Orbit Boundaries — The Rhombus-Band Result

An orbit’s outer envelope can be compared with a familiar family of curves, from a four-cusped astroid through a rhombus to an ellipse and a rounded box. The rhombus is the most frequent clean fit in this census.

Of 613 canon entities in the census, 405 produced bounded, fittable outer envelopes. Of those, 116 were described cleanly by a superellipse. Within that quality-selected subset, 90 of 116 landed in the rhombus band around $n=1$.

That makes the rhombus the modal clean-fit boundary family. It does not make the rhombus the universal boundary of the corpus.

From an orbit to an exponent

The experiment asks one precise question: when an orbit’s outer envelope is well represented by a superellipse, which member of that family fits best?

For each entity, the procedure is:

$$\left|\frac{x}{a}\right|^n+\left|\frac{y}{b}\right|^n=1$$

step	operation
1	iterate for 15,000 steps after a 2,000-step warmup
2	center the orbit and rotate it onto its own PCA principal axes
3	divide the plane into 60 angular sectors and take the 95th-percentile radius in each sector
4	fit the equation above on log-scale residuals
5	retain the fit as clean when $R^2\geq0.5$

The exponent $n$ moves through a familiar family:

exponent	boundary
$n=2/3$	four-cusped astroid
$n=1$	rhombus, or $L^1$ ball
$n=2$	ellipse
$n=4$	squircle
$n\to\infty$	square limit

This is a shape-family fit, not a proof that every orbit has an $n$-norm boundary. In fact, most of the census does not pass that description cleanly.

The denominator ladder

The safest way to read the result is to keep every denominator attached:

population	count	rhombus-band share
full canon census	613	90/613 = 14.7%
bounded and fittable	405	90/405 = 22.2%
clean superellipse fits	116	90/116 = 77.6%

The 77.6% figure answers a conditional question:

Given that a canon orbit has a clean superellipse outer-envelope fit, how often is its best-fit exponent in the rhombus band?

It does not answer how often every canon orbit is rhombic. The conditional result is the measured one.

What the clean subset says

The complete seven-bin distribution is:

$n$ range	shape family	count	share of clean fits
$n<0.5$	sub-astroid	0	0.0%
$0.5\leq n<0.85$	astroid band	1	0.9%
$0.85\leq n<1.5$	rhombus band	90	77.6%
$1.5\leq n<2.5$	ellipse band	12	10.3%
$2.5\leq n<3.5$	rounded-square band	5	4.3%
$3.5\leq n<5$	squircle band	3	2.6%
$n\geq5$	square approach	5	4.3%

The clean-fit median is $n=1.214$ and the mean is $n=1.717$. The distance between them is informative: the central mass lies close to the rhombus, while a thinner high-$n$ tail pulls the mean upward.

The earlier comparison to a uniform distribution in $\log n$ is not used as the headline here. That baseline is a chosen reference measure, not a natural law. The directly measured statement is enough: the astroid appears once in the clean subset, while the rhombus band appears ninety times.

The rejected fits are part of the result

Of the 405 fittable boundaries, 289 failed the $R^2$ threshold. A filter can manufacture a pattern if the shapes it rejects gather in one omitted family, so those failures were re-fit and inspected.

They do not hide a large astroid population. The failed fits gather mainly at the high-$n$ end (about 44%) and in the rhombus band (about 30%). Only about 4% lean toward the astroid band. This makes a simple “the threshold rejected the cusps” explanation unlikely.

But the failures still matter: they say that the superellipse exponent is not a sufficient vocabulary for much of the corpus. Ragged envelopes, lobed boundaries, disconnected radial sectors, or multiscale structure can all be compressed badly into one value of $n$.

Outer envelope is not inner caustic

The astroid occupies two very different roles in the corpus:

geometry	how it is obtained	what it describes
outer orbit envelope	95th-percentile radius of iterated orbit points	the outer occupied boundary
fold caustic	forward image of the critical set $\det J=0$	where folded sheets concentrate or become singular

The first is what this census fits. The second belongs to the engine’s folding geometry. An astroid can therefore be rare as a best-fit outer envelope and still be structurally important as an inner caustic. There is no contradiction: they are different objects at different stages of the map.

The Mirror Frame is a testable bridge

A rhombus is the unit ball of the $L^1$ metric:

$$\frac{|x|}{a}+\frac{|y|}{b}=1.$$

That makes the result suggestive beside the engine’s fixed diagonal coordinates. But the present experiment rotates each orbit separately onto its PCA axes. The Mirror Frame is instead one specific $45^\circ$ frame derived from the kernel.

The decisive follow-up is simple: retain each orbit’s PCA rotation angle and test whether the clean rhombus fits cluster near the Mirror Frame’s diagonal directions. If they do, the two structures genuinely meet. If the angles are broad or uniform, the rhombus is an intrinsic boundary class without a shared global orientation.

What the next pass should measure

The present census earns one durable claim and opens four clean extensions:

• orientation: compare PCA angles with the fixed $\pm45^\circ$ axes;
• model competition: compare the superellipse against Fourier-radial, spline, convex-hull, and multi-lobed boundary models;
• fit stability: bootstrap orbit samples and report uncertainty in $n$;
• stratification: ask whether rhombus-band fits gather by kernel regime, Lyapunov class, basin, or observer window.

The measured result is compact:

Among canon orbits with clean PCA-aligned superellipse envelopes, the rhombus band is overwhelmingly modal. Across the whole corpus, the boundary vocabulary remains open.

:: APPARATUS ::

Sources for this page

Full bibliography →

1. §

   Cosmopoiesis · living edition

   The public kernel definition, parameter meanings, iteration protocol, and dynamical vocabulary used throughout the scholia.

   Read the live canon ↗
2. §

   Mathematica · living edition

   The canonical constants, algebraic identities, and computed closures used by the engine and its derived findings.

   Read the live canon ↗
3. §

   Principia · living edition

   The proposition-level canon, including the Archetypal Lattice, its address generator, morphology classes, and parameter-space discontinuity surfaces.

   Read the live canon ↗