Pillar IV — Dissipative: arrangement sets dissipation

Events carry a nonnegative dissipation measure with the exact budget ⟨ε_inj⟩=⟨ε_ν⟩+⟨ε_events⟩; as ν→0 dissipation migrates to the event channel and saturates at the injection rate. At substrate level the events are local unjamming avalanches of the marginal arrangement; driven marginality (G-SOC) is the gated reason the channel is always available. Their sizes obey a scale-free law with the marginal-stability exponent (τ→1.46), measured directly in developed turbulence. On release into decay the dissipation coefficient C_ε follows the Vassilicos non-equilibrium law ∝Rey_λ⁻¹ and relaxes onto the fixed-point plateau (the zeroth law).

The last pillar addresses the fact most often presented as a property of a limit rather than of the medium: anomalous dissipation. We give it a configurational origin. Rearrangement events carry energy out of the resolved flow as a nonnegative measure, and in a forced steady state energy conservation forces this event dissipation to take over from the viscous channel as viscosity vanishes. The dissipation rate is then set by the large-scale arrangement (the injection), not by the microscopic viscosity.

Events as a dissipation measure

Event dissipation (locked record, derived budget)

[LOCK] (A0, event part). Each rearrangement event at $(\xv_k,t_k)$ releases a binding energy $\Delta E_{{\rm bind},k}\ge0$ . The space–time event measure

\[ \mu_{\rm events}=\sum_k\Delta E_{{\rm bind},k}\,\delta_{(\xv,t)=(\xv_k,t_k)}\ge0 \]

defines a windowed event dissipation rate $\eps_{\rm events}=\langle d\mu_{\rm events}/(dt\,d\xv)\rangle_W$ . [DERIVE] (budget). In a steady window with periodic or controlled boundaries and a fixed injection $\langle\eps_{\rm inj}\rangle=I$ , energy conservation gives the exact budget

\begin{equation} \langle\eps_{\rm inj}\rangle=\langle\eps_\nu\rangle+\langle\eps_{\rm events}\rangle, \qquad \langle\eps_\nu\rangle=2\nu\langle Z\rangle. \label{eq:budget_window} \end{equation}

Equation (eq.) is the windowed version of the exact identity (eq.) verified in Pillar I, with the single addition of a nonnegative event channel on the right. It is bookkeeping: whatever energy enters must leave, through one channel or the other.

What the events are: local unjamming of the marginal substrate

Up to here, "event" has been operational: a rearrangement that releases binding energy, $\eps_{\rm bind}=\Theta\sigma\ge0$ . The substrate section gives the operational definition a microscopic face, at no new cost.

[DERIVE] (geometry) — events are local unjamming avalanches

At the margin the shear reserve of the packing is its overlap distribution, and it vanishes with the distance to isostaticity (Section §3; unjam_inflow.py). A local shear demand exceeding the local reserve must break contacts; broken contacts release stored contact (binding) energy; and because the medium is marginal, one break lowers neighbouring reserves and propagates—an avalanche of unjamming that re-jams when the released stress is absorbed. That is an event in precisely this pillar's sense: localized in space–time, dissipative, with $\eps_{\rm bind}\ge0$ guaranteed because contact energy is nonnegative and breaking only releases it. The identification is geometric bookkeeping, not a new postulate.

Why the event channel is always available in a driven medium is the substrate gate G-SOC (\S§3): drive unjams, rest re-jams, and the companion observes the resulting pinning at marginal coordination ( $g^\star/g_0\approx1.1$ – 1.3 ). If G-SOC fails—a steady state drifting with drive instead of pinning—this identification loses its "permanently marginal" premise; the falsification route is stated there.

The anomaly as channel migration

Onsager-type anomaly via arrangement

[DERIVE] (budget level). Consider a one-parameter family in which the viscous channel is weakened ( $\nu\to0$ ) while the event-core scale r_c shrinks accordingly. If the steady budget (eq.) holds and the event channel can absorb the flux, then

\[ \langle\eps_\nu\rangle\to0,\qquad \langle\eps_{\rm events}\rangle\to I>0, \]

so the total dissipation does not vanish with viscosity but saturates at the injection rate

. [GATE] (mechanism). The premise that the event channel physically carries the inertial flux down to r_c

—that event activity concentrates where velocity increments violate the 1/3

H\"older threshold, and that this concentration persists under resolution—is exact in 1D Burgers, resolution-stably demonstrated in 2D, and resolution-converged in developed 3D turbulence (Section below and ns3d.py); only its infinite-Reynolds asymptotic form remains an open gate.

The split is the content. In the standard field account, anomalous dissipation is the statement that the inviscid limit retains a defect; here the defect is named—it is the event measure—and the budget shows why its magnitude is fixed by the large scale. The dissipation is configured: it is what the arrangement must shed to stay steady, independent of how small the viscosity is.

The metriplectic decomposition, derived

The saturation is not an assumption; it follows from the structure of the evolution once dissipation is admitted alongside the Hamiltonian core of Section §4. Write the rate of change of any observable as a reversible (Poisson) part plus an irreversible (metric) part,

\begin{equation} \frac{dA}{dt}=\{A,H\}+[A,S], \label{eq:generic} \end{equation}

with the degeneracy conditions $\{A,S\}=0$ and [A,H]=0

(the metriplectic, or GENERIC, conditions). Here

is the total energy: the reversible bracket conserves entropy, the irreversible bracket conserves the total energy, and $G\nabla C=0$ keeps the Casimirs (enstrophy in

D, helicity in

D) constant. The observable that actually dissipates is the resolved (kinetic) energy

, which is not

: the metric bracket conserves the total

by transferring resolved energy into unresolved internal and binding modes. Its reversible transport—exactly the $U_2{+}U_3$ Hamiltonian flow of (eq.)–(eq.)—conserves

, while its irreversible part can only drain it, $dE/dt|_{\rm irr}\le0$ , with $dS/dt\ge0$ by positive semidefiniteness.

Apply this to the resolved energy of a forced steady state, with the two dissipative transfer channels—a viscous one with coefficient $a\nu$ and an event (binding) one with coefficient , both nonnegative by the semidefiniteness—and an injection :

$\begin{equation} \frac{dE}{dt}=\underbrace{\{E,H\}}_{=\,0}\;+\;I-a\nu E-bE \;\Longrightarrow\; E_\star=\frac{I}{a\nu+b}. \label{eq:Estar} \end{equation}$ The two dissipation rates are then $\eps_\nu=a\nu E_\star$ and $\eps_{\rm bind}=bE_\star$ , and they close the budget exactly, $\eps_\nu+\eps_{\rm bind}=I$ , recovering (eq.). The inviscid limit is now immediate:

$\begin{equation} \nu\to0:\quad E_\star\to\frac{I}{b},\qquad \eps_\nu\to0,\qquad \eps_{\rm bind}\to I . \label{eq:saturation} \end{equation}$ The viscous channel switches off and the event channel takes over the entire injection—channel migration, not a coincidence of limits. The binding rate is the event law of Section §4, $\eps_{\rm bind}=\Theta\sigma$ with $\sigma\ge0$ , so the same nonnegativity that makes the metric bracket dissipative makes $\eps_{\rm bind}\ge0$ . Crucially, a naive closure that lets cores merge by contact without the metric (binding) channel fails to close the budget—it leaves a residual of order the injection—which is exactly why the event channel is necessary and is the gate the numerics probe.

What reproduces, and what is deliberately left open

We separate cleanly the part that is a theorem (the budget and its saturation) from the part that is a gate (that a microscopic event rule realizes the flux). An honest negative result sharpens the distinction: a naive position-based merger rule, in which like-sign vortices simply bind on contact, fails to close the budget (residual $\approx0.49$ ). Saturation is therefore not automatic; it requires the event channel to be energy-consistent, which is exactly what the gate demands.

Pillar IV — budget closure and saturation (metriplectic_vortex.py, validate_all.py)

Energy-consistent reduced metriplectic vortex-gas model: forced steady state with a smooth viscous sink and a binding-event sink, $\dot E=I-a\nu E-bE$ , giving $E_\star=I/(a\nu+b)$ .

Budget closes exactly. $\langle\eps_{\rm inj}\rangle=\langle\eps_\nu\rangle+\langle\eps_{\rm bind}\rangle=I=0.21$ for every $\nu$ . PASS
Saturation. As $\nu$ falls from to $5\times10^{-4}$ , $\langle\eps_\nu\rangle\to0$ (from to $4\times10^{-4}$ ) and $\langle\eps_{\rm bind}\rangle\to I$ (from to ). PASS\\ The plateau equals the injection rate by construction: the dissipation is set by the large-scale arrangement, not by $\nu$ .
Reference viscous branch. The viscous curve $(\Rey_{\rm eff},\max_t\eps_\nu)$ that the event branch overlays is the validated output of Pillar I's solver, with $\max_t\eps_\nu=2\nu Z(0)\propto\nu$ exactly. PASS
Negative control. A naive contact-merger rule does not close the budget (residual $\approx0.49$ ): saturation is conditional, matching the [GATE] status of the mechanism. (documented, not a PASS)

Run python validate_all.py; checks 4–5 are the budget and saturation.

The mechanism gate—that the event channel physically carries the inertial flux to the dissipation scale—is the heaviest open claim. In one dimension it is exactly tractable, and we test it directly: in Burgers turbulence the events are shocks, and the Duchon–Robert defect is exact.

Pillar IV mechanism — events carry the flux (1D Burgers, event_flux.py)

Decaying Burgers from $u=\sin x$ ; once the shock forms it is the rearrangement event. The inviscid energy defect of a shock of jump $\Delta u$ is $(\Delta u)^3/12$ , localized at the shock.

Dissipation event flux. The viscous dissipation $\int\eps_\nu$ equals $(\Delta u)^3/12$ to $\sim10\%$ (the residual is the finite-window jump estimate), across $\nu$ . PASS
$\nu$ -independence. $\int\eps_\nu=0.61,0.64,0.65,0.66$ as $\nu$ falls $0.02\to0.0025$ (a factor ): the dissipation is fixed by the event, not by viscosity. PASS
Flux concentrates at the event. The Duchon–Robert inter-scale flux $\Pi_\ell(x)=\tfrac14\!\int\phi_\ell'(r)\,\delta u(r)^3\,dr$ is $\sim100\%$ localized at the shock, and matches the dissipation at the inertial scale. PASS

$\Rightarrow$ in the canonical (1D) case the event does carry the inertial flux to the dissipation scale. The same Duchon–Robert structure governs 2D/3D Navier–Stokes (vortex sheets and stretching events); the next box tests it directly in two dimensions.

Pillar IV (flux-carrying), 2D Navier–Stokes (multid_flux.py)

A forced-scale vorticity field is evolved to a filamented snapshot. In 2D the forward cascade is the enstrophy cascade: enstrophy is carried to the small (dissipation) scale, where the energy dissipation $\eps_\nu=2\nu Z$ lives, and the events are the thin intense vorticity-gradient structures. Filtering at scale $\ell$ (Germano), the local inter-scale enstrophy flux $\Pi(x)=-\sigma_j\,\partial_j\bar\omega$ , with subfilter vorticity flux $\sigma_j=\overline{u_j\omega}-\bar u_j\,\bar\omega$ , is compared against the palinstrophy density $|\nabla\bar\omega|^2$ (proportional to the local dissipation).

Events carry the flux. The top $20\%$ most intense gradient regions carry $\approx60\%$ of the forward enstrophy flux ( $\langle\Pi\rangle>0$ ); only $\approx12$ – $16\%$ of the area carries half of it. PASS
Flux co-locates with dissipation. $\mathrm{corr}(\Pi, |\nabla\bar\omega|^2)\approx0.28$ : the inter-scale flux concentrates where the dissipation is. PASS
Resolution-stable. Across at fixed physical $\ell$ , the top- $20\%$ fraction holds at $63\%,61\%,58\%$ and the half-flux area at $12\%,14\%,16\%$ . PASS

$\Rightarrow$ in 2D the events do carry the inter-scale flux to the small scale—as in 1D, but shown by spatial concentration rather than an exact identity. This narrows the gate from "multidimensional" to the high-Reynolds asymptotic question—whether the concentration sharpens onto a measure-zero set as $\mathrm{Re}\to\infty$ —which is beyond reach at these resolutions and is stated honestly as the residual [GATE].

Pillar IV (flux-carrying), 3D Navier–Stokes (ns3d.py)

The genuine Onsager case: in 3D the forward cascade is the energy cascade. A 3D pseudo-spectral solver (velocity/rotational form, 2/3 dealiasing, IF-RK4) is first validated—in the inviscid limit it conserves energy to machine precision ( $\Delta E/E\sim10^{-16}$ , the rotational nonlinearity being energy-orthogonal to $\uv$ ) and stays divergence-free to $\sim10^{-14}$ . Low- forcing then drives a developed cascade, confirmed by the velocity-derivative skewness reaching the canonical $\approx-0.5$ . The filtered (Germano) local energy flux $\Pi=-\tau_{ij}\bar S_{ij}$ is compared with the strain magnitude $|\bar S|^2\propto$ the local dissipation.

Flux is forward. $\langle\Pi\rangle>0$ , about – $0.6\,\eps$ in the inertial range and rising toward $\eps$ near the dissipation scale; – $95\%$ of $|\Pi|$ is forward. PASS
Events carry it. At a fixed physical filter scale the top $20\%$ of the strain field carries $\approx60$ – $66\%$ of the forward flux; only $\approx10$ – $14\%$ of the volume carries half. The flux is strain-aligned: $\mathrm{corr}(\Pi,|\bar S|^2)\approx0.80$ (and only weakly vorticity-aligned), exactly as 3D theory predicts. PASS
Resolution-converged. Across (identical physics, fixed physical filter): top- $20\%=63\%,66\%,60\%,60\%$ ; half-flux volume $=13\%,10\%,14\%,14\%$ ; correlation . PASS

$\Rightarrow$ in 3D, as in 1D and 2D, the events carry the inter-scale energy flux to the dissipation scale, and the concentration is resolution-converged. The only residual is the infinite-Reynolds asymptotic—whether the support tends to measure zero as $\mathrm{Re}\to\infty$ —which is the genuine open Onsager/Duchon–Robert question, beyond any finite simulation and stated honestly as the surviving [GATE].

A second, independent test of the same identification sharpens it from geometry to a number. If the events are genuinely unjamming avalanches of a marginal substrate, their sizes cannot carry an intrinsic scale: the thresholded dissipation structures must be distributed as a power law, with the exponent fixed by the marginal-stability avalanche relation. Measuring the local dissipation field $\eps(\xv)=2\nu\,S_{ij}S_{ij}$ of developed 3D turbulence, thresholding it, and recording the size distribution of the connected high-dissipation structures is therefore a direct, falsifiable test — and an exponential (any intrinsic scale) would refute the avalanche reading outright.

Pillar IV (events are marginal-stability avalanches) — the dissipation-event size law (dissipation_avalanche.py)

From the same validated solver, developed turbulence (skewness $\approx-0.5$ ) is sampled over decorrelated snapshots; the full-resolution dissipation field is thresholded at $h\langle\eps\rangle$ and the connected structures are labelled with periodic boundaries; their volume distribution P(s) is fitted by maximum likelihood (Clauset, discrete, KS-selected cutoff) and cross-checked by logarithmic binning. The marginal-stability (Lin–Wyart) relation $\tau=2-\tfrac{\theta}{1+\theta}\tfrac{d}{d_f}$ , with stationary $\theta\approx0.5$ – 0.6 and sheet-like $d_f\approx2$ , predicts in advance $\tau\approx1.4$ – 1.5 .

Scale-free, not exponential. is a power law over more than a decade (log-binned $R^2\approx0.97$ –), and a fitted exponential is rejected at every threshold (positive power-law/exponential log-likelihood ratio). PASS
Sheet-like structures. The gyration-radius–volume scaling gives $d_f\approx2.0$ –, resolution-stable — the geometric input the prediction assumes. PASS
Predicted exponent, with the right trend. The measured exponent falls with resolution into the predicted band: $\tau=1.60\pm0.01$ at $N=64\Rightarrow\tau=1.46\pm0.01$ at (log-binned $\approx1.4$ ), landing inside along a clean finite-Reynolds trend. PASS

$\Rightarrow$ the dissipation events of developed turbulence are distributed as marginal-stability avalanches with the predicted exponent, tying Pillar IV's events to the substrate of Section §3 by a quantitative law and not only by a shared name. This is a genuine prediction (not a postdiction): the band [1.4,1.5]

was fixed by the substrate before the turbulence was measured. The residual is the one carried throughout—the precise $\mathrm{Re}\to\infty$ exponent, beyond any finite resolution, remains a [GATE].

A third developed-turbulence test closes the dissipation story by its magnitude. The budget box showed the dissipation saturates at the injection rate (the anomaly); the event-RG (\S§5) reads that saturated, scale-independent state as the $\Pi_L$ fixed point. The sharp consequence concerns leaving that state: out of equilibrium the dissipation coefficient $C_\eps=\eps L/u'^3$ must not stay constant but flow as the non-plateau RG transient. Releasing a forced field into free decay (the Goto–Vassilicos protocol) and tracking $C_\eps$ against the Taylor-scale Reynolds number $\Rey_\lambda$ tests exactly this — and a constant $C_\eps$ throughout would refute the "non-plateau transient" reading.

Pillar IV (dissipation magnitude) — non-equilibrium $C_\eps$ and the fixed-point plateau (nonequilibrium_dissipation.py)

A developed field (skewness $\approx-0.5$ ) is released into free decay; the integral scale (from the energy spectrum), , $\eps=2\nu Z$ , and $\Rey_\lambda=u'\lambda/\nu$ are logged along the decay, giving the trajectory $C_\eps(\Rey_\lambda)$ . Equilibrium (Taylor's zeroth law the $\Pi_L$ fixed point) predicts $C_\eps=$ const (slope ); Vassilicos's non-equilibrium law predicts $C_\eps\propto\Rey_\lambda^{-1}$ .

Non-equilibrium law, not constant. In the high- $\Rey_\lambda$ (early-decay) window $C_\eps$ rises as $\Rey_\lambda$ falls, with slope $C_\eps\propto\Rey_\lambda^{-1.01\pm0.02}$ at the well-resolved ( $k_{\max}\eta>1.3$ throughout, ) and at — the Vassilicos , opposite to the equilibrium constant. PASS
Relaxation to the fixed point. As the decay becomes self-similar (low $\Rey_\lambda$ ) the slope flattens to $\approx-0.3$ and $C_\eps$ approaches a plateau ( $\approx1.0$ ): the non-plateau transient relaxes onto the $\Pi_L$ fixed point, i.e. the zeroth law itself. PASS

$\Rightarrow$ the framework's reading of anomalous dissipation is borne out dynamically: the constant- $C_\eps$ state is the event-RG fixed point (\S§5), and departure from it follows the predicted non-plateau flow with the Vassilicos exponent. The residual is the universality of the precise exponent and plateau value at higher $\Rey$ , beyond these resolutions — the same infinite-Reynolds [GATE].

Pillar IV, in one line. Rearrangement events carry a nonnegative dissipation measure whose steady-state budget is exact ([DERIVE]) and saturates at the injection rate as $\nu\to0$ (Onsager-type anomaly); that the event channel physically carries the inertial flux to the dissipation scale is exact in 1D Burgers, resolution-stably concentrated on events in 2D, and resolution-converged on strain events in developed 3D turbulence, leaving only the infinite-Reynolds asymptotic limit as a [GATE]. Independently, those events are distributed as marginal-stability avalanches: the thresholded dissipation structures follow a scale-free size law with the pre-registered Lin–Wyart exponent ( $\tau\to1.46\in[1.4,1.5]$ ), not an exponential. And by magnitude: on release into decay the dissipation coefficient flows as the Vassilicos non-equilibrium law $C_\eps\propto\Rey_\lambda^{-1}$ ( -1.01 at the resolved N=96 ) and then relaxes onto the constant- $C_\eps$ plateau — the $\Pi_L$ fixed point, i.e. the zeroth law.