Artian Nuclear, Electronic-Contact, and Hyperfine Hamiltonian Framework

One spectral line sees a product; a clock constellation tests the factors An alkali hyperfine line does not separately measure a nuclear magnetic response and an electronic contact density. It measures their product. This paper constructs both as finite source operators and proves exactly what a family of clock lines can identify before any spectroscopy is used as a constructor. For an alkali \(S1/2\) ground state, the leading source Hamiltonian is \[ { HA, hfsˢʳᶜ =h cR_∞QTT 8/3I_A {α_λ^2}{μₚₑQTT} G_Aⁿᵘᶜ C_Aᵉˡ \, I_A\!·\! J_e . } \] Here \( G_Aⁿᵘᶜ\) is the gauge derivative of a finite nuclear Hamiltonian, while \( C_Aᵉˡ\) is the finite co-address overlap of the electronic and nuclear-magnetization states. Their joint-address composition is defined on a balanced real- \(J\) tensor product. The point-contact delta is a continuum readout limit, not the source object. Taking logarithms turns a family of product rows into a bipartite incidence problem, \[ z=Bθ, { rankB=v-q, B=q, } \] where \(v\) is the number of nuclear/contact source vertices and \(q\) is the number of connected components. Every component retains one multiplicative gauge, \[ G_a↦ e^tG_a, C_b↦ e⁻ᵗC_b. \] Thus additional clock lines can create cycle tests, but precision alone cannot separate all source factors. One independently derived nuclear or contact source value per connected component is necessary and sufficient for identifiability. For every alternating cycle, \[ { ∏ᵢ₌₁ᵏ {ya_i b_i}{y_{aᵢ₊₁b_i}}=1. } \] The first non-definitional clock-constellation gate is the rubidium/caesium ratio: \[ { {ν_{87{ Rb}}}{ν_{133{ Cs}}} =76 { G₈₇ⁿᵘᶜ C87,5sᵉˡ} { G₁₃₃ⁿᵘᶜ C133,6sᵉˡ}. } \] The Artian tick, endpoint energy, Rydberg scale, fine-structure rail, and electron-proton mass ratio cancel. The BIPM 2025 adjustment gives \[ { ({ν_{87{ Rb}}}{ν_{133{ Cs}}})BIPM =0.7434957454957769, u_r=3.4×10⁻¹⁶. } \] The previously frozen caesium identity branch gives \(1.1694963419043314\), a \(+57.296978%\) transfer residual, and is rejected as a universal alkali source law. The finite composition theorem, rank/nullity theorem, cycle locks, angular spectrum, and completed no-go tests are closed in their stated scopes. The numerical heavy-alkali nuclear and many-electron source matrices remain the next calculation. Public anchors Concept DOI: 10.5281/zenodo.21309105 Main Book v10.01: 10.5281/zenodo.17527179 Artian Hamiltonian Framework: 10.5281/zenodo.20484906 Electron/Rydberg source theorem: 10.5281/zenodo.20800371 K2 caesium source-packet theorem: 10.5281/zenodo.21249436 Corpus Tree · Lexicon · Two Universes