Tier 4 Specification: Somatic, Neuro-Attentional & Affective Interiority¶

Document ID: tier_4_somatic_contemplative_interiority.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 4 maps the mechanics and nutrition of load-bearing connective tissue, the physiology of balance and support contact, neuro-attentional and autonomic regulation, and the regulation of affect through the microcirculation onto Spine $\mathcal{T}$. Three apertures (4.1, 4.2, 4.4) 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 (dissolved oxygen, or blood held in a compliant capillary bed) spreading through a living tissue, with supply crossing the boundary and consumption or loss, where declared, by a named interior sink. Aperture 4.3 is provisionally homological in this epoch: it declares no domain, maps all nine coordinates as conceptual correspondences, like Tier 1, and makes no empirical claim (§0.1 item 12).

The wider phenomena named in the aperture titles (posture, balance, attention, autonomic regulation, affect, resilience) enter the empirical mappings as boundary conditions, through the declared parameters, through the hysteresis record $\mathcal{H}$, or as illustrative context, and each §1 states which. No Tier 4 aperture claims that attention, affect, or any interior state obeys a spine law. Each aperture is formulated at the level of general physiology, biomechanics, and neuroscience, for vertebrate and mammalian tissues and for human populations studied at cohort level, without presupposing a single practitioner, practice, or site. No human-subject test is included in this epoch (empirical_validation_protocol.md §3.3).

Slot & Functional Role	Aperture Designation	Declared Domain and Diffusing Quantity	Titled Phenomena Outside the Domain	Vertical Homology Across Tiers
Slot 4.1: Structural Substrate	Myofascial Biotensegrity & Postural Mechanics ($\mathcal{E}_{\mathrm{somatic}}$)	Oxygen diffusing through an explant of avascular articular cartilage of a single zone, from a supply chamber to a receiving chamber held at the datum, consumed by chondrocytes.	The myofascial continuum and its tensegrity, posture and joint loading, synovial lubrication, remodelling, the intervertebral disc: as boundary data, through the parameters, through $\mathcal{H}$, or context.	$1.1 \leftrightarrow 2.1 \leftrightarrow 3.1 \leftrightarrow \mathbf{4.1} \leftrightarrow 5.1$
Slot 4.2: Dynamic Flow & Waves	Vestibular Equilibrium & Sensorimotor Contact ($\mathcal{E}_{\mathrm{kinesthetic}}$)	Oxygen diffusing through the viable epidermis at an occluded support contact, from the dermal–epidermal junction toward the contact, consumed by keratinocytes.	Vestibular sensing, balance control and postural sway, contact loading, cutaneous mechanoreception, callus formation: as boundary data, through $\mathcal{H}$, or context.	$1.2 \leftrightarrow 2.2 \leftrightarrow 3.2 \leftrightarrow \mathbf{4.2} \leftrightarrow 5.2$
Slot 4.3: Boundary Gating	Neuro-Attentional & Autonomic Regulation ($\mathcal{E}_{\mathrm{neuro}}$)	None declared: provisionally homological (quasi-static bioelectric potentials).	All of its phenomena are mapped as conceptual correspondences only.	$1.3 \leftrightarrow 2.3 \leftrightarrow 3.3 \leftrightarrow \mathbf{4.3} \leftrightarrow 5.3$
Slot 4.4: Coalescence & Hysteresis	Affective Regulation & Psychophysiological Resilience ($\mathcal{E}_{\mathrm{affective}}$)	Blood pressure energy in a capillary bed treated as a porous medium, from the feeding arterioles to the draining venules, with transcapillary filtration as the sink.	Affect and its regulation, autonomic outflow and arteriolar tone, cardiac and respiratory rhythms, vascular adaptation, recovery after stressors: as boundary data, through $\mathcal{H}$, or context.	$1.4 \leftrightarrow 2.4 \leftrightarrow 3.4 \leftrightarrow \mathbf{4.4} \leftrightarrow 5.4$

0.1 Conventions Common to the Tier 4 Apertures¶

The following conventions apply to every empirical aperture of this tier (4.1, 4.2, 4.4); item 12 states how the provisionally homological Aperture 4.3 is mapped. 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_3_ecological_living_networks.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 4 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 every empirical aperture the transient is a declared step of the boundary data in the direction that raises $\mathcal{F}$: a step increase of the supply-chamber oxygen (4.1), of the oxygen supplied at the dermal–epidermal junction (4.2), or of the pressure at the arteriolar boundary (4.4). Each declared balance law is a scalar parabolic equation whose sink is Lipschitz and non-decreasing in $\mathcal{F}$ (Michaelis–Menten, first order), 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 the in-vivo supply of 4.2 and the arteriolar dilation of 4.4), the same conclusion follows without time translation: $\partial_\tau\mathcal{F}$ obeys the linearized equation with zero initial data and non-negative boundary data, so it stays non-negative. Robin data (4.2's outflow face) do not change the argument, and the duct lumens of 4.2, which receive oxygen only from the tissue, form with it a cooperative system, for which the comparison principle also holds. The argument is exact on rigid domains. Under the optional torsional perturbation of 4.1 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 4.1 checks it from its oxygen record.
Direction. The opposite direction is excluded in each aperture, because it would break a declaration. In 4.4 a fall of capillary pressure shrinks the tissue ($\theta < 0$), which alone excludes it; physically it can also turn filtration into reabsorption, an interior source, which the declared datum form of the sink cannot represent (4.4 §1). In 4.2 a change of contact load compresses the epidermis ($\theta < 0$) or, on unloading, is followed by a reactive hyperaemia whose overshoot makes the supply data non-monotone. In 4.1 an oxygen step is osmotically negligible, so either direction keeps the volume; the raising direction is declared for the comparison argument, and a glucose step is not declared because it changes the bath's osmolarity, and a rise of osmolarity deswells cartilage ($\theta < 0$).
Boundary and lateral exchange. Every exchange of the declared field across an internal or lateral boundary is one-signed, out of $\mathcal{M}$: transcapillary filtration in 4.4 is a sink and never reabsorption (4.4 §1), and exchange across the walls of the epidermal ducts in 4.2 runs from the tissue into lumens that receive oxygen from nowhere else (4.2 §1). No compartment outside $\mathcal{M}$ discharges into it within a window, except across the declared supply faces.
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. 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 4 aperture declares a non-Euclidean material metric in this epoch: the growth or remodelling material metric that the master table admits for 4.1 is weighed in 4.1 §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 4 aperture; with Tiers 2 and 3, it is exercised by no empirical aperture of Tiers 2–4.
Rigid domains (4.2, 4.4). In a Cartesian chart $g_{\mu\nu} = \delta_{\mu\nu}$, so $\Omega \equiv 1$, $\hat{h} \equiv 0$, $\theta \equiv 0$, and $\sigma^\mu_{\;\nu} \equiv 0$. The tissue is declared rigid within a pre-declared tolerance on its measured volumetric strain. In the declared window direction any residual volume change is non-negative (epidermal thickness under held load in 4.2; vascular distension and filtration in 4.4), and windows in which the measured volume decreases beyond the 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 (4.1). The volumetric strain of the explant 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 4.1 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) 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 $u$): for dissolved oxygen in 4.1 and 4.2, the tissue's water volume fraction times the partition coefficient of oxygen between tissue and bath water, with $u$ the bath-equivalent oxygen concentration (oxygen tension times its solubility in bath water), which is continuous across the tissue's faces; in 4.4, the compliance of the capillary bed per unit tissue volume. 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 a single storage class: $s$ uniform on $\mathcal{M}$ within a tolerance (a declared, checkable precondition). Adjacent media with a different coefficient (other zones of the cartilage in 4.1; the stratum corneum and the dermis in 4.2; arterioles, venules, and the interstitium in 4.4) lie outside $\mathcal{M}$ and enter as boundary data or compartments, because a jump in $s$ would make $J_W$ discontinuous where the physical flux is continuous, which the unit-storage law reads as an interface source or sink.
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 4 medium has an identified global control field in this epoch, so Tier 4 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 4 domains) they read $\mathbb{L}_{\mathrm{Ein}}[\sigma] = 0$ and $\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} = 0$. The mechanical stresses of tissues (the swelling prestress of cartilage, myofascial tension, contact pressure, vessel-wall stress) 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 the longitudinal sound speed of its tissue. Rays of that signal are null for the declared cone and, like all null rays, are not trajectories of $O_2$ Eq. [3]. Every material speed in $\mathcal{M}$ (diffusive speeds, twist-recovery speeds, blood velocities) is $\ll c_*$, so $u^2 \ll 1$. 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. All three empirical apertures instantiate Eq. [3] with $a^\mu_{\mathrm{d}} = 0$ and map no tracer. Bodily movement, postural sway, and the motion of blood cells are forced motions that no declared drive describes in this epoch.
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 $\nabla_\mu J_E^\mu = -p_0^{-1}(\partial_\tau\mathcal{F} + \mathcal{F}\theta + \mathcal{R}_{\mathcal{F}}) \neq \theta$. Breathing volume, cardiac stroke volume, and the pulsation of vessels are cyclic volume changes outside the declared fields and are never mapped to $\theta$.
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 $\mathcal{F}_{\mathrm{ref}} = 0$ (4.1's receiving face, 4.4's venular boundary), it contributes nothing. The outflow face of 4.2 (the base of the stratum corneum) is not at the datum, and its term is evaluated from the measured outflow and the measured $\mathcal{F}$ there; the terms of its duct walls are bounded, not measured (4.2 §1).
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; 4.2 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, 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 4 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 4.1, 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 tissue is a property in time and is not the spatial gate. Attentional filtering is not a property of these domains; it is mapped homologically in 4.3 and never identified with the gate of an empirical aperture.
Cover, overlaps, and coupling ($O_5$, $O_6$).
The cover $\{\mathcal{U}_i\}$ consists of $\tau$-independent material subregions of $\mathcal{M}$ with Lipschitz boundaries, and every overlap $\mathcal{U}_{ij}$ is convex at $\tau_0$. Where $\partial\mathcal{M}$ has concave parts (the undulating dermal–epidermal junction and the duct walls in 4.2; the walls of the excised vessels in 4.4), each concave part lies in a single cover set, which need not be convex, and every overlap stays clear of the concave parts: 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 tissue lives in $\mathfrak{K}$, never in $g$. Effective-medium averages of diffusivities or conductances, including geometric-mean averages, are statements about $\mathfrak{K}$ and are not mapped to $O_5$ Eq. [1].
Partner fields. In every empirical aperture 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 $\eta_{[i\to j]}^\lambda = \partial_\tau J_W^\lambda$. They decay with the transient, which gives the integrable coupling that Branch II requires; a steady flux would give $\eta \equiv 0$, and a sustained coupling field is never declared.
Zero sets. $\partial_\tau J_W$ is not one-signed in general. Where the supply boundary holds a value of $\mathcal{F}$ and an outflow boundary holds the datum, 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 slab, a precondition that 4.1 declares for this purpose, that surface is the level at which the slowest relaxation mode of the linearized problem peaks; without lateral uniformity (4.2, with its ducts and undulating junction; 4.4, with its excised vessels) it 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 from the predicted solution.
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 4.1, 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 4 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 4 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.
On a connected domain the declared transient reaches every overlap for $\tau > \tau_0$, so every edge of $\mathcal{K}_0$ lies in $\mathcal{K}$ and $\phi_{\mathrm{perc}} = 1 > \phi_c$; empirically, an edge is counted when 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 perfusion. In Branch II the transient decays as the domain settles, so each coupling-tensor boundary flux decays and is integrable on $[\tau_0, \infty)$; no sustained coupling is asserted. No Tier 4 aperture is among the systems of the current coupling test.
Topology. The explant of 4.1 is declared simply connected, so $H^1_{\mathrm{dR}}(\mathcal{M}) = 0$ and $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$. The epidermis of 4.2 is pierced by ducts (and, at hair-bearing sites, by follicles), and the capillary bed of 4.4 by the excised vessel segments that pass through it, so their first Betti numbers $b_1(\mathcal{M})$ are in general positive and the spaces of harmonic Neumann fields $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}}$ of r7 $O_5$ Eq. [5] are non-trivial: a property of each domain's topology. Tier 4 identifies no transport current or measured field with them and makes no claim about them.
Gauge sector and hysteresis ($O_7$, $O_0$). Tier 4 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 4 aperture makes a synchronization, alignment, or coherence claim (of tissues, of bodily rhythms, or of persons): 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 4 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 4.1 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. The memory variables of each tissue (the loading and remodelling history of cartilage, epidermal thickening under habitual load, vascular adaptation) 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 empirical aperture's 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 4.1 is exercised by the shear-relaxation family, and 4.2 and 4.4 by no test (empirical_validation_protocol.md §3.3). No human-subject test is included, so the in-vivo forms that 4.2 and 4.4 admit are declared but not exercised. Everything else is stated as untested, not as confirmed.
The provisionally homological aperture (4.3). Aperture 4.3 maps neuro-attentional and autonomic regulation onto all nine coordinates as conceptual correspondences, like the Tier 1 grammars (aperture_generation_guidance.md §2.3). It declares no domain, metric, storage, free energy, transport tensor, sink, drive, or Branch II evidence, and makes no empirical claim. The bioelectric potentials of neural tissue are quasi-static: in volume conduction the potential follows its sources without a storage term, so r7's storage–diffusion balance law cannot represent it literally. For uniformity of the tier, each 4.3 block carries the same r7 spine-equation line as the empirical blocks, and its correspondences are read against the Branch II form, because the regulated systems it describes are bounded (a homological aperture may choose a branch form with a stated reason, aperture_generation_guidance.md §2.3); as a homology it declares no branch in the sense of closure (v), because nothing is evaluated. The shared line is read schematically: 4.3 declares no cone speed, so the r7 form with $c_* = 1$ applies to its trajectory equation, and the entry $J_E^\mu = J_W^\mu/p_0$ is the Branch II form being read, not a declared current. The same reading applies to any later homological aperture written on this pattern. Its correspondences are mapped via Spine $\mathcal{T}$ and are never reductive identities; none is evidence for a spine law, and none is used by an empirical aperture. Coordinates that hold only trivially or by construction in the empirical apertures carry no content in 4.3 either.

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. Aperture 4.3 makes no claim at any coordinate and is not listed.

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	4.1, 4.2, 4.4	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	4.2, 4.4; 4.1 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	4.1 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 $2\gamma_\sigma + 2\,d\ln\langle\Theta_E\rangle/d\tau$ (4.1 $O_7$).
$J_E^\mu\nabla_\mu\mathcal{F} < 0$, $\dot{s}_{\mathrm{prod}} > 0$, $\dot{W}_{\mathrm{diss}} > 0$	4.1, 4.2, 4.4	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	4.1, 4.2, 4.4	The named dissipation is non-negative by the second law (item 7).
Stress sector: $T^{\mathrm{mat}} \equiv 0$, the $O_4$ conservation equivalence, and the compatibility law (Saint-Venant form)	4.1, 4.2, 4.4	Flat metric and declared $\eta \equiv 0$ (items 2 and 4).
$O_2$ Eq. [3]	4.1, 4.2, 4.4	Default drive and no tracer mapped.
$O_5$ gluing, $g^{(i)} = G^{(i)}_\infty$, and overlap convexity	4.1, 4.2, 4.4 (convexity is checked for 4.1 under the torsional perturbation, and holds there exactly because its cover consists of full-width layers, which the twist maps onto themselves)	Restricted metrics on a rigid or volume-rigid body (item 9).
$\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$	4.1, 4.2, 4.4	$\theta \equiv 0$ on every declared domain.
$\phi_{\mathrm{perc}} = 1 > \phi_c$	4.1, 4.2, 4.4	A connected domain whose transient reaches every overlap (item 9).
$\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$	4.1	Declared simple connectivity; for 4.2 and 4.4 the harmonic space is in general non-trivial and no claim is made about it.
$O_7$ Eq. [3] identities: divergence-free $J_{\mathrm{geom}}$ and $u_{\mathrm{exp}}$, and $\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0$	4.2, 4.4; 4.1 without the torsional perturbation	$\theta \equiv 0$ and $\hat{h} \equiv \hat{h}_\infty$ on rigid domains.
$\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$	4.1, 4.2	A zero datum for a non-negative concentration. For 4.4 it is a checkable precondition (capillary pressure above venular pressure).
$\mathcal{I} \to s_0V_\infty$	4.1, 4.2, 4.4	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 $\hat{\mathcal{E}}_\mu \equiv 0$	4.1, 4.2, 4.4; the $\chi_E$ statements except 4.1 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 $D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0$	4.1, 4.2, 4.4	Declared but not identified with any field; stated and not tested.

The empirical content of Tier 4 therefore sits in the $O_1$/$O_4$ flux predictions; the $O_3$ shear law of 4.1 under the torsional perturbation (the only part exercised by a current test); the $O_6$ standing hypotheses on the declared overlaps; the sign of the measured-dissipation estimate; and the checkable preconditions of each domain.

Aperture 4.1: Myofascial Biotensegrity & Postural Mechanics ($\mathcal{E}_{\mathrm{somatic}}$)¶

1. Epistemological & Physiological Scope¶

Aperture 4.1 maps the mechanics of the body's load-bearing tissues onto Spine $\mathcal{T}$: the tensioned myofascial continuum in which compression-bearing skeletal elements are suspended (biotensegrity), the joints and their articular cartilage through which posture transmits load, and the adaptation of these tissues to their loading history. Its empirical domain is the nutrition of avascular load-bearing tissue: oxygen supplied across a boundary diffuses through articular cartilage, which has no blood supply of its own, and is consumed by its sparse cells, the chondrocytes. Lexicon is strictly restricted to biomechanics, cartilage and connective-tissue physiology, mechanobiology, myofascial anatomy, and kinesiology.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A cylindrical explant of articular cartilage, cut parallel to the articular surface from a single zone of the tissue so that its water volume fraction is uniform within tolerance (§0.1 item 3), and mounted in a two-chamber diffusion cell with its lateral surface sealed. $\partial\mathcal{M}$ consists of:
the articular-side face (supply), bathed by a well-stirred supply chamber whose medium stands in for synovial fluid, at a held oxygen concentration;
the deep-side face (outflow at the datum), bathed by a receiving chamber perfused with deoxygenated medium, so that its oxygen is held at the datum within tolerance;
the sealed lateral surface, closed to oxygen.

All supply crosses the articular-side face; cartilage contains no vessels and no oxygen-producing process, so there are no interior sources. The explant is declared, rather than the joint in situ, because in situ the deep boundary of adult cartilage is nearly closed: the oxygen field would then have a zero-flux face, on which $\nabla\mathcal{F} = 0$, and no measurable outflow. The other zones of the cartilage, the subchondral bone, and the joint lie outside $\mathcal{M}$; mechanical loading enters as a boundary condition (a held axial grip load) and through $\mathcal{H}$. * Intervertebral disc (not declared in this epoch). The disc, the other avascular load-bearing tissue of the body, is not a declared domain: in situ the disc is supplied from both endplates and from its periphery, so its nutrient field has an interior minimum at which $\nabla\mathcal{F} = 0$, and its nucleus, annulus, and cartilage endplates are distinct storage classes (§0.1 item 3). * Window. A relaxation window (§0.1 item 1), opened from a prior steady state by a declared step increase of the supply chamber's oxygen tension, after which it is held within tolerance, as are the receiving-chamber datum, temperature, osmolarity, and the axial grip load. The grip load is applied before the prior steady state, and the explant's consolidation under it is complete within tolerance before $\tau_0$. An oxygen step is osmotically negligible, so the explant's volume stays within tolerance. A glucose window is not declared: a step of glucose changes the bath's osmolarity, and a rise of osmolarity deswells cartilage ($\theta < 0$). * Optional torsional perturbation. The explant, gripped between porous platens, may be twisted about its axis and released at a declared time $\tau_{\mathrm{r}}$ within such a window. The recovery of the twist then supplies a shear transient ($O_3$), while oxygen diffusion and uptake continue to supply the throughput. Releasing the twist late in the window, once the measured dissipation is constant within tolerance, lets the $\chi_E$ statement of $O_7$ be evaluated. * Metric. Flat and volume-rigid (§0.1 item 2): the explant's thickness, measured by ultrasound or laser displacement, stays within a declared tolerance, 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 twist angle $\Phi_{\mathrm{tw}}(z, \tau)$ about the axis, the deformation $(r, \varphi, z) \mapsto (r, \varphi + \Phi_{\mathrm{tw}}, z)$ is isochoric and maps each cross-section and the lateral cylinder onto themselves. * Cone speed. $c_*$ is the longitudinal sound speed of cartilage. Diffusive speeds and twist-recovery speeds are $\ll c_*$. * Storage. $s = \phi_{\mathrm{w}}\kappa_{\mathrm{O}}$: the explant's water volume fraction times the partition coefficient of oxygen between its interstitial water and bath water. It is declared uniform over $\mathcal{M}$ within tolerance (§0.1 item 3). In intact cartilage the water fraction falls with depth from the articular surface; that is why the explant is cut from a single zone. * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = c\,\epsilon_{\mathrm{O}}$ (J m⁻³), with $c$ the bath-equivalent oxygen concentration (oxygen tension times its solubility in bath water, mol m⁻³), continuous across the explant's faces, and $\epsilon_{\mathrm{O}} > 0$ the declared standard-state free-energy yield per mole of oxygen reduced in the oxidation of glucose (J mol⁻¹), from tabulated standards. This is the declared linear form of aperture_generation_guidance.md §2.3 item 3. The datum is zero ($\mathcal{F}_{\mathrm{ref}} = 0$), held at the deep-side face, and $\mathcal{F} \ge 0$ holds trivially. The oxygen stored per unit volume is $s\,c$, so $\mathcal{F}$ is that stored oxygen times the declared conversion factor $\epsilon_{\mathrm{O}}/s$. The relaxation time $\ell^2/D$ across the explant's thickness $\ell$ is finite and resolvable. * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D^{\alpha\beta}$, the effective oxygen diffusivity tensor of the explant (its oxygen conductance over the storage coefficient $s$), parametric and depth-dependent: transversely isotropic about the face normal, with a declared linear dependence on depth across the explant, and the uniform isotropic diffusivity as the default. With $j$ the molar oxygen flux per unit bulk area, $J_W = \epsilon_{\mathrm{O}}j/s$; the conversion factor $\epsilon_{\mathrm{O}}/s$ is declared, with $\phi_{\mathrm{w}}$ measured on the explant itself after the window (wet and dry weights) and $\kappa_{\mathrm{O}}$ from an oxygen equilibration of the explant. * Sink $\mathcal{R}_{\mathcal{F}}$. Chondrocyte oxygen consumption with Michaelis–Menten kinetics, $\mathcal{R}_{\mathcal{F}} = V_{\mathrm{m}}\mathcal{F}/(K_{\mathcal{F}} + \mathcal{F})$, with $K_{\mathcal{F}} = K_{\mathrm{m}}\epsilon_{\mathrm{O}}$ and $V_{\mathrm{m}}$ in the storage-normalized units of the declared law. $\mathcal{R}_{\mathcal{F}}$ is continuous, non-negative, non-decreasing in $\mathcal{F}$, zero at the datum, and spread over the explant. $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$ are estimated per closure (ii) from the same explant: from its oxygen uptake in a sealed respirometry chamber after the window (a different modality), or from the difference of supply and receiving fluxes at several held supply levels before the window (a declared perturbation regime). Cell density is declared uniform within tolerance. * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). * Growth or remodelling material metric (not declared in this epoch). The master table admits an optional growth or remodelling material metric for this aperture. It has been weighed against the reasons for which 3.1 declined one, and it is not declared, for five reasons. 1. Within a fixed window remodelling is excluded (the turnover of the collagen network takes years), so a remodelling metric would be $\tau$-independent: $\theta = \sigma = 0$ in it. The imposed twist, a viscoelastic deformation, would not appear in it, so the shear-relaxation test of this aperture (Annex B Test 1) 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 connective-tissue field is identified as the global control field $\eta$; myofascial tension and the swelling prestress are stresses, not control fields. A $\xi$ constructed from a declared material metric so as to satisfy this equation would satisfy the stress sector by construction. 3. The incompatibility of cartilage's natural (stress-free) metric shows as residual strain, which is measured by cutting strips and observing their curl, a destructive method. Osmotic unloading in a hypertonic bath estimates it without cutting, but it changes the tissue's swelling state and would itself need validation as an estimator. 4. That incompatibility arises mainly because swelling varies with depth across the zones. The declared single-zone explant has uniform composition within tolerance, so on this domain a declared material metric would be flat to the order of the composition tolerance: the single-storage-class restriction and a non-trivial material curvature pull in opposite directions. 5. r7 uses one metric $g$ in every coordinate. Oxygen diffuses over physical distances, so a non-Euclidean material metric in the $O_4$ Laplacian, the Neumann spectrum, and the flux test would misstate the physical transport law unless $\mathfrak{K}$ were re-declared to compensate.

A future declaration would need a material metric whose incompatibility is estimated by an independent, non-destructive modality on a domain where it is not flat to tolerance; 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, and the flat stress-sector reduction applies to it. * 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 explant thickness, 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 oxygen balance of the supply chamber (the oxygen delivered to it minus that leaving it, from in-line sensors on its gas or medium supply) and $\mathcal{F}$ at the articular-side face from the chamber's own sensor; * the deep-side 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, independent profiling record; * the storage term uses repeated profiles.

The $O_1$ test uses the deep-side face, a different surface, and its own instruments.

The physical dissipation is the entropy production of oxygen diffusion, which is non-negative by the second law, so the comparison tests only the sign of the estimate.
Admissible domains (declared, checkable preconditions):
a viable, avascular explant with no interior oxygen source, and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$;
a single storage class: water volume fraction and oxygen partition coefficient uniform within tolerance, checked after the window by wet and dry weights of sub-slices from two or more depths; cell density uniform within tolerance, checked histologically after the window;
supply-face concentration held at its new level and deep-side face at the datum within tolerance, with the resistances of the chambers' unstirred layers and of the porous platens negligible against the explant's, by a declared estimate (stirring calibration);
lateral surface sealed, with oxygen leakage across it below the flux tolerance;
volumetric strain within the volume tolerance, with chamber osmolarity and grip load held, and consolidation under the grip load complete before $\tau_0$. 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 oxygen record (§0.1 item 1);
$V_{\mathrm{m}}$, $K_{\mathrm{m}}$ constant within tolerance: glucose supplied from both chambers at a held, non-limiting level, pH and temperature held, and no change in the cells' metabolic state beyond tolerance within the window;
no mechanical loading within the window other than the held grip load and the optional twist, so that transport is purely diffusive (no load-driven convection of interstitial fluid);
for the one-dimensional statements of $O_6$ only: a laterally uniform explant and supply.
Titled phenomena outside the domain.
Myofascial biotensegrity. The tensioned continuum of fascia, muscle, and ligament in which skeletal elements are suspended is context. Its tensions set the loads that joints transmit; these enter the explant only through its loading history ($\mathcal{H}_0$) and the held grip load.
Posture and joint loading. Habitual postural loading shapes cartilage composition and thickness between windows and enters through the declared parameters and $\mathcal{H}$. Cyclic loading pumps interstitial fluid and adds convective transport; it is excluded within windows (precondition 7) and is context.
Synovial fluid and lubrication. The supply chamber stands in for synovial fluid at the articular boundary; lubrication and the joint's synovial lining are context.
Remodelling. Adaptation of connective tissues to load (as summarized in Davis's and Wolff's laws) lies between windows and is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$, not identified with it.
Proprioception. Sensing of posture and joint position 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 explant 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 oxygen entering at the articular-side face and leaving at the deep-side face is throughput and is never mapped to it (§0.1 item 6). Openness is carried by throughput: oxygen diffusing down its gradient and being consumed produces entropy at every point, so $\Theta_E > 0$ (by $O_1$), $\Theta_{\mathrm{tot}} = \Theta_E$, and $\tau_\Theta$ counts entropy production per $s_0$, 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 tissue's loading and remodelling history (its composition, collagen architecture, and cell density) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it. Within the window such changes are excluded, 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 in oxygen between the supply and receiving chambers, together with consumption, holds the explant far from equilibrium. With $J_E^\mu = -p_0^{-1}D^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 1), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. Oxygen diffuses down its gradient from the articular-side face toward the deep-side face, 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). $D^{\alpha\beta}$ is estimated on the same explant per closure (ii), from the transient uptake of an inert dissolved gas tracer before the window (a different modality), scaled by the declared ratio of the free diffusivities of oxygen and the tracer in water (tabulated), and $s$ independently as above. The declared law must predict $J_W^{\mathrm{meas}} = \epsilon_{\mathrm{O}}j^{\mathrm{meas}}/s$ across the deep-side face, with $j^{\mathrm{meas}}$ from the receiving chamber's perfusion rate and outflow oxygen, and $\nabla\mathcal{F}$ from oxygen profiles across the explant by a declared optical or microsensor method; neither uses the other or the declared $D$, $V_{\mathrm{m}}$, or $K_{\mathrm{m}}$. 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: 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. The swelling prestress of cartilage, the tension of its collagen network, and the tensions of the myofascial continuum are Cauchy stresses, not $T_{\mu\nu}^{\mathrm{mat}}$; biotensegrity is a statement about how such stresses are distributed, and it is context, not $G_{\mu\nu} = \kappa_GT_{\mu\nu}^{\mathrm{mat}}$. Collagen 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. The paths of material points under the twist are prescribed by the deformation, not by Eq. [3], and postural movement of the body is forced motion outside $\mathcal{M}$.

$O_3$ — 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 relaxation of the oxygen field 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 explant, 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). Recovery better described by several rates or by a power law would falsify the law for this system; whether it does is the content of the test. This is the aperture's contribution to the shear-relaxation family (empirical_validation_protocol.md §3.3; Annex B Test 1, system 1). As the twist recovers, $\hat{h} \to \hat{h}_\infty$, the logarithmic strain of any residual twist, 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 the tissue to loading (changes of composition or of collagen architecture) 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 $\partial_\tau\mathcal{F} = \nabla_\alpha(D^{\alpha\beta}\nabla_\beta\mathcal{F}) - V_{\mathrm{m}}\mathcal{F}/(K_{\mathcal{F}} + \mathcal{F})$: the reaction–diffusion equation of oxygen in avascular cartilage, normalized by its storage coefficient. Its finite relaxation time makes the field stored, not quasi-static. The anisotropic, depth-dependent $D$ 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)). 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 explant 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 oxygen field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The selective partitioning of solutes into the matrix by size and charge is a property of the declared parameters, not the spatial gate. The stress-conservation equivalence holds identically (§0.1 item 4); stresses in cartilage 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 layers of the explant, each a full-width layer between two cross-sections, with layer-shaped overlaps $\mathcal{U}_{ij}$. Full width is required: the twist maps a full-width layer onto itself as a set, but it would not map a partial-width cover set onto itself, and that set's convexity would then hold only for small twists. Each overlap is the intersection of a slab with the explant's cylinder, so it is convex at $\tau_0$. The twist maps each cross-section and the lateral cylinder onto themselves, so it maps each layer onto the same set in space; a layer is convex for $g(\tau)$ exactly when its image is convex, so the overlaps stay convex for every twist, and convexity is checked on the measured metric (§0.1 item 9). 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 oxygen transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The connectivity of the matrix's interstitial water is a property of the tissue that enters through $D$: a matrix whose water space did not percolate would have no continuum diffusivity and would not be 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 connectivity of the myofascial continuum across the body is context. The declared explant 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 adjacent layers that share an overlap $\mathcal{U}_{ij}$. The partner fields are the rate of change of the oxygen 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 articular-side face holds a value of $\mathcal{F}$ and the deep-side face holds the datum, so after the step the flux near the supply decays from an initial overshoot, while deeper it rises. There $\partial_\tau J_W$ points toward the supply and away from it, respectively, and the zero set between them travels toward the deep face during the early window.
In a laterally uniform explant (precondition 8) 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.

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 (computed with $g(\tau)$ under the twist), 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 tension and compression in a tensegrity, and not of enhanced or suppressed exchange. 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 oxygen transient decays as the explant settles, so the coupling flux is integrable on $[\tau_0, \infty)$. The overlaps stay convex under any twist ($O_5$), 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, and no threshold of twist 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 (a late release of the twist makes these available), $\chi_E - 1$ 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 oxygen flow with collagen orientation beyond what $\mathfrak{K}$ declares, and no postural or myofascial coherence.

$O_\Omega$ — 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 explant approaches a steady oxygen profile that carries a steady flux from the supply chamber, partly consumed by the chondrocytes and partly drained into the receiving chamber: a bounded non-equilibrium steady state. $\theta \equiv 0$ (so $\int\langle\theta\rangle\,d\tau = 0$ and $V_{\mathcal{M}} = V_\infty$). $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while supply is maintained. $\sigma \to 0$ 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 uniform oxygen concentration with no consumption and no flux, nor a periodic return: cycles of joint loading lie between windows and are never mapped to $\theta$ or read as a return. The limit belongs to the declared idealization of an explant with constant cells under held data; the physical tissue leaves it as its cells adapt to the chamber, 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.
In this epoch only the shear-relaxation family exercises this aperture (Annex B Test 1, system 1).

Aperture 4.2: Vestibular Equilibrium & Sensorimotor Contact ($\mathcal{E}_{\mathrm{kinesthetic}}$)¶

1. Epistemological & Physiological Scope¶

Aperture 4.2 maps the physiology of balance and of the body's contact with its support onto Spine $\mathcal{T}$: vestibular sensing of head motion and gravity, the sensorimotor control that keeps the centre of mass over the base of support, cutaneous sensing of contact, and the load that balance places on the skin of the contact. Its empirical domain is the oxygen supply of the skin under that load: oxygen from the dermal capillaries diffuses across the dermal–epidermal junction into the avascular epidermis and is consumed by keratinocytes, while the support occludes the skin from the air. Lexicon is strictly restricted to vestibular physiology, posturography, sensorimotor control, cutaneous physiology, and contact mechanics.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. The viable epidermis (the basal, spinous, and granular layers) of a load-bearing skin site in contact with a support, over a declared contact area, in one of two forms:
in vivo, at a support contact under a held load;
ex vivo, an epidermal sheet separated from its dermis, whose basal face is bathed by a medium at held oxygen and whose outer face rests on the support under a held load.

$\partial\mathcal{M}$ consists of: * the dermal–epidermal junction (supply), across which oxygen enters from the papillary capillaries (in vivo) or the bathing medium (ex vivo); it undulates around the dermal papillae; * the outer face of the viable epidermis, at the base of the stratum corneum (outflow), across which oxygen passes through the stratum corneum to the support; * the walls of the epidermal ducts of sweat glands (and, at hair-bearing sites, of follicles) that pierce the epidermis (outflow, see below); * lateral faces at the edge of the declared contact area, closed within tolerance (ex vivo by a seal; in vivo by lateral uniformity of the contact, precondition 3).

The stratum corneum is a different storage class (its water fraction is far lower) and lies outside $\mathcal{M}$ as a thin barrier compartment. The support is occlusive and holds the skin-surface oxygen at the datum (for example a contact plate flushed with an oxygen-free gas). Occlusion removes the supply of oxygen from the air, so all supply crosses the dermal–epidermal junction; the datum at the surface gives the field an outflow, without which the occluded face would be a zero-flux face on which $\nabla\mathcal{F} = 0$. The dermis, its capillaries, and the support lie outside $\mathcal{M}$. The epidermis contains no capillaries and no oxygen-producing process, so there are no interior sources. Balance control and contact loading enter as boundary conditions and through $\mathcal{H}$ (below). * Duct walls. The duct lumens lie outside $\mathcal{M}$. With no sweat secretion within the window (precondition 5), the lumen fluid is static and receives oxygen only from the surrounding epidermis, and it drains toward the surface at the datum; exchange across a duct wall is therefore one-signed, out of $\mathcal{M}$, and never a source. These fluxes are not measured. Their total is bounded by a declared estimate (duct density, lumen cross-section, and the lumen's diffusive conductance to the surface) and must lie below the flux tolerance. * Window. A relaxation window (§0.1 item 1), opened from a prior steady state by a declared increase of the oxygen supplied at the dermal–epidermal junction, after which the supply data, the contact load, and skin temperature are held within tolerance. The contact load is applied before the prior steady state, and the skin's creep under it is complete within tolerance before $\tau_0$ (precondition 6). Ex vivo the increase is a step of the bathing medium's oxygen. In vivo it is a monotone rise of capillary oxygen, for example after a step increase of the inspired oxygen fraction, which reaches the junction as a ramp (§0.1 item 1); windows in which the junction data overshoot (for example through hyperoxic vasoconstriction) are excluded. Windows that open with a change of contact load are excluded: loading compresses the epidermis ($\theta < 0$) and lowers the supply by compressing the capillaries, and unloading is followed by reactive hyperaemia, whose overshoot makes the supply data non-monotone. * Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, with the epidermal thickness under the held load, measured by optical coherence tomography or high-frequency ultrasound, within the rigid tolerance. * Cone speed. $c_*$ is the longitudinal sound speed of skin. Diffusive speeds are $\ll c_*$. * Storage. $s = \phi_{\mathrm{w}}\kappa_{\mathrm{O}}$: the viable epidermis's water volume fraction times the partition coefficient of oxygen between the tissue and water (which includes oxygen dissolved in cell lipids). It is declared uniform over $\mathcal{M}$ within tolerance (§0.1 item 3). The water fraction is roughly constant through the viable layers and falls steeply in the stratum corneum, which is why the stratum corneum lies outside $\mathcal{M}$. * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = c\,\epsilon_{\mathrm{O}}$ (J m⁻³), with $c$ the bath-equivalent oxygen concentration and $\epsilon_{\mathrm{O}}$ the declared standard-state yield per mole of oxygen, as in 4.1. The datum is zero ($\mathcal{F}_{\mathrm{ref}} = 0$), held at the skin surface, and $\mathcal{F} \ge 0$ holds trivially. The relaxation time $\ell^2/D$ across the viable epidermis is finite and resolvable. * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D^{\alpha\beta}$, the layered effective oxygen diffusivity tensor of the viable epidermis (its conductance over $s$): transversely isotropic about the local normal to the skin surface, with declared values in the basal, spinous, and granular layers joined smoothly across declared transition zones, so $\mathfrak{K} \in C^1$. A uniform isotropic diffusivity is the default. With $j$ the molar flux per unit area, $J_W = \epsilon_{\mathrm{O}}j/s$; the conversion factor $\epsilon_{\mathrm{O}}/s$ is declared, with $\phi_{\mathrm{w}}$ from a confocal Raman water profile of the same site (non-invasive) and $\kappa_{\mathrm{O}}$ from tabulated or separately measured partition data. * Outflow boundary. The stratum corneum is represented as a thin barrier of declared oxygen conductance $h_{\mathrm{SC}}$ between the outer face of $\mathcal{M}$ and the surface held at the datum: the outflow obeys the Robin condition $-J_W\cdot\hat{n} = (h_{\mathrm{SC}}/s)\,\mathcal{F}$ there, with the barrier's own storage neglected (precondition 4). $h_{\mathrm{SC}}$ is estimated independently, from the surface oxygen flux and the oxygen beneath the stratum corneum in a declared perturbation regime before the window. * Sink $\mathcal{R}_{\mathcal{F}}$. Keratinocyte oxygen consumption with Michaelis–Menten kinetics, $\mathcal{R}_{\mathcal{F}} = V_{\mathrm{m}}\mathcal{F}/(K_{\mathcal{F}} + \mathcal{F})$, with $K_{\mathcal{F}} = K_{\mathrm{m}}\epsilon_{\mathrm{O}}$, continuous, non-negative, non-decreasing in $\mathcal{F}$, and zero at the datum. $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$ are estimated per closure (ii): ex vivo from respirometry of the same sheet after the window; in vivo from the response of the surface oxygen flux of the same site to declared steps of the support's oxygen level before the window (a declared perturbation regime). * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). Postural sway and balance corrections move the body, not material of $\mathcal{M}$ along trajectories; no balance-control drive is declared in this epoch. * 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 epidermal thickness, which must stay within the rigid tolerance. * $\lim\langle\Theta_E\rangle > 0$ is estimated from measured dissipation (§0.1 item 7): * the outflow-face term uses the oxygen taken up by the support (from a second, independent sensor set on its flushing gas), which equals the outflow across the outer face of $\mathcal{M}$ within the barrier-storage bound, with $\mathcal{F}$ there from an optical oxygen measurement beneath the stratum corneum; * the supply term uses the junction inflow: ex vivo from the depletion of the bathing medium; in vivo from the oxygen balance of $\mathcal{M}$ (outflow plus declared consumption plus storage change), with $\mathcal{F}$ at the junction from an optical measurement at its depth; * the duct-wall terms are bounded by the declared estimate, not measured; * the consumption term uses the declared, independently estimated $V_{\mathrm{m}}$ and $K_{\mathrm{m}}$ with $\mathcal{F}$ from a second profiling record, and the storage term uses repeated profiles. * Weaker independence. The $O_1$ test uses the surface flux into the support. Ex vivo the evidence's supply term uses a different surface (the bathing medium). In vivo the evidence uses the same surface and quantity, measured by a second, independent sensor set: it is independent of the $O_1$ instruments but not of the measured quantity, as in 3.3, and the duct terms are bounded rather than measured. * The physical dissipation is the entropy production of oxygen 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. viable, avascular epidermis with no interior oxygen source, and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$; 2. a single storage class: the water fraction of the viable epidermis uniform within tolerance, checked by its confocal Raman profile at two or more depths; 3. contact load, skin temperature, and (in vivo) systemic oxygenation held within tolerance after the step; supply data monotone and then held, checked by the optical junction measurement; contact load uniform over the declared area, so that lateral faces carry no net flux within tolerance. During balance tasks, postural sway shifts the centre of pressure and the local contact load; windows in which the load at the declared area varies beyond tolerance are excluded; 4. a support that both carries the declared contact load and holds the skin surface at the datum (for example a porous contact plate flushed with an oxygen-free gas), checked under the declared load by an oxygen sensor at the contact face, with skin-surface oxygen at the datum within tolerance; and the storage change of the stratum corneum below the flux tolerance, by a declared bound (its thickness, water fraction, and oxygen partition relative to those of $\mathcal{M}$); 5. no sweat secretion within the window (thermoneutral, resting conditions), and the duct-wall bound below the flux tolerance; 6. epidermal thickness within the rigid tolerance, with creep of the skin under the held contact load complete within tolerance before $\tau_0$ (checked from the thickness record before the window), so that no thinning ($\theta < 0$) continues into it; 7. $V_{\mathrm{m}}$, $K_{\mathrm{m}}$ constant within tolerance: isothermal conditions and no change of the epidermis's metabolic state within the window. * Titled phenomena outside the domain. * Vestibular equilibrium. The vestibular organs, their sensing of head acceleration and gravity, and the reflexes they drive lie outside $\mathcal{M}$ and are context. * Balance control and postural sway. Sensorimotor control keeps the centre of mass over the base of support by shifting the centre of pressure. Within windows the contact load at the declared area is held (precondition 3), so balance control enters through the held level of the contact data, which sets the supply at the junction by compressing the dermal capillaries. Sway oscillations are never mapped to $\theta$ or $\sigma$. * Contact sensing. Cutaneous mechanoreceptors (including Merkel cells within the basal layer) transduce contact; their sensory function is context and is not mapped. * Habitual loading. Thickening of the epidermis and callus formation under habitual load lie between windows; they enter through the declared parameters and are mapped via Spine $\mathcal{T}$ to $\mathcal{H}$, not identified with it. * Heat exchange at the contact. Heat flow through the contact is not the declared field in this epoch; skin temperature is held as a boundary datum.

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 epidermis is rigid within tolerance under the held load, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$. The oxygen supplied across the dermal–epidermal junction is throughput and is never mapped to $J_{\mathrm{geom}}$ (§0.1 item 6). Diffusion of oxygen from the junction to the respiring keratinocytes and toward the support 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 skin's record of its loading history (epidermal thickness, callus, the distribution of contact pressure it has borne) and the sensorimotor system's learned balance strategies 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 oxygen between the dermal capillaries and the occluding support, together with consumption, holds the epidermis far from equilibrium. With $J_E^\mu = -p_0^{-1}D^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 1), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. Oxygen diffuses down its gradient from the junction toward the support and into the duct lumens, 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). $D^{\alpha\beta}$ is estimated per closure (ii) from the transient of surface oxygen flux after a declared step of the support's oxygen level before the window (a declared perturbation regime), and $s$ independently as above. The declared law must predict $J_W^{\mathrm{meas}} = \epsilon_{\mathrm{O}}j^{\mathrm{meas}}/s$ across the outer face of $\mathcal{M}$, with $j^{\mathrm{meas}}$ from the oxygen taken up by the support (corrected by the bounded duct and barrier-storage terms), and $\nabla\mathcal{F}$ from optical oxygen profiles across the viable epidermis; neither uses the other or the declared $D$, $V_{\mathrm{m}}$, or $K_{\mathrm{m}}$. No current test exercises this aperture (empirical_validation_protocol.md §3.3).

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

Spine Equations: $g_{\mu\nu}(\tau, \mathbf{x}) = \Omega^2(\tau, \mathbf{x})\hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}(\tau, \mathbf{x}))^\alpha_{\;\nu}$ with $\det\hat{g} = 1$, $\mathrm{Tr}(\hat{h}) = 0 \implies \det g = \Omega^{2n}$, $\frac{\partial\ln\Omega}{\partial\tau} = \frac{1}{n}\theta$; $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$; $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$, $R = \kappa_G\nabla_\alpha\xi^\alpha$; $\frac{2(n-1)}{n}\nabla^2\theta + \frac{2R}{n}\theta = -\kappa_G\frac{\partial(\nabla_\alpha\xi^\alpha)}{\partial\tau} + 2\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} - 2\sigma^{\alpha\beta}R_{\alpha\beta}$; $\left.\left(\frac{dv^\mu}{d\tau} + \Gamma^\mu_{\rho\epsilon}v^\rho v^\epsilon + g^{\mu\lambda}\left(\frac{\partial g_{\lambda\nu}}{\partial\tau}\right)v^\nu - \frac{1}{2}\left(\frac{\partial g_{\rho\epsilon}}{\partial\tau}v^\rho v^\epsilon\right)v^\mu\right)\right\vert_{N=1,\,\beta^\mu=0} = f^\mu_{\mathrm{drive}}$, $f^\mu_{\mathrm{drive}} = (1-u^2)\left(a^\mu_{\mathrm{d}} - c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu\right)$, $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2 < 1$, with the default $a^\mu_{\mathrm{d}} = 0$.
Physical Instantiation: On the rigid epidermis 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 contact pressure and the stresses it induces in the skin are Cauchy stresses, not $T_{\mu\nu}^{\mathrm{mat}}$; they enter through the supply data. The layered structure of the epidermis enters through $D$, and its undulating junction through the declared geometry of $\mathcal{M}$, 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. The trajectory of the body's centre of mass and the path of the centre of pressure are trajectories of the body outside $\mathcal{M}$ under forces (gravity, support reaction, muscle action); no drive is declared for them, and they are not geodesics of $g$.

$O_3$ — 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 epidermis $\sigma^\mu_{\;\nu} \equiv 0$, 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 oxygen profile toward its new steady state is a property of the $O_4$ balance law and is not mapped to $\sigma$. The recovery of balance after a perturbation, and the viscoelastic recovery of soft tissue after unloading, lie outside the declared windows: the first in $\mathcal{H}$ and context, the second excluded with the load changes that cause it. $V_{\mathrm{local}}$ is the hysteresis content of a cover set.

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

Spine Equations: $\frac{\partial\mathcal{F}}{\partial\tau} + \mathcal{F}\theta = -\nabla_\alpha J_W^\alpha - \mathcal{R}_{\mathcal{F}}$, $J_W^\alpha \equiv -\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}$, $\mathfrak{K}^{\alpha\beta} = \mathfrak{K}^{\beta\alpha} \succ 0$, $\mathcal{R}_{\mathcal{F}} \ge 0$; $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}} \equiv \mathbf{1}_{(-\nabla^2 \le \Lambda_{\mathrm{gate}}^2)}$; $Q(\mathbf{x}, \tau) = \langle\mathbf{x}\vert\rho_{\mathrm{local}}(\tau)\vert\mathbf{x}\rangle > 0$; $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P) > 0$; $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$.
Physical Instantiation: With $\theta \equiv 0$ the balance law reads $\partial_\tau\mathcal{F} = \nabla_\alpha(D^{\alpha\beta}\nabla_\beta\mathcal{F}) - V_{\mathrm{m}}\mathcal{F}/(K_{\mathcal{F}} + \mathcal{F})$: the reaction–diffusion equation of oxygen in the viable epidermis, normalized by its storage coefficient, with Robin data at its outer face. Its finite relaxation time makes the field stored, not quasi-static. The layered $D$ 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 epidermis are fixed, so the gate does not drift, and the faster decay of fine spatial structure in the oxygen field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The barrier function of the stratum corneum, which selects what passes between the body and its surroundings, is a boundary property outside $\mathcal{M}$, 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 subregions of the viable epidermis: one cover set around each duct, containing its wall (a concave part of $\partial\mathcal{M}$); one cover set along the undulating junction, containing its concave parts; and further sets that are intersections of $\mathcal{M}$ with strictly convex bodies clear of those concave parts. Every overlap is declared clear of the concave parts and convex on the rigid metric (§0.1 item 9). 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 $\tau$ ($H_0 = 0$). Every overlap is crossed by the transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The connectivity of the tissue's intercellular and intracellular paths for oxygen enters through $D$; it is mapped via Spine $\mathcal{T}$ to the condition $\phi_{\mathrm{perc}} > \phi_c$ and is not identified with it. Each duct that passes through the epidermis adds a non-contractible loop, so $b_1(\mathcal{M})$ equals the number of ducts and follicles in the declared area, and the harmonic Neumann fields that circulate around them are non-trivial. No transport current is identified with them, and no claim is made about them (§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 that share an overlap $\mathcal{U}_{ij}$, for example a deeper and a more superficial region of the spinous layer between ducts. The partner fields are the rate of change of the oxygen 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 such as sway and muscle activity.
The junction holds the raised supply and the outer face drains through the stratum corneum to the datum, so after the increase the flux near the junction decays from an initial overshoot while nearer the surface it rises, and the zero set between them travels outward during the early window.
The ducts and the undulating junction break lateral uniformity, so the zero set is located only by the predicted solution, from the independently estimated parameters, the declared junction data, and $h_{\mathrm{SC}}$.
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 postural stability or instability. 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 epidermis 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 $\hat{h} \equiv \hat{h}_\infty$. The coordination of vestibular, visual, and somatosensory signals in balance, and any coupling of sway between persons, are not mapped to the gauge connection; they are context, and no coordination 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 held supply, contact load, and temperature, the epidermis approaches a steady oxygen profile that carries a steady flux from the junction, partly consumed by the keratinocytes and partly drained through the stratum corneum and the ducts to the support: 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 oxygen concentration with no consumption and no flux, nor a periodic return: the alternation of loading and unloading in standing and walking lies between windows and is never read as a return. A balanced stance is a regulated state of the body outside $\mathcal{M}$; it is not mapped to this limit. The limit belongs to the declared idealization of held load and supply; the physical skin leaves it as load shifts, 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 epidermis.
No current test exercises this aperture.

Aperture 4.3: Neuro-Attentional & Autonomic Regulation ($\mathcal{E}_{\mathrm{neuro}}$; provisionally homological)¶

1. Epistemological & Physiological Scope¶

Aperture 4.3 maps how nervous systems select and regulate: attention, which selects among competing signals and sets their gain; arousal and its neuromodulatory control; and the autonomic nervous system, whose sympathetic and parasympathetic branches regulate the internal milieu through reflex loops such as the baroreflex, and whose afferents carry interoceptive signals to the brain. Lexicon is strictly restricted to cognitive neuroscience, electrophysiology, neurodynamics, and autonomic and interoceptive physiology.

Status (aperture_generation_guidance.md §2.3; provisionally homological). This aperture declares no empirical domain and makes no empirical claim in this epoch (§0.1 item 12). The systems it describes do not fit r7's balance law literally. The electrical potentials of neural tissue are quasi-static: in volume conduction the potential follows its current sources without a storage term, so it is not a stored field with a finite relaxation time, which aperture_generation_guidance.md §2.3 item 3 requires. Attention and autonomic regulation are, besides, properties of whole circuits and of the body, not fields on one declared tissue domain. Each coordinate is therefore mapped as a conceptual correspondence, mapped via Spine $\mathcal{T}$ and never a reductive identity. Populations are described at cohort level; practices that train attention or regulate arousal are context and are not mapped. Test 2 (spatial gating) remains deferred while this aperture is homological (Annex B).

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); $\frac{\partial\mathcal{H}}{\partial\tau} \equiv J_E^\mu\nabla_\mu q$, $q(\mathbf{x}, \tau) \equiv -\ln(v_0Q(\mathbf{x}, \tau))$.
Homological Mapping: Neural tissue never rests in equilibrium: it is sustained by a continuous supply of oxygen and glucose, and most of its energy is spent restoring the ionic gradients that signalling runs down. That sustained metabolic throughput corresponds to the throughput rate $\Theta_E$ of a Branch II state, in which openness is carried by throughput and not by expansion ($\theta \to 0$). Brain volume is bounded; no expansion of tissue is mapped to $\theta$, and the cyclic swelling of tissue with the pulse is not $\theta$. The accumulation of learned expectations from surprising input corresponds to the record law $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$, with prediction error read as a correspondence of the surprisal gradient $\nabla q$; this is a correspondence with predictive-processing accounts of perception, not an identity.

$O_1$ — 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); $J_E^\mu\nabla_\mu\mathcal{F} < 0$; $\dot{s}_{\mathrm{prod}} = -\beta_{\mathrm{th}}J_E^\mu\nabla_\mu\mathcal{F} > 0$.
Homological Mapping: The energy budget of signalling (ion pumping against the leak of action and synaptic potentials) corresponds to the dissipative power $\dot{W}_{\mathrm{diss}} > 0$ and to positive entropy production. In neural tissue the potential that drives currents is an electrical potential in volts, which is not an energy density and is never $\mathcal{F}$; in the homology, $\mathcal{F}$ corresponds to the free energy stored in ionic and metabolic gradients. The sign condition $J_E^\mu\nabla_\mu\mathcal{F} < 0$ corresponds to currents that run down those gradients and must be restored by metabolism. No flux prediction is made.

$O_2$ — 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 $a^\mu_{\mathrm{d}} = 0$.
Homological Mapping: In predictive-processing accounts, attention is described as the weighting of prediction errors by their expected precision. That notion corresponds to the precision Hessian $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$ and its trace-free part $\hat{\Pi}$, which weights the control 1-form $\xi_\alpha = \hat{\Pi}_{\alpha\lambda}\eta^\lambda$; the correspondence is between roles, not between quantities, and no precision potential is estimated from neural data. The Weyl split's algebraic isolation of the volume element corresponds to the distinction between the overall level of arousal and the selective shaping of what is attended, and, as in every aperture, the traced elliptic $\tau$-compatibility equation keeps the two coupled (r10 Decision 2). The trajectory equation with a declared drive corresponds to directed movements of gaze and attention toward goals: a forced, purposeful motion, which r7 admits only through a declared drive under the binding closure. No drive is declared and no trajectory is mapped.

$O_3$ — Shear Damping & Hysteresis Consolidation¶

Spine Equations: $\left.\frac{\partial g_{\mu\nu}}{\partial\tau}\right\vert_{N=1,\,\beta^\mu=0} = \frac{2}{n}\theta g_{\mu\nu} + 2\sigma_{\mu\nu}$; $\nabla_\tau\sigma^\mu_{\;\nu} \equiv \partial_\tau\sigma^\mu_{\;\nu} = -\gamma_\sigma\sigma^\mu_{\;\nu}$ ($\gamma_\sigma = \mathrm{const} > 0$, $\sigma^\mu_{\;\mu} = 0$) $\implies \lim_{\tau\to\infty}\Vert\sigma^\mu_{\;\nu}\Vert_{C^2(\mathcal{M})} = 0$; $\hat{h}(\tau, \mathbf{x}) \xrightarrow{C^2(\mathcal{M})} \hat{h}_\infty(\mathbf{x})$, $(\hat{g}_\infty(\mathbf{x}))_{\mu\nu} \equiv \hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}_\infty(\mathbf{x}))^\alpha_{\;\nu}$ ($\det\hat{g}_\infty = \det\hat{g} = 1$); $V_{\mathrm{local}}(\tau) \equiv \int_{\mathcal{U}_i}\mathcal{H}(\mathbf{x}, \tau)\sqrt{\vert g\vert}\,d^nx$.
Homological Mapping: Habituation, the decline of a response to a repeated stimulus that carries no new information, and sensory adaptation correspond to the decay of a transient at a constant rate, with what is learned retained in the record $\mathcal{H}$, as $\hat{h}$ settles to $\hat{h}_\infty$ while $V_{\mathrm{local}}$ holds the consolidated content. The correspondence concerns the pairing of decay with retention. Neural and autonomic responses often relax with oscillation, which the mixed-form shear law excludes, and no rate of a neural response is mapped to $\gamma_\sigma$.

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

Spine Equations: $\frac{\partial\mathcal{F}}{\partial\tau} + \mathcal{F}\theta = -\nabla_\alpha J_W^\alpha - \mathcal{R}_{\mathcal{F}}$, $J_W^\alpha \equiv -\mathfrak{K}^{\alpha\beta}\nabla_\beta\mathcal{F}$, $\mathfrak{K}^{\alpha\beta} = \mathfrak{K}^{\beta\alpha} \succ 0$, $\mathcal{R}_{\mathcal{F}} \ge 0$; $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}} \equiv \mathbf{1}_{(-\nabla^2 \le \Lambda_{\mathrm{gate}}^2)}$; $Q(\mathbf{x}, \tau) = \langle\mathbf{x}\vert\rho_{\mathrm{local}}(\tau)\vert\mathbf{x}\rangle > 0$; $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P) > 0$; $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$.
Homological Mapping: Attentional selection, which passes some signals and suppresses others, corresponds to the gate $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}}$ that forms the gated density $Q$ from $\rho_{\mathrm{total}}$, with the variational power $\dot{W}_{\mathrm{var}}$ as the cost of keeping $Q$ apart from the reference $P$. The breadth of attentional scope corresponds to the spatial cutoff $\Lambda_{\mathrm{gate}}$: a broad scope to a low cutoff that keeps coarse structure, a narrow scope to a high cutoff. The gate is a spatial projector onto Neumann eigenmodes; it does not correspond to temporal frequency bands of neural rhythms, and the smoothing of conducted potentials by volume conduction is a physical property of the conductor, not the gate. The balance law corresponds to the restoration of ionic gradients against their dissipation. The autonomic baroreflex, which holds blood pressure near a set point by opposing deviations, corresponds to the regulation of $Q$ toward $P$ and is not mapped to the stress-conservation equivalence $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$, which has no counterpart here.

$O_5$ — 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 $H_0 = 0$).
Homological Mapping: Large-scale brain networks join specialized regions through shared connections without erasing their specializations; that pattern corresponds to the gluing of local metrics over overlaps by the geometric mean, $g^{(ij)} = g^{(i)} \# g^{(j)}$, which keeps each cover set's metric. The integration of a network whose connections exceed a threshold of density corresponds to the percolation condition $\phi_{\mathrm{perc}}(\mathcal{K}) > \phi_c$. Recurrent loops of connectivity correspond to non-trivial first cohomology and to harmonic circulation around closed cycles, as a correspondence between a graph's cycles and a manifold's topology, not an identity. No connectivity estimate is mapped to $\mathcal{K}$.

$O_6$ — 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).
Homological Mapping: Directed influence between brain regions (effective connectivity, in which one region's activity shapes another's more than the reverse) corresponds to the odd phase-lead 1-form $J_{[i\to j]} = -J_{[j\to i]}$, which reverses sign when the pair is read the other way. Temporal lead–lag statistics between scalar recordings are not spatial 1-form fields, and the correspondence implies no estimator. The paired antagonism of the sympathetic and parasympathetic branches, which act in opposite senses on the same organs, corresponds to the opposite-signed extreme eigenvalues $m_+ > 0 > m_-$ of the trace-free operator $M^{(ij)}$ under the non-degeneracy hypothesis, with $m_+ \neq -m_-$ in general, as autonomic balance is not a symmetric see-saw. The growth of shared information between coupled regions corresponds to the flux-positivity hypothesis $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$. Both remain standing hypotheses of r7, and in the Branch II form the coupling must be transient (integrable), so the correspondence is with episodes of coupling that resolve, not with sustained coupling.

$O_7$ — 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); $D_{\mathrm{KL}}(\hat{\rho}_g(\tau) \parallel \hat{\rho}_{\mathcal{H}}(\tau)) \to 0$.
Homological Mapping: Synchronization of neural populations across distant regions corresponds to gauge transport, with the decay of the covariant derivative $\hat{\mathcal{E}}_\mu$ as the relaxation of mismatches between local frames; synchrony corresponds to gauge transport and never to the anisotropy index $\chi_E$, which concerns the shape of deformation only. The alignment $D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0$ corresponds to the convergence of current processing with what has been learned, as expectations come to match the statistics of the environment.

$O_\Omega$ — 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 $\tau \to \infty$).
Homological Mapping: Homeostatic and allostatic regulation hold a nervous system and its body in a bounded, sustained state of activity: never static equilibrium, which for living tissue is death, and never a strict periodic return, although rhythms of sleep and waking recur. That state corresponds to the Branch II limit: bounded volume, sustained throughput ($\langle\Theta_E\rangle \to \Theta_\infty > 0$), decaying shear and coupling transients, and a record whose growth stops when expectations no longer meet surprise ($J_E^\lambda\nabla_\lambda q \to 0$). The correspondence makes no claim about any state of attention or awareness, and no test exercises this aperture in this epoch.

Aperture 4.4: Affective Regulation & Psychophysiological Resilience ($\mathcal{E}_{\mathrm{affective}}$)¶

1. Epistemological & Physiological Scope¶

Aperture 4.4 maps the bodily side of affect onto Spine $\mathcal{T}$: the autonomic expression of emotion and arousal, its regulation, and the capacity of physiological systems to recover after stressors (psychophysiological resilience). A principal route by which affective and autonomic states act on tissues is the tone of small arteries and arterioles, which sets how blood is distributed through the microcirculation. Its empirical domain is the perfusion of a capillary bed: blood enters from the feeding arterioles, spreads through the capillary network, which stores blood in its compliance, and leaves through the draining venules, while part of the plasma filters into the tissue. Lexicon is strictly restricted to affective neuroscience, autonomic psychophysiology, and microvascular physiology.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. A tissue region containing a capillary bed, treated at the network scale as a porous medium, with the lumens and walls of the feeding arterioles and draining venules that run through it excised from $\mathcal{M}$, together with any larger vessels. The capillaries and the post-capillary segments of a single compliance class form the medium. $\partial\mathcal{M}$ consists of:
the arteriolar boundary (supply): the walls of the terminal arterioles where the capillaries arise, across which blood enters at a pressure set by arteriolar tone;
the venular boundary (outflow at the datum): the walls of the collecting venules into which the capillaries drain, at venular pressure $p_{\mathrm{ven}}$, held within tolerance;
the remaining boundary, closed: tissue surfaces, or surfaces across which no blood crosses within tolerance, located by a declared perfusion measurement.

All supply crosses the arteriolar boundary; the capillary bed contains no other inflow, so there are no interior sources. The arterioles, venules, and interstitium have compliance or storage of a different class (§0.1 item 3) and lie outside $\mathcal{M}$; the interstitium receives the filtrate, which lymphatics drain outside $\mathcal{M}$. The domain may be observed in vivo by intravital microscopy, or in an isolated perfused tissue preparation. * Window. A relaxation window (§0.1 item 1), opened from a prior steady state by a declared step (or monotone ramp) increase of the pressure at the arteriolar boundary, after which arteriolar pressure, venular pressure, and temperature are held within tolerance. The increase may come from dilation of the feeding arterioles (withdrawal of vasoconstrictor tone, or a vasodilator applied to the feeding arterioles only) or, in a perfused preparation, from a raised perfusion pressure. Capillary pressure then rises at every point, so filtration rises and remains a sink, and the tissue's volume change is non-negative (vascular distension and filtrate). Windows that open with a fall of arteriolar pressure are excluded, because falling capillary pressure shrinks the tissue ($\theta < 0$); that alone justifies the exclusion. Physically, a fall can also turn filtration into reabsorption from the interstitium, an interior source. The declared datum form cannot show this: with $p_{\mathrm{f}} = p_{\mathrm{ven}}$ and $p \ge p_{\mathrm{ven}}$ the declared filtration never reverses. This second reason therefore rests on the physical Starling balance, not on the declared law, and it is why the declared form would misrepresent such a window. Windows in which myogenic or metabolic autoregulation returns the arteriolar pressure beyond tolerance are excluded. * Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, with the tissue volume, measured by plethysmography or tissue thickness, within the rigid tolerance; any residual change is non-negative in the declared direction. * Cone speed. $c_*$ is the longitudinal sound speed of the tissue. Blood velocities in the microcirculation are $\ll c_*$. * Storage. $C_{\mathrm{v}}$: the compliance of the capillary bed per unit tissue volume, the blood volume stored per unit volume per unit rise of capillary pressure (m³ m⁻³ Pa⁻¹). It is declared uniform over $\mathcal{M}$ within tolerance and linear over the window's range of pressures (§0.1 item 3). * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = p - p_{\mathrm{ven}}$ (J m⁻³), with $p$ the network-scale capillary blood pressure. The datum is the venular pressure, held at the venular boundary. Blood flows from the arteriolar to the venular boundary, so $p \ge p_{\mathrm{ven}}$ throughout $\mathcal{M}$ is a checkable precondition (precondition 2), not a triviality. $\mathcal{F}$ is linear in the stored blood: the blood stored per unit volume above the datum is $C_{\mathrm{v}}\mathcal{F}$, so $\mathcal{F}$ is that stored blood times the declared conversion factor $1/C_{\mathrm{v}}$. The relaxation time $\ell^2C_{\mathrm{v}}/k_{\mathrm{b}}$ is finite, and it is required to exceed the time resolution of the pressure instruments by a declared factor (precondition 8). * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = k_{\mathrm{b}}^{\alpha\beta}/C_{\mathrm{v}}$, with $k_{\mathrm{b}}^{\alpha\beta}$ the network-scale (Darcy) conductance tensor of the capillary bed for blood at its effective viscosity, parametric: for example one conductance along and one across the dominant capillary orientation (as along muscle fibres), with isotropic conductance as the default. With $q$ the Darcy blood flux (m s⁻¹), $J_W = q/C_{\mathrm{v}}$; $1/C_{\mathrm{v}}$ is the declared conversion factor. * Sink $\mathcal{R}_{\mathcal{F}}$. Transcapillary filtration, first order relative to the datum: $\mathcal{R}_{\mathcal{F}} = k_{\mathrm{f}}(\mathcal{F} - \mathcal{F}_{\mathrm{ref}}) \ge 0$, with $k_{\mathrm{f}} = L_{\mathrm{p}}S_{\mathrm{v}}/C_{\mathrm{v}}$ ($L_{\mathrm{p}}$ the hydraulic conductivity of the capillary wall, $S_{\mathrm{v}}$ the capillary surface area per unit tissue volume). The physical filtration follows the Starling balance of hydrostatic and oncotic pressures across the wall, which vanishes at some capillary pressure $p_{\mathrm{f}}$; the first-order datum form declares $p_{\mathrm{f}} = p_{\mathrm{ven}}$ within tolerance, 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)). $L_{\mathrm{p}}S_{\mathrm{v}}$ is estimated per closure (ii) by a declared method on the same tissue (for example the filtration response to venous pressure steps before the window, a declared perturbation regime), and $p_{\mathrm{f}}$ is checked against $p_{\mathrm{ven}}$ in the same regime. * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). Red cells move with the blood under pressure and drag; no drag drive is declared, and no cell is mapped as a tracer. * 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 tissue volume, 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 inflow through the feeding arterioles (red-cell velocimetry times lumen cross-section in vivo, or the perfusion flow of a perfused preparation) and $\mathcal{F}$ at the arteriolar boundary from servo-null micropressure measurement; * the venular boundary, at the datum, contributes nothing; * the consumption term uses the declared, independently estimated $k_{\mathrm{f}}$ with $\mathcal{F}$ from a second set of capillary pressure measurements; * the storage term uses repeated pressure measurements.

The $O_1$ test uses an interior section and its own instruments.

The physical dissipation is the viscous dissipation of blood flow, $q\cdot(-\nabla p)$ per unit volume. Under the declarations it equals the r7 dissipation density $-J_E^\mu\nabla_\mu\mathcal{F}$ times the positive constant $p_0C_{\mathrm{v}}$, and its measured sign must agree with that of the estimate.
Admissible domains (declared, checkable preconditions):
no interior inflow (no arteriovenous shunts or feeding vessels inside $\mathcal{M}$ beyond the excised ones), and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$;
capillary pressure above venular pressure throughout $\mathcal{M}$, and net filtration non-negative everywhere (no reabsorption), with interstitial hydrostatic and oncotic pressures constant within tolerance over the window;
a single compliance class: $C_{\mathrm{v}}$ uniform within tolerance, checked by the blood-volume response to a pressure step in two or more subregions;
no capillary recruitment or derecruitment within the window (perfused capillary density constant within tolerance, by intravital microscopy), and a steady distribution of haematocrit, so that $k_{\mathrm{b}}$ is constant;
arteriolar and venular boundary pressures held within tolerance after the step, with residual cardiac pulsation, respiratory modulation, and vasomotion at the arteriolar boundary within tolerance; windows with vasomotion beyond tolerance are excluded;
tissue volume within the rigid tolerance, isothermal conditions within tolerance;
the closed parts of the boundary carry no blood flux within tolerance;
a relaxation time $\ell^2C_{\mathrm{v}}/k_{\mathrm{b}}$ that exceeds the time resolution of the pressure instruments by a declared factor, so that the field is resolvably stored and not quasi-static.
Titled phenomena outside the domain.
Affect and its regulation. Emotions, their appraisal, and their regulation are processes of the person, outside $\mathcal{M}$; they are context. Their autonomic expression reaches the bed through sympathetic vasoconstrictor tone on the feeding arterioles, which sets the arteriolar boundary data. Within windows those data are held, so affect enters only through their held level and through the step that opens the window.
Autonomic rhythms. Heart-rate variability, respiratory modulation, and vasomotion are temporal quantities; they are excluded from windows by tolerance and are never mapped to $\theta$, $\sigma$, or any spatial operator.
Psychophysiological resilience. Recovery of physiological variables after a stressor is a property of the person's regulatory systems, outside $\mathcal{M}$. It is not mapped to $\gamma_\sigma$, to the relaxation of the pressure field, or to any rate of this aperture. Repeated stress shapes the microvasculature between windows (remodelling, changes of capillary density), which enters through the declared parameters and is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$, not identified with it.
Lymphatic drainage and oedema. Drainage of the filtrate and the accumulation of interstitial fluid lie outside $\mathcal{M}$; windows end before interstitial pressure changes beyond tolerance (precondition 2).

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 within tolerance, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$. The blood entering across the arteriolar boundary is throughput and is never mapped to $J_{\mathrm{geom}}$ (§0.1 item 6); the cyclic pulsation of vessels with the heartbeat is not $\theta$. Viscous flow of blood down its pressure 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 $s_0$. The record accumulates by $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput current. The bed's structural adaptation (capillary density, vessel calibre, the history of its tone) 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: The maintained arteriovenous pressure difference, together with filtration, holds the bed far from equilibrium. With $J_E^\mu = -p_0^{-1}(k_{\mathrm{b}}^{\mu\nu}/C_{\mathrm{v}})\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 1), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$. Blood flows down its pressure gradient from the arteriolar to the venular 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). $k_{\mathrm{b}}$ is estimated per closure (ii) from the bed's pressure–flow relation at several steady perfusion pressures before the window (a declared perturbation regime), and $C_{\mathrm{v}}$ from the blood-volume response to a pressure step measured optically (a different modality). The declared law must predict $J_W^{\mathrm{meas}} = q^{\mathrm{meas}}/C_{\mathrm{v}}$ across a declared interior section $S$, with $q^{\mathrm{meas}}$ from the summed flows of the capillaries crossing $S$ (red-cell velocity and lumen cross-section by intravital microscopy, per unit area), and $\nabla\mathcal{F}$ from micropressure measurements on both sides of $S$; neither uses the other or the declared $k_{\mathrm{b}}$ or $k_{\mathrm{f}}$. No current test exercises this aperture (empirical_validation_protocol.md §3.3).

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

Spine Equations: $g_{\mu\nu}(\tau, \mathbf{x}) = \Omega^2(\tau, \mathbf{x})\hat{g}_{\mu\alpha}(\mathbf{x})(\exp\hat{h}(\tau, \mathbf{x}))^\alpha_{\;\nu}$ with $\det\hat{g} = 1$, $\mathrm{Tr}(\hat{h}) = 0 \implies \det g = \Omega^{2n}$, $\frac{\partial\ln\Omega}{\partial\tau} = \frac{1}{n}\theta$; $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$; $G_{\mu\nu} = \kappa_G T_{\mu\nu}^{\mathrm{mat}}$, $R = \kappa_G\nabla_\alpha\xi^\alpha$; $\frac{2(n-1)}{n}\nabla^2\theta + \frac{2R}{n}\theta = -\kappa_G\frac{\partial(\nabla_\alpha\xi^\alpha)}{\partial\tau} + 2\nabla^\alpha\nabla^\beta\sigma_{\alpha\beta} - 2\sigma^{\alpha\beta}R_{\alpha\beta}$; $\left.\left(\frac{dv^\mu}{d\tau} + \Gamma^\mu_{\rho\epsilon}v^\rho v^\epsilon + g^{\mu\lambda}\left(\frac{\partial g_{\lambda\nu}}{\partial\tau}\right)v^\nu - \frac{1}{2}\left(\frac{\partial g_{\rho\epsilon}}{\partial\tau}v^\rho v^\epsilon\right)v^\mu\right)\right\vert_{N=1,\,\beta^\mu=0} = f^\mu_{\mathrm{drive}}$, $f^\mu_{\mathrm{drive}} = (1-u^2)\left(a^\mu_{\mathrm{d}} - c_*^{-2}(g_{\alpha\beta}a^\alpha_{\mathrm{d}}v^\beta)v^\mu\right)$, $u^2 = g_{\mu\nu}v^\mu v^\nu/c_*^2 < 1$, with the default $a^\mu_{\mathrm{d}} = 0$.
Physical Instantiation: On the rigid 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). Vessel-wall stresses and the tone of vascular smooth muscle are Cauchy stresses and active tensions, not $T_{\mu\nu}^{\mathrm{mat}}$; tone acts through the arteriolar boundary data. The architecture of the network enters through $k_{\mathrm{b}}$ and the declared geometry of $\mathcal{M}$, 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.

$O_3$ — 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 pressure field toward its new steady state is a property of the $O_4$ balance law and is not mapped to $\sigma$. The haemodynamic shear stress on vessel walls is a stress, not the shear rate $\sigma$ of the metric. Recovery after a stressor lies outside $\mathcal{M}$ and is not mapped here. $V_{\mathrm{local}}$ is the hysteresis content of a cover set.

$O_4$ — 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 $\partial_\tau\mathcal{F} = \nabla_\alpha((k_{\mathrm{b}}^{\alpha\beta}/C_{\mathrm{v}})\nabla_\beta\mathcal{F}) - k_{\mathrm{f}}\mathcal{F}$: the storage–conductance equation of a compliant porous capillary bed with first-order filtration, in pressure form. Compliance gives the finite relaxation time, so the field is stored, not quasi-static (precondition 8). The anisotropic $k_{\mathrm{b}}$ and the filtration 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 pressure field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The selective permeability of the capillary wall to water and solutes is a property of the sink's parameters, 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 subregions of the tissue: one cover set around each excised vessel segment, containing its wall (a concave part of $\partial\mathcal{M}$), and further sets that are intersections of $\mathcal{M}$ with strictly convex bodies clear of those walls. Every overlap is declared clear of the concave parts and convex on the rigid metric (§0.1 item 9). 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 $\tau$ ($H_0 = 0$). Every overlap is crossed by the transient, so $\phi_{\mathrm{perc}} = 1 > \phi_c$. The connectivity of the capillary network is a property of the bed that enters through $k_{\mathrm{b}}$: a network below its percolation threshold has no network-scale conductance 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; anastomoses between vascular territories are context. Each excised vessel segment that passes through $\mathcal{M}$ adds a non-contractible loop, so $b_1(\mathcal{M}) \ge 1$ in general, and harmonic Neumann fields circulate around these segments. No blood flow or measured field is identified with them, and no claim is made about them (§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 between a feeding arteriole and a draining venule, sharing an overlap $\mathcal{U}_{ij}$. The partner fields are the rate of change of the free-energy flux of blood 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 such as heart rate and blood pressure.
The arteriolar boundary holds the raised pressure and the venular boundary the datum, so after the step the flux near the arterioles decays from an initial overshoot while nearer the venules it rises, and the zero set between them travels toward the venules during the early window.
The excised vessels make the geometry far from a slab, so the zero set is located only by the predicted solution, from the independently estimated $k_{\mathrm{b}}$, $C_{\mathrm{v}}$, and $k_{\mathrm{f}}$ and the declared boundary data.
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 sympathetic and parasympathetic influences, and not directions of enhanced or suppressed perfusion. 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 perfusion. In Branch II the transient decays as the bed 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. No range of autonomic perturbation, clinical or otherwise, is mapped to a convexity threshold.

$O_7$ — 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 $\hat{h} \equiv \hat{h}_\infty$. Coherence between cardiac, respiratory, and vasomotor rhythms, and physiological synchrony between persons, are not mapped to the gauge connection; they are context, and no coherence 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 held arteriolar and venular pressures, the bed approaches a steady pressure field that carries a steady flow of blood from the arterioles to the venules, with steady filtration: a bounded non-equilibrium steady state. $\theta \equiv 0$, $V_{\mathcal{M}} = V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while perfusion 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 uniform pressure with no flow and no filtration (the arrest of circulation), nor a periodic return: the cardiac and respiratory cycles are excluded from windows by tolerance, and successive changes of autonomic tone form a sequence of windows. A calm or regulated affective state of the person is not mapped to this limit. The limit belongs to the declared idealization of held tone and constant interstitial pressures; the physical bed leaves it as tone changes or oedema develops, 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 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	4.1, 4.2, 4.4	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	4.2, 4.4; 4.1 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	4.1 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 \(2\gamma_\sigma + 2\,d\ln\langle\Theta_E\rangle/d\tau\) (4.1 \(O_7\)).
\(J_E^\mu\nabla_\mu\mathcal{F} < 0\), \(\dot{s}_{\mathrm{prod}} > 0\), \(\dot{W}_{\mathrm{diss}} > 0\)	4.1, 4.2, 4.4	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	4.1, 4.2, 4.4	The named dissipation is non-negative by the second law (item 7).
Stress sector: \(T^{\mathrm{mat}} \equiv 0\), the \(O_4\) conservation equivalence, and the compatibility law (Saint-Venant form)	4.1, 4.2, 4.4	Flat metric and declared \(\eta \equiv 0\) (items 2 and 4).
\(O_2\) Eq. [3]	4.1, 4.2, 4.4	Default drive and no tracer mapped.
\(O_5\) gluing, \(g^{(i)} = G^{(i)}_\infty\), and overlap convexity	4.1, 4.2, 4.4 (convexity is checked for 4.1 under the torsional perturbation, and holds there exactly because its cover consists of full-width layers, which the twist maps onto themselves)	Restricted metrics on a rigid or volume-rigid body (item 9).
\(\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0\)	4.1, 4.2, 4.4	\(\theta \equiv 0\) on every declared domain.
\(\phi_{\mathrm{perc}} = 1 > \phi_c\)	4.1, 4.2, 4.4	A connected domain whose transient reaches every overlap (item 9).
\(\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0\)	4.1	Declared simple connectivity; for 4.2 and 4.4 the harmonic space is in general non-trivial and no claim is made about it.
\(O_7\) Eq. [3] identities: divergence-free \(J_{\mathrm{geom}}\) and \(u_{\mathrm{exp}}\), and \(\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0\)	4.2, 4.4; 4.1 without the torsional perturbation	\(\theta \equiv 0\) and \(\hat{h} \equiv \hat{h}_\infty\) on rigid domains.
\(\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}\)	4.1, 4.2	A zero datum for a non-negative concentration. For 4.4 it is a checkable precondition (capillary pressure above venular pressure).
\(\mathcal{I} \to s_0V_\infty\)	4.1, 4.2, 4.4	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 \(\hat{\mathcal{E}}_\mu \equiv 0\)	4.1, 4.2, 4.4; the \(\chi_E\) statements except 4.1 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 \(D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0\)	4.1, 4.2, 4.4	Declared but not identified with any field; stated and not tested.

Tier 4 Specification: Somatic, Neuro-Attentional & Affective Interiority¶

0. Tier Architecture & Four-Part Functional Homology¶

0.1 Conventions Common to the Tier 4 Apertures¶

0.2 Coordinates That Hold Trivially or by Construction¶

Aperture 4.1: Myofascial Biotensegrity & Postural Mechanics (\(\mathcal{E}_{\mathrm{somatic}}\))¶

1. Epistemological & Physiological 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 4.2: Vestibular Equilibrium & Sensorimotor Contact (\(\mathcal{E}_{\mathrm{kinesthetic}}\))¶

1. Epistemological & Physiological 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 4.3: Neuro-Attentional & Autonomic Regulation (\(\mathcal{E}_{\mathrm{neuro}}\); provisionally homological)¶

1. Epistemological & Physiological 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 4.4: Affective Regulation & Psychophysiological Resilience (\(\mathcal{E}_{\mathrm{affective}}\))¶

1. Epistemological & Physiological 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¶