The difficulty in WP-110 was that the roughness of the object was controlled through a spectrum on a surface, while the quantity the variance theory is parameterised by is the smoothness of the area function A(z). The two are related by the geometry in a way nobody computed. So build A(z) directly:
Amplitude b−nH against frequency bn gives Hölder exponent exactly H, and ab > 1 makes it nowhere differentiable for every H < 1. The regularity is now an input to the experiment, not something estimated from the object afterwards.
∫01 cos(bnπz + φ) dz = (sin(bnπ + φ) − sin φ)/(bnπ), and for b odd every bn is odd, so sin(bnπ + φ) = −sin φ and the integral is −2 sin(φ)/(bnπ) — known in closed form. Centring each term therefore makes ∫W exactly zero, and a flat base gives volume exactly 1. With b = 3 there is no numerical ground truth to converge to; there is an identity.
Any non-negative A(z) is the area function of a real body — the solid of revolution of radius √(A/π) — and block [3] checks A > 0 throughout, so nothing here is an abstraction that no object could realise.
Fitting a single line across all section counts mixes two regimes and reports an exponent belonging to neither. At H = 0.1 the all-n fit gives 2.596 where the asymptotic value is 2.003. Every exponent below is fitted to the tail — the largest five section counts — with the correlation of that fit printed beside it.
A = 1 + cW: a cylinder with a rough cross-section, so A(0) and A(1) are not zero.
| H | tail γ | r | all-n γ |
|---|---|---|---|
| 0.10 | 0.611 | −0.99015 | 1.000 |
| 0.20 | 0.609 | −0.97915 | 1.005 |
| 0.35 | 0.759 | −0.97819 | 1.065 |
| 0.50 | 0.980 | −0.99903 | 1.085 |
| 0.65 | 1.012 | −0.99986 | 1.019 |
| 0.80 | 1.004 | −0.99993 | 0.997 |
| 0.90 | 1.002 | −0.99995 | 0.994 |
γ = 1 wherever the tail fit is a straight line. The three roughest rows are not asymptotic even at n = 3203, and their correlations say so; they are reported and not used.
A = sin(πz)·(1 + cW) — the same interior, now vanishing at both ends the way a closed body does at its poles. For the unit sphere A(z) = π(1 − z²) ≈ 2π(1 − z) near the pole: linear.
| H | tail γ | r | all-n γ |
|---|---|---|---|
| 0.10 | 2.003 | −0.99998 | 2.596 |
| 0.20 | 2.003 | −0.99997 | 2.472 |
| 0.35 | 2.003 | −0.99997 | 2.254 |
| 0.50 | 2.003 | −0.99997 | 2.101 |
| 0.65 | 2.003 | −0.99996 | 2.027 |
| 0.80 | 2.003 | −0.99996 | 2.005 |
| 0.90 | 2.003 | −0.99996 | 2.003 |
Seven Hölder exponents spanning nearly the whole admissible range, and the asymptotic decay exponent is 2.003 in every one of them — identical to three decimals, with every tail correlation better than −0.9999. Changing what the area function does at the two ends of its support moved the exponent by a full unit. Changing its regularity everywhere in between moved it by nothing.
It is not inert; it just does not act on the exponent. Read the two fits side by side:
The rough object obeys the same law and takes longer to get there. Interior roughness sets the width of the pre-asymptotic regime — how many sections are needed before the law applies — and nothing else. For anyone actually cutting sections, that is the more useful of the two statements: the question a convoluted object raises is not which error law but from how many sections onward.
A closed triangulated surface is the p = 1 case: its area function vanishes linearly at both poles. WP-110 measured γ = 1.82, 2.00, 2.19, 1.98 across four roughness spectra by slicing a mesh — an apparatus sharing no code with this one — and found no dependence on roughness. Both numbers and both nulls now have the same cause.
So WP-110’s stated reason was wrong. Its null was not an artefact of surfaces that were not folded enough; corrugating the interior of a closed body could not have moved that exponent at any amplitude, because the poles fix it. The restriction that paper named — an isoperimetric ratio of 6.23 against a cortical scale near 15 — is a true statement about its test objects and was the wrong diagnosis of its result.
No published variance predictor is implemented or tested here. This identifies which property of the object the exponent tracks; whether a practitioner’s default smoothness constant gets the constant right on a folded object is the question WP-110 left open and it is still open. Two further limits are structural. At p ≥ 2 the coefficients of variation reach 10−10 and the fits measure double precision rather than sampling, so the vanishing order cannot be pushed further with this arithmetic. And the design is sensitive to the arithmetic of the sampling: with b = 3 and n a power of two the aliasing resonates and the fitted exponent is meaningless — an earlier version of this experiment reported γ near 1 for exactly that reason. The section counts used here are odd and coprime to b, with phases randomised across four realisations, which is a mitigation and not a proof of independence.