Fluid Dynamics from Volume Particles (VP)

This is consistent with the trace of the Irving--Kirkwood interaction stress (Appendix~A) after averaging. In the jammed (stiff) VP regime the interaction sector typically dominates, so $p\simeq p_{\rm int}+p_{\rm kin}$ with a smaller kinetic/thermal part $p_{\rm kin}$.

2.3.2 The Tait--Murnaghan equation of state (recommended \LOCK)

For a quasi-incompressible ``stiff fluid'' regime with small density fluctuations $\delta\rho/\rho_0\ll 1$, the virial pressure is well approximated by a Tait/Murnaghan-type EOS. We \LOCK\ the following compressible closure for use when a stiff-pressure model is desired:

Here $\rho_0$ is a reference density (in VP, $\rho_0\sim m/a^3$ with lattice spacing $a$ and particle mass $m$), $\gamma>1$ is a stiffness exponent (often $\gamma\simeq 7$ for water-like quasi-incompressible modeling, but it can be matched to the curvature of the chosen VP repulsive potential), $B>0$ is a bulk-modulus parameter, and $p_{\rm back}$ is a background offset pressure.

2.3.3 Numerical sound speed and stability (engineering constraint)

The isentropic speed of sound implied by \eqref{eq:tait_eos} is

To remain in a quasi-incompressible regime one enforces a small Mach number $\mathrm{Ma}:=\max|\uv|/c_s\lesssim 0.1$ (rule of thumb). A practical design choice is therefore

which ties the stiffness parameter $B$ to the target velocity scale. In explicit time stepping one must also respect the usual acoustic CFL stability constraint $\Delta t \lesssim \mathrm{CFL}\,\Delta x/(|\uv|+c_s)$.

2.4 Incompressible Navier--Stokes

In the incompressible limit, $\rho=\rho_0$ is constant and $\nabla\!\cdot\uv=0$. Then

and \eqref{eq:NS_incompressible} follows.

2.5 Relating viscosity to VP: a minimal, falsifiable mapping

This white paper separates what can be derived exactly from what is a modeling identification.

• \DERIVE: $\mu$ and $\lambda$ are not arbitrary once VP is fixed: they are summaries of the coarse-grained stress response.
• \GATE: the map from VP microparameters and event statistics $(\phi,\text{event rate},\text{core size},\dots)$ to $(\mu,\lambda)$ must be empirically or numerically identified.

A useful minimal mapping is via a Maxwell-type relaxation picture:

where $G$ is a local elastic modulus of the transient jammed network and $\nu_{\rm tot}$ is the total event rate. This predicts that higher unjamming activity (higher $\phi\nu_{\rm tot}$) lowers viscosity.

\begin{remark}[Physical anchor: a mesoscopic bridge scale $D^\ast$] To physically ground the effective viscosity $\mu_{\rm eff}$ in a way that is relevant to nano--micro hardware, it is useful to introduce a mesoscopic bridge scale $D^\ast$---the correlation length of VP ``unjamming chains'' (or, more generally, the length beyond which the VP lattice can be treated as a continuum).

• Continuum criterion (VP Knudsen number). Define
$$ \mathrm{Kn}_{\rm VP}:=\frac{D^\ast}{L}, $$
• A practical numerical anchor (67\,nm). For ordinary air at STP the molecular mean free path is $\lambda_{\rm air}\approx 67\,\mathrm{nm}$, which is a familiar benchmark for the onset of non-continuum effects in micro/nano flows. In this white paper we do not equate $D^\ast$ with $\lambda_{\rm air}$ as a matter of derivation; rather, $\lambda_{\rm air}$ provides an order-of-magnitude reference scale for designing and interpreting calibration experiments.
• Hardware implication. If a device feature size $L_{\rm feat}\lesssim D^\ast$, the VP medium is expected to exhibit discrete transport (slip, ballistic segments, event-dominated dissipation). If $L_{\rm feat}\gg D^\ast$, the same medium should admit a classical continuum description with effective $(\mu,\lambda)$.

where $L$ is the characteristic system length. The Navier--Stokes closure is expected to be accurate when $\mathrm{Kn}_{\rm VP}\ll 1$, while $\mathrm{Kn}_{\rm VP}\sim \mathcal O(1)$ signals a slip/ballistic regime where non-continuum corrections may be required.

\noindentGate note. $D^\ast$ is a measurable parameter: it can be inferred from the crossover between no-slip and slip behavior (or from the onset of scale-dependent $\mu_{\rm eff}$) in a controlled geometry. \end{remark}

\begin{remark}[Wall protocol (Option A locked): frozen-VP no-slip boundary]\label{rem:wall_protocol} The bulk derivation of Navier--Stokes in Sections~1--2 is an interior statement. However, any hardware-facing calibration of $\mu_{\rm eff}$ or inference of the bridge scale $D^\ast$ necessarily involves walls, and therefore requires an explicit boundary protocol.

\medskip \noindent\LOCK\ BC-W1 (Declare the boundary regime). Whenever $\mu_{\rm eff}$ is inferred from a wall-driven experiment or simulation (Couette/Poiseuille), we must either: (i) enforce a no-slip regime (full tangential momentum accommodation), or (ii) measure the slip length $b$ and use a partial-slip (Navier) boundary condition. Otherwise wall slip can masquerade as a reduced $\mu_{\rm eff}$.

\medskip \noindent\LOCK\ BC-W2A (Option A, chosen standard in this volume: frozen-VP wall). We represent a solid wall by one or more layers of fixed VPs that act as a rough, momentum-accommodating surface. This is the VP analog of a diffuse-reflection / full-accommodation wall and is intended to enforce macroscopic no-slip ($b\approx 0$) in the Newtonian window.

• Geometry. Choose an integer $n_w\ge 2$ and build a wall of thickness $h_w:=n_w a$, where $a$ is the VP lattice spacing. Place wall VPs on a near-lattice arrangement and add a small random jitter (roughness) of amplitude $\delta\in[0.05,0.2]a$ to break specular symmetries.
• Kinematics. Prescribe a wall velocity $\uv_w$ (e.g.\ $\pm \tfrac12\Delta U\,\hat{\mathbf x}$ for Couette). Wall VPs are treated as infinite-mass carriers: they do not accelerate under collisions. Operationally this means that collisions transfer momentum to the fluid while the wall velocity remains imposed.
• Interaction. Fluid VPs interact with wall VPs via the same repulsive/contact law as the bulk, optionally with a stiffer wall coefficient $\kappa_w\ge 1$ to suppress penetration. Because the wall is rough and momentum-accommodating, tangential momentum is efficiently exchanged and slip is reduced.

\medskip \noindent\GATE\ BC-W3 (No-slip verification; separate bulk $\mu$ from wall slip). Even with BC-W2A, finite slip can occur if the wall is too smooth or the interaction is too specular. Therefore, in any Couette/Poiseuille calibration we measure the slip length $b$ from the near-wall profile. A convenient estimator is obtained by linear fitting of the tangential velocity in the boundary layer:

A ``no-slip'' calibration regime is declared when $b/h\le b_{\rm tol}$ with $b_{\rm tol}\sim 10^{-2}$, or (more VP-natively) when $b\le 0.1\,D^\ast$. If this gate fails, one must either report a partial-slip model or redesign the wall roughness/stickiness.

\medskip \noindent\GATE\ BC-W2B (Option B, optional: force-field wall). A smooth wall can be represented by an external repulsive potential. If near-wall interactions are predominantly specular, tangential momentum is only weakly exchanged and partial slip occurs. This option is useful for deliberately exploring slip regimes ($b\gtrsim D^\ast$), but it must not be used to infer bulk $\mu_{\rm eff}$ unless $b$ is measured and included in the fit.

\medskip \noindentContinuum parametrization (Navier slip). In the partial-slip case, a minimal macroscopic description is

where $\uv_w$ is the wall velocity, $(\cdot)_t$ denotes the tangential component, and $\partial_n$ is the outward normal derivative.

For a plane Couette cell of gap $h$ with equal slip length $b$ at both walls, the shear rate is $\partial_y u=\Delta U/(h+2b)$ (instead of $\Delta U/h$). Consequently, the thermal calibration in Gate~G5-A modifies to

so wall slip must be controlled or measured to identify $\mu$.

\medskip \noindentBridge-scale diagnostic. In micro/nano devices one expects $b=\mathcal O(D^\ast)$ in the absence of intentional roughness, so the ratio $b/D^\ast$ provides an additional diagnostic of the bridge scale. \end{remark}

2.5.1 Viscosity identification: Green--Kubo (formal) and dissipation matching (operational)

A direct (and falsifiable) way to connect VP event statistics to Navier--Stokes transport coefficients is to match the irreversible power sink.

Green--Kubo / linear-response identification of $\mu$ (formal).

Equation \eqref{eq:newtonian_from_memory} already suggests a first-principles route: $\mu$ is the area under the stress-relaxation kernel. Under a local-equilibrium assumption with an effective temperature $\Theta$, linear response yields an equivalent Green--Kubo representation for the shear viscosity:

where $\langle\cdot\rangle_{\rm le(\xv,t)}$ denotes a local-equilibrium ensemble conditioned on the coarse state at $(\xv,t)$, and $\tau_{xy}$ may be taken as the interaction-dominated shear stress in the jammed regime. In practice \eqref{eq:green_kubo_mu} is useful in two ways: (i) it shows that $\mu$ is intrinsic (not an arbitrary fit parameter) once the VP microdynamics is fixed, and (ii) it provides an explicit calibration protocol if stress autocorrelations can be measured in simulation.

Consistency with the Maxwell picture.

If the stress relaxation is approximately single-exponential, $G(s)\approx G_0 e^{-s/\tau_M}$, then $\mu=\int_0^\infty G(s)\,ds \approx G_0\tau_M$, which recovers the Maxwell mapping \eqref{eq:maxwell_mapping} (with $G_0$ interpreted as a transient shear modulus).

Local event rate and event dissipation density.

Fix a coarse-graining volume $V_{\rm cg}\sim \ell_{\rm cg}^3$ and a local time window of length $\Delta t_{\rm cg}$. Let $N_{\rm unjam}(W_{\xv,t})$ be the number of unjamming-type events recorded in the local window $W_{\xv,t}$. Define the corresponding event rate

and let $\langle \Delta E_{\rm bind}\rangle_{\xv,t}$ denote the mean energy release per event in the same window (protocol-locked). Then the event dissipation density is

Incompressible shear-dominated regime.

If $\nabla\!\cdot\uv\approx 0$ locally, the Newtonian viscous dissipation density reduces to

In a regime where VP dissipation is dominated by events and the Newtonian closure applies, a minimal matching principle is

This yields an operational viscosity estimator:

The Newtonian gate condition is that $\mu_{\rm eff}$ is approximately rate-independent and isotropic over the declared window and forcing class.

Relation to fluidity $\phi$.

Under a fixed VP update protocol with a micro time step $\Delta t_{\rm micro}$, the windowed fluidity $\phi=N_{\rm unjam}/N_{\rm total}$ implies $\lambda_{\rm unjam}\sim \phi/(V_{\rm cg}\,\Delta t_{\rm micro})$ up to a protocol-dependent conversion between ``update steps'' and physical time. This makes the VP notion of fluidity directly measurable in the continuum dissipation balance.

\begin{remark}[Why this is still ``Navier--Stokes''] Navier--Stokes uses $\mu,\lambda$ as phenomenological inputs. Equation \eqref{eq:maxwell_mapping} is an optional identification that becomes part of the VP-to-NS bridge only after it passes a declared gate (stress--strain measurement, see Section~6). \end{remark}

2.6 Viscous dissipation and thermodynamic admissibility

The Navier--Stokes closure is only meaningful as a VP bridge if it is thermodynamically admissible: the irreversible mechanical power density must be nonnegative.

\begin{proposition}[Viscous dissipation identity and positivity]\label{prop:visc_pos} Assume the compressible Newtonian closure \eqref{eq:newtonian_stress} with coefficients $\mu,\lambda\in\mathbb R$. Define the bulk viscosity

Then the viscous dissipation density is the identity

where $S$ is the trace-free strain-rate tensor in \eqref{eq:newtonian_stress}. Moreover, $\epsilon_\nu(\xv,t)\ge 0$ for all velocity gradients $\nabla\uv$ if and only if

\end{proposition}

\begin{proof} Write $\nabla\uv = S + \frac13(\nabla\!\cdot\uv)\Id + A$ with $A:=\frac12(\nabla\uv-(\nabla\uv)^\top)$ skew. Because $\tau$ is symmetric, $\tau:A=0$ and only the symmetric part contributes. Using $\tau=2\mu S+\lambda(\nabla\!\cdot\uv)\Id$ and $S:\Id=0$ gives $\tau:\nabla\uv = 2\mu\,S:S + (\lambda+\frac23\mu)(\nabla\!\cdot\uv)^2$. If $\mu<0$, choose a pure shear with $\nabla\!\cdot\uv=0$ and $S\neq 0$ to make $\epsilon_\nu<0$. If $\zeta<0$, choose a pure dilation $\nabla\uv=\alpha\Id$ to make $\epsilon_\nu=3\zeta\alpha^2<0$. Thus \eqref{eq:admissibility_conditions} is necessary. If \eqref{eq:admissibility_conditions} holds, both terms in \eqref{eq:visc_dissipation_density} are nonnegative, so $\epsilon_\nu\ge 0$ pointwise, proving sufficiency. \end{proof}

\noindent\GATE\ (Thermodynamic check). When fitting $(\mu,\lambda)$ from VP data, report $\zeta=\lambda+\frac23\mu$ and verify \eqref{eq:admissibility_conditions}. Violations indicate either (i) a closure breakdown (non-Newtonian response) or (ii) a mismatch between the measurement window and the assumed continuum regime.

\begin{remark}[Stokes hypothesis is optional] Some texts impose $\lambda=-\tfrac{2}{3}\mu$ (so that $\zeta=0$). This is an additional modeling choice, not a derivation. In the VP-to-continuum bridge, $\lambda$ should be treated as an empirical summary of volumetric relaxation until proven otherwise by a dedicated calibration. \end{remark}

3. Energy and entropy budgets with event dissipation

3.0 Budget bookkeeping: what is ``dissipated''

In this document, ``dissipation'' is always meant with respect to the resolved flow (mechanical) energy. Both classical viscosity and VP events remove energy from the resolved flow and transfer it to unresolved/internal degrees of freedom (heat, core deformation, binding rearrangements, \dots). This is the correct level of description for the Onsager-type anomaly question: can the resolved-flow energy decay remain nonzero as $\nu\to 0$?

\medskip \noindent\LOCK\ Resolved flow energy. For an incompressible constant-density regime ($\rho=\rho_0$), define the resolved kinetic energy density

and its spatial mean and total energy

\medskip \noindent\LOCK\ Dissipation rates (mean form). We use the following spatial-mean power densities (per unit volume):

(Integral forms are obtained by multiplying by $|\Omega|$.)

3.1 Classical kinetic-energy budget (incompressible)

Assume $\uv$ is a smooth solution of incompressible Navier--Stokes on $\Omega$ with either periodic boundaries or boundary conditions such that the mechanical energy flux through the boundary vanishes. Multiply \eqref{eq:NS_incompressible} by $\rho_0\uv$, integrate over $\Omega$, and use $\nabla\!\cdot\uv=0$ together with integration by parts to obtain the exact identity

where

so that $\varepsilon_\nu(t)\ge 0$.

\begin{remark}[Weak solutions and energy inequality] For Leray weak solutions the equality \eqref{eq:energy_NS} becomes an inequality

with a possible additional nonnegative ``defect'' term when taking inviscid limits. Section~5 reviews the Duchon--Robert defect language for this purpose. \end{remark}

3.1.1 Compressible energy and entropy equations (reference form)

While this white paper focuses on the incompressible resolved-energy balance \eqref{eq:energy_NS}, it is useful to record the standard compressible reference form, because VP ``event heat'' naturally enters through an internal-energy channel.

Let $e(\xv,t)$ be the internal energy per unit mass and define the total specific energy $E:=e+\frac12|\uv|^2$. With heat flux $\mathbf q$ (e.g.\ Fourier $\mathbf q=-\kappa\nabla T$) the compressible Navier--Stokes total-energy equation can be written as

Equivalently, the internal-energy equation (first law) reads

Thermodynamic admissibility requires the viscous contribution $\tau:\nabla\uv=\epsilon_\nu$ to be nonnegative, which holds under the conditions of Proposition~\ref{prop:visc_pos}. A compatible entropy balance reads schematically

where $s$ is the specific entropy. Equation \eqref{eq:entropy_balance_reference} is the continuum ``second law'' target that any VP projection must respect in the macroscopic limit.

3.1.2 Micro--macro thermal mapping (VP $\leftrightarrow$ temperature and heat flux)

Section~3.1.1 recorded the standard compressible energy/entropy forms with temperature $T$ and heat flux $\mathbf q$. For VP-based engineering, it is useful to state explicitly how $T$ and $\mathbf q$ arise from VP statistics.

Kinetic temperature from peculiar velocity.

Define the peculiar velocity of particle $i$ relative to the coarse flow as

where $\uv(\xv,t)$ is the coarse-grained velocity at the evaluation point. We then define the local kinetic temperature by the standard kinetic-theory mapping

where $\langle\cdot\rangle_{\xv,t}$ denotes the kernel/window average and $k_B$ is Boltzmann's constant (or its simulation-unit analogue). (For strictly 2D simulations, replace the factor $3/2$ by $d/2$ with $d=2$.)

Heat flux and the Fourier limit.

Let $T_i$ denote a particle-level temperature proxy (peculiar kinetic energy). A minimal VP ``mixing'' model is a relaxation/exchange step of the form

where $W_{ij}$ is a symmetric nonnegative weight induced by the coarse-graining kernel $W_\ell$. In the continuum limit, a Taylor expansion $T_j\simeq T_i+(\xv_j-\xv_i)\cdot\nabla T$ and symmetry of $W_{ij}$ yield a diffusion operator, leading to the macroscopic heat equation

Positivity $\kappa>0$ is guaranteed when the mixing step is entropy producing (a metric/second-law sector), so that the discrete exchange \eqref{eq:discrete_temp_exchange} is a contraction in variance.

Where VP events enter.

Event dissipation $\epsilon_{\rm events}$ (Section~3.2) is an additional internal-energy source. In a compressible closure one may therefore write schematically

consistent with the reference first-law form in Section~3.1.1.

3.2 VP internal reservoir and event dissipation

VP includes discrete events (unjam/merge/reconnect) that release a well-defined energy $\Delta E_n\ge 0$ into unresolved degrees of freedom (core deformation, filamentation, sub-grid heat, \dots). The key point is that these transfers are recorded.

Event log.

We assume an event list $\{(t_n,\xv_n,\Delta E_n)\}_{n=1}^{N_{\rm total}}$, where $t_n$ is the event time, $\xv_n$ is a representative event location (e.g.\ centroid of participants), and $\Delta E_n$ is the released energy (protocol-locked).

Internal reservoir (global).

Define the cumulative reservoir energy

The (mean) event-dissipation rate is then the distributional derivative

\begin{remark}[Event dissipation as a positive measure in time] The map $t\mapsto h_{\rm int}(t)$ is nondecreasing and of bounded variation. It therefore defines a positive Radon measure $\mu_{\rm events}$ on time such that $dh_{\rm int}=d\mu_{\rm events}$ in the sense of distributions. Concretely, for any test function $\varphi\in C_c^\infty(\mathbb R)$,

In this language, $\varepsilon_{\rm events}$ is not assumed to be an ordinary function of time; it is the (possibly singular) measure density of $\mu_{\rm events}$, and may contain Dirac spikes. \end{remark}

Internal reservoir (local density).

A compatible coarse-grained density is obtained by spreading each event over space with $W_\ell$:

By construction, $\langle \epsilon_{\rm events}(\cdot,t)\rangle_\Omega=\varepsilon_{\rm events}(t)$.

Window averages.

For a time window $W=[t_0,t_1]$,

3.3 Extended resolved-flow budget and the steady-window closure

\noindent\LOCK\ Event bookkeeping (first-law at event times). At each event time $t_n$, the resolved flow loses energy and the reservoir gains the same amount:

This rule ensures that any ``event dissipation'' measured in the resolved flow budget is not a numerical artifact; it is a recorded physical transfer.

\medskip \noindent\DERIVE\ Resolved-flow budget (mean form). Combining the smooth viscous evolution \eqref{eq:energy_NS} with the jump losses at events yields the following exact identity. It is understood in the distributional sense in time: between events it reduces to \eqref{eq:energy_NS}, while at event times it contains Dirac spikes that encode the jump losses (Appendix~H).

Integrating \eqref{eq:resolved_flow_budget} over a window $W=[t_0,t_1]$ gives the exact identity

Therefore, on a steady window (Section~0.3) the drift term is negligible and we recover the closure \eqref{eq:budget_window}:

\begin{remark}[What survives as $\nu\to 0$] Equation \eqref{eq:resolved_flow_budget} shows that, for the resolved-flow energy, the total dissipation is $\varepsilon_{\rm tot}:=\varepsilon_\nu+\varepsilon_{\rm events}$. In a vanishing-viscosity scan, $\varepsilon_\nu$ may go to zero, but a nonzero limit $\langle\varepsilon_{\rm events}\rangle_W\to\varepsilon_*>0$ is precisely an Onsager-type anomalous dissipation channel carried by events. \end{remark}

4. Metriplectic / GENERIC structure for VP-based fluid dynamics

4.1 Algebraic template

A metriplectic (GENERIC-like) template writes the evolution of any functional $F[\Psi]$ as

where $H$ is total energy and $S$ is entropy. The brackets satisfy:

• $\{\cdot,\cdot\}$ is a Poisson bracket generated by a skew operator $\PoissonOp$.
• $(\cdot,\cdot)$ is a metric bracket generated by a symmetric positive semi-definite operator $\MetricOp$.

Written in operator form:

4.1.1 Incompressible Euler as a Lie--Poisson system (orientation)

The advective (inviscid) part of fluid mechanics is naturally Hamiltonian. For incompressible Euler, one may formulate the dynamics on the Lie algebra of divergence-free vector fields. In two dimensions, writing the state as vorticity $\omega=\nabla\times\uv$ (a scalar), a standard Lie--Poisson bracket is

where $[\cdot,\cdot]$ is the Jacobian (Poisson bracket) on the plane. Choosing the Hamiltonian $H(\omega)=\frac12\int |\uv|^2\,d\xv$ (with $\uv$ obtained from $\omega$ via the Biot--Savart law) yields the Euler equation.

This section is included to emphasize a point that is central for VP: advection is reversible and should be modeled by a skew (Poisson) operator, whereas irreversibility (viscosity, event dissipation) must enter through a symmetric positive semi-definite sector.

4.2 Degeneracy conditions (thermodynamic consistency)

Thermodynamic consistency is ensured by degeneracy:

The second condition is precisely the statement that ``dissipation'' redistributes energy into internal modes without changing the total.

4.3 Event channels as jump terms

In VP, events are naturally modeled as jump processes on top of smooth advection. A hybrid description is:

where $N_r$ are counting processes (merger/unjam channels) and $\zeta_r$ are jump maps.

4.4.1 H-theorem via generalized detailed balance

A standard route to an H-theorem is to impose generalized detailed balance (GDB) on the effective transition rates of the event kernel. Then entropy production takes a relative-entropy form

with $J^\pm$ forward/backward fluxes. This structure guarantees that net reactions proceed in the direction of increasing entropy.

4.4.2 A priori parameter scaling (the engineering bridge)

The VP$\to$NS bridge is not complete for engineering unless one can choose VP parameters to target macroscopic transport coefficients before large simulations. This subsection records order-of-magnitude scaling laws linking VP microparameters to continuum coefficients. These relations are not exact identities; they are intended as design heuristics that can be validated (or rejected) by the Gate suite in Section~6.

Microparameters.

We write $m$ for a representative particle mass, $a$ for lattice spacing, and $k_{\rm spr}$ for a repulsive/contact stiffness (spring constant in a local quadratic approximation of the pair potential near contact). We use $\tau_M$ for the Maxwell relaxation time of the jammed network and $\lambda_{\rm event}$ for a local unjamming rate (Section~2.5.1), so that $\tau_M\sim 1/\lambda_{\rm event}$ in the simplest Poisson-reset picture.

\subsubsection*{4.4.2.1 Regime A: unjammed / collisional (gas-like)}

In a high-fluidity regime where momentum transport is dominated by particle translation and mixing, a kinetic-theory estimate gives

where $\alpha$ is a geometric prefactor (order $1/3$ in 3D) and $v_{\rm th}$ is the thermal velocity associated with the kinetic temperature \eqref{eq:kinetic_temperature}. This scaling is the analog of $\nu\sim v_{\rm th}\ell_{\rm mfp}$ with mean free path $\ell_{\rm mfp}\sim a$.

\subsubsection*{4.4.2.2 Regime B: jammed / network (liquid/solid-like Maxwell regime)}

In the jammed limit (force-chain dominated), the effective shear viscosity is controlled by stress relaxation:

with an effective shear modulus scaling as

If unjamming events randomize anisotropy with rate $\lambda_{\rm event}$, then a minimal estimate is

where $\beta$ is a protocol-dependent efficiency factor (order unity).

\subsubsection*{4.4.2.3 ``Knobs'' for targeting Reynolds number}

Combining \eqref{eq:mu_jam_scaling}--\eqref{eq:tauM_scaling} with $\nu=\mu/\rho$ and $\rho_0\sim m/a^3$ gives the jammed-regime kinematic viscosity scaling

where $\Delta t_{\rm event}\sim\tau_M$ is the mean time between unjamming events. Equation \eqref{eq:nu_eff_knob} is the practical ``design knob'': to target a Reynolds number $\Rey=UL/\nu_{\rm eff}$ one may tune either (i) stiffness $k_{\rm spr}$ or (ii) event sensitivity (which sets $\lambda_{\rm event}$ and hence $\tau_M$), while keeping the bridge-scale gate $D^\ast\ll L$ satisfied.

\begin{remark}[Dimensional note and simulation units] The scalings above are written in physical dimensions. In nondimensional simulation units with $a=1$ and $m=1$, \eqref{eq:nu_eff_knob} reduces to $\nu_{\rm jam}\sim k_{\rm spr}\tau_M$ up to order-one factors. \end{remark}

5. Inviscid limit, defect measures, and Onsager-type anomalous dissipation

5.1 Continuum language: weak Euler, coarse graining, and the Duchon--Robert defect

In the inviscid limit, the Euler equations may admit weak solutions that do not conserve kinetic energy. A precise way to encode this is through coarse graining and the associated nonlinear energy flux.

Filtered energy balance.

Let $G$ be a smooth mollifier with $\int G=1$ and define $G_\ell(\rv)=\ell^{-3}G(\rv/\ell)$. For a velocity field $\uv$ define the filtered field $\uv_\ell:=G_\ell*\uv$ and the subscale stress

For a weak Euler solution $\uv$ (incompressible, $\nabla\!\cdot \uv=0$), the filtered kinetic-energy balance reads

The scalar $\Pi_\ell(\xv,t)$ is the local energy flux across the scale $\ell$.

Duchon--Robert defect.

Duchon and Robert showed that for suitable weak solutions (e.g.\ $\uv\in L^3_{\rm loc}$), $\Pi_\ell$ admits a distributional limit as $\ell\to 0$,

and the local energy balance becomes

A convenient increment representation is

\begin{lemma}[H\"older bound: no defect above $1/3$]\label{lem:holder_DR} If $\uv$ is locally H\"older continuous with exponent $h>1/3$, i.e.\ $|\delta\uv(\rv)|\le C|\rv|^h$ for $|\rv|$ small, then $\mathcal D(\uv)=0$. \end{lemma} \begin{proof} Using $|\nabla G_\ell(\rv)|\lesssim \ell^{-4}$ supported on $|\rv|\lesssim \ell$, \eqref{eq:DR_increment_formula} yields $|\Pi_\ell|\lesssim \ell^{-4}\int_{|\rv|\lesssim \ell}|\rv|^{3h}\,d\rv \lesssim \ell^{3h-1}\to 0$ as $\ell\to 0$ when $h>1/3$. \end{proof}

Therefore an Onsager-type anomalous dissipation channel ($\mathcal D(\uv)\not\equiv 0$) is compatible only with velocity roughness at or below the $1/3$ threshold.

5.2 VP language: event dissipation as a discrete defect

VP provides an explicit, discrete candidate for the defect measure: the cumulative binding-energy release in unjamming/merger-type events.

Event dissipation as a measure in space--time.

If an event $k$ occurs at $(\xv_k,t_k)$ with released energy $\Delta E_{{\rm bind},k}\ge 0$, define the space--time event measure

Its time-marginal is Appendix~C.2. A local event dissipation density $\epsilon_{\rm events}(\xv,t)$ exists only after choosing a coarse-graining volume and window; the measure formulation avoids ambiguity when events are discrete.

Consistency statement (limit of energy identities).

Let $\uv^\nu$ denote a family of VP-informed flows with viscosity $\nu$ (possibly $\nu\to 0$), with an energy balance of the form (in distributions)

where $\mu_{\rm events}^\nu$ is a nonnegative space--time measure. Assume $\uv^\nu\to \uv$ in $L^3_{\rm loc}$ and that the measures $\nu|\nabla\uv^\nu|^2\,d\xv dt \rightharpoonup \mu_{\rm visc}$ and $\mu_{\rm events}^\nu \rightharpoonup \mu_{\rm evt}$ weakly. Then the limiting $\uv$ satisfies a weak Euler energy balance with defect

In particular, if $\mu_{\rm visc}\to 0$ (vanishing viscous channel) but $\mu_{\rm evt}\not\equiv 0$, then the remaining defect is purely event-driven.

Support localization as a falsifiable gate.

A stronger claim sometimes desired is that the spatial support of event activity converges to the singular set of the limiting Euler solution. In this white paper that statement is treated as a \GATE\ item: it is accepted only if diagnostics show that event locations concentrate in regions where velocity increments violate the $1/3$ H\"older threshold (Lemma~\ref{lem:holder_DR}), and that this concentration persists under resolution/seed scans. A discrete--continuum dictionary is summarized in Appendix~C.

5.3 Practical anomaly statement (falsifiable)

A typical falsifiable claim is:

\medskip \noindentAnomaly claim. In a steady window with fixed injection $\langle \varepsilon_{\rm inj}\rangle=I>0$, there exists a scan parameter (e.g.\ viscosity $\nu\to 0$ or an event-core size $r_c\to 0$) such that

while the budget closure \eqref{eq:budget_window} holds within tolerance.

5.3.1 Robustness: point-vortex number and seed ensembles (metriplectic reference)

The dissipation-saturation plot (Figure~\ref{fig:dissipation_saturation}) is generated by the reproducibility package shipped in the all-in-one DOI bundle (metriplectic\_onsager\_repro/). In that package, the event-driven dissipation is reported as $\varepsilon_{\rm bind}\equiv\langle \varepsilon_{\rm events}\rangle_W$ (binding/merger events).

Two minimal robustness diagnostics are provided:

• Point-vortex number scan (``$N_{\rm v$-scan'').} Vary the number of discrete point vortices $N_{\rm v}$ (degrees of freedom) in the discrete-vortex metriplectic model and report the resulting $\varepsilon_{\rm bind}(Re_{\rm eff})$.
• Seed ensemble spread. At fixed model parameters (e.g.\ merger core $r_c$), repeat over random initial seeds to quantify the intrinsic scatter.

\begin{remark}[Avoiding symbol collision: $N_{\rm v}$ vs grid resolution $N$] In Section~6, $N$ denotes the spectral grid resolution used in Navier--Stokes reference simulations. In this subsection, $N_{\rm v}$ denotes the number of discrete point vortices in the metriplectic event model. \end{remark}

5.4 Example: 2D vortex-merger scenario and an effective Reynolds number

A concrete realization of the event-defect idea is a 2D vortex model in which the inviscid limit corresponds not to $\nu\to 0$ directly, but to a shrinking event-core scale $r_c\to 0$ that plays the role of an effective viscosity. One introduces an effective Reynolds number

where $U_{\rm rms}$ is the rms velocity and $L$ is the box size. Increasing $\Rey_{\rm eff}$ corresponds to shrinking $r_c$ while keeping large-scale statistics fixed.

In such models one often observes two regimes:

• Region 1 (low/moderate $\Rey_{\rm eff$):} event dissipation follows the classical viscous trend.
• Region 2 (large $\Rey_{\rm eff$):} event dissipation saturates to a nonzero plateau, consistent with Onsager-type anomaly.

5.5 Reference viscous trend from 2D Navier--Stokes (vorticity form)

For comparison and calibration, one may generate a ``reference'' viscous trend using the 2D vorticity equation

on a periodic box of side $L=2\pi$ with pseudo-spectral discretization and dealiasing.

Define enstrophy and viscous dissipation rate by

and define $\Rey_{\rm eff}(t):=U_{\rm rms}(t)\,L/\nu$. A convenient summary statistic is then the pair $\big(\max_t\varepsilon_\nu(t),\ \langle\Rey_{\rm eff}\rangle_{t\in W}\big)$ for each viscosity $\nu$.

This diagnostic is deliberately ``laptop-scale'': it is intended as a reproducible reference curve, not an asymptotic high-$\Rey$ test.

6. Verification and reproducibility program (Gate suite)

6.1 Gate G1: balance-law sanity

\noindent\GATE\ G1 (mass). Verify that the discretization of VP (or any derived solver) conserves total mass and that \eqref{eq:conti} holds in weak form.

\noindent\GATE\ G1 (momentum). Verify that internal forces satisfy Newton's third law (or the appropriate antisymmetry) so that a stress representation exists (Appendix~A).

6.2 Gate G2: Newtonian regime identification

\noindent\GATE\ G2 (stress--strain test). Measure $\tau$ and $S$ in homogeneous shear (or oscillatory shear) and test:

• linearity: $\tau_{ij}\propto S_{ij}$,
• isotropy: the proportionality is the same under rotated frames,
• locality: no strong dependence on higher gradients in the tested regime,
• Maxwell time: measure the stress relaxation kernel $G(s)$ (step strain or oscillatory shear), extract $\tau_M$, and confirm $\mathrm{De}=\tau_M/T_{\rm flow}\ll 1$ in the declared window,
• Anisotropy decay: measure a fabric tensor $Q$ (or equivalent anisotropy index) and verify that $Q$ is small after window averaging ($\Delta t_{\rm cg}\gg\tau_M$).
• Viscosity cross-check: in the same window, compare $\mu_{\rm GK}$ from \eqref{eq:green_kubo_mu}, $\mu_{\rm events}$ from \eqref{eq:mu_eff_from_events}, and the Maxwell estimate $\mu_{\rm Maxwell}\approx G\,\tau_M$ from \eqref{eq:maxwell_mapping}. Require agreement within a declared tolerance; disagreement flags non-Newtonian response or mis-windowing.

Declare the regime of validity and the fitted $\mu_{\rm eff},\lambda_{\rm eff}$.

6.3 Gate G3: anomaly scan and budget closure

\noindent\GATE\ G3 (scan). Perform $\nu\to 0$ (or core-size) scans and compute window averages of $\varepsilon_\nu$, $\varepsilon_{\rm events}$, $\varepsilon_{\rm inj}$. Require the resolved-flow budget \eqref{eq:resolved_flow_budget} in window form:

On a steady window the drift term is negligible, reducing to \eqref{eq:budget_window}, and then test whether $\langle\varepsilon_{\rm events}\rangle$ saturates.

6.4 Gate G4: reference Navier--Stokes curve (optional, but strongly recommended)

This gate constructs a reproducible viscous benchmark that the VP/event model can be compared against.

Reference PDE (2D, periodic).

Solve the vorticity form \eqref{eq:2D_vorticity_NS} on a doubly periodic square domain of side $L=2\pi$. Recover $\uv$ from $\omega$ by solving $\nabla^2\psi=\omega$ and setting $\uv=\nabla^\perp\psi$.

Numerics (minimal).

A standard choice is pseudo-spectral discretization with $N^2$ collocation points and $2/3$ dealiasing. Time stepping may be RK4 or an exponential-time-differencing scheme.

Initial condition (turbulence-like, laptop-friendly).

A simple reproducible option is a sum of a small number of Gaussian vortices (random positions and signs), or a band-limited random vorticity field. The goal is not to engineer an asymptotic inertial range, but to obtain a stable decay curve.

Parameter scan.

Choose a small set of viscosities (e.g.\ $\nu\in\{0.05,0.02,0.01,0.005\}$) and for each run compute:

• enstrophy $Z(t)=\frac12\langle\omega^2\rangle$,
• viscous dissipation $\varepsilon_\nu(t)=2\nu Z(t)$,
• effective Reynolds number $\Rey_{\rm eff}(t)=U_{\rm rms}(t)L/\nu$.

A representative pair for the curve is $\big(\max_t\varepsilon_\nu(t),\ \langle\Rey_{\rm eff}\rangle_{t\in W}\big)$.

Resolution check (optional).

Repeat one or two viscosities at higher resolutions (e.g.\ $256^2$ and $512^2$) to verify that the extracted pairs are insensitive to grid resolution in the overlap range.

6.5 Included Navier--Stokes reference data and what ``low resolution'' means

This white paper is primarily analytical, but it also ships a minimal Navier--Stokes reference dataset and scripts (via the companion reproducibility archive) to support the Gate logic of Section~6.4.

\begin{remark}[Notation note] In this section, $N$ denotes the spectral grid resolution (collocation points per direction) of the Navier--Stokes solver. It is unrelated to the point-vortex number $N_{\rm v}$ used in the metriplectic robustness plots of Section~5.3.1. \end{remark}

Meaning of ``low resolution''.

In pseudo-spectral 2D Navier--Stokes, ``resolution'' typically means the number of collocation points $N^2$. With standard $2/3$ dealiasing, the highest retained wavenumber is $k_{\max}\approx N/3$, so that $N=64$ implies $k_{\max}\approx 21$ active Fourier modes per direction. This is far below what would be required for precision inertial-range turbulence studies. However, for the specific diagnostic used here---the extracted pair \( (Re_{\rm eff}, \max_t\varepsilon_\nu(t)) \) in a modest viscosity scan---$N=64$ can serve as a fast, laptop-scale calibration curve.

Reference pairs and a resolution overlay.

Table~\ref{tab:NS_ref_pairs} lists the extracted $(Re_{\rm eff},\max_t\varepsilon_\nu)$ pairs. Figure~\ref{fig:NS_resolution_overlay} overlays $N=64$ with high-resolution checks at $N=256$ and $N=512$. In the overlapping point $\nu=0.005$, the relative differences between $N=64$ and $N=512$ are small (\(\approx 1.15\%\) in $Re_{\rm eff}$ and \(\approx 0.72\%\) in $\max_t\varepsilon_\nu\)), indicating that the baseline curve is effectively converged for this limited purpose.

6.6 Gate G5: Hardware-in-the-loop mapping (optional)

This gate connects the continuum parameters of this white paper to measurable observables in a hardware testbed (e.g.\ ``Black Copper'' heat extraction, photonic/solid-state VQC elements, microchannel rigs). It is not required to validate the derivation of Navier--Stokes, but it makes the bridge engineering-ready.

\noindent\GATE\ G5-A (Viscosity $\to$ viscous heating / thermal signature). In any Newtonian regime, the irreversible mechanical power density is

A minimal thermal scaling (steady conduction with effective conductivity $k_{\rm eff}$ and thermal length $L_T$) is

In a simple plane-Couette calibration cell of gap $h$ and wall speed jump $\Delta U$, one has $S:S=\tfrac12(\Delta U/h)^2$ and the exact conduction solution with isothermal walls gives

\noindentWall note. The Couette scaling above assumes a no-slip regime. In this volume the default micro protocol is BC-W2A (frozen-VP wall), designed to enforce $b\approx 0$; nevertheless BC-W3 requires measuring $b$ to confirm the regime. If partial slip occurs with slip length $b$, use the slip-corrected formula \eqref{eq:couette_slip_temp} (or fit $(\mu,b)$ jointly from velocity-profile / flow-rate data); see Remark~\ref{rem:wall_protocol}.

Pass criteria (example). Fit $\mu_{\rm eff}$ from the measured $\Delta T(\Delta U)$ and verify that $\mu_{\rm eff}$ is rate-independent over the declared Newtonian window. A design target such as $\Delta T<10\,\mathrm{mK}$ can be used to flag an operational ``low-$\mu_{\rm eff}$'' regime, but the numerical threshold must be declared per device and thermal path.

\medskip \noindent\GATE\ G5-B (Anomaly $\to$ event-burst signature). If an event channel exists beyond classical viscosity, then after subtracting the viscous contribution from the power budget, the residual

should remain nonnegative and can exhibit bursty statistics (intermittent spikes) rather than Gaussian noise. Operationally this can be detected by: (i) budget closure in a steady window, and (ii) non-Gaussian increment statistics or thresholded ``burst'' events in temperature/noise/power traces. Pass criteria (example). Under a control scan that reduces the fitted $\mu_{\rm eff}$ (or $\nu_{\rm eff}$), test whether the time-averaged residual $\langle \varepsilon_{\rm events}\rangle_W$ saturates to a nonzero plateau while $\langle\varepsilon_\nu\rangle_W\to 0$. This is the hardware analog of an Onsager-type anomalous dissipation claim.

\medskip \noindentEnergy-accounting note. Claims of ``self-cooling'' must be stated with a full energy balance: a decrease in a measured temperature can occur if energy is redirected to an internal reservoir or exported through an unmonitored channel, but the total budget must still close.

7. $\Pi$-invariants and RG contact diagnostics (optional but recommended)

This section imports a set of dimensionless invariants that are useful for diagnosing quasi-steady turbulent windows and for organizing event-dissipation claims in a scale-aware way. They are not required to derive Navier--Stokes, but they are recommended if this white paper is used as a ``final'' fluid-mechanics closure document.

7.1 Definitions: $\Pi_T$, $\Pi_L$, and $\Pi_{ST

Let $\varepsilon_{\rm bind}$ denote a window-averaged event/binding dissipation rate (we use $\varepsilon_{\rm bind}\equiv \langle \varepsilon_{\rm events}\rangle_W$ when the event channel is binding-like). Let $\Delta F_{\rm crit}$ be a declared ``critical'' free-energy gap associated with a transition cycle, and let $\sigma_{\rm eff}$ be an effective entropy-production scale (power$\times$length$^{1+\alpha}$).

Define the time and length invariants

where $T$ is a window length, $L$ is the system or outer scale, $\alpha$ is a scaling exponent, and $C_\alpha$ is a dimensionless calibration constant.

A combined invariant is

with user-chosen weights $a,b>0$. A small $\delta_{ST}$ indicates a straight ``contact line'' in $(\log\Pi_T,\log\Pi_L)$ space.

7.2 RG contact condition

Consider a scale transformation $L\mapsto \lambda L$ with $\lambda>1$ and assume scaling laws

Then invariance of $\Pi_L$ at a fixed point implies the contact (fixed point) condition

The sign of $(1+\alpha)-\psi-\chi$ away from the fixed point indicates the monotone trend of $\Pi_L$ under coarse-graining.

7.3 Anomaly closure theorem (window form)

The following proposition summarizes the event-dissipation anomaly statement in a budget-closed quasi-steady window.

\begin{proposition}[Event anomaly in the vanishing-viscosity limit (window form)] Assume there exists a family of VP/flow states indexed by $(\nu,\gamma)$ (viscosity/friction-like parameters) such that:

1. Positive injection: $\langle\varepsilon_{\rm inj}\rangle_W \to I>0$ as $(\nu,\gamma)\to (0,0)$.
2. Energy tightness: $\sup_{\nu,\gamma}\sup_{t\in W}\mathbb{E}[H_{\rm flow}(t)]<\infty$.
3. Budget closure: $\langle\varepsilon_{\rm inj}\rangle_W= \langle\varepsilon_\nu+\varepsilon_\gamma+\varepsilon_{\rm events}\rangle_W$ within declared tolerance.

Then if $\langle\varepsilon_\nu+\varepsilon_\gamma\rangle_W\to 0$ as $(\nu,\gamma)\to(0,0)$, it follows that

i.e.\ total dissipation is carried by the event channel in the inviscid/frictionless limit. \end{proposition}

7.4 Gate usage

\noindent\GATE\ (G6) $\Pi$-diagnostics. If this white paper claims quasi-steady universality or RG contact, report $(\Pi_T,\Pi_L,\Pi_{ST},\delta_{ST})$ as standard diagnostics and enforce pass/fail criteria such as

The tolerances $\varepsilon_T,\varepsilon_L,\varepsilon_{ST}$ must be declared per experiment/simulation suite.

\appendix

Appendix A. Irving--Kirkwood stress for VP coarse graining (detailed derivation)

This appendix provides the step-by-step coarse-graining derivation of Proposition~\ref{prop:momentum}. The result is an identity: no continuum closure is assumed. The only inputs are (i) differentiable VP trajectories, and (ii) a Newton-type VP balance with internal forces.

\subsection*{A.0 Shorthand}

For brevity write

Recall

\subsection*{A.1 Time derivative of momentum density}

Start from $\mathbf j(\xv,t)=\sum_i m_i\vv_i W_i$ and differentiate:

where we used $\partial_t W_i = -\dot{\xv}_i\cdot\nabla W_i = -\vv_i\cdot\nabla W_i$ and the identity $\nabla\cdot(\vv_i\otimes\vv_i W_i)=\vv_i(\vv_i\cdot\nabla W_i)$ because $\vv_i$ is independent of $\xv$.

Now impose the VP translational dynamics $m_i\dot{\vv}_i = \Fv_i^{\rm int}+ m_i\mathbf f(\xv_i,t)$ to obtain

\subsection*{A.2 Convective flux and kinetic stress}

Define the microscopic momentum-flux tensor

Introduce the peculiar velocity $\mathbf c_i:=\vv_i - \uv(\xv,t)$. Then $\sum_i m_i\mathbf c_i W_i = \mathbf j - \rho\uv = 0$, and we obtain the exact decomposition

Define the kinetic stress

so that $\Pi = \rho\uv\otimes\uv - \sigma^{\rm kin}$. Substituting into \eqref{eq:appA_preflux} yields

\subsection*{A.3 Pairwise internal forces and interaction stress}

Assume the internal forces decompose into pair forces

Then

For fixed $i,j$, set $\mathbf r_{ij}:=\xv_i-\xv_j$ and consider the segment $\xv_j+s\mathbf r_{ij}$, $s\in[0,1]$. Using the fundamental theorem of calculus,

Because $\Fv_{ij}$ and $\mathbf r_{ij}$ do not depend on $\xv$, we can rewrite

Insert this into \eqref{eq:appA_force_sym} and define the Irving--Kirkwood interaction stress

Then the internal-force term becomes an exact divergence:

\subsection*{A.4 Completion of the momentum identity}

Define the total stress

\begin{remark}[Pressure from the stress trace (kinetic + interaction contributions)] Given the Irving--Kirkwood stress $\sigma=\sigma^{\rm kin}+\sigma^{\rm int}$, it is natural to define the mechanical pressure as the isotropic part of $\sigma$:

Then $\sigma$ admits the exact split

where $\tau$ is the deviatoric (shear) stress. In VP language, this pressure is the sum of a kinetic (velocity-fluctuation) part and an interaction (contact-network) part:

In a jammed/contact regime, $p_{\rm int}$ typically dominates and is the continuum counterpart of isotropic compressive force-chain stress.

\medskip \noindent2D note. For the 2D reference simulations (periodic $[0,2\pi]^2$), the isotropic part is defined with $1/2$ instead of $1/3$, namely $p:=-(1/2)\,\mathrm{tr}\,\sigma$. \end{remark}

Combining \eqref{eq:appA_kinetic_form} with the previous identity yields

which is exactly Proposition~\ref{prop:momentum}.

\subsection*{A.5 Jammed/contact regime and a contact-network approximation}

In a jammed (solid-like) VP regime the kinetic contribution $\sigma^{\rm kin}$ is often subdominant: macroscopic stress is transmitted primarily through interlocked contacts (``force chains''). The identity \eqref{eq:appA_sigma_int} still holds, but a simpler approximation is often used.

\noindentContact-network approximation (useful in simulations). When contacts are well localized, the line-integral coarse graining can be approximated (on a coarse-graining volume $V_{\rm cg}$) by a discrete contact sum

where $\mathbf f_c$ is the contact force and $\mathbf \ell_c$ is the branch vector between the contact partners.

\noindentVP interpretation. In VP, $\Fv_{ij}$ typically represents repulsive contact forces between volume particles in the jammed network. A localized ``Qu-Vortex'' can be interpreted as a region with reduced compressive contact stress, i.e.\ an effective low-pressure pocket that can steer transport pathways.

Appendix B. From linear response to Newtonian stress: representation, Green--Kubo, and Maxwell time

This appendix records (i) the representation-theorem step used to justify the Newtonian form once local isotropic linear response is declared, and (ii) a controlled route to that declaration via memory kernels (Maxwell time) and Green--Kubo formulas.

\subsection*{B.1 Representation theorem (isotropic linear map, explicit proof)}

The Newtonian stress \eqref{eq:newtonian_stress} is not an arbitrary guess once one declares local, objective, isotropic linear response at the coarse scale: it is the only possible instantaneous tensor form.

Let $\mathcal F$ be a linear map that assigns to each velocity gradient $A:=\nabla\uv$ a symmetric tensor $\tau=\mathcal F(A)$. Assume $\mathcal F$ is isotropic in the sense that for every rotation $R\in\mathrm{SO}(3)$,

Because $\mathcal F$ is linear, there exists a fourth-order tensor $L$ such that $\tau_{ij}=L_{ijkl}A_{kl}$. Condition \eqref{eq:isotropy_F} implies that $L$ is an isotropic fourth-order tensor. The standard classification of isotropic fourth-order tensors gives

for some scalars $a,b,c$. Contracting with $A_{kl}$ yields

If $\tau$ is symmetric for all $A$, then \eqref{eq:tau_general_isotropic_linear} forces $b=c$, hence

Writing $\mathrm{sym}(A)=S+\tfrac13(\nabla\!\cdot\uv)\Id$ with the trace-free strain-rate $S$ in \eqref{eq:newtonian_stress}, we obtain

Identifying $\mu:=b$ and $\lambda:=a+\tfrac23b$ recovers \eqref{eq:newtonian_stress}. In particular, the antisymmetric part $\mathrm{skew}(A)$ cancels automatically in \eqref{eq:tau_isotropic_symmetric}, so rigid-body rotation does not contribute to the viscous stress (objectivity).

\subsection*{B.2 Linear response with memory and the Markovian limit}

The representation theorem above explains why the Newtonian form is the only possibility once one accepts a local, isotropic, linear dependence on $\nabla\uv$. What remains is to explain why VP should ever fall into that class.

A general linear-response closure has memory:

with $G(s)$ a shear relaxation kernel. If $G(s)$ decays on a Maxwell time $\tau_M$ and if the macroscopic strain-rate varies slowly on that time scale, then a Taylor expansion $S(t-s)\approx S(t)$ over the support of $G$ yields the Newtonian approximation

which is \eqref{eq:newtonian_from_memory}. The parameter controlling the approximation is the Deborah number $\mathrm{De}=\tau_M/T_{\rm flow}$.

\subsection*{B.3 Green--Kubo formula and $\mu\ge 0$}

Under a local-equilibrium assumption with an effective temperature $\Theta$, linear response yields the Green--Kubo formula

which is \eqref{eq:green_kubo_mu}. This makes two points mathematically transparent:

• Intrinsic transport. Once the VP microdynamics (including event rules) is fixed, the equilibrium (or local-equilibrium) stress correlations determine $\mu$.
• Positivity. If the underlying equilibrium measure is stationary and the stress autocorrelation is nonnegative in the PSD sense, then $\mu\ge 0$, which is the key ingredient for nonnegative viscous dissipation.

\subsection*{B.4 Jamming, force-chain anisotropy, and isotropization}

In a jammed medium the interaction stress is transmitted through a contact/force-chain network and is instantaneously anisotropic. A convenient diagnostic is the fabric tensor $Q$ introduced in \eqref{eq:fabric_relax}. If $Q$ relaxes rapidly (small $\tau_M$) compared to the observation window, then time-averaged stresses become effectively isotropic even though instantaneous stresses are not. If $Q$ relaxes slowly, the correct continuum description is viscoelastic (or more generally non-Newtonian), and the Navier--Stokes closure is not expected.

\subsection*{B.5 Chapman--Enskog / relaxation-time derivation (BGK/RTA model, optional but explicit)}

Section~2.1 used the most general (and safest) closure logic: memory $\to$ Markovian limit $\to$ Newtonian stress. This subsection gives a concrete, fully worked toy derivation showing how the same structure appears from a fast relaxation operator, in the spirit of Chapman--Enskog. It is optional because it introduces an intermediate kinetic variable that is not part of the minimal VP contract.

B.5.1 A minimal BGK-style kinetic proxy (unjammed / gas-like window).

Assume there exists a mesoscopic one-particle distribution $f(\xv,\vv,t)$ whose moments reproduce the coarse fields, and write the peculiar velocity $\cv:=\vv-\uv(\xv,t)$. Consider a BGK relaxation model

where $f^{(0)}$ is a local equilibrium that is isotropic in $\cv$. In particular, assume

so that $f^{(0)}$ carries pressure but no deviatoric stress.

Introduce a small parameter $\varepsilon\sim\mathrm{Kn}_{\rm VP}$ and expand $f=f^{(0)}+\varepsilon f^{(1)}+\cdots$. At order $\varepsilon^0$ one obtains the Euler equations. At order $\varepsilon^1$, the linearized relaxation equation gives schematically $f^{(1)}\approx-\tau_{\rm rel}(\partial_t+\vv\cdot\nabla_\xv)f^{(0)}$. The deviatoric stress (kinetic part) is the traceless second moment

Evaluating the moment (standard Chapman--Enskog algebra) yields the Newtonian law

Thus $\mu>0$ is intrinsic provided $p>0$ and the relaxation time $\tau_{\rm rel}>0$.

B.5.2 A relaxation-time model for jammed force chains (Maxwell/Fabric route).

In the jammed VP regime, the interaction stress dominates, and a more appropriate mesoscopic variable is the fabric/anisotropy tensor $Q$. A minimal linear relaxation-time approximation is

which says: (i) anisotropy relaxes on $\tau_M$, and (ii) a slowly varying strain-rate drives a small steady anisotropy proportional to $S$. Assume the deviatoric stress is proportional to fabric,

where $G_0$ is a transient shear modulus of the jammed network. In the Markovian limit ($\mathrm{De}\ll 1$), solving \eqref{eq:RTA_fabric} gives $Q\approx\kappa\tau_M S$ and therefore

This explicitly reconciles ``jammed force chains'' with a Newtonian law: Newtonian viscosity is the product of a (transient) shear modulus and a relaxation time.

B.5.3 Relation to the memory kernel and Green--Kubo.

The relaxation-time models above correspond to an exponential memory kernel, $G(s)\propto e^{-s/\tau_M}$, so that $\mu=\int_0^\infty G(s)ds$. More generally, whenever a fast relaxation operator exists and the dynamics admits a local equilibrium, the viscosity can be written either as an inverse linearized-collision operator (Chapman--Enskog view) or as a stress autocorrelation integral (Green--Kubo view), which is why \eqref{eq:green_kubo_mu} and \eqref{eq:newtonian_from_memory} are the structurally stable statements used in the main text.

\medskip \noindent\GATE\ Interpretation. Equations \eqref{eq:bgk_mu} and \eqref{eq:maxwell_mu_positive} are not claimed as universal VP identities. They are explicit model examples that show how the Newtonian closure emerges once a fast relaxation time exists and local isotropy is recovered by averaging. The Gate suite (Section~6) determines which (if any) such window is realized in a concrete VP implementation.

Appendix C. Discrete--continuum dictionary for anomalous dissipation (refined)

This appendix refines the informal ``dictionary'' used to interpret VP event dissipation as a discrete counterpart of the Duchon--Robert defect.

\subsection*{C.1 Continuum: defect as a small-scale flux}

For a weak Euler solution, the defect $\mathcal D(\uv)$ is defined as the distributional limit of the filtered nonlinear flux $\Pi_\ell$ in \eqref{eq:filtered_energy_balance}:

or equivalently by the increment formula \eqref{eq:DR_increment_formula}. The key features are: (i) $\mathcal D(\uv)\ge 0$ for suitable weak solutions, and (ii) $\mathcal D(\uv)=0$ if $\uv$ is H\"older with exponent $>1/3$ (Lemma~\ref{lem:holder_DR}).

\subsection*{C.2 VP: event dissipation as a positive measure}

In the VP event-extended model, internal energy increments are recorded at discrete event times. If events occur at times $\{t_k\}$ with released energies $\Delta E_{{\rm bind},k}\ge 0$, define the event dissipation measure in time

Its absolutely continuous part (if any) corresponds to a smooth ``rate''; otherwise it is purely atomic (Dirac spikes).

For any observation window $W=[t_0,t_1]$,

and the window-averaged event dissipation rate is $\langle \varepsilon_{\rm events}\rangle_W = (|\Omega|\,|W|)^{-1}\int_W d\mu_{\rm events}$.

\subsection*{C.3 Matching principle and the inviscid limit}

The intended bridge is:

A rigorous identification requires a two-scale limit (spatial $\ell\to 0$ and vanishing micro-correlation scale $D^\ast$) together with a convergence statement for the coarse-grained velocity fields. In this white paper we treat this as a \GATE\ item: the identification is accepted only if simulations show that the spatial concentration of event activity tracks the regions where the resolved velocity field violates the $1/3$ H\"older threshold.

\subsection*{C.4 Practical co-localization test (an explicit gate protocol)}

The strongest (and most falsifiable) version of the ``event defect'' identification is not merely a budget match in time, but a spatial co-localization statement: event activity should concentrate where the continuum flux $\Pi_\ell$ concentrates as $\ell\to 0$.

A minimal protocol that can be implemented on simulation data is:

1. Choose a scale ladder. Pick a sequence $\ell_1>\ell_2>\cdots>\ell_m$ within the resolved range ($\ell_m$ above the grid scale, but below the outer scale).
2. Compute a windowed flux field. For each $\ell_k$, compute the positive part of the Duchon--Robert flux
$$ \Pi_{\ell_k}^+(\xv,t):=\max(\Pi_{\ell_k}(\xv,t),0), $$

and its time average on the analysis window $W$:

$$ D_{\ell_k}(\xv):=\langle \Pi_{\ell_k}^+(\xv,\cdot)\rangle_W . $$
3. Compute a windowed event-power density. Using the same spatial kernel width $\ell_k$, define
$$ E^{\rm evt}_{\ell_k}(\xv) := \frac{1}{|W|}\sum_{t_n\in W}\Delta E_n\,W_{\ell_k}(\xv-\xv_n), $$

so that $\langle E^{\rm evt}_{\ell_k}\rangle_\Omega=\langle\varepsilon_{\rm events}\rangle_W$.

4. Measure scale-by-scale overlap. Define the normalized $L^2$ overlap
$$ \mathrm{OL}(\ell_k) := \frac{\langle D_{\ell_k}\,E^{\rm evt}_{\ell_k}\rangle_\Omega} {\|D_{\ell_k}\|_{L^2(\Omega)}\,\|E^{\rm evt}_{\ell_k}\|_{L^2(\Omega)}}\ \in[0,1]. $$

A co-localization claim requires $\mathrm{OL}(\ell_k)$ to remain bounded away from $0$ as $\ell_k\downarrow 0$, and to be robust under seed/resolution changes.

5. Optional H\"older-threshold filter. Define a ``roughness set'' at scale $\ell_k$ by
$$ A_{\ell_k}:=\Big\{\xv:\ \sup_{|\rv|\le \ell_k}\frac{|\uv(\xv+\rv)-\uv(\xv)|}{|\rv|^{1/3}} > C\Big\}, $$

with a declared threshold $C>0$. A stronger gate is that most event energy occurs inside $A_{\ell_k}$ for small $\ell_k$:

$$ \frac{\int_{A_{\ell_k}} E^{\rm evt}_{\ell_k}(\xv)\,d\xv}{\int_\Omega E^{\rm evt}_{\ell_k}(\xv)\,d\xv} \ \ge\ 1-\delta, $$

with a small declared $\delta$.

\noindentInterpretation. If the overlap vanishes as $\ell\to 0$, then events may still be a global dissipation bookkeeping device, but they are not a good candidate for the local Duchon--Robert defect support. If the overlap persists and tightens with $\ell$, the event channel provides a concrete discrete model for the continuum defect localization.

Appendix D. Boussinesq / convection example (global identity with event dissipation)

This appendix records one representative ``complex fluid'' example where an event channel can be inserted without breaking classical global identities.

\subsection*{D.1 Classical Boussinesq Rayleigh--B\'enard identity (reference)}

For Rayleigh--B\'enard convection (Boussinesq approximation) with height $H$, kinematic viscosity $\nu$, thermal diffusivity $\kappa$, Rayleigh number $\Ra$ and Prandtl number $\Pran=\nu/\kappa$, a classical global identity relates the Nusselt number $Nu$ to the kinetic-energy dissipation rate. In one common nondimensionalization, the dimensional statement can be written schematically as

where $\varepsilon_u$ is the viscous dissipation per unit mass (or per unit volume, depending on convention) and $\langle\cdot\rangle$ denotes a space--time average in a statistically steady regime.

\subsection*{D.2 Event-extended identity}

If VP injects an additional event dissipation channel $\varepsilon_{\rm events}$ that transfers flow energy into unresolved internal modes (but does not violate the total-energy bookkeeping), then a natural extension is

In words: at fixed $(\Ra,\Pran)$, a nonzero event fraction increases the required $Nu-1$, i.e.\ increases heat transport relative to a purely viscous budget.

\subsection*{D.3 Event fraction and a minimal predictive parameterization}

Define the event fraction

Then \eqref{eq:RBC_event_global} implies

where $Nu_{\rm classic}$ is the Nusselt number predicted by the classical (viscous-only) theory at the same control parameters.

A minimal monotone-saturation ansatz for $\xi$ as a function of a dimensionless event-coupling parameter $\chi$ is

to be identified from data.

\subsection*{D.4 Length-scale implication under $\Pi_L$ contact}

If the $\Pi_L$ contact condition (Section~7) is enforced with $\varepsilon_{\rm bind}\equiv\langle\varepsilon_{\rm events}\rangle$, then the characteristic ``cell'' length scale satisfies

providing a direct scaling prediction connecting event dissipation to pattern size.

Appendix E. A structure-preserving time integrator template (ADG-style)

This appendix summarizes a practical numerical template for \eqref{eq:metriplectic_template} that is aligned with the VP philosophy: avoid numerical drift in budgets so that any observed dissipation can be interpreted physically.

\subsection*{E.1 Two-step split: conservative then dissipative}

For metriplectic dynamics

a robust approach is to integrate the skew and symmetric sectors in two substeps per time step $\Delta t$.

Given state z^n at time t_n

Step 1 (Poisson / conservative):
  z* solves  (z* - z^n)/dt = J(z_bar) * gradbar_H(z^n, z*)

Step 2 (metric / dissipative):
  z^{n+1} solves (z^{n+1} - z*)/dt = G(z_bar) * gradbar_S(z*, z^{n+1})

Here:

• $z_{\rm bar}$ is a suitable midpoint/average state,
• $\mathrm{gradbar}_H$ and $\mathrm{gradbar}_S$ are discrete gradients satisfying a discrete chain rule:
$$ H(z^*)-H(z^n)=\langle \mathrm{gradbar}_H(z^n,z^*),\, z^*-z^n\rangle, $$

and similarly for $S$.

\subsection*{E.2 Discrete thermodynamic guarantees}

If $\PoissonOp$ is skew-symmetric, then Step~1 preserves $H$ (and Casimirs, when included) to solver tolerance. If $\MetricOp$ is symmetric positive semi-definite, then Step~2 guarantees

To enforce the ``energy degeneracy'' condition $(H,S)=0$ at the discrete level, one additionally imposes

so that $H$ is conserved while $S$ increases. This is the numerical analog of the VP requirement that event dissipation be a physical channel, not a numerical artifact.

Appendix F. Turbulence diagnostics (flux, third-order laws, and budget closure)

This appendix collects practical diagnostics that connect classical turbulence measurements to the event-extended budget used in this white paper.

\subsection*{F.1 3D: third-order law as an $\varepsilon$ estimator}

In homogeneous isotropic 3D turbulence, the longitudinal third-order structure function

obeys, in the inertial range, the Kolmogorov $4/5$ law

where $\varepsilon$ is the mean energy dissipation rate. In an event-extended model one interprets $\varepsilon$ as the total dissipation,

so that a viscosity scan with fixed forcing should show a slope that remains constant while the split between $\varepsilon_\nu$ and $\varepsilon_{\rm events}$ changes.

\subsection*{F.2 2D and QG: energy/enstrophy flux plateaus}

In 2D turbulence (and quasi-geostrophic analogs), a useful spectral diagnostic is the flux of energy and enstrophy. Denoting by $\Pi_E(k)$ the cumulative nonlinear transfer of energy across wavenumber $k$, a negative plateau $\Pi_E(k)\approx -\varepsilon_E$ indicates an inverse energy cascade. Similarly, a positive plateau of enstrophy flux $\Pi_\Omega(k)\approx +\varepsilon_\Omega$ indicates a forward enstrophy cascade.

A robust ``cascade window'' is characterized by (i) simultaneous plateaus where applicable, and (ii) budget closure with event dissipation included.

\subsection*{F.3 Budget residual as a pass/fail metric}

For any statistically steady window $W$, define the relative residual

On a steady window the drift term is negligible and $\mathrm{Res}(W)$ reduces to the relative mismatch in \eqref{eq:budget_window}. A typical gate is $\mathrm{Res}(W)\le 0.02$ (2\%), but the tolerance must be declared.

Appendix G. $\Pi$-pipeline pseudocode (windowing, RG exponents, and fail-fast logic)

This appendix sketches an implementation-level view of the $\Pi$-diagnostics of Section~7. It is included so that the rotation/verification logic can live inside this white paper (without requiring a separate ``rotation dynamics'' document).

Given a time window W, length scale series L(t), and measured series:
  eps_inj(t), eps_nu(t), eps_events(t), sigma_eff(t)

Compute:
  eps_bind(t) := eps_events(t)   # binding/event dissipation
  Pi_T := T * mean(eps_bind) / DeltaF_crit
  Pi_L(t) := (L(t)^(1+alpha) * eps_bind(t)) / (C_alpha * sigma_eff(t))

Combined:
  Pi_ST := Pi_T^a * (mean over W of Pi_L)^b
  delta_ST := | d/dt log Pi_ST | (window derivative)

Budget residual:
  Res := | mean(eps_inj - eps_nu - eps_events - dH_flow/dt) | / mean(eps_inj)

Decision logic (fail-fast):
  if delta_ST > thr_ST:     SPLIT window
  if Res > thr_budget:      REFINE budget / resolution
  else:                     ACCEPT window for anomaly/RG tests

The goal is to enforce that any claim about $\nu\to 0$ or event saturation is made only on windows that are: (i) quasi-steady (small $\delta_{ST}$) and (ii) budget-closed (small residual).

Appendix H. Distributional formulation of the resolved-flow energy with events

This appendix justifies the event-extended balance \eqref{eq:resolved_flow_budget} as an exact identity for piecewise-smooth dynamics with jump events. The main point is that when events are discrete, the correct mathematical object is a positive measure in time, not a smooth function.

\subsection*{H.1 Setup: càdlàg energy, jump sizes, and the event measure}

Let $E_{\rm flow}(t)=\langle e_{\rm flow}(\cdot,t)\rangle_\Omega$ be the spatial-mean resolved kinetic energy. Assume:

1. Between event times $\{t_n\}$, the resolved flow evolves smoothly (e.g.\ by Navier--Stokes) so that $E_{\rm flow}$ is absolutely continuous and satisfies
$$ \frac{d}{dt}E_{\rm flow}(t)=\varepsilon_{\rm inj}(t)-\varepsilon_\nu(t) \qquad (t\neq t_n). $$
2. At each event time $t_n$, the resolved flow loses a recorded amount of energy $\Delta E_n\ge 0$, so that the jump in the mean energy is
$$ E_{\rm flow}(t_n^+)-E_{\rm flow}(t_n^-)= -\frac{\Delta E_n}{|\Omega|}. $$

(Equivalently, $H_{\rm flow}(t_n^+)-H_{\rm flow}(t_n^-)= -\Delta E_n$ in total-energy form.)

Define the positive Radon measure on time

where $\delta_{t_n}$ is the Dirac measure at $t_n$. For any test function $\varphi\in C_c^\infty(\mathbb R)$,

\subsection*{H.2 Distributional identity}

Consider the distributional time derivative of $E_{\rm flow}$. Using the standard calculus of functions of bounded variation, the derivative decomposes into:

Substituting the jump size yields

Equivalently,

where $\varepsilon_{\rm events}$ denotes the measure $\mu_{\rm events}$ (possibly containing Dirac spikes). Equation \eqref{eq:dist_resolved_budget} is precisely \eqref{eq:resolved_flow_budget}.

\subsection*{H.3 Proof by testing (one line)}

Fix $\varphi\in C_c^\infty(\mathbb R)$. By definition of distributional derivative,

Split the integral into disjoint intervals between events and integrate by parts on each open interval. The boundary terms telescope and produce the jump sum $\sum_n (E_{\rm flow}(t_n^+)-E_{\rm flow}(t_n^-))\varphi(t_n)$. Using the smooth evolution away from events gives

which becomes \eqref{eq:dist_resolved_budget} after inserting the jump size formula and identifying the measure $\mu_{\rm events}$ via \eqref{eq:mu_events_test}.

\subsection*{H.4 Window averages are exact}

For a finite window $W=[t_0,t_1]$ with length $|W|$, the window-averaged event dissipation is

which is exactly the discrete sum used in Section~3.2. Thus the budget closure \eqref{eq:budget_window} can be checked without approximating derivatives at event times.

\begin{remark}[Dense-event limit] If events become dense while individual releases $\Delta E_n$ become small, the measure $\mu_{\rm events}$ may converge to an absolutely continuous measure $\varepsilon_{\rm events}(t)\,dt$. In that case the event channel becomes indistinguishable, at the level of time-averaged budgets, from a classical continuum dissipation density, even though its microscopic origin is discrete. \end{remark}

Appendix I. The Navier--Stokes Millennium problem in the VP frame

This appendix addresses a recurring question: does the VP event-dissipation picture ``solve'' the Navier--Stokes existence and smoothness Millennium problem?

\subsection*{I.0 VP position statement (physics vs.\ mathematics)}

The Clay Millennium question is a statement about an idealized continuum PDE that implicitly assumes one may resolve the velocity field to arbitrarily small spatial and temporal scales. In the VP framework this assumption is not part of the model: continuum fields are only interpreted after coarse graining above a finite cutoff scale.

\medskip \noindentVP claim (physical). Within VP, the ``blow-up'' mechanism that worries analysts (unbounded $|\nabla \uv|$ in finite time) is not a physically admissible state:

• Hard cutoff. VP carriers have finite volume (scale $a$) and the continuum projection is only used for $\ell_{\rm cg}\gtrsim a$ (and, more strongly, $\ell_{\rm cg}\gg D^\ast$). At any fixed $\ell_{\rm cg}$, the measured/coarse-grained fields are automatically smooth with explicit derivative bounds (Section~I.3).
• Safety valve (events). When local strain approaches a \LOCK\ lattice/binding threshold, VP triggers a discrete event (unjam/merge/reconnect) that transfers resolved-flow energy into an internal reservoir $H_{\rm int}$. This produces a nonnegative defect measure $\mu_{\rm events}$ in the resolved-flow energy budget (Appendix~H / Section~3), and preempts any ``infinite'' concentration of resolved energy in a point-like region.

\noindentConsequence. From the VP standpoint, ``global smoothness of the classical PDE down to $\ell\to 0$'' is a modeling extrapolation beyond the VP continuum window. The physically relevant target is instead the existence and stability of a coarse-grained, event-weak description with a recorded defect measure.

\medskip \noindentImportant limitation. This is not a proof of the Clay Millennium statement for the classical Navier--Stokes PDE. It is a declaration about what is (and is not) a well-posed physical question once a finite-volume ontology (VP) is assumed.

\subsection*{I.1 What the Clay problem asks (mathematical statement)}

The Clay Millennium problem concerns the classical 3D incompressible Navier--Stokes PDE, posed on $\mathbb R^3$ (or $\mathbb T^3$), with smooth divergence-free initial data $\uv_0$:

The question is purely mathematical: either prove that a smooth solution exists for all time, or produce an example of finite-time blow-up from smooth data.

\subsection*{I.2 VP standpoint: ``smoothness to arbitrarily small scales'' is a modeling extrapolation}

In VP theory the continuum hypothesis has a declared domain of validity. There is a finite microscopic scale $a$ (VP carrier scale) and a mesoscopic bridge scale $D^\ast$ (Section~2.5). Continuum fields are only interpreted physically after coarse graining with $\ell_{\rm cg}\gg D^\ast$.

Thus, the limit $\ell_{\rm cg}\to 0$---which is implicit in the continuum idealization---is not a physical limit inside VP. In that sense, the Clay question is asking about an extrapolated regime where the VP ontology is no longer ``continuum''.

\subsection*{I.3 Hard cutoff: coarse-grained fields are automatically smooth at finite $\ell_{\rm cg}$}

Let $u_\ell = W_\ell * u^{\rm micro}$ be any coarse-grained scalar (or vector) field built from a bounded micro field and a smooth kernel $W_\ell$ (Section~1.1). Then for any multiindex $\alpha$,

In particular, at a \LOCK\ coarse-graining length $\ell_{\rm cg}\gtrsim a$ there is a finite, explicit upper bound on gradients and higher derivatives of the observed fields. Within VP language, ``$\nabla\uv\to\infty$'' is not an admissible physical statement because it requires resolving below the carrier scale.

This bound does not prove global smoothness of the classical PDE as $\ell\to 0$; it only states that the VP-measurable fields are smooth for all finite coarse-graining.

\subsection*{I.4 Events: physical realization of the defect measure (and why blow-up is preempted)}

VP introduces discrete events with recorded released energy $\Delta E_n\ge 0$. Appendix~H shows that the correct mathematical object is the positive Radon measure

and that the resolved-flow energy is a càdlàg function whose distributional derivative contains a jump part. In the inviscid limit $\nu\to 0$, it is consistent (and testable) that a nonzero dissipation channel remains, encoded either as a dense-event limit $\mu_{\rm events}\to \varepsilon_{\rm events}(t)\,dt$ or as a singular measure (bursty spikes).

In this language, the ``blow-up fear'' is reinterpreted as follows:

• A continuum blow-up scenario is a scenario of unbounded local strain (or vorticity stretching), usually accompanied by a breakdown of classical calculus in the energy budget.
• In VP, when a local strain indicator approaches a \LOCK\ critical threshold (a binding/unjamming limit), an event occurs before any divergence. The excess resolved-flow energy is transferred to $H_{\rm int}$ and appears as a positive contribution to $\mu_{\rm events}$.
• Therefore the physically relevant statement is: the global resolved-flow budget stays closed (with a defect measure) even when $\nu\to 0$.

\subsection*{I.5 Replacement statement (VP-appropriate well-posedness target)}

A VP-appropriate replacement of the classical ``existence and smoothness'' target is:

\medskip \noindentEvent-weak solution (informal). A triple $(\uv,p,\mu_{\rm events})$ is a VP-consistent solution if: (i) $(\uv,p)$ satisfy the incompressible momentum equation in the sense of distributions; (ii) the resolved-flow energy balance holds in $\mathcal{D}'(t)$ with an additional nonnegative measure $\mu_{\rm events}$ as in Appendix~H; and (iii) $\mu_{\rm events}$ is recordable from VP micro events.

\medskip \noindent In this formulation, ``singularity'' is replaced by ``nonzero defect measure'' and the existence question becomes: do global event-weak solutions exist and are they stable under refinement? The VP micro-dynamics are ordinary differential equations with finite carrier volume and therefore exist globally (by construction), and the coarse-grained fields are smooth at any fixed $\ell_{\rm cg}$. The only nontrivial step is to characterize the continuum limit and the statistics/geometry of $\mu_{\rm events}$.

\begin{remark}[About claiming a Clay-prize ``solution''] Nothing in this appendix is a proof of (or a counterexample to) the Clay Millennium problem as stated for the classical Navier--Stokes PDE. Rather, it gives a VP interpretation: the Clay question probes an extrapolation of the continuum hypothesis beyond the VP cutoff scales, while VP replaces that extrapolation by an explicit event-measure channel that keeps budgets well-defined. \end{remark}

\subsection*{I.6 Toy demonstration: a finite-time gradient blow-up prevented by a VP-style event valve}

As a minimal demonstration of the mechanism (not of 3D Navier--Stokes), consider inviscid 1D Burgers. For $u(x,0)=\sin x$ the classical solution forms a shock at $t^\ast=1$ where $|u_x|$ diverges. Below we compare a baseline pseudo-spectral run (no events) with a VP-style run where an ``event'' triggers when $\max|u_x|>S_{\rm crit}$ and applies a discrete smoothing step. The energy removed is recorded as $H_{\rm int}$.

\medskip \noindentBaseline (no events): the strain grows rapidly as $t\to 1^+$.

+-----------------------------------------------------------------------------+
| BASELINE (Toy inviscid Burgers, no events)                                 |
+-----------------------------------------------------------------------------+
| STEP | TIME   | MAX|u_x|  | E_flow    | dE_event | H_int    | STATUS       |
+-----------------------------------------------------------------------------+
|    0 | 0.0000 |     1.000 | 0.250000 | 0.000000 | 0.000000 | BASELINE    |
| 1800 | 0.9000 |    10.000 | 0.250000 | 0.000000 | 0.000000 | BASELINE    |
| 2000 | 1.0000 |   110.153 | 0.250000 | 0.000000 | 0.000000 | BASELINE    |
| 2100 | 1.0500 |   316.072 | 0.250000 | 0.000000 | 0.000000 | BASELINE    |
+-----------------------------------------------------------------------------+

\noindentVP-style event valve: events occur near the would-be singular time and keep $\max|u_x|$ bounded while recording the removed energy in $H_{\rm int}$.

+-----------------------------------------------------------------------------+
| VP-STYLE SAFETY VALVE (Toy Burgers: event dissipation)                     |
+-----------------------------------------------------------------------------+
| STEP | TIME   | MAX|u_x|  | E_flow    | dE_event | H_int    | STATUS       |
+-----------------------------------------------------------------------------+
|    0 | 0.0000 |     1.000 | 0.250000 | 0.000000 | 0.000000 | Stable      |
| 1800 | 0.9000 |    10.000 | 0.250000 | 0.000000 | 0.000000 | Stable      |
| 1986 | 0.9930 |    54.790 | 0.249997 | 0.000003 | 0.000003 | EVENT       |
| 1999 | 0.9995 |    53.959 | 0.249988 | 0.000008 | 0.000012 | EVENT       |
| 2000 | 1.0000 |    55.455 | 0.249988 | 0.000000 | 0.000012 | Stable      |
| 2012 | 1.0060 |    53.525 | 0.249973 | 0.000015 | 0.000027 | EVENT       |
| 2025 | 1.0125 |    55.093 | 0.249953 | 0.000021 | 0.000047 | EVENT       |
| 2352 | 1.1760 |    53.581 | 0.244740 | 0.000361 | 0.005260 | EVENT       |
| 2365 | 1.1825 |    54.318 | 0.244368 | 0.000372 | 0.005632 | EVENT       |
| 2377 | 1.1885 |    54.763 | 0.244004 | 0.000364 | 0.005996 | EVENT       |
| 2389 | 1.1945 |    55.342 | 0.243631 | 0.000372 | 0.006369 | EVENT       |
| 2400 | 1.2000 |    79.557 | 0.243631 | 0.000000 | 0.006369 | Approaching |
+-----------------------------------------------------------------------------+

\noindentReproducibility. The full script and logs are included in the DOI bundle under toy\_singularity\_demo/ (file vp\_burgers\_safety\_valve\_demo.py).

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