Pairing Lifts the Price and Blurs the Step: The Weak Closures Sharpen as Excitations and Soften as Counts
Year: 2026
Type: preprint
Abstract: AF-27's computed Woods–Saxon spectrum put the record's gaps at the record's addresses and added a spectrum fact: the light closures are the weak ones — 28's gap is 0.78/1.14/1.54 MeV and 50's 1.26/1.84/2.50 across κ_WS = 0.30/0.44/0.60, against 82's 1.85/2.71/3.69 and 126's 2.54/3.20/3.73. The charge was to put pairing on that spectrum and ask whether the weakness survives it. This paper applies the declared BCS occupation smearing — v²(E) = ½[1 − (E−λ)/√((E−λ)² + Δ²)], λ solved per closure so the smeared count is exact, Δ scanned over {0.50, 0.832, 1.25} MeV with 0.832 = 12/√208 the empirical odd-even class value — to AF-27's bench verbatim (its calibration table reproduces AF-27's gaps digit for digit first). Two measures, pre-registered. The excitation value G_qp = √(d₊² + Δ²) + √(d₋² + Δ²), d₊/d₋ the distances from λ to the nearest levels: pairing's quadrature floor lifts every closure, the weak ones proportionally most — at κ_WS = 0.30, Δ = 0.832: ×2.371 (28), ×1.660 (50), ×1.345 (82), ×1.197 (126), ×1.131 (184) — so the spectrum's robustness ordering flattens toward uniformity: G(28)/G(126) = 0.305 → 0.605, G(50)/G(126) = 0.495 → 0.686, and the flattening grows with Δ at all three strengths (at κ_WS = 0.60: 0.415 → 0.554 at the empirical Δ, 0.651 at Δ = 1.25). The occupation step, counted in states with 0.1 < v² < 0.9 at λ: at κ_WS = 0.30 the same Δ blurs 14 states at 28, 18 at 50, 22 at 82 — and 0 at 126 and 0 at 184, whose gaps (2.54, 3.18 MeV) exceed Δ widely enough that no level sits inside the smearing window; the weak closures are precisely the ones whose steps dissolve. The control (§3) keeps the pair of verdicts honest: with the spin–orbit term off, where no level ends at 28 or 50 at all, pairing still quotes a G_qp there (3.640/4.377) — an excitation value across a closure the spectrum does not close, λ jammed against a level (d₋ = 0.095/0.078). Pairing evaluates; it does not place. What AF-27 called the light closures' weakness is thus two facts wearing one name: their excitation values were the spectrum's lowest and pairing raises them toward the heavy closures' (sharpening, real, Δ-monotone), and their count-steps were the spectrum's softest and pairing is exactly the agent that softens them (the smearing reaches them first). The record's own closures — the empirical ones, in real nuclei — live with pairing on: their observed robustness is the paired object, and this bench's paired object says the light closures are less exceptional than the unpaired spectrum made them look, in value, and more fragile, in count.
Author: Aldred, William Alexander
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225425083