Spatial Displacement Theory · Paper 05 · Prepared for Submission

A Constitutional Construction Grammar for the Nuclides: α-Core Factorisation, the Packing Radius R(A) = R_p(A/η)^1/3, and Binding as Shared Occlusion

J. C. Harvey

Melbourne, Australia · July 2026 · Domain 05 — Nuclear Physics

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 + n_d + 2n_t. 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 n_d < 0. A free neutron is an exposed triton remnant, and neutron decay (Q_β = 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 (n_t, n_d, 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 The superseded R = Z·R_p is retracted in canon (RMS 890.8%) and retained only as a compile-compatibility stub. He-4 check: R_He = 2R_p = 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. The mass defect as shared occlusion — with its residual

Packed boundaries overlap their occlusion cross-sections; the assembled nuclide shadows less than its free parts, and the deficit is the mass defect. One least-squares coefficient over 217 nuclides yields total binding energy at R² = 0.988, mean error 7%. Limitation: the per-nucleon curve BE/A is not reproduced (the α-cluster and Fe-peak fine structure require the shared-electron well depth of §3, not area geometry alone), and the volume-price hypothesis E = P·ΔV is EXCLUDED (NP05; 45× spread across reactions once a units slip was removed). The successor E_bind = ℏ·Δω (meshed-circulation frequency change) is under test.

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 supra-threshold ejection of a deuteron or triton; exposure of an incomplete core opens the EC channel thermally inaccessible otherwise. Beam kinematics (²³⁸U + p, Bernas dataset): the 1.83c surface circulation is CONFIRMED zero-fit — fragment systematics reproduce with β = 0.875, slowdown v_circ = 1/γ = 0.484, and a 309° sweep; the fine-tuned Ekin(Z) discriminant is EXCLUDED as Coulomb-degenerate (both frameworks predict it; it cannot discriminate). Central impact yields fission (core–shell separation), grazing impact yields spallation; fission neutron multiplicity follows ν = 100 − A(Zr) at r = 0.973. The Golden Boundary Z = 79 — first n_t > n_d — 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

References