In Dynamic Closure Theory, the coarse-grained energy dissipation rate ${\varepsilon }_r$ is equipped with a two-state coherence closure fraction $C\left(\varepsilon \right)\in (0,1)$—the fraction that is cascade committed (forward flux) at dissipation level $\varepsilon$—and its complementary open branch decays as a power law, $1-C\left(\varepsilon \right)\sim {(\varepsilon /{\varepsilon }_0)}^{-\lambda }$, with a singled-out, untuned default $\lambda =1$. Crucially, $C$ is neither the dissipation density $p\left(\varepsilon \right)$ nor a deterministic function of $\varepsilon$: we give it an operational meaning as the dissipation-conditioned backscatter fraction, $1-C\left(\varepsilon \right)=P({\mathrm{\Pi }}_r<0\mid {\varepsilon }_r=\varepsilon )$, with ${\mathrm{\Pi }}_r$ the signed filtered interscale energy flux, so that by construction, the one-point $\varepsilon$-statistics carry no information about it. This observable is measured directly on three Johns Hopkins Turbulence Database forced isotropic fields: the ${8192}^3$ simulation at ${\mathrm{Re}}_{\lambda }\approx 1300$ (five snapshots, six disjoint ${512}^3$ subvolumes, $k_{max}\eta \approx 1.9$), its finely resolved ${\mathrm{Re}}_{\lambda }=613$ snapshot ($k_{max}\eta =5.3$), and the $32{768}^3$ simulation at ${\mathrm{Re}}_{\lambda }\approx 2556$ ($k_{max}\eta \approx 2.8$), using Gaussian filter widths $\mathrm{\Delta }/\eta \approx 4–100$, together with a hierarchical (subvolume and spatial-block) bootstrap, a phase-randomized surrogate control, and controlled resolution-degradation experiments. All three datasets show the same directional association: the backscatter fraction falls monotonically with ${\varepsilon }_r$, from $\approx \frac{1}{2}$ (random flux sign) in quiescent regions to a few percent in the most intensely dissipating ones, and at fixed Reynolds number, the conditional curve is scale invariant—written in absolute units of ${\varepsilon }_r$, it collapses across filter widths, which is the sense the term carries throughout. The functional form, however, is not the one written down: the data display a plateau near $\frac{1}{2}$ below $\langle \varepsilon \rangle$ that the logistic closure (which plateaus at unity) cannot capture, so the global fit rules out $\lambda =1$ at every scale, and at high Reynolds number, the curvature of the knee and the steepness of the tail emerge as separate exponents. The estimated upper-branch steepness varies across the three Reynolds-number datasets and is not consistent with a Reynolds-number-independent unit exponent. Across 20 disjoint subvolumes at ${\mathrm{Re}}_{\lambda }=613$ (each binned relative to its own mean), the upper branch (${\varepsilon }_r\ge \langle \varepsilon \rangle$) has fitted steepness 0.48–0.74 (95% intervals excluding unity), with a wide between-region scatter; the four subvolumes of the full analysis lie in the steep half of that distribution, so once between-region variance is included, the default $\lambda =1$ is not supported. Before the physical size of the analysis units is matched, the ${\mathrm{Re}}_{\lambda }=613$ upper branch appears two or more times steeper than at the higher Reynolds numbers, where the fitted $\lambda$ is 0.20–0.32 at ${\mathrm{Re}}_{\lambda }\approx 2556$, excluding both unity and zero, and $-0.02–0.18$ at ${\mathrm{Re}}_{\lambda }\approx 1300$. The steepness is, however, sensitive to the size of the region used to normalize ${\varepsilon }_r$ (the analyzed cores span about $200\eta , 390\eta$, and $560\eta$ in the three fields); after size matching, the remaining difference is only $10–20\%$ and is not resolved by the present intervals, so the apparent Reynolds-number dependence of the tail steepness is dominated by analysis-unit size. Two resolution controls—degrading the resolved ${\mathrm{Re}}_{\lambda }=613$ field to the resolution of the ${\mathrm{Re}}_{\lambda }\approx 1300$ run, and low-passing the ${\mathrm{Re}}_{\lambda }\approx 2556$ field on its own grid—shift the upper-branch steepness by less than 0.1 when compared on the same subvolumes, so neither the missing dissipative-range modes nor the coarse-grid gradient error tested here reproduces the flatter branch at high Reynolds number on the subvolumes as analyzed. The steepness of the intense-dissipation branch thus depends on the scale of the region against which the dissipation is normalized and, beyond that, its Reynolds-number or inertial-range-width dependence is, at most, $10–20\%$ within the sampled range, while the shoulder of the curve stays approximately unchanged across the three fields. A $Q$-criterion coherence proxy computed on the same fields shows no tail whatsoever (slope $\approx 0$) since a positive-definite gradient invariant does not encode the flux sign and is not an interchangeable proxy for flux-direction occupancy. We explain why one-point dissipation statistics cannot test the closure, and we state precisely what the measurement does and does not establish.

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• Fluctuation-dissipation theorem
• Homogeneous turbulence
• Intermittency
• Isotropic turbulence
• Turbulence
• Turbulence modeling