Do nuclear shell closures shift with neutron–proton asymmetry?
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
README.md
# Do nuclear shell closures shift with neutron–proton asymmetry?
**Hypothesis H141, Volume XII — "Shell closures shift with asymmetry"**
<https://incoprea.com/research-library/original-theories-vol12-20261030/> · 30 October 2026 · not peer reviewed · AI-assisted
## Result
The gap at a magic neutron number does vary between isotopes — ranges of 1.6 to
3.98 MeV in the five shells tested — so half the claim survives. It is **not**
systematic in asymmetry (N−Z)/A: with N fixed effects and a proton-magic control,
b = −2.14 ± 1.02 MeV per unit asymmetry, bootstrap CI −4.66 … +0.62, within-N
permutation p = 0.094, partial R² 0.061. The decisive fact is the sign flip: the
within-shell slope runs from −7.1 MeV at N = 20 to +24.4 MeV at N = 126. The
third prediction holds — the five proton-magic isotones sit **+1.38 ± 0.28 MeV**
high (p = 0.00015). Verdict: the asymmetry mechanism is refuted; the phenomenon is
not.
## Design
AME2020 masses, parsed by fixed-width column after hand-inspecting the header
(`SOURCE.md`). Of 3,558 data rows, 1,008 (28.3 %) carry the `#` estimated-mass flag
and were discarded; 2,550 remain. The brief's statistic, exactly as written,
S2n(Z,N) = M(Z,N−1) + Mn − M(Z,N), D2n = S2n(Z,N) − S2n(Z,N+2), Mn = 8.0713171 MeV,
for N ∈ {20, 28, 50, 82, 126} and every isotone with all four masses measured:
**n = 75** (12, 12, 19, 20, 12 per shell). Five rows would have needed an imputed
N−1 mass (6.25 %, over the 5 % gate): none was imputed. Families, each
Benjamini–Hochberg adjusted once: **A** five within-N Spearman tests, **B** four
within-shell proton-magic vs rest permutation tests, **C** two slopes.
## Table 1 — spread at fixed N
| N | min | median | max | IQR | range | IQR, proton-magic removed |
|---|---|---|---|---|---|---|
| 20 | 0.17 | 1.71 | 4.15 | 1.10 | 3.98 | 0.83 |
| 28 | 1.82 | 2.99 | 4.43 | 0.90 | 2.60 | 0.71 |
| 50 | 1.95 | 2.74 | 3.56 | 0.38 | 1.61 | 0.38 |
| 82 | 1.86 | 2.10 | 3.72 | 0.49 | 1.87 | 0.40 |
| 126 | 0.70 | 1.48 | 2.32 | 0.88 | 1.62 | 0.82 |
MeV (`spread_by_shell.csv`). **P1 — supported on the range** (1.61–3.98 MeV,
all five shells), **not on the IQR bar**: 0 of 5 exceed 1.5 MeV.
## P2 — refuted: the trend changes sign
| Model | n | b | SE | p (perm) | 95 % bootstrap CI | R² | partial R² |
|---|---|---|---|---|---|---|---|
| N fixed effects | 75 | −2.267 | 1.172 | 0.128 | −5.10 … +0.93 | 0.381 | — |
| + proton-magic | 75 | −2.141 | 1.016 | 0.094 (q = 0.128) | −4.66 … +0.62 | 0.543 | 0.061 |
b in MeV per unit (N−Z)/A; OLS, within-N permutation p (20,000 permutations, seed
7), row bootstrap (20,000 draws, seed 11); family C, m = 2.
Within-shell Spearman rho: −0.63 at N = 20 (q = 0.054), −0.02 at N = 28
(q = 0.96), −0.44 at N = 50 (q = 0.072), **+0.82** at N = 82 (q = 0.0003),
**+0.92** at N = 126 (q = 0.0006); family A, m = 5. Pooled *without* N fixed
effects, r = −0.34 (p = 0.0035) — between-shell structure.
## P3 — supported, on thin data
The five proton-magic isotones sit **+1.384 ± 0.282 MeV** above the rest at the
same N (t = 4.90, permutation p = 0.00015, analytic CI 0.82 … 1.95), spanning
asymmetry 0.00 to 0.24 — not an asymmetry trend in disguise. Within-shell, with
one or two magic rows each, the tests are powerless: p = 0.084, 0.048, 0.050,
0.250; q = 0.112, 0.099, 0.099, 0.250.
## Definitional fork (post-hoc)
The brief's S2n is algebraically the *one*-neutron separation energy, so its D2n is
a **first** difference of S_n and keeps the smooth trend in it; the conventional
gap is a **second** difference and removes it (r = 0.89 between them, means
differing by 2.1 MeV). With Δ2n = M(Z,N−2) − 2M(Z,N) + M(Z,N+2), n = 73:
b = **−7.46 ± 1.79**, p = 0.0022, CI −11.47 … −2.14, IQR above 1.5 MeV in 3 of 5
shells — which would **meet** the support criterion. The briefed formula stands as
primary and refuted; the gap definition, not the data, decides P2.
## What the test cannot see
A mass table measures mass differences, not shell structure: a large D2n fits a
closed shell and a deformed intruder equally, so nothing here says whether N = 50
is "magic" in a given nucleus. Mass uncertainties (1–100 keV) were not propagated;
they are one to two orders below the spreads reported, but that is an argument,
not a measurement. Asymmetry varies little inside a shell (0.06 at N = 126 against
0.34 at N = 20), so the slope rests mainly on the light shells.
## What would change the answer
A clean second-difference gap definition — it exists, and reverses P2 — plus more
proton-magic isotones with measured masses, which AME2020 does not supply.
## Files
`run.py` · `results.json` · `data/` · `figures/` (Figures 1–3) · `hypothesis.md` · `SOURCE.md`.
**Disclosure:** designed and executed with AI assistance (DeepSeek, via its coding
agent), not peer reviewed. Observational associations in published nuclear mass
data only; no medical, clinical, financial or safety advice.
hypothesis.md
# The hypothesis, as published
Source: <https://incoprea.com/research-library/original-theories-vol12-20261030/>
Volume XII, hypothesis H141 — "Shell closures shift with asymmetry".
## Claim (verbatim)
> Nuclear shell closures shift with neutron-proton asymmetry, so a number that is
> magic in one isotope is not magic in another.
## Predictions (verbatim)
- P1 — the two-neutron shell gap at fixed N varies substantially across isotones;
- P2 — the variation is systematic in the asymmetry (N−Z)/A, not scatter;
- P3 — the proton-magic isotones (Z = 20, 28, 50, 82) sit apart from the rest.
## What the brief fixes, and what it leaves to be fixed here
The design (below) is taken from `briefs/08-nuclear-shells.md` verbatim. The brief
specifies the *form* of the decisive test but not every operational detail, so the
following choices are fixed here, on 30 October 2026, **before the first
computation of any D2n value**. n is known at this point (it is a property of
which masses exist, not of the outcome): 75 isotones with complete measured
quadruples — 12 at N = 20, 12 at N = 28, 19 at N = 50, 20 at N = 82, 12 at N = 126.
Five of those 75 are proton-magic isotones (Z ∈ {20, 28, 50, 82}).
### Design (verbatim from the brief)
1. Masses: AME2020 analysis table
`https://www-nds.iaea.org/amdc/ame2020/mass_1.mas20.txt`. Fixed-width; header
inspected, column positions and estimated-mass flag identified by hand, in
`SOURCE.md`, before parsing. Estimated masses discarded, and the number
removed reported.
2. For N ∈ {20, 28, 50, 82, 126} and every isotone with the needed neighbours
present as measured masses, compute
`S2n(Z,N) = M(Z,N-1) + Mn - M(Z,N)` and `D2n(Z,N) = S2n(Z,N) - S2n(Z,N+2)` in
MeV, with Mn = 8.0713171 MeV.
3. Fit `D2n ~ a + b·(N-Z)/A` with N fixed effects, and separately with
proton-magic dummies; report b, its standard error, partial R², and n. Also
report the within-N Spearman rho between D2n and asymmetry.
4. Report the spread of D2n at fixed N (min, median, max, IQR) and the same after
removing the proton-magic isotones.
5. Sensitivity: impute the N−1 isotone from the smooth mass surface **only if
fewer than 5 % of entries need it** — otherwise do not impute at all and
report the loss.
### Operational definitions fixed in advance
- **Which separation energy.** The brief's `S2n(Z,N) = M(Z,N-1) + Mn − M(Z,N)` is
algebraically the *one*-neutron separation energy S_n; the brief names it S2n
and then takes `D2n = S2n(Z,N) − S2n(Z,N+2)`. The test is run exactly as the
brief writes it, so the primary statistic is the four-point second difference
`D2n = M(Z,N−1) − M(Z,N) − M(Z,N+1) + M(Z,N+2)`. The conventional two-neutron
shell gap `Δ2n = M(Z,N−2) − 2M(Z,N) + M(Z,N+2)` is reported as a **post-hoc
sensitivity**, labelled as such, because it is not the brief's formula.
- **Masses.** M is the atomic mass excess Δ (MeV) from the AME2020 analysis
table. Using atomic rather than nuclear masses is exact here: the Z electrons
cancel in both M(Z,N−1)+Mn−M(Z,N) and in the four-point difference, and Mn is
the neutron mass excess.
- **Measured** means the mass-excess field carries no `#` (estimated) and no `*`
(not calculable). One entry per (Z, A); the file has no duplicate (Z, A).
- **Spread is measured by the IQR** — this is the decisive measure for the
support/refute call, chosen in advance because the range of 12–20 points is
sensitive to one poorly measured mass. Range (max−min) and the standard
deviation are reported alongside and the verdict is also tabulated under the
range, so the choice can be audited rather than argued about.
- **b is estimated by ordinary least squares**, with its classical
heteroskedasticity-free standard error, `df = n − p`. The reported CI is a
percentile bootstrap over rows (20,000 draws, seed 11, `stats.bootstrap_ci`).
The reported p-value is a seeded within-N permutation p (20,000 permutations,
seed 7): asymmetry is shuffled among isotones *within* each shell, which holds
the N fixed effects fixed and tests only the asymmetry association.
- **Partial R² for b** is (RSS of the model without asymmetry − RSS of the full
model) / RSS of the model without asymmetry, computed inside the N-fixed-effects
model that also contains the proton-magic dummy. The semipartial (R² change) is
reported too.
- **Family for the Benjamini–Hochberg adjustment.** Three families are declared,
each adjusted once, and no further correction is applied anywhere:
- **Family A** — the five within-N Spearman tests of P2.
- **Family B** — the four within-shell proton-magic vs rest mean-difference
permutation tests of P3 (N = 50 has no proton-magic isotone among the 75 and
cannot enter).
- **Family C** — the two decisive regression slopes: b with N fixed effects
only, and b with N fixed effects plus the proton-magic dummy.
### Support / refute criteria (verbatim from the brief, made numerical)
The brief:
> support if the spread of D2n at fixed N exceeds 1.5 MeV in at least three of the
> five shells AND b is negative with a CI excluding zero after proton-magic
> control.
> refute if the spread is under 0.3 MeV or b is indistinguishable from zero.
Operationally, with "spread" = IQR as fixed above:
- **Supported** if IQR(D2n) > 1.5 MeV in ≥ 3 of the 5 shells **and** the
proton-magic-controlled b is negative with a 95 % bootstrap CI excluding zero.
- **Refuted** if the median across-shell IQR is < 0.3 MeV **or** the
proton-magic-controlled b has a 95 % CI containing zero.
- **Partial** otherwise, and then reported shell by shell: which magic numbers
carry the effect and which do not is the interesting outcome, per the brief.
- **P1** is additionally called *supported* in a given shell if that shell's IQR
exceeds 1.5 MeV on its own; **P2** if that shell's within-N Spearman rho is
negative with a BH-adjusted q < 0.05 in the pre-specified (negative) direction;
**P3** if the proton-magic dummy in the full model is non-zero with a
permutation p < 0.05. A positive rho is a directional failure for P2, which
predicts a *systematic* trend; its magnitude is still reported.
### The imputation gate
The N−1 mass is missing or estimated for 5 of the 80 otherwise-complete candidate
rows (6.2 %, computed from the mass table before any D2n): 2 at N = 20, 1 at
N = 28, 1 at N = 50, 1 at N = 82, 0 at N = 126. Because 6.2 % > 5 %, the brief's
rule forbids imputation. **No imputation is performed**; the analysis uses the 75
rows with all four masses measured, and the 5 lost rows are named in
`results.json`. This gate is checked in code (`run.py`), not by hand, so a
different mass-table release cannot silently switch it on.
### Analysis choices fixed in advance
Primary data: measured masses only, all four neighbours required. Mn = 8.0713171
MeV (as briefed; the table's own neutron mass excess is 8.07131806 MeV — the
0.96 eV difference shifts every S_n by a constant and cancels exactly in D2n, so
it cannot affect any result here; both are reported). Asymmetry (N−Z)/A at the
fixed magic N, A = N+Z. Proton-magic set {20, 28, 50, 82}. Seeds 7 (permutation)
and 11 (bootstrap). No optional stopping: one pass over the pre-specified set of
five shells, and the per-shell table is reported in full.
## Addendum, written after the run — one ambiguity, reported not repaired
The pre-registration above fixes the IQR as the decisive spread measure for the
aggregate support/refute call, but its per-shell P1 sentence ("supported in a given
shell if that shell's IQR exceeds 1.5 MeV") turned out to be a stricter bar than
the brief's wording ("the spread of D2n at fixed N exceeds 1.5 MeV"), which reads
as a total spread. Both are therefore reported: the range exceeds 1.5 MeV in 5 of
5 shells (1.61–3.98 MeV), the IQR in 0 of 5 (0.38–1.10 MeV). Nothing was
re-specified — the sentence above has been left as it was written — and the
aggregate call does not depend on the choice, because it fails on the second
clause of the brief's support criterion under either measure: the
proton-magic-controlled slope has a 95 % bootstrap CI of −4.66 … +0.62 MeV.