§2.5 — Nuclear and Atomic Derived Values

Nuclear and atomic derived values define the corridor spacings, shell radii, wavelength products, and binding references used across post Freeze-Out structure.

Nuclear Corridor Spacings

The nuclear framework uses three standard corridor spacings. These define the admissible proton-neutron interaction grammar used for nuclear closure and binding calculations.

\[ d_4 = 1024\,L_0 \] \[ d_5 = 1024 + 2(16) = 1056\,L_0 \] \[ d_6 = 1024 + 43 = 1067\,L_0 \]

These values represent the fixed corridor spacing family used to count productive nuclear binding corridors.

Nuclear Corridor Energies

Each corridor spacing has a corresponding binding contribution. Nuclear binding is counted from alpha closure plus the engaged \(d_4\), \(d_5\), and \(d_6\) corridor populations.

\[ E_{d4}(1024) = 2.0780644\,\mathrm{MeV} \] \[ E_{d5}(1056) = 0.9892874\,\mathrm{MeV} \] \[ E_{d6}(1067) = 0.6735958\,\mathrm{MeV} \]

The total binding expression is:

\[ E_{\mathrm{bind}} = N_{\alpha}E_{\alpha} + n_4E_{d4} + n_5E_{d5} + n_6E_{d6} \]

Here \(N_{\alpha}\) is the alpha-closure count, while \(n_4\), \(n_5\), and \(n_6\) count the engaged productive corridor classes.

Region Reach Values

Stable Regions and composite nuclear structures are assigned effective reach values in native \(L_0\) units. These values provide the post Freeze-Out scale references used in Region, nucleus, atom, and curvature calculations.

Region Structure \(R_{\mathrm{eff}}\) \(R_{\mathrm{bary}}\)
Electron \(EZZZ\) \(816\,L_0\) \(128\,L_0\)
Proton \(UUDZZZ\) \(1836\,L_0\) \(288\,L_0\)
Neutron \(p+e\) \(5100\,L_0\) \(800\,L_0\)
Deuterium nucleus \(p+n\) \(13056\,L_0\) \(2048\,L_0\)
Alpha / He-4 nucleus \(2p+2n\) \(52224\,L_0\) \(8192\,L_0\)

These values preserve the distinction between effective curvature reach and baryonic participation reach.

Atomic Shell Radii

The atomic and bonding frameworks use derived shell radii. These values define the admissible shell geometry for ionization, bonding, and higher atomic structure.

\[ S_1 = 29{,}447{,}706\,L_0 \] \[ S_2 = 105{,}124{,}156\,L_0 \] \[ S_3 = 121{,}091{,}309\,L_0 \] \[ S_4 = 153{,}576{,}343\,L_0 \]

These shell values are used as structural radii rather than orbital paths. They define admissible Region separations and shared shell geometry.

Fundamental Wavelength Products

Region wavelength products define the interaction grammar used by atoms, ionization, bonding, and compact curvature response.

\[ \lambda_{pe} = \lambda_p\lambda_e = 1836 \times 816 = 1{,}498{,}176\,L_0 \] \[ \lambda_{ee} = \lambda_e^2 = 816^2 = 665{,}856\,L_0 \] \[ \lambda_{pp} = \lambda_p^2 = 1836^2 = 3{,}370{,}896\,L_0 \] \[ \lambda_{np} = \lambda_n\lambda_p = 5100 \times 1836 = 9{,}363{,}600\,L_0 \]

The proton-electron product governs atomic spacing and ionization. The electron-electron product constrains bonding participation. The proton-proton product enters compact baryonic curvature response. The neutron-proton product appears in nuclear and composite Region comparisons.

Use in Post Freeze-Out Structure

These derived values form the numerical grammar of the post Freeze-Out framework. Nuclear binding uses the corridor values. Atomic structure uses shell radii and proton-electron wavelength products. Bonding uses shared shell geometry and electron-electron constraints. Curvature uses Region inventory and compact carrier products.

\[ d_4,d_5,d_6 \rightarrow E_{\mathrm{bind}} \] \[ S_i,\lambda_{pe},\lambda_{ee} \rightarrow \text{ionization and bonding} \] \[ R_{\mathrm{eff}},\lambda_{pp} \rightarrow \text{compact curvature response} \]

Summary

§2.5 collects the derived nuclear and atomic values used by the post Freeze-Out framework. The nuclear corridor spacings are \(d_4=1024\,L_0\), \(d_5=1056\,L_0\), and \(d_6=1067\,L_0\), with corresponding corridor energies \(2.0780644\,\mathrm{MeV}\), \(0.9892874\,\mathrm{MeV}\), and \(0.6735958\,\mathrm{MeV}\).

The atomic structure ledger uses shell radii \(S_1=29{,}447{,}706\,L_0\), \(S_2=105{,}124{,}156\,L_0\), \(S_3=121{,}091{,}309\,L_0\), and \(S_4=153{,}576{,}343\,L_0\). The primary wavelength products are \(\lambda_{pe}=1{,}498{,}176\,L_0\), \(\lambda_{ee}=665{,}856\,L_0\), \(\lambda_{pp}=3{,}370{,}896\,L_0\), and \(\lambda_{np}=9{,}363{,}600\,L_0\).