Spatial Displacement Theory · Paper 05 · Prepared for Submission
A Constitutional Construction Grammar for the Nuclides: α-Core Factorisation, the Packing Radius R(A) = Rp(A/η)1/3, and Tests of Contact Binding
Melbourne, Australia · August 2026 · Domain 05 — Nuclear Physics
Abstract. Every nuclide with Z ≥ 2 factorises uniquely as one α core plus nd deuterons plus nt tritons, nt = A − 2Z, nd = 3Z − A − 2; electron-capture nuclides obey the conjugate grammar on a He-3 core. Free neutrons are forbidden in stable nuclides — a neutron is a proton with an internal electron, and the shared electron is the binding agent: the parameter-free interleaved-trefoil energy E = −Σqiqj·αℏc/rij reproduces the deuteron binding (2.200 vs 2.224 MeV) with no fitted scale. The close-packing radius R(A) = Rp(A/η)1/3, η = π/√18, is zero-fit and grades at 4.96% RMS over 908 measured radii, beating the empirical rival on isotope chains by 5×. The shadow-overlap account of the mass defect is presented with its calibration and its per-nucleon residual; the volume-price hypothesis is reported EXCLUDED. The magic numbers are derived: two closed forms — bipartite-triangular tier capacity F(n) = (n+1)(n+2) and rolling-equator triton belt B(n) = 2(n+1) — produce the full closure sequence 2, 8, 20, 28, 50, 82, 126 and predict the next at N = 184; the closure compaction is measured to be an isotone invariant. Grammar-count degeneracy discipline (NP24-A) is enforced throughout.
Keywords: construction grammar · α core · triton ledger · packing radius · shadow overlap · shared-electron binding · beam-peel kinematics
1. The grammar
The counts are fixed by (Z, N) arithmetic — the factorisation is unique wherever both are non-negative — and ATOMICUS characterises 290 isotopes on it. The neutron ledger closes as N = 2 + nd + 2nt. Electron economy: a nuclide of mass number A carries A protons and A electrons, N of them stabled inside neutron structures and Z fielded; "neutronium" is the grammar edge nd < 0. A free neutron is an exposed triton remnant, and neutron decay (Qβ = 0.782 MeV, exact in the ledger) is regime-II metastability of a composite, not the disintegration of an elementary object.
Degeneracy discipline (NP24-A). The map (nt, nd, v) = (N−Z, 2Z−N−2, Z−2) is an invertible re-coordinatisation of (Z, N). Any regularity expressible purely in the counts — including every magnetism rule so written — restates the neutron ledger and is claimed as bookkeeping, never as independent evidence. What the grammar earns is structure: lineages, the two-grammar split, and the scission algebra (2d → α untouched by the demotion).
2. The packing radius (NP12, canon)
Zero-fit close packing of nucleon boundaries. Graded against 908 measured boundary radii: RMS 4.96%, statistically degenerate with the one-parameter empirical rival (5.00%) in aggregate but 5× better along isotope chains (Ca flat at 1.67% where the rival drifts +8.63%; Sn 3.20%; Pb 3.28%). The superseded R = Z·Rp is retracted in canon (RMS 890.8%) and retained only as a compile-compatibility stub. He-4 check: RHe = 2Rp = 1.683 fm vs 1.6755 fm measured (0.43%). A standing structural rule: adding a triton draws the boundary radius inward (88% of isotope chains, 90% of isobar steps, maximal at shell closures) — the interleave takes the least-resistance path.
3. Binding: the shared electron, not a new force
Nucleons pack as interleaved 6π trefoils; the neutron's internal electron is shared between adjacent protons, and the handed occlusion sum (4) — with no fitted scale — delivers the deuteron binding to 1.1% (NP17, RESOLVED; α-cluster case PENDING). There is no separate strong force: nuclear binding is the same handed occlusion of Paper 03 at contact range, with the connection law of §10 (six connections per nucleon) setting the mesh.
4. Projected shadow area — tested and not promoted
NSEQ04 applied the projected-union rule to the active dual-tetra packing rather than the superseded opposition geometry. The registered corpus gives R²(total BE) = 0.7918, mean absolute relative error 40.4%, and R²(BE/A) = −161.94. The per-nucleon gate is not met, so projected area is not promoted as the binding law. The separate volume-price hypothesis E = P·ΔV is also EXCLUDED (NP05). The active target-free work is the contact/lock wall and circulation-reorganisation route; the deuteron result in §3 is a separate handed-occlusion computation and does not validate equation (5).
5. Decay as geometry: EC, spallation, fission
An He-3-cored nuclide is an incomplete tetrahedron; inner-shell capture effects p → n and completes it — deterministic self-repair, transferring the nuclide between lineages (2). Spallation is a shallow flay: supra-threshold ejection of a deuteron or triton. Fission is a deep flay: unfrozen outer shells unwrap around sealed T12 masonry (NH = 82); the jacket locks a second core by 2d → α. Stop-at-N=126 (Pb-like remnant) is excluded against eight IAEA independent-yield tables (NP35). Beam kinematics (²³⁸U + p, Bernas): the 1.83c surface circulation is CONFIRMED zero-fit; the fine-tuned Ekin(Z) discriminant is EXCLUDED as Coulomb-degenerate. Central impact yields fission (deep flay); grazing yields spallation (shallow flay). The Golden Boundary Z = 79 — first nt > nd — locates the onset of natural radioactivity from construction geometry alone.
6. The shell schedule: where the magic numbers come from
Every closure ("magic number") is a completed seat tier on which the next tier rests, and the capacities are forced by four mesh facts. The contact graph is bipartite — a proton gears only with neutrons — so every tier is a double layer, a p-face and an n-face; the packing's two-dimensional order is triangular, so one layer holds the (n+1)-th triangular number of rod seats. Together these give the tier capacity F(n) = (n+1)(n+2): 2, 6, 12, 20, 30 — and F(0) = 2 is the α core itself, obeying the same formula. Tritons are radial rods (the lone rod seats inward — measured, 83.1% whole-range census), and the assembly must roll without jamming, so a radial rod can co-rotate slip-free only on the equator of the collective roll: a one-dimensional ring, capacity linear in tier girth, antipodally paired — the belt B(n) = 2(n+1): 8, 10, 12, 14. From n = 3 the belt separates and descends to seal the previous closure; the surface remainder R(n) = n(n+1) completes at its own. Two closed forms produce all seven magic numbers and all nine capacities. The forward prediction comes free: closure(7) = 126 + R(6) + B(7) = N = 184.
The descent is not narration — it is measured. A pre-registered single-pass sweep over the full radius dataset finds the compaction kink at a magic-N closure to be an isotone invariant: constant along N = 28, 50, 82 while the triton count varies two- to four-fold, beating the occupancy-proportional alternative on every adjudicable isotone (RMS 4.4 vs 15.5; 1.7 vs 5.7; 1.4 vs 3.2), with 19 of 19 computable closure kinks negative against a zero-centred background. The freeze does the compacting; the ledger only decides who sits in the seats. Seat law: positions fixed by tier geometry, filling order fixed (antipodal pairs, lone rod inward), occupancy set by the ledger and frozen at closure. The parity corollary: a stable odd-Z nuclide carries an odd triton count — the lone rod on the inward seat — and the odd-Z, even-triton misfits are precisely the five naturally occurring odd-odd quasi-stables (K-40, V-50, La-138, Lu-176, Ta-180m), every one a long-lived radioactive oddity.
Disclosure. F(n) equals the harmonic-oscillator degeneracy and B(n) the intruder-orbit capacity of the prevailing shell model — shared counting, which cannot discriminate at sequence level. The native content is the bipartite doubling and the rolling-equator constraint: the prevailing account requires a fitted spin–orbit strength to place its intruders; this law carries no strength parameter — the belt's location is geometric. The descent onset n = 3 is read, not derived (the field-cost pricing of a triton against a deuteron at given Z is the open residual).
7. Certification
| Result | Value | Status |
|---|---|---|
| Grammar factorisation | unique for nd, nt ≥ 0; 290 isotopes characterised | CONSTITUTIONAL — counts are (Z,N) arithmetic |
| Count-rule degeneracy | (nt,nd) ↔ (Z,N) | NOT INDEPENDENT — bookkeeping, enforced (NP24-A) |
| R(A) packing law | 4.96% RMS, N = 908 | DERIVED — zero-fit; chain test 5× vs rival |
| Shared-electron deuteron | 2.200 vs 2.224 MeV | DERIVED — no fitted scale (NP17); α PENDING |
| BE ≈ κ·ΔA | R²(total) = 0.7918; R²(BE/A) = −161.94 | CALIBRATED(1) — per-nucleon binding law excluded (NSEQ04) |
| E = P·ΔV volume price | — | EXCLUDED (NP05) |
| 1.83c surface (beam peel) | β = 0.875; 309° | CONFIRMED zero-fit (NP27); Ekin(Z) EXCLUDED as degenerate |
| Triton-contraction rule | 88% chains / 90% isobars | RULE — native mechanism; convergent numbers |
| Neutron Qβ | 0.782 MeV | EXACT in two-ledger accounting |
| Shell schedule capacities | F(n) = (n+1)(n+2), B(n) = 2(n+1); 2, 8, 20, 28, 50, 82, 126 sequence-exact | DERIVED — two closed forms; onset n = 3 READ; oscillator convergence disclosed |
| Closure-kink isotone invariant | constant along N = 28/50/82; 19/19 negative | SUPPORTED — pre-registered single pass, 2026-07-30 |
| Parity lock (odd Z, even t) | the five natural odd-odd quasi-stables, no false positives on the seven lone-rod grips | RULE — measured stability record, 12/12 |
| Next neutron closure | N = 184 | PREDICTION — from the same two forms |
Methodological declaration. No quark or gluon degrees of freedom, no shell-model potential, no pairing term, and no semi-empirical mass formula enters any derivation. Binding is the handed occlusion of Paper 03 at contact range; the single calibrated coefficient (κ) is labelled and its domain of failure stated. Where a discriminant proved degenerate with the rival framework's prediction (Ekin(Z)), it was excluded rather than claimed. Measured radii (Angeli–Marinova), masses, and the Bernas fragment dataset enter as verification targets only.
References
- ATOMICUS/rules/On the Nature of Atomicus Rules.md — the construction constitution; ATOMICUS/isotopes/ (284 enriched characterisations).
- Engine/include/sdt/laws.hpp — namespace sdt::laws::nuclear (nuclear_boundary_radius; retraction stub in situ); measured:: nuclear anchors.
- Investigations/05_Nuclear_Physics — NP05 (kill), NP12 (radius law), NP17 (interleaved trefoils, shared-electron binding), NP24-A (degeneracy audit), NP27 (beam-peel kinematics), NP30 (invariant v = c√(re/r)).
- Angeli & Marinova (2013), IAEA compilation — 908 measured boundary radii (verification target).
- Bernas et al., ²³⁸U + p fragment yields (verification target).