Artian Inertial Mass Operator and Three-Readout Spectral Equivalence Theorem

Abstract

When does the coefficient in Newton's second law become a finite spectrum rather than an inserted input? The paper constructs a positive finite source operator from funded completed-event classes. For a finite region \(\mathcal R\), \[ \widehat B_{\mathcal R} = \sum_{p\in\mathfrak P_{\mathcal R}} \iota_p\!\left(\frac{\widehat Q^{\rm vis}_p}{2\pi}\right), \qquad 0\preceq \widehat B_{\mathcal R} \preceq |\mathfrak P_{\mathcal R}|\,\mathsf I. \] The quotient set \(\mathfrak P_{\mathcal R}\) counts funded physical completions once. Copying a record does not change the operator. An observed mass can test an eigenvalue after construction; it cannot create one. For a source eigenstate \(|b\rangle\), the source rest branch is \[ H_b(\mathbf p) = \sqrt{c^2|\mathbf p|^2+b^2E_*^2}, \qquad \left. \frac{\partial^2H_b}{\partial p_i\partial p_j} \right|_{\mathbf p=0} = \frac{\delta_{ij}}{b m_*}. \] Its inverse curvature therefore supplies the inertial coefficient rather than presupposing it: \[ \mathbf F=bm_*\,\mathbf a+O(v^2/c^2). \] The same A1 source action in a weak external lapse gives \[ L_b = -bm_*c^2 +\frac12bm_*v^2 -bm_*\Phi+\cdots, \qquad M_{\rm pass}=M_{\rm in}=bm_*. \] With the independently declared A2 endurance-source map, the central three-readout identity is \[ \boxed{ \widehat B = \frac{\widehat H_{\rm rest}}{E_*} = \frac{\widehat M_{\rm in}}{m_*} = \frac{\widehat M_{\rm pass}}{m_*} = \frac{\widehat M_{\rm active}}{m_*} = \widetilde t\,\widehat\Gamma_{A2} }. \] For a full interacting state, binding energy is transported through all three mass readouts with one sign: \[ \Delta M_{\rm in} = \Delta M_{\rm pass} = \Delta M_{\rm active} = \frac{\Delta E_{\rm bind}}{c^2}. \] The action ledger is explicitly typed. A degree-one canonical phase turn and one positive A6 capacity token are related but are not the same object: \[ S_{\rm can}=2\pi\hbar\nu, \qquad \mathcal C_A=\hbar\sum_j|\nu_j|. \] The theorem also fixes its own boundary: A6 and A7 define the admissible finite operator class, but do not by themselves choose the complete particle-mass population. That remaining sector-population problem is exposed as a typed theorem interface rather than hidden in a fitted coefficient. Machine certificate: 11/11 deterministic checks pass, including the funding quotient, finite spectrum, free-branch Hessian, weak-lapse equality, binding transport, local-versus-extensive ceiling, constructor firewall, typed action/capacity distinction, and nine negative mutations. Stable anchors: paper concept DOI; Main Book v10.01; Inertia Theorem lexicon entry; Derivation Atlas; source/readout ontology.