Software · Nuclear Engine
V4 Nuclear Binding Engine
A zero-parameter nuclear binding energy model derived from torus knot topology. Predicts binding energies, decay modes, and drip lines across the entire periodic table with 99.29% accuracy.
Live calculator
Enter any nucleus by proton and neutron number. The engine computes the binding energy in real time using the same algorithm as the Python script below. Where AME2020 data exists, the prediction is compared automatically.
Fe-56
Z=26, N=3013α + 4n · FCC-13The 5-term additive model
The total binding energy is the sum of five physically motivated terms. No parameters are fitted — every constant traces to the torus knot winding numbers (p, q) or measured few-body binding energies.
BE(Z, N) = α-core + ring-bonds + Coulomb + appendage + couplingα-core
nα × 28.296 MeVEach α-particle contributes its measured binding energy. The number of α-clusters nα = ⌊min(Z,N)/2⌋.
Ring bonds
nBonds × b(nα) × S(nα)Inter-alpha quasi-deuteron bonds. Bond energy b(nα) is geometry-specific for nα ≤ 25, then follows the Coulomb erosion law: b = BE_BOND × (1 + ln(15/nα)/√3).
Coulomb dilution
0.7 × Z(Z−1) × [1/A_NZ^⅓ − 1/A^⅓]Bond energies come from N=Z nuclei (maximum Coulomb). Real isotopes with N > Z have less Coulomb repulsion, so bonds are effectively stronger.
Appendage internal
BE(d), BE(t), or BE(³He)After packing α-clusters, leftover nucleons form an appendage (d, t, or ³He). This term counts its internal binding energy.
Appendage coupling
C(type, nα)How the appendage binds to the α-cluster core. Four coupling mechanisms: neutron (¾×BE(d)), deuteron (linear), triton (saturating), and ³He (1/4 of triton).
Free neutron BE
nContacts × BE_BOND × fN^(−1/7)Excess neutrons occupy spare lobes on the cage surface. Each makes up to 5 contacts with α-particles, with 1/7 packing dilution per neutron added.
Fundamental constants
Every constant either derives from the torus knot winding numbers (p, q) or is a measured few-body binding energy. Zero fitted parameters.
| Constant | Value | Derivation |
|---|---|---|
p_p, q_p | 2, 3 | Proton torus knot T(2,3) winding numbers |
p_n, q_n | 3, 2 | Neutron torus knot T(3,2) winding numbers |
BE(d) | 2.2246 MeV | Measured deuteron binding energy (AME 2020) |
BE(t) | 8.4818 MeV | Measured triton binding energy (AME 2020) |
BE(³He) | 7.7180 MeV | Measured helion binding energy (AME 2020) |
BE(α) | 28.2957 MeV | Measured alpha binding energy (AME 2020) |
BE_BOND | 2.4718 MeV | BE(d) × 10/9 — ring closure resonance (q²+1)/q² |
SHELL_STEP | 0.8618 | (4+√5)/(5+√5) — icosahedral spectral factor 1 − 1/λ_max |
SHELL_DECAY | 0.02293 | 1/(42+φ) — golden coherence length |
FREE_N_DIL | 1/7 | 1/(p_n × q_n + 1) — Thurston-Bennequin invariant |
WEBER_S4D | 3/4 | SU(2) Casimir C₂(j=½) = j(j+1) |
A_COULOMB | 0.7 MeV | Standard Coulomb coefficient |
Cluster decomposition
Every nucleus is decomposed into α-clusters plus an appendage. The rule is simple: pack as many α-particles as possible, then classify the leftover nucleons.
nα = ⌊min(Z, N) / 2⌋remainder = (Z − 2nα, N − 2nα)| Remainder (rZ, rN) | Appendage | Example |
|---|---|---|
| (0, 0) | None | C-12 → 3α |
| (1, 1) | Deuteron (d) | N-14 → 3α + d |
| (1, 2) | Triton (t) | F-19 → 4α + t |
| (2, 1) | Helion (³He) | He-3 → 0α + ³He |
| (1, 0) | Proton (p) | Na-23 → 5α + p |
| (0, k) | k neutrons | Fe-54 → 13α + 2n |
| (1, k>2) | Triton + (k−2)n | Au-197 → 39α + t + 38n |
Appendage coupling mechanisms
The appendage couples to the α-cluster core through “through-alpha waveguide” channels. The mechanism depends on the appendage type and cage size.
Neutron
C = ¾ × BE(d) = 1.669 MeVNo phase-locking partner, so no through-alpha waveguide. Direct Weber overlap only. The ¾ is the SU(2) Casimir C₂(½).
Deuteron
C = D_DIRECT × (1 + min(nα−2, 18)/p_n)Phase-locked pair creates a cohesive even harmonic. Linear growth with backing vertices until nα ≈ 20, where the cage transitions to multi-shell geometry.
Triton
C = T_DIRECT × (p+q)/q (nα ≥ 5)Phase-locked triplet creates an odd harmonic. Saturates at nα ≥ 5 when the polyhedral cage closes fully. The saturation value is set by the p/q mode ratio.
³He
C = C(triton) × 1/4Charge mirror of the triton. Same face-site geometry but only 1 neutron spare lobe (vs 4 for triton), giving exactly 1/4 the coupling strength.
Shell attenuation
For nuclei beyond the Z=82 shell closure (nα > 42), additional α-particles sit outside the icosidodecahedral cage. The ring bond energy and free neutron binding are attenuated by the icosahedral spectral factor:
S(nα) = SHELL_STEP × exp(−SHELL_DECAY × (nα − 42))
where
SHELL_STEP = (4+√5)/(5+√5) ≈ 0.8618
SHELL_DECAY = 1/(42+φ) ≈ 0.02293Validation results
Binding energy predictions compared against AME2020 measured values for the primary isotope of each element (Z = 3–118).
| Period | Range | Count | Avg Error | Worst |
|---|---|---|---|---|
| Period 2 | Z = 3–10 | 8 | 0.23% | F-19 (0.79%) |
| Period 3 | Z = 11–18 | 8 | 0.27% | P-31 (0.68%) |
| Period 4 | Z = 19–36 | 18 | 0.58% | Se-74 (1.55%) |
| Period 5 | Z = 37–54 | 18 | 0.62% | Zr-90 (1.75%) |
| Period 6 | Z = 55–86 | 32 | 0.64% | Au-197 (2.11%) |
| Period 7 | Z = 87–118 | 32 | 1.16% | Fm-257 (1.94%) |
Download and explore
Python Replication Script
v4_replication.py · 636 lines · Zero dependencies
Self-contained script that reproduces the V4 engine. Run it anywhere with Python 3.10+. Contains all constants, all formulas, and the full 118-element validation scan.
Download v4_replication.pyInteractive Explorer
atomic-structure.com
Visual, interactive nuclear binding engine with isotope-by-isotope exploration, cluster topology diagrams, and full decay mode prediction.
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