Index. A small group of identities relating the exponents of the post, with numerical checks against the data tables. The most useful are the change-of-variable identities between size-distribution exponents ($\tau_w = D_i^{f}\tau_n$, $\tau_w = 2\tau_a$, $\tau_w = D_i^{uf}\tau_p$, and $\tau_p/\tau_n = D_i^f/D_i^{uf}$); a box-counting derivation of the ensemble dimension giving $D_e = \max(D_i^f,\tau_w)$, with self-similarity recovering Thomas's $D_e = D_i^f\tau_n$ and the cleaner $D_e = \tau_w = 2\tau_a$ in the small-cloud regime; a relation $D_\text{cloudy} = 2\,D_{i,uf}^{p,w}/D_{i,uf}^{p,a}$ linking the two "unfilled individual dimensions" to the dimension of cloudy points within one cluster; $D_e = 2-H$ for fBm from the appendix, and how it tiles into the size-distribution chain; the Kondev–Henley–Salinas closure $D_i^f = (3-H)/2$ for fBm, which collapses the entire mean-threshold geometric-exponent chain onto $H$ alone; three statistical-exponent identities ($\xi_h(q) = qH - K(q)$, $\beta = 1+\xi_h(2) = 1+2H-K(2)$, $K(2) = 2\xi_h(1)-\xi_h(2)$); a cross-link $2D_e + \beta = 5$ for fBm obtained by eliminating $H$ from the geometrical and statistical identities; a summary identity diagram; and an honest list of what's still open.
Conventions
I follow the post throughout. Cloud size distributions are reported as
$n(s)\,ds \propto s^{-(1+\tau)}\,ds$, equivalently
$N(>s)\propto s^{-\tau}$, with $\tau>0$. The size_dist_*
columns in the binary tables store $\tau$ in this convention (the script
negates the raw OLS slope so the stored value is positive).
$K(q)$ is the cumulant-generating exponent of the cascade flux, $\langle\varphi_\varepsilon^q\rangle \propto \varepsilon^{-K(q)}$, with $K(1)=0$ by construction (canonical normalisation). For universal multifractals $K(q) = \tfrac{C_1}{\alpha-1}(q^\alpha-q)$. The field $F$ is the order-$H$ fractional integral of $\varphi$, so its difference moments obey $\langle|\Delta F(r)|^q\rangle\propto r^{qH-K(q)}$ and the Haar fluctuation exponent is $\xi_h(q) = qH - K(q)$.
Geometrical exponents are taken at the mean threshold and
nearest-neighbour connectivity unless stated otherwise. $D_i^f$ and
$D_i^{uf}$ are the individual fractal dimensions of the cloud edge with
holes filled or not. I use the perimeter–width variants
(D_ind_filled_pw, D_ind_unfilled_pw) as the
canonical values; the perimeter–area variants are equivalent under
$a_f \sim w^2$ for the filled case but differ from them for the unfilled
case in a way that itself turns out to be informative
(§D_cloudy).
$D_e$ is shorthand for the box dimension of the level set
D_sandbox_q0.
Change-of-variable identities among size-distribution exponents
Different choices of size metric $s$ — area $a$, bounding-box width $w$, summed perimeter $p_s$, nested perimeter $p_n$ — give different size-distribution exponents $\tau_s$. But because the various $s$ values for the same cloud are related by power laws, the $\tau$'s are related by change-of-variable.
Two size measures $s_1$ and $s_2$ for the same set of objects, related by $s_2 = c\,s_1^{\rho}$ for some exponent $\rho$ and prefactor $c$. Counting objects must agree, $n(s_2)\,ds_2 = n(s_1)\,ds_1$, so
$$ n(s_2) \;=\; n(s_1)\,\Bigl|\frac{ds_1}{ds_2}\Bigr| \;\propto\; s_1^{-(1+\tau_1)}\,s_2^{(1-\rho)/\rho} \;\propto\; s_2^{-(1+\tau_1)/\rho}\,s_2^{(1-\rho)/\rho} \;=\; s_2^{-(1+\tau_1/\rho)}.$$Hence
$$\boxed{\,\tau_2 \;=\; \tau_1/\rho\,}.$$The three special cases relevant to the post's geometrical exponents:
Width and area, filled clouds — $\tau_w = 2\,\tau_a$
For hole-filled clouds $a_f = c\,w^2$ (the shape factor $c$ does not carry scale information), so $\rho = 2$ and
$$\boxed{\,\tau_w \;=\; 2\,\tau_a\,}\qquad\text{(filled).}$$Width and nested perimeter — $\tau_w = D_i^f\,\tau_n$
The nested perimeter of a single contour scales with that contour's bounding-box width as $p_n \sim w^{D_i^f}$ by definition of the individual filled fractal dimension. With $\rho = D_i^f$,
$$\boxed{\,\tau_w \;=\; D_i^f\,\tau_n\,}.$$This is the relation Thomas wrote $D_e = D_i^f\,\tau_n$ in DeWitt et al. 2026. Note that it is at face value a relation among geometrical exponents per se, not directly an expression for $D_e$. The connection to $D_e$ comes via the box-counting argument in §Ensemble dimension together with this identity.
Multi-contour subtlety. Nested perimeter assigns one value per contour, not one per cloud — a donut-shaped cloud contributes two nested-perimeter values, one for its outer boundary and one for its hole. So $n(p_n)$ is a distribution over contours while $n(w)$ above is a distribution over clouds. They are nonetheless related by the same change-of-variable, provided the field is statistically self-similar: in that case the population of all contours (exterior + interior) carries the same power-law exponent as the population of cloud bounding-box widths. The self-similarity argument is sketched in §Ensemble dimension; empirically the identity holds to ~1 % across the data, vindicating it.
Width and summed perimeter — $\tau_w = D_i^{uf}\,\tau_p$
Summed perimeter sums all contour lengths within a single cloud (exterior + interior holes) into one number per cloud, and by definition of $D_i^{uf}$, this scales with the cloud's bounding-box width as $p_s \sim w^{D_i^{uf}}$. So $\rho = D_i^{uf}$ and
$$\boxed{\,\tau_w \;=\; D_i^{uf}\,\tau_p\,}.$$Because summed perimeter is one value per cloud (not per contour), this derivation does not need the self-similarity step that nested perimeter does — it is a straight change of variable on the cloud distribution.
Corollary: ratio of $\tau_p/\tau_n$ recovers $D_i^f/D_i^{uf}$
Dividing the nested-perimeter and summed-perimeter identities,
$$\boxed{\,\frac{\tau_p}{\tau_n} \;=\; \frac{D_i^f}{D_i^{uf}}\,}.$$Useful in practice: if you have measured both size-distribution exponents, the ratio of filled to unfilled individual dimensions comes for free, without doing any perimeter–width fitting. In our data the ratio is generally 0.85–0.95, reflecting that filled contours are smoother than the unfilled ones (which include the geometry of all interior hole boundaries).
| case | $H$ | $C_1$ | $\tau_p/\tau_n$ | $D_i^f/D_i^{uf}$ | diff |
|---|---|---|---|---|---|
| fBm, H=0 | 0.0 | 0 | 0.862 | 0.850 | 0.012 |
| fBm, H=0.3 | 0.3 | 0 | 0.892 | 0.895 | -0.002 |
| fBm, H=0.7 | 0.7 | 0 | 0.965 | 0.953 | 0.013 |
| fBm, H=1.0 | 1.0 | 0 | 0.991 | 0.992 | -0.001 |
| H0.3, α=2, C1=0.03 | 0.3 | 0.03 | 0.911 | 0.912 | -0.001 |
| H0.3, α=2, C1=0.10 | 0.3 | 0.10 | 0.938 | 0.938 | 0.000 |
| H0.3, α=2, C1=0.20 | 0.3 | 0.20 | 0.956 | 0.964 | -0.008 |
| H0, α=2, C1=0.20 | 0.0 | 0.20 | 0.870 | 0.862 | 0.008 |
| H0.7, α=2, C1=0.20 | 0.7 | 0.20 | 1.001 | 0.996 | 0.005 |
Numerical check
| case | $H$ | $C_1$ | measured $\tau_w$ | $D_i^{f}\,\tau_n$ | $D_i^{uf}\,\tau_p$ | $2\,\tau_a$ |
|---|---|---|---|---|---|---|
| fBm 8192 H=0 | 0.0 | 0 | 1.920 | 1.922 | 1.951 | 2.033 |
| fBm 8192 H=0.1 | 0.1 | 0 | 1.871 | 1.880 | 1.923 | 1.972 |
| fBm 8192 H=0.3 | 0.3 | 0 | 1.743 | 1.734 | 1.749 | 1.774 |
| fBm 8192 H=0.5 | 0.5 | 0 | 1.519 | 1.514 | 1.556 | 1.535 |
| fBm 8192 H=0.7 | 0.7 | 0 | 1.243 | 1.220 | 1.258 | 1.357 |
| fBm 8192 H=0.9 | 0.9 | 0 | 1.050 | 1.014 | 1.002 | 1.024 |
| fBm 8192 H=1.0 | 1.0 | 0 | 0.980 | 0.971 | 1.134 | 1.017 |
| H0.3, α=2, C1=0.03 | 0.3 | 0.03 | 1.748 | 1.717 | 1.715 | 1.766 |
| H0.3, α=2, C1=0.10 | 0.3 | 0.10 | 1.715 | 1.717 | 1.717 | 1.747 |
| H0.3, α=2, C1=0.20 | 0.3 | 0.20 | 1.719 | 1.678 | 1.664 | 1.702 |
| H0, α=2, C1=0.20 | 0.0 | 0.20 | 1.965 | 1.976 | 1.994 | 2.060 |
| H0.7, α=2, C1=0.20 | 0.7 | 0.20 | 1.389 | 1.332 | 1.339 | 1.359 |
The $2\tau_a$ column is systematically a bit larger than $\tau_w$ — the bias is ~0.03–0.05 and constant across the table. This reflects that the shape factor $a_f/w^2$ is not strictly a scale-invariant constant: small clouds are systematically less square than large ones because of pixel quantisation of small bounding boxes. The other two columns agree with $\tau_w$ within 1–3 %.
None of these identities require the field to be a multiplicative cascade — they apply to any set of objects whose size–size relationships are power laws. This is why they hold equally well for fBm ($C_1=0$) and FIF ($C_1>0$).
Ensemble dimension from individual dimension and size distribution
DeWitt et al. 2026 proposed $D_e = D_i^f \cdot \tau_n$ as a relation between the level-set fractal dimension and the individual cloud-edge dimension, mediated by the nested-perimeter size-distribution exponent. The derivation below recovers it — but only in one of two regimes, with an explicit condition for which regime applies, and a cleaner companion identity $D_e = \tau_w = 2\tau_a$.
Direct box-counting argument
At scale $\varepsilon$ the box count of all cloud edges in the image splits as a sum over clouds. A cloud of bounding-box width $w$ contributes $(w/\varepsilon)^{D_i^f}$ boxes to its exterior contour (definition of $D_i^f$). Integrating against the cloud-size distribution $n(w)\propto w^{-(1+\tau_w)}$,
$$ N(\varepsilon) \;=\; \int_\varepsilon^L n(w)\,(w/\varepsilon)^{D_i^f}\,dw \;\propto\; \varepsilon^{-D_i^f}\int_\varepsilon^L w^{\,D_i^f - 1 - \tau_w}\,dw . $$With $p \equiv D_i^f - 1 - \tau_w$ the integral $\int_\varepsilon^L w^p\,dw$ has three regimes:
| regime | dominated by | integral $\sim$ | $N(\varepsilon) \sim$ | $\Rightarrow D_e =$ |
|---|---|---|---|---|
| $D_i^f > \tau_w$ | upper cutoff $L$ | $L^{p+1}/(p+1)$ | $\varepsilon^{-D_i^f}\,L^{D_i^f-\tau_w}$ | $D_i^f$ (largest cloud dominates) |
| $D_i^f = \tau_w$ | balanced | $\log(L/\varepsilon)$ | $\varepsilon^{-D_i^f}\log(L/\varepsilon)$ | $D_i^f$ with log correction |
| $D_i^f < \tau_w$ | lower cutoff $\varepsilon$ | $\varepsilon^{p+1}/|p+1|$ | $\varepsilon^{-\tau_w}$ | $\tau_w$ (small clouds dominate) |
The two regimes can be packaged as the upper-bound expression
$$\boxed{\,D_e \;=\; \max(D_i^f,\,\tau_w)\,}.$$The intuition. If individual cloud edges are rough ($D_i^f$ large) and the size distribution is flat enough that one big cloud dominates the image ($\tau_w$ small), the ensemble inherits the big cloud's roughness. If the distribution is steep ($\tau_w$ large) so that many small clouds populate the image, the ensemble dimension is set by how those small clouds pile up rather than by any one cloud's contour roughness.
Self-consistency derivation (no $D_i^f$ needed in the small-cloud regime)
A complementary argument bypasses $D_i^f$ entirely and is perhaps the cleanest version of the result. Assume the field is statistically self-similar, so that the interior of one cloud of width $W$ "looks like" a miniature copy of the full image rescaled by $W/L$ — both contain the same statistical population of smaller clouds, holes, and contours.
Then the total box count of the level set within a cloud of width $W$ at scale $\varepsilon \ll W$ is $(W/\varepsilon)^{D_e}$, with the same exponent $D_e$ as for the global level set. Summing over all clouds,
$$ N(\varepsilon) \;=\; A\int_\varepsilon^L n(w)\,(w/\varepsilon)^{D_e}\,dw \;\propto\; \varepsilon^{-D_e}\int_\varepsilon^L w^{D_e - 1 - \tau_w}\,dw . $$For this to be self-consistent — left side scales as $\varepsilon^{-D_e}$, right side must too — the integral on the right has to converge (i.e. not introduce additional $\varepsilon$-dependence). The integral $\int w^{D_e-1-\tau_w}dw$ converges at the lower cutoff iff $D_e - 1 - \tau_w > -1$, i.e. $D_e \ge \tau_w$.
The two solutions of the self-consistency:
- $D_e = \tau_w$ (marginal): the integral is logarithmic and the self-similar ansatz is consistent. This is the small-cloud-dominated regime.
- $D_e > \tau_w$: the integral is dominated by the upper cutoff $L$, and the self-similar ansatz is consistent for any $D_e$ between $\tau_w$ and 2. Then $D_e$ is set by something other than the size distribution — specifically, by the dimension of the single largest cloud's contour, $D_e = D_i^f$.
Combining the self-similarity argument with the change-of-variable identities of §change-of-variable, in the small-cloud regime
$$\boxed{\,D_e \;=\; \tau_w \;=\; D_i^f\,\tau_n \;=\; 2\,\tau_a\,}.$$Three equivalent expressions for the ensemble dimension. The right-hand form, $D_e = 2\tau_a$, is the simplest: in the small-cloud regime the ensemble fractal dimension is simply twice the area-distribution exponent. No fractal-dimension or sandbox calculation is needed — only counts of small clouds at the smallest resolvable scales.
Where does the regime crossover sit?
The crossover is at $D_i^f = \tau_w$. For fBm the closure of §KH gives $D_i^f = (3-H)/2$ and $\tau_w = 2-H$, which are equal only at the boundary $H = 1$ (both $\to 1$). So in the idealized infinite-domain limit the small-cloud regime holds for every $H < 1$, and $D_e = \tau_w = 2-H$ throughout. At finite resolution the picture shifts: near $H = 1$ the measured $D_i^f$ is biased above $(3-H)/2$ (pixel discretization forces nominally-smooth contours to wiggle along grid edges — see the KH table below), while $\tau_w$ stays close to $2-H$, so the two measured exponents meet earlier, near $H \approx 0.85$–$0.9$. The new $H = 0.9$ fBm field confirms this: there the measured $D_i^f = 1.106$ exceeds $\tau_w = 1.050$, and $D_e = 1.112$ tracks $\max = D_i^f$ rather than $\tau_w$ — the large-cloud regime has set in just below $H = 1$, as predicted. For $H$ below the crossover, small clouds dominate and $D_e \approx \tau_w$; above it the largest cloud dominates and the size distribution loses predictive power over $D_e$.
Numerical check
| case | $H$ | $C_1$ | $D_e$ | $D_i^f$ | $\tau_w$ | $\max$ | $D_i^f\tau_n$ | $2\tau_a$ |
|---|---|---|---|---|---|---|---|---|
| fBm 8192 H=0 | 0.0 | 0 | 1.966 | 1.416 | 1.920 | 1.920 | 1.922 | 2.033 |
| fBm 8192 H=0.1 | 0.1 | 0 | 1.925 | 1.391 | 1.871 | 1.871 | 1.880 | 1.972 |
| fBm 8192 H=0.3 | 0.3 | 0 | 1.681 | 1.339 | 1.743 | 1.743 | 1.734 | 1.774 |
| fBm 8192 H=0.5 | 0.5 | 0 | 1.456 | 1.276 | 1.519 | 1.519 | 1.514 | 1.535 |
| fBm 8192 H=0.7 | 0.7 | 0 | 1.253 | 1.163 | 1.243 | 1.243 | 1.220 | 1.357 |
| fBm 8192 H=0.9 | 0.9 | 0 | 1.112 | 1.106 | 1.050 | 1.106 | 1.014 | 1.024 |
| fBm 8192 H=1.0 | 1.0 | 0 | 1.069 | 1.125 | 0.980 | 1.125 | 0.971 | 1.017 |
| H0.3, α=2, C1=0.03 | 0.3 | 0.03 | 1.672 | 1.317 | 1.748 | 1.748 | 1.717 | 1.766 |
| H0.3, α=2, C1=0.10 | 0.3 | 0.10 | 1.662 | 1.285 | 1.715 | 1.715 | 1.717 | 1.747 |
| H0.3, α=2, C1=0.20 | 0.3 | 0.20 | 1.606 | 1.229 | 1.719 | 1.719 | 1.678 | 1.702 |
| H0, α=2, C1=0.20 | 0.0 | 0.20 | 1.892 | 1.404 | 1.965 | 1.965 | 1.976 | 2.060 |
| H0.7, α=2, C1=0.20 | 0.7 | 0.20 | 1.162 | 1.097 | 1.389 | 1.389 | 1.332 | 1.359 |
For fBm, $\max(D_i^f,\tau_w)$ matches $D_e$ within ~0.05 across all $H$ — comparable to the discretization bias on $D_e$ itself. For intermittent FIF the formula over-predicts $D_e$ by 0.05–0.20 in the worst cases. The derivation assumed independent contributions from each cloud in the box-counting integral; intermittency clusters cascade-active regions spatially, so the box-count of the union is less than the sum of per-cloud box-counts. The formula therefore degrades to an upper bound in the intermittent regime. Quantifying the correction is open question 2.
Practical implication. If you can measure the cloud width distribution cleanly (which you usually can — counts of small clouds extend to the smallest resolvable scales), $\tau_w$ alone gives you $D_e$. Equivalently $2\tau_a$. Either is substantially easier to estimate than a true box-count or sandbox dimension. Even without measuring $D_i^f$ you can decide which regime you are in by checking whether the cumulative cloud count $N(>w)$ has a clear power-law tail — if it does, you are in the small-cloud regime and the relation applies.
Caveat: this is a mean-threshold statement. The $D_e = \tau_w = 2\tau_a$ chain holds near the mean threshold but degrades quickly as the binarizing threshold moves away from it. On the $8192^2$ fBm fields binarized at percentile thresholds (data added after the original derivation), $D_e$ and $\tau_w$ agree to within $\sim 0.06$ at the mean but diverge by up to $0.73$ at the $5$th-percentile threshold. The mechanism is the regime structure above: at a low threshold the exceedance set percolates into one space-filling cluster — the large-cloud regime, where the size distribution no longer controls $D_e$ — and at a high threshold the clouds are too sparse for the width distribution to resolve a clean power law. The change-of-variable identity $\tau_w = 2\tau_a$ is hardier, being pure geometry rather than a regime statement: it survives across thresholds, though its shape-factor bias grows at the extremes. Use $D_e = \tau_w$ only when the threshold sits near the field median and $N(>w)$ shows a clean tail.
Unfilled perimeter–area gives the cluster cloudy-set dimension
The post argues that $D_i^{p,a}_{uf}$, the slope of $\log\langle p_s\rangle$ vs $\log\sqrt{a_{uf}}$ for unfilled isolated clouds, is "not interpretable as a clean fractal dimension" because the denominator $\sqrt{a_{uf}}$ is itself fractal — unfilled area is smaller than $w^2$ by the presence of holes.
But this is the very fact that makes the ratio of the unfilled perimeter–area and perimeter–width exponents interpretable. By definition,
$$ \langle p_s\rangle \;\propto\; w^{D_{i,uf}^{p,w}} . $$If the cloudy points (the interior of one isolated unfilled cluster) fill the bounding box as a fractal of dimension $D_\text{cloudy}$, $a_{uf} \sim w^{D_\text{cloudy}}$, then $\sqrt{a_{uf}} \sim w^{D_\text{cloudy}/2}$ and
$$ \langle p_s\rangle \;\propto\; w^{D_{i,uf}^{p,w}} \;=\; \bigl(\sqrt{a_{uf}}\bigr)^{2\,D_{i,uf}^{p,w}/D_\text{cloudy}} . $$Hence the unfilled perimeter–area exponent stored by the analysis is $D_{i,uf}^{p,a} = 2\,D_{i,uf}^{p,w}/D_\text{cloudy}$, giving
$$\boxed{\,D_\text{cloudy} \;=\; \frac{2\,D_{i,uf}^{p,w}}{D_{i,uf}^{p,a}}\,}.$$This is a clean identity that turns the "unphysical mix" called out in the post into a direct measurement of a third dimension: the dimension of the cloudy points inside a single isolated cloud (a fractal generally distinct from both $D_i^{uf}$ and $D_e$, though for mean-thresholded fields with no exceedance-set rarefaction it equals the topological dim 2).
Numerical check
| case | $H$ | $C_1$ | $D_{i,uf}^{p,a}$ | $D_{i,uf}^{p,w}$ | $D_\text{cloudy}$ |
|---|---|---|---|---|---|
| fBm canonical, H=0 | 0.0 | 0 | 1.752 | 1.665 | 1.901 |
| fBm canonical, H=0.3 | 0.3 | 0 | 1.552 | 1.472 | 1.896 |
| fBm canonical, H=0.7 | 0.7 | 0 | 1.253 | 1.232 | 1.967 |
| fBm canonical, H=1.0 | 1.0 | 0 | 1.083 | 1.122 | 2.073 |
| H0.3, α=2, C1=0.03 | 0.3 | 0.03 | 1.490 | 1.444 | 1.939 |
| H0.3, α=2, C1=0.10 | 0.3 | 0.10 | 1.389 | 1.371 | 1.974 |
| H0.3, α=2, C1=0.20 | 0.3 | 0.20 | 1.287 | 1.275 | 1.981 |
| H0, α=2, C1=0.20 | 0.0 | 0.20 | 1.737 | 1.628 | 1.873 |
| H0.7, α=2, C1=0.20 | 0.7 | 0.20 | 1.110 | 1.102 | 1.984 |
For the FIF cases with substantial intermittency the result is more uniformly close to 2 than for fBm — likely because the isolated-cluster filter selects compact, hole-free clouds preferentially in the intermittent case (fewer of the cloud's interior pixels are "missing"). The slight deficit at fBm is the only weak hint of any deviation from $D_\text{cloudy}=2$ in this dataset.
$D_e = 2-H$ for fBm at the mean threshold
Re-derived in the post appendix; I include the chain consequence here because it ties cleanly to the change-of-variable identities above. The box-counting argument: view a 2D fBm field as a graph (surface) in 3D. At scale $\varepsilon$, the field fluctuates vertically by $\varepsilon^H$, so each $\varepsilon\times\varepsilon$ spatial column needs $\varepsilon^H/\varepsilon = \varepsilon^{H-1}$ stacked boxes to cover the graph. There are $L^2/\varepsilon^2$ columns, so the total box count of the graph in 3D is $\varepsilon^{-2}\cdot\varepsilon^{H-1} = \varepsilon^{-(3-H)}$, giving graph box-dim $3-H$. Slicing with a horizontal plane (the level set) reduces dimension by 1, so $D_e = 2-H$.
Combining with the size-distribution chain (small-cloud regime),
$$\tau_w = 2-H, \qquad \tau_a = 1-H/2, \qquad \tau_n = (2-H)/D_i^f, \qquad \tau_p = (2-H)/D_i^{uf}.$$| case | $H$ | $2-H$ | $D_e$ | $\tau_w$ | $2\tau_a$ |
|---|---|---|---|---|---|
| fBm 8192 H=0 | 0.0 | 2.000 | 1.966 | 1.920 | 2.033 |
| fBm 8192 H=0.1 | 0.1 | 1.900 | 1.925 | 1.871 | 1.972 |
| fBm 8192 H=0.3 | 0.3 | 1.700 | 1.681 | 1.743 | 1.774 |
| fBm 8192 H=0.5 | 0.5 | 1.500 | 1.456 | 1.519 | 1.535 |
| fBm 8192 H=0.7 | 0.7 | 1.300 | 1.253 | 1.243 | 1.357 |
| fBm 8192 H=0.9 | 0.9 | 1.100 | 1.112 | 1.050 | 1.024 |
| fBm 8192 H=1.0 | 1.0 | 1.000 | 1.069 | 0.980 | 1.017 |
$D_e$ slightly exceeds the theoretical $2-H = 1$ at $H=1$; the size distribution itself becomes pathological at this boundary of the parameter space, as the post discusses.
Kondev–Henley–Salinas closure: $D_i^f(H)$ for fBm
The chain identities above relate the geometric exponents to one another but leave $D_i^f$ — the individual filled fractal dimension — as a free input. For fBm at the mean threshold there is an external closure that fixes $D_i^f$ in terms of $H$ alone, from the Coulomb-gas analysis of 2D Gaussian contour loops by Kondev & Henley (1995), expanded in Kondev, Henley & Salinas (1999/2000). They derive the hyperscaling relation
$$ D_f\,(\tau_\mathrm{KH} - 1) \;=\; 2 - H , $$where $\tau_\mathrm{KH}$ is the loop-length exponent in the cumulative convention $N(>\ell)\sim \ell^{-(\tau_\mathrm{KH}-1)}$. The hyperscaling alone matches the post's identity $D_e = D_i^f\,\tau_n$ once one identifies $\tau_n = \tau_\mathrm{KH} - 1$ — neither paper predicts $D_i^f$ from $H$ by itself. The further input is a separate relation between the loop fractal dimension and a loop-correlation exponent $x_l$ (the exponent governing the probability that two points lie on the same contour loop):
$$ D_f \;=\; 2 - x_l - \tfrac{H}{2} . $$The Coulomb-gas mapping of 2D Gaussian surfaces conjectures $x_l = \tfrac{1}{2}$ (a super-universal value, exact at $H = 1$ where the level-set support has dim $2-H = 1$ and each contour loop must also have dim 1, fixing $x_l = 1/2$ via the formula above). Substituting $x_l = 1/2$ and combining with the hyperscaling:
$$ \boxed{\,D_i^f \;=\; \frac{3 - H}{2}, \qquad \tau_n \;=\; \frac{4 - 2H}{3 - H}\,} \quad\text{(fBm, mean threshold).} $$This closes open question 3 for fBm. (KH is a published result, reached for here while looking for the closed form — not an invention of these derivations.) It does not apply to FIF — Gaussianity is the condition the Coulomb-gas argument uses — so for intermittent fields $D_i^f(H, C_1, \alpha)$ remains open.
Numerical check
| case | $H$ | $(3-H)/2$ | $D_i^f$ measured | diff |
|---|---|---|---|---|
| fBm 8192 H=0 | 0.0 | 1.500 | 1.416 | $-0.084$ |
| fBm 8192 H=0.1 | 0.1 | 1.450 | 1.391 | $-0.059$ |
| fBm 8192 H=0.3 | 0.3 | 1.350 | 1.339 | $-0.011$ |
| fBm 8192 H=0.5 | 0.5 | 1.250 | 1.276 | $+0.026$ |
| fBm 8192 H=0.7 | 0.7 | 1.150 | 1.163 | $+0.013$ |
| fBm 8192 H=0.9 | 0.9 | 1.050 | 1.106 | $+0.056$ |
| fBm 8192 H=1.0 | 1.0 | 1.000 | 1.125 | $+0.125$ |
Sign of the bias is informative. At $H=0$ the empirical $D_i^f$ is below the KH prediction by $\sim 0.08$ — fractal contours flagged by a finite-resolution algorithm are systematically straightened, dragging $D_i^f$ down from $1.5$ toward something closer to $1.4$. At $H=1$ the empirical $D_i^f$ is above the prediction by $\sim 0.12$ — at $H=1$ the underlying fBm contour is nominally a smooth curve with $D=1$, but pixel discretization forces it to wiggle along the grid, padding $D_i^f$ upward. Both biases are expected and both are larger than the bias on $D_e$ itself at the same array size, consistent with the individual-fractal-dim estimator being noisier (fewer pixels per single contour than per ensemble).
Closed-form chain for fBm at the mean threshold
With KH plus the change-of-variable identities and $D_e = 2 - H$, the entire suite of mean-threshold geometric exponents collapses onto $H$:
$$ \begin{array}{lll} D_e &=& 2 - H \\ \tau_w &=& 2 - H \\ \tau_a &=& (2 - H)/2 \\ D_i^f &=& (3 - H)/2 \\ \tau_n &=& (4 - 2H)/(3 - H) \\ D_\text{cloudy} &=& 2 \end{array} $$Six exponents, one parameter. $D_i^{uf}$ and $\tau_p$ are not predicted by KH (the loop-correlation argument is for filled, simply-connected contours), but they remain linked by $\tau_p/\tau_n = D_i^f/D_i^{uf}$, so measuring any one fixes the others.
Statistical exponent identities
Three identities that follow immediately from the FIF construction and the isotropic spectrum convention. None are new derivations — they reduce to definitions and to one calculation of a Fourier transform — but the empirical agreement they show is the cleanest in the entire data set, and they cross-link the four statistical exponents ($H$, $K(q)$, $\xi_h(q)$, $\beta$) tightly.
$\xi_h(q) = qH - K(q)$
The Haar fluctuation $\Delta_H F(r)$ is a difference of consecutive $r/2$-block averages of $F$. The FIF construction defines $F$ as the order-$H$ fractional integral of a cascade flux $\varphi_r$ whose moments scale as $\langle\varphi_r^q\rangle \propto r^{-K(q)}$ (small-$r$ direction). The block average over a region of size $r/2$ is a smoothing of $F$ at scale $r$, so by the order-$H$ integration, $\Delta_H F(r) \sim r^H \varphi_r$ in distribution. Therefore
$$\langle|\Delta_H F(r)|^q\rangle \;\sim\; r^{qH}\langle\varphi_r^q\rangle \;\propto\; r^{qH - K(q)},$$giving
$$\boxed{\,\xi_h(q) \;=\; qH - K(q)\,}.$$$K(1) = 0$ by construction (canonical normalisation of $\varphi$), so $\xi_h(1) = H$ exactly — even for intermittent fields. The "correction" to a fBm-style Hurst exponent kicks in only at orders $q \ne 1$.
Spectral slope: $\beta = 1 + \xi_h(2) = 1 + 2H - K(2)$ (isotropic 2D)
For an isotropic 2D field $F$ with second-order structure function $\langle|\Delta F(r)|^2\rangle \propto r^{\xi(2)}$, the 2D power spectrum has a power-law radial profile. The Wiener–Khinchin theorem relates the variogram $V(r) = \langle|\Delta F(r)|^2\rangle$ to the spectrum $P_2(\mathbf{k})$ via $V(r) = 2\int(1-\cos(\mathbf{k}\cdot\mathbf{r}))P_2(\mathbf{k})\,d^2\mathbf{k}$. For a self-affine field, the radial part of $P_2$ must scale as $P_2(k)\propto k^{-(2+\xi(2))}$ so that $V(r)\propto r^{\xi(2)}$ — the exponent $-(d+\xi(2))$ in $d$ dimensions, here $d=2$, is the standard self-similarity result.
The 1D radial spectrum $E(k)$ that the analysis computes (the isotropic average of the 2D spectrum integrated over an angular shell of radius $k$) is $E(k) = 2\pi k\,P_2(k) \propto k^{-(1+\xi(2))}$. Hence
$$\boxed{\,\beta \;=\; 1 + \xi_h(2) \;=\; 1 + 2H - K(2)\,}.$$
Empirical agreement is striking: $\beta - (1+\xi_h(2))$ is within
$\pm 0.01$ across nearly every case in
exponents_canonical.bin. Both quantities are biased
together by finite-size discretization so their difference is much
cleaner than either is on its own. The only exception is the
$H=1$ block, where the spectrum itself has a logarithmic divergence at
the largest scales and discretization bias is large (the boundary of
the fBm parameter space).
$K(2) = 2\xi_h(1) - \xi_h(2)$: intermittency from one Haar analysis
Algebraic rearrangement of the first identity at $q=1$ and $q=2$. In principle this lets you read off $K(2)$ — the intermittency at second order — from a Haar fluctuation analysis evaluated at two orders only, with no separate cascade analysis required.
In practice this estimator overshoots $K(2)$ by ~50 % at $C_1 = 0.1$ on the FIF data (see the post). Both $\xi_h(1)$ and $\xi_h(2)$ are biased low by the FIF numerical generation; the biases compound rather than cancel in the combination $2\xi_h(1) - \xi_h(2)$. This is a real limitation of estimating intermittency from intermediate-$q$ moment-scaling exponents, well-known in the multifractal literature under various names.
Cross-link between $D_e$ and $\beta$ for fBm
The two purely-input relations $D_e = 2 - H$ and $\beta = 1 + 2H$ (the fBm versions of the statistical and geometrical identities above) can be combined to eliminate $H$:
$$\boxed{\,2\,D_e + \beta \;=\; 5 \qquad\text{(fBm)}\,}.$$Equivalently $D_e = (5-\beta)/2$ — a purely geometric-to-statistical relation with no input parameter on the right-hand side. I check it on the $4096^2$ fBm pool rather than the $8192^2$ set, because the latter's measured $\beta$ saturates toward $2$ for $H > 0.5$: those fields are generated aperiodic, so the FFT power spectrum picks up a boundary-discontinuity leakage tail $\sim k^{-2}$ that overtakes the true $k^{-\beta}$ spectrum precisely once $\beta > 2$ — an estimator artifact on aperiodic data, not physics. (The $4096^2$ fields are periodic by construction and carry no such leakage: their $\beta$ tracks $1 + 2H$ to within $0.02$ at every $H$, whereas the $8192^2$ $\beta$ falls $0.26$ short at $H = 0.7$ and nearly $1.0$ short at $H = 1$.) On the $4096^2$ pool, $2D_e + \beta$ stays within $\pm 0.07$ of $5$ for $H \le 0.7$ and reaches $+0.20$ at $H = 1$, where $D_e$ itself carries the most finite-domain bias.
For FIF the relation generalises to $2D_e + \beta = 5 - K(2)$ if the analytical argument for $D_e = 2-H$ holds at $q=0$ regardless of intermittency (which it does — see open question 1). In the data $2D_e + \beta$ trends downward from 5 with increasing $C_1$; this is dominated by the $-K(2)$ correction from the spectrum. The residual gap on top of that is finite-size bias on $D_e$, which grows with $C_1$ because the FIF $16384\to 4096$ block-averaging is known not to fully converge at $4\times$. The infinite-domain prediction is exactly $2D_e + \beta = 5 - K(2)$ with no additional level-set correction.
Summary identity diagram
INPUT ─► STATISTICAL ─► GEOMETRICAL (mean threshold)
───── ─────────── ────────────────────────────
H, C1, α ─► ξ_h(q) = qH − K(q)
─► β = 1 + 2H − K(2)
─► D_e = 2 − H (q=0; fBm and FIF)
(q>0 picks up cascade modulation)
Small-cloud regime (D_i^f < τ_w):
─► τ_w = D_e
─► τ_a = D_e / 2 (filled)
─► τ_n = D_e / D_i^f
─► τ_p = D_e / D_i^uf
─► τ_p/τ_n = D_i^f / D_i^uf
fBm closure (Kondev–Henley):
─► D_i^f = (3 − H)/2
─► τ_n = (4 − 2H)/(3 − H)
Cluster-internal density:
─► D_cloudy = 2 D_i^uf,pw / D_i^uf,pa
( = 2 at mean thresh.)
Large-cloud regime (D_i^f > τ_w):
─► D_e = D_i^f
─► size dist no longer informative
For fBm at the mean threshold and in the small-cloud regime, the combined KH + change-of-variable + $D_e=2-H$ chain reduces $\{D_e, \tau_w, \tau_a, D_i^f, \tau_n, D_\text{cloudy}\}$ to a single parameter $H$. $D_i^{uf}$ and $\tau_p$ remain unpredicted but mutually linked. For FIF the same chain holds at $q=0$ in the infinite-domain limit, but the higher-$q$ Rényi spectrum $D_q^\text{lev}(q)$ of the level set picks up additional cascade-modulation content not in $H$ alone. So the post's "too many exponents" thesis is sharpest at $q=0$: there, the apparent geometric zoo at mean threshold collapses to one number for fBm and to one number plus a cascade spectrum for FIF.
Open questions / what's still open
-
Rényi spectrum $D_q^\text{lev}(q)$ for FIF level sets, and the
$D_e$ drift with $C_1$. The post's box-counting argument for
$D_e = 2-H$ uses the mean-order field fluctuation
$\sim\varepsilon^{\,\xi_h(1)}$ with $\xi_h(1) = H - K(1) = H$. Because
$K(1) = 0$ identically, the argument carries through unchanged for FIF:
at $q = 0$ it predicts no intermittency correction,
$D_0^\text{lev} = 2 - H$ for fBm and FIF alike. Thomas's revised text
instead proposes, from an empirical intermittency-correction slider,
$D_e = 2 - H - \tfrac{1}{2}C_1$. The data sits between these two
readings, and says something more specific than either.
Three facts about the measured drift $D_e - (2-H)$ on the $4096^2$ canonical pool. (i) It is already nonzero at $C_1 = 0$ in an $H$-dependent way — fBm undershoots $2 - H$ by $0.04$ at $H = 0$ (but by only $\sim 0$ at $H = 0.3$), pure discretization bias on $D_e$ with no intermittency present. Where this floor is large it already accounts for part of the apparent "$C_1$ drift". (ii) At fixed $(H, C_1)$ the drift depends strongly on $\alpha$, growing as $\alpha$ falls: at $H = 0.3$, $C_1 = 0.1$ it runs $-0.038,\ -0.062,\ -0.109$ for $\alpha = 2,\ 1.8,\ 1.3$ — a threefold spread that a correction in $C_1$ alone cannot see. (iii) At $\alpha = 2$, $-\tfrac{1}{2}C_1$ is a decent effective fit at larger $C_1$ ($-0.094$ observed vs $-0.10$ predicted at $C_1 = 0.2$, $H = 0.3$) but overshoots at small $C_1$ ($-0.038$ vs $-0.05$ at $C_1 = 0.1$).
So $-\tfrac{1}{2}C_1$ is a serviceable engineering correction for $\alpha = 2$ fields at this resolution, but it cannot be the infinite-domain law: that law, if it exists, must carry $\alpha$-dependence and sit on top of the $H$-dependent fBm discretization floor. And whether the $C_1$-and-$\alpha$-dependent part is genuine infinite-domain physics or FIF-generation bias cannot be settled from these data, because both grow the same way — with $C_1$, and as $\alpha$ falls (lower $\alpha$ means more extreme cascade values, which both broaden the true singularity spectrum and make the $16384 \to 4096$ block-average converge worse). The decisive test is to redo the high-$C_1$, low-$\alpha$ cases at $128\times$ coarsening — the convergence threshold the post identifies for Haar fluctuations — and see whether the drift collapses toward $2 - H$ (generation bias) or survives with its $\alpha$-dependence intact (a real correction). My lean is mostly bias with a small real tail, but I would not publish either as settled.
The real $q$-dependence shows up at $q \ne 0$. Bootstrap data at $H=0.3$, $C_1=0.05$ gives ensemble-converged $D_2 = 1.632$ vs. fBm $D_2 = 1.681$ at the same $H$: a $0.05$ gap that doesn't shrink with ensemble size. So the FIF level set is genuinely multifractal as a measure — contour density along the level set varies with the cascade modulation — even though the underlying support has the same fBm-like dimension $2-H$. The dependence on $q$ should be expressible as a Legendre transform of $K(q)$ restricted to the $(2-H)$-dimensional level-set support, analogous to the dim/codim duality for cascade measures, but a closed form remains out of reach. Empirically $D_q^\text{lev}$ drifts by $\sim 0.07$ over $q \in [0, 5]$ for $H=0.3, C_1=0.1$, with a smooth concave shape — the right qualitative behaviour for a multifractal projection.
Bottom line for the post's $D_e = 2-H$: it holds for both fBm and FIF at $q = 0$ in the infinite-domain limit; the canonical-table drift is dominated by finite-size and generation bias, with at most a small real $\alpha$-dependent correction that $-\tfrac{1}{2}C_1$ approximates for $\alpha = 2$. The genuine intermittency physics lives in $D_{q \ne 0}^\text{lev}$, a separate quantity from $D_e$.
- Intermittency correction to the ensemble-from-size-distribution formula. The integral $\int n(w)(w/\varepsilon)^{D_i^f}\,dw$ assumed independent contributions from different clouds — exactly the assumption broken by intermittency, which clusters cascade-active regions in space. Empirically $\max(D_i^f, \tau_w)$ degrades to an upper bound on intermittent fields, with the gap growing with $C_1$. A cleaner statement would weight the integral by the local cascade value, so cloud density inside intermittent regions is correctly suppressed, but a closed-form result has not been written down.
- Closed form for $D_i^{uf}$ from $H$ (and the FIF version of $D_i^f$). The KH closure (§KH) fixes $D_i^f = (3-H)/2$ for fBm at the mean threshold via the Coulomb-gas loop-correlation argument, closing what was previously open for fBm. The unfilled individual dimension $D_i^{uf}$ is a different geometric quantity — it measures the boundary of a cluster including interior hole boundaries — and KH says nothing about it directly. Empirically $D_i^{uf}$ tracks $D_i^f$ closely (ratio 0.85–0.95 across the fBm pool) but is not identical, and no published closure is known to either pass of these derivations. For FIF the KH argument also fails because the Coulomb-gas mapping uses Gaussianity; an intermittency-corrected version would need to track the cascade's $K(q)$ through the loop-correlation step and is open.
- Threshold dependence for FIF. Across moment thresholds $0.5$, $1.0$, $1.5$ the canonical $D_e$ drifts a few percent. A clean derivation would use the multifractal codimension function $c(\gamma)$ to predict the sign and magnitude as a function of threshold, $C_1$, and $\alpha$. I can predict the sign (more-singular thresholds should give smaller $D_e$, which matches the data at $q=1.5$ vs $q=0.5$) but not the magnitude.
- Cohen-style formula for sandbox $D_q$ on level sets of fBm. For fBm the level-set $D_q$ should be exactly $q$-independent (no intermittency, no multifractal modulation). On our finite data it drifts by ~0.03–0.05 per unit $q$ — this is discretization bias only, with no analytical content, but a sharper finite-domain-corrected formula for $D_q$ on fBm-like fields would be useful for distinguishing "real" $q$-dependence (multifractal modulation) from artefactual $q$-dependence (numerical bias).
- Threshold-set Korcak law. If the level-set dimension depends on threshold (which it must for multifractals, even if weakly at the mean), then by the size-distribution chain $\tau_w(T)$ also depends on threshold. A first-pass derivation via the multifractal codimension function would give $\tau_w(T)$ as a function of the cascade $c(\gamma)$ evaluated at the singularity $\gamma_T$ corresponding to threshold $T$. The cascade codimension is itself a Legendre transform of $K$, so this threshold-dependent Korcak's law is in principle a function only of $H, C_1, \alpha$ and $T$. I did not work out the form.
Derivations and numerical checks by Claude Opus 4.7 and Opus 4.8.
Opus 4.7 wrote the original identities and numerical checks in May 2026,
working from an earlier draft of the post. Opus 4.8 revised the page against
Thomas's later edits and the expanded data: re-verifying every identity on
the current tables, repairing the verification script, extending the checks
to the new $H = 0.1$ and $0.9$ fBm fields, validating the regime crossover,
and reworking the $D_e$–$C_1$ drift discussion below in response to Thomas's
empirical $-\tfrac{1}{2}C_1$ correction. Algebra that didn't make the cut and
the punted-questions list live in scratch.md. All numerical
checks are reproducible via verify.py against the exponent
tables in site/too-many-exponents-assets/.