2.1.0.3.1 (i) Injective symbol registry.

\[\sigma:\Sigma\to\mathcal{M}\ \text{injective}, \qquad \sigma(s_1)=\sigma(s_2)\Rightarrow s_1=s_2.\] Violation of injectivity is a LOCK failure and halts derivations until repaired.

2.1.0.3.2 (ii) Dimensional consistency.

Let \([\cdot]\) denote physical dimensions (units). For every equation \(E(\cdot)=0\) in the document, \[\text{all additive terms in }E\text{ must share the same dimension.}\] Any equation with mismatched dimensions is invalid.

2.1.0.3.3 (iii) Ledger admissibility.

For state variables interpreted as normalized fractions, the admissibility set is \[\mathcal{S}:=\Big\{(e_{\mathrm{bg}},\rho,e_{\mathrm{a}}):\ 0\le e_{\mathrm{bg}},\rho,e_{\mathrm{a}}\le 1,\ \ e_{\mathrm{bg}}+\rho+e_{\mathrm{a}}=1\Big\}.\] Any derivation that produces values outside \(\mathcal{S}\) without explicitly declaring a different meaning is invalid.

2.1.0.3.4 (iv) Locked parameter subset.

Parameters are partitioned as \[\Theta = \Theta_{\mathrm{LOCK}}\times \Theta_{\mathrm{HYP}}\times \Theta_{\mathrm{SPEC}},\] where \(\Theta_{\mathrm{LOCK}}\) are constants declared fixed across all analyses of a given version. If a quantity is in \(\Theta_{\mathrm{LOCK}}\), it must not be estimated from evaluation data (see §2.4).

2.2.0.5.1 (F1) LOCK rescue.

Any transformation with \(\mathrm{LOCK}'\neq \mathrm{LOCK}\) performed in response to a FAIL is forbidden: \[g(\Pi(M),X)=\mathrm{FAIL}\ \Rightarrow\ \mathrm{LOCK}'=\mathrm{LOCK}.\]

2.2.0.5.2 (F2) Evaluation-data parameter fitting (leakage).

Let \(X_{\mathrm{eval}}\) be evaluation data for a gate. Adjusting \(\theta\) using \(X_{\mathrm{eval}}\) and then re-evaluating on the same \(X_{\mathrm{eval}}\) is forbidden. Formally, if \[\theta' = \mathcal{F}(X_{\mathrm{eval}},\theta) \quad\text{and}\quad g(\Pi(\mathrm{LOCK},H,\theta'),X_{\mathrm{eval}})\] is used as evidence, then this is disallowed unless \(X_{\mathrm{eval}}\) is explicitly redefined as training data and a distinct evaluation set is introduced.

2.2.0.5.3 (F3) Closure selection by evaluation performance without re-registration.

Choosing \(H\) (closure/regime) by trying multiple candidates on the evaluation gate and selecting the best is forbidden unless:

1. the candidate set is pre-registered,

2. model selection is performed on training/validation splits,

3. evaluation is done only once on a held-out test gate.

2.2.0.5.4 (F4) Post-hoc threshold manipulation.

Changing gate thresholds (acceptance criteria) after seeing outcomes is forbidden: \[g \text{ is fixed at version time; thresholds cannot be altered post hoc.}\]

2.2.0.5.5 (F5) Redefining observables.

Altering the mapping from model variables to observables after evaluation is forbidden unless it is declared as a new HYP and re-gated from scratch.

2.2.0.5.6 (F6) Silent regime narrowing.

Declaring a regime after a FAIL in a way that excludes the failing case, without stating the restriction as a new claim, is forbidden. Regime restrictions must be explicit, versioned, and justified by independent reasoning or separate evidence.

2.3.0.3.1 (G1) Internal consistency gates.

These gates test self-consistency independent of external data, including:

• symbol registry injectivity;

• positivity and admissibility constraints (e.g. \(f\ge 0\), \((e_{\mathrm{bg}},\rho,e_{\mathrm{a}})\in\mathcal{S}\));

• ledger conservation identities (local and integral forms);

• well-posedness checks (e.g. closure yields closed PDE system).

Failure of (G1) is a hard stop: no observational comparisons are permitted until repaired.

2.3.0.3.2 (G2) Dimensional-analysis gates.

These verify that every equation is dimensionally consistent and that each parameter has declared units: \[[\partial_t e_{\mathrm{tot}}]=[\nabla\cdot S],\qquad [T]=[v\otimes v][f]\,dv,\ \text{etc.}\] A mismatch is a FAIL.

2.3.0.3.3 (G3) Limiting-regime gates.

These ensure the model behaves correctly in specified limits (as declared by the document), such as:

• low-flux or weak-gradient limit producing diffusion-like behavior if claimed;

• isotropic limit of an anisotropic closure reproducing the isotropic closure;

• symmetry reductions (spherical/axial) matching the reduced equations.

These are mathematical/structural gates, not empirical ones.

2.3.0.3.4 (G4) Observational gates.

These compare predictions to observational datasets using fixed metrics and thresholds, e.g.:

• residual norms with error bars,

• likelihood ratios with pre-registered priors and degrees of freedom,

• cross-dataset consistency constraints (one parameter set across multiple phenomena).

Observational gates must declare data provenance and preprocessing steps as part of \(\mathcal{X}\).

2.3.0.3.5 (G5) Numerical gates.

These ensure results are not numerical artifacts:

• grid/time-step convergence (solution changes below tolerance),

• sensitivity to random seeds within confidence intervals,

• stability checks and conservation error bounds.

2.4.0.3.1 (P1) Globality (one set across modules).

A single parameter set must be used across all gates in its declared scope: \[\theta_{\mathrm{HYP}}^{\mathrm{(module\ A)}}=\theta_{\mathrm{HYP}}^{\mathrm{(module\ B)}}=\cdots\] unless the document explicitly declares a regime-dependent parameterization with independent justification and separate gates.

2.4.0.3.2 (P2) Data separation.

Datasets are partitioned into disjoint roles: \[\mathcal{X}=\mathcal{X}_{\mathrm{train}}\ \dot{\cup}\ \mathcal{X}_{\mathrm{val}}\ \dot{\cup}\ \mathcal{X}_{\mathrm{test}}.\] Parameter estimation may use \(\mathcal{X}_{\mathrm{train}}\) (and possibly \(\mathcal{X}_{\mathrm{val}}\) for model selection), but final PASS/FAIL must be computed on \(\mathcal{X}_{\mathrm{test}}\) exactly once per model version.

2.4.0.3.3 (P3) Degrees-of-freedom budget.

Let \(k:=\dim(\theta_{\mathrm{HYP}})\) be the number of fitted parameters used for a gate, and let \(n\) be the effective number of independent data points used in that gate. A gate must declare \((k,n)\) and enforce that \(k\ll n\) in a quantified sense. A minimal admissibility rule is: \[\frac{k}{n} \le \epsilon,\] for a small pre-registered \(\epsilon\) appropriate to the application, together with a penalty-aware metric (e.g. information criteria or cross-validation). The specific criterion must be fixed by the gate definition.

2.4.0.3.4 (P4) Identifiability requirement.

If multiple parameter sets yield indistinguishable predictions within uncertainty, the parameters are not identifiable. Formally, if the map \[\theta_{\mathrm{HYP}} \mapsto \Pi(\mathrm{LOCK},H,\theta_{\mathrm{HYP}})\] is not injective on the admissible set (up to observational noise), then the document must either:

1. reduce parameterization (lower \(k\)),

2. add independent gates that break degeneracy,

3. or restrict claims to prediction-level statements that do not interpret the parameters.

2.6.0.3.1 (R1) Semantic/notation failure.

Examples: symbol collision; undefined symbol; inconsistent meaning; registry injectivity violation.

2.6.0.3.2 (R2) Dimensional/unit failure.

Examples: unit mismatch; inconsistent scaling; hidden unit conversions.

2.6.0.3.3 (R3) Ledger/admissibility violation.

Examples: negativity (\(f<0\)); fractions leaving \([0,1]\); normalization failure.

2.6.0.3.4 (R4) Mathematical closure failure.

Examples: moment hierarchy not closed under declared closure; missing boundary conditions; ill-posed PDE type under assumptions.

2.6.0.3.5 (R5) Numerical failure.

Examples: non-convergence; instability; results sensitive to grid/time step or seed; conservation error above tolerance.

2.6.0.3.6 (R6) Observational mismatch.

Examples: residuals exceed threshold; cross-dataset inconsistency for a single parameter set; falsified discriminating prediction.

2.6.0.3.7 (R7) Overfitting/model-selection leakage.

Examples: parameter chosen using evaluation data; closure selected on the test gate; threshold changed post hoc.

2.6.0.4.1 If (R1)–(R2) occurs:

this is typically a Patch fix (typos, symbol registry repair, unit correction), provided the claim statement is unchanged. After patching, re-run all affected internal gates.

2.6.0.4.2 If (R3) occurs:

either:

1. tighten admissibility enforcement (e.g. revise numerical scheme or impose gates), or

2. restrict the regime where the model is claimed to apply,

without changing LOCK meanings.

2.6.0.4.3 If (R4) occurs:

the derivation is incomplete; either add missing closure assumptions as HYP with gates, or retract the DERIVE claim.

2.6.0.4.4 If (R5) occurs:

fix the SPEC (solver, resolution, stability), add numerical gates, and re-evaluate. Numerical repair must not alter physical claims without explicit declaration.

2.6.0.4.5 If (R6) occurs:

the primary allowed responses are:

1. replace or revise HYP closures/constitutive laws (Minor version),

2. restrict regime validity explicitly,

3. retract the claim if no admissible modification exists.

LOCK rescue is forbidden.

2.6.0.4.6 If (R7) occurs:

invalidate the evaluation as non-admissible; redesign the protocol with proper train/validation/test separation and re-register gates.

3.1.1.1 Symbol set.

Let \(\Sigma\) denote the set of all symbols (tokens) that can appear in the document, including Latin/Greek letters, decorated forms (subscripts/superscripts), and indexed objects: \[\Sigma \;=\; \{ s : s \text{ is a syntactic symbol allowed in this document}\}.\]

3.1.1.2 Meaning set.

Let \(\mathcal{M}\) denote the set of meanings (semantic objects): a meaning can be a scalar field, vector field, tensor field, rate parameter, operator, or a well-defined map.

3.1.1.3 Symbol\(\rightarrow\)meaning registry (injective).

A valid symbol registry is an injective mapping \[\sigma:\Sigma_{\mathrm{can}}\to \mathcal{M}, \qquad \sigma \text{ injective},\] where \(\Sigma_{\mathrm{can}}\subseteq\Sigma\) is the set of canonical symbols (the ones that are allowed to appear in final equations as primary names). Injectivity means: \[\sigma(s_1)=\sigma(s_2)\ \Longrightarrow\ s_1=s_2.\] Any need for synonyms must be handled by aliases (see below) rather than breaking injectivity.

3.1.1.4 Alias map (non-canonical tokens).

If alternative spellings are convenient, introduce an alias map \[\alpha:\Sigma_{\mathrm{alias}} \to \Sigma_{\mathrm{can}},\] where \(\Sigma_{\mathrm{alias}}\subseteq\Sigma\setminus\Sigma_{\mathrm{can}}\). Aliases are editorial conveniences only; all formal statements and derivations must be written in canonical symbols.

3.1.2.1 Dimension group.

Let the base-dimension set be \[\mathbb{B}=\{L,T,M,\ldots\},\] where at minimum \(L\) (length) and \(T\) (time) are included, and \(M\) (mass) is included if/when a mass-carrying mapping is declared in later Parts. Define the dimension group \(\mathrm{Dim}\) as the free abelian group generated by \(\mathbb{B}\), i.e. elements are exponent vectors. For concreteness, if \(\mathbb{B}=\{L,T,M\}\) then \[\mathrm{Dim}\cong \mathbb{Z}^3,\qquad d = (\ell,\tau,m)\ \leftrightarrow\ [L]^\ell [T]^\tau [M]^m.\]

3.1.2.2 Dimension registry.

Every canonical symbol must have a declared dimension: \[\delta:\Sigma_{\mathrm{can}}\to \mathrm{Dim}.\] Dimensional consistency of an equation is defined by \(\delta\) and the standard rules:

• \(\delta(uv)=\delta(u)+\delta(v)\) (multiplication adds exponents),

• \(\delta(u/v)=\delta(u)-\delta(v)\),

• \(\delta(u+v)\) is defined only if \(\delta(u)=\delta(v)\).

3.1.2.3 Dimensional well-formedness of expressions.

An expression is dimensionally well-formed if every addition/subtraction occurs only between terms of identical dimension. An equation is valid only if both sides have identical dimension.

3.1.3.1 Scalars, vectors, tensors.

• Scalars: italic (e.g. \(e_{\mathrm{a}}\), \(\rho\), \(\mu\)).

• Vectors: bold (e.g. \(\mathbf{S}\)) or arrow notation; in this document we prefer bold: \[\mathbf{S}(x,t)\in\mathbb{R}^3.\]

• Second-order tensors: bold uppercase or explicit components, e.g. \(\mathbf{T}(x,t)\) with components \(T_{ij}\).

3.1.3.2 Identity tensor and Kronecker symbols.

\[\mathbf{I}\ \text{denotes the identity tensor},\qquad (\mathbf{I})_{ij}=\delta_{ij}.\]

3.1.3.3 Differential operators.

\[\nabla := (\partial_{x_1},\partial_{x_2},\partial_{x_3}),\quad \nabla\cdot \mathbf{S} := \sum_{i=1}^3 \partial_{x_i} S_i,\quad \nabla\cdot \mathbf{T} := \big(\sum_{j=1}^3 \partial_{x_j} T_{ij}\big)_{i=1}^3.\]

3.1.3.4 Norms and inner products.

\[\|\mathbf{u}\| := \sqrt{\mathbf{u}\cdot\mathbf{u}},\qquad \mathbf{u}\cdot\mathbf{v}:=\sum_{i=1}^3 u_i v_i.\] For tensors, \(\|\mathbf{T}\|\) denotes a specified matrix norm (default: Frobenius), which must be declared where used.

3.1.3.5 Big-O notation (regime declarations).

For a small parameter \(\varepsilon>0\), \[f(\varepsilon)=\mathcal{O}(\varepsilon^p)\quad(\varepsilon\to 0)\] means there exist \(C,\varepsilon_0>0\) such that \(|f(\varepsilon)|\le C\varepsilon^p\) for \(0<\varepsilon<\varepsilon_0\).

3.1.5.1 Reserved canonical symbols.

Define a reserved set \(\Sigma_{\mathrm{res}}\subseteq \Sigma_{\mathrm{can}}\) of symbols whose meaning is locked and may never be reassigned. At minimum, the following are reserved in this document: \[\Sigma_{\mathrm{res}}\supseteq \{e_{\mathrm{bg}},\ \rho,\ e_{\mathrm{a}},\ e_{\mathrm{tot}},\ f,\ \mathbf{S},\ \mathbf{T},\ \mu,\ \Gamma,\ B,\ a,\ \Delta t,\ c,\ v^\ast\}.\] Additional reserved symbols will be declared as the document expands.

3.1.5.2 Forbidden (overloaded) symbols.

Define a forbidden set \(\Sigma_{\mathrm{forb}}\subseteq\Sigma\) that must not appear as a standalone canonical symbol because it is historically overloaded or ambiguous. This Part imposes, at minimum: \[\Sigma_{\mathrm{forb}}\supseteq \{\ e,\ \kappa\ \},\] meaning:

• the bare symbol \(e\) is forbidden (must be replaced by \(e_{\mathrm{bg}}\), \(e_{\mathrm{a}}\), \(e_{\mathrm{tot}}\), etc.);

• the bare symbol \(\kappa\) is forbidden (must be replaced by decorated symbols such as \(\kappa_{\mathrm{opt}}\), \(\kappa_T\), etc.).

3.1.5.3 Collision repair rule (mandatory).

If a legacy source uses a forbidden symbol, every occurrence must be mapped to a canonical symbol by an explicit mapping table (see §3.7). Any unmapped occurrence is an editorial failure and blocks DERIVE claims until fixed.

3.2.1.1 Integrability requirement.

At any \((x,t)\) for which \(e_{\mathrm{a}}(x,t)\) is defined, require \[f(x,\cdot,t)\in L^1(\mathbb{R}^3_v),\] and for defining \(\mathbf{S}\) and \(\mathbf{T}\) require additional moment integrability: \[\int_{\mathbb{R}^3}\|v\|\,f(x,v,t)\,dv<\infty, \qquad \int_{\mathbb{R}^3}\|v\|^2\,f(x,v,t)\,dv<\infty.\] These integrability assumptions are part of the regime declaration whenever invoked.

3.2.2.1 Active fraction (moving/transport-capable actor).

Define the active fraction as the velocity integral of \(f\): \[e_{\mathrm{a}}(x,t) := \int_{\mathbb{R}^3} f(x,v,t)\,dv.\] Dimension rule: \(e_{\mathrm{a}}\) is dimensionless: \[\delta(e_{\mathrm{a}})=0\in\mathrm{Dim}.\] Therefore \(f\) must have the reciprocal dimension of \(dv\): \[\delta(f) = -\delta(dv) = -3\,\delta(v),\] so that \(\delta\!\left(\int f\,dv\right)=0\). In an \(L,T\) dimension basis where \(\delta(v)=(1,-1)\), this corresponds to \(\delta(f)=(-3,3)\).

3.2.2.2 Flux (first moment).

Define the flux vector \[\mathbf{S}(x,t) := \int_{\mathbb{R}^3} v\, f(x,v,t)\,dv \in \mathbb{R}^3.\] Since \(e_{\mathrm{a}}\) is dimensionless, \(\mathbf{S}\) has dimension of velocity: \[\delta(\mathbf{S})=\delta(v).\]

3.2.2.3 Second moment tensor.

Define the second moment tensor \[\mathbf{T}(x,t) := \int_{\mathbb{R}^3} (v\otimes v)\, f(x,v,t)\,dv,\] with dimension \[\delta(\mathbf{T})=2\,\delta(v).\]

3.2.2.4 Stored fraction (locally stored actor).

Introduce a stored fraction \[\rho:\Omega\times\mathbb{R}\to [0,1], \qquad \rho=\rho(x,t),\] interpreted as actor content in a non-transporting (stored) state. It is dimensionless: \[\delta(\rho)=0.\]

3.2.2.5 Total actor fraction.

Define \[e_{\mathrm{tot}}(x,t) := \rho(x,t) + e_{\mathrm{a}}(x,t), \qquad \delta(e_{\mathrm{tot}})=0.\]

3.2.2.6 Background (stage) fraction.

Define the background fraction by normalization: \[e_{\mathrm{bg}}(x,t) := 1 - e_{\mathrm{tot}}(x,t) = 1-\rho(x,t)-e_{\mathrm{a}}(x,t).\] It is dimensionless: \[\delta(e_{\mathrm{bg}})=0.\]

3.2.3.1 Pointwise bounds (LOCK).

The three-phase interpretation requires \[0\le \rho(x,t)\le 1,\qquad 0\le e_{\mathrm{a}}(x,t)\le 1,\qquad 0\le e_{\mathrm{bg}}(x,t)\le 1.\]

3.2.3.2 Normalization (LOCK).

\[e_{\mathrm{bg}}(x,t)+\rho(x,t)+e_{\mathrm{a}}(x,t)=1 \quad \text{for all } (x,t).\]

3.2.3.3 Admissibility set.

Define \[\mathcal{S} := \Big\{ (e_{\mathrm{bg}},\rho,e_{\mathrm{a}})\in\mathbb{R}^3: 0\le e_{\mathrm{bg}},\rho,e_{\mathrm{a}}\le 1,\ \ e_{\mathrm{bg}}+\rho+e_{\mathrm{a}}=1 \Big\}.\] All solutions and intermediate approximations used for physical claims must remain in \(\mathcal{S}\); leaving \(\mathcal{S}\) is a FAIL unless the meaning is explicitly redefined (which requires Major version changes).

3.3.1.1 (C1) Closure constants.

Constants that appear in moment closures (e.g. relating \(\mathbf{T}\) to \(e_{\mathrm{a}}\)). Example: \[\mathbf{T} = \kappa_T\, e_{\mathrm{a}}\, \mathbf{I}.\] Here \(\kappa_T\) must have dimension of \(v^2\): \[\delta(\kappa_T)=2\,\delta(v).\]

3.3.1.2 (C2) Transport/relaxation rates.

Coefficients that damp or relax fluxes, typically with dimension \(T^{-1}\). Example (a generic relaxation form): \[\partial_t \mathbf{S} + \nabla\cdot \mathbf{T} = -B\,\mathbf{S} + \cdots,\] so \[\delta(B)= -\delta(t) = T^{-1}.\]

3.3.1.3 (C3) Conversion rates.

Coefficients converting between stored and active fractions, also typically with dimension \(T^{-1}\), e.g. \(\mu\).

3.3.1.4 (C4) Observable-mapping coefficients.

Coefficients mapping model variables to observational quantities. Example (optical attenuation for redshift-like mapping): \[\frac{dE}{E} = -\kappa_{\mathrm{opt}}\, ds,\] so \[\delta(\kappa_{\mathrm{opt}})=L^{-1}.\] This is dimensionally and semantically distinct from \(\kappa_T\).

3.3.1.5 (C5) Geometric or dimensionless factors.

Dimensionless constants (e.g. lattice-geometry factors) must be explicitly marked as dimensionless.

3.3.2.1 No-bare-\(\kappa\) (LOCK editorial rule).

The symbol \(\kappa\) without a qualifier is forbidden: \[\kappa \notin \Sigma_{\mathrm{can}}.\] Any appearance of \(\kappa\) must be rewritten as a qualified constant with a role-specific subscript: \[\kappa_{\mathrm{opt}},\ \kappa_T,\ \kappa_{\mathrm{lens}},\ \kappa_{\mathrm{mix}},\ \ldots\] and each must have a distinct registry entry with distinct meaning and (typically) distinct dimension.

3.3.2.2 General separation rule.

If two constants have different roles or different dimensions, they must not share the same base symbol. Formally, if constants \(c_1,c_2\in\Sigma_{\mathrm{can}}\) satisfy either \[\delta(c_1)\neq \delta(c_2) \quad\text{or}\quad \sigma(c_1)\neq \sigma(c_2)\ \text{(different meaning)},\] then they must be written as distinct symbols and must not be conflated by aliasing.

3.3.3.1 Optical/interaction-cost coefficient.

\[\frac{dE}{E} = -\kappa_{\mathrm{opt}}\, ds \quad\Longrightarrow\quad 1+z = e^{\kappa_{\mathrm{opt}} s}.\] Registry requirements: \[\delta(\kappa_{\mathrm{opt}})=L^{-1},\qquad \sigma(\kappa_{\mathrm{opt}})=\text{``attenuation per unit path length''}.\]

3.3.3.2 Moment-closure coefficient.

\[\mathbf{T} = \kappa_T\, e_{\mathrm{a}}\, \mathbf{I}.\] Registry requirements: \[\delta(\kappa_T)=2\,\delta(v)=L^2 T^{-2},\qquad \sigma(\kappa_T)=\text{``isotropic second-moment scale (variance-like)''}.\]

3.3.3.3 Consistency check.

If an equation uses both, dimensional analysis alone forbids identifying them: \[\delta(\kappa_{\mathrm{opt}})\neq \delta(\kappa_T) \quad\Rightarrow\quad \kappa_{\mathrm{opt}}\neq \kappa_T \quad\text{(as symbols and as meanings)}.\]

3.4.1.1 Reference length \(a\) (cell/lattice spacing).

Let \(a>0\) be the reference microscopic length scale (e.g. lattice spacing). Declare: \[\delta(a)=L.\]

3.4.1.2 Reference time step \(\Delta t\).

Let \(\Delta t>0\) be the reference microscopic time increment. Declare: \[\delta(\Delta t)=T.\]

3.4.1.3 Reference speed \(c\) (derived).

Define the reference speed \[c := \frac{a}{\Delta t}, \qquad \delta(c)=L\,T^{-1}.\] In later Parts, \(c\) may be interpreted as a throughput-limited characteristic speed; in this Part it is purely a unit-derived quantity.

3.4.1.4 Reference volume \(v^\ast\).

Define the reference volume by \[v^\ast := \zeta\, a^3, \qquad \delta(v^\ast)=L^3,\] where \(\zeta>0\) is a dimensionless geometry factor (e.g. \(\zeta=1\) for cubic cells). If the lattice geometry changes, \(\zeta\) must be updated, but \(\zeta\) remains dimensionless.

3.5.1.1 Micro scale.

Micro-scale variables resolve cell/lattice structure: \[\ell_{\mu} \sim a,\qquad \tau_{\mu} \sim \Delta t,\qquad v_{\mu}\sim c.\] The kinetic distribution \(f(x,v,t)\) may be interpreted as defined at micro resolution.

3.5.1.2 Meso scale.

Meso-scale variables describe environments near astrophysical objects (e.g. disk/jet regions, halos) where continuum fields vary on scales \[\ell_{\mathrm{me}} \gg a, \qquad \tau_{\mathrm{me}} \gg \Delta t.\] At this scale, coarse-grained fields (e.g. \(e_{\mathrm{a}}\), \(\rho\), \(\mathbf{S}\)) are treated as smooth.

3.5.1.3 Macro scale.

Macro-scale variables describe cosmological-scale behavior with characteristic scale \[\ell_{\mathrm{ma}} \gg \ell_{\mathrm{me}} \gg a, \qquad \tau_{\mathrm{ma}} \gg \tau_{\mathrm{me}} \gg \Delta t.\] At this scale, additional symmetries (statistical homogeneity/isotropy, or controlled anisotropy) may be declared as regime assumptions.

3.5.2.1 Smoothing kernel.

Let \(W_\ell:\mathbb{R}^3\to[0,\infty)\) be a smoothing kernel with characteristic width \(\ell\), normalized by \[\int_{\mathbb{R}^3} W_\ell(r)\,dr = 1,\qquad W_\ell(r)=\ell^{-3}W_1(r/\ell).\] Define coarse-grained fields for any micro field \(q(x,t)\) by convolution: \[\overline{q}^{(\ell)}(x,t):=\int_{\mathbb{R}^3} W_\ell(x-y)\, q(y,t)\,dy.\]

3.5.2.2 Coarse-graining consistency.

A model is scale-consistent if:

1. the coarse-grained fields remain admissible: \[(\overline{e}_{\mathrm{bg}}^{(\ell)},\overline{\rho}^{(\ell)},\overline{e}_{\mathrm{a}}^{(\ell)})\in\mathcal{S} \quad\text{for all }\ell \text{ in the declared scale range};\]

2. the ledger form is preserved under coarse-graining up to controlled error: \[\partial_t \overline{e}_{\mathrm{tot}}^{(\ell)} + \nabla\cdot \overline{\mathbf{S}}^{(\ell)} = \mathcal{O}(\varepsilon_\ell),\] where \(\varepsilon_\ell\) quantifies unresolved subgrid contributions and must be bounded or modeled.

3.6.3.1 Transport vs. conversion (Damköhler-type numbers).

A conversion-to-transport comparison naturally appears as: \[\hat{\mu}=\mu T_0 = \mu \frac{L_0}{c_0}, \qquad \hat{\Gamma}_0=\Gamma_0 T_0 = \Gamma_0 \frac{L_0}{c_0}.\] Large \(\hat{\mu}\) indicates conversion dominates within one transport time \(L_0/c_0\).

3.6.4.1 Isotropic closure and diffusion limit (dominant balance).

Assume isotropic closure: \[\mathbf{T}=\kappa_T e_{\mathrm{a}} \mathbf{I}, \qquad \delta(\kappa_T)=L^2T^{-2}.\] Define dimensionless closure coefficient: \[\hat{\kappa}_T:=\frac{\kappa_T}{c_0^2}.\] Then \(\hat{\mathbf{T}}=\hat{\kappa}_T \hat{e}_{\mathrm{a}}\mathbf{I}\) and the flux equation becomes \[\partial_{\hat{t}}\hat{\mathbf{S}} + \hat{\kappa}_T \hat{\nabla}\hat{e}_{\mathrm{a}} = -\hat{B}\,\hat{\mathbf{S}}.\] In the strong-relaxation regime \[\hat{B}\gg 1,\qquad \partial_{\hat{t}}\hat{\mathbf{S}}=\mathcal{O}(1),\] dominant balance yields \[\hat{\mathbf{S}} \approx -\frac{\hat{\kappa}_T}{\hat{B}}\,\hat{\nabla}\hat{e}_{\mathrm{a}}.\] Returning to dimensional variables gives Fick-like form: \[\mathbf{S}\approx -D\,\nabla e_{\mathrm{a}}, \qquad D:=\frac{\kappa_T}{B}, \qquad \delta(D)=L^2T^{-1}.\] Dimensionless diffusion number: \[\hat{D}:=\frac{D T_0}{L_0^2}=\frac{\hat{\kappa}_T}{\hat{B}}.\] This makes the diffusion regime a transparent ordering statement.

3.7.1.1 Legacy and upgraded symbol sets.

Let \(\Sigma_{\mathrm{leg}}\) be the set of legacy symbols found in prior manuscripts. Let \(\Sigma_{\mathrm{can}}\) be the canonical set in the upgraded document.

3.7.1.2 Mapping function.

Define a mapping \[\Phi:\Sigma_{\mathrm{leg}}\to \big(\Sigma_{\mathrm{can}}\cup \mathcal{E}(\Sigma_{\mathrm{can}})\big),\] where \(\mathcal{E}(\Sigma_{\mathrm{can}})\) denotes expressions built from canonical symbols (to allow cases where a legacy symbol maps to a composite expression).

3.7.1.3 Context-dependence policy.

If a legacy symbol is ambiguous (maps differently depending on context), then \(\Phi\) must be extended to a context-tagged mapping: \[\Phi:\Sigma_{\mathrm{leg}}\times \mathcal{K}\to \Sigma_{\mathrm{can}}\cup \mathcal{E}(\Sigma_{\mathrm{can}}),\] where \(\mathcal{K}\) is a finite set of context labels (e.g. “cosmology”, “transport”, “background”). Any context-tag use must be explicitly documented. Silent context-dependent remapping is forbidden.

3.7.1.4 Mapping correctness gates (mandatory).

After applying \(\Phi\) to any legacy equation:

1. Registry gate: all resulting symbols must be in \(\Sigma_{\mathrm{can}}\) and have entries.

2. Dimension gate: the transformed equation must be dimensionally consistent under \(\delta\).

3. Admissibility gate: if the equation concerns \((e_{\mathrm{bg}},\rho,e_{\mathrm{a}})\), it must preserve membership in \(\mathcal{S}\) or explicitly state when it can be violated (which requires redefining meaning and thus a Major change).

3.7.4.1 Step 1 (tokenization).

Extract the set of legacy symbols \(\Sigma_{\mathrm{leg}}(\mathcal{L})\subseteq\Sigma_{\mathrm{leg}}\).

3.7.4.2 Step 2 (registry and forbiddance check).

Identify forbidden legacy symbols: \[\Sigma_{\mathrm{leg}}(\mathcal{L})\cap \Sigma_{\mathrm{forb}}.\] Every element in this intersection must be remapped; otherwise the upgrade fails.

3.7.4.3 Step 3 (define \(\Phi\) rows).

For each \(s\in\Sigma_{\mathrm{leg}}(\mathcal{L})\), define \(\Phi(s)\) (or \(\Phi(s,k)\) with context).

3.7.4.4 Step 4 (substitution).

Replace all occurrences by: \[\ell_i \mapsto \ell_i^\Phi \quad \text{(apply $\Phi$ to all symbols in $\ell_i$)}.\]

3.7.4.5 Step 5 (dimension gate).

Verify dimensional consistency of every transformed equation \(\ell_i^\Phi\) using \(\delta\).

3.7.4.6 Step 6 (semantic gate).

Verify that each transformed statement uses symbols consistent with their registry meanings \(\sigma\) and constraints.

3.7.4.7 Step 7 (claim-tier reassignment).

Assign tiers to upgraded claims:

• if the legacy step used an unstated closure, mark it as HYP and declare the closure explicitly;

• if the legacy step used only LOCK items and admissible inference, mark as DERIVE;

• if the legacy step was an implementation choice, mark as SPEC.

3.7.4.8 Step 8 (archive).

Store the mapping table, transformed derivation, and dimension-check results as artifacts linked to the relevant claim identifiers.

4.1.2.1 Primitive object (VP).

A volume particle (VP) is the minimal capacity unit of the stage: it is not assumed to be a standard point particle, but rather the atomic unit with respect to which the model tracks occupancy fractions. The VP concept is implemented by a reference volume \(v^\ast\) and the normalization constraints on occupancy fields.

4.1.2.2 Reference volume \(v^\ast\).

Let \[v^\ast>0,\qquad \delta(v^\ast)=L^3,\] denote a fixed reference volume. In a lattice/cell realization, one may set \[v^\ast := \zeta\, a^3,\] where \(a>0\) is a reference length scale (cell spacing) and \(\zeta>0\) is a dimensionless geometric factor (e.g. \(\zeta=1\) for cubic cells). The values of \((a,\zeta)\) belong to the unit-realization infrastructure (Part 03), while \(v^\ast\) is the primitive physical reference for capacity accounting.

4.1.2.3 Capacity interpretation.

Occupancy fields in this document are dimensionless fractions of local capacity per reference volume \(v^\ast\). Thus, “\(1\)” means “full local capacity” and “\(0\)” means “no occupancy” in the relevant phase.

4.1.3.1 Primitive unit axis field.

Introduce a unit vector field \[k:\Omega\times\mathbb{R}\to \mathbb{S}^2:=\{u\in\mathbb{R}^3:\|u\|=1\}, \qquad (x,t)\mapsto k(x,t),\] with the LOCK constraint \[\|k(x,t)\|=1 \quad \forall(x,t).\] The unit axis \(k\) represents a preferred local direction used to describe alignment-dominated regimes (e.g. channeling/jet-like transport). The origin of \(k\) (e.g. from rotation/spin or boundary geometry) is not fixed in this Part; later modules may prescribe \(k\) by additional hypotheses. In this Part, \(k\) is a primitive field used to define alignment measures and axisymmetric regimes.

4.1.4.1 Primitive defect quantity with canonical definition.

Introduce an alignment defect scalar field \[a_k:\Omega\times\mathbb{R}\to[0,1], \qquad a_k=a_k(x,t),\] interpreted as a normalized measure of how far the active phase departs from perfect alignment with the axis \(k\). The canonical definition of \(a_k\) is given in §4.5 in terms of the alignment moment \(m_b\) and the axis \(k\); here we only fix its type, range, and meaning.

4.1.4.2 Range constraint (LOCK).

\[0\le a_k(x,t)\le 1\quad \forall(x,t).\] The limiting cases have the intended interpretation:

• \(a_k=0\): perfect alignment with \(k\) (no transverse defect),

• \(a_k=1\): maximal transverse misalignment compatible with the bounded bias constraint (defined precisely in §4.5).

4.2.1.1 Option A (bounded support).

There exists a finite bound \(c_{\max}>0\) such that \[\mathcal{V}=\{v\in\mathbb{R}^3:\|v\|\le c_{\max}\}.\] If the document chooses to identify \(c_{\max}\) with the unit-realization speed \(c=a/\Delta t\) (Part 03), then \(c_{\max}=c\) is treated as a LOCK identification within the declared regime.

4.2.1.2 Option B (unbounded support with finite moments).

\[\mathcal{V}=\mathbb{R}^3,\] but \(f\) must satisfy integrability conditions sufficient to define the moments used in the theory (below). In this option, any later assumption of bounded speed must be declared as a gate or additional hypothesis.

For the rest of this Part, \(\mathcal{V}\) is treated as fixed by the chosen regime; all integrals in \(v\) are over \(\mathcal{V}\) unless otherwise stated.

4.2.2.1 Primitive distribution.

Define \[f:\Omega\times\mathcal{V}\times\mathbb{R}\to [0,\infty), \qquad (x,v,t)\mapsto f(x,v,t).\]

4.2.2.2 Positivity (LOCK).

\[f(x,v,t)\ge 0\quad \forall (x,v,t)\in\Omega\times\mathcal{V}\times\mathbb{R}.\]

4.2.3.1 Active fraction (moving/transport-capable actor content).

Define \[e_{\mathrm{a}}(x,t) := \int_{\mathcal{V}} f(x,v,t)\,dv.\] This is dimensionless and interpreted as a fraction of capacity: \[\delta(e_{\mathrm{a}})=0.\]

4.2.3.2 Flux (first moment).

Define the flux vector \[\mathbf{S}(x,t) := \int_{\mathcal{V}} v\, f(x,v,t)\,dv\in\mathbb{R}^3, \qquad \delta(\mathbf{S})=LT^{-1}.\]

4.2.3.3 Second moment tensor.

Define \[\mathbf{T}(x,t) := \int_{\mathcal{V}} (v\otimes v)\, f(x,v,t)\,dv, \qquad \delta(\mathbf{T})=L^2T^{-2}.\]

4.2.3.4 Moment existence requirements (admissibility).

At each \((x,t)\) where these quantities are invoked, require: \[\int_{\mathcal{V}} f(x,v,t)\,dv<\infty,\] and additionally for \(\mathbf{S},\mathbf{T}\): \[\int_{\mathcal{V}} \|v\|\, f(x,v,t)\,dv<\infty, \qquad \int_{\mathcal{V}} \|v\|^2\, f(x,v,t)\,dv<\infty.\] These conditions are part of the regime declaration (§4.7) for any claim that uses \(\mathbf{S}\) or \(\mathbf{T}\).

4.2.3.5 Capacity bound for the active fraction (LOCK admissibility).

Because \(e_{\mathrm{a}}\) is a capacity fraction, we impose: \[0\le e_{\mathrm{a}}(x,t)\le 1\quad \forall(x,t).\] This is a LOCK admissibility constraint. Any model or numerical scheme producing \(e_{\mathrm{a}}>1\) (without redefining meaning) fails the admissibility gate.

4.2.4.1 Stored fraction.

\[\rho:\Omega\times\mathbb{R}\to[0,1], \qquad \rho=\rho(x,t), \qquad \delta(\rho)=0.\]

4.2.4.2 Total actor fraction.

\[e_{\mathrm{tot}}(x,t) := \rho(x,t) + e_{\mathrm{a}}(x,t), \qquad 0\le e_{\mathrm{tot}}\le 1.\]

4.2.4.3 Background fraction (stage occupancy complement).

\[e_{\mathrm{bg}}(x,t) := 1 - e_{\mathrm{tot}}(x,t)=1-\rho(x,t)-e_{\mathrm{a}}(x,t).\] The admissibility constraints are: \[0\le \rho(x,t)\le 1,\qquad 0\le e_{\mathrm{bg}}(x,t)\le 1, \qquad e_{\mathrm{bg}}+\rho+e_{\mathrm{a}}=1.\] These are treated as LOCK normalization/admissibility constraints.

4.3.2.1 Axiom (Ledger / conservation of total actor fraction).

For every admissible control volume \(V\subseteq\Omega\) and for all times \(t\) where the integrals exist, \[\frac{d}{dt}\int_V e_{\mathrm{tot}}(x,t)\,dx \;+\; \int_{\partial V} \mathbf{S}(x,t)\cdot \mathbf{n}(x)\,dA \;=\; 0. \label{eq:ledger_integral}\] This expresses accumulation \(+\,\) outflow \(=0\) (no net creation of actor content).

4.3.2.2 Closed-system boundary condition (special case).

If \(\Omega\) is bounded and the system is closed, impose the no-through-flow condition: \[\mathbf{S}(x,t)\cdot \mathbf{n}(x)=0 \quad \text{for } x\in\partial\Omega,\] which implies global conservation \[\frac{d}{dt}\int_\Omega e_{\mathrm{tot}}(x,t)\,dx = 0.\]

4.3.2.3 Open-system boundary specification (alternative).

If the system is open, then \(\mathbf{S}\cdot \mathbf{n}\) on \(\partial\Omega\) must be specified as a boundary input. In this case, the ledger axiom still holds on all sub-volumes \(V\), and global conservation is replaced by boundary-flux accounting.

4.3.3.1 Ledger consistency with normalization.

Because \(e_{\mathrm{bg}}=1-e_{\mathrm{tot}}\), the ledger axiom implies \[\partial_t e_{\mathrm{bg}}(x,t) = -\partial_t e_{\mathrm{tot}}(x,t) = \nabla\cdot \mathbf{S}(x,t),\] so the background fraction changes exactly as required to maintain \(e_{\mathrm{bg}}+\rho+e_{\mathrm{a}}=1\).

4.4.1.1 Activation rate \(\mu\).

Introduce \[\mu:\Omega\times\mathbb{R}\to[0,\infty), \qquad \delta(\mu)=T^{-1}.\] \(\mu\) represents the rate of conversion from stored fraction \(\rho\) to active fraction \(e_{\mathrm{a}}\).

4.4.1.2 Deactivation/return rate \(\Gamma\).

Introduce a nonnegative return functional \[\Gamma:\Omega\times\mathbb{R}\times[0,1]\to[0,\infty), \qquad (x,t,u)\mapsto \Gamma(x,t;u), \qquad \delta(\Gamma)=T^{-1},\] and interpret \(\Gamma(x,t;e_{\mathrm{a}}(x,t))\) as the rate of conversion from active to stored content. We require (minimal admissibility): \[\Gamma(x,t;u)\ge 0\ \ \forall u\in[0,1],\qquad \Gamma(x,t;0)=0.\]

4.4.1.3 Optional saturation bound (admissible hypothesis class).

If saturation is imposed (often used as a gate-ready hypothesis), require: \[0\le \Gamma(x,t;u)\le \Gamma_{\max}(x,t)\quad \forall u\in[0,1],\] for some measurable \(\Gamma_{\max}\ge 0\) with \(\delta(\Gamma_{\max})=T^{-1}\). Whether saturation is required is a model choice; if used in claims, it must be declared explicitly.

4.4.2.1 Axiom (Conversion / phase exchange without net creation).

The stored and active fractions satisfy: \[\begin{aligned} \partial_t \rho(x,t) &= -\mu(x,t)\,\rho(x,t) + \Gamma\big(x,t; e_{\mathrm{a}}(x,t)\big), \label{eq:conversion_rho}\\ \partial_t e_{\mathrm{a}}(x,t) + \nabla\cdot \mathbf{S}(x,t) &= +\mu(x,t)\,\rho(x,t) - \Gamma\big(x,t; e_{\mathrm{a}}(x,t)\big). \label{eq:conversion_ea}\end{aligned}\] Equations [eq:conversion_rho]–[eq:conversion_ea] fix the meaning and sign conventions of \(\mu\) and \(\Gamma\).

4.4.2.2 Ledger compatibility.

Summing [eq:conversion_rho] and [eq:conversion_ea] yields: \[\partial_t(\rho+e_{\mathrm{a}})+\nabla\cdot \mathbf{S} = 0,\] which is exactly the local ledger axiom [eq:ledger_local] with \(e_{\mathrm{tot}}=\rho+e_{\mathrm{a}}\).

4.5.1.1 Primitive mixing intensity.

Introduce a mixing intensity field \[\lambda:\Omega\times\mathbb{R}\to[0,\infty), \qquad \delta(\lambda)=0,\] interpreted as a nonnegative, dimensionless measure of local mixing strength (randomization/isotropization tendency) relative to alignment tendency. Larger \(\lambda\) corresponds to stronger mixing dominance; smaller \(\lambda\) corresponds to stronger alignment dominance. The exact dynamical role of \(\lambda\) is model-dependent and is specified in later Parts (closures/regime maps), but \(\lambda\) itself is a primitive scalar field with fixed meaning and range.

4.5.1.2 Convenient bounded mixing weight.

For later use it is often convenient to define a bounded weight \[w_\lambda(x,t):=\frac{\lambda(x,t)}{1+\lambda(x,t)}\in[0,1).\] This is a deterministic re-parameterization; it does not add content, but it avoids unbounded coefficients in interpolations.

4.5.2.1 Bias/orientation function.

Introduce a measurable function \[b:\Omega\times\mathcal{V}\times\mathbb{R}\to \mathbb{R}^3, \qquad (x,v,t)\mapsto b(x,v,t),\] interpreted as a bounded “orientation bias” associated to each velocity state \(v\) at position \(x\) and time \(t\).

4.5.2.2 Boundedness (LOCK admissibility).

Require \[\|b(x,v,t)\|\le 1\quad \forall(x,v,t).\] This bound is crucial: it implies that the alignment moment defined below is controlled by the active fraction \(e_{\mathrm{a}}\).

4.5.3.1 Definition.

Define the alignment moment (a vector field) \[\mathbf{m}_b(x,t):=\int_{\mathcal{V}} b(x,v,t)\, f(x,v,t)\,dv \in \mathbb{R}^3.\]

4.5.3.2 Bound (implied by admissibility).

By \(\|b\|\le 1\) and \(f\ge 0\), \[\|\mathbf{m}_b(x,t)\| \le \int_{\mathcal{V}}\|b(x,v,t)\| f(x,v,t)\,dv \le \int_{\mathcal{V}} f(x,v,t)\,dv = e_{\mathrm{a}}(x,t).\] Hence, whenever \(e_{\mathrm{a}}(x,t)\le 1\), we also have \(\|\mathbf{m}_b(x,t)\|\le 1\).

4.5.3.3 Normalized alignment vector and degree.

When \(e_{\mathrm{a}}(x,t)>0\), define: \[\mathbf{u}_b(x,t):=\frac{\mathbf{m}_b(x,t)}{e_{\mathrm{a}}(x,t)}\in \mathbb{R}^3, \qquad A(x,t):=\|\mathbf{u}_b(x,t)\|=\frac{\|\mathbf{m}_b(x,t)\|}{e_{\mathrm{a}}(x,t)}\in[0,1].\] When \(e_{\mathrm{a}}(x,t)=0\), set \(\mathbf{u}_b(x,t):=\mathbf{0}\) and \(A(x,t):=0\) by convention.

\(A(x,t)\) is the degree of alignment of the active fraction: \(A=0\) means no net alignment; \(A=1\) means maximal alignment consistent with \(\|b\|\le 1\).

4.5.4.1 Definition (alignment defect).

Define \[a_k(x,t):= \begin{cases} \dfrac{\|\mathbf{m}_{\perp}(x,t)\|}{e_{\mathrm{a}}(x,t)}, & e_{\mathrm{a}}(x,t)>0,\\[1.2ex] 0, & e_{\mathrm{a}}(x,t)=0. \end{cases}\] Then \(a_k(x,t)\in[0,1]\) follows from \(\|\mathbf{m}_{\perp}\|\le \|\mathbf{m}_b\|\le e_{\mathrm{a}}\).

4.5.4.2 Interpretation.

• \(a_k=0\) iff \(\mathbf{m}_b\) is parallel to \(k\) (perfect axis alignment),

• larger \(a_k\) indicates a larger transverse component of the alignment moment.

4.5.5.1 Axiom (Axisymmetric admissibility under alignment).

In an alignment-dominated regime (declared explicitly in §4.7 for each claim), we require that the bias is \(k\)-aligned in the sense that there exists a measurable scalar function \(\beta(x,v,t)\) such that \[b(x,v,t)=\beta(x,v,t)\,k(x,t), \qquad \text{with}\quad |\beta(x,v,t)|\le 1.\] This implies that \(\mathbf{m}_b\) is parallel to \(k\) and therefore \(a_k=0\) in the aligned limit, while allowing partial alignment through the magnitude of \(\beta\).

4.5.5.2 Axiom (Mixing-dominated admissibility under isotropy).

In a mixing-dominated regime, the bias is required to be sufficiently isotropic so that no preferred direction is selected by \(b\) alone. A minimal mathematically explicit condition is: \[\mathbf{m}_b(x,t)=\mathbf{0} \quad \text{for all $(x,t)$ in the declared mixing-dominated region.}\] Equivalently, \(A(x,t)=0\) and \(a_k(x,t)=0\) there. More refined mixing closures may relax this to small-but-nonzero \(\mathbf{m}_b\) under explicit hypotheses; any such refinement is declared outside this Part as a closure hypothesis.

4.6.1.1 Axiom (Finite local capacity).

At each \((x,t)\) there exists a finite local capacity per reference volume \(v^\ast\), normalized to unity, such that the phase fractions satisfy: \[e_{\mathrm{bg}}(x,t)+\rho(x,t)+e_{\mathrm{a}}(x,t)=1, \qquad 0\le e_{\mathrm{bg}},\rho,e_{\mathrm{a}}\le 1.\] This is the mathematical expression of the jammed-lattice (capacity-limited stage) assumption. It does not specify microscopic geometry beyond the existence of a fixed capacity.

4.6.1.2 Background variable introduction.

Under this capacity postulate, the background fraction is not independent; it is defined by the complement: \[e_{\mathrm{bg}}:=1-\rho-e_{\mathrm{a}}.\] Therefore \(e_{\mathrm{bg}}\) is meaningful precisely when (i) the normalization is meaningful (capacity is well-defined) and (ii) the actor phases \((\rho,e_{\mathrm{a}})\) are meaningful as fractions of the same capacity.

4.6.2.1 Condition (S1) Coarse-grained capacity.

There exists a coarse-graining length \(\ell\) (Part 03) such that capacity fluctuations below \(\ell\) are averaged out and the local capacity per reference volume is effectively constant. Formally, there exists \(\ell\) such that for coarse-grained fields, \[\overline{e}_{\mathrm{bg}}^{(\ell)}+\overline{\rho}^{(\ell)}+\overline{e}_{\mathrm{a}}^{(\ell)}=1,\] and each term lies in \([0,1]\).

4.6.2.2 Condition (S2) No net creation of actor content.

The ledger axiom [eq:ledger_local] applies for \(e_{\mathrm{tot}}=\rho+e_{\mathrm{a}}\) in the intended regime. If a later module proposes exchange between actor and stage beyond this ledger, it must explicitly modify the ledger structure and re-declare the normalization meaning (Major change if it alters core semantics).

4.6.2.3 Condition (S3) Compatible units and dimensions.

The unit-realization \((a,\Delta t,c,v^\ast)\) is fixed and dimensionally consistent with the chosen regime (Part 03). In particular, \(e_{\mathrm{a}},\rho,e_{\mathrm{bg}}\) remain dimensionless fractions.

4.7.1.1 (R0) Geometric regularity.

Control volumes \(V\subseteq\Omega\) used in ledger statements have Lipschitz boundaries, and the divergence theorem applies.

4.7.1.2 (R1) Kinetic admissibility.

The distribution \(f\) is measurable and satisfies \[f\ge 0,\qquad \int_{\mathcal{V}} f\,dv <\infty,\] and whenever \(\mathbf{S}\) or \(\mathbf{T}\) is used, the corresponding moment integrals are finite: \[\int_{\mathcal{V}} \|v\| f\,dv <\infty,\qquad \int_{\mathcal{V}} \|v\|^2 f\,dv <\infty.\]

4.7.1.3 (R2) Phase admissibility and normalization.

\[(e_{\mathrm{bg}},\rho,e_{\mathrm{a}})\in\mathcal{S} \quad \text{pointwise, where}\quad \mathcal{S}:=\{(u_1,u_2,u_3)\in[0,1]^3:\ u_1+u_2+u_3=1\}.\]

4.7.1.4 (R3) Flux integrability.

\(\mathbf{S}\) is integrable enough to define boundary fluxes: \[\int_{\partial V} |\mathbf{S}\cdot\mathbf{n}|\,dA <\infty\] for admissible control volumes \(V\) and times \(t\) in question.

4.7.1.5 (R4) Conversion admissibility.

\(\mu\ge 0\) is measurable with \(\delta(\mu)=T^{-1}\), and \(\Gamma(x,t;u)\ge 0\) is measurable in \((x,t)\) and measurable (or continuous) in \(u\in[0,1]\), with \(\delta(\Gamma)=T^{-1}\) and \(\Gamma(x,t;0)=0\).

4.7.1.6 (R5) Mixing/alignment admissibility.

\(\lambda\ge 0\) is measurable and dimensionless, \(k\) is measurable with \(\|k\|=1\), and \(b\) is measurable with \(\|b\|\le 1\), so that \(\mathbf{m}_b\) exists and satisfies \(\|\mathbf{m}_b\|\le e_{\mathrm{a}}\).

4.7.1.7 (R6) Scale separation for continuum usage (when invoked).

When continuum differential statements are used as approximations to micro dynamics, a scale-separation parameter \[\varepsilon:=\frac{a}{L}\ll 1\] must be declared (Part 03), where \(L\) is the characteristic variation scale of the fields in the claim. If no such \(L\) is declared, claims must be restricted to integral (control-volume) statements only.

5.1.2.1 Stored fraction (stored actor content).

\[\rho:\Omega\times\mathbb{R}\to[0,1],\qquad \rho=\rho(x,t).\]

5.1.2.2 Active fraction (moving/transport-capable actor content).

The active fraction is defined kinetically as the zeroth velocity moment of the distribution \(f\): \[e_{\mathrm{a}}(x,t) := \int_{\mathcal{V}} f(x,v,t)\,dv, \qquad 0\le e_{\mathrm{a}}(x,t)\le 1.\]

5.1.2.3 Total actor fraction.

\[e_{\mathrm{tot}}(x,t) := \rho(x,t)+e_{\mathrm{a}}(x,t), \qquad 0\le e_{\mathrm{tot}}(x,t)\le 1.\]

5.1.2.4 Background fraction (stage complement).

\[e_{\mathrm{bg}}(x,t) := 1-e_{\mathrm{tot}}(x,t)=1-\rho(x,t)-e_{\mathrm{a}}(x,t), \qquad 0\le e_{\mathrm{bg}}(x,t)\le 1.\] Thus \(e_{\mathrm{bg}}\) is a defined quantity once \((\rho,e_{\mathrm{a}})\) are known, but it is still convenient to treat it as a state component because it appears in constraints and (in later Parts) may gate regime transitions.

5.1.3.1 Interpretation (informal but fixed).

• \(\rho\) measures actor content that is stored (non-transporting at the continuum level).

• \(e_{\mathrm{a}}\) measures actor content that is active (transport-capable) and is represented kinetically by \(f\).

• \(e_{\mathrm{bg}}\) measures remaining background/stage capacity fraction (the complement to unity).

5.1.3.2 Bookkeeping invariants (must be preserved).

\[e_{\mathrm{bg}}+\rho+e_{\mathrm{a}}=1, \qquad e_{\mathrm{tot}}=\rho+e_{\mathrm{a}}, \qquad e_{\mathrm{bg}}=1-e_{\mathrm{tot}}.\] Any model closure or numerical method must preserve these identities (at least within declared tolerances); otherwise it fails the admissibility gate.

5.2.2.1 Zeroth moment: active fraction.

\[e_{\mathrm{a}}(x,t):=\int_{\mathcal{V}} f(x,v,t)\,dv.\]

5.2.2.2 First moment: flux vector.

\[\mathbf{S}(x,t):=\int_{\mathcal{V}} v\,f(x,v,t)\,dv \in \mathbb{R}^3.\] \(\mathbf{S}\) is the canonical actor flux associated with the total actor content (see §5.2.4).

5.2.2.3 Second moment: transport tensor.

\[\mathbf{T}(x,t):=\int_{\mathcal{V}} (v\otimes v)\,f(x,v,t)\,dv.\] \(\mathbf{T}\) is a symmetric, positive semidefinite tensor whenever the integral exists (see §5.2.6).

5.2.4.1 Outward flux is positive.

The boundary flux of actor content is \[\int_{\partial V} \mathbf{S}\cdot\mathbf{n}\,dA.\] By convention:

• \(\mathbf{S}\cdot\mathbf{n}>0\) contributes to outflow from \(V\),

• \(\mathbf{S}\cdot\mathbf{n}<0\) contributes to inflow into \(V\).

5.2.4.2 Inflow/outflow decomposition (useful identity).

Define pointwise on \(\partial V\): \[(\mathbf{S}\cdot\mathbf{n})_{+}:=\max(\mathbf{S}\cdot\mathbf{n},0), \qquad (\mathbf{S}\cdot\mathbf{n})_{-}:=\max(-\mathbf{S}\cdot\mathbf{n},0).\] Then \[\mathbf{S}\cdot\mathbf{n} = (\mathbf{S}\cdot\mathbf{n})_{+} - (\mathbf{S}\cdot\mathbf{n})_{-},\] and the net outward flux equals outflow minus inflow.

5.2.6.1 Symmetry and positive semidefiniteness.

For any \(\xi\in\mathbb{R}^3\), \[\xi^\top \mathbf{T}(x,t)\,\xi = \int_{\mathcal{V}} (\xi\cdot v)^2\, f(x,v,t)\,dv \ \ge\ 0,\] so \(\mathbf{T}(x,t)\) is symmetric and positive semidefinite.

5.2.6.2 Cauchy–Schwarz inequality for the flux.

Using \(f\ge 0\), \[\|\mathbf{S}\|^2 = \Big\|\int_{\mathcal{V}} v f\,dv\Big\|^2 \le \Big(\int_{\mathcal{V}} f\,dv\Big)\Big(\int_{\mathcal{V}}\|v\|^2 f\,dv\Big) = e_{\mathrm{a}}\ \mathrm{tr}(\mathbf{T}),\] where \[\mathrm{tr}(\mathbf{T})=\int_{\mathcal{V}}\|v\|^2 f\,dv.\] Equivalently, \[\|\mathbf{u}\|^2 \le \frac{\mathrm{tr}(\mathbf{T})}{e_{\mathrm{a}}}\quad (e_{\mathrm{a}}>0).\]

5.2.6.3 Positive semidefiniteness of the covariance tensor.

For any \(\xi\in\mathbb{R}^3\) and \(e_{\mathrm{a}}>0\), \[\xi^\top \boldsymbol{\Sigma}\,\xi = \frac{1}{e_{\mathrm{a}}}\int_{\mathcal{V}} \big(\xi\cdot (v-\mathbf{u})\big)^2 f\,dv \ \ge\ 0,\] so \(\boldsymbol{\Sigma}\succeq 0\).

5.2.6.4 Bounded-speed consequences (candidate constraint, if \(\mathcal{V}\) is bounded).

If the regime assumes \(\|v\|\le c_{\max}\) for all \(v\in\mathcal{V}\), then: \[\|\mathbf{S}(x,t)\| \le \int_{\mathcal{V}}\|v\| f\,dv \le c_{\max}\int_{\mathcal{V}} f\,dv = c_{\max} e_{\mathrm{a}}(x,t),\] and \[\mathrm{tr}(\mathbf{T})(x,t) = \int_{\mathcal{V}}\|v\|^2 f\,dv \le c_{\max}^2 e_{\mathrm{a}}(x,t).\] Moreover, in positive semidefinite order, \[\mathbf{T}(x,t) = \int_{\mathcal{V}} (v\otimes v)\,f\,dv \preceq \int_{\mathcal{V}} \|v\|^2 \mathbf{I}\,f\,dv \preceq c_{\max}^2 e_{\mathrm{a}}(x,t)\,\mathbf{I}.\] If additionally one identifies \(c_{\max}=c\) (Part 03), then the strongest bound becomes the canonical “flux limit” \[\|\mathbf{S}\|\le c\, e_{\mathrm{a}}.\]

5.3.3.1 Alignment defect.

Define the transverse (axis-misalignment) defect: \[a_k(x,t):= \begin{cases} \dfrac{\|\mathbf{m}_{\perp}(x,t)\|}{e_{\mathrm{a}}(x,t)}, & e_{\mathrm{a}}(x,t)>0,\\[1.0ex] 0, & e_{\mathrm{a}}(x,t)=0. \end{cases}\] Then \(a_k(x,t)\in[0,1]\).

5.3.3.2 Pythagorean relation (normalized).

For \(e_{\mathrm{a}}>0\), let \(\mathbf{u}_b=\mathbf{m}_b/e_{\mathrm{a}}\). Decompose \[\mathbf{u}_b = (\mathbf{u}_b\cdot k)\,k + \big(\mathbf{u}_b-(\mathbf{u}_b\cdot k)\,k\big).\] Then \[A(x,t)^2 = \|\mathbf{u}_b(x,t)\|^2 = \underbrace{(\mathbf{u}_b\cdot k)^2}_{=\xi(x,t)^2} + \underbrace{\|\mathbf{u}_b-(\mathbf{u}_b\cdot k)\,k\|^2}_{=a_k(x,t)^2},\] so \[A^2 = \xi^2 + a_k^2\qquad (e_{\mathrm{a}}>0).\] This relation is useful as an internal consistency gate.

5.3.4.1 Axisymmetry/alignment condition (canonical form).

In an aligned (axisymmetric) regime, require that there exists a scalar \(\beta(x,v,t)\) such that \[b(x,v,t)=\beta(x,v,t)\,k(x,t), \qquad |\beta(x,v,t)|\le 1.\] Then \(\mathbf{m}_b\) is parallel to \(k\) and thus \[a_k(x,t)=0, \qquad A(x,t)=|\xi(x,t)|.\]

5.3.4.2 Mixing-dominated (isotropic) condition (minimal form).

In a strongly mixing-dominated regime, impose \[\mathbf{m}_b(x,t)=\mathbf{0},\] so that \[A(x,t)=0,\qquad \xi(x,t)=0,\qquad a_k(x,t)=0.\] Any weaker mixing statement (e.g. “\(\|\mathbf{m}_b\|\) is small but nonzero”) must be declared explicitly as a hypothesis and must be gated quantitatively (thresholds fixed by pre-registration).

5.4.1.1 Integral (control-volume) form.

For any control volume \(V\subseteq\Omega\), \[\frac{d}{dt}\int_V q(x,t)\,dx \;+\; \int_{\partial V} \mathbf{J}_q(x,t)\cdot \mathbf{n}(x)\,dA \;=\; \int_V R_q(x,t)\,dx. \label{eq:generic_balance_integral}\] Here \(\mathbf{J}_q\cdot\mathbf{n}>0\) counts as outward flux (loss from \(V\)), consistent with §5.2.4.

5.4.1.2 Local (differential) form (under regularity).

If \(q\) and \(\mathbf{J}_q\) have sufficient regularity, [eq:generic_balance_integral] corresponds to \[\partial_t q(x,t) + \nabla\cdot \mathbf{J}_q(x,t) = R_q(x,t) \label{eq:generic_balance_local}\] in the appropriate weak/strong sense.

5.4.4.1 (BC1) No-through-flow (closed boundary).

\[\mathbf{S}\cdot\mathbf{n}=0\quad \text{on }\partial\Omega.\]

5.4.4.2 (BC2) Prescribed normal flux (Neumann-type).

\[\mathbf{S}\cdot\mathbf{n}=s_N(x,t)\quad \text{on }\partial\Omega,\] where \(s_N\) is a given function (positive = net outflow, negative = net inflow).

5.4.4.3 (BC3) Mixed/feedback flux.

\[\mathbf{S}\cdot\mathbf{n}=F\big(x,t;\mathbf{e}(x,t)\big) \quad \text{on }\partial\Omega,\] with \(F\) declared as a hypothesis/implementation choice and gated.

5.5.1.1 Sign convention (canonical).

• \(\Xi_{\mathrm{EV}}(x,t)>0\): background capacity is converted into actor capacity (stage \(\rightarrow\) actor),

• \(\Xi_{\mathrm{EV}}(x,t)<0\): actor capacity is absorbed into background (actor \(\rightarrow\) stage).

This sign convention is locked as a bookkeeping rule.

5.5.1.2 Canonical placement in the actor ledger.

The total actor fraction equation becomes \[\partial_t e_{\mathrm{tot}}(x,t) + \nabla\cdot \mathbf{S}(x,t) = \Xi_{\mathrm{EV}}(x,t). \label{eq:etot_with_Xi}\] Equivalently, in control-volume form: \[\frac{d}{dt}\int_V e_{\mathrm{tot}}\,dx + \int_{\partial V}\mathbf{S}\cdot\mathbf{n}\,dA = \int_V \Xi_{\mathrm{EV}}\,dx.\] Setting \(\Xi_{\mathrm{EV}}\equiv 0\) recovers the closed actor ledger axiom.

5.5.1.3 Canonical induced placement in the background equation.

Since \(e_{\mathrm{bg}}=1-e_{\mathrm{tot}}\), [eq:etot_with_Xi] implies \[\partial_t e_{\mathrm{bg}}(x,t) = \nabla\cdot \mathbf{S}(x,t) - \Xi_{\mathrm{EV}}(x,t). \label{eq:ebg_with_Xi}\] This is not an independent postulate; it is the bookkeeping identity required by normalization.

5.5.1.4 How \(\Xi_{\mathrm{EV}}\) splits between stored and active (optional split parameter).

If one needs a canonical split of \(\Xi_{\mathrm{EV}}\) between \(\rho\) and \(e_{\mathrm{a}}\), introduce a split weight \[\eta(x,t)\in[0,1]\] and place the terms as: \[\begin{aligned} \partial_t \rho &= -\mu\rho + \Gamma(x,t;e_{\mathrm{a}}) + (1-\eta)\,\Xi_{\mathrm{EV}}, \label{eq:rho_with_Xi}\\ \partial_t e_{\mathrm{a}} + \nabla\cdot \mathbf{S} &= +\mu\rho - \Gamma(x,t;e_{\mathrm{a}}) + \eta\,\Xi_{\mathrm{EV}}. \label{eq:ea_with_Xi}\end{aligned}\] Then summing [eq:rho_with_Xi] and [eq:ea_with_Xi] reproduces [eq:etot_with_Xi], and using \(e_{\mathrm{bg}}=1-\rho-e_{\mathrm{a}}\) reproduces [eq:ebg_with_Xi]. The choice of \(\eta\) is HYP (or SPEC if purely algorithmic) and must be declared and gated if it affects predictions.

5.5.2.1 Canonical placement (phase-resolved).

A standard placement consistent with the sign convention is: \[\begin{aligned} \partial_t u_{\mathrm{a}} + \nabla\cdot \mathbf{J}_{u,\mathrm{a}} &= \cdots - Q_{\mathrm{EV}} + \cdots, \label{eq:ua_with_Q}\\ \partial_t u_{\mathrm{bg}} + \nabla\cdot \mathbf{J}_{u,\mathrm{bg}} &= \cdots + Q_{\mathrm{EV}} + \cdots, \label{eq:ubg_with_Q}\end{aligned}\] so that (ignoring other terms) the exchange cancels in the total: \[(\partial_t u_{\mathrm{a}}+\partial_t u_{\mathrm{bg}})+\nabla\cdot(\mathbf{J}_{u,\mathrm{a}}+\mathbf{J}_{u,\mathrm{bg}})=0.\] If the background is assumed non-transporting in energy, one may set \(\mathbf{J}_{u,\mathrm{bg}}=\mathbf{0}\) as a modeling choice; this must be declared.

5.5.2.2 Relationship between Type V and Type E (optional coupling).

If occupancy exchange \(\Xi_{\mathrm{EV}}\) carries an energy cost/credit \(\chi(x,t)\) (energy per unit occupancy fraction), a consistent coupling is: \[Q_{\mathrm{EV}} = \chi(x,t)\,\Xi_{\mathrm{EV}}.\] Then \(\Xi_{\mathrm{EV}}>0\) (stage\(\rightarrow\)actor) corresponds to \(Q_{\mathrm{EV}}<0\) if \(\chi>0\) and the actor must receive energy to create actor capacity; conversely \(\Xi_{\mathrm{EV}}<0\) corresponds to \(Q_{\mathrm{EV}}>0\) if actor capacity is absorbed and releases energy to the stage. Which of these couplings is physically intended is a hypothesis, but the sign logic must be consistent with the above conventions.

5.5.2.3 Critical bookkeeping rule.

Type E terms must not be placed in the occupancy ledger unless they truly change occupancy fractions. Energy loss (e.g. attenuation/redshift-like effects) that does not change the amount of actor capacity is a Type E term and belongs in energy equations only. Conversely, any mechanism that changes \((e_{\mathrm{bg}},\rho,e_{\mathrm{a}})\) must appear as a Type V term (or within conversion terms) and must preserve normalization.

5.6.1.1 Kinetic positivity.

\[f(x,v,t)\ge 0.\]

5.6.1.2 Phase bounds and normalization.

\[0\le e_{\mathrm{bg}}(x,t)\le 1,\quad 0\le \rho(x,t)\le 1,\quad 0\le e_{\mathrm{a}}(x,t)\le 1, \quad e_{\mathrm{bg}}+\rho+e_{\mathrm{a}}=1.\] Equivalently, \(\mathbf{e}(x,t)\in\mathcal{S}\).

5.6.1.3 Conversion nonnegativity constraints (for inward-pointing dynamics).

\[\mu(x,t)\ge 0,\qquad \Gamma(x,t;u)\ge 0,\qquad \Gamma(x,t;0)=0.\] These are minimal for preventing the conversion terms from forcing negativity at the boundaries \(\rho=0\) and \(e_{\mathrm{a}}=0\).

5.6.2.1 Finite-moment requirements.

Whenever used, require: \[\int_{\mathcal{V}} f\,dv<\infty,\qquad \int_{\mathcal{V}}\|v\| f\,dv<\infty,\qquad \int_{\mathcal{V}}\|v\|^2 f\,dv<\infty.\]

5.6.2.2 Transport tensor positivity.

\[\mathbf{T}(x,t)\succeq 0.\] This is automatically true if \(\mathbf{T}\) is defined as a velocity second moment of a nonnegative \(f\), but it becomes a nontrivial gate when closures approximate \(\mathbf{T}\).

5.6.2.3 Covariance positivity.

For \(e_{\mathrm{a}}>0\), \[\boldsymbol{\Sigma}=\frac{\mathbf{T}}{e_{\mathrm{a}}}-\mathbf{u}\otimes\mathbf{u}\succeq 0.\] This is a strong internal-consistency gate for closure models.

5.6.3.1 Candidate A (bounded support).

Assume \(\mathcal{V}\) is bounded: \[\mathcal{V}\subseteq \{v:\|v\|\le c_{\max}\}.\] Then signal/transport speed is bounded by \(c_{\max}\) at the kinetic level. If the model identifies \(c_{\max}=c=a/\Delta t\), this identification must be declared in the regime.

5.6.3.2 Candidate B (bounded mean speed).

If unbounded \(\mathcal{V}\) is used, one may impose a gate on the mean speed: \[\|\mathbf{u}(x,t)\|=\frac{\|\mathbf{S}(x,t)\|}{e_{\mathrm{a}}(x,t)} \le c_{\mathrm{eff}}(x,t),\] where \(c_{\mathrm{eff}}\) is a declared effective bound. This is a closure/gate choice, not an automatic consequence.

5.6.4.1 Flux bound from bounded support (candidate).

If \(\|v\|\le c_{\max}\) holds in the regime, then \[\|\mathbf{S}\|\le c_{\max}\,e_{\mathrm{a}}.\] In dimensionless units where \(c_{\max}=1\), this reads \(\|\hat{\mathbf{S}}\|\le e_{\mathrm{a}}\).

5.6.4.2 Flux bound from second moment (always valid when moments exist).

Even without bounded support, the Cauchy–Schwarz inequality yields: \[\|\mathbf{S}\|^2 \le e_{\mathrm{a}}\ \mathrm{tr}(\mathbf{T}).\] This inequality is useful for gating approximate closures: if a closure predicts \((e_{\mathrm{a}},\mathbf{S},\mathbf{T})\) that violates this inequality, it cannot correspond to any nonnegative distribution \(f\).

5.6.5.1 Bias bound (hard admissibility).

\[\|b(x,v,t)\|\le 1.\]

5.6.5.2 Alignment moment bound (implied gate).

\[\|\mathbf{m}_b(x,t)\|\le e_{\mathrm{a}}(x,t).\]

5.6.5.3 Normalized alignment bounds.

For \(e_{\mathrm{a}}>0\): \[A(x,t)=\frac{\|\mathbf{m}_b\|}{e_{\mathrm{a}}}\in[0,1], \qquad \xi(x,t)=\frac{\mathbf{m}_b\cdot k}{e_{\mathrm{a}}}\in[-1,1], \qquad a_k(x,t)\in[0,1], \qquad A^2=\xi^2+a_k^2.\] Any closure or numerical approximation that violates these bounds fails internal consistency.

6.2.4.1 No reverse injection (rule).

A closure is not allowed to inject observational targets directly by tuning \(\theta\) post hoc. The closure must be:

• specified before gate evaluation,

• dimensionally consistent,

• internally realizable (must not violate inequalities that follow from \(f\ge 0\)),

• associated with a declared regime of validity.

6.2.5.1 (T0) Scalar-only diffusion-level truncation (order \(N=0\)).

One evolves \(e_{\mathrm{a}}\) (and \(\rho\) through conversion) and assumes a constitutive flux law \[\mathbf{S}=\mathcal{J}(e_{\mathrm{a}},\nabla e_{\mathrm{a}},\dots).\] This is the drift-diffusion level.

6.2.5.2 (T1) First-moment truncation (order \(N=1\)).

One evolves \((e_{\mathrm{a}},\mathbf{S})\) (and \(\rho\) through conversion) and closes with a constitutive law for \(\mathbf{T}\): \[\mathbf{T}=\mathcal{T}(e_{\mathrm{a}},\mathbf{S};\theta_T).\] This is the minimal hyperbolic/relaxation level.

6.2.5.3 (T2) Second-moment truncation (order \(N=2\)).

One evolves \((e_{\mathrm{a}},\mathbf{S},\mathbf{T})\) and closes with \(\mathbf{M}^{(3)}\) (third moment). This is higher-fidelity and more expensive; it is optional and not required for the core mapping in this Part.

The remainder of PART 06 focuses on the (T1) core as the default minimal expressive closure, because it cleanly supports both isotropic (spherical) and axisymmetric (jet) regimes.

6.2.6.1 Phase equations (conversion + transport).

\[\begin{aligned} \partial_t \rho &= -\mu\rho + \Gamma(x,t;e_{\mathrm{a}}), \label{eq:core_rho_T1}\\ \partial_t e_{\mathrm{a}} + \nabla\cdot \mathbf{S} &= +\mu\rho - \Gamma(x,t;e_{\mathrm{a}}). \label{eq:core_ea_T1}\end{aligned}\]

6.2.6.2 Flux equation (momentum-like balance with relaxation).

\[\partial_t \mathbf{S} + \nabla\cdot \mathbf{T} = -\mathbf{B}\,\mathbf{S} + e_{\mathrm{a}}\,\mathbf{F}_{\mathrm{eff}} + \mathbf{R}_S^{\mathrm{ext}}. \label{eq:core_S_T1}\] Here:

• \(\mathbf{B}(x,t)\) is a (possibly scalar) relaxation-rate operator with dimension \(T^{-1}\); if isotropic, \(\mathbf{B}=B\,\mathbf{I}\),

• \(\mathbf{F}_{\mathrm{eff}}(x,t)\) is an effective “driving” field (dimension \(LT^{-2}\) if interpreted as acceleration),

• \(\mathbf{R}_S^{\mathrm{ext}}\) collects any declared external flux sources (optional; usually set to \(0\) in the closed core).

Equation [eq:core_S_T1] is a canonical place where “deficit” (effective potential) forcing enters (see §6.5).

6.2.6.3 Closure location.

At (T1), closure is precisely: \[\mathbf{T}=\mathcal{T}(e_{\mathrm{a}},\mathbf{S};\theta_T),\] subject to internal realizability constraints (PSD, Cauchy–Schwarz, limiting-regime consistency).

6.2.7.1 (G1) Dimensional consistency.

\[\delta(\mathbf{T})=L^2T^{-2}.\] If the closure uses a parameter \(\kappa_T\), then \(\delta(\kappa_T)=L^2T^{-2}\) and it must be distinct from any \(\kappa_{\mathrm{opt}}\) with \(\delta(\kappa_{\mathrm{opt}})=L^{-1}\) (Part 03).

6.2.7.2 (G2) PSD gate (realizability).

\[\mathbf{T}(x,t)\succeq 0.\]

6.2.7.3 (G3) Flux-moment inequality gate.

For any realizable triple \((e_{\mathrm{a}},\mathbf{S},\mathbf{T})\) (coming from some \(f\ge 0\)), \[\|\mathbf{S}\|^2 \le e_{\mathrm{a}}\,\mathrm{tr}(\mathbf{T}).\] A closure must not systematically violate this inequality.

6.2.7.4 (G4) Limiting-regime gates.

The closure must reduce to the correct limiting forms in regimes that are declared within its validity:

• isotropic mixing limit: \(\mathbf{T}\to p(e_{\mathrm{a}})\mathbf{I}\),

• aligned axisymmetric limit: \(\mathbf{T}\to p_\perp(e_{\mathrm{a}})(\mathbf{I}-k\otimes k)+p_\parallel(e_{\mathrm{a}})(k\otimes k)\),

• low-speed/strong-relaxation diffusion limit: \(\mathbf{S}\to\) drift-diffusion form (see §6.7).

6.3.4.1 No-flux static baseline.

If the solution is regular at \(r=0\) (or if one imposes \(S_r(0)=0\)), then necessarily \(C=0\) and \[S_r(r)\equiv 0. \label{eq:static_no_flux}\] In that case, the phase equations reduce to local conversion equilibrium: \[\mu(r)\,\rho(r)=\Gamma\big(r; e_{\mathrm{a}}(r)\big), \qquad e_{\mathrm{tot}}(r)=\rho(r)+e_{\mathrm{a}}(r)\ \text{is time-independent.} \label{eq:static_conversion_balance}\]

6.3.4.2 Static balance with deficit driving.

If a nonzero effective driving \(F_r\) exists, the steady flux equation [eq:spherical_Sr] yields \[\partial_r p(r) = -B(r)\,S_r(r) + e_{\mathrm{a}}(r)\,F_r(r) \quad (\text{if }R_{S,r}^{\mathrm{ext}}=0).\] In a strongly relaxing steady regime where \(B\) is large and \(S_r\) adjusts quasi-instantaneously, one obtains the drift-diffusion-type approximation: \[S_r(r)\approx -\frac{1}{B(r)}\partial_r p(r) + \frac{e_{\mathrm{a}}(r)}{B(r)}F_r(r). \label{eq:spherical_drift_diffusion}\] This expresses that the flux is driven by a “pressure” gradient and by an effective body-force term.

6.4.3.1 Divergence of the axisymmetric closure.

Writing \(\mathbf{T}\) in components, \[\mathbf{T}=p_\perp\,\mathbf{I} + (p_\parallel-p_\perp)\,k\otimes k,\] so \[\nabla\cdot \mathbf{T} = \nabla p_\perp + \nabla\cdot\big((p_\parallel-p_\perp)\,k\otimes k\big).\] If \(k=\hat{z}\) is constant in space, then \(\nabla\cdot(k\otimes k)=0\) and \[\nabla\cdot \mathbf{T} = \nabla p_\perp + \partial_z(p_\parallel-p_\perp)\,k.\] Thus the radial component is driven by \(\partial_R p_\perp\), and the axial component is driven by \(\partial_z p_\parallel\): \[(\nabla\cdot\mathbf{T})\cdot \hat{R}=\partial_R p_\perp, \qquad (\nabla\cdot\mathbf{T})\cdot k=\partial_z p_\parallel.\]

6.4.5.1 Axisymmetry and alignment connection.

In a strongly aligned jet regime one declares \(a_k\approx 0\) (PART 05) and may additionally assume that \(\mathbf{S}\) is predominantly parallel to \(k\): \[S_R=\mathcal{O}(\varepsilon_{\perp})\ll S_z,\] with an ordering parameter \(\varepsilon_{\perp}\) declared in the regime.

6.5.2.1 (i) Pressure-like gradients from the closure \(\nabla\cdot\mathbf{T}\).

If \(\mathbf{T}\) has an isotropic or anisotropic “pressure” structure (e.g. [eq:isotropic_closure] or [eq:axisym_T_closure]), then \(\nabla\cdot\mathbf{T}\) produces a gradient of a scalar (or two scalars) that behaves like a pressure gradient term.

6.5.2.2 (ii) Potential-like gradients from the deficit field \(-\nabla \Phi_{\mathrm{eff}}\).

This is an externally specified (or self-consistently computed in later Parts) gradient that produces drift.

The “volume-pressure” viewpoint is the statement that what appears as “effective gravity” in a reduced equation can be interpreted as the combined effect of:

• a closure-induced pressure-like gradient \(\nabla p\) (or \(\nabla p_\perp\) and \(\partial_z p_\parallel\)),

• a deficit-induced potential gradient \(e_{\mathrm{a}}(-\nabla \Phi_{\mathrm{eff}})\).

In a steady, strongly relaxing regime, the drift-diffusion approximation reads: \[\mathbf{S}\approx -\mathbf{B}^{-1}\big(\nabla\cdot\mathbf{T}\big) - \mathbf{B}^{-1}\big(e_{\mathrm{a}}\nabla\Phi_{\mathrm{eff}}\big) + \mathbf{B}^{-1}(e_{\mathrm{a}}\mathbf{F}_{\mathrm{other}}). \label{eq:drift_diffusion_general}\] For isotropic \(\mathbf{B}=B\mathbf{I}\) and isotropic closure \(\mathbf{T}=p(e_{\mathrm{a}})\mathbf{I}\) this becomes: \[\mathbf{S}\approx -\frac{1}{B}\nabla p(e_{\mathrm{a}}) - \frac{e_{\mathrm{a}}}{B}\nabla\Phi_{\mathrm{eff}} + \frac{e_{\mathrm{a}}}{B}\mathbf{F}_{\mathrm{other}}. \label{eq:drift_diffusion_isotropic}\] This is the mathematically exact location of the deficit term in the core: it is a drift term in the flux, not a source term in the continuity equation.

6.6.2.1 (S1) Replace bare occupancy symbol \(e\).

If a legacy core uses \[\partial_t e + \nabla\cdot \mathbf{S}=0,\] then in upgraded notation it must be rewritten as \[\partial_t e_{\mathrm{tot}} + \nabla\cdot\mathbf{S}=0,\] unless the legacy context explicitly indicates \(e=e_{\mathrm{a}}\) (active only). In that case, the correct upgraded equation is [eq:core_ea_T1] (with conversion terms), not a closed continuity equation.

6.6.2.2 (S2) Replace bare \(\kappa\).

Bare \(\kappa\) is forbidden. Every occurrence must be rewritten with a role-specific symbol and correct dimension: \[\kappa \mapsto \kappa_T,\ \kappa_\perp,\ \kappa_\parallel,\ \kappa_{\mathrm{opt}},\ldots\] with explicit statements such as \[\delta(\kappa_T)=L^2T^{-2},\qquad \delta(\kappa_{\mathrm{opt}})=L^{-1}.\]

6.6.2.3 (S3) Replace ambiguous “pressure” or “potential” terms.

Legacy “deficit” terms must be mapped into either:

• a closure pressure-like scalar \(p(e_{\mathrm{a}})\) inside \(\mathbf{T}\), or

• an effective potential \(\Phi_{\mathrm{eff}}\) entering as \(-\nabla\Phi_{\mathrm{eff}}\) in \(\mathbf{F}_{\mathrm{eff}}\),

not into a source term in the continuity equation.

6.6.2.4 (S4) Enforce phase bookkeeping.

Any legacy “reaction” that converts between stored and active must be expressed with \(\mu\) and \(\Gamma\) (or explicitly declared alternatives) in the phase equations [eq:core_rho_T1]–[eq:core_ea_T1]. It must cancel from the total continuity [eq:core_continuity_etot].

6.7.2.1 Closed actor conservation.

If \(\mathbf{S}\cdot\mathbf{n}=0\) on \(\partial\Omega\) (closed boundary), then integrating [eq:core_ledger_integral] over \(V=\Omega\) yields: \[\frac{d}{dt}\int_\Omega e_{\mathrm{tot}}\,dx = 0.\]

6.7.2.2 Conversion cancellation.

Summing [eq:core_conversion_rho] and [eq:core_conversion_ea] cancels \(\mu\rho\) and \(\Gamma\) exactly. Any model that fails this cancellation (e.g. by inconsistent signs) violates the meaning layer and fails the bookkeeping gate.

7.1.1.1 Isotropic linear closure (CL-ISO).

Define \[\mathbf{T}(x,t)=\kappa_T(x,t)\,e_{\mathrm{a}}(x,t)\,\mathbf{I}, \qquad \kappa_T(x,t)\ge 0, \qquad \delta(\kappa_T)=L^2T^{-2}. \label{eq:part07_T_iso}\] This closure is admissible (PSD) because \(\mathbf{I}\succeq 0\) and \(\kappa_T e_{\mathrm{a}}\ge 0\) implies \(\mathbf{T}\succeq 0\).

7.1.1.2 Divergence of the isotropic closure.

Using \(\nabla\cdot(p\mathbf{I})=\nabla p\) for a scalar field \(p\), we obtain: \[\nabla\cdot\mathbf{T}=\nabla\big(\kappa_T e_{\mathrm{a}}\big) =\kappa_T\nabla e_{\mathrm{a}} + e_{\mathrm{a}}\nabla \kappa_T. \label{eq:part07_div_T_iso}\]

7.1.1.3 Relaxation structure for isotropic regimes.

In mixing-dominated isotropic regimes it is standard to use an isotropic relaxation operator \[\mathbf{B}(x,t)=B(x,t)\,\mathbf{I}, \qquad B(x,t)>0, \qquad \delta(B)=T^{-1}. \label{eq:part07_B_iso}\] Anisotropic \(\mathbf{B}\) is treated in §7.4–§7.5.

7.1.2.1 Diffusion coefficient and drift velocity.

Define the effective diffusion coefficient \[D(x,t):=\frac{\kappa_T(x,t)}{B(x,t)}, \qquad \delta(D)=\frac{L^2T^{-2}}{T^{-1}}=L^2T^{-1}, \label{eq:part07_D_def}\] and define the effective drift velocity \[\mathbf{u}_{\mathrm{eff}}(x,t):= \frac{1}{B(x,t)}\Big(\mathbf{F}_{\mathrm{eff}}(x,t)-\nabla\kappa_T(x,t)\Big), \qquad \delta(\mathbf{u}_{\mathrm{eff}})=LT^{-1}. \label{eq:part07_ueff_def}\] Then [eq:part07_S_iso_expanded] can be written as the standard drift-diffusion form: \[\mathbf{S}\approx -D\,\nabla e_{\mathrm{a}} + e_{\mathrm{a}}\,\mathbf{u}_{\mathrm{eff}}. \label{eq:part07_S_drift_diffusion}\] If \(\kappa_T\) is spatially constant, \(\mathbf{u}_{\mathrm{eff}}=\mathbf{F}_{\mathrm{eff}}/B\).

7.1.3.1 Gate implication: positivity and maximum principle (model-dependent).

Equation [eq:part07_ea_drift_diffusion_reaction] is not automatically positivity-preserving at the PDE level unless further conditions hold (e.g. appropriate boundary conditions, \(D\ge 0\), inward-pointing reaction terms). Therefore, positivity must be enforced as a gate at the solution/implementation level.

7.2.2.1 Lemma 7.1 (Axisymmetry \(\Rightarrow \mathbf{m}_b\parallel k\)).

Fix \((x,t)\) and a unit axis \(k=k(x,t)\). Assume:

1. (Axisymmetric distribution) For all \(Q\in G_k\) and all \(v\in\mathcal{V}\), \[f(x,Qv,t)=f(x,v,t). \label{eq:part07_axisym_f}\]

2. (Rotation-covariant bias) For all \(Q\in G_k\) and all \(v\in\mathcal{V}\), \[b(x,Qv,t)=Q\,b(x,v,t). \label{eq:part07_axisym_b_covariant}\]

Then the alignment moment \[\mathbf{m}_b(x,t)=\int_{\mathcal{V}} b(x,v,t)\,f(x,v,t)\,dv\] is parallel to \(k\), i.e. \[\mathbf{m}_b(x,t)=m_\parallel(x,t)\,k(x,t)\quad \text{for some scalar }m_\parallel. \label{eq:part07_mb_parallel}\]

7.2.2.2 Proof.

Fix any \(Q\in G_k\). Perform the change of variables \(w:=Qv\) (Jacobian \(=1\) because \(Q\in SO(3)\)). Then \[\begin{aligned} \mathbf{m}_b &=\int_{\mathcal{V}} b(x,v,t)\,f(x,v,t)\,dv\\ &=\int_{\mathcal{V}} b(x,Q^{-1}w,t)\,f(x,Q^{-1}w,t)\,dw\\ &=\int_{\mathcal{V}} Q^{-1}b(x,w,t)\,f(x,w,t)\,dw \quad \text{(by \eqref{eq:part07_axisym_b_covariant} and \eqref{eq:part07_axisym_f})}\\ &=Q^{-1}\int_{\mathcal{V}} b(x,w,t)\,f(x,w,t)\,dw =Q^{-1}\mathbf{m}_b.\end{aligned}\] Thus \(Q\mathbf{m}_b=\mathbf{m}_b\) for all \(Q\in G_k\). The only vectors fixed by all rotations about \(k\) are scalar multiples of \(k\). Therefore \(\mathbf{m}_b\parallel k\). \(\square\)

7.2.2.3 Corollary (alignment defect vanishes).

If \(e_{\mathrm{a}}(x,t)>0\), then \(a_k(x,t)=\|\mathbf{m}_b-(\mathbf{m}_b\cdot k)k\|/e_{\mathrm{a}}=0\).

7.3.2.1 (D1) Alignment-smallness diagnostic.

For \(e_{\mathrm{a}}>0\) define \(A=\|\mathbf{m}_b\|/e_{\mathrm{a}}\). Mixing dominance expects \[A(x,t)\le \varepsilon_A^{\mathrm{mix}} \quad\text{(small alignment)}. \label{eq:part07_mix_A_small}\] If \(e_{\mathrm{a}}=0\) set \(A=0\).

7.3.2.2 (D2) Tensor anisotropy diagnostic (if \(\mathbf{T}\) is available or reconstructed).

Define the isotropic projection \[\mathbf{T}_{\mathrm{iso}}:=\frac{1}{3}\mathrm{tr}(\mathbf{T})\,\mathbf{I}.\] Define the dimensionless anisotropy index \[\mathcal{A}_T(x,t) := \begin{cases} \dfrac{\|\mathbf{T}(x,t)-\mathbf{T}_{\mathrm{iso}}(x,t)\|_F}{\mathrm{tr}(\mathbf{T}(x,t))}, & \mathrm{tr}(\mathbf{T})>0,\\[1.2ex] 0, & \mathrm{tr}(\mathbf{T})=0, \end{cases} \label{eq:part07_AT_def}\] where \(\|\cdot\|_F\) is the Frobenius norm. Mixing dominance expects \[\mathcal{A}_T(x,t)\le \varepsilon_T^{\mathrm{mix}}. \label{eq:part07_mix_T_isotropic}\]

7.3.2.3 (D3) Diffusion-residual diagnostic (if the diffusion approximation is claimed).

If one claims the diffusion-limit flux law [eq:part07_S_drift_diffusion], define the residual \[\mathbf{R}_{\mathrm{DD}}:=\mathbf{S} + D\nabla e_{\mathrm{a}} - e_{\mathrm{a}}\mathbf{u}_{\mathrm{eff}}, \label{eq:part07_R_DD_def}\] and the normalized error \[\eta_{\mathrm{DD}}(x,t):= \frac{\|\mathbf{R}_{\mathrm{DD}}(x,t)\|}{\|\mathbf{S}(x,t)\| + \eta_0}, \qquad \eta_0>0\ \text{(small fixed regularizer)}. \label{eq:part07_eta_DD_def}\] Mixing-dominated diffusion expects \[\eta_{\mathrm{DD}}(x,t)\le \varepsilon_{\mathrm{DD}}^{\mathrm{mix}} \quad \text{in the declared mixing-dominated region.} \label{eq:part07_mix_DD_good}\]

7.3.3.1 Default closure choice in \(\mathcal{R}_{\mathrm{mix}}\).

Use CL-ISO [eq:part07_T_iso] and (optionally) the diffusion reduction [eq:part07_S_drift_diffusion] if [eq:part07_strong_relax_ordering] and [eq:part07_mix_DD_good] are satisfied.

7.4.1.1 Flux-alignment cosine.

Define \[c_S(x,t):= \frac{\mathbf{S}(x,t)\cdot k(x,t)}{\|\mathbf{S}(x,t)\|+\eta_0}, \qquad c_S\in[-1,1], \label{eq:part07_cS_def}\] where \(\eta_0>0\) is a fixed regularizer. Large \(c_S\) indicates a flux aligned with \(k\).

7.4.1.2 Transverse leakage ratio.

Define \[\ell_\perp(x,t):= \frac{\|\mathbf{S}_\perp(x,t)\|}{\|\mathbf{S}(x,t)\|+\eta_0}\in[0,1]. \label{eq:part07_leak_ratio}\] In a perfectly collimated regime \(\ell_\perp\approx 0\); in mixing-dominated isotropy \(\ell_\perp\) is not systematically small.

7.4.1.3 Partial-alignment range (diagnostic).

A partial-alignment region is characterized by intermediate alignment: \[\varepsilon_A^{\mathrm{mix}} < A(x,t) < 1-\varepsilon_A^{\mathrm{jet}} \quad\text{or}\quad a_k(x,t)\gtrsim \varepsilon_k^{\mathrm{jet}}, \label{eq:part07_partial_alignment_range}\] together with a nontrivial but imperfect flux alignment (typical but not mandatory): \[c_S\ \text{moderate-to-large},\qquad \ell_\perp\ \text{non-negligible}.\]

7.4.2.1 (M1) Channel (solid-core tube) diagnostic.

Define the radial profile at fixed \((z,t)\): \[\bar{e}_{\mathrm{a}}(R;z,t):=\frac{1}{2\pi}\int_0^{2\pi} e_{\mathrm{a}}(R,\phi,z,t)\,d\phi \quad (\text{axisymmetry proxy}).\] A solid-core channel at \((z,t)\) is diagnosed if \[R_{\mathrm{peak}}(z,t):=\arg\max_{R\in[0,R_{\max}]}\bar{e}_{\mathrm{a}}(R;z,t) \approx 0 \quad\text{and}\quad \bar{e}_{\mathrm{a}}(0;z,t)\ \text{is significantly above background}. \label{eq:part07_channel_peak}\] Operationally one enforces \[R_{\mathrm{peak}}(z,t)\le \varepsilon_R^{\mathrm{core}}, \qquad \bar{e}_{\mathrm{a}}(0;z,t)\ge e_{\min}^{\mathrm{core}}.\]

7.4.2.2 (M2) Hollow tube (channel-with-hole) diagnostic.

A hollow tube is diagnosed if the radial profile is not maximal at the axis: \[R_{\mathrm{peak}}(z,t)\ge R_{\min}^{\mathrm{hole}}>0, \label{eq:part07_hollow_peak}\] and the axis value is suppressed: \[H(z,t):=1-\frac{\bar{e}_{\mathrm{a}}(0;z,t)}{\max_{R}\bar{e}_{\mathrm{a}}(R;z,t)+\eta_0} \ge H_{\min}^{\mathrm{hole}}. \label{eq:part07_hollowness}\] Here \(H\) is a “hollowness index” (\(H\approx 1\) means a deep hole).

7.4.2.3 (M3) Corona / shell diagnostic (spherical geometry).

In nearly spherical settings, define the spherical average: \[\langle e_{\mathrm{a}}\rangle(r,t):=\frac{1}{4\pi}\int_{\mathbb{S}^2} e_{\mathrm{a}}(r\omega,t)\,d\omega.\] A corona/shell is diagnosed if \(\langle e_{\mathrm{a}}\rangle(r,t)\) has a maximum at \(r=r_c(t)>0\) and is suppressed near \(r=0\): \[r_c(t):=\arg\max_{r\in[0,r_{\max}]}\langle e_{\mathrm{a}}\rangle(r,t), \qquad r_c(t)\ge r_{\min}^{\mathrm{shell}}, \label{eq:part07_shell_peak}\] \[H_{\mathrm{sph}}(t):=1-\frac{\langle e_{\mathrm{a}}\rangle(0,t)}{\max_{r}\langle e_{\mathrm{a}}\rangle(r,t)+\eta_0} \ge H_{\min}^{\mathrm{shell}}. \label{eq:part07_shell_hollowness}\]

7.4.3.1 Axisymmetric anisotropic form (baseline).

Use the standard axisymmetric tensor structure (PART 06): \[\mathbf{T} = p_\perp\,(\mathbf{I}-k\otimes k) + p_\parallel\,(k\otimes k), \qquad p_\perp\ge 0,\ p_\parallel\ge 0. \label{eq:part07_T_axis_form}\] To parameterize anisotropy while keeping the trace controlled, define a baseline scalar “pressure”: \[p(x,t):=\kappa_T(x,t)\,e_{\mathrm{a}}(x,t), \qquad \kappa_T\ge 0,\ \delta(\kappa_T)=L^2T^{-2},\] and introduce a dimensionless anisotropy parameter \(\alpha(x,t)\) via \[p_\perp:=p\,(1-\alpha),\qquad p_\parallel:=p\,(1+2\alpha). \label{eq:part07_pperp_ppar_alpha}\] Then \[\mathrm{tr}(\mathbf{T})=2p_\perp+p_\parallel=3p,\] so \(\alpha\) redistributes stress between transverse and longitudinal directions while preserving the total trace scale.

7.4.3.2 PSD constraints on \(\alpha\).

Because \(p\ge 0\), the PSD requirement is equivalent to \[1-\alpha\ge 0,\qquad 1+2\alpha\ge 0 \quad\Longleftrightarrow\quad -\frac{1}{2}\le \alpha\le 1. \label{eq:part07_alpha_bounds}\]

7.4.3.3 Interpolation rule: suppress anisotropy under strong mixing.

Define the mixing weight \(w_\lambda=\lambda/(1+\lambda)\) and define a bounded alignment-to-anisotropy map \(g_A:[0,1]\to[0,1]\). A simple gate-friendly choice is: \[g_A(A):= \begin{cases} 0, & A\le A_0,\\[0.5ex] \dfrac{A-A_0}{1-A_0}, & A_0< A < 1,\\[1.0ex] 1, & A=1, \end{cases} \qquad A_0\in[0,1). \label{eq:part07_gA_def}\] Then define \[\alpha(x,t):=\alpha_{\max}\,\big(1-w_\lambda(x,t)\big)\,g_A\big(A(x,t)\big), \qquad \alpha_{\max}\in[0,1]. \label{eq:part07_alpha_interp}\] This ensures:

• if \(\lambda\) is large (\(w_\lambda\approx 1\)), then \(\alpha\approx 0\) and \(\mathbf{T}\approx p\mathbf{I}\) (isotropy);

• if alignment is weak (\(A\le A_0\)), then \(\alpha=0\) (isotropy);

• if alignment is strong and mixing is weak, \(\alpha\) can approach \(\alpha_{\max}\) (anisotropy).

Because \(\alpha_{\max}\le 1\), [eq:part07_alpha_bounds] holds (for nonnegative \(\alpha\)) and \(\mathbf{T}\succeq 0\) is guaranteed.

7.4.3.4 Resulting closure (CL-PA: partial-alignment closure).

Combine [eq:part07_T_axis_form]–[eq:part07_alpha_interp] to define: \[\mathbf{T} = \kappa_T e_{\mathrm{a}} \Big[(1-\alpha)(\mathbf{I}-k\otimes k)+(1+2\alpha)(k\otimes k)\Big], \qquad \alpha=\alpha_{\max}(1-w_\lambda)g_A(A). \label{eq:part07_T_partial_alignment_closure}\] This closure is a HYP unless \(\alpha_{\max},A_0\) and the functional form are locked for a version.

7.4.3.5 Optional anisotropic relaxation (consistent with partial alignment).

Similarly one may allow \[\mathbf{B}=B_\perp(\mathbf{I}-k\otimes k)+B_\parallel(k\otimes k), \qquad B_\perp>0,\ B_\parallel>0. \label{eq:part07_B_aniso}\] This is a modeling choice that affects jet collimation diagnostics in §7.5.

7.5.1.1 (J1) High alignment degree.

\[A(x,t)\ge 1-\varepsilon_A^{\mathrm{jet}}. \label{eq:part07_jet_A_high}\]

7.5.1.2 (J2) Small alignment defect.

\[a_k(x,t)\le \varepsilon_k^{\mathrm{jet}}. \label{eq:part07_jet_ak_small}\] Note that [eq:part07_A_identity] implies then \(|\xi|\approx A\approx 1\).

7.5.1.3 (J3) Flux collimation.

Using [eq:part07_cS_def] and [eq:part07_leak_ratio], \[c_S(x,t)\ge 1-\varepsilon_S^{\mathrm{jet}} \quad \text{and/or}\quad \ell_\perp(x,t)\le \varepsilon_\perp^{\mathrm{jet}}. \label{eq:part07_jet_flux_collimation}\]

7.5.1.4 (J4) Stress anisotropy (if \(\mathbf{T}\) is available).

For the axisymmetric form [eq:part07_T_axis_form], define the anisotropy ratio \[r_T(x,t):=\frac{p_\parallel(x,t)}{p_\perp(x,t)+\eta_0}. \label{eq:part07_rT_def}\] Strong alignment typically expects \(r_T\) to be significantly larger than \(1\): \[r_T(x,t)\ge r_{T,\min}^{\mathrm{jet}}>1. \label{eq:part07_rT_gate}\] (If \(p_\perp\) is very small, one must ensure numerical regularity and PSD is maintained.)

7.5.1.5 (J5) Jet-tube geometry gate (if a tube model is used).

If a jet radius \(R_j(z,t)\) and axial scale \(L_z\) are defined, require slenderness: \[\varepsilon_{\mathrm{geom}}:=\frac{\sup_{z,t}R_j(z,t)}{L_z}\ll 1, \label{eq:part07_slenderness}\] and require small radial leakage at the tube boundary: \[|S_R(R_j,z,t)| \le \varepsilon_{\mathrm{leak}}^{\mathrm{jet}}\ |S_z(R_j,z,t)|. \label{eq:part07_leak_boundary_gate}\]

7.5.3.1 Transverse collimation condition.

A collimated jet has \(S_\perp\approx 0\) (or \(\ell_\perp\ll 1\)). In a steady/strong-relaxation transverse balance (neglecting \(\partial_t S_\perp\)), [eq:part07_Sperp_balance] yields: \[\nabla_\perp p_\perp \approx e_{\mathrm{a}}F_\perp. \label{eq:part07_transverse_equilibrium}\] This is the mathematically explicit statement that collimation requires either:

• a transverse “pressure” gradient \(\nabla_\perp p_\perp\) balancing the transverse driving \(e_{\mathrm{a}}F_\perp\) (equilibrium collimation), and/or

• strong transverse relaxation \(B_\perp\) that suppresses \(S_\perp\) for given forcing (damped collimation).

Equation [eq:part07_transverse_equilibrium] is a diagnostic: if measured fields violate it strongly while claiming \(S_\perp\approx 0\), the jet-collimation regime claim fails.

7.6.1.1 Mixing-dominated region \(\mathcal{R}_{\mathrm{mix}}\).

Given by [eq:part07_Rmix_def].

7.6.1.2 Strong-alignment (jet) region \(\mathcal{R}_{\mathrm{jet}}\).

Given by [eq:part07_jet_A_high]–[eq:part07_jet_flux_collimation] (and optionally [eq:part07_rT_gate], [eq:part07_slenderness]–[eq:part07_leak_boundary_gate] if geometry is used).

7.6.1.3 Partial-alignment region \(\mathcal{R}_{\mathrm{pa}}\).

Define \[\mathcal{R}_{\mathrm{pa}}:=\big(\Omega\times\mathbb{R}\big)\setminus\big(\mathcal{R}_{\mathrm{mix}}\cup\mathcal{R}_{\mathrm{jet}}\big),\] together with the additional requirement that \(A\) is not negligible (otherwise it would be mixing) and not near-saturated with \(a_k\approx 0\) (otherwise it would be jet). Operationally one can enforce [eq:part07_partial_alignment_range] inside \(\mathcal{R}_{\mathrm{pa}}\).

7.6.2.1 Mixing \(\rightarrow\) partial alignment.

Occurs when either mixing weakens or alignment grows: \[w_\lambda \downarrow \ \text{below }w_{\lambda,\min}^{\mathrm{mix}} \quad\text{and/or}\quad A \uparrow \ \text{above }\varepsilon_A^{\mathrm{mix}} \quad\text{and/or}\quad \mathcal{A}_T\uparrow \ \text{above }\varepsilon_T^{\mathrm{mix}}.\]

7.6.2.2 Partial alignment \(\rightarrow\) strong alignment (jet).

Occurs when alignment saturates and defect collapses: \[A \uparrow \ \text{to }\ge 1-\varepsilon_A^{\mathrm{jet}} \quad\text{and}\quad a_k \downarrow \ \text{to }\le \varepsilon_k^{\mathrm{jet}} \quad\text{and}\quad \ell_\perp \downarrow \ \text{to }\le \varepsilon_\perp^{\mathrm{jet}}\] (with optional geometry gates if using tube reduction).

7.6.2.3 Strong alignment \(\rightarrow\) partial/mixing (de-collimation).

Occurs if any of the jet gates fail persistently: \[a_k\uparrow\ \text{or}\ \ell_\perp\uparrow\ \text{or}\ A\downarrow,\] possibly accompanied by increased mixing \(\lambda\).

7.6.2.4 Hysteresis note (allowed but must be declared).

One may allow different up/down thresholds (hysteresis) to avoid regime chatter. If so, all hysteresis parameters must be locked and reported as part of the regime specification; otherwise regime switching becomes post hoc tuning.

7.6.3.1 (BG1) Boundary flux compatibility.

If the domain boundary is closed: \[\mathbf{S}\cdot\mathbf{n}=0 \quad \text{on }\partial\Omega,\] then any regime that assumes sustained net outflow through \(\partial\Omega\) is invalid. Conversely, a jet tube exiting a domain must be modeled as an open boundary with prescribed flux or inflow distribution.

7.6.3.2 (BG2) Axis-field regularity (curvature/variation gate).

Axisymmetric closures assume that \(k\) varies slowly compared to the closure scale. Define the transverse gradient magnitude \[\mathcal{K}_k(x,t):=\|(\mathbf{I}-k\otimes k)\,\nabla k\|_F,\] and require (for axisymmetric closure validity): \[\mathcal{K}_k(x,t)\,L_{\mathrm{cl}} \le \varepsilon_k^{\mathrm{curv}}, \label{eq:part07_k_curv_gate}\] where \(L_{\mathrm{cl}}\) is the declared closure length scale (e.g. the coarse-graining length) and \(\varepsilon_k^{\mathrm{curv}}\) is a fixed small threshold. If [eq:part07_k_curv_gate] fails, one must either (i) avoid axisymmetric closures there or (ii) explicitly include \(k\)-variation terms in the closure model (a different closure class).

7.6.3.3 (BG3) Axisymmetry consistency gate (if axisymmetry is claimed).

If axisymmetry is claimed in cylindrical coordinates, require small \(\phi\)-variation: \[\frac{\|\partial_\phi e_{\mathrm{a}}\|}{\|e_{\mathrm{a}}\|+\eta_0}\le \varepsilon_\phi^{\mathrm{axi}}, \qquad \frac{\|\partial_\phi \rho\|}{\|\rho\|+\eta_0}\le \varepsilon_\phi^{\mathrm{axi}},\] (with appropriate norms, e.g. local \(L^2\) over a neighborhood). Otherwise the axisymmetric regime label is invalid.

7.7.1.1 Inputs (data or evolving fields).

At minimum: \[\rho,\ e_{\mathrm{a}},\ \mathbf{S},\ k,\ \lambda,\ \mu,\ \Gamma.\] If available, also: \[\mathbf{T}\ (\text{measured or reconstructed}),\quad \mathbf{m}_b\ (\text{or }A,a_k,\xi).\] If \(\mathbf{m}_b\) is unavailable, one must specify a proxy for alignment (SPEC) and gate it carefully.

7.7.1.2 Computed diagnostics.

Compute: \[w_\lambda=\frac{\lambda}{1+\lambda},\quad A=\frac{\|\mathbf{m}_b\|}{e_{\mathrm{a}}+\eta_0},\quad a_k=\frac{\|\mathbf{m}_b-(\mathbf{m}_b\cdot k)k\|}{e_{\mathrm{a}}+\eta_0},\] \[c_S=\frac{\mathbf{S}\cdot k}{\|\mathbf{S}\|+\eta_0},\quad \ell_\perp=\frac{\|\mathbf{S}- (\mathbf{S}\cdot k)k\|}{\|\mathbf{S}\|+\eta_0},\] and if \(\mathbf{T}\) is present, \[\mathcal{A}_T=\frac{\|\mathbf{T}-\frac{1}{3}\mathrm{tr}(\mathbf{T})\mathbf{I}\|_F}{\mathrm{tr}(\mathbf{T})+\eta_0}.\]

7.7.2.1 Gate G0: Phase admissibility (simplex).

\[0\le \rho\le 1,\quad 0\le e_{\mathrm{a}}\le 1,\quad 0\le e_{\mathrm{bg}}:=1-\rho-e_{\mathrm{a}}\le 1.\]

7.7.2.2 Gate G1: Alignment identity (if \(A,a_k,\xi\) are used).

If \(e_{\mathrm{a}}>0\) and both \(A\) and \(\xi\) and \(a_k\) are computed, require \[\big|A^2-\xi^2-a_k^2\big|\le \varepsilon_{\mathrm{id}}.\] This is an internal consistency check.

7.7.2.3 Gate G2: Realizability gates for closures involving \(\mathbf{T}\).

Any proposed closure must satisfy: \[\mathbf{T}\succeq 0,\qquad \|\mathbf{S}\|^2\le e_{\mathrm{a}}\ \mathrm{tr}(\mathbf{T})\quad (\text{if }\mathbf{T}\text{ is used}).\]

7.7.2.4 Gate G3: Geometry gates (if axisymmetric/jet closure is chosen).

Require [eq:part07_k_curv_gate] and, if axisymmetry is claimed, small \(\phi\)-variation.

If any mandatory gate fails, the procedure outputs FAIL with the failing gate label; no closure is selected.

8.1.1.1 Hard (on/off) gate.

The idealized on/off gate is the Heaviside step: \[G_{\mathrm{hard}}(x,t) := H\!\big(q(x,t)-q_c\big) = \begin{cases} 0,& q<q_c,\\ 1,& q\ge q_c. \end{cases} \label{eq:part08_hard_gate}\] Hard gates are analytically convenient but introduce discontinuities.

8.1.1.2 Soft (smoothed) gate.

For numerical stability and weak-solution well-posedness, a smooth approximation is usually preferred. A standard choice is the logistic gate: \[G_{\mathrm{soft}}(x,t) := \sigma\!\left(\frac{q(x,t)-q_c}{\Delta_q}\right), \qquad \sigma(z):=\frac{1}{1+e^{-z}}, \qquad \Delta_q>0, \label{eq:part08_soft_gate_logistic}\] or equivalently a \(\tanh\)-gate: \[G_{\mathrm{soft}}(x,t) := \frac{1}{2}\left[1+\tanh\!\left(\frac{q(x,t)-q_c}{2\Delta_q}\right)\right]. \label{eq:part08_soft_gate_tanh}\] The gate width \(\Delta_q\) must be declared (SPEC) and locked for a given version to avoid post hoc tuning.

8.1.1.3 Gate surface and critical radius.

The gate surface is the set where \(q=q_c\). In spherical settings, when \(q=q(r,t)\) depends only on \(r=\|x\|\), the gate surface becomes a sphere at a critical radius \(r=r_c(t)\) defined implicitly by: \[q(r_c(t),t)=q_c. \label{eq:part08_critical_radius_def}\] If \(q(r,t)\) is monotone in \(r\) for fixed \(t\), the critical radius is unique.

8.1.2.1 NOT.

\[\neg G := 1-G.\]

8.1.2.2 AND.

For \(G_1,\dots,G_n\in[0,1]\), \[G_{\mathrm{AND}}:=\prod_{i=1}^n G_i. \label{eq:part08_gate_and}\] For hard gates this is exact logical AND; for soft gates it is a smooth “all conditions nearly satisfied” operator.

8.1.2.3 OR.

\[G_{\mathrm{OR}}:=1-\prod_{i=1}^n (1-G_i). \label{eq:part08_gate_or}\]

8.1.2.4 Usage rule.

Gate algebra is SPEC: the exact form ([eq:part08_gate_and]–[eq:part08_gate_or]) and the thresholds used must be fixed before any evaluation. Otherwise the gating system becomes a tuning knob rather than a falsifiable structure.

8.1.3.1 (i) Saturation decision variable.

For conversion saturation we use a dimensionless loading ratio: \[\chi_\Gamma(x,t):=\frac{e_{\mathrm{a}}(x,t)}{K_\Gamma(x,t)+\eta_0}, \qquad K_\Gamma>0,\qquad \eta_0>0 \text{ (small fixed regularizer)}. \label{eq:part08_chi_gamma}\] Saturation typically corresponds to \(\chi_\Gamma\gg 1\).

8.1.3.2 (ii) Choking decision variable (flux ratio).

Given a speed limit \(c>0\) (defined in §8.4), define: \[\chi_S(x,t):=\frac{\|\mathbf{S}(x,t)\|}{c\,e_{\mathrm{a}}(x,t)+\eta_0}, \qquad \chi_S\in[0,\infty). \label{eq:part08_chi_S}\] Choking corresponds to \(\chi_S\to 1\) (or exceeding 1 in an unconstrained demand model).

8.1.3.3 (iii) Potential/drive decision variable.

If \(\mathbf{F}_{\mathrm{eff}}=-\nabla\Phi_{\mathrm{eff}}+\cdots\) (PART 06), define a drive-to-capacity ratio: \[\chi_F(x,t):=\frac{\|\mathbf{F}_{\mathrm{eff}}(x,t)\|}{B_{\mathrm{ref}}(x,t)\,c+\eta_0}, \label{eq:part08_chi_F}\] where \(B_{\mathrm{ref}}\) is a declared scalar reference relaxation (e.g. \(B\) in isotropic regimes). In diffusion-like balances, choking is often triggered when \(\chi_F\gtrsim 1\) (see §8.3.3).

8.1.3.4 (iv) Throughput decision variables (surface flux).

For a surface \(\Sigma\) with unit normal \(\mathbf{n}\), define the local normal throughput density: \[j_\Sigma(x,t):=\mathbf{S}(x,t)\cdot \mathbf{n}(x),\] and the integrated throughput: \[\mathcal{J}_\Sigma(t):=\int_\Sigma \mathbf{S}\cdot \mathbf{n}\,dA. \label{eq:part08_throughput_surface}\]

8.1.3.5 (v) Geometric decision variable (radius).

In spherical problems, \(q=r\) itself is used with a threshold \(r_c\) to declare zones (inner/outer) where different mechanisms/closures apply.

8.2.2.1 (A1) Dimension and non-negativity.

\[\delta(\Gamma)=T^{-1}, \qquad \Gamma(x,t;e_{\mathrm{a}})\ge 0 \ \ \text{for } e_{\mathrm{a}}\in[0,1]. \label{eq:part08_gamma_dim_pos}\]

8.2.2.2 (A2) Zero-input consistency.

No active content implies no active\(\rightarrow\)stored conversion: \[\Gamma(x,t;0)=0. \label{eq:part08_gamma_zero}\]

8.2.2.3 (A3) Boundedness (saturation).

There exists a declared capacity \(\Gamma_{\max}(x,t)\ge 0\) such that \[0\le \Gamma(x,t;e_{\mathrm{a}})\le \Gamma_{\max}(x,t) \quad \text{for all } e_{\mathrm{a}}\in[0,1]. \label{eq:part08_gamma_bounded}\] This encodes processing saturation.

8.2.2.4 (A4) Regularity (for well-posedness and reproducibility).

For each \((x,t)\), the map \(e_{\mathrm{a}}\mapsto \Gamma(x,t;e_{\mathrm{a}})\) is locally Lipschitz on \([0,1]\): \[|\Gamma(e_1)-\Gamma(e_2)|\le L_\Gamma |e_1-e_2| \quad \text{for some }L_\Gamma<\infty. \label{eq:part08_gamma_lipschitz}\] Hard saturation (piecewise linear) is allowed but must be handled carefully in numerics.

8.2.2.5 (A5) Monotonicity (optional but common).

Many physical interpretations require \(\Gamma\) to increase with \(e_{\mathrm{a}}\) (more active content leads to more conversion up to capacity): \[\partial_{e_{\mathrm{a}}}\Gamma(x,t;e_{\mathrm{a}})\ge 0. \label{eq:part08_gamma_monotone}\] If non-monotone \(\Gamma\) is used, the mechanism and the gate tests must be explicitly stated.

8.2.4.1 (G1) Hill/Michaelis–Menten saturation (smooth, tunable sharpness).

For \(K_\Gamma>0\) and Hill exponent \(n\ge 1\): \[g_{\mathrm{Hill}}(e_{\mathrm{a}}):= \frac{e_{\mathrm{a}}^n}{K_\Gamma^n+e_{\mathrm{a}}^n}. \label{eq:part08_gamma_hill}\] Properties: \[g_{\mathrm{Hill}}(0)=0,\qquad g_{\mathrm{Hill}}(e_{\mathrm{a}})\uparrow 1 \text{ as } e_{\mathrm{a}}\gg K_\Gamma, \qquad 0\le g_{\mathrm{Hill}}\le 1.\] Derivative: \[g'_{\mathrm{Hill}}(e_{\mathrm{a}})= \frac{n K_\Gamma^n e_{\mathrm{a}}^{n-1}}{(K_\Gamma^n+e_{\mathrm{a}}^n)^2}\ge 0.\]

8.2.4.2 (G2) Exponential saturation (smooth, fast approach).

\[g_{\exp}(e_{\mathrm{a}}):=1-\exp\!\left(-\frac{e_{\mathrm{a}}}{K_\Gamma}\right). \label{eq:part08_gamma_exp}\] Properties: \[g_{\exp}(0)=0,\qquad 0\le g_{\exp}<1,\qquad g'_{\exp}(e_{\mathrm{a}})=\frac{1}{K_\Gamma}e^{-e_{\mathrm{a}}/K_\Gamma}\ge 0.\]

8.2.4.3 (G3) Rational saturator (smooth, low-order).

\[g_{\mathrm{rat}}(e_{\mathrm{a}}):=\frac{e_{\mathrm{a}}}{K_\Gamma+e_{\mathrm{a}}}. \label{eq:part08_gamma_rational}\] This is the \(n=1\) Hill case.

8.2.4.4 (G4) Piecewise-linear hard saturation (non-smooth at \(K_\Gamma\)).

\[g_{\mathrm{lin}}(e_{\mathrm{a}}):=\min\left\{\frac{e_{\mathrm{a}}}{K_\Gamma},\,1\right\}. \label{eq:part08_gamma_piecewise}\] This is admissible but non-differentiable at \(e_{\mathrm{a}}=K_\Gamma\).

8.2.4.5 (G5) Shifted logistic (smooth, tunable threshold) with exact \(g(0)=0\).

Define \[\tilde{g}(e_{\mathrm{a}}):=\sigma\!\left(\frac{e_{\mathrm{a}}-K_\Gamma}{\Delta_\Gamma}\right),\qquad \tilde{g}_0:=\tilde{g}(0),\] and renormalize: \[g_{\mathrm{log}}(e_{\mathrm{a}}):= \frac{\tilde{g}(e_{\mathrm{a}})-\tilde{g}_0}{1-\tilde{g}_0}, \qquad \Delta_\Gamma>0. \label{eq:part08_gamma_logistic_shifted}\] Then \(g_{\mathrm{log}}(0)=0\) and \(g_{\mathrm{log}}\to 1\) as \(e_{\mathrm{a}}\to 1\) (if \(K_\Gamma<1\) and \(\Delta_\Gamma\) small enough). This form is useful when the onset of conversion is itself gated near a threshold.

8.3.1.1 Proposition 8.1 (flux bound).

If [eq:part08_velocity_support_bound] holds and \(f\ge 0\), then \[\|\mathbf{S}(x,t)\|\le c\,e_{\mathrm{a}}(x,t) \quad \text{for all }(x,t). \label{eq:part08_flux_bound_main}\]

8.3.1.2 Proof.

By definition \(\mathbf{S}=\int_{\mathcal{V}_c} v f\,dv\). Then \[\|\mathbf{S}\| = \left\|\int_{\mathcal{V}_c} v f\,dv\right\| \le \int_{\mathcal{V}_c}\|v\| f\,dv \le \int_{\mathcal{V}_c} c\,f\,dv = c\int_{\mathcal{V}_c} f\,dv = c\,e_{\mathrm{a}}.\] \(\square\)

8.3.1.3 Second-moment refinement and closure constraint.

Under [eq:part08_velocity_support_bound], the trace of the second moment satisfies: \[\mathrm{tr}(\mathbf{T})(x,t) = \int_{\mathcal{V}_c} \|v\|^2 f\,dv \le c^2 \int_{\mathcal{V}_c} f\,dv = c^2 e_{\mathrm{a}}(x,t). \label{eq:part08_Ttrace_bound}\] Combined with the general realizability inequality (PART 06), \[\|\mathbf{S}\|^2\le e_{\mathrm{a}}\ \mathrm{tr}(\mathbf{T}),\] we again obtain [eq:part08_flux_bound_main]. Importantly, [eq:part08_Ttrace_bound] imposes a gate constraint on closures: any closure implying \(\mathrm{tr}(\mathbf{T})>c^2 e_{\mathrm{a}}\) violates the speed-support assumption and must be rejected.

8.3.2.1 Choking ratio.

Using [eq:part08_chi_S], define: \[\chi_S(x,t)=\frac{\|\mathbf{S}\|}{c e_{\mathrm{a}}+\eta_0}.\] In the speed-limited theory, admissibility requires \(\chi_S\le 1\) (up to the small regularizer). Therefore:

8.3.2.2 Definition (choking).

A region is choked if the flux is close to its maximal admissible magnitude: \[\chi_S(x,t)\ge \theta_{\mathrm{choke}}, \qquad \theta_{\mathrm{choke}}\in(0,1)\ \text{locked}. \label{eq:part08_choking_def}\] A natural choice is \(\theta_{\mathrm{choke}}\approx 0.9\).

8.3.3.1 Choking condition (generic).

Choking is expected wherever \[\|\mathbf{S}_{\mathrm{dem}}\| > c\,e_{\mathrm{a}}. \label{eq:part08_choking_condition_generic}\] In that case the physical system cannot realize the demanded flux without violating the speed limit, so a flux limiter (or a different dynamical regime) must intervene.

8.3.3.2 Choking condition under isotropic closure (CL-ISO).

With \(\mathbf{T}=\kappa_T e_{\mathrm{a}}\mathbf{I}\) and \(\mathbf{B}=B\mathbf{I}\), \[\nabla\cdot\mathbf{T}=\nabla(\kappa_T e_{\mathrm{a}}),\] so \[\mathbf{S}_{\mathrm{dem}} = -\frac{1}{B}\nabla(\kappa_T e_{\mathrm{a}}) +\frac{e_{\mathrm{a}}}{B}\mathbf{F}_{\mathrm{eff}}. \label{eq:part08_S_demand_iso}\] Thus choking occurs where \[\left\| -\nabla(\kappa_T e_{\mathrm{a}}) +e_{\mathrm{a}}\mathbf{F}_{\mathrm{eff}} \right\| > B\,c\,e_{\mathrm{a}}. \label{eq:part08_choking_iso_condition}\] If gradients of \((\kappa_T e_{\mathrm{a}})\) are negligible compared to the driving, \[\|\mathbf{F}_{\mathrm{eff}}\| > Bc\] is a simple trigger; this connects to the drive ratio \(\chi_F\) in [eq:part08_chi_F].

8.3.4.1 Hard limiter.

\[\mathcal{L}_{\mathrm{hard}}(\mathbf{S}_{\mathrm{dem}};c e_{\mathrm{a}}) := \begin{cases} \mathbf{S}_{\mathrm{dem}}, & \|\mathbf{S}_{\mathrm{dem}}\|\le c e_{\mathrm{a}},\\[0.75ex] c e_{\mathrm{a}}\dfrac{\mathbf{S}_{\mathrm{dem}}}{\|\mathbf{S}_{\mathrm{dem}}\|}, & \|\mathbf{S}_{\mathrm{dem}}\|> c e_{\mathrm{a}}. \end{cases} \label{eq:part08_limiter_hard}\]

8.3.4.2 Smooth limiter (recommended for numerics).

\[\mathcal{L}_{\mathrm{smooth}}(\mathbf{S}_{\mathrm{dem}};c e_{\mathrm{a}}) := c e_{\mathrm{a}}\, \tanh\!\left(\frac{\|\mathbf{S}_{\mathrm{dem}}\|}{c e_{\mathrm{a}}+\eta_0}\right)\, \frac{\mathbf{S}_{\mathrm{dem}}}{\|\mathbf{S}_{\mathrm{dem}}\|+\eta_0}. \label{eq:part08_limiter_smooth}\] Then \(\|\mathcal{L}_{\mathrm{smooth}}\|\le c e_{\mathrm{a}}\) and the map is continuous.

8.3.4.3 Gate view.

The limiter implements the gate \[G_{\mathrm{choke}} = H\!\left(\|\mathbf{S}_{\mathrm{dem}}\|-c e_{\mathrm{a}}\right)\] by switching between \(\mathbf{S}_{\mathrm{dem}}\) and a clamped magnitude.

8.3.5.1 Example: inverse-square drive.

If \(|F_r(r)|=K_F/r^2\) and \(B\) is constant, then \[r_{\mathrm{ch}}=\sqrt{\frac{K_F}{B c}}. \label{eq:part08_rch_inverse_square}\] This is a concrete formula for an “emergent critical radius” produced solely by a flux capacity bound.

8.4.1.1 Local transport velocity.

When \(e_{\mathrm{a}}>0\), define the mean transport velocity of the active phase: \[\mathbf{u}(x,t):=\frac{\mathbf{S}(x,t)}{e_{\mathrm{a}}(x,t)}. \label{eq:part08_u_def}\] The flux bound [eq:part08_flux_bound_main] implies an emergent speed bound: \[\|\mathbf{u}(x,t)\|\le c. \label{eq:part08_u_bound}\]

8.4.1.2 Surface throughput bound.

For any surface \(\Sigma\), \[|\mathbf{S}\cdot\mathbf{n}|\le \|\mathbf{S}\|\le c e_{\mathrm{a}},\] so \[|\mathcal{J}_\Sigma(t)| = \left|\int_\Sigma \mathbf{S}\cdot\mathbf{n}\,dA\right| \le \int_\Sigma |\mathbf{S}\cdot\mathbf{n}|\,dA \le c\int_\Sigma e_{\mathrm{a}}\,dA. \label{eq:part08_throughput_bound_surface}\] In a spherical setting, \[\mathcal{J}(r,t)=4\pi r^2 S_r(r,t),\] and the pointwise bound \(|S_r|\le c e_{\mathrm{a}}\) yields: \[|\mathcal{J}(r,t)|\le 4\pi r^2 c\,e_{\mathrm{a}}(r,t). \label{eq:part08_throughput_spherical_bound}\]

8.4.2.1 (M1) Velocity-support postulate.

Postulate [eq:part08_velocity_support_bound]. This yields [eq:part08_flux_bound_main] and [eq:part08_u_bound] directly and is the cleanest analytic path.

8.4.2.2 (M2) Lattice step-speed realization.

If the micro-kinematics is realized by a coarse lattice with length step \(a\) and time step \(\Delta t\) (PART 03 unit realization), then the maximal speed is \[c:=\frac{a}{\Delta t}. \label{eq:part08_c_from_lattice}\] If the active phase moves at most one cell per \(\Delta t\), then \(\|v\|\le c\) holds effectively.

Both mechanisms imply the same macroscopic gate: \(\|\mathbf{S}\|\le c e_{\mathrm{a}}\).

8.4.3.1 Isotropic closure constraint.

For CL-ISO, \(\mathbf{T}=\kappa_T e_{\mathrm{a}}\mathbf{I}\), hence \(\mathrm{tr}(\mathbf{T})=3\kappa_T e_{\mathrm{a}}\). The trace bound [eq:part08_Ttrace_bound] requires: \[3\kappa_T e_{\mathrm{a}}\le c^2 e_{\mathrm{a}} \quad \Longrightarrow \quad \kappa_T\le \frac{c^2}{3}. \label{eq:part08_kappaT_bound}\] Thus \(\kappa_T\) cannot be arbitrarily large if \(c\) is finite.

8.4.3.2 Axisymmetric closure constraint.

For \(\mathbf{T}=p_\perp(\mathbf{I}-k\otimes k)+p_\parallel(k\otimes k)\) with \(p_\perp,p_\parallel\ge 0\), \[\mathrm{tr}(\mathbf{T})=2p_\perp+p_\parallel.\] The speed bound requires: \[2p_\perp+p_\parallel \le c^2 e_{\mathrm{a}}. \label{eq:part08_axis_trace_bound}\] Any parameterization (e.g. \(p_\perp=p(1-\alpha)\), \(p_\parallel=p(1+2\alpha)\) with \(p=\kappa_T e_{\mathrm{a}}\)) must obey [eq:part08_axis_trace_bound] to be compatible with the same \(c\).

8.5.1.1 Gate rule.

All decision variables used in hard gates (§8.1.1) may be evaluated on coarse-grained fields \(q_{\ell_{\min}}\) to avoid pathological triggering from sub-resolution fluctuations.

8.6.1.1 Additional delay from choking.

In a choked region, the flux saturates: \[\|\mathbf{S}\|\approx c e_{\mathrm{a}}, \qquad \chi_S\approx 1,\] and further increases in driving do not increase throughput. If the external driving increases abruptly, the system responds by accumulating gradients (and/or active density) rather than increasing flux. This creates a plateau in signal speed and a delayed response at larger radii.

8.6.1.2 Spherical travel-time estimate (baseline).

If the effective radial transport speed is \(u_r(r,t)=S_r/e_{\mathrm{a}}\) (for \(e_{\mathrm{a}}>0\)), then a quasi-1D travel-time estimate from \(r_1\) to \(r_2\) is: \[t_{\mathrm{travel}}\approx \int_{r_1}^{r_2} \frac{dr}{u_r(r,\cdot)}, \qquad |u_r|\le c. \label{eq:part08_travel_time_integral}\] Choking corresponds to \(|u_r|\approx c\) in some region, fixing a minimal local travel time density \(dr/c\).

8.6.2.1 Operational gate for jet onset (summary).

Define the strong-alignment diagnostics (PART 07): \[A=\frac{\|\mathbf{m}_b\|}{e_{\mathrm{a}}},\qquad a_k=\frac{\|\mathbf{m}_b-(\mathbf{m}_b\cdot k)k\|}{e_{\mathrm{a}}}, \qquad \ell_\perp=\frac{\|\mathbf{S}-(\mathbf{S}\cdot k)k\|}{\|\mathbf{S}\|+\eta_0}.\] Then a minimal jet-onset gate is: \[G_{\mathrm{jet}} = H\!\big(A-(1-\varepsilon_A^{\mathrm{jet}})\big)\; H\!\big(\varepsilon_k^{\mathrm{jet}}-a_k\big)\; H\!\big(\varepsilon_\perp^{\mathrm{jet}}-\ell_\perp\big). \label{eq:part08_jet_gate}\] When \(G_{\mathrm{jet}}=1\), the closure-selection tree chooses an axisymmetric jet closure (CL-JET) and enables jet-tube diagnostics.

8.6.2.2 Drive-assisted onset (throughput gate).

Additionally, define an axial throughput ratio (if \(k\) is the axis): \[\chi_{\parallel}(x,t):= \frac{\mathbf{S}(x,t)\cdot k(x,t)}{c\,e_{\mathrm{a}}(x,t)+\eta_0}. \label{eq:part08_chi_parallel}\] A strong-drive onset can be declared when \(\chi_{\parallel}\) approaches capacity while \(\ell_\perp\) becomes small: \[\chi_{\parallel}\to 1,\qquad \ell_\perp\to 0,\] indicating that the system is saturating axial throughput while suppressing transverse leakage—a robust signature of collimation.

8.6.3.1 Reduced local model in a choked/isolated core.

Assume that inside a region \(\Omega_c\): \[\nabla\cdot\mathbf{S}\approx 0.\] This can occur either because \(\mathbf{S}\approx 0\) or because it is capacity-limited and nearly divergence-free locally.

Then [eq:part08_core_rho]–[eq:part08_core_ea] reduce to the local ODE system [eq:part08_phase_ode_rho]–[eq:part08_phase_ode_ea]. If \(e_{\mathrm{tot}}=E\) is approximately constant locally, then \(\rho=E-e_{\mathrm{a}}\) and \[\dot{e}_{\mathrm{a}} = \mu(E-e_{\mathrm{a}})-\Gamma(e_{\mathrm{a}}). \label{eq:part08_core_reduced_ode}\]

8.6.3.2 Equilibrium and saturation-induced plateaus.

An equilibrium \(e_{\mathrm{a}}^\star\) satisfies \[\mu(E-e_{\mathrm{a}}^\star)=\Gamma(e_{\mathrm{a}}^\star). \label{eq:part08_equilibrium_condition}\] If \(\Gamma\) saturates at \(\Gamma_{\max}\), then for large \(E\) or large \(\mu E\) one may reach a regime where \(\Gamma(e_{\mathrm{a}})\approx \Gamma_{\max}\) over a wide range of \(e_{\mathrm{a}}\). This creates a plateau in the relaxation: the conversion cannot accelerate beyond its capacity, so the core relaxes on the slower of the timescales \(\mu^{-1}\) and the effective slope of \(\Gamma\).

8.6.3.3 Linearized relaxation rate.

Linearize [eq:part08_core_reduced_ode] around \(e_{\mathrm{a}}^\star\): \[\dot{\delta e} = -\mu\,\delta e - \Gamma'(e_{\mathrm{a}}^\star)\,\delta e =-(\mu+\Gamma'(e_{\mathrm{a}}^\star))\,\delta e.\] Thus the relaxation time is \[\tau_{\mathrm{core}}=\frac{1}{\mu+\Gamma'(e_{\mathrm{a}}^\star)}. \label{eq:part08_core_relax_time}\] If \(\Gamma\) is strongly saturated at the operating point, \(\Gamma'(e_{\mathrm{a}}^\star)\) is small, and \(\tau_{\mathrm{core}}\approx 1/\mu\) (slow, storage-limited relaxation). If \(\Gamma\) is in its steep response region, \(\Gamma'\) is large and the relaxation can be significantly faster.

8.6.3.4 Observable signature.

A saturated/choked core tends to produce:

• delayed response to increased external drive (because throughput cannot increase beyond capacity),

• a transient build-up of gradients or local active fraction,

• and a characteristic exponential (or near-exponential) relaxation with time constant [eq:part08_core_relax_time] once driving decreases.

8.6.4.1 Time delay.

Bounded by \(L/c\) and amplified by choking. Diagnostics: \(\chi_S\) near 1 in an inner zone, plus delayed changes in \(e_{\mathrm{a}}\) at outer radii.

8.6.4.2 Jet onset.

A gate-crossing event in alignment and leakage (and often in axial throughput). Diagnostics: \(G_{\mathrm{jet}}\) in [eq:part08_jet_gate] switches on; simultaneously \(\chi_{\parallel}\) increases while \(\ell_\perp\) decreases.

8.6.4.3 Core relaxation.

Dominated by conversion with saturated \(\Gamma\) and/or suppressed transport. Diagnostics: \(G_{\mathrm{sat}}\approx 1\) (conversion saturated), \(\chi_S\approx 1\) (transport choked), and relaxation time consistent with [eq:part08_core_relax_time].

9.2.2.1 Regime declaration (MIXING, thin disk, quasi-steady).

Assume:

1. (HYP/SPEC) Outside a core region \(R\ge R_0\), the VP active field is in a mixing-dominated regime (PART 07.3) with an isotropic closure and strong relaxation reduction, so the active dynamics reduces to drift-diffusion.

2. (SPEC) The relevant deficit field is proportional to \(e_{\mathrm{a}}\) in the outer disk plane: \[\Delta(R,0)\approx \Delta_0\,e_{\mathrm{a}}(R,0), \qquad \Delta_0>0\ \text{(LOCK)}. \label{eq:part09_Delta_propto_ea}\] (This is consistent with \(\Delta=e_{\mathrm{bg}}^\infty-e_{\mathrm{bg}}\) and \(e_{\mathrm{bg}}=1-\rho-e_{\mathrm{a}}\) when \(\rho\) is negligible in the outer disk.)

3. (SPEC) The disk is effectively thin compared to the radii of interest so that transport is effectively 2D in \((R,\phi)\) on scales \(\gg\) thickness.

4. (SPEC) Quasi-steady outer region: \(\partial_t e_{\mathrm{a}}\approx 0\) for the time window relevant to rotation-curve measurement.

9.2.2.2 Outer-region equation.

From PART 07.1, in a diffusion-dominated regime and neglecting drift and reaction in the outer region, we obtain an approximately harmonic equation: \[\nabla_\perp\cdot\big(D\,\nabla_\perp e_{\mathrm{a}}\big)\approx 0 \quad\text{for } R\ge R_0, \label{eq:part09_outer_harmonic}\] where \(\nabla_\perp\) is the 2D gradient in the disk plane, and \(D=\kappa_T/B\) is the effective diffusion coefficient (PART 07.1).

For constant \(D\) this becomes \(\nabla_\perp^2 e_{\mathrm{a}}=0\) in the outer region.

9.2.2.3 Axisymmetric solution in 2D.

In cylindrical coordinates with axisymmetry (\(\partial_\phi=0\)), the 2D Laplacian is \[\nabla_\perp^2 e_{\mathrm{a}}=\frac{1}{R}\partial_R(R\partial_R e_{\mathrm{a}}).\] Solving \(\nabla_\perp^2 e_{\mathrm{a}}=0\) gives \[e_{\mathrm{a}}(R,0)=A\ln\!\left(\frac{R}{R_0}\right)+B, \qquad R\ge R_0, \label{eq:part09_ea_log_solution}\] with constants \(A,B\) fixed by boundary/throughput conditions.

9.2.2.4 Coupling to potential and flat speed.

Using [eq:part09_Phi_def_coupling] and [eq:part09_Delta_propto_ea] we obtain \[\Phi_{\mathrm{def}}(R,0) \approx -\alpha_\Phi c^2 \Delta_0 e_{\mathrm{a}}(R,0) = -\alpha_\Phi c^2 \Delta_0 \Big[A\ln(R/R_0)+B\Big].\] Thus \[\partial_R\Phi_{\mathrm{def}}(R,0) = -\alpha_\Phi c^2 \Delta_0 \frac{A}{R},\] and the deficit contribution to the circular speed is \[v_{\mathrm{def}}^2(R):=R\partial_R\Phi_{\mathrm{def}}(R,0) = -\alpha_\Phi c^2 \Delta_0 A \equiv V_f^2\quad (\text{constant}). \label{eq:part09_flat_speed_from_A}\] Therefore a flat rotation curve emerges if \(A<0\) (so that \(V_f^2>0\) with \(\alpha_\Phi,\Delta_0\ge 0\)), i.e. if \(e_{\mathrm{a}}\) (hence \(\Delta\)) decreases with radius.

9.3.1.1 GR-matching policy (to be fixed).

If one enforces GR matching at leading post-Newtonian order, one sets \[\Phi=\Psi=\Phi_{\mathrm{eff}}. \label{eq:part09_PhiPsi_equals_Phi_eff}\] If not enforced, one must treat the slip \(\Psi-\Phi\) as an additional field and gate it observationally (see §9.5).

9.3.4.1 Einstein ring (axisymmetric lens).

For an axisymmetric lens, the Einstein ring angle \(\theta_E\) (perfect alignment) satisfies \[\theta_E = \frac{D_{LS}}{D_S}\,\alpha(\theta_E), \label{eq:part09_Einstein_condition}\] and equivalently in terms of enclosed projected mass \[\theta_E^2 = \frac{4G}{c^2}\frac{D_{LS}}{D_LD_S}\,M_{\mathrm{proj}}(<\theta_E), \label{eq:part09_thetaE_mass}\] where \[M_{\mathrm{proj}}(<\theta) := \int_{\|\boldsymbol{\xi}\|\le D_L\theta}\Sigma_{\mathrm{eff}}(\boldsymbol{\xi})\,d^2\xi. \label{eq:part09_Mproj_def}\]

9.3.4.2 Deficit contribution to lensing.

In this framework, \(\Sigma_{\mathrm{def}}\) is generated by the VP deficit field: \[\Sigma_{\mathrm{def}}(\boldsymbol{\xi}) = \int \varrho_{\mathrm{def}}(\boldsymbol{\xi},z)\,dz = \frac{1}{4\pi G}\int \nabla^2\Phi_{\mathrm{def}}(\boldsymbol{\xi},z)\,dz = -\frac{\alpha_\Phi c^2}{4\pi G}\int \nabla^2\Delta(\boldsymbol{\xi},z)\,dz,\] using [eq:part09_rho_def_eff]. This is a derived comparison object, not a fundamental mass density.

9.4.1.1 Relaxation time scale.

Linearizing the net sink \(L_\Delta-Q_\Delta\) around the operating point yields an effective relaxation time \(\tau_\Delta\): \[L_\Delta - Q_\Delta \approx \frac{1}{\tau_\Delta}\,(\Delta-\Delta_{\mathrm{eq}}), \qquad \tau_\Delta>0. \label{eq:part09_tau_Delta_def}\] In VP language, \(\tau_\Delta\) is controlled by conversion/reaction rates (\(\mu,\Gamma'\)) and any background restoration gates (PART 08.2, 08.6).

9.4.2.1 (C1) Slow relaxation (no rapid reattachment).

\[\tau_\Delta \gg \tau_{\mathrm{coll}}. \label{eq:part09_condition_slow_relax}\]

9.4.2.2 (C2) Small diffusion during the collision (no smearing).

\[\sqrt{D_\Delta\,\tau_{\mathrm{coll}}}\ll L_{\mathrm{off}}. \label{eq:part09_condition_small_diffusion}\]

9.4.2.3 (C3) Advection follows a collisionless component rather than shocked gas.

Let \(\mathbf{u}_{\mathrm{gal}}\) be the bulk velocity of the collisionless galaxy component and \(\mathbf{u}_{\mathrm{gas}}\) that of the shocked gas. Separation requires that \(\mathbf{u}_\Delta\) tracks the former: \[\|\mathbf{u}_\Delta-\mathbf{u}_{\mathrm{gal}}\|\ll \|\mathbf{u}_{\mathrm{gal}}-\mathbf{u}_{\mathrm{gas}}\|. \label{eq:part09_condition_advection_tracking}\] This is the VP analogue of “collisionless” behavior: not because a new particle is collisionless, but because the deficit-carrying structure is transported with the collisionless component and does not dissipate rapidly.

9.5.2.1 Slip as a falsifiable extension (optional).

If the framework allows a potential slip \(\Psi\ne\Phi\), then dynamics and lensing constrain different combinations: \[\text{dynamics}\sim \nabla\Phi,\qquad \text{lensing}\sim \nabla(\Phi+\Psi).\] This can be used as a gate: the same parameter set must fit both rotation curves and lensing without introducing an arbitrary slip function.

9.5.3.1 Matching order (explicit).

The weak-field matching claims in this document are limited to: \[\text{metric: } \mathcal{O}\!\left(\frac{\Phi}{c^2}\right),\qquad \text{dynamics: } \mathcal{O}\!\left(\frac{v^2}{c^2}\right),\qquad \text{lensing deflection: leading order in }\frac{\Phi}{c^2}. \label{eq:part09_matching_order}\] Any higher-order PPN claims require an explicit extension beyond the present scope.

9.6.3.1 Weak lensing.

Given observed convergence/shear data \(\{\kappa_{\mathrm{obs}}(\theta_j),\sigma_{\kappa,j}\}\) (or shear \(\gamma\)), compute \[\chi^2_{\mathrm{WL}} := \sum_j \frac{\big(\kappa_{\mathrm{pred}}(\theta_j)-\kappa_{\mathrm{obs}}(\theta_j)\big)^2}{\sigma_{\kappa,j}^2}, \qquad \chi^2_{\mathrm{WL,red}}:=\frac{\chi^2_{\mathrm{WL}}}{N_{\mathrm{WL}}-\nu_{\mathrm{WL}}}. \label{eq:part09_chi2_WL}\]

9.6.3.2 Strong lensing (Einstein ring).

If an Einstein ring angle \(\theta_E\) is observed with uncertainty \(\sigma_E\), define \[\chi^2_{\mathrm{SL}}:=\frac{\big(\theta_{E,\mathrm{pred}}-\theta_{E,\mathrm{obs}}\big)^2}{\sigma_E^2}, \label{eq:part09_chi2_SL}\] with \(\theta_{E,\mathrm{pred}}\) computed from [eq:part09_thetaE_mass] under the chosen policy.

9.6.3.3 Lensing PASS/FAIL.

\[\mathrm{PASS}_{\mathrm{lens}} \Longleftrightarrow \chi^2_{\mathrm{WL,red}}\le \Theta_{\mathrm{WL}} \ \ \wedge\ \ \chi^2_{\mathrm{SL}}\le \Theta_{\mathrm{SL}} \ \ \wedge\ \ \text{the GR-matching policy gates (e.g.\ }\Phi=\Psi\text{ if claimed) are satisfied.} \label{eq:part09_PASS_lens}\]

9.6.5.1 Artifact requirement.

Every PASS/FAIL evaluation must archive:

• the exact \(\theta\) (parameter hash),

• the predicted fields (\(\Phi_{\mathrm{eff}}\), \(v_c\), \(\kappa\) or \(\alpha\)),

• all \(\chi^2\) values and margins to thresholds,

• and all intermediate gate diagnostics (mixing regime, diffusion residual, speed-limit/trace constraints).

This is mandatory to prevent post hoc narrative fitting.

10.1.1.1 Choking gate.

Fix a choke threshold \(\theta_{\mathrm{choke}}\in(0,1)\) (LOCK). Define \[G_{\mathrm{choke}}(x,t):=H\!\big(\chi_S(x,t)-\theta_{\mathrm{choke}}\big), \label{eq:part10_gate_choke}\] with \(\chi_S\) given by [eq:part10_chiS].

10.1.1.2 Choking surface and critical radius.

In a spherically symmetric baseline (\(\rho=e_{\mathrm{a}}=\rho(r,t),e_{\mathrm{a}}(r,t)\), \(\mathbf{S}=S_r(r,t)\hat r\)), the choking surface is the sphere where \(G_{\mathrm{choke}}\) switches. Define the choking radius \(r_{\mathrm{ch}}(t)\) by the implicit condition \[\chi_S(r_{\mathrm{ch}}(t),t)=\theta_{\mathrm{choke}}. \label{eq:part10_rch}\] For \(r<r_{\mathrm{ch}}\) the flux is near its maximal admissible magnitude. In the hard-limiter idealization, \[\|\mathbf{S}\| \le c e_{\mathrm{a}} \quad\Longrightarrow\quad \text{in the fully choked limit: }\ \ \mathbf{S}(x,t)\approx -c\,e_{\mathrm{a}}(x,t)\,\hat{\mathbf{d}}_{\mathrm{in}}(x,t), \label{eq:part10_choked_flux}\] where \(\hat{\mathbf{d}}_{\mathrm{in}}\) is the inward direction (for spherical inflow, \(\hat{\mathbf{d}}_{\mathrm{in}}=\hat r\) and \(S_r\approx -c e_{\mathrm{a}}\)).

10.1.1.3 Interpretation.

If in the inner region the effective driving is inward (e.g. \(\mathbf{F}_{\mathrm{eff}}\approx -\nabla\Phi_{\mathrm{eff}}\) is inward) then outward transport cannot exceed the same capacity bound, producing an effective trapping in which signals/outflows are throughput-limited. This is the reactor analogue of an “apparent horizon.”

10.1.2.1 Saturation gate for \(\Gamma\).

Let \(\Gamma_{\max}(x,t)\) be the declared conversion capacity and \(\theta_{\mathrm{sat}}\in(0,1)\) a LOCK threshold. Define: \[G_{\mathrm{sat}}(x,t):= H\!\left(\frac{\Gamma(x,t;e_{\mathrm{a}})}{\Gamma_{\max}(x,t)+\eta_0}-\theta_{\mathrm{sat}}\right). \label{eq:part10_gate_sat}\]

10.1.2.2 Why a decomposition channel is needed.

If inflow is choked ([eq:part10_choked_flux]) and conversion is saturated ([eq:part10_gate_sat]), then the inner region cannot increase throughput nor processing rate to accommodate additional accumulation. A third mechanism is required to prevent indefinite concentration: a channel that reduces the actor content (here represented by \(\rho\) and/or \(e_{\mathrm{a}}\)) by transferring it into background-like reservoir(s), i.e. “primordial volume”.

We therefore introduce in §10.2 a decomposition rate \(\Lambda_{\mathrm{pv}}(x,t)\) that converts a portion of stored content \(\rho\) into \(e_{\mathrm{pv}}\), triggered by a reactor gate.

10.1.3.1 Reactor region indicator.

For radius \(r=\|x\|\), define \[G_R(x):=H(r_R-r) \qquad \text{(hard reactor gate)}. \label{eq:part10_GR}\] A soft version can replace \(H\) by a logistic gate; the hard form suffices for analytic statements.

10.1.3.2 Minimum scale regularization.

To avoid unphysical delta-like gradients, evaluate decision variables on coarse-grained fields at scale \(\ell_{\min}\): \[q_{\ell_{\min}}(x,t):=\frac{1}{|B_{\ell_{\min}}|}\int_{B_{\ell_{\min}}(x)} q(y,t)\,dy, \qquad |B_{\ell_{\min}}|=\frac{4\pi}{3}\ell_{\min}^3. \label{eq:part10_coarse_grain}\] We will use this to bound \(\nabla\Delta\) and hence \(\mathbf{g}_{\mathrm{def}}\).

10.1.4.1 Proposition 10.1 (finite deficit acceleration under minimum scale).

Assume:

1. \(0\le \Delta(x,t)\le \Delta_{\max}\le 1\) (automatic if \(e_{\mathrm{bg}}\in[0,1]\)),

2. \(\Delta\) is evaluated as \(\Delta_{\ell_{\min}}\) from [eq:part10_coarse_grain], so that \(\Delta_{\ell_{\min}}\) is Lipschitz with constant \(\le \Delta_{\max}/\ell_{\min}\) in the sense \[\|\nabla \Delta_{\ell_{\min}}(x,t)\|\le \frac{\Delta_{\max}}{\ell_{\min}},\]

3. the deficit potential coupling is [eq:part10_Phi_def].

Then the deficit acceleration is uniformly bounded: \[\|\mathbf{g}_{\mathrm{def}}(x,t)\| = \alpha_\Phi c^2 \|\nabla\Delta_{\ell_{\min}}(x,t)\| \le \alpha_\Phi c^2\,\frac{\Delta_{\max}}{\ell_{\min}}. \label{eq:part10_gdef_bound}\]

10.1.4.2 Proof.

Immediate from [eq:part10_Phi_def] and the Lipschitz bound on \(\Delta_{\ell_{\min}}\). \(\square\)

10.1.4.3 Interpretation.

The VP deficit field cannot generate an infinite inward acceleration if (i) \(\Delta\) is bounded by construction and (ii) the theory declares a minimum operational scale \(\ell_{\min}\). This is the core mathematical reason the “central singularity” is replaced by an effective core.

10.1.4.4 Additional bound from choking (finite flux and finite mean speed).

In any regime where the speed-support postulate holds (PART 08), the flux bound \[\|\mathbf{S}\|\le c\,e_{\mathrm{a}} \label{eq:part10_flux_bound}\] implies the mean active transport velocity \(\mathbf{u}:=\mathbf{S}/(e_{\mathrm{a}}+\eta_0)\) satisfies \(\|\mathbf{u}\|\lesssim c\). Thus no “infinite inflow speed” singularity can appear in the active transport channel.

10.1.5.1 Reactor-modified phase equations (steady, radial).

Inside the reactor, the steady phase equations take the form \[\begin{aligned} 0 &= -\mu(r)\rho(r) + \Gamma(e_{\mathrm{a}}(r)) - \Lambda_{\mathrm{pv}}(r)\rho(r), \label{eq:part10_steady_rho}\\ \frac{1}{r^2}\frac{d}{dr}\big(r^2 S_r(r)\big) &= \mu(r)\rho(r)-\Gamma(e_{\mathrm{a}}(r)) + \mu_{\mathrm{pv}}(r)\,e_{\mathrm{pv}}(r)-\Lambda_{\mathrm{a}}(r)e_{\mathrm{a}}(r), \label{eq:part10_steady_ea_div}\end{aligned}\] where \(\mu_{\mathrm{pv}}\) is an optional activation of primordial volume into the active channel and \(\Lambda_{\mathrm{a}}\) is an optional active\(\to\)PV sink; both are defined later and can be set to \(0\) in the minimal reactor.

10.1.5.2 Eliminating \(\rho\) in the simplest minimal case.

Take the minimal case \(\mu_{\mathrm{pv}}=\Lambda_{\mathrm{a}}=0\). Then [eq:part10_steady_rho] gives \[\rho(r)=\frac{\Gamma(e_{\mathrm{a}}(r))}{\mu(r)+\Lambda_{\mathrm{pv}}(r)}. \label{eq:part10_rho_from_ea}\] Substituting into [eq:part10_steady_ea_div] yields \[\frac{1}{r^2}\frac{d}{dr}\big(r^2 S_r(r)\big) = \mu(r)\frac{\Gamma(e_{\mathrm{a}}(r))}{\mu(r)+\Lambda_{\mathrm{pv}}(r)}-\Gamma(e_{\mathrm{a}}(r)) = -\Gamma(e_{\mathrm{a}}(r))\,\frac{\Lambda_{\mathrm{pv}}(r)}{\mu(r)+\Lambda_{\mathrm{pv}}(r)}\le 0. \label{eq:part10_divS_sign}\] In the fully choked approximation \(S_r\approx -c e_{\mathrm{a}}\) this becomes \[-\frac{c}{r^2}\frac{d}{dr}\big(r^2 e_{\mathrm{a}}(r)\big) = -\Gamma(e_{\mathrm{a}}(r))\,\frac{\Lambda_{\mathrm{pv}}(r)}{\mu(r)+\Lambda_{\mathrm{pv}}(r)},\] or equivalently \[\frac{c}{r^2}\frac{d}{dr}\big(r^2 e_{\mathrm{a}}(r)\big) = \Gamma(e_{\mathrm{a}}(r))\,\frac{\Lambda_{\mathrm{pv}}(r)}{\mu(r)+\Lambda_{\mathrm{pv}}(r)}\ge 0. \label{eq:part10_ea_ode_choked}\] Thus \(r^2 e_{\mathrm{a}}(r)\) is nondecreasing outward. Regularity at the center requires \(r^2 e_{\mathrm{a}}(r)\to 0\) as \(r\to 0\) (since the flux through a vanishing sphere must vanish), which is consistent with \(e_{\mathrm{a}}(r)\) remaining finite. This is a concrete mechanism: decomposition (\(\Lambda_{\mathrm{pv}}>0\)) forces a sign structure that avoids pathological inward growth.

10.2.2.1 Algebraic background.

Given [eq:part10_normalization], \(e_{\mathrm{bg}}^{\mathrm{reg}}\) is determined as \[e_{\mathrm{bg}}^{\mathrm{reg}}(x,t)=1-\rho(x,t)-e_{\mathrm{a}}(x,t)-e_{\mathrm{pv}}(x,t), \qquad e_{\mathrm{bg}}(x,t)=1-\rho(x,t)-e_{\mathrm{a}}(x,t). \label{eq:part10_bg_reg_def}\]

10.2.2.2 Positivity and admissibility constraints (GATE).

The system must satisfy: \[0\le \rho,\ e_{\mathrm{a}},\ e_{\mathrm{pv}},\ e_{\mathrm{bg}}^{\mathrm{reg}} \le 1, \qquad \rho+e_{\mathrm{a}}+e_{\mathrm{pv}}\le 1, \label{eq:part10_positivity}\] and all rates must be nonnegative and locally Lipschitz: \[\mu,\ \Lambda_{\mathrm{pv}},\ \Lambda_{\mathrm{a}},\ \mu_{\mathrm{pv}}\ge 0, \qquad \Gamma(\cdot)\ge 0,\quad \Gamma(0)=0,\quad \Gamma\le \Gamma_{\max}. \label{eq:part10_rate_admissibility}\] Any closure for \(\mathbf{T}\) must respect the speed-limit realizability constraints (e.g. \(\mathrm{tr}(\mathbf{T})\le c^2 e_{\mathrm{a}}\)).

10.2.3.1 Defining a decomposition gate.

Choose thresholds: \[\Delta_c\in(0,1),\quad \theta_{\mathrm{choke}}\in(0,1),\quad \theta_{\mathrm{sat}}\in(0,1),\] and a reactor radius \(r_R\). Define: \[G_{\mathrm{dec}}(x,t) := G_R(x)\; H\!\big(\Delta(x,t)-\Delta_c\big)\; G_{\mathrm{choke}}(x,t)\; G_{\mathrm{sat}}(x,t), \label{eq:part10_Gdec}\] with \(G_R\) from [eq:part10_GR], \(G_{\mathrm{choke}}\) from [eq:part10_gate_choke], and \(G_{\mathrm{sat}}\) from [eq:part10_gate_sat].

10.2.3.2 Decomposition rate.

A minimal choice is \[\Lambda_{\mathrm{pv}}(x,t) := \Lambda_0\,G_{\mathrm{dec}}(x,t), \qquad \Lambda_0>0\ \text{(\textsf{LOCK})}. \label{eq:part10_Lambda_pv_def}\] A smooth version may replace \(H\) by soft gates; the functional form must be LOCKed.

10.2.3.3 Interpretation.

The decomposition channel is not a free knob: it requires simultaneous evidence of (i) large deficit, (ii) choked throughput, and (iii) saturated conversion. This makes the “reactor” falsifiable: if these conditions are not realized (or not realized where needed), the decomposition does not turn on.

10.2.4.1 Activation gate for \(\mu\).

Define a gate (one minimal option): \[G_\mu(x,t) := G_R(x)\; H\!\big(\Delta(x,t)-\Delta_\mu\big), \qquad \Delta_\mu\in(0,1). \label{eq:part10_Gmu}\] Then set \[\mu(x,t):=\mu_{\mathrm{out}} + (\mu_{\mathrm{in}}-\mu_{\mathrm{out}})\,G_\mu(x,t), \qquad 0\le \mu_{\mathrm{out}}\le \mu_{\mathrm{in}}, \label{eq:part10_mu_profile}\] with \(\mu_{\mathrm{out}},\mu_{\mathrm{in}}\) LOCKed per version.

10.2.4.2 Interpretation.

If \(\mu\) is small outside but large inside, stored content is “activated” into the mobile channel primarily in the reactor core. This converts a static accumulation problem into a throughput-limited transport problem, which then couples naturally to choking/jet production.

10.2.5.1 Axis and jet gate.

Let \(k(x,t)\) be the local axis direction (unit vector). Define a jet gate \(G_{\mathrm{jet}}(x,t)\in[0,1]\) using alignment/leakage diagnostics (as in PART 08.6). We will reuse the hard form: \[G_{\mathrm{jet}} = H\!\big(A-(1-\varepsilon_A^{\mathrm{jet}})\big)\; H\!\big(\varepsilon_k^{\mathrm{jet}}-a_k\big)\; H\!\big(\varepsilon_\perp^{\mathrm{jet}}-\ell_\perp\big), \label{eq:part10_Gjet}\] where \(A,a_k,\ell_\perp\) are alignment/leakage diagnostics and \(\varepsilon_*^{\mathrm{jet}}\) are LOCK thresholds.

10.2.5.2 Recycling/activation rate.

A minimal model is \[\mu_{\mathrm{pv}}(x,t) := \mu_{\mathrm{pv},0}\,G_R(x)\,G_{\mathrm{jet}}(x,t), \qquad \mu_{\mathrm{pv},0}\ge 0\ \text{(\textsf{LOCK})}. \label{eq:part10_mu_pv_def}\] This forces PV\(\to\)active activation to occur only in the reactor and only when jet conditions are satisfied, avoiding an unconstrained global reactivation.

10.3.1.1 Local occupancy conservation.

By definition [eq:part10_normalization], \[\rho+e_{\mathrm{a}}+e_{\mathrm{bg}}^{\mathrm{reg}}+e_{\mathrm{pv}} \equiv 1,\] so local occupancy is a LOCK constraint.

10.3.1.2 Actor content and its balance.

Define the actor content \[e_{\mathrm{act}}(x,t):=\rho(x,t)+e_{\mathrm{a}}(x,t). \label{eq:part10_eact}\] Adding [eq:part10_rho_PDE] and [eq:part10_ea_PDE] gives the local balance \[\partial_t e_{\mathrm{act}} + \nabla\cdot\mathbf{S} = -\Lambda_{\mathrm{pv}}\,\rho -\Lambda_{\mathrm{a}}\,e_{\mathrm{a}} +\mu_{\mathrm{pv}}\,e_{\mathrm{pv}}. \label{eq:part10_actor_balance}\] Thus, decomposition terms reduce actor content while PV reactivation increases it.

10.3.1.3 Primordial volume balance.

From [eq:part10_epv_PDE], \[\partial_t e_{\mathrm{pv}} = +\Lambda_{\mathrm{pv}}\,\rho +\Lambda_{\mathrm{a}}\,e_{\mathrm{a}} -\mu_{\mathrm{pv}}\,e_{\mathrm{pv}}. \label{eq:part10_pv_balance}\]

10.3.1.4 Conserved “reactor charge.”

Summing [eq:part10_actor_balance] and [eq:part10_pv_balance] yields \[\partial_t(\rho+e_{\mathrm{a}}+e_{\mathrm{pv}})+\nabla\cdot\mathbf{S}=0. \label{eq:part10_reactor_charge_conservation}\] Therefore the quantity \[q_R(x,t):=\rho(x,t)+e_{\mathrm{a}}(x,t)+e_{\mathrm{pv}}(x,t) \label{eq:part10_qR}\] is transported by \(\mathbf{S}\) with no net sources/sinks; it is the reactor charge.

10.3.1.5 Integral form on a control volume.

Let \(\Omega\) be any fixed control volume with boundary \(\partial\Omega\) and outward normal \(\mathbf{n}\). Integrating [eq:part10_reactor_charge_conservation] yields \[\frac{d}{dt}\int_{\Omega} q_R\,dV + \int_{\partial\Omega}\mathbf{S}\cdot\mathbf{n}\,dA =0. \label{eq:part10_integral_qR}\] Thus all net change in reactor charge is due to boundary throughput, not internal conversion. Internal conversion only repartitions \(q_R\) between \((\rho,e_{\mathrm{a}},e_{\mathrm{pv}})\).

10.3.2.1 Energy scales and definitions (SPEC, version LOCK).

Let \(u_*>0\) be a reference energy density (units \(ML^{-1}T^{-2}\)). Define dimensionless specific energies (per unit phase fraction): \[\epsilon_\rho,\ \epsilon_{\mathrm{a}},\ \epsilon_{\mathrm{pv}}\ \ge 0,\] and define the dimensionless internal energy density of VP phases: \[\varepsilon_{\mathrm{VP}}(x,t) := \epsilon_\rho\,\rho + \epsilon_{\mathrm{a}}\,e_{\mathrm{a}} + \epsilon_{\mathrm{pv}}\,e_{\mathrm{pv}}. \label{eq:part10_eps_VP}\] The corresponding physical energy density is \(u_{\mathrm{VP}}:=u_*\varepsilon_{\mathrm{VP}}\).

10.3.2.2 Energy flux carried by the active channel.

Since only the active channel is transported by \(\mathbf{S}\), define the VP advective energy flux \[\mathbf{J}_{\mathrm{VP}} := u_*\,\epsilon_{\mathrm{a}}\,\mathbf{S}. \label{eq:part10_J_VP}\] This is the minimal consistent choice: energy per unit active fraction times active flux.

10.3.2.3 Radiation reservoir.

Let \(u_\gamma(x,t)=u_*\varepsilon_\gamma(x,t)\) be the radiation energy density with flux \(\mathbf{J}_\gamma\): \[\partial_t \varepsilon_\gamma + \nabla\cdot\mathbf{j}_\gamma = \mathcal{P}_\gamma, \qquad \mathbf{J}_\gamma := u_*\,\mathbf{j}_\gamma, \label{eq:part10_radiation_ledger}\] where \(\mathcal{P}_\gamma\) is the dimensionless power density injected into radiation.

10.3.2.4 Energy exchange from reactor conversions.

Multiply [eq:part10_rho_PDE] by \(\epsilon_\rho\), [eq:part10_ea_PDE] by \(\epsilon_{\mathrm{a}}\), and [eq:part10_epv_PDE] by \(\epsilon_{\mathrm{pv}}\), add them, and use \(\nabla\cdot(\epsilon_{\mathrm{a}}\mathbf{S})\) from the \(e_{\mathrm{a}}\) equation. One obtains \[\partial_t \varepsilon_{\mathrm{VP}} + \nabla\cdot(\epsilon_{\mathrm{a}}\mathbf{S}) = (\epsilon_{\mathrm{a}}-\epsilon_\rho)\,\mu\rho +(\epsilon_\rho-\epsilon_{\mathrm{a}})\,\Gamma +(\epsilon_{\mathrm{pv}}-\epsilon_\rho)\,\Lambda_{\mathrm{pv}}\rho +(\epsilon_{\mathrm{pv}}-\epsilon_{\mathrm{a}})\,\Lambda_{\mathrm{a}}e_{\mathrm{a}} +(\epsilon_{\mathrm{a}}-\epsilon_{\mathrm{pv}})\,\mu_{\mathrm{pv}}e_{\mathrm{pv}}. \label{eq:part10_energy_exchange_VP}\] The right-hand side represents energy not carried by the advective VP energy flux. To enforce total energy conservation, we define the radiation injection as \[\mathcal{P}_\gamma := -\Big[ (\epsilon_{\mathrm{a}}-\epsilon_\rho)\,\mu\rho +(\epsilon_\rho-\epsilon_{\mathrm{a}})\,\Gamma +(\epsilon_{\mathrm{pv}}-\epsilon_\rho)\,\Lambda_{\mathrm{pv}}\rho +(\epsilon_{\mathrm{pv}}-\epsilon_{\mathrm{a}})\,\Lambda_{\mathrm{a}}e_{\mathrm{a}} +(\epsilon_{\mathrm{a}}-\epsilon_{\mathrm{pv}})\,\mu_{\mathrm{pv}}e_{\mathrm{pv}} \Big], \label{eq:part10_Pgamma_def}\] i.e. radiation receives the residual energy not accounted for by advective VP transport.

10.3.2.5 Total energy conservation (local).

Define the total dimensionless energy density \[\varepsilon_{\mathrm{tot}}:=\varepsilon_{\mathrm{VP}}+\varepsilon_\gamma, \qquad \mathbf{j}_{\mathrm{tot}}:=\epsilon_{\mathrm{a}}\mathbf{S}+\mathbf{j}_\gamma. \label{eq:part10_total_energy_def}\] Then [eq:part10_energy_exchange_VP] and [eq:part10_radiation_ledger] with [eq:part10_Pgamma_def] imply the conservation law \[\partial_t \varepsilon_{\mathrm{tot}} + \nabla\cdot \mathbf{j}_{\mathrm{tot}} = 0. \label{eq:part10_total_energy_conservation}\] In physical units: \[\partial_t u_{\mathrm{tot}} + \nabla\cdot \mathbf{J}_{\mathrm{tot}} = 0, \qquad u_{\mathrm{tot}}:=u_*\varepsilon_{\mathrm{tot}}, \quad \mathbf{J}_{\mathrm{tot}}:=u_*\mathbf{j}_{\mathrm{tot}}.\]

10.3.2.6 Interpretation.

The reactor may convert stored content into primordial volume in an exothermic or endothermic way depending on \((\epsilon_\rho,\epsilon_{\mathrm{pv}})\). Any difference is automatically transferred into radiation via [eq:part10_Pgamma_def], yielding a well-defined luminosity output on a control surface.

10.4.1.1 (J1) Strong alignment and low transverse leakage.

The jet gate \(G_{\mathrm{jet}}\) from [eq:part10_Gjet] must be ON: \[G_{\mathrm{jet}}=1. \label{eq:part10_J1}\]

10.4.1.2 (J2) Available power source.

There must be a nonzero conversion/decomposition power density in the core, i.e. at least one of \[\mu\rho,\ \Lambda_{\mathrm{pv}}\rho,\ \mu_{\mathrm{pv}}e_{\mathrm{pv}}\] must be non-negligible in the jet-launch region. In the energy ledger, this is the requirement that the total outward power on \(\Sigma\) is positive: \[L_{\mathrm{rad}}(t)+P_{\mathrm{VP}}(t)>0. \label{eq:part10_J2}\]

10.4.1.3 (J3) Open axial transport channel (anisotropic drag/permeability).

Let \(\mathbf{B}\) be the drag tensor in the \(\mathbf{S}\) equation. Define its axial and transverse components: \[B_\parallel := k\cdot \mathbf{B}k, \qquad \mathbf{B}_\perp := (\mathbf{I}-k\otimes k)\mathbf{B}(\mathbf{I}-k\otimes k).\] A minimal “open axial channel” condition is \[B_\parallel \ll \|\mathbf{B}_\perp\|, \label{eq:part10_open_channel}\] within the launching region. This is a SPEC closure/gate statement: jets require anisotropic permeability.

10.4.1.4 (J4) Throughput capacity not exceeded (axial choking consistency).

Jet throughput is limited by the same speed constraint: \[|\mathbf{S}\cdot k|\le \|\mathbf{S}\|\le c e_{\mathrm{a}}. \label{eq:part10_jet_capacity_pointwise}\]

10.4.3.1 Interpretation.

Equation [eq:part10_RSreac] is the minimal statement: decomposition acts as a source of directed flux along the jet axis. In a more detailed closure, \(\zeta_{\mathrm{jet}}\) could depend on alignment and geometry; in this minimal model it is a LOCK coefficient.

10.5.2.1 Energy cap.

Using [eq:part10_Pvp], the VP-carried power through \(\Sigma\) is bounded: \[|P_{\mathrm{VP}}(t)| = u_*\,\epsilon_{\mathrm{a}}\left|\int_\Sigma \mathbf{S}\cdot\mathbf{n}\,dA\right| \le u_*\,\epsilon_{\mathrm{a}}\,c\,\mathrm{Area}(\Sigma). \label{eq:part10_Pvp_cap}\] This is an emergent, purely throughput-based cap (distinct in principle from the Eddington limit). Its observational relevance is a gate question, not assumed.

10.5.3.1 Recycling-center interpretation.

In steady state, the reactor maintains a nontrivial \(e_{\mathrm{pv}}\) reservoir in the core by balancing creation \(\Lambda_{\mathrm{pv}}\rho\) and reactivation \(\mu_{\mathrm{pv}}e_{\mathrm{pv}}\). The latter can directly fuel the mobile channel and hence jets, while the energy mismatch is carried by radiation via [eq:part10_total_energy_conservation].

10.6.1.1 Relevance.

Reactor operation typically increases background-like content (\(e_{\mathrm{bg}}^{\mathrm{reg}}\) and/or \(e_{\mathrm{pv}}\)) while redistributing \(\rho\) and \(e_{\mathrm{a}}\). The mixture entropy provides a quantitative record of how “mixed” the local composition becomes.

10.6.2.1 Proposition 10.2 (entropy production for diffusion).

Define \(F(t):=\int_\Omega e_{\mathrm{a}}\ln e_{\mathrm{a}}\,dV\). Then \[\frac{dF}{dt} = -\int_{\Omega} D\,\frac{\|\nabla e_{\mathrm{a}}\|^2}{e_{\mathrm{a}}}\,dV \le 0. \label{eq:part10_diffusion_entropy_ineq}\] Equivalently, \(\int_\Omega (-e_{\mathrm{a}}\ln e_{\mathrm{a}})\,dV\) is nondecreasing.

10.6.2.2 Proof.

Differentiate: \[\frac{d}{dt}\int e_{\mathrm{a}}\ln e_{\mathrm{a}}\,dV = \int (1+\ln e_{\mathrm{a}})\,\partial_t e_{\mathrm{a}}\,dV = \int (1+\ln e_{\mathrm{a}})\,\nabla\cdot(D\nabla e_{\mathrm{a}})\,dV.\] Integrate by parts using the no-flux boundary condition: \[= -\int D\nabla(1+\ln e_{\mathrm{a}})\cdot\nabla e_{\mathrm{a}}\,dV = -\int D\,\frac{\|\nabla e_{\mathrm{a}}\|^2}{e_{\mathrm{a}}}\,dV \le 0.\] \(\square\)

10.6.2.3 Interpretation.

Mixing/diffusion irreversibly dissipates gradients and increases a natural entropy functional. This provides a rigorous monotonicity statement usable as a GATE in numerical implementations.

10.6.3.1 Gated claim (preview).

Under detailed-balance-like constraints on the reaction rates (to be stated in PART 17), \(I(t)\) is a Lyapunov functional that is nonincreasing: \[\frac{dI}{dt}\le 0, \label{eq:part10_I_decay_claim}\] with strict decay unless the system is at equilibrium. In this Part we do not assert detailed balance as a fact; we define the information ledger \(I(t)\) so that a future Part can specify and gate the conditions under which [eq:part10_I_decay_claim] holds.

10.6.4.1 Link forward (PART 17).

PART 17 will formalize:

1. the exact definition of the information ledger appropriate to the VP distribution-level description,

2. the conditions for entropy production and the analog of an \(H\)-theorem in the LOCK/DERIVE regime,

3. and how the reactor core changes the accounting (e.g. by adding PV reservoirs and gating transport).

The present Part provides the minimal variables and conservation structure needed for that completion.

11.1.1.1 (A) Global spin direction from angular momentum.

Let a central object (or inner control volume) have physical angular momentum vector \(\mathbf{L}(t)\neq 0\) (units \(ML^2T^{-1}\)). Define the unit spin direction \[\hat{\mathbf{s}}(t):=\frac{\mathbf{L}(t)}{\|\mathbf{L}(t)\|}. \label{eq:part11_spin_global}\]

11.1.1.2 (B) Local spin direction from VP vorticity.

Define the VP mean transport velocity \(\mathbf{u}\) by [eq:part11_speedlimit]. Its vorticity is \[\boldsymbol{\omega}(x,t):=\nabla\times \mathbf{u}(x,t), \qquad \hat{\mathbf{s}}(x,t):= \frac{\boldsymbol{\omega}(x,t)}{\|\boldsymbol{\omega}(x,t)\|+\eta_0}. \label{eq:part11_spin_local}\] Here \(\hat{\mathbf{s}}(x,t)\) is well-defined wherever \(\|\boldsymbol{\omega}\|\) is not too small; \(\eta_0\) prevents division by zero.

11.1.1.3 Spin gate (rotation present).

Jets in this Part are spin-induced. We declare a LOCK threshold \(\omega_{\min}>0\) and define \[G_{\mathrm{spin}}(x,t):=H\!\big(\|\boldsymbol{\omega}(x,t)\|-\omega_{\min}\big) \quad\text{or}\quad G_{\mathrm{spin}}(t):=H\!\big(\|\mathbf{L}(t)\|-L_{\min}\big), \label{eq:part11_Gspin}\] depending on whether a local vorticity or global angular momentum diagnostic is used. When \(G_{\mathrm{spin}}=0\), this Part predicts that persistent, narrow, spin-locked jets should not be forced (see §11.6).

11.1.2.1 Distribution-level objects (minimal).

Let \(f(x,v,t)\ge 0\) be the active-phase velocity distribution (PART 04), normalized so that \[e_{\mathrm{a}}(x,t)=\int_{\mathbb{R}^3} f(x,v,t)\,dv. \label{eq:part11_ea_from_f}\] Let \(b(x,v,t)\) be the (dimensionless) alignment kernel entering the alignment moment(s). The LOCK axisymmetric kernel assumption is: \[b(x,v,t)=\tilde b\!\big(x,\,v\cdot k(x,t),\,t\big), \label{eq:part11_axisymmetric_b}\] i.e. \(b\) is invariant under rotations of \(v\) about \(k\).

11.1.2.2 Alignment moment.

Define the (first) alignment moment vector \[\mathbf{m}_b(x,t):=\int_{\mathbb{R}^3} v\, b(x,v,t)\, f(x,v,t)\,dv. \label{eq:part11_mb_def}\] (Other moments can be defined similarly; the present lemma uses only [eq:part11_mb_def].)

11.1.2.3 Lemma 11.1 (axisymmetry forces \(\mathbf{m}_b\parallel k\)).

Assume:

1. \(k(x,t)\) is a unit vector,

2. \(b\) is axisymmetric about \(k\) as in [eq:part11_axisymmetric_b],

3. \(f(x,v,t)\) is itself axisymmetric about \(k\) in the sense that \(f(x,Rv,t)=f(x,v,t)\) for any rotation \(R\) with \(Rk=k\) (this is the “axisymmetric kernel regime”).

Then \(\mathbf{m}_b(x,t)\) is parallel to \(k(x,t)\): \[\mathbf{m}_b(x,t) = m_b(x,t)\,k(x,t) \quad\text{for some scalar }m_b(x,t)\in\mathbb{R}. \label{eq:part11_mb_parallel}\]

11.1.2.4 Proof.

Fix \((x,t)\) and choose coordinates so that \(k=\hat z\). Write \(v=(v_\perp\cos\phi,\,v_\perp\sin\phi,\,v_\parallel)\) in cylindrical velocity coordinates, where \(v_\parallel=v\cdot k\) and \(v_\perp=\|v-(v\cdot k)k\|\). By axisymmetry, both \(f\) and \(b\) depend on \(\phi\) only through invariants, hence are independent of \(\phi\): \[f=f(v_\perp,v_\parallel),\qquad b=\tilde b(v_\parallel).\] Compute \(\mathbf{m}_b\): \[\mathbf{m}_b=\int v\,b\,f\,dv.\] The \(x\)-component is proportional to \(\int v_\perp\cos\phi \,\tilde b(v_\parallel)\,f(v_\perp,v_\parallel)\,dv\), and similarly the \(y\)-component to \(\int v_\perp\sin\phi \,\tilde b(v_\parallel)\,f\,dv\). Since \(\int_0^{2\pi}\cos\phi\,d\phi=0\) and \(\int_0^{2\pi}\sin\phi\,d\phi=0\), the transverse components vanish: \[(\mathbf{m}_b)_x=(\mathbf{m}_b)_y=0.\] Only the \(z\)-component can survive: \[(\mathbf{m}_b)_z=\int v_\parallel \,\tilde b(v_\parallel)\,f(v_\perp,v_\parallel)\,dv.\] Therefore \(\mathbf{m}_b\) is parallel to \(k=\hat z\). Returning to general coordinates gives [eq:part11_mb_parallel]. \(\square\)

11.1.2.5 Interpretation.

In the axisymmetric kernel regime, the only distinguished direction is \(k\). Therefore any odd-in-\(v\) vector moment must align with \(k\). This is the minimal mathematical content behind “a jet is forced to be axial” once a strong axis \(k\) exists.

11.1.3.1 Spin-alignment energy.

Define the misalignment energy density (dimensionless) as \[\mathcal{E}_{\mathrm{spin}}(k,\hat{\mathbf{s}}) := \frac{\kappa_s}{2}\,\|k\times \hat{\mathbf{s}}\|^2 = \frac{\kappa_s}{2}\,\big(1-(k\cdot \hat{\mathbf{s}})^2\big), \qquad \kappa_s\ge 0\ \text{(\textsf{LOCK})}. \label{eq:part11_Espin}\] Minima occur at \(k=\pm \hat{\mathbf{s}}\).

11.1.3.2 Projected relaxation equation.

To preserve \(\|k\|=1\), we evolve \(k\) on the unit sphere using the projection operator \(P_\perp(k):=I-k\otimes k\). A minimal transport+relaxation law is \[\partial_t k + (\mathbf{u}\cdot\nabla)k = -\frac{1}{\tau_k}\,P_\perp(k)\,\nabla_k \mathcal{E}_{\mathrm{spin}}(k,\hat{\mathbf{s}}) = -\frac{\kappa_s}{\tau_k}\,P_\perp(k)\,\big(-(k\cdot\hat{\mathbf{s}})\hat{\mathbf{s}}\big), \label{eq:part11_k_evolution_preproj}\] where \(\tau_k>0\) is a LOCK relaxation time and \(\nabla_k\) denotes the gradient with respect to \(k\) in \(\mathbb{R}^3\). Using \(P_\perp(k)\hat{\mathbf{s}}=\hat{\mathbf{s}}-(k\cdot\hat{\mathbf{s}})k\), this becomes \[\partial_t k + (\mathbf{u}\cdot\nabla)k = \frac{\kappa_s}{\tau_k}(k\cdot\hat{\mathbf{s}})\big(\hat{\mathbf{s}}-(k\cdot\hat{\mathbf{s}})k\big) = -\frac{\kappa_s}{\tau_k}\,k\times\big(k\times\hat{\mathbf{s}}\big). \label{eq:part11_k_evolution}\] The last form uses the vector identity \(k\times(k\times\hat{\mathbf{s}})=(k\cdot\hat{\mathbf{s}})k-\hat{\mathbf{s}}\).

11.1.3.3 Unit-length invariance.

Dot [eq:part11_k_evolution] with \(k\): \[k\cdot(\partial_t k + (\mathbf{u}\cdot\nabla)k)= -\frac{\kappa_s}{\tau_k}\,k\cdot\big(k\times(k\times\hat{\mathbf{s}})\big)=0,\] so \(\partial_t(\|k\|^2)+(\mathbf{u}\cdot\nabla)(\|k\|^2)=0\). If \(\|k\|=1\) initially, it remains \(1\) along characteristics.

11.1.3.4 Misalignment angle dynamics (local, in a frame where \(\hat{\mathbf{s}}\) is constant).

Define \(\cos\theta:=k\cdot \hat{\mathbf{s}}\). Differentiate along the advective derivative \(D_t:=\partial_t+(\mathbf{u}\cdot\nabla)\): \[D_t(\cos\theta)=D_t(k\cdot\hat{\mathbf{s}})= (D_t k)\cdot \hat{\mathbf{s}},\] and use [eq:part11_k_evolution]: \[(D_t k)\cdot \hat{\mathbf{s}} = -\frac{\kappa_s}{\tau_k}\big(k\times(k\times\hat{\mathbf{s}})\big)\cdot \hat{\mathbf{s}} = -\frac{\kappa_s}{\tau_k}\big((k\cdot\hat{\mathbf{s}})^2-1\big) = \frac{\kappa_s}{\tau_k}\sin^2\theta.\] Thus \[D_t(\cos\theta)=\frac{\kappa_s}{\tau_k}\sin^2\theta, \qquad D_t\theta=-\frac{\kappa_s}{\tau_k}\sin\theta\cos\theta. \label{eq:part11_theta_dynamics}\] For \(0<\theta<\pi/2\) we have \(D_t\theta<0\) so the misalignment decreases. The stable equilibria are \(\theta=0\) (alignment) and \(\theta=\pi\) (anti-alignment); \(\theta=\pi/2\) is unstable.

11.1.3.5 Explicit solution in the simplest case (no advection, constant \(\hat{\mathbf{s}}\)).

If \(D_t=\partial_t\) and \(\hat{\mathbf{s}}\) constant, then from [eq:part11_theta_dynamics]: \[\frac{d\theta}{dt}=-\frac{\kappa_s}{\tau_k}\sin\theta\cos\theta =-\frac{\kappa_s}{2\tau_k}\sin(2\theta).\] Using \(\frac{d}{dt}\ln(\tan\theta)=\frac{1}{\sin\theta\cos\theta}\frac{d\theta}{dt}\), we obtain \[\frac{d}{dt}\ln(\tan\theta)=-\frac{\kappa_s}{\tau_k} \quad\Longrightarrow\quad \tan\theta(t)=\tan\theta(0)\,\exp\!\left(-\frac{\kappa_s}{\tau_k}t\right). \label{eq:part11_theta_solution}\] Therefore the spin-locking time is \(\tau_k/\kappa_s\).

11.1.4.1 Alignment defect \(a_k\) (moment-based definition).

Given \(f(x,v,t)\) and axis \(k(x,t)\), define the longitudinal/transverse second moments \[M_\parallel(x,t):=\int (v\cdot k)^2\,f(x,v,t)\,dv, \qquad M_\perp(x,t):=\int \|v-(v\cdot k)k\|^2\,f(x,v,t)\,dv. \label{eq:part11_Mpar_Mperp}\] Define the alignment defect as the transverse fraction of the kinetic second moment: \[a_k(x,t):=\frac{M_\perp(x,t)}{M_\parallel(x,t)+M_\perp(x,t)}\in[0,1], \qquad 1-a_k=\frac{M_\parallel}{M_\parallel+M_\perp}. \label{eq:part11_ak_def}\] Thus \(a_k\to 0\) means nearly all velocity variance is along \(k\) (strong alignment), while \(a_k\to 1\) means variance is mostly transverse (anti-jet).

11.1.4.2 Alignment quality gate.

Fix a LOCK threshold \(a_{\max}\in(0,1)\) and define \[G_{\mathrm{align}}(x,t):=H(a_{\max}-a_k(x,t)). \label{eq:part11_Galign}\]

11.1.4.3 Spin-locking gate (axis defined and stable).

Fix a LOCK misalignment tolerance \(\theta_{\max}^{(k)}\in(0,\pi/2)\) and define \[G_{k\parallel s}(x,t):= H\!\big(\cos\theta(x,t)-\cos\theta_{\max}^{(k)}\big), \qquad \cos\theta(x,t)=k(x,t)\cdot \hat{\mathbf{s}}(x,t). \label{eq:part11_Gkpar}\]

11.1.4.4 Composite axis gate.

\[G_{\mathrm{axis}}(x,t):=G_{\mathrm{spin}}(x,t)\,G_{k\parallel s}(x,t)\,G_{\mathrm{align}}(x,t). \label{eq:part11_Gaxis}\] When \(G_{\mathrm{axis}}=1\), the system is in a regime where a sharp, spin-locked axis exists and axial moments are forced to be parallel to \(k\) (Lemma 11.1), creating the structural precondition for a jet.

11.2.3.1 Axis coordinate and local cross-sections.

Assume in the thin-tube regime that \(k\) varies slowly along the tube, so a local axial coordinate \(z\) can be defined with \(\partial_z\) approximately equal to \(k\cdot\nabla\). Each cross-section \(\Sigma(z,t)\) is approximately orthogonal to \(k\) at that \(z\).

11.2.3.2 No-leak condition (defining the tube surface).

We define the tube side surface as a (nearly) material surface for the transported charge: \[\big(\mathbf{S}-q\mathbf{w}\big)\cdot \mathbf{n}=0 \quad\text{on }\Sigma_{\mathrm{side}}(t), \label{eq:part11_no_leak}\] i.e. there is no net charge flux through the side boundary in the frame moving with the tube boundary.

In steady tubes with fixed boundaries (\(\mathbf{w}=0\)), [eq:part11_no_leak] reduces to \[\mathbf{S}\cdot \mathbf{n}=0\quad\text{on }\Sigma_{\mathrm{side}}.\]

11.2.4.1 Steady, fixed tube (canonical jet-tube invariant).

If the tube is steady and fixed, then \(\mathbf{w}=0\) and \(\frac{d}{dt}\int_{\Omega_T} q\,dV=0\), so \[\int_{\Sigma_+}\mathbf{S}\cdot\mathbf{n}\,dA + \int_{\Sigma_-}\mathbf{S}\cdot\mathbf{n}\,dA =0. \label{eq:part11_inout_balance}\] Choosing \(\mathbf{n}\) to point outward, the inlet has \(\mathbf{n}\approx -k\) and the outlet has \(\mathbf{n}\approx +k\). Therefore, defining the axial throughput (axial flux invariant) \[\mathcal{J}(z) := \int_{\Sigma(z)} \mathbf{S}\cdot k\,dA, \label{eq:part11_J_def}\] equation [eq:part11_inout_balance] implies \[\mathcal{J}(z_2)=\mathcal{J}(z_1) \quad\text{for any two cross-sections }z_1,z_2 \text{ in the steady no-leak tube.} \label{eq:part11_J_invariant}\] This is the jet-tube axial invariant: axial throughput is constant along the tube.

11.2.4.2 Non-steady or leaky tubes (diagnostic form).

If the tube is not steady or leakage is nonzero, [eq:part11_tube_balance_full] provides a diagnostic: \[\mathcal{J}(z_2)-\mathcal{J}(z_1) = -\frac{d}{dt}\int_{\Omega(z_1,z_2)} q\,dV - \int_{\Sigma_{\mathrm{side}}(z_1,z_2)}(\mathbf{S}-q\mathbf{w})\cdot\mathbf{n}\,dA, \label{eq:part11_J_diagnostic}\] where \(\Omega(z_1,z_2)\) is the sub-tube between cross-sections \(z_1\) and \(z_2\). Thus deviations from invariance directly measure unsteadiness and side leakage.

11.3.1.1 Proposition 11.2 (Cauchy–Schwarz collimation bound).

Let \(f(x,v,t)\ge 0\) be such that \(e_{\mathrm{a}}=\int f\,dv>0\), and define \(\mathbf{S}\) as the first velocity moment: \[\mathbf{S}(x,t):=\int v\, f(x,v,t)\,dv. \label{eq:part11_S_from_f}\] Let \(M_\parallel,M_\perp\) be defined by [eq:part11_Mpar_Mperp]. Then: \[\|\mathbf{S}_\perp(x,t)\|^2 \le e_{\mathrm{a}}(x,t)\,M_\perp(x,t), \qquad |\mathbf{S}_\parallel(x,t)\cdot k(x,t)|^2 \le e_{\mathrm{a}}(x,t)\,M_\parallel(x,t). \label{eq:part11_CS_bounds}\] Consequently, if \(M_\parallel>0\), the flux anisotropy ratio satisfies \[\frac{\|\mathbf{S}_\perp\|}{|\mathbf{S}_\parallel\cdot k|} \le \sqrt{\frac{M_\perp}{M_\parallel}} = \sqrt{\frac{a_k}{1-a_k}}. \label{eq:part11_opening_angle_bound}\]

11.3.1.2 Proof.

Fix \((x,t)\). Write \(v=v_\parallel k+v_\perp\) with \(v_\parallel=v\cdot k\) and \(v_\perp=v-(v\cdot k)k\). Then \[\mathbf{S}_\perp=\int v_\perp f\,dv, \qquad \mathbf{S}_\parallel\cdot k = \int v_\parallel f\,dv.\] By Cauchy–Schwarz, \[\|\mathbf{S}_\perp\|^2 = \left\|\int v_\perp f\,dv\right\|^2 \le \left(\int f\,dv\right)\left(\int \|v_\perp\|^2 f\,dv\right)=e_{\mathrm{a}} M_\perp,\] and similarly \[|\mathbf{S}_\parallel\cdot k|^2 = \left(\int v_\parallel f\,dv\right)^2 \le \left(\int f\,dv\right)\left(\int v_\parallel^2 f\,dv\right)=e_{\mathrm{a}} M_\parallel.\] Divide the square-roots to obtain [eq:part11_opening_angle_bound]. \(\square\)

11.3.1.3 Operational consequence: opening angle \(\leftrightarrow a_k\).

If \(\theta_j\) is small and \(\eta_0\) negligible compared to \(|\mathbf{S}_\parallel|\), then \(\tan\theta_j\approx \|\mathbf{S}_\perp\|/|\mathbf{S}_\parallel|\), and [eq:part11_opening_angle_bound] implies \[\tan\theta_j \lesssim \sqrt{\frac{a_k}{1-a_k}} \quad\Longrightarrow\quad a_k \gtrsim \frac{\tan^2\theta_j}{1+\tan^2\theta_j}=\sin^2\theta_j. \label{eq:part11_ak_from_theta}\] Thus observed opening angles directly constrain the alignment defect: narrower jets require smaller \(a_k\).

11.3.2.1 Projectors.

Define the parallel/perpendicular projectors: \[P_\parallel := k\otimes k, \qquad P_\perp := I - k\otimes k. \label{eq:part11_projectors}\]

11.3.2.2 Anisotropic (jet) closure for \(\mathbf{T}\).

In the strong-alignment regime (PART 07.5), a minimal symmetric closure is \[\mathbf{T} = \kappa_\parallel\,e_{\mathrm{a}}\,P_\parallel + \kappa_\perp\,e_{\mathrm{a}}\,P_\perp, \qquad \kappa_\parallel\ge 0,\ \kappa_\perp\ge 0\ \text{(\textsf{LOCK})}. \label{eq:part11_T_aniso}\] Realizability gate. If \(\mathbf{T}\) represents the second velocity moment, then the speed limit implies \[\mathrm{tr}(\mathbf{T}) = (\kappa_\parallel+2\kappa_\perp)e_{\mathrm{a}} \le c^2 e_{\mathrm{a}}, \qquad\Rightarrow\qquad \kappa_\parallel+2\kappa_\perp\le c^2, \label{eq:part11_T_trace_gate}\] a GATE constraint.

11.3.2.3 Anisotropic drag.

A minimal anisotropic drag tensor is \[\mathbf{B} = B_\parallel\,P_\parallel + B_\perp\,P_\perp, \qquad B_\parallel\ge 0,\ B_\perp\ge 0\ \text{(\textsf{LOCK})}. \label{eq:part11_B_aniso}\] The “open axial channel” condition (PART 10.4) is the collimation requirement \[B_\parallel \ll B_\perp, \label{eq:part11_open_channel}\] meaning transverse transport is strongly damped while axial transport is relatively free.

11.3.3.1 Perpendicular projected equation (with anisotropic closure).

Assume \(k\) varies slowly (thin-tube approximation) so derivatives of \(P_\parallel,P_\perp\) are subleading. Using [eq:part11_T_aniso] and [eq:part11_B_aniso], we obtain approximately \[\partial_t \mathbf{S}_\perp + \nabla_\perp(\kappa_\perp e_{\mathrm{a}}) \approx - B_\perp\,\mathbf{S}_\perp + e_{\mathrm{a}}\mathbf{F}_\perp + \mathbf{R}_\perp, \label{eq:part11_Sperp_eq}\] where \(\nabla_\perp:=P_\perp\nabla\).

11.3.3.2 Quasi-steady transverse balance.

In a steady or slowly varying jet core, take \(\partial_t \mathbf{S}_\perp\approx 0\) and \(\mathbf{R}_\perp\approx 0\) (or treat it as small). Then \[\mathbf{S}_\perp \approx \frac{1}{B_\perp}\Big(e_{\mathrm{a}}\mathbf{F}_\perp-\nabla_\perp(\kappa_\perp e_{\mathrm{a}})\Big). \label{eq:part11_Sperp_balance}\] Therefore a sufficient condition for strong collimation is \[\|\mathbf{S}_\perp\| \ll |\mathbf{S}_\parallel\cdot k| \quad\Leftarrow\quad \frac{1}{B_\perp}\Big(\|e_{\mathrm{a}}\mathbf{F}_\perp\|+\|\nabla_\perp(\kappa_\perp e_{\mathrm{a}})\|\Big) \ll |\mathbf{S}_\parallel\cdot k|. \label{eq:part11_collimation_condition_general}\] This condition becomes easier to satisfy when \(B_\perp\) is large (strong transverse damping), \(\kappa_\perp\) is small (weak transverse “pressure”), and the transverse driving force is small or inward-restoring.

11.3.3.3 Collimation gate (practical).

Fix a LOCK collimation ratio \(\epsilon_{\mathrm{col}}\ll 1\) and define \[G_{\mathrm{col}}(x,t):= H\!\left(\epsilon_{\mathrm{col}}-\frac{\|\mathbf{S}_\perp(x,t)\|}{|\mathbf{S}_\parallel(x,t)\cdot k(x,t)|+\eta_0}\right). \label{eq:part11_Gcol}\] In strong-alignment regimes, Proposition 11.2 provides a purely kinematic sufficient condition: \[\sqrt{\frac{a_k}{1-a_k}}\le \epsilon_{\mathrm{col}} \quad\Rightarrow\quad G_{\mathrm{col}}=1\ \text{(in the idealized moment model)}.\]

11.3.4.1 No-leak stationary profile (transverse equilibrium).

If the tube boundary is chosen so that \(\mathbf{S}_\perp\approx 0\) in the jet core (no transverse net flux), then [eq:part11_Sperp_balance] reduces to \[\nabla_\perp(\kappa_\perp e_{\mathrm{a}}) \approx e_{\mathrm{a}}\mathbf{F}_\perp. \label{eq:part11_no_leak_balance}\] If \(\kappa_\perp\) is approximately constant across the core, then \[\kappa_\perp \nabla_\perp e_{\mathrm{a}} \approx e_{\mathrm{a}}\mathbf{F}_\perp \quad\Rightarrow\quad \nabla_\perp \ln e_{\mathrm{a}} \approx \frac{1}{\kappa_\perp}\mathbf{F}_\perp.\] Using [eq:part11_confining_force] and taking radial symmetry in the transverse plane (\(r:=\|\mathbf{r}_\perp\|\)), \[\frac{d}{dr}\ln e_{\mathrm{a}}(r) \approx -\frac{\Omega_c^2}{\kappa_\perp}\,r.\] Integrating gives the Gaussian core profile: \[e_{\mathrm{a}}(r) \approx e_{\mathrm{a}}(0)\, \exp\!\left(-\frac{\Omega_c^2}{2\kappa_\perp}\,r^2\right). \label{eq:part11_gaussian_ea}\]

11.3.4.2 Jet radius estimate.

Define the jet radius \(R_j\) as the \(1/e\) radius of \(e_{\mathrm{a}}\): \[e_{\mathrm{a}}(R_j)=e_{\mathrm{a}}(0)\,e^{-1}.\] From [eq:part11_gaussian_ea], \[R_j \approx \sqrt{\frac{2\kappa_\perp}{\Omega_c^2}}. \label{eq:part11_Rj}\] This expresses a basic design principle: \[\begin{aligned} \text{strong confinement (large }\Omega_c\text{)} &\Rightarrow \text{narrow jet},\\ \text{strong transverse ``pressure'' (large }\kappa_\perp\text{)} &\Rightarrow \text{wider jet}. \end{aligned}\]

11.4.3.1 Minimal “string with confinement” model.

Define the effective line inertia \(I\) and effective tension \(T\) by cross-section integrals: \[I:=\int_{\Sigma} e_{\mathrm{a}}\,dA \approx A e_0, \qquad T:=\int_{\Sigma} \kappa_\parallel e_{\mathrm{a}}\,dA \approx A \kappa_\parallel e_0, \label{eq:part11_I_T_def}\] in a uniform core approximation. Let transverse damping be \(\nu_\perp>0\) (modeled as proportional to \(B_\perp\)). Let confinement stiffness be \(K_c:=I\Omega_c^2\) with \(\Omega_c>0\) from [eq:part11_confining_force].

Then a minimal linear transverse dynamics is \[I\,\partial_t^2 \boldsymbol{\xi} +\nu_\perp I\,\partial_t \boldsymbol{\xi} +K_c\,\boldsymbol{\xi} - T\,\partial_z^2 \boldsymbol{\xi} =0. \label{eq:part11_transverse_string}\] Plane-wave perturbations \(\boldsymbol{\xi}\propto e^{i(kz-\omega t)}\) yield \[-\omega^2 + i\nu_\perp \omega + \Omega_c^2 + c_T^2 k^2 = 0, \qquad c_T^2:=\frac{T}{I}\approx \kappa_\parallel. \label{eq:part11_transverse_dispersion}\] Equivalently, \[\omega^2 - i\nu_\perp \omega - (\Omega_c^2 + c_T^2 k^2)=0. \label{eq:part11_transverse_dispersion_alt}\] The roots satisfy \(\mathrm{Im}(\omega)\le 0\) when \(\nu_\perp>0\) and \(\Omega_c^2+c_T^2 k^2\ge 0\), so no exponential growth occurs.

11.4.3.2 Destabilizing external shear (Kelvin–Helmholtz analog in minimal form).

Represent external shear as a negative stiffness term \(-\sigma_{\mathrm{sh}}^2 I\,\boldsymbol{\xi}\) (a minimal linearization of shear-driven displacement growth), yielding \[I\,\partial_t^2 \boldsymbol{\xi} +\nu_\perp I\,\partial_t \boldsymbol{\xi} +I(\Omega_c^2-\sigma_{\mathrm{sh}}^2)\boldsymbol{\xi} - T\,\partial_z^2 \boldsymbol{\xi} =0. \label{eq:part11_transverse_shear}\] Then the stability requirement becomes \[\Omega_c^2-\sigma_{\mathrm{sh}}^2 \ge 0 \quad\Longleftrightarrow\quad \Omega_c \ge \sigma_{\mathrm{sh}}. \label{eq:part11_shear_stability}\] This is a GATE: if measured/environment-inferred shear exceeds confinement, the jet should bend and disrupt.

11.4.4.1 (i) Sausage-like instability in a 1D variable-area model.

Let the cross-sectional area be \(A(z,t)\) and define axial throughput \(J(z,t):=A(z,t)\bar S(z,t)\). Assume a conserved charge \(q=e_{\mathrm{a}}\) with cross-section mean \(\bar e_{\mathrm{a}}\) and no side leakage. Then mass/charge conservation becomes \[\partial_t\big(A\bar e_{\mathrm{a}}\big) + \partial_z\big(A\bar S\big)=0 \quad\Longleftrightarrow\quad \partial_t(A\bar e_{\mathrm{a}})+\partial_z J=0. \label{eq:part11_variable_area_cont}\] For the axial momentum/flux equation, use [eq:part11_1D_S] in conservative form multiplied by \(A\) (neglecting \(z\)-derivatives of coefficients): \[\partial_t J + \partial_z\big(A\kappa_\parallel \bar e_{\mathrm{a}}\big)= -B_\parallel J. \label{eq:part11_variable_area_S}\] Linearize about a steady base state \((A,\bar e_{\mathrm{a}},J)=(A_0,e_0,J_0)\) with constant coefficients and \(B_\parallel\ge 0\). Set \(B_\parallel=0\) for conservative stability. Let perturbations \(\delta A,\delta e,\delta J\propto e^{i(kz-\omega t)}\). Then [eq:part11_variable_area_cont] gives \[-i\omega(A_0\delta e+e_0\delta A)+ik\,\delta J=0. \label{eq:part11_sausage_lin1}\] Equation [eq:part11_variable_area_S] gives \[-i\omega\,\delta J + ik\kappa_\parallel(A_0\delta e+e_0\delta A)=0. \label{eq:part11_sausage_lin2}\] Eliminate \(\delta J\) by substituting \(\delta J=\frac{\omega}{k}(A_0\delta e+e_0\delta A)\) from [eq:part11_sausage_lin1] into [eq:part11_sausage_lin2], obtaining \[-i\omega\left(\frac{\omega}{k}\right)(A_0\delta e+e_0\delta A)+ik\kappa_\parallel(A_0\delta e+e_0\delta A)=0.\] If \(A_0\delta e+e_0\delta A\neq 0\), this simplifies to \[\omega^2=\kappa_\parallel k^2. \label{eq:part11_sausage_dispersion}\] Thus the variable-area mode propagates as a wave with speed \(\sqrt{\kappa_\parallel}\) and shows no exponential growth as long as \(\kappa_\parallel\ge 0\). Including \(B_\parallel>0\) adds damping, not growth (as in §11.4.2).

11.4.4.2 (ii) Diffusive broadening gate.

Even without exponential instability, transverse diffusion/mixing can broaden the jet. In the strong-alignment closure [eq:part11_T_aniso], transverse “pressure” corresponds to \(\kappa_\perp\) and transverse damping \(B_\perp\) controls the effective transverse diffusivity \[D_\perp := \frac{\kappa_\perp}{B_\perp+\eta_0}. \label{eq:part11_Dperp}\] A rough broadening time over radius \(R_j\) is \[t_{\mathrm{diff},\perp}\sim \frac{R_j^2}{D_\perp}. \label{eq:part11_tdiff}\] A necessary condition for a jet to remain collimated over an axial distance \(L\) with axial transport speed \(u_\parallel\) is \[t_{\mathrm{diff},\perp}\gg t_{\mathrm{adv}} \quad\text{where}\quad t_{\mathrm{adv}}\sim \frac{L}{u_\parallel}, \qquad u_\parallel:=\frac{\bar S}{\bar e_{\mathrm{a}}+\eta_0}. \label{eq:part11_diffusive_survival}\] This provides a practical GATE: if mixing is too strong (\(D_\perp\) large), the jet becomes a wide wind on the scale \(L\).

12.1.1.1 Flux equation under isotropic closure.

Start from the generic flux dynamics (PART 06 form): \[\partial_t \mathbf{S} + \nabla\cdot\mathbf{T} = -\mathbf{B}\mathbf{S} + e_{\mathrm{a}}\mathbf{F}_{\mathrm{eff}} + \mathbf{R}_S, \label{eq:part12_S_eq_general}\] where \(\mathbf{B}\) is a (possibly anisotropic) drag/resistance tensor, \(\mathbf{F}_{\mathrm{eff}}\) is the effective driving force (including deficit, boundary forcing, etc.), and \(\mathbf{R}_S\) collects optional sources.

Using [eq:part12_isotropic_T], \(\nabla\cdot\mathbf{T}=\nabla(\kappa_T e_{\mathrm{a}})=\nabla p_v\), so: \[\partial_t \mathbf{S} + \nabla p_v = -\mathbf{B}\mathbf{S} + e_{\mathrm{a}}\mathbf{F}_{\mathrm{eff}} + \mathbf{R}_S. \label{eq:part12_S_eq_isotropic}\]

12.1.3.1 Alignment defect \(a_k\) (recalled).

Using the distribution \(f(x,v,t)\) for the active phase and axis field \(k(x,t)\), define \[M_\parallel:=\int (v\cdot k)^2 f\,dv, \qquad M_\perp:=\int \|v-(v\cdot k)k\|^2 f\,dv, \qquad a_k:=\frac{M_\perp}{M_\parallel+M_\perp}\in[0,1]. \label{eq:part12_ak_def}\] Strong jets require \(a_k\ll 1\) (PART 11). Sprinkler outflow uses partial alignment: not too small, not too large.

12.1.3.2 Partial alignment gate (LOCK).

Fix LOCK thresholds \(0<a_{\mathrm{jet}}<a_{\mathrm{spr}}<1\) and define \[G_{\mathrm{part}}(x,t) := H\!\big(a_k(x,t)-a_{\mathrm{jet}}\big)\, H\!\big(a_{\mathrm{spr}}-a_k(x,t)\big). \label{eq:part12_Gpart}\] Thus \(G_{\mathrm{part}}=1\) means alignment is present but not in the “jet-strong” window.

12.1.3.3 Low confinement gate (LOCK).

Let \(\Omega_c(x,t)\ge 0\) be a confinement strength diagnostic (PART 11 used it as a transverse restoring frequency). Fix a LOCK upper bound \(\Omega_{\mathrm{spr,max}}\) and define \[G_{\mathrm{lowconf}}(x,t):=H\!\big(\Omega_{\mathrm{spr,max}}-\Omega_c(x,t)\big). \label{eq:part12_Glowconf}\] This gate enforces that the sprinkler is not in the strong geometric confinement regime that creates narrow jet tubes.

12.1.3.4 Non-jet gate.

Let \(G_{\mathrm{JET}}\) denote the jet gate from PART 11 (composite of axis locking, collimation, confinement, channel, and power gates). Define \[G_{\neg\mathrm{JET}} := 1 - \min(1,G_{\mathrm{JET}}), \label{eq:part12_Gnotjet}\] so that \(G_{\neg\mathrm{JET}}=1\) when the system is not in the jet-forcing regime.

12.1.4.1 Normal drag and openness.

Let \(\mathbf{B}(x,t)\) be symmetric positive semidefinite. Define the normal resistance on a surface with outward normal \(\mathbf{n}\) as \[B_n(x,t):=\mathbf{n}(x,t)\cdot \mathbf{B}(x,t)\,\mathbf{n}(x,t)\ \ge 0. \label{eq:part12_Bn}\] Small \(B_n\) means the surface is permeable (open); large \(B_n\) means the surface is sealed (closed).

12.1.4.2 Geometric “open-line” condition.

In many stellar/disk contexts, outflow is preferentially along a local direction \(k\) (magnetic/rotationally organized). For a surface, open escape requires \(k\) to have a significant outward component. Fix a LOCK tolerance angle \(\theta_{\mathrm{open}}\in(0,\pi/2)\) and define \[G_{k\cdot n}(x,t):=H\!\big(k(x,t)\cdot \mathbf{n}(x,t)-\cos\theta_{\mathrm{open}}\big). \label{eq:part12_Gkdotn}\]

12.1.4.3 Aperture (hole) gate (LOCK).

Fix a LOCK normal-resistance threshold \(B_{n,\mathrm{open}}>0\) and define \[G_{\mathrm{open}}(x,t):= H\!\big(B_{n,\mathrm{open}}-B_n(x,t)\big)\,G_{k\cdot n}(x,t). \label{eq:part12_Gopen}\] This declares a surface point open if (i) normal resistance is small, and (ii) the local alignment axis is sufficiently outward.

12.1.4.4 Outward-drive diagnostic.

From [eq:part12_S0], the predicted normal flux (before limiting) is \[S_{0n}(x,t):=\mathbf{S}_0(x,t)\cdot \mathbf{n}(x,t) = \mathbf{n}\cdot \mathbf{B}^{-1}\Big(e_{\mathrm{a}}\mathbf{F}_{\mathrm{eff}} - \nabla p_v + \mathbf{R}_S\Big). \label{eq:part12_S0n}\] We define an outward-drive gate requiring positive outward tendency. Fix a LOCK minimal outward flux \(S_{n,\min}\ge 0\): \[G_{\mathrm{drive}}(x,t):=H\!\big(S_{0n}(x,t)-S_{n,\min}\big). \label{eq:part12_Gdrive}\]

12.1.5.1 Composite sprinkler gate.

We define the sprinkler gate on the launching surface \(\Sigma\) as \[G_{\mathrm{SPR}}(x,t) := G_{\neg\mathrm{JET}}\, G_{\mathrm{part}}(x,t)\, G_{\mathrm{lowconf}}(x,t)\, G_{\mathrm{open}}(x,t)\, G_{\mathrm{drive}}(x,t), \qquad x\in\Sigma. \label{eq:part12_GSPR}\]

12.1.5.2 Sprinkler regime (definition).

A system is in the sprinkler regime at time \(t\) if the set \[\Sigma_{\mathrm{open}}(t):=\{x\in\Sigma(t):\ G_{\mathrm{SPR}}(x,t)=1\} \label{eq:part12_Sigma_open}\] has positive area measure: \[\mathrm{Area}\big(\Sigma_{\mathrm{open}}(t)\big)>0. \label{eq:part12_sprinkler_regime_def}\] The sprinkler outflow rate is then defined by integrating the limited flux [eq:part12_S_limited] over the open set (see §12.2).

12.2.1.1 Hole.

A hole is a connected component \(\mathcal{H}_i(t)\) of \(\Sigma_{\mathrm{open}}(t)\) that is (topologically) simply connected and not long/thin: \[\mathcal{H}_i(t)\ \text{ is a connected component of }\Sigma_{\mathrm{open}}(t)\ \text{ with a bounded aspect ratio.}\]

12.2.1.2 Channel.

A channel is a connected component \(\mathcal{C}_j(t)\) of \(\Sigma_{\mathrm{open}}(t)\) that has large aspect ratio, i.e. it forms a narrow corridor connecting regions: \[\mathcal{C}_j(t)\ \text{ is a connected component of }\Sigma_{\mathrm{open}}(t)\ \text{ with large aspect ratio.}\]

12.2.1.3 Remark (geometry-free operationalization).

Since aspect ratio requires geometric measurement, the implementation can use a LOCK morphological criterion (e.g. based on geodesic thickness vs length) to classify each component as hole or channel. The theory requires only that \(\Sigma_{\mathrm{open}}(t)\) decomposes into disjoint connected components: \[\Sigma_{\mathrm{open}}(t)=\left(\bigsqcup_i \mathcal{H}_i(t)\right)\ \sqcup\ \left(\bigsqcup_j \mathcal{C}_j(t)\right), \label{eq:part12_open_decomposition}\] up to a null-measure boundary set.

12.2.3.1 Proposition 12.1 (sprinkler throughput bound).

For any surface \(\Sigma\) and any open set \(\Sigma_{\mathrm{open}}(t)\subset\Sigma\), the outflow rate satisfies: \[\dot Q_{\mathrm{spr}}(t) \le \int_{\Sigma_{\mathrm{open}}(t)} \|\mathbf{S}(x,t)\|\,dA \le c\int_{\Sigma_{\mathrm{open}}(t)} e_{\mathrm{a}}(x,t)\,dA. \label{eq:part12_Qspr_bound}\] In particular, if \(e_{\mathrm{a}}(x,t)\le 1\), then \[\dot Q_{\mathrm{spr}}(t) \le c\,\mathrm{Area}\big(\Sigma_{\mathrm{open}}(t)\big). \label{eq:part12_Qspr_area_bound}\]

12.2.3.2 Proof.

Since \(\mathbf{n}\) is unit length, \(S_n=\mathbf{S}\cdot\mathbf{n}\le \|\mathbf{S}\|\). Thus \[\dot Q_{\mathrm{spr}}=\int_{\Sigma_{\mathrm{open}}} S_n\,dA \le \int_{\Sigma_{\mathrm{open}}}\|\mathbf{S}\|\,dA.\] By the speed limit [eq:part12_speedlimit], \(\|\mathbf{S}\|\le c e_{\mathrm{a}}\), giving [eq:part12_Qspr_bound]. Finally \(e_{\mathrm{a}}\le 1\) gives [eq:part12_Qspr_area_bound]. \(\square\)

12.2.3.3 Open-area fraction scaling.

Let \(A_{\mathrm{tot}}:=\mathrm{Area}(\Sigma)\) and define \[f_{\mathrm{open}}(t):=\frac{\mathrm{Area}\big(\Sigma_{\mathrm{open}}(t)\big)}{A_{\mathrm{tot}}}\in[0,1]. \label{eq:part12_fopen}\] Then [eq:part12_Qspr_area_bound] implies \[\dot Q_{\mathrm{spr}}(t)\le c\,f_{\mathrm{open}}(t)\,A_{\mathrm{tot}}. \label{eq:part12_Qspr_fopen_bound}\] For a spherical star surface of radius \(R_*\), \(A_{\mathrm{tot}}=4\pi R_*^2\): \[\dot Q_{\mathrm{spr}}(t)\le 4\pi R_*^2\,c\,f_{\mathrm{open}}(t). \label{eq:part12_Qspr_sphere_bound}\]

12.2.4.1 Surface normal resistance field as a function of defect and background.

Define a LOCK constitutive map \[B_n(x,t)=\mathcal{B}_n\!\big(a_k(x,t),\,e_{\mathrm{bg}}(x,t),\,\chi(x,t)\big), \qquad \mathcal{B}_n\ge 0, \label{eq:part12_Bn_constitutive}\] where \(\chi(x,t)\) is an optional surface-activity marker (e.g. heating/forcing index) that is treated as an input observable or an internal diagnostic.

A minimal monotone choice is: \[\mathcal{B}_n(a,e_{\mathrm{bg}},\chi) := B_{n0}\,\big(1+\beta_a a+\beta_{\mathrm{bg}} e_{\mathrm{bg}}\big)\,\frac{1}{1+\beta_\chi \chi}, \qquad B_{n0}>0,\ \beta_a,\beta_{\mathrm{bg}},\beta_\chi\ge 0\ \text{(\textsf{LOCK})}. \label{eq:part12_Bn_minimal}\] Then:

• larger disorder (\(a_k\) large) increases resistance (fewer holes),

• stronger background/jamming (\(e_{\mathrm{bg}}\) large) increases resistance,

• stronger activity/heating (\(\chi\) large) decreases resistance (more holes).

With [eq:part12_Bn_minimal], the open set \(\Sigma_{\mathrm{open}}(t)\) becomes the superlevel set of \(\chi\) (and/or sublevel set of \(a_k,e_{\mathrm{bg}}\)), producing holes/channels as connected components of a thresholded scalar field, which is mathematically well-defined.

12.3.2.1 Case 1: pressure-driven diffusion wind (\(F_r\approx 0\)).

If the effective force is negligible compared to pressure-gradient drive in the outer region (\(F_r\approx 0\)), then \[\kappa_T \frac{d e_{\mathrm{a}}}{dr} = - B_r(r)\,\frac{\dot Q}{4\pi r^2}. \label{eq:part12_ea_ODE_diff}\] Integrate from the base radius \(R_*\) to \(r\): \[\kappa_T\big(e_{\mathrm{a}}(r)-e_{\mathrm{a}}(R_*)\big) = -\frac{\dot Q}{4\pi}\int_{R_*}^{r}\frac{B_r(s)}{s^2}\,ds. \label{eq:part12_ea_integral}\] If \(e_{\mathrm{a}}(r)\to 0\) as \(r\to\infty\) (active phase dilutes far away), then taking \(r\to\infty\) yields a throughput formula: \[\dot Q = 4\pi\,\kappa_T\,e_{\mathrm{a}}(R_*)\left[\int_{R_*}^{\infty}\frac{B_r(s)}{s^2}\,ds\right]^{-1}. \label{eq:part12_Q_from_drag_integral}\] This is a concrete scaling: wind throughput increases with base active fraction \(e_{\mathrm{a}}(R_*)\) and with the pressure coefficient \(\kappa_T\), and decreases with the integrated drag barrier \(\int B_r/s^2\).