is about $\chi_{\mathrm{P}}\Delta b/D_{\mathrm{P}}$, with $\Delta b$ the range of prey density over $\mathcal{M}$. This number is computed before the window from the declared parameters and must lie below the uniformity tolerance;
$\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$: prey density decreases from the ecotone boundary toward the predation zones and the outflow boundary;
prey density at the outflow boundary at the datum within tolerance;
habitat structure, and hence $D_{\mathrm{p}}$, fixed within the window;
for the one-dimensional statement of $O_6$ only: patches and data laterally uniform across the transition zone, and taxis drift negligible across it ($\chi_{\mathrm{P}}\vert\nabla b\vert\ell_{ij}/D_{\mathrm{P}}$ below a declared tolerance, with $\ell_{ij}$ the width of the zone), because a spatially varying drift can break the monotonicity of the label densities.
Titled phenomena outside the domain.
Foraging decisions (patch choice, giving-up rules, the ideal free distribution) redistribute predators between windows. Within the window predator density is declared uniform, and the decisions enter only through the drive's parameters and as context.
Predator–prey population cycles and prey reproduction lie between windows. Their oscillations are never mapped to $\theta$ or $\sigma$.
Migration, home ranges, and learned routes are context, and enter each window through $\mathcal{H}_0$.
The ecotone is the supply boundary only; the boundaries between patches inside $\mathcal{M}$ are called internal transition zones, never ecotones. Edge effects on prey supply enter as boundary data.

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 domain is a fixed region of space, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$ by construction. The prey supply across the ecotone is throughput and is never mapped to $J_{\mathrm{geom}}$ (§0.1 item 6). Openness is carried by throughput: prey disperse down their density gradient and are consumed, so $\Theta_E > 0$ (by $O_1$), $\Theta_{\mathrm{tot}} = \Theta_E$, and $\tau_\Theta$ counts the entropy production of the declared mapping per $s_0$. The record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput current. The spatial memory of predators (learned routes, remembered patch quality) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it; it enters each window through $\mathcal{H}_0$.

$O_1$ — 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: Prey supply across the ecotone, set against consumption by predators and loss across the outflow boundary, holds the volume far from equilibrium; equilibrium would be a uniform prey density with no supply, no flux, and no predation. With $J_E^\mu = -p_0^{-1}D_{\mathrm{p}}^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 4), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. The prey biomass flux runs down the density gradient, from the ecotone toward the predation zones and the outflow boundary, in the direction of $J_E^\mu$. $\dot{W}_{\mathrm{ext}} = 0$, 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). $D_{\mathrm{p}}^{\alpha\beta}$ is estimated per closure (ii) from marked prey (mark–release–recapture or tracking) released in a declared perturbation regime before the window. It must predict $J_W^{\mathrm{meas}} = \epsilon_{\mathrm{p}}j_b^{\mathrm{meas}}$ across a declared interior section $S$, with $j_b^{\mathrm{meas}}$ from bidirectional interception or counts of prey crossing $S$, and $\nabla\mathcal{F}$ from prey census on both sides of $S$; neither uses the other or the declared $D_{\mathrm{p}}$. This aperture is not among the systems of the current flux-prediction test.

$O_2$ — 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 fixed region 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 and 4). Vegetation structure and water layering enter through $D_{\mathrm{p}}$, the drive's parameters, and the geometry of $\mathcal{M}$, not through $g$. Least-cost movement paths are geodesics of a separate cost-of-transport metric, which is not $g_{\mu\nu}$. $\Pi_{\mu\nu}$ is the Hessian of a declared precision potential and enters only the coupling sector ($O_6$). Eq. [3] is instantiated with the declared taxis drive. On the static flat metric, in the Cartesian chart, $\Gamma = 0$ and $\partial_\tau g = 0$, so Eq. [3] reduces to $dv^\mu/d\tau = f^\mu_{\mathrm{drive}} \approx a^\mu_{\mathrm{d}} = \tau_v^{-1}(\chi_{\mathrm{P}}\,\partial^\mu b - v^\mu)$, with $u^2 \ll 1$: predators relax toward a taxis velocity up the prey gradient. This conditional mean acceleration is the aperture's trajectory claim, and the driven trajectory equation is untested in this epoch (empirical_validation_protocol.md §3.3). Prey paths are random dispersal, described by the $O_4$ balance law, and are not trajectories 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 fixed region $\sigma^\mu_{\;\nu} \equiv 0$, so the 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). Behavioural recovery after a disturbance, and the settling of the prey field, are not mapped to $\sigma$: the first lies in $\mathcal{H}$ and context, the second in the $O_4$ balance law. $V_{\mathrm{local}}$ is the hysteresis content of a patch $\mathcal{U}_i$.

$O_4$ — 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 $\theta \equiv 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha(D_{\mathrm{p}}^{\alpha\beta}\nabla_\beta\mathcal{F}) - aP\mathcal{F}/(1 + (ah/\epsilon_{\mathrm{p}})\mathcal{F})$: the reaction–diffusion equation of dispersing prey under type II predation, steady within the window. Its finite relaxation time makes the field stored, not quasi-static. Both departures from the default (anisotropic dispersal, predation) 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 8: the Neumann eigenmodes of $-\nabla^2$ on the fixed region are fixed, so the gate does not drift, and the faster decay of fine spatial structure in the prey field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The stress-conservation equivalence holds identically (§0.1 item 4).

$O_5$ — 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 fixed subregions of the volume: the patches and their surroundings, with convex overlaps $\mathcal{U}_{ij}$ along the internal transition zones. Each $g^{(i)}$ is the restriction of the flat metric, so the gluing holds identically, each overlap stays convex, $\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$, and $g^{(i)} = G_\infty^{(i)}$ at every $\tau$ ($H_0 = 0$; §0.1 item 9). The labelled predator groups mix across every overlap of the connected volume, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. Habitat connectivity (gaps, corridors, and barriers to movement) enters through $D_{\mathrm{p}}$ and the predator movement parameters. It is mapped via Spine $\mathcal{T}$ to the condition $\phi_{\mathrm{perc}} > \phi_c$ and is not identified with it. The declared volume is simply connected, so $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$ (§0.1 item 9). A volume pierced by an impassable obstacle lies outside this declaration.

$O_6$ — 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 adjacent patches $\mathcal{U}_i$ and $\mathcal{U}_j$ that share a transition zone $\mathcal{U}_{ij}$. The partner fields are the movement-flux fields of the two labelled predator groups: $\eta_{i\to j}^\lambda$ is the flux density of the predators labelled $i$ at $\tau_0$ (animals per unit area per unit time), and $\eta_{j\to i}^\lambda$ that of the predators labelled $j$. Both are estimated from individual tracks with a declared smoothing kernel, so they are defined on all of $\mathcal{M}$ and lie in $C^2$. They are spatially resolved flux fields, not temporal lead–lag statistics between scalar signals.
The two fields are not antisymmetric by construction. Their sum is the net predator flux, which is small under precondition 3, so $\eta_{[i\to j]} = \frac{1}{2}(\eta_{i\to j} - \eta_{j\to i})$ is close to each group's own flux.
As the groups mix after $\tau_0$, predators labelled $i$ cross the transition zone toward $\mathcal{U}_j$ and those labelled $j$ toward $\mathcal{U}_i$, so on the zone $\eta_{[i\to j]}$ points from $\mathcal{U}_i$ toward $\mathcal{U}_j$. The zero set of $\eta_{[i\to j]}$, where the two group fluxes balance, is predicted before the window from $D_{\mathrm{P}}$ and the declared drive. The declared overlap is one that the predicted zero set does not reach, with the detection threshold of §0.1 item 9.
Where the patches and data are laterally uniform across the zone (precondition 7), each label density starts as a step across the zone and, with taxis drift negligible across the zone, stays monotone under diffusive mixing with closed ends. The label flux then keeps one direction in the interior, and the zero set lies on the closed faces. Without lateral uniformity no such argument is claimed.

The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is the image of $\eta_{[i\to j]}$ under the declared non-singular precision map, with distinct Hessians (§0.1 item 9), and is non-zero on the overlap. 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 $M^{(ij)}$, not of aggregation or spreading of animals. 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 movement, and this pair is the aperture's system in the current coupling test (empirical_validation_protocol.md §3.3, family 3).

In Branch II the labelled fluxes decay as the groups mix, at the rate of the slowest mixing mode, so the coupling flux is integrable on $[\tau_0, \infty)$ and no sustained coupling is asserted. What decays is the difference between the groups' fluxes, not movement itself: the predators keep moving and foraging. The overlaps are fixed and convex, so the r10 §5.12 constraint $a_\infty = 2s_\sigma/\gamma_\sigma < a_c(E)$, a design constraint of a deforming witness overlap, 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 10, 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 $\hat{h} \equiv \hat{h}_\infty$. Collective alignment of moving animals (schooling, flocking) is not mapped to the gauge connection; it is context, and no alignment or synchrony claim is made.

$O_\Omega$ — 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 prey supply and a held predator density, the volume holds a steady prey distribution that carries a steady prey flux from the ecotone into the predation zones and across the outflow boundary: a bounded non-equilibrium steady state. $\theta \equiv 0$, $V_{\mathcal{M}} = V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while supply is maintained, $\sigma \equiv 0$, and $\partial_\tau g \equiv 0$. The labelled-group coupling decays as the groups mix, 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 uniform prey density with no supply, flux, or predation, nor a periodic return: predator–prey cycles lie between windows and are never read as a return. The limit belongs to the declared idealization of constant, uniform predators and non-reproducing prey; the physical community leaves it as prey reproduce or predators respond, 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 the fixed region.

Aperture 3.3: Benthic & Edaphic Biogeochemical Metabolism ($\mathcal{E}_{\mathrm{metabolic}}$)¶

1. Epistemological & Biological Scope¶

Aperture 3.3 maps the metabolism of sediments and water-saturated soils onto Spine $\mathcal{T}$: microbial respiration, the sequence of terminal electron acceptors, redox stratification, the decomposition of detritus, and the exchange of solutes across the sediment–water interface. Its empirical domain is the supply of a dissolved oxidant from overlying water, across the diffusive boundary layer (DBL), into a sediment column where microbes consume it. Lexicon is strictly restricted to biogeochemistry, microbial ecology, aquatic and soil metabolism, and bioenergetics.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A column of sediment or water-saturated soil, for example an intact core, with the DBL of water above its surface. $\mathcal{M}$ is the DBL together with the sediment down to a declared lower face. $\partial\mathcal{M}$ consists of:
the top of the DBL, where the overlying water is well mixed (supply), held at a declared oxidant concentration $c_w$;
the lower face, at the base of the zone that the oxidant penetrates, where it is consumed by reduced species diffusing up from below in a thin reaction zone outside $\mathcal{M}$, so that the oxidant concentration is held at zero there (outflow at the datum);
closed lateral walls.

The sediment–water interface lies inside $\mathcal{M}$, with a declared smooth transition zone. There are no interior sources: all oxidant crosses the supply face. * Window. A relaxation window (§0.1 item 1), opened by a declared step increase of $c_w$ from a prior steady state, after which $c_w$ is held within tolerance. The window ends before the lower face moves beyond a declared tolerance, and before microbial activity changes beyond tolerance. * Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, with compaction and swelling of the sediment within the rigid tolerance. * Cone speed. $c_*$ is the compressional wave speed of the water-saturated sediment. Diffusive speeds are $\ll c_*$. * Storage. In the sediment the oxidant's storage coefficient is the porosity $\varphi_{\mathrm{s}}$, declared uniform over the sediment part of $\mathcal{M}$ within tolerance; in the DBL it is unity. The declared law is normalized by $\varphi_{\mathrm{s}}$ (§0.1 item 3). It is then exact in the sediment and keeps $J_W$ continuous across the interface, as the physical flux is. Its only departure from the physical law is in the DBL, where it stores oxidant with the coefficient $\varphi_{\mathrm{s}}$ instead of $1$. Relative to the oxidant stored in the sediment, about $\varphi_{\mathrm{s}}c\,\ell_{\mathrm{p}}$ per unit area against the excess $(1 - \varphi_{\mathrm{s}})c\,\delta$, that departure is of relative order $(1 - \varphi_{\mathrm{s}})\delta/(\varphi_{\mathrm{s}}\ell_{\mathrm{p}})$ in the storage rate, with $\delta$ the DBL thickness and $\ell_{\mathrm{p}}$ the oxidant penetration depth, and the pre-declared flux tolerance must exceed it (precondition 3). * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = c\,(-\Delta G_r^{\circ\prime})$ (J m⁻³), with $c$ the pore-water (or, in the DBL, water) concentration of the oxidant (mol m⁻³) and $-\Delta G_r^{\circ\prime} > 0$ the declared standard-state free-energy yield per mole of oxidant reduced in the declared respiration reaction (J mol⁻¹), taken from tabulated standards. This is the declared linear form of aperture_generation_guidance.md §2.3 item 3: the actual reaction free energy depends on the concentrations, and the standard-state yield is used as a declared constant. The datum is zero ($\mathcal{F}_{\mathrm{ref}} = 0$), and $\mathcal{F} \ge 0$ holds trivially. The relaxation time $\ell_{\mathrm{p}}^2/D_{\mathrm{s}}$ is finite and resolvable. * Transport $\mathfrak{K}$. In the sediment, $\mathfrak{K}^{\alpha\beta} = D_{\mathrm{s}}^{\alpha\beta}$, the effective (tortuosity-corrected) pore-water diffusion tensor, parametric with layering: $\mathrm{diag}(D_{\mathrm{H}}, D_{\mathrm{H}}, D_{\mathrm{V}})$ in a chart aligned with the bedding for laminated sediment, with isotropic $D_{\mathrm{s}}$ as the default. In the DBL, $\mathfrak{K} = (D_0/\varphi_{\mathrm{s}})\,g^{\alpha\beta}$, with $D_0$ the molecular diffusivity of the oxidant in water, tabulated at the declared temperature and salinity. The two are joined smoothly across the declared transition zone, so $\mathfrak{K} \in C^1$. With $j$ the molar oxidant flux per unit bulk area, $J_W = (-\Delta G_r^{\circ\prime})\,j/\varphi_{\mathrm{s}}$. The conversion factor $(-\Delta G_r^{\circ\prime})/\varphi_{\mathrm{s}}$ is declared, with $\varphi_{\mathrm{s}}$ measured independently on the same column, for example from the water content and density of its sediment after the window, or from electrical resistivity. * Sink $\mathcal{R}_{\mathcal{F}}$. Microbial consumption with Michaelis–Menten kinetics, $\mathcal{R}_{\mathcal{F}} = V_{\mathrm{m}}\mathcal{F}/(K_{\mathcal{F}} + \mathcal{F})$, with $K_{\mathcal{F}} = K_{\mathrm{m}}(-\Delta G_r^{\circ\prime})$ and $V_{\mathrm{m}}$ in the storage-normalized units of the declared law. $\mathcal{R}_{\mathcal{F}}$ is continuous, non-negative, non-decreasing in $\mathcal{F}$, and spread over the oxic sediment, with no sharp front inside $\mathcal{M}$. $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$ are estimated per closure (ii): from the consumption response of the same column to declared changes of overlying-water oxidant before the window (a declared perturbation regime), or from incubations of its own sediment after the window. * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). * Branch II evidence (closure (v)). * The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau$ come from the measured elevation of the sediment surface and the column length, which must stay within the rigid tolerance. * $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 7): * the supply term uses the oxidant supply to the overlying water and $\mathcal{F}$ there, from a second, independent set of overlying-water sensors (or discrete samples by a different analytical method); * the lower face, at the datum, contributes nothing; * the consumption term uses the declared, independently estimated $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$ with $\mathcal{F}$ from a second set of microsensor profiles; * the storage term uses repeated profiles. * Weaker independence. The only boundary at which $\mathcal{F} \neq 0$ is the supply face, and the $O_1$ test measures the flux across the sediment–water interface. The two surfaces are separated only by the DBL, so the two flux records differ only by the DBL's storage change. The evidence is therefore independent of the $O_1$ measurement (different instruments) but not of the quantity that it measures, which is weaker than the separation of surfaces in Tier 2. * The physical dissipation is the entropy production of oxidant diffusion, which is non-negative by the second law, so the comparison tests only the sign of the estimate. * Admissible domains (declared, checkable preconditions): 1. no bioturbation, bioirrigation, ebullition, or advective pore-water flow within the window (a defaunated or naturally quiescent column, checked by inert-tracer profiles); 2. no oxidant production in $\mathcal{M}$ (no benthic photosynthesis or other oxidant-producing process), and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$; 3. porosity uniform over the sediment part of $\mathcal{M}$ within tolerance, and the DBL departure of relative order $(1 - \varphi_{\mathrm{s}})\delta/(\varphi_{\mathrm{s}}\ell_{\mathrm{p}})$ within the flux tolerance; 4. a fixed lower face: oxidant at the lower face below a pre-declared detection threshold, and its depth constant within tolerance, throughout the window. A step increase of $c_w$ pushes the oxic boundary down unless the upward flux of reduced species absorbs the extra oxidant, so the step size is chosen before the window such that the shift of the lower face, predicted from the declared parameters and a declared, independently measured upward flux of reduced species, stays within that tolerance; and $V_{\mathrm{m}}$, $K_{\mathrm{m}}$ constant (no growth or decline of microbial activity beyond tolerance); 5. isothermal conditions within tolerance, so that $D_0$, $D_{\mathrm{s}}$, $V_{\mathrm{m}}$, and $K_{\mathrm{m}}$ are constant; 6. compaction and swelling within the rigid tolerance; 7. for the one-dimensional statements of $O_6$ only: a laterally uniform supply and sediment, so that the problem is one-dimensional in depth. * Titled phenomena outside the domain. * Redox sequence. The other electron acceptors (nitrate, manganese and iron oxides, sulfate) and methanogenesis lie below the lower face, or enter as context; each window declares one oxidant. The reduced species that consume it at the lower face enter as the outflow boundary condition. * Organic matter. Extracellular enzymatic hydrolysis and the humification of organic matter set the supply of electron donors, and so $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$; they enter through the declared parameters and $\mathcal{H}$. * Fauna and resuspension. Bioturbation, bioirrigation, and resuspension are excluded within windows (precondition 1). Between windows they rework the sediment and enter through $\mathcal{H}$. * Unsaturated soils. Gas-phase diffusion in unsaturated soils is outside this declaration; it would need its own storage coefficient (the air-filled porosity) and is context here. * Persistence. Long-term persistence of buried metabolism below the lower face is 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: The column is rigid, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$. The oxidant supplied across the top of the DBL is throughput and is never mapped to $J_{\mathrm{geom}}$ (§0.1 item 6). Diffusion of oxidant from the overlying water to the respiring sediment and the lower face 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. The sediment's chemical and biological record (layers of oxides and sulfides, organic content, the structure of the microbial community) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it. Within the window it is held fixed (precondition 4), and it enters through $\mathcal{H}_0$. The oxidant 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 maintained difference in oxidant between the overlying water and the lower face, together with consumption, holds the column far from equilibrium. With $J_E^\mu = -p_0^{-1}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 2), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. The oxidant flux runs down its gradient, from the overlying water into the sediment, in the direction of $J_E^\mu$. $\dot{W}_{\mathrm{ext}} = 0$, 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), which exercises this aperture in the current epoch (empirical_validation_protocol.md §3.3). $D_{\mathrm{s}}^{\alpha\beta}$ is estimated on the same column per closure (ii), from an inert-tracer diffusion experiment (a different modality), and $\varphi_{\mathrm{s}}$ independently as above. The declared law must predict $J_W^{\mathrm{meas}} = (-\Delta G_r^{\circ\prime})\,j^{\mathrm{meas}}/\varphi_{\mathrm{s}}$ across a declared section $S$ just beneath the sediment–water interface. Here $j^{\mathrm{meas}}$ is the oxidant uptake measured from the decline of overlying-water oxidant in a chamber (with the DBL's storage change negligible, precondition 3), and $\nabla\mathcal{F}$ comes from microsensor profiles across $S$; neither uses the other or the declared $D_{\mathrm{s}}$ or $V_{\mathrm{m}}$. The redox potential $E_h$, a potential in volts, is not mapped to $\mathcal{F}$, and electron transport within the reactions is not $J_E^\mu$.

$O_2$ — 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 column 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 and 4). The fabric and tortuosity of the sediment enter through $D_{\mathrm{s}}$, not through $g$, and the topography of the sediment surface enters through the declared geometry of $\mathcal{M}$ and its DBL. $\Pi_{\mu\nu}$ 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. Solute transport is diffusive and is described by the $O_4$ balance law, and microbes are not mapped as tracers.

$O_3$ — 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 column $\sigma^\mu_{\;\nu} \equiv 0$, so the 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 the oxidant profile toward its new steady state is a property of the $O_4$ balance law and is not mapped to $\sigma$. Compaction and consolidation of sediment lie outside the declared windows. $V_{\mathrm{local}}$ 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 $\theta \equiv 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha(\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}) - V_{\mathrm{m}}\mathcal{F}/(K_{\mathcal{F}} + \mathcal{F})$: the reaction–diffusion (early-diagenesis) equation of oxidant uptake without advection, normalized by the sediment porosity. Its finite relaxation time makes the field stored, not quasi-static. The layered $D_{\mathrm{s}}$ and the Michaelis–Menten 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 8: the Neumann eigenmodes of $-\nabla^2$ on the rigid column are fixed, so the gate does not drift, and the faster decay of fine spatial structure in the oxidant profile is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The selective uptake of particular solutes by microbial membranes and enzymes is a property of the reactions, set through $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$, and is not the spatial gate. The stress-conservation equivalence holds identically (§0.1 item 4).

$O_5$ — 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 horizontal layers of the column (the DBL, the uppermost sediment, and deeper oxic layers), 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 9). 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 sediment that enters through $D_{\mathrm{s}}$: a sediment whose pore space does not percolate has no continuum diffusivity 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 9). A column pierced by tubes or burrows lies outside this declaration (precondition 1). Cycles in the reaction network are not $H^1_{\mathrm{dR}}(\mathcal{M})$.

$O_6$ — 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 that share a transition zone $\mathcal{U}_{ij}$, for example the DBL and the uppermost sediment across the interface transition, or two sediment layers of differing tortuosity. The partner fields are the rate of change of the oxidant free-energy flux in the two orientations (§0.1 item 9), so $\eta_{[i\to j]} = \partial_\tau J_W$: a spatially resolved flux field, not a temporal lag between scalar signals.
The supply face holds a value of $\mathcal{F}$ and the lower face holds the datum, so after the step the flux near the supply decays from an initial overshoot, while farther down it rises. There $\partial_\tau J_W$ points toward the supply and away from it, respectively, and the zero set between them travels downward during the early window.
In a laterally uniform column (precondition 7) the late zero set is the level at which the slowest relaxation mode of the problem linearized about its new steady state peaks. Without lateral uniformity it is located only by the predicted solution.
The declared overlap is one that the predicted zero set does not reach during the overlap's window, with the detection threshold of §0.1 item 9; on it $\eta_{[i\to j]}$ keeps one direction, which the declaration states from the predicted solution. Where the sediment is laterally heterogeneous, the transition zones between patches of differing demand are overlaps too, with partner fields from the lateral components of the same $\partial_\tau J_W$.

The odd 1-form $J_{[i\to j]} = -J_{[j\to i]}$ is its image under the declared non-singular precision map, with distinct Hessians, 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 a laterally uniform column 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, not consequences of diffusion, and this aperture is not among the systems of the current coupling test. In Branch II the transient decays as the column 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.

$O_7$ — 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 10, 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 oxidant field with the sediment's stored record 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 a held overlying-water concentration the column approaches a steady oxidant profile that carries a steady flux from the overlying water into the respiring sediment and the lower face: a bounded non-equilibrium steady state. $\theta \equiv 0$, $V_{\mathcal{M}} = V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while supply is maintained, $\sigma \equiv 0$, and $\partial_\tau g \equiv 0$. 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 concentration with no consumption and no flux, nor a periodic return: natural cycles of supply form a sequence of windows and are never read as a return. The limit belongs to the declared idealization of fixed microbial activity and a fixed lower face; the physical column leaves it as the community and the reduced pools change, 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 the rigid column.

Aperture 3.4: Chronobiology & Phenological Synchrony ($\mathcal{E}_{\mathrm{chrono}}$)¶

1. Epistemological & Biological Scope¶

Aperture 3.4 maps the temporal organization of living systems onto Spine $\mathcal{T}$: circadian and other endogenous oscillators, their entrainment by environmental cues, the humoral and paracrine signals by which pacemaker tissues synchronize peripheral tissues, and the seasonal timing of development (dormancy, vernalization, flowering, mast seeding). Its empirical domain is the spatial delivery of an entraining signal into a target tissue: a signal molecule secreted by pacemaker cells outside the tissue diffuses into it and is degraded there. The oscillators and their phase dynamics lie outside the domain, and temporal cycles are never mapped to spatial operators. Lexicon is strictly restricted to chronobiology, the endocrinology of timing signals, phenology, nonlinear oscillator theory, and dynamical systems ecology.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. An avascular layer of target tissue, or a tissue slice or explant, that receives a humoral or paracrine entraining signal secreted by cells outside $\mathcal{M}$. $\partial\mathcal{M}$ consists of:
a supply interface (vascular or fluid) at which the signal concentration is set by the secretory output of the pacemaker cells and held during the window;
a drainage interface (vascular, lymphatic, or fluid) that holds the signal at the datum within tolerance (in a two-chamber explant preparation, the receiving medium);
closed faces.

The tissue contains no cells that secrete the signal and no capillaries, so there are no interior sources and all supply crosses $\partial\mathcal{M}$. * Window. A relaxation window (§0.1 item 1), opened by a declared step increase of the signal concentration at the supply interface from a prior steady state, after which the concentration is held at its new level within tolerance. In vivo the step is the rise of secretion at a declared phase of the pacemaker's cycle, which is a ramp with a finite rise time; $\partial_\tau\mathcal{F} \ge 0$ then follows from the ramp form of the comparison argument (§0.1 item 1), not from time translation. In vitro it is a change of the supplying medium. The window ends before the pacemaker's output changes again. Each window therefore covers one monotone segment of the signal's cycle. The cycle itself is a sequence of windows and is never mapped to $\theta$ or read as a return. * Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, with swelling or shrinkage of the tissue within the rigid tolerance. * Cone speed. $c_*$ is the longitudinal sound speed of the tissue. Diffusive speeds are $\ll c_*$. * Storage. $s = \alpha(1 + B)$: the extracellular volume fraction $\alpha$ times the retardation factor $1 + B$ of fast, linear, reversible binding of the signal (to extracellular matrix or to receptors). It is declared uniform over $\mathcal{M}$ within tolerance (§0.1 item 3); saturable binding is excluded. * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = RT\,c$ (J m⁻³), with $c$ the extracellular concentration of the free signal (mol m⁻³) and the declared constant yield $RT$ (J mol⁻¹) at the declared tissue temperature $T$. $RT\,c$ is the van 't Hoff osmotic pressure of the signal, an energy density linear in $c$. It is the declared linear form of aperture_generation_guidance.md §2.3 item 3; the chemical potential itself is logarithmic in $c$. The datum is zero ($\mathcal{F}_{\mathrm{ref}} = 0$), and $\mathcal{F} \ge 0$ holds trivially. The relaxation time $\ell^2/D^*$ across the layer is finite and resolvable. * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D^{*\alpha\beta}/(1 + B)$, with $D^{*\alpha\beta}$ the effective extracellular diffusivity tensor (the free diffusivity reduced by the tortuosity of the extracellular space). It is parametric along declared tissue axes (for example a fibre direction), with isotropic diffusivity as the default. With $j$ the molar flux per unit tissue area, $J_W = RT\,j/s$. The conversion factor $RT/s$ is declared. $\alpha$ and $D^*$ are estimated per closure (ii) on the same tissue, for example from the real-time diffusion of an inert probe ion (a different modality), and $B$ from binding assays of the same tissue after the window. * Sink $\mathcal{R}_{\mathcal{F}}$. Enzymatic degradation, first order: $\mathcal{R}_{\mathcal{F}} = k_{\mathrm{d}}\mathcal{F}$, with $k_{\mathrm{d}} = \ln 2/t_{1/2}$ from the measured half-life of the signal in the same tissue, after its supply is removed in a declared perturbation regime. $k_{\mathrm{d}}$ is the effective rate in the storage-normalized law: the degradation rate of free signal divided by $1 + B$ if bound signal is protected. Cellular uptake and receptor internalization are included in $k_{\mathrm{d}}$ where they are first order, and otherwise must be negligible. * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). * Branch II evidence (closure (v)). * The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau$ come from measurements of tissue thickness (optical or acoustic), which must stay within the rigid tolerance. * $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 7): * the supply term uses the depletion record of the supplying compartment (or, in vivo, the difference between arterial and venous concentrations times the flow), with $\mathcal{F}$ at the supply interface; * the drainage interface, at the datum, contributes nothing; * the consumption term uses the declared $k_{\mathrm{d}}$ with $\mathcal{F}$ from tissue concentration measurements that are not those of the $O_1$ profile; * the storage term uses repeated measurements.

The $O_1$ test uses the drainage interface, a different surface.

The physical dissipation is the entropy production of the signal's diffusion, which is non-negative by the second law, so the comparison tests only the sign of the estimate.
Admissible domains (declared, checkable preconditions):
an avascular tissue with no secretion of the signal and no other interior source;
$\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$;
uniform $\alpha$ and fast, linear, reversible binding with constant $B$ (or none), and no saturable uptake, within tolerance;
supply-interface concentration held at its new level, and drainage interface at the datum, within tolerance;
tissue dimensions within the rigid tolerance, and isothermal conditions within tolerance;
purely diffusive extracellular transport, with interstitial convection below tolerance;
for the one-dimensional statements of $O_6$ only: a laterally uniform supply and tissue.
Titled phenomena outside the domain.
Pacemakers. The pacemaker oscillators and their cycles lie outside $\mathcal{M}$, and their output sets the supply data window by window. Oscillator phase, period, and amplitude are temporal quantities and are not mapped to $\theta$, $\sigma$, or any spatial operator.
Entrainment. The entrainment of the target cells' own clocks by the delivered signal (phase response, phase locking) is a response in the target cells. It enters through $\mathcal{H}$ and as context, and no synchronization claim is made (§0.1 item 10).
Environmental cues. Cues that entrain the pacemakers act outside $\mathcal{M}$ and are context.
Phenology. Dormancy, vernalization, flowering time, and mast seeding are seasonal developmental timing, between windows. The memory of accumulated cues (for example accumulated chilling) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it.
Long cycles. Succession and multi-year cycles are 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: The tissue is rigid, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$. The signal supplied across the supply interface is throughput and is never mapped to $J_{\mathrm{geom}}$ (§0.1 item 6). Diffusion of the signal from the supply interface toward the drainage interface, with degradation along the way, produces entropy at every point, so $\Theta_E > 0$ (by $O_1$), $\Theta_{\mathrm{tot}} = \Theta_E$, and $\tau_\Theta$ counts entropy production per $s_0$. $\tau$ remains laboratory proper time, and the oscillators' cycles run outside $\mathcal{M}$ within this foliation; they are not $\theta$, which never changes sign. The record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput current. The entrainment state of the target tissue (the phase of its cellular clocks) and its memory of accumulated seasonal cues are mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and are not identified with it; they enter each window through $\mathcal{H}_0$.

$O_1$ — 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 difference in signal concentration between the supply and drainage interfaces, together with degradation, holds the tissue far from equilibrium. With $J_E^\mu = -p_0^{-1}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 2), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. The signal flux runs down its gradient from the supply interface toward the drainage interface, in the direction of $J_E^\mu$. $\dot{W}_{\mathrm{ext}} = 0$, 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). $D^*$, $\alpha$, and $B$ are estimated on the same tissue as declared above. The declared law must predict $J_W^{\mathrm{meas}} = RT\,j^{\mathrm{meas}}/s$ at the drainage interface, with $j^{\mathrm{meas}}$ from the accumulation of the signal in the receiving compartment, and $\nabla\mathcal{F}$ from a profiling method within the window that uses neither $j^{\mathrm{meas}}$ nor the declared $D^*$ or $k_{\mathrm{d}}$. No current test exercises this aperture (empirical_validation_protocol.md §3.3). Each oscillator outside $\mathcal{M}$ is a dissipative limit cycle that requires a continuous supply of free energy; that is context, not a coordinate of the declared domain.

$O_2$ — 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 tissue 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 and 4). Tissue architecture enters through the axes of $D^*$, not through $g$. $\Pi_{\mu\nu}$ 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: signal transport is diffusive and is described by the $O_4$ balance law. Developmental and life-history trade-offs are not mapped to $\hat{h}$.

$O_3$ — 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 tissue $\sigma^\mu_{\;\nu} \equiv 0$, so the 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 the signal profile toward its new steady state is a property of the $O_4$ balance law and is not mapped to $\sigma$. The phase relationships that the target tissue retains after entrainment lie in $\mathcal{H}$. $V_{\mathrm{local}}$ 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 $\theta \equiv 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha(\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}) - k_{\mathrm{d}}\mathcal{F}$: the reaction–diffusion equation of a signal delivered across a boundary and degraded in the tissue, normalized by its storage coefficient. Its finite relaxation time makes the field stored, not quasi-static. The anisotropic $D^*$ and the degradation 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 8: the Neumann eigenmodes of $-\nabla^2$ on the rigid tissue are fixed, so the gate does not drift, and the faster decay of fine spatial structure in the signal field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The tissue also attenuates fast temporal fluctuations of its supply; that is a filter in time, distinct from the spatial gate, and it is not mapped to it. The stress-conservation equivalence holds identically (§0.1 item 4).

$O_5$ — 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 the tissue at increasing distance from the supply interface, 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 9). Every overlap is crossed by the transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The connectivity of the extracellular space is a property of the tissue that enters through $D^*$: a tissue whose extracellular space does not percolate has no continuum diffusivity 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 tissue is simply connected, so $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$ (§0.1 item 9).

$O_6$ — 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 that share a transition zone $\mathcal{U}_{ij}$. The partner fields are the rate of change of the signal's free-energy flux in the two orientations (§0.1 item 9), so $\eta_{[i\to j]} = \partial_\tau J_W$: a spatially resolved flux field. Temporal phase lags between oscillators are not spatial 1-forms and are not used as partner fields.
The supply interface holds a value of $\mathcal{F}$ and the drainage interface holds the datum, so after the step the flux near the supply decays from an initial overshoot, while farther in it rises. There $\partial_\tau J_W$ points toward the supply and away from it, respectively, and the zero set between them travels inward during the early window.
In a laterally uniform layer (precondition 7) the late zero set is the level at which the slowest relaxation mode peaks. Without lateral uniformity it is located only by the predicted solution.
The declared overlap is one that the predicted zero set does not reach during the overlap's window, with the detection threshold of §0.1 item 9; on it $\eta_{[i\to j]}$ 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, with distinct Hessians, 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 directions of enhanced or suppressed signalling. 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 tissue 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.

$O_7$ — 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 10, 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 $\hat{h} \equiv \hat{h}_\infty$. Synchrony between the pacemaker and the target tissue's clocks is a phase property of oscillators outside the declared field. Synchronization claims would belong to gauge transport, which is trivial and untested here, so the aperture makes no synchrony claim; the "synchrony" of its title enters as context.

$O_\Omega$ — 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 held supply concentration the tissue approaches a steady signal profile that carries a steady flux from the supply interface, partly degraded within the tissue and partly drained: a bounded non-equilibrium steady state. $\theta \equiv 0$, $V_{\mathcal{M}} = V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while supply is maintained, $\sigma \equiv 0$, and $\partial_\tau g \equiv 0$. 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 concentration with no degradation and no flux, nor a periodic return. While the pacemaker keeps cycling, successive windows settle toward successive held levels, and that sequence is never mapped onto the spine as a 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 tissue.
No current test exercises this aperture.

Statement	Apertures	Why it holds
\(\theta \equiv 0\), \(dV_{\mathcal{M}}/d\tau = 0\), \(V_{\mathcal{M}} = V_\infty\), \(\int\langle\theta\rangle\,d\tau = 0\), \(\dot{W}_{\mathrm{ext}} = 0\), and the \(\theta\) parts of the Branch II evidence	All	3.2 by construction (a fixed region of space); 3.1, 3.3, and 3.4 by their rigid or volume-rigid declarations, so the evidence reduces to a tolerance check on measured volume.
\(\sigma \equiv 0\), the \(O_3\) shear law, \(\hat{h} = \hat{h}_\infty\), and the tests (1)–(3) of `aperture_generation_guidance.md` §5.5	3.2, 3.3, 3.4; 3.1 without the torsional perturbation	Rigid declaration.
The \(\partial_\tau g\) residual test (3); the \(\chi_E\) rate test (2) where \(\langle\Theta_E\rangle\) is constant	3.1 under the torsional perturbation	With \(\theta \equiv 0\), \(\partial_\tau g = 2\sigma\), so (3) is equivalent to the shear-rate test (1) unconditionally. (2) is equivalent to (1) only on sub-intervals of constant measured dissipation, because the decay rate of \(\chi_E - 1\) is \(2\gamma_\sigma + 2\,d\ln\langle\Theta_E\rangle/d\tau\) (3.1 \(O_7\)).
\(J_E^\mu\nabla_\mu\mathcal{F} < 0\), \(\dot{s}_{\mathrm{prod}} > 0\), \(\dot{W}_{\mathrm{diss}} > 0\)	All	Identities of the fixed Branch II current wherever \(\nabla\mathcal{F} \neq 0\); the empirical content of \(O_1\) is the flux prediction.
Sign agreement with a measured physical dissipation	All	The named dissipation is non-negative by the second law or by metabolism (item 7). In 3.2, \(\dot{s}_{\mathrm{prod}}\) is the entropy production of the declared mapping, not a measured physical entropy production (3.2 §1).
Stress sector: \(T^{\mathrm{mat}} \equiv 0\), the \(O_4\) conservation equivalence, and the compatibility law (Saint-Venant form)	All	Flat metric and declared \(\eta \equiv 0\) (items 2 and 4).
\(O_2\) Eq. [3]	3.1, 3.3, 3.4	Default drive and no tracer mapped.
\(O_5\) gluing, \(g^{(i)} = G^{(i)}_\infty\), and overlap convexity	All (convexity is non-trivial only for 3.1 under the torsional perturbation)	Restricted metrics on a rigid or volume-rigid body (item 9).
\(\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0\)	All	\(\theta \equiv 0\) on every declared domain.
\(\phi_{\mathrm{perc}} = 1 > \phi_c\)	All	A connected domain whose transient reaches every overlap (item 9).
\(\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0\)	3.2, 3.3, 3.4	Declared simple connectivity; for 3.1 the harmonic space is non-trivial and no claim is made about it.
\(O_7\) Eq. [3] identities: divergence-free \(J_{\mathrm{geom}}\) and \(u_{\mathrm{exp}}\), and \(\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0\)	3.2, 3.3, 3.4; 3.1 without the torsional perturbation	\(\theta \equiv 0\) and \(\hat{h} \equiv \hat{h}_\infty\) on rigid domains.
\(\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}\)	3.2, 3.3, 3.4	A zero datum for a non-negative concentration or density.
\(\mathcal{I} \to s_0V_\infty\)	All	The Branch II limit with which `r7` defines the finite capacity \(s_0V_\infty\); it follows from the integrable coupling and is not tested.
\([F^{\mathrm{YM}}_{\mu\nu}, \hat{\chi}] = 0\); \(\chi_E \equiv 1\) and \(\hat{\mathcal{E}}_\mu \equiv 0\)	All; the \(\chi_E\) statements except 3.1 under the torsional perturbation	Trivial gauge sector and \(\sigma \equiv 0\) (item 10).
The \(O_4\) gate, the \(O_\Omega\) generator condition, and the alignment \(D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0\)	All	Declared but not identified with any field; stated and not tested.

Tier 3 Specification: Ecological, Metabolic & Phenological Networks¶

0. Tier Architecture & Four-Part Functional Homology¶

0.1 Conventions Common to the Four Tier 3 Apertures¶

0.2 Coordinates That Hold Trivially or by Construction¶

Aperture 3.1: Vascular Plant Physiology & Rhizosphere Coupling (\(\mathcal{E}_{\mathrm{botanical}}\))¶

1. Epistemological & Biological 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 3.2: Movement Ecology & Ecotone Foraging Dynamics (\(\mathcal{E}_{\mathrm{trophic}}\))¶

1. Epistemological & Biological 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 3.3: Benthic & Edaphic Biogeochemical Metabolism (\(\mathcal{E}_{\mathrm{metabolic}}\))¶

1. Epistemological & Biological 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 3.4: Chronobiology & Phenological Synchrony (\(\mathcal{E}_{\mathrm{chrono}}\))¶

1. Epistemological & Biological 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¶