Tier 5 Specification: Acoustic, Poetic & Cross-Substrate Symbiosis¶

Document ID: tier_5_expressive_collective_symbiosis.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 5 maps the physics of sound in the environment and its relation to rhythm, the prosody and comprehension of verse, the movement of persons in space and in relation to one another, and the association of heterogeneous agents onto Spine $\mathcal{T}$. One aperture (5.1) is an empirical aperture declared in Branch II of r7, the bounded non-equilibrium steady state (aperture_generation_guidance.md §2.3). It 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: the energy of a diffuse sound field, supplied across the surface of an excised source region, spreading through the air of an enclosure by repeated reflection and scattering, and absorbed by the enclosure's bounding surfaces and, as a declared interior sink, by the air. Apertures 5.2, 5.3, and 5.4 are provisionally homological in this epoch: they declare no domain, map all nine coordinates as conceptual correspondences, like Tier 1, and make no empirical claim (§0.1 item 12).

The wider phenomena named in the aperture titles (rhythm and entrainment, verse and its meaning, choreography and the coordination of movement, inter-subjective and cross-substrate association) enter the empirical mapping of 5.1 as boundary conditions, through the declared parameters, through the hysteresis record $\mathcal{H}$, or as illustrative context, and its §1 states which. No Tier 5 aperture claims that an expressive, aesthetic, or collective phenomenon obeys a spine law. Each aperture is formulated at the level of general physics, linguistics, movement science, and systems theory, for populations studied at cohort level, without presupposing a single practitioner, practice, work, or site. No human-subject test is included in this epoch (empirical_validation_protocol.md §3.3), and the declared domain of 5.1 is unoccupied within its windows.

Slot & Functional Role	Aperture Designation	Declared Domain and Diffusing Quantity	Titled Phenomena Outside the Domain	Vertical Homology Across Tiers
Slot 5.1: Structural Substrate	Environmental Acoustics & Rhythmic Entrainment ($\mathcal{E}_{\mathrm{acoustic}}$)	The energy density of a diffuse sound field in the air of an unoccupied enclosure, supplied across the surface of an excised source region and absorbed by the bounding surfaces and the air.	Sound sources and their resonance, direct sound, early reflections and low-frequency modes, outdoor propagation, hearing, rhythm and entrainment: as boundary data, through the parameters, through $\mathcal{H}$, or context.	$1.1 \leftrightarrow 2.1 \leftrightarrow 3.1 \leftrightarrow 4.1 \leftrightarrow \mathbf{5.1}$
Slot 5.2: Dynamic Flow & Waves	Poetic Prosody & Psycholinguistic Condensation ($\mathcal{E}_{\mathrm{poetic}}$)	None declared: provisionally homological (inertial phonatory airflow).	All of its phenomena are mapped as conceptual correspondences only.	$1.2 \leftrightarrow 2.2 \leftrightarrow 3.2 \leftrightarrow 4.2 \leftrightarrow \mathbf{5.2}$
Slot 5.3: Boundary Gating	Spatial Choreography & Relational Kinesiology ($\mathcal{E}_{\mathrm{choreographic}}$)	None declared: provisionally homological (advective goal-directed movement).	All of its phenomena are mapped as conceptual correspondences only.	$1.3 \leftrightarrow 2.3 \leftrightarrow 3.3 \leftrightarrow 4.3 \leftrightarrow \mathbf{5.3}$
Slot 5.4: Coalescence & Hysteresis	Inter-Subjective & Cross-Substrate Symbiosis ($\mathcal{E}_{\mathrm{symbiotic}}$)	None declared: provisionally homological (quasi-static signal media).	All of its phenomena are mapped as conceptual correspondences only.	$1.4 \leftrightarrow 2.4 \leftrightarrow 3.4 \leftrightarrow 4.4 \leftrightarrow \mathbf{5.4}$

0.1 Conventions Common to the Tier 5 Apertures¶

The following conventions apply to the empirical aperture of this tier (5.1); item 12 states how the provisionally homological Apertures 5.2, 5.3, and 5.4 are mapped. Aperture 5.1's §1 and coordinate blocks state its specific values and cite these items where they apply. Where an item repeats a convention of tier_4_somatic_contemplative_interiority.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 5.1 §1, and the domain of 5.1 is never 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.
The transient is a declared step increase of the source's acoustic power, from a held non-zero level to a higher held level. The supply data are the net inflow of acoustic power across the source surface (flux data, not a held value of $\mathcal{F}$); the bounding surfaces absorb through a Robin exchange (5.1 §1); and the sink, air absorption, is first order, Lipschitz, and non-decreasing in $\mathcal{F}$. The declared balance law is a scalar parabolic equation, so the comparison principle holds. The prior steady state is a subsolution for the raised inflow, 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. The inflow reaches its held level over a short rise time (the response of the source and the travel of sound across the excised region), a non-decreasing ramp, for which 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. The Robin data do not change the argument. The argument is exact on the declared rigid domain.
Direction. The opposite direction is excluded. When the source is switched off, the field relaxes to the datum: this is the decay from which reverberation time is measured, and it is not a Branch II window, because the throughput stops and $\lim\langle\Theta_E\rangle = 0$. A decrease to a lower non-zero power would keep every declaration; the raising direction is declared for the comparison argument, as for 4.1 in Tier 4. Sources whose power varies within the window (periodic, rhythmic, modulated, or intermittent signals) are excluded, because their supply data are not monotone.
Boundary exchange. Every exchange of acoustic energy across the bounding surfaces is one-signed, out of $\mathcal{M}$: absorption by the surfaces and transmission through them to the outside. No compartment outside $\mathcal{M}$ discharges into it within a window, except across the declared supply surface: sound entering from outside the enclosure is held below a declared fraction of $\mathcal{F}$ everywhere in $\mathcal{M}$, the enclosure has no opening to another volume that could store sound and return it, the linings of the bounding surfaces store negligible energy, and the bounding surfaces are rigid within tolerance (5.1 preconditions 1, 3, 4, and 5).
The rate of change of the flux, $\partial_\tau J_W$, is not one-signed in general (item 9).
Metric. The domain is a fixed region of space, the air of a rigid enclosure minus the excised source region, with the Euclidean metric. 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). 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 enclosure is declared rigid within a pre-declared tolerance on its volume. The air in it has no reference configuration whose deformation $g$ could record: the compressions and rarefactions of sound are oscillations at acoustic frequencies with zero mean, and the vibration of the bounding surfaces is cyclic; neither is $\theta$ (item 6). With $\sigma \equiv 0$ the $O_3$ shear law holds trivially, and 5.1 supports no shear-relaxation test. No non-Euclidean material metric is declared: a gas has no stress-free reference configuration from which one could be built. The curved configurations of the r10 §5.11 and §5.13 witnesses are therefore realized by no Tier 5 aperture, and r7's curvature sector, which no empirical aperture of Tiers 2–4 exercises, is exercised by no empirical aperture in this epoch.
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. The diffusing quantity of 5.1 is the acoustic energy density $u$ of the air, in a declared band and over a declared averaging time (5.1 §1). Where the enclosure contains distributed scatterers, which are treated at the continuum scale as part of the medium (5.1 §1), acoustic energy is stored only in the air between them, and the storage coefficient is $s = \phi_{\mathrm{a}}$, the air volume fraction; without scatterers $s = 1$. With $s$ uniform, the physical balance $s\,\partial_\tau u = \nabla_\alpha(k^{\alpha\beta}\nabla_\beta u) - r$ becomes the r7 law for $\mathcal{F} = u$ ($u$ is itself an energy density, so the declared yield is $1$), $\mathfrak{K} = k/s$, and $\mathcal{R}_{\mathcal{F}} = r/s$, and the r7 flux is $J_W = j/s$, with $j$ the net acoustic intensity per unit bulk area. The factor $1/s$ 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. Aperture 5.1 therefore declares a single storage class: $s$ uniform on $\mathcal{M}$ within a tolerance (a declared, checkable precondition). Media that store acoustic energy differently (porous or resonant linings of the bounding surfaces, other volumes coupled to the enclosure, the excised source region) lie outside $\mathcal{M}$ and enter as boundary data (the Robin exchange at the bounding surfaces, the inflow at the source surface) or are excluded, because a jump in $s$ would make $J_W$ discontinuous where the physical intensity 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 5 medium has an identified global control field in this epoch, so Tier 5 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 $\frac{n-1}{n}\nabla^2\theta = \nabla^\alpha\nabla^\beta\sigma_{\alpha\beta}$: the Saint-Venant compatibility conditions of the strain rate. On the rigid domain of 5.1 both sides vanish. The pressures and stresses of the domain (the static pressure of the air, the oscillating acoustic pressure, the radiation pressure of sound, and the stresses in the bounding surfaces) 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. Aperture 5.1 declares $c_*$ as the speed of sound in the air at the declared temperature. In this aperture the declared quantity itself travels on the cone at the scale of the mean free path: between reflections and scatterings, sound energy propagates along rays at $c_*$, which are null for the declared cone and, like all null rays, are not trajectories of $O_2$ Eq. [3]. The diffusion law describes the energy only at scales large against the mean free path $\lambda$, where the energy transport velocity $\vert j\vert/(s\,u)$ is $\ll c_*$; that is the condition $\lambda\vert\nabla\mathcal{F}\vert \ll \mathcal{F}$ (5.1 precondition 2). The law's instantaneous propagation is outside its validity at points that sound from a change at the source surface could not yet have reached, which item 9 takes into account. Every material speed in $\mathcal{M}$ (drifts of the air) 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. Aperture 5.1 instantiates Eq. [3] with $a^\mu_{\mathrm{d}} = 0$ and maps no tracer. The movement of sources, persons, and objects is forced motion outside $\mathcal{M}$, and within windows nothing moves (5.1 precondition 5).
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: the acoustic power entering at the source surface is throughput. In general $\nabla_\mu J_E^\mu = -p_0^{-1}(\partial_\tau\mathcal{F} + \mathcal{F}\theta + \mathcal{R}_{\mathcal{F}}) \neq \theta$. The cyclic volume changes of sound (the condensation of the air in each period) and of vibrating surfaces are outside the declared fields and are never mapped to $\theta$. The oscillating particle velocity of the air is not a current of the spine: $J_W$ is built from the net intensity, averaged over the declared averaging time.
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 $$p_0\dot{W}_{\mathrm{diss}} = \oint_{\partial\mathcal{M}}\mathcal{F}\,(-J_W^\mu)\,d\Sigma_\mu - \int_{\mathcal{M}}\mathcal{F}\,\mathcal{R}_{\mathcal{F}}\sqrt{\vert g\vert}\,d^nx - \frac{1}{2}\frac{d}{d\tau}\int_{\mathcal{M}}\mathcal{F}^2\sqrt{\vert g\vert}\,d^nx - \frac{1}{2}\int_{\mathcal{M}}\mathcal{F}^2\theta\sqrt{\vert g\vert}\,d^nx$$
No face of $\mathcal{M}$ is held at the datum. The source surface carries the held inflow, and its term is evaluated from the source's calibrated acoustic power and the measured $\mathcal{F}$ there. The bounding surfaces absorb in proportion to $\mathcal{F}$, and their term is evaluated from the measured normal intensity into them and the measured $\mathcal{F}$ there (5.1 §1).
Each term is evaluated from measured $\mathcal{F}$, measured boundary fluxes, and measured or independently declared consumption; none uses the declared $\mathfrak{K}$. 5.1 §1 names instruments and surfaces different from those of the $O_1$ flux-prediction test, so that the evidence is independent of that test.
Because the identity uses the balance law, the estimate is conditional on it. 5.1 §1 also names a directly measured physical dissipation whose sign the estimate must share: the conversion of acoustic energy into heat by the bounding surfaces and the air. It 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.
Reflection and scattering conserve acoustic energy: no heat is produced where the energy density merely spreads. $\dot{s}_{\mathrm{prod}} = -\beta_{\mathrm{th}}J_E^\mu\nabla_\mu\mathcal{F}$ is therefore the entropy production of the declared mapping (the spreading of an incoherent field's energy down its gradient), not a measured heat production, as for Aperture 3.2 in Tier 3. Heat is produced only by the absorption.
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 5 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 the domain's physics supplies is, first, the spectral setting: the Neumann eigenmodes of $-\nabla^2_g$ on $\mathcal{M}$, whose eigenvalues are fixed on the rigid domain. On this domain these eigenmodes coincide in shape with the acoustic modes of a rigid-walled enclosure of the same shape. The gate is nonetheless a spatial projector acting on the declared state sector, never a filter in frequency: the declared field is an energy density averaged over a band above the frequency at which the modes of the enclosure overlap (5.1 precondition 2), so individual modes are not resolved in $\mathcal{F}$, and $\Lambda_{\mathrm{gate}}$ is never read in hertz. Second, the physics 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. The stronger absorption of higher frequencies by surfaces and air is a property of the declared parameters across bands, not the spatial gate.
Cover, overlaps, and coupling ($O_5$, $O_6$).
The cover $\{\mathcal{U}_i\}$ consists of $\tau$-independent subregions of $\mathcal{M}$ with Lipschitz boundaries, and every overlap $\mathcal{U}_{ij}$ is convex at $\tau_0$. $\partial\mathcal{M}$ has concave parts: the source surface, which $\mathcal{M}$ surrounds, and any re-entrant parts of the bounding surface. 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 or inhomogeneity of a medium lives in $\mathfrak{K}$, never in $g$. Effective-medium averages of diffusion coefficients, including the combination of the mean free paths of the enclosure and of its scatterers, are statements about $\mathfrak{K}$ and are not mapped to $O_5$ Eq. [1].
Partner fields. 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. The supply of 5.1 is a held inflow, not a held value of $\mathcal{F}$: after the step the normal flux at the source surface is held, and the flux near the supply does not decay from an initial overshoot, as it does under the held-value supplies of Tiers 3 and 4. In a one-dimensional configuration (an inflow uniform across a whole cross-section) and in a radially symmetric one, the flux obeys a parabolic equation of the same kind as $\mathcal{F}$, with its held value at the supply and a non-decreasing value at the absorbing surface (the Robin outflow is proportional to $\mathcal{F}$, which does not decrease); by the comparison principle the flux then rises at every interior point, so $\partial_\tau J_W$ points away from the supply throughout the interior of $\mathcal{M}$ and vanishes only on the supply surface, where the held inflow fixes it. In general geometry the held inflow fixes the normal component of $J_W$ on the supply surface, so the normal component of $\partial_\tau J_W$ vanishes there. Every overlap therefore stays clear of the supply surface, whatever its shape; in the declared enclosure it is also a concave part. The declared enclosure is in general neither one-dimensional nor radially symmetric, so any zero set of $\partial_\tau J_W$ in its interior is located only by the predicted solution. On a 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.
5.1 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 in the diffusion law 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 no earlier than a declared margin after the time sound needs to travel from the source surface to the overlap plus several mean free times (item 5); the data must then show $\vert\eta_{[i\to j]}\vert$ above the threshold there (a declared, checkable precondition). On such an overlap $\eta_{[i\to j]}$ keeps one direction throughout the overlap's window, which 5.1 states from the predicted solution.
Precision potentials. The potentials $\Phi^{(i)}$ are declared before data collection as quadratic potentials in a Cartesian chart. 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 5 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 5 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 this domain.
$M^{(ij)} \neq 0$ and $\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$ are standing hypotheses of r7 $O_6$, not consequences of diffusion. In Branch II the transient decays as the domain settles, so each coupling-tensor boundary flux decays and is integrable on $[\tau_0, \infty)$; no sustained coupling is asserted. Aperture 5.1 is not among the systems of the current coupling test.
Topology. The harmonic field of r7 $O_5$ Eq. [5] carries absolute boundary conditions (it is tangential to $\partial\mathcal{M}$), so its space has dimension $b_1(\mathcal{M})$, the first Betti number. The excised source region does not by itself make $b_1$ positive: attached to a bounding surface through its mounting, it is a concave dent in $\partial\mathcal{M}$; free-standing, it is an interior cavity, which adds to the second Betti number, not the first. $b_1(\mathcal{M}) > 0$ where a fixed body in the enclosure (the source mounting among them) joins two separate parts of the bounding surface, or where the enclosure's own shape has a handle; then $\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}}$ is non-trivial, a property of the domain's topology. The scatterers, treated at the continuum scale, are not holes of $\mathcal{M}$. Tier 5 identifies no transport current or measured field with the harmonic field and makes no claim about it. With the declared uniform, isotropic $\mathfrak{K}$, the r7 flux is the exact 1-form $J_W = -D\,d\mathcal{F}$, which is $L^2$-orthogonal to every tangential harmonic field (the Hodge–Morrey–Friedrichs decomposition), so the declared current has no harmonic component.
Gauge sector and hysteresis ($O_7$, $O_0$). Tier 5 declares the trivial gauge sector for 5.1: $A_\mu \equiv 0$ 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 5 aperture makes a synchronization, alignment, entrainment, or coherence claim (of sound fields, of rhythms, of persons, or of agents): such claims would belong to gauge transport (aperture_generation_guidance.md §5.5), not to $\chi_E$, and the homological apertures map such phenomena to gauge transport as correspondences only. 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 5 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 $\tau \ge \tau_0$. 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 acoustic history of the enclosure (changes of its surfaces, linings, and contents between windows) is mapped via Spine $\mathcal{T}$ to $\mathcal{H}$ and is not identified with it. $\hat{\rho}_g = \frac{1}{n}\delta^\mu_\nu$ at every $\tau$ on the rigid domain, 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 5.1 states and does not test.
Finite windows and coverage. No finite window exhibits a $\tau \to \infty$ limit. The claims of 5.1 are the within-window tests of aperture_generation_guidance.md §5.5 that its domain supports, listed in its $O_\Omega$ block. In this epoch Aperture 5.1 is exercised by no test (empirical_validation_protocol.md §3.3), and no human-subject test is included. Everything else is stated as untested, not as confirmed.
The provisionally homological apertures (5.2, 5.3, 5.4). Apertures 5.2, 5.3, and 5.4 map their phenomena onto all nine coordinates as conceptual correspondences, like the Tier 1 grammars and Aperture 4.3 (aperture_generation_guidance.md §2.3). They declare no domain, metric, storage, free energy, transport tensor, sink, drive, or Branch II evidence, and make no empirical claim. Each §1 states why r7's storage–diffusion balance law cannot represent its system literally: the airflow of phonation is inertial (5.2), goal-directed movement is advective (5.3), and the media through which agents signal are quasi-static or carry waves (5.4). For uniformity of the tier, each of their blocks carries the same r7 spine-equation line as the empirical blocks. The shared line is read as tier_4_somatic_contemplative_interiority.md §0.1 item 12 states: schematically, with no cone speed declared, so that the r7 form with $c_* = 1$ applies to the trajectory equation, and with the entry $J_E^\mu = J_W^\mu/p_0$ as the Branch II form being read, not a declared current. Their correspondences are read against the Branch II form, because the systems they describe (a speaker's airflow, a moving group, an association of agents) are bounded, are sustained by throughput, and do not expand without bound (a homological aperture may choose a branch form with a stated reason, aperture_generation_guidance.md §2.3); as homologies they declare no branch in the sense of closure (v), because nothing is evaluated. Their 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 5.1 carry no content in them either.

0.2 Coordinates That Hold Trivially or by Construction¶

On the declared domain of 5.1 several coordinates hold identically and carry no empirical content. They are listed here so that no block below is read as evidence for them. Apertures 5.2, 5.3, and 5.4 make no claim at any coordinate and are 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	5.1	Rigid declaration of a fixed region, so the evidence reduces to a tolerance check on the enclosure's 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	5.1	Rigid declaration.
$J_E^\mu\nabla_\mu\mathcal{F} < 0$, $\dot{s}_{\mathrm{prod}} > 0$, $\dot{W}_{\mathrm{diss}} > 0$	5.1	Identities of the fixed Branch II current wherever $\nabla\mathcal{F} \neq 0$; $\dot{s}_{\mathrm{prod}}$ is the entropy production of the declared mapping (item 7). The empirical content of $O_1$ is the flux prediction.
Sign agreement with a measured physical dissipation	5.1	The absorbed acoustic power 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)	5.1	Flat metric and declared $\eta \equiv 0$ (items 2 and 4).
$O_2$ Eq. [3]	5.1	Default drive and no tracer mapped.
$O_5$ gluing, $g^{(i)} = G^{(i)}_\infty$, and overlap convexity	5.1	Restricted metrics on a rigid domain (item 9).
$\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0$	5.1	$\theta \equiv 0$.
$\phi_{\mathrm{perc}} = 1 > \phi_c$	5.1	A connected domain whose transient reaches every overlap (item 9).
$\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0$	5.1, where $b_1(\mathcal{M}) = 0$	A property of the declared enclosure and its excised region (an interior source cavity does not raise $b_1$); where $b_1 > 0$ the harmonic space is non-trivial and no claim is made about it.
The declared current has no harmonic component	5.1	$J_W = -D\,d\mathcal{F}$ is exact for the declared uniform, isotropic $\mathfrak{K}$, hence $L^2$-orthogonal to the tangential harmonic fields (item 9).
$O_7$ Eq. [3] identities: divergence-free $J_{\mathrm{geom}}$ and $u_{\mathrm{exp}}$, and $\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0$	5.1	$\theta \equiv 0$ and $\hat{h} \equiv \hat{h}_\infty$ on the rigid domain.
$\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}$	5.1	A zero datum for a non-negative energy density.
$\mathcal{I} \to s_0V_\infty$	5.1	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 $\hat{\mathcal{E}}_\mu \equiv 0$	5.1	Trivial gauge sector and $\sigma \equiv 0$ (item 10).
The $O_4$ gate, the $O_\Omega$ generator condition, and the alignment $D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0$	5.1	Declared but not identified with any field; stated and not tested.

The empirical content of Tier 5 therefore sits in the $O_1$/$O_4$ flux prediction of 5.1; the $O_6$ standing hypotheses on its declared overlaps; the sign of its measured-dissipation estimate; and the checkable preconditions of its domain. No current test exercises it.

Aperture 5.1: Environmental Acoustics & Rhythmic Entrainment ($\mathcal{E}_{\mathrm{acoustic}}$)¶

1. Epistemological & Physical Scope¶

Aperture 5.1 maps the physics of sound in the environment onto Spine $\mathcal{T}$: the radiation of sound by sources; its reflection, scattering, and absorption by surfaces and objects; the reverberant field that builds up where sound is enclosed; and the relation between rhythmic sound and the timing of the organisms that hear it (rhythmic entrainment). Its empirical domain is the reverberant field in its diffuse regime. Acoustic power supplied by a source spreads through the air of an enclosure by repeated reflection and scattering, which at scales large against the mean free path of sound is a diffusion of acoustic energy, and is absorbed by the enclosure's surfaces and by the air. Lexicon is strictly restricted to physical acoustics, room and environmental acoustics, psychoacoustics, and the science of rhythm and timing.

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

Domain $\mathcal{M}$ ($n = 3$) and boundary $\partial\mathcal{M}$. The air of a closed, rigid enclosure (a test enclosure, a scale model, or any enclosed space that meets the preconditions below), unoccupied within windows, minus an excised source region: a closed region around the sound source and its mounting that contains the source's near field and every point within a declared number of mean free paths of the source, where the diffusion law does not hold. Diffuse reflection by the bounding surfaces and, where present, scattering by a uniform random distribution of small, fixed, rigid scatterers make the field diffuse; the scatterers are treated at the continuum scale as part of the medium, through $\mathfrak{K}$ and the storage coefficient, and are not holes of $\mathcal{M}$. $\partial\mathcal{M}$ consists of:
the source surface (supply), the boundary of the excised region, across which the source's acoustic power enters $\mathcal{M}$;
the bounding surfaces of the enclosure (outflow), which absorb part of the energy incident on them, in proportion to $\mathcal{F}$ there (a Robin exchange, below), and transmit part to the outside.

Every part of the bounding surface absorbs (precondition 4): a perfectly reflecting part would be a zero-flux face, on which the normal derivative of $\mathcal{F}$ vanishes. All supply crosses the source surface; the air contains no other source, and sound entering from outside is held below tolerance (precondition 1), so there are no interior sources. The excised region is either attached through its mounting to a bounding surface, so that its surface is a concave part of $\partial\mathcal{M}$, or free-standing, an interior cavity whose surface $\mathcal{M}$ surrounds and which is concave as seen from $\mathcal{M}$. The source and its signal, the linings of the bounding surfaces (a different storage class, §0.1 item 3), and the space outside the enclosure lie outside $\mathcal{M}$. * Window. A relaxation window (§0.1 item 1), opened from a prior steady state at a held non-zero source power by a declared step increase of that power, after which the source power, the temperature, humidity, and pressure of the air, and the state of the enclosure are held within tolerance. The source radiates stationary random noise filtered to the declared band, never a periodic, rhythmic, or modulated signal. Windows in which the source is switched off are excluded: the decay of the field after switch-off, from which reverberation time is measured, relaxes to the datum with no throughput, so it is not a Branch II window. It may serve as a declared perturbation regime before the window. * Metric. Flat and rigid (§0.1 item 2): $g_{\mu\nu} = \delta_{\mu\nu}$ in a Cartesian chart, with the enclosure's volume, surveyed before and after the window, within the rigid tolerance. The vibration of the bounding surfaces is cyclic and is never $\theta$. * Cone speed. $c_*$ is the speed of sound in the air at the declared temperature (§0.1 item 5). Drifts of the air are $\ll c_*$, and in the diffusion regime the transport velocity of acoustic energy is $\ll c_*$ (precondition 2). * Storage. $s = \phi_{\mathrm{a}}$: the volume fraction of the enclosure's interior occupied by air, which is $1$ without scatterers. It is declared uniform over $\mathcal{M}$ within tolerance (§0.1 item 3), with the scatterers' number and volume taken from a survey of the enclosure. * Free energy $\mathcal{F}$ and datum. $\mathcal{F} = u$ (J m⁻³), the acoustic energy density of the air (kinetic plus potential) in a declared frequency band, averaged over a declared time $\tau_{\mathrm{avg}}$ that is long against the reciprocal of the bandwidth and short against the relaxation time. $u$ is already an energy density, so the declared yield is $1$, and $\mathcal{F}$ is the acoustic energy stored per unit volume times the declared conversion factor $1/s$. This is the declared linear form of aperture_generation_guidance.md §2.3 item 3: an incoherent diffuse field carries entropy, so its thermodynamic free energy is smaller than its energy, and $\mathcal{F}$ is the energy density above the zero datum, not that free energy. The datum is zero ($\mathcal{F}_{\mathrm{ref}} = 0$, silence), and $\mathcal{F} \ge 0$ holds trivially. The relaxation time $\ell^2/D$ across the enclosure's extent $\ell$ is finite and resolvable: it exceeds $\tau_{\mathrm{avg}}$ and the mean free time $\lambda/c_*$ by declared factors. * Transport $\mathfrak{K}$. $\mathfrak{K}^{\alpha\beta} = D\,g^{\alpha\beta}$, with $D = \lambda c_*/3$ the acoustic diffusion coefficient of the mean-free-path model (the medium's energy conductance over $s$). $\lambda$ is the mean free path of sound in the enclosure, combining by the sum of reciprocals the enclosure's own mean free path (four times its volume over its bounding area) with the scattering mean free path set by the scatterers' number density and mean cross-section. $D$ is uniform and isotropic, the default form of aperture_generation_guidance.md §2.3 item 4, with its value declared: $\lambda$ is estimated per closure (ii) from a geometric survey of the enclosure and its scatterers (a different modality), and $c_*$ from the measured air temperature. With $j$ the net acoustic intensity per unit bulk area (W m⁻²), $J_W = j/s$, and $1/s$ is the declared conversion factor. * Outflow boundary. At the bounding surfaces the outward normal component of $J_W$ is $(A/s)\,\mathcal{F}$, with the exchange coefficient $A = c_*\alpha/4$ of a diffusely reflecting surface of absorption coefficient $\alpha$ (the low-absorption form, the default) or the declared modified form for larger $\alpha$. $\alpha$ is measured in situ on the enclosure's own surfaces before the window, by a pressure–velocity probe or an in-situ impedance method, in the declared band. The linings' own storage is neglected (precondition 3). * Supply boundary. After a short rise time, the net inflow across the source surface equals the source's acoustic power $P$ within the flux tolerance: the source is omnidirectional within tolerance in the declared band, so the inflow is spread uniformly over the surface, and absorption within the excised region and the change of energy stored in it are below tolerance (precondition 7). $P$ is the source's calibrated acoustic power at its held drive, measured on the source itself in a declared calibration regime before the window, a different modality from any measurement in the enclosure. * Sink $\mathcal{R}_{\mathcal{F}}$. Air absorption, first order relative to the datum: $\mathcal{R}_{\mathcal{F}} = k_{\mathrm{a}}\mathcal{F}$, with $k_{\mathrm{a}} = m\,c_*$ and $m$ the energy attenuation coefficient of air in the declared band, from the tabulated standard for atmospheric absorption at the measured temperature, humidity, and pressure. $\mathcal{R}_{\mathcal{F}}$ is continuous, non-negative, non-decreasing in $\mathcal{F}$, zero at the datum, and uniform. It departs from the default and satisfies the binding closure, and it is rejected if the default fits at least as well (closure (iv)), as is expected in bands where air absorption is negligible. Absorption at the scatterers' surfaces is bounded below tolerance (precondition 4) and is not declared. * Drive. Default, $a^\mu_{\mathrm{d}} = 0$ (§0.1 item 5). * Branch II evidence (closure (v)). * The trend of $\langle\theta\rangle$ and a bound on $\int\langle\theta\rangle\,d\tau$ come from the surveyed volume of the enclosure, which must stay within the rigid tolerance; for this 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 calibrated source power and $\mathcal{F}$ on the source surface from a set of microphones on it; * the bounding-surface term uses the normal intensity into the surfaces, measured by a second, independent set of pressure–velocity probes at declared sampling positions over them, with $\mathcal{F}$ there from the same probes; its sampling error over a large surface is part of the declared tolerance; * the consumption term uses the tabulated $k_{\mathrm{a}}$ at the measured air conditions with $\mathcal{F}$ from a second, independent microphone record; * the storage term uses repeated maps of $\mathcal{F}$.

The $O_1$ test uses an interior section, a different surface, and its own instruments.

The physical dissipation is the conversion of acoustic energy into heat by the bounding surfaces and the air, which is non-negative by the second law, so the comparison tests only the sign of the estimate. $\dot{s}_{\mathrm{prod}}$ is the entropy production of the declared mapping, not of that heat (§0.1 item 7).
Admissible domains (declared, checkable preconditions):
no interior source: no other sound source in or near the enclosure in the declared band, with the background energy density, measured with the test source silent before the prior steady state, below a declared fraction of the prior steady-state $\mathcal{F}$ everywhere in $\mathcal{M}$; and $\nabla\mathcal{F} \neq 0$ throughout $\bar{\mathcal{M}}$, checked against the predicted steady solution before the window and against the measured maps of $\mathcal{F}$ in it, at the resolution of precondition 8;
the diffuse regime: a declared band above the frequency at which the modes of the enclosure overlap, wide enough that $\mathcal{F}$ averages over many modes; every point of $\mathcal{M}$ at least a declared number of mean free paths from the source, and the error of the diffusion law within about one mean free path of the bounding surfaces within the declared tolerance; a mean absorption coefficient of the bounding surfaces below a declared bound; and $\lambda\vert\nabla\mathcal{F}\vert \ll \mathcal{F}$ throughout $\mathcal{M}$;
a single storage class: scatterers, where present, distributed uniformly within tolerance, checked by survey in two or more subregions; and the energy stored in the linings of the bounding surfaces below the flux tolerance, by a declared bound (their thickness and porosity, and their relaxation time against the field's);
an absorbing, closed boundary: every part of the bounding surface absorbs, with in-situ absorption coefficients between declared positive bounds; the enclosure has no opening to another volume; and absorption by the scatterers is below a declared bound;
a rigid, still enclosure: volume within the rigid tolerance, bounding surfaces and scatterers fixed, the enclosure unoccupied, and nothing moving within the window;
still, uniform air: mean air speeds low enough that advection of acoustic energy is negligible against its diffusion (a declared bound on their ratio across $\ell$), no ventilation within the window, and temperature, humidity, and pressure uniform and held within tolerance, so that $c_*$, $D$, and $k_{\mathrm{a}}$ are constant;
the source: omnidirectional within tolerance in the declared band, radiating stationary random noise at held power, with its acoustic power calibrated; absorption within the excised region and the change of energy stored in it below the flux tolerance, so that the inflow across the source surface rises to its new level without overshoot within a declared rise time, checked from the microphones on that surface;
the instruments: the dynamic capability of the intensity and pressure–velocity probes exceeds, by a declared margin, the field's ratio of mean-square pressure to net intensity, which is large in a diffuse field; the estimate of $u$ from mean-square pressure ($p^2/\rho_0c_*^2$, valid in the diffuse regime) agrees with pressure–velocity estimates at declared check points; and the random error of each map of $\mathcal{F}$, after averaging over a declared number of repeated steps, is below a declared fraction of the predicted change of $\mathcal{F}$ across the spacing of the microphones, not of $\mathcal{F}$ itself: in the diffuse regime $\mathcal{F}$ is nearly uniform ($\lambda\vert\nabla\mathcal{F}\vert \ll \mathcal{F}$), so precondition 1 and the $O_1$ test rest on small differences between large mean-square pressures.
Titled phenomena outside the domain.
Sound sources and resonance. The source and its resonances lie in the excised region and enter through the supply data (the held power and its spectrum in the declared band). The resonance of vibrating bodies and air columns is context.
Direct sound, early reflections, and low-frequency modes. The direct sound, early reflections, and echoes precede the diffuse regime, and below the frequency at which they overlap the modes of an enclosure are resolved individually. Both lie outside the declared field and are context.
Outdoor propagation and transmission. Sound outdoors, without an enclosure, builds no stored diffuse field and is context. Sound transmitted out through the bounding surfaces leaves $\mathcal{M}$ and is counted in their exchange.
Hearing and the perception of space. Hearing, and the perception of reverberance and of the size of a space, are processes of listeners outside $\mathcal{M}$ and are context; no listener is present within windows.
Rhythmic entrainment. The alignment of the timing of organisms, human or other, with a periodic sound is a property of the organisms, outside $\mathcal{M}$; it is described at cohort level and is context. Within windows the source signal is stationary noise, so no rhythm is present in the declared field. A tempo or a meter is a temporal quantity and is never mapped to $\theta$, $\sigma$, $\Lambda_{\mathrm{gate}}$, or any spatial operator. A claim of entrainment would be a claim of synchronization, which would belong to gauge transport (§0.1 item 10); none is made.
Acoustic history. Changes of the enclosure's surfaces, linings, and contents lie between windows; they enter through the declared parameters and are mapped via Spine $\mathcal{T}$ to $\mathcal{H}$, not identified with it.

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 enclosure is rigid, so $\theta \equiv 0$, $dV_{\mathcal{M}}/d\tau = 0$, and $V_{\mathcal{M}} = V_\infty$ at every $\tau$. The expansion current enters only through its vanishing divergence and boundary flux. The acoustic power entering at the source surface and leaving through the bounding surfaces is throughput and is never mapped to it, and the oscillating compressions of the air are not $\theta$ (§0.1 item 6). Openness is carried by throughput: acoustic energy spreading down its gradient gives $\Theta_E > 0$ (by $O_1$), the entropy production of the declared mapping (§0.1 item 7), so $\Theta_{\mathrm{tot}} = \Theta_E$ and $\tau_\Theta$ counts that 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 acoustic history of the enclosure (changes of its surfaces, linings, and contents) 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 acoustic energy density between the source surface and the absorbing surfaces, together with air absorption, holds the field far from equilibrium. With $J_E^\mu = -p_0^{-1}D\,g^{\mu\nu}\nabla_\nu\mathcal{F}$, $J_E^\mu\nabla_\mu\mathcal{F} < 0$ holds identically wherever $\nabla\mathcal{F} \neq 0$ (precondition 1), so $\dot{s}_{\mathrm{prod}} > 0$ and $\dot{W}_{\mathrm{diss}} > 0$, as identities of the declared mapping (§0.1 item 7). Acoustic energy spreads down its gradient from the source surface toward the absorbing surfaces, 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). $D$ is estimated per closure (ii) from the surveyed mean free path and the speed of sound at the measured air temperature, and $s$ from the survey of the scatterers. The declared law must predict $J_W^{\mathrm{meas}} = j^{\mathrm{meas}}/s$ across a declared interior section $S$, with $j^{\mathrm{meas}}$ the band-limited, averaged net intensity normal to $S$ from a set of intensity probes on $S$, and $\nabla\mathcal{F}$ from the energy density measured by an array of pressure microphones on both sides of $S$; neither uses the other or the declared $D$ or $k_{\mathrm{a}}$. In a diffuse field the net intensity is small against $c_*\mathcal{F}$, which is why the probes' dynamic capability is a precondition (precondition 8). 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: In the rigid enclosure 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 static pressure of the air, the oscillating acoustic pressure, and the radiation pressure of sound are not $T_{\mu\nu}^{\mathrm{mat}}$. The shape of the enclosure and its scatterers enter through $D$, the Robin exchange, 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. Sound rays, including the paths of reflection and scattering on which the mean-free-path model is built, are null for the declared cone and are not trajectories. The movements of sources, persons, and objects are forced motions outside $\mathcal{M}$, and none occurs within windows.

$O_3$ — 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: In the rigid enclosure $\sigma^\mu_{\;\nu} \equiv 0$, 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 rise of the field toward its new steady state is a property of the $O_4$ balance law and is not mapped to $\sigma$. The decay of a reverberant field after its source stops is likewise a relaxation of $\mathcal{F}$, not shear, and is not a Branch II window (§0.1 item 1); reverberation time is never mapped to $\gamma_\sigma$. The vibrations of the bounding surfaces and the oscillations of the sound field are periodic and are never mapped to $\sigma$. $V_{\mathrm{local}}$ 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 $\partial_\tau\mathcal{F} = D\nabla^2\mathcal{F} - k_{\mathrm{a}}\mathcal{F}$, with the held inflow at the source surface and the Robin exchange at the bounding surfaces: the diffusion equation of acoustic energy in a reverberant enclosure, normalized by its storage coefficient. Its finite relaxation time makes the field stored, not quasi-static. The air-absorption sink departs from the default and satisfies the binding closure, and it 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 domain are fixed, so the gate does not drift. They coincide in shape with the acoustic modes of a rigid-walled enclosure of the same shape, but the gate is a spatial projector on the declared state sector, not a filter in frequency, and the declared field does not resolve modes. The faster decay of fine spatial structure in the energy field is mapped via Spine $\mathcal{T}$ to the gate without being the gate. The dependence of absorption on frequency is a property of the declared parameters across bands, 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 $\mathcal{M}$: one cover set around the source surface, containing it (a concave part of $\partial\mathcal{M}$); one cover set around each re-entrant part of the bounding surface, containing it; 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 connectedness of the air between the scatterers is a property of the medium that enters through $D$: a packing of scatterers that closed the paths of sound would have no continuum diffusion coefficient and would not be an admissible domain. That connectedness is mapped via Spine $\mathcal{T}$ to the condition $\phi_{\mathrm{perc}} > \phi_c$ and is not identified with it. Whether the excised source region is a concave dent or an interior cavity, it does not raise $b_1(\mathcal{M})$; $b_1 > 0$ only where a fixed body joins two separate parts of the bounding surface or the enclosure has a handle, and then harmonic Neumann fields circulate around those bodies. No transport current or measured field is identified with them, and no claim is made about them; the declared current $J_W = -D\,d\mathcal{F}$ is exact and has no harmonic component (§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 $\mathcal{U}_{ij}$ along a path from the source surface toward an absorbing surface. The partner fields are the rate of change of the acoustic 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 the source signal and a microphone record.
The supply is a held inflow, so after the step the flux at the source surface does not overshoot. In a one-dimensional or radially symmetric configuration the flux would rise at every interior point, and $\partial_\tau J_W$ would point away from the source throughout the interior of $\mathcal{M}$, vanishing only on the source surface, which every overlap stays clear of (§0.1 item 9).
The declared enclosure, with its excised region placed where the preconditions allow and its own shape, is in general neither, so the direction of $\partial_\tau J_W$ and any zero set are located only by the predicted solution, from the independently estimated $D$, $A$, and $k_{\mathrm{a}}$ and the declared source power.
The declared overlap is one that the predicted zero set does not reach during the overlap's window, with the detection threshold and the travel-time margin 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 in which sound propagates or is reflected, and not directions of enhanced or suppressed transmission. Under the flux-positivity hypothesis the coupling transmits correlation between the subsystem marginals of the declared state sector ($\frac{d}{d\tau}\mathcal{I}(Q_i; Q_j) > 0$). Both are standing hypotheses, not consequences of diffusion, and this aperture is not among the systems of the current coupling test. In Branch II the transient decays as the field 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 phase relations of sound waves (their interference), which the band- and time-averaged field does not carry, and the alignment of the timing of organisms with a periodic sound, are not mapped to the gauge connection; they are context, and no coherence, synchronization, or entrainment 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 a held source power the enclosure approaches a steady diffuse field that carries a steady flux of acoustic energy from the source surface, partly absorbed by the air and partly by the bounding surfaces: a bounded non-equilibrium steady state. $\theta \equiv 0$, $V_{\mathcal{M}} = V_\infty$, $\langle\Theta_E\rangle \to \Theta_\infty > 0$ while the source sounds, $\sigma \equiv 0$, 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 be silence (a uniform zero field with no flux), nor a periodic return: rhythmic and modulated signals are excluded from windows, and successive steps of source power form a sequence of windows. The limit belongs to the declared idealization of held power in a still, unoccupied enclosure under held air conditions; the physical field leaves it as those conditions drift, 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 in the rigid enclosure.
No current test exercises this aperture.

Aperture 5.2: Poetic Prosody & Psycholinguistic Condensation ($\mathcal{E}_{\mathrm{poetic}}$; provisionally homological)¶

1. Epistemological & Linguistic Scope¶

Aperture 5.2 maps how verse organizes sound and meaning: the production of speech, in which air driven from the lungs is set into vibration at the larynx and shaped by the vocal tract; prosody, the patterning of stress, duration, pitch, and phrasing; the metrical structure of verse, which regularizes that patterning; and the comprehension of language, including condensation, the packing of much meaning into few words in figurative and compressed expression. Lexicon is strictly restricted to phonetics and speech science, prosody and metrics, poetics, and psycholinguistics.

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 airflow of speech is inertial: through the glottis and the vocal tract it is governed by the momentum of the air (jet formation, flow separation, turbulence), not by a conductance acting on a stored quantity, and the sound it generates is a propagating wave. Neither is a stored field relaxing by diffusion, which aperture_generation_guidance.md §2.3 item 3 requires. Prosody, meter, and meaning are, besides, properties of languages, texts, and their users, not fields on one declared domain. Each coordinate is therefore mapped as a conceptual correspondence, mapped via Spine $\mathcal{T}$ and never a reductive identity. Speakers, readers, and listeners are described at cohort level; practices of speaking or reading verse are context and are not mapped.

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: Speech is sustained by a supply: the respiratory muscles maintain the pressure that drives air through the larynx, and metabolism maintains the muscles that shape the tract. That sustained throughput corresponds to the throughput rate $\Theta_E$ of a Branch II state, in which openness is carried by throughput and not by expansion ($\theta \to 0$); the vocal tract is bounded, and the cyclic changes of volume of the respiratory system are not $\theta$. The conventions a language community holds (its stock of words, idioms, figures, and metrical patterns) grow with use, and they change most where usage meets the unexpected, as when a new figure is coined and taken up. That growth corresponds to the record law $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$ along the throughput, as a correspondence between a community's accumulated conventions and the record $\mathcal{H}$, 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 aerodynamic power of speech is mostly dissipated: only a small fraction of the power delivered by the airflow leaves the mouth as sound, and the rest is lost to viscous and turbulent dissipation in the larynx and the vocal tract. That loss corresponds to the dissipative power $\dot{W}_{\mathrm{diss}} > 0$ and to positive entropy production. The pressure drop from the lungs to the lips is a difference of energy density and corresponds to the gradient of $\mathcal{F}$; the flow it drives, however, is set by the inertia of the air, through a Bernoulli relation between pressure drop and velocity, and not by a conductance, which is one reason no transport tensor is declared. 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: A vocal tract has an overall size and a shape, the distribution of cross-sectional area along it, which sets vowel quality; listeners identify the same vowel produced by tracts of different sizes. That separation of scale from shape corresponds to the Weyl split of the volume factor $\Omega$ from the unimodular shape $\hat{h}$, and, as in every aperture, the traced elliptic $\tau$-compatibility equation keeps the two coupled (r10 Decision 2), as the size of a tract constrains the shapes it can reach. In task-dynamic models of speech production, each articulatory gesture moves toward a target with a declared stiffness, high for the constrictions of consonants and lower for vowels; the stiffness of a gesture's target corresponds to the precision Hessian $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$, as a correspondence between roles, not between quantities. The movement of the articulators toward their targets corresponds to the trajectory equation with a declared drive: a forced, goal-directed 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: After a reader or listener has processed a sentence of a given structure, the same structure is used again with raised probability (structural priming); the raised probability declines as further material intervenes, while repeated exposure leaves a lasting change in the probabilities a speaker uses, a form of implicit learning. The pairing of a decaying transient with a retained change corresponds to the decay of shear at a constant rate as $\hat{h}$ settles to $\hat{h}_\infty$ while $V_{\mathrm{local}}$ holds the consolidated content. The correspondence concerns that pairing. The decline of priming is not known to follow a single exponential, and no rate of a linguistic process 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: In psycholinguistics, the surprisal of a word, the negative logarithm of its probability given its context, predicts the effort of processing it. That measure corresponds to the surprisal $q = -\ln(v_0Q)$ of r7, with the distribution over continuations that a reader builds corresponding to the gated density $Q$, and the divergence $D_{\mathrm{KL}}(Q \parallel P)$ from a reference distribution to the cost $\dot{W}_{\mathrm{var}}$ of holding an interpretation apart from what was expected. Condensation keeps the coarse structure of what is expressed and leaves fine detail implicit; that corresponds to the gate $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}}$, which keeps low spatial modes and removes fine structure. The gate is a spatial projector onto Neumann eigenmodes, and the correspondence is between the roles of keeping coarse and removing fine structure; it corresponds neither to the frequency content of speech nor to the temporal regularity of meter. The balance law corresponds to the replenishment of the pressure below the larynx by the respiratory muscles against its loss through the glottis, as a correspondence only, since that pressure is not a diffusing stored field. The stress-conservation equivalence $\nabla^2\xi_\nu + R_\nu^{\;\alpha}\xi_\alpha = 0 \iff \nabla^\mu T_{\mu\nu}^{\mathrm{mat}} = 0$ 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: Verse lines are units of their own, and a phrase that runs on across a line break (enjambment) joins two lines while each keeps its metrical shape. That joining corresponds to the gluing of local metrics over an overlap by the geometric mean, $g^{(ij)} = g^{(i)} \# g^{(j)}$, which keeps each cover set's metric. A text becomes a connected whole once enough links between its parts (repetitions, rhymes, recurring images) are present, which corresponds to the percolation condition $\phi_{\mathrm{perc}}(\mathcal{K}) > \phi_c$. A text that returns to an earlier point, as a refrain does, corresponds to a non-trivial cycle and to harmonic circulation around it, as a correspondence between a sequence's returns and a manifold's topology, not an identity.

$O_6$ — 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: The dependency between two constituents of a sentence is directed: one governs the other, and read the other way the relation reverses. That direction corresponds to the odd phase-lead 1-form $J_{[i\to j]} = -J_{[j\to i]}$. Word order is temporal, not a spatial 1-form, and the correspondence implies no estimator. In stress, prominence and reduction are paired: a stressed syllable is lengthened and given full quality while its unstressed neighbours are shortened and reduced, by amounts that need not match. That pairing 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. The growth of the information that two paired lines share as the second unfolds, as when a rhyme is completed, 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 pairings that are completed, 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: Each line of metrical verse is read against the same template, which is carried from line to line, and mismatches between a line's natural stresses and the template are accommodated as reading proceeds. Carrying a frame of reference from place to place corresponds to gauge transport, and the reduction of mismatches to the decay of the covariant derivative $\hat{\mathcal{E}}_\mu$; the correspondence concerns the transport of a frame, never 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 usage with accumulated convention, as a coined figure becomes conventional.

$O_\Omega$ — 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: A language community sustains its prosodic and poetic conventions through continuous use: never static equilibrium, which for a language would be the end of its use, and never a strict periodic return, although forms recur. That sustained state corresponds to the Branch II limit: bounded extent, sustained throughput ($\langle\Theta_E\rangle \to \Theta_\infty > 0$), decaying shear and coupling transients, and a record whose growth stops when usage no longer meets surprise ($J_E^\lambda\nabla_\lambda q \to 0$). The correspondence makes no claim about any language, text, or reader, and no test exercises this aperture in this epoch.

Aperture 5.3: Spatial Choreography & Relational Kinesiology ($\mathcal{E}_{\mathrm{choreographic}}$; provisionally homological)¶

1. Epistemological & Kinesiological Scope¶

Aperture 5.3 maps how persons move in space and in relation to one another: the collective motion of groups and crowds, from the spacing that individuals keep to the flows and lanes that form among them; the coordination of movement between persons in joint action (relational kinesiology); and choreographic composition, the arrangement of moving persons in space, treated through its general principles (the distribution of movers over a space, their spacing, direction, and density). Lexicon is strictly restricted to pedestrian and crowd dynamics, collective-motion theory, movement science, and the theory of choreographic composition.

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). Persons move by self-propulsion toward goals, so the density of a moving group is carried by velocities that the movers choose (advection), not spread down its own gradient by a conductance; r7's balance law has no advective term, and a regime dominated by random dispersal, in which it would hold, is the exception rather than the phenomenon. Coordination and composition are, besides, properties of persons and groups, not fields on one declared domain. Each coordinate is therefore mapped as a conceptual correspondence, mapped via Spine $\mathcal{T}$ and never a reductive identity. Groups are described at population level; practices of movement and works of choreography are context and are not mapped.

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: A moving group is sustained by metabolic throughput: each mover converts chemical energy into motion and heat, continuously. That throughput corresponds to the throughput rate $\Theta_E$ of a Branch II state, in which openness is carried by throughput and not by expansion ($\theta \to 0$): a group moves within a bounded space, and its spreading and gathering are not $\theta$. Populations develop conventions for moving among others, such as a preferred side for passing, which persist and change where movement meets the unexpected. That accumulation corresponds to the record law $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$, as a correspondence between a population's conventions of movement and the record $\mathcal{H}$, 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: Movement dissipates: the work of the muscles ends as heat, so a moving group corresponds to the dissipative power $\dot{W}_{\mathrm{diss}} > 0$ and to positive entropy production. The density of persons is not an energy density, and no gradient of it drives a flux of the form $J_E^\mu = -p_0^{-1}\mathfrak{K}^{\mu\nu}\nabla_\nu\mathcal{F}$; this is one reason no free energy is declared. 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: A group occupies an overall extent and has a shape, compact or elongated, dense in some parts and sparse in others. That separation corresponds to the Weyl split of the volume factor $\Omega$ from the unimodular shape $\hat{h}$, and the traced elliptic $\tau$-compatibility equation keeps the two coupled (r10 Decision 2), as a group that spreads within the limits of a space also changes shape. In models of pedestrian motion each person accelerates toward a desired velocity and away from others at short range; that law corresponds to the trajectory equation with a declared drive, a forced, goal-directed motion, which r7 admits only through a declared drive under the binding closure. The stiffness with which movers keep their distance at short range corresponds to the precision Hessian $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$, as a correspondence between roles. 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: When neighbouring movers slide past one another, the configuration of a group is sheared; as relative motion dies down, the shear decays, while the spacing the group has adopted persists. That pairing corresponds to the decay of $\sigma$ at a constant rate as $\hat{h}$ settles to $\hat{h}_\infty$, with $V_{\mathrm{local}}$ holding the consolidated content. Relative motion in groups often relaxes with oscillation or in stop-and-go waves, which the mixed-form shear law excludes, and no rate of a group's motion 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: Continuum descriptions of crowds use a density of persons averaged over areas larger than one person, which keeps the large-scale distribution of a group and removes the detail of individual positions. That coarse-graining corresponds to the gate $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}}$, which keeps low spatial modes, and the coarse-grained density to the gated density $Q$. A composition, an intended distribution of movers over a space, corresponds to a reference $P$, and the effort of keeping a group's actual distribution near it against dispersal to the cost $\dot{W}_{\mathrm{var}} = \frac{1}{t_0\beta_{\mathrm{th}}}D_{\mathrm{KL}}(Q \parallel P)$. The conservation of persons, with persons crossing the boundary of a space, corresponds to the balance law only in its bookkeeping: it is a continuity equation with advection, which is why this aperture is homological. The stress-conservation equivalence 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: Two groups that move in neighbouring regions and share a region in which they intermix, each keeping its own spacing, correspond to two cover sets glued over an overlap by the geometric mean, $g^{(ij)} = g^{(i)} \# g^{(j)}$, which keeps each set's metric. A group becomes a connected body, in which every mover is linked to every other through a chain of neighbours within interaction range, once its density exceeds a threshold; that transition corresponds to the percolation condition $\phi_{\mathrm{perc}}(\mathcal{K}) > \phi_c$. No pattern of movement is mapped to harmonic circulation: harmonic fields belong to the topology of a domain, and this aperture declares none.

$O_6$ — 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: When two persons coordinate their movements, one person's movement usually anticipates the other's more than the reverse, and read the other way the relation reverses. That direction corresponds to the odd phase-lead 1-form $J_{[i\to j]} = -J_{[j\to i]}$. Lead–lag statistics between movement records are temporal, not spatial 1-form fields, and the correspondence implies no estimator. Models of collective motion pair attraction toward others at a distance with repulsion at short range, opposite in sign and unequal in strength; that pairing 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. The growth of the information that each person's movement carries about the other's during a coordinated episode 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 coordination 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: In models of collective motion, each mover turns toward the average heading of its neighbours, so that a direction is carried from mover to mover across a group and mismatches of heading decay. That transport of a local direction corresponds to gauge transport, and the decay of mismatches to the decay of the covariant derivative $\hat{\mathcal{E}}_\mu$; it never corresponds 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 a group's current movement with its accumulated conventions.

$O_\Omega$ — 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: A population that moves through a space under a steady demand reaches a sustained, bounded pattern of flow: never stillness, which would be static equilibrium, and never a strict periodic return, although patterns of use recur. That state corresponds to the Branch II limit: bounded extent, sustained throughput ($\langle\Theta_E\rangle \to \Theta_\infty > 0$), decaying shear and coupling transients, and a record whose growth stops when movement no longer meets surprise ($J_E^\lambda\nabla_\lambda q \to 0$). The correspondence makes no claim about any group, space, or composition, and no test exercises this aperture in this epoch.

Aperture 5.4: Inter-Subjective & Cross-Substrate Symbiosis ($\mathcal{E}_{\mathrm{symbiotic}}$; provisionally homological)¶

1. Epistemological & Systems Scope¶

Aperture 5.4 maps how heterogeneous agents, biological and artificial, form associations in which each keeps its own organization while they depend on one another: biological symbiosis, in which organisms of different kinds exchange resources across shared interfaces; communication between persons and the building of common ground; and the coupling of artificial agents with one another and with persons in multi-agent systems. Lexicon is strictly restricted to multi-agent systems theory, information theory, social cognition, and symbiosis biology.

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 media through which agents exchange signals carry them quasi-statically or as waves: a potential conducted through a medium follows its sources without a storage term, and a propagating signal is not a stored field relaxing by diffusion, so r7's balance law cannot represent either literally (aperture_generation_guidance.md §2.3 item 3). The exchanges themselves are discrete messages and transfers between agents, not fields on one declared domain. Each coordinate is therefore mapped as a conceptual correspondence, mapped via Spine $\mathcal{T}$ and never a reductive identity. Agents are described at the level of populations and classes of systems. The internal representations of language models, which Level 2 of the validation protocol concerns, are not mapped here (empirical_validation_protocol.md §3.2).

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); $\frac{\partial\mathcal{H}}{\partial\tau} \equiv J_E^\mu\nabla_\mu q$, $q(\mathbf{x}, \tau) \equiv -\ln(v_0Q(\mathbf{x}, \tau))$.
Homological Mapping: An association of agents is sustained by throughput: organisms by metabolism, artificial agents by electrical power, and both by a continuing exchange of signals or resources. That throughput corresponds to the throughput rate $\Theta_E$ of a Branch II state, in which openness is carried by throughput and not by expansion ($\theta \to 0$). The common ground that partners accumulate, the conventions and shared knowledge on which later exchanges rely, grows most where an exchange brings something unexpected. That growth corresponds to the record law $\partial_\tau\mathcal{H} = J_E^\mu\nabla_\mu q$, as a correspondence between accumulated common ground and the record $\mathcal{H}$, 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: Every physical exchange of signals dissipates energy, and erasing information has a minimum thermodynamic cost; a coupled association therefore corresponds to the dissipative power $\dot{W}_{\mathrm{diss}} > 0$ and to positive entropy production. The potentials that drive signal currents in conductors are electrical potentials in volts, which are not energy densities and are never $\mathcal{F}$. 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: An association has a size (the number of its members and the resources they hold) and a shape (the pattern of who exchanges with whom). That separation corresponds to the Weyl split of the volume factor $\Omega$ from the unimodular shape $\hat{h}$, and the traced elliptic $\tau$-compatibility equation keeps the two coupled (r10 Decision 2), as a growing association also changes its pattern of exchange. The strength with which an agent holds to its own course of action against deviation corresponds to the precision Hessian $\Pi_{\mu\nu} = \nabla_\mu\nabla_\nu\Phi$, as a correspondence between roles. Agents pursuing goals under a policy correspond to the trajectory equation with a declared drive, a forced, goal-directed 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: Agents that repeatedly exchange their estimates of a common quantity converge on a shared value (consensus); in linear consensus models the disagreement decays exponentially, at a rate set by the structure of the exchange network, while the agreed value is retained. That pairing of a decaying disagreement with a retained agreement corresponds to the decay of $\sigma$ at a constant rate as $\hat{h}$ settles to $\hat{h}_\infty$, with $V_{\mathrm{local}}$ holding the consolidated content. No rate of a collective process 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: The shared model of the environment that an association maintains corresponds to the gated density $Q$, and the environment's own statistics to the reference $P$, with the divergence $D_{\mathrm{KL}}(Q \parallel P)$ and its cost $\dot{W}_{\mathrm{var}}$ as the cost of the mismatch. Only features that many members can observe and confirm enter common ground, while the detail that each agent alone observes stays with it; that retention of broadly shared, coarse structure corresponds to the gate $\hat{\mathcal{P}}_{\Lambda_{\mathrm{gate}}}$, which keeps low spatial modes, as a correspondence between roles. The balance law corresponds to the resources of a symbiosis, replenished across the association's boundary and drawn down by its members. The stress-conservation equivalence 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: In a symbiosis each partner keeps its own organization while the partners share an interface across which they exchange; in a team of persons and artificial agents, each keeps its own model while they maintain a shared one. That association without annexation 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. A network of agents becomes able to pass information across its whole extent once its density of links exceeds a threshold, which corresponds to the percolation condition $\phi_{\mathrm{perc}}(\mathcal{K}) > \phi_c$. Closed chains of dependence, as when each member of a community of microorganisms feeds on what the next produces around a cycle (cross-feeding), correspond to non-trivial cycles and to harmonic circulation around them, as a correspondence between a network's cycles and a manifold's topology, not an identity.

$O_6$ — 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: In most partnerships one partner's state shapes the other's more than the reverse, so that it is the other that adapts. That direction corresponds to the odd phase-lead 1-form $J_{[i\to j]} = -J_{[j\to i]}$; a perfectly symmetric exchange carries no directional lead, as fully symmetric partner fields remove a pair from $\mathcal{K}$ in r7. An association can benefit a partner in one respect and cost it in another, by unequal amounts; that pairing 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. The growth of the information that partners share during an exchange 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 exchange 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: Agents that communicate must read one another's signals in their own frames of reference, and populations of communicating agents converge on shared names and conventions. Carrying a frame of reference from agent to agent corresponds to gauge transport, and the decay of mismatches between frames to the decay of the covariant derivative $\hat{\mathcal{E}}_\mu$; it never corresponds 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 exchanges with accumulated common ground.

$O_\Omega$ — 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: A symbiotic association persists through continuing exchange: never static equilibrium, which for an association is its dissolution, and never a strict periodic return, although cycles of exchange recur. That state corresponds to the Branch II limit: bounded extent, sustained throughput ($\langle\Theta_E\rangle \to \Theta_\infty > 0$), decaying shear and coupling transients, and a record whose growth stops when exchanges no longer bring surprise ($J_E^\lambda\nabla_\lambda q \to 0$). The correspondence makes no claim about any association, agent, or collective, and no test exercises this aperture in this epoch.

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	5.1	Rigid declaration of a fixed region, so the evidence reduces to a tolerance check on the enclosure's 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	5.1	Rigid declaration.
\(J_E^\mu\nabla_\mu\mathcal{F} < 0\), \(\dot{s}_{\mathrm{prod}} > 0\), \(\dot{W}_{\mathrm{diss}} > 0\)	5.1	Identities of the fixed Branch II current wherever \(\nabla\mathcal{F} \neq 0\); \(\dot{s}_{\mathrm{prod}}\) is the entropy production of the declared mapping (item 7). The empirical content of \(O_1\) is the flux prediction.
Sign agreement with a measured physical dissipation	5.1	The absorbed acoustic power 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)	5.1	Flat metric and declared \(\eta \equiv 0\) (items 2 and 4).
\(O_2\) Eq. [3]	5.1	Default drive and no tracer mapped.
\(O_5\) gluing, \(g^{(i)} = G^{(i)}_\infty\), and overlap convexity	5.1	Restricted metrics on a rigid domain (item 9).
\(\partial_\tau\sqrt{\vert g^{(i)}\vert} = 0\)	5.1	\(\theta \equiv 0\).
\(\phi_{\mathrm{perc}} = 1 > \phi_c\)	5.1	A connected domain whose transient reaches every overlap (item 9).
\(\mathbf{J}_{\mathcal{M}}^{\mathrm{harm}} \equiv 0\)	5.1, where \(b_1(\mathcal{M}) = 0\)	A property of the declared enclosure and its excised region (an interior source cavity does not raise \(b_1\)); where \(b_1 > 0\) the harmonic space is non-trivial and no claim is made about it.
The declared current has no harmonic component	5.1	\(J_W = -D\,d\mathcal{F}\) is exact for the declared uniform, isotropic \(\mathfrak{K}\), hence \(L^2\)-orthogonal to the tangential harmonic fields (item 9).
\(O_7\) Eq. [3] identities: divergence-free \(J_{\mathrm{geom}}\) and \(u_{\mathrm{exp}}\), and \(\mathcal{L}_{\mathbf{J}_E}(\hat{h} - \hat{h}_\infty) \equiv 0\)	5.1	\(\theta \equiv 0\) and \(\hat{h} \equiv \hat{h}_\infty\) on the rigid domain.
\(\mathcal{F} \ge \mathcal{F}_{\mathrm{ref}}\)	5.1	A zero datum for a non-negative energy density.
\(\mathcal{I} \to s_0V_\infty\)	5.1	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 \(\hat{\mathcal{E}}_\mu \equiv 0\)	5.1	Trivial gauge sector and \(\sigma \equiv 0\) (item 10).
The \(O_4\) gate, the \(O_\Omega\) generator condition, and the alignment \(D_{\mathrm{KL}}(\hat{\rho}_g \parallel \hat{\rho}_{\mathcal{H}}) \to 0\)	5.1	Declared but not identified with any field; stated and not tested.

Tier 5 Specification: Acoustic, Poetic & Cross-Substrate Symbiosis¶

0. Tier Architecture & Four-Part Functional Homology¶

0.1 Conventions Common to the Tier 5 Apertures¶

0.2 Coordinates That Hold Trivially or by Construction¶

Aperture 5.1: Environmental Acoustics & Rhythmic Entrainment (\(\mathcal{E}_{\mathrm{acoustic}}\))¶

1. Epistemological & Physical Scope¶

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

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

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

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

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

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

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

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

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

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

Aperture 5.2: Poetic Prosody & Psycholinguistic Condensation (\(\mathcal{E}_{\mathrm{poetic}}\); provisionally homological)¶

1. Epistemological & Linguistic 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 5.3: Spatial Choreography & Relational Kinesiology (\(\mathcal{E}_{\mathrm{choreographic}}\); provisionally homological)¶

1. Epistemological & Kinesiological 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 5.4: Inter-Subjective & Cross-Substrate Symbiosis (\(\mathcal{E}_{\mathrm{symbiotic}}\); provisionally homological)¶

1. Epistemological & Systems 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¶