Nuclear Magic Numbers in the TCC Framework: The Four-Node Dirac Identification and Its Consequences

We present a conditional analysis of nuclear magic numbers within Tick Compression Cosmology (TCC). The two-node TCC system with harmonic remainder potential generates shell closure numbers 2⋅T(n) — the harmonic oscillator sequence 2, 8, 20, 40, 70, 112, 168 — agreeing with observation for n ≤ 3 and predicting 40 instead of 28 for n = 4. We argue that the three-dimensional generalization of TCC’s Postulate 1 requires a fourth node corresponding to the temporal direction of causal propagation. The resulting four-node TCC Hamiltonian in remainder space is structurally identical to the Dirac equation. This identification is adopted as an unproven hypothesis (Postulate 2), motivated by a Clifford algebra node-counting argument but not derived from TCC axioms; its derivation from first principles is Open Problem OP1. Physical nuclear interpretation additionally requires the 𝜌→x correspondence mapping remainder space to coordinate space (Open Problem OP4). All nuclear results below are conditional on both Postulate 2 and OP4. Conditional on Postulate 2, the standard Foldy–Wouthuysen reduction [Foldy and Wouthuysen 1950] applied to the four-node system produces spin-orbit coupling ΔH₃ = −κ⋅L⋅S where κ = ℏ²ω²/(2mv²). This generates a triangular number correction n(n−1) to the tetrahedral baseline, yielding the formula: Magic(n) = 2·T(n) for n = 1, 2, 3 Magic(n) = 2·T(n) − n(n−1) for n ≥ 4 This formula reproduces all seven known nuclear magic numbers exactly (2, 8, 20, 28, 50, 82, 126) and predicts Magic(8) = 184. The correction activates at n = 4 because the f₇₂ state (N = 3, j = 7/2) is the lowest-shell state with sufficient L⋅S eigenvalue to cross the harmonic oscillator gap; for N < 3 no such state exists. The spin-orbit coefficient κ is a free parameter of TCC. Matching κ to the empirical nuclear spin-orbit coupling strength provides the first measurement of TCC’s remainder propagation speed v from nuclear data; this is a domain-specific measurement of a free parameter, not a TCC-wide prediction. The formula is algebraically equivalent to Borg’s (2006) n(n²+5)/3 for n ≥ 4 but emerges from a conditional-dynamical Hamiltonian structure, covers all n in a single parameterization, and embeds the spin-orbit coefficient in the structural relationship κ = ℏ²ω²/(2mv²) — a cross-domain falsifiability test if v is independently measured from other TCC results.