# ABACUS R8 mathematical descent — the minimum-action triaxial curvature surface

## 1. Why R7 was one level too high

R7 was globally analytic, but it minimized the Fourier complexity of the **support
function**.  The design criterion in R8 is stricter and more physical: the visible
curvature comb itself must depart from a circle by the least possible triaxial mode.

Let s be arclength, L the perimeter, q=s/L, and k(q) curvature.  A circle has

\[
k_0=\frac{2\pi}{L}.
\]

Total turning fixes the mean curvature, so every perturbation delta=k/k0-1 has zero
mean.  Threefold symmetry requires delta(q+1/3)=delta(q).  Reflection symmetry fixes
an even phase.

For any derivative order m>=1, minimize

\[
J_m=\int_0^1 |\partial_q^m\delta|^2\,dq
\]

at fixed RMS noncircularity.  The permitted Fourier frequencies are 3,6,9,...;
the first frequency uniquely minimizes every J_m.  Therefore

\[
\boxed{k(q)=\frac{2}{W}[1+\epsilon\cos(6\pi q)]}
\]

where W=L/pi is the mean width.  This is the same answer whether the fairness metric
penalizes curvature rate, curvature acceleration, or any higher derivative.  It is
therefore the precise mathematical version of "start from equal combs and lengthen
three poles with the least possible acceleration field."

Strict convexity is exactly |epsilon|<1.

## 2. Exact closure

The tangent angle is

\[
\psi(q)=2\pi q+\frac{\epsilon}{3}\sin(6\pi q).
\]

Because psi(q+1/3)=psi(q)+2pi/3, the displacement of each third is the 120-degree
rotation of the previous third.  Their vector sum is zero, so the curve closes exactly
and has exact triaxial symmetry.

The boundary is the single analytic integral

\[
\gamma(q)=\pi W\int_0^q
\exp\left[i\left(2\pi t+\frac{\epsilon}{3}\sin6\pi t\right)\right]dt-c.
\]

No circular arcs, fillets, splines, or joins are part of the mathematical source.
The curve is real analytic.  G3/G4 continuity is exceeded because no join exists.

Using the Jacobi-Anger expansion, the same curve has the exact global Fourier-Bessel
representation

\[
\gamma(q)=\frac{W}{2i}\sum_{n\in\mathbb Z}
\frac{J_n(\epsilon/3)}{1+3n}
\exp\left[2\pi i(1+3n)q\right]-c.
\]

No denominator vanishes for integer n.  This makes the one-function, all-derivative
continuity explicit; numerical CAD samples approximate this analytic authority.

## 3. Why combs cannot only get longer

For every simple closed convex curve,

\[
\int k\,ds=2\pi.
\]

Lengthening the three curvature comb poles must be balanced by shortening curvature
between them.  The cosine is the least-action redistribution.  In this study the
curvature ratio is 1.500:1, hence

\[
\epsilon_0=\frac{r-1}{r+1}=0.200000.
\]

The comb differs from a circle by only plus/minus 20.0%.

## 4. The same descent along Z

Let u=z/H.  The mean width W(u) is the unique cubic minimizing integral (W'')^2
subject to the bottom/top widths and the foundry minimum draw at both rims:

\[
W(u)=W_b+m u+(\Delta W-m)(3u^2-2u^3),
\]

where m=-2pH and p is minimum draft per wall.

The triaxial curvature amplitude is the unique minimum-bending cubic carrying the
bottom phase to its 60-degree-inverted top phase with zero endpoint rate:

\[
\epsilon(u)=\epsilon_0(1-6u^2+4u^3).
\]

At u=1/2, epsilon=0, so the section is an exact circle.  The top epsilon is negative;
for this curvature law that is exactly the bottom curve rotated by 60 degrees.

The complete shell is one parametric function:

\[
\boxed{S(q,u)=\left(\Re\gamma(q,u),\Im\gamma(q,u),Hu\right)}.
\]

Both scalar fields are polynomials and gamma is analytic in their parameters.  The
surface is one real-analytic manifold, not a CAD loft patched between profiles.

## 5. Canonical R8 values

- bottom mean width: **275.0 mm**
- top mean width: **180.0 mm**
- height: **285.0 mm**
- curvature ratio: **1.50:1**
- epsilon0: **0.200**
- wall: **4.0 mm**
- minimum rim draft per wall: **0.50%**
- board plane: **75.9 mm**

The amplitude remains the single semantic/physical scalar.  If it tends to zero, the
circle is recovered.  R8 does not pretend mathematics can select a nonzero identity
amplitude without a packaging or product-identity constraint.
