Asymmetry-Term Ladder Toy Model

$$\text{total cost} = \frac{N(N+1)+Z(Z+1)}{2} = \frac{A^2}{4} + d^2 + \frac{A}{2}, \quad \text{excess} = d^2 = \frac{(N-Z)^2}{4}$$

Status: Unproven overall — but with one real, exact partial result.

The asymmetry term in the semi-empirical mass formula, $-a_A(N-Z)^2/A^2$, is normally justified by quantum statistics (the Pauli exclusion principle) without a mechanical account of why it takes that specific form. Paper 13 (unpublished draft) attempts one: model protons and neutrons as each filling their own stack of discrete equilibrium levels, one nucleon per level, so each additional same-species nucleon must occupy the next level up. Writing $N=A/2+d$, $Z=A/2-d$ (so $N-Z=2d$), the total “cost” of filling both stacks works out to exactly $d^2=(N-Z)^2/4$ above the symmetric case — an exact match to the real term’s quadratic dependence on $(N-Z)$.

What’s genuinely proven here: the quadratic form, $(N-Z)^2$, is exactly recovered by this simple discrete-filling argument — a real, checkable piece of algebra, not asserted.

What’s not proven, stated plainly by the paper itself: the model does not reproduce the real term’s $1/A$ dependence (an earlier version of this argument claimed level spacing shrinks as $1/A$, which doesn’t match the standard nuclear picture — real single-particle level spacing near the Fermi surface scales closer to $A^{-1/3}$ or $A^{-2/3}$). It also accounts for, at most, the kinetic (Pauli-exclusion) contribution to the real coefficient $a_A$ — not the interaction (isospin-dependence of the nuclear force) contribution modern nuclear-matter calculations attribute to roughly half or more of the real symmetry energy.

Why this is labeled Unproven rather than Proven despite the exact partial match: a formula is proven when it’s shown to correctly account for what it claims to explain, not just one piece of it. This model explicitly doesn’t derive the full real term — only its exponent, not its scale or its full physical origin. That’s honest, real partial progress, not nothing — but “unproven” is the accurate label for the term as a whole.

Used in: Paper 13 (unpublished draft) only.