Maria's V_EM Explained

Mechanism first, images second

What Changes

Barbour's Variety uses pairwise separations only. Maria's V_EM keeps the same scale-invariant spirit but adds an explicitly orientation-sensitive three-body term.

V EM = \sqrtI ( α Σ i<j |q i q j | / r ij + β Σ i<j<k |q i q j q k | (A ijk \cdot n̂) / (r ij r jk r ki) )

Read it in three pieces.

The pair term is radial and separation-based.
The triangle term depends on oriented area, so it is sensitive to collective rotation-like geometry.
The coefficient β tells you how strongly that rotational organization matters.

The Geometric Mechanism

For each triple (i,j,k), Maria uses the oriented area vector A_ijk = (1/2)(r_ij × r_ik). Summing over all triples creates a total oriented-area vector N = Σ(r_ij × r_ik). The unit vector n̂ = N / |N| is then an axis generated by the configuration itself.

Instructional point: the system does not receive an external axis. The axis is built from the geometry of the configuration. The magnetic-like term then rewards triangles whose oriented areas align with that emergent axis.

How to Read the Figures

Panel	Meaning
Top view	Projection onto the in-plane directions. If the shape is oblate, this is where the full spread remains visible.
Side views	Projections onto planes containing the emergent axis. Flattening shows up here first.
Aspect	Reported as the ratio of the smallest to largest principal scale. Aspect near 1 means nearly spherical. Smaller aspect means flatter, more oblate.
α	Weight of the pairwise electric-like term.
β	Weight of the triangle-based magnetic-like term.

Maria's Functional by Itself

These three figures keep α = 1 fixed and vary β. In the current implementation all charges are +1 and all masses are equal. The only question being tested is: does the triangle term create an emergent flattening axis?

β = 0.0: only the pair term survives. Aspect = 1.00. The configuration is close to isotropic.

β = 2.0: flattening begins. Aspect = 0.90. The side views visibly compress relative to the top view.

β = 10.0: stronger flattening. Aspect = 0.85. The magnetic-like term is now setting the global organization.

What we understand from these three figures:

Maria's triangle term does not merely perturb the pairwise solution. It creates a self-organised preferred axis and drives the minimiser away from spherical symmetry.

Our Controlled Extension

We also tested a cheaper proxy that does not reproduce Maria's full V_EM term, but asks the same qualitative question: can an orientation-sensitive correction flatten the configuration into a Jupiter/Saturn-like oblate shape?

V oblate = V S (1 + λ oblate A oblate)

This proxy is not a replacement for Maria's functional. It is a controlled extension: lower cost, easier sweeps, same geometric target.

λ_oblate = 0.0: baseline shape. z/xy = 1.99. Nearly spherical in three dimensions.

λ_oblate = 2.0, equal masses: strong flattening. z/xy = 0.39. Same qualitative effect as Maria's magnetic-like term.

λ_oblate = 2.0, mass ratio 1:80: flattening survives unequal masses. The heavy particles retain their own visible substructure inside the oblate envelope.

What This Shows

Claim	Status
Maria's triangle term can generate an emergent axis	Shown
Increasing β produces stronger oblateness in the current equal-charge tests	Shown
A cheaper oblate proxy reproduces the same qualitative flattening target	Shown
The proxy is mathematically equivalent to Maria's full triangle term	Not claimed
Charge heterogeneity changes the resulting oblate families in an interpretable way	Not yet tested here
Maria's functional yields molecule-like local motifs and global oblateness simultaneously	Open

Best Next Extensions

β sweep at fixed N: map aspect ratio against the magnetic coefficient and identify where oblateness first appears.
Charge patterns: move beyond all-ones charges and test whether sign structure changes the emergent axis or local clustering.
Mass-ratio comparison: hold α and β fixed while varying masses, to see whether gravitational-molecule motifs survive under the electromagnetic extension.
Proxy vs full functional: compare the cheap oblate proxy to Maria's full V_EM at matched flattening, so the extension reads as controlled rather than ad hoc.

The right narrative for the site:

First show that Maria's functional is understood on its own terms.
Then show one extension that preserves the geometric question while making larger exploratory sweeps feasible.

Maria's VEM Explained