Common Mass-Charge Matching and Universal Inertial–Active Proportionality under AASC

Overview Common Mass–Charge Matching is Module B of the post–structural-unification gravity–quantum continuation of the AASC unification ARC. Module A, Operational Metric–Source Attachment, constructs the lawful source-bearing bridge from common-interior operational data to the Einstein metric projection. Module B addresses the next unresolved question: Do the gravitational source and the inertial coefficient in Schrödinger dynamics arise from one common standing-bearing quantity, or are they merely independent parameters identified by assumption? Within the manuscript’s frozen stable-sector comparison class, Module B reconstructs the relevant total stress tensor, identifies its integrated conserved charge with the translation generator, and derives gravitational and inertial mass as two projections of the same Poincaré Casimir invariant. The resulting equality is therefore not imposed as an empirical coincidence or primitive equivalence postulate. It follows from common source, generator, sector, and calibration structure. Central Result The manuscript establishes the chain normalized-ray operational source⟶T^μν⟶P^μ⟶P^μP^μ⟶mCasimir⟶{mg,mi.\text{normalized-ray operational source} \longrightarrow \widehat T_{\mu\nu} \longrightarrow \widehat P_\mu \longrightarrow \widehat P^\mu\widehat P_\mu \longrightarrow m_{\mathrm{Casimir}} \longrightarrow \begin{cases} m_g,\\ m_i. \end{cases}normalized-ray operational source⟶Tμν⟶Pμ⟶PμPμ⟶mCasimir⟶{mg,mi. The operator-valued total stress tensor determines the conserved translation generator through the appropriate hypersurface charge: P^ν=∫ΣT^μν dΣμ.\widehat P_\nu = \int_{\Sigma} \widehat T_{\mu\nu}\,d\Sigma^\mu .Pν=∫ΣTμνdΣμ. On a stable massive sector, the Poincaré Casimir fixes the invariant mass: P^μP^μ=−mCasimir 2c2\widehat P^\mu\widehat P_\mu = -m_{\mathrm{Casimir}}^{\,2}c^{2}PμPμ=−mCasimir2c2 with sign and normalization understood according to the manuscript’s declared conventions. The same invariant then appears in two lawful downstream roles: as the monopole charge sourcing the weak-field gravitational response; as the inertial coefficient governing the nonrelativistic Schrödinger kinetic term. Under the common calibration fixed by the manuscript, the result is mg=mi=mCasimir.m_g=m_i=m_{\mathrm{Casimir}}.mg=mi=mCasimir. This is a common-origin matching theorem: the two masses agree because they are projections of one conserved standing-bearing charge. Principal Contributions Imports the operational source architecture constructed in Module A. Reconstructs the operator-valued total stress tensor from admissible normalized-ray source presentations. Separates ray-invariant source content from phase, normalization, basis, coordinate, and representative skin. Establishes the conserved hypersurface charge associated with translations. Identifies that charge with the Poincaré four-momentum generator. Restricts the mass theorem to the declared stable massive sector in which the relevant Casimir is well defined. Derives invariant sector mass from the translation-generator Casimir. Identifies the gravitational monopole charge as a projection of that invariant. Identifies the Schrödinger inertial coefficient through the controlled nonrelativistic expansion of the same invariant energy. Proves that gravitational and inertial mass are not independently standing-bearing quantities within the fixed construction. Excludes arbitrary rescaling or relabeling from creating a second physical mass charge. Makes the equality calibration-sensitive without reducing it to conventional notation. Supplies the common mass coordinate required by Module C’s upstream continuation construction. Locks the mass-bearing entries later used in the exact FEP-1.0 certificate. The Matching Problem The appearance of mass in the Einstein and Schrödinger projections does not by itself prove that both occurrences refer to the same physical quantity. The gravitational branch contains a source charge governing metric response, while the quantum branch contains an inertial coefficient in the kinetic operator: H^NR=P^ 22mi+V.\widehat H_{\mathrm{NR}} = \frac{\widehat{\mathbf P}^{\,2}}{2m_i}+V.HNR=2miP2+V. A genuine matching theorem must establish: that both quantities descend from the same admissible source construction; that the relevant conserved charge is representation-independent; that the translation generator belongs to the same stable physical sector; that gravitational extraction and nonrelativistic reduction preserve the common reference; that calibration is shared rather than adjusted independently after the fact; that no additional standing-bearing mass parameter has been introduced. Module B closes these obligations within its declared comparison class. Common Generator and Casimir Structure The decisive bridge is not a bare numerical identification mg=mim_g=m_img=mi. It is the common generator architecture. The source tensor supplies the translation charge. The translation charge fixes the four-momentum. The four-momentum Casimir fixes the stable-sector invariant mass. Gravitational and inertial mass then arise through different lawful projections of that same invariant. Accordingly, the theorem has the dependency order T^μν→P^μ→mCasimir→mg,  mi,\widehat T_{\mu\nu} \rightarrow \widehat P_\mu \rightarrow m_{\mathrm{Casimir}} \rightarrow m_g,\;m_i,Tμν→Pμ→mCasimir→mg,mi, rather than mg=postulatemi.m_g\stackrel{\text{postulate}}{=}m_i.mg=postulatemi. This distinction is load-bearing. A numerical equality obtained by independently choosing two parameters would not establish common standing, common reference, or common physical origin. Gravitational Projection In the gravitational branch, the stable-sector invariant appears through the long-range monopole component of the operationally attached source. The gravitational mass is therefore not an unrelated scalar appended to the Einstein equation. It is extracted from the same conserved source-and-generator structure already admitted by the common interior. Within the module’s declared weak-field or asymptotic calibration regime, this yields mg=mCasimir.m_g=m_{\mathrm{Casimir}}.mg=mCasimir. The claim is deliberately confined to the exact source, sector, and calibration interface proved in the manuscript. It is not a claim of generic local gravitational-energy localization on arbitrary curved spacetimes. Inertial Projection In the quantum branch, the same Casimir invariant enters the stable-sector energy relation. Its controlled nonrelativistic expansion takes the form E=mCasimirc2+p22mCasimir+⋯ .E = m_{\mathrm{Casimir}}c^2 + \frac{\mathbf p^2}{2m_{\mathrm{Casimir}}} +\cdots .E=mCasimirc2+2mCasimirp2+⋯. After removal of the common rest-energy phase, the surviving first-response Hamiltonian contains P^ 22mCasimir.\frac{\widehat{\mathbf P}^{\,2}} {2m_{\mathrm{Casimir}}}.2mCasimirP2. Comparison with the Schrödinger kinetic term identifies mi=mCasimir.m_i=m_{\mathrm{Casimir}}.mi=mCasimir. The inertial coefficient is therefore fixed by the same stable-sector invariant that determines the gravitational source charge. Calibration and No-Rescaling Discipline The equality theorem requires common calibration. This is not a weakness or an arbitrary convention; it is what prevents two independently normalized projections from being compared as though their numerical coordinates already carried a shared reference. The manuscript therefore distinguishes: a change of units or representative presentation; a lawful common recalibration preserving the shared charge; an independent rescaling that changes the relation between the two projections; the introduction of a genuinely new coupling or source coordinate. Only the first two preserve the fixed matching theorem. An independent projection-specific rescaling cannot create a new physical mass while leaving the original certificate unchanged. Thus the equality mg=mi=mCasimirm_g=m_i=m_{\mathrm{Casimir}}mg=mi=mCasimir is exact relative to the common source, stable sector, projection maps, and calibration package. Place in the Completed Unification ARC The domain-general AASC kernel and Unus Solus Possibilis Est remain upstream machinery used across the wider research program. The gravity–quantum sequence proceeds as follows: Gravity as Structural Necessity — establishes the necessary physical constraint role; Einstein Dynamics — derives the unique minimal first-response metric projection; Schrödinger Dynamics — derives the corresponding minimal Hamiltonian quantum projection; Structural Unification — proves that the gravitational and quantum descriptions are projections of one admissible physical interior; Module A — completes operational metric–source attachment. Module B — proves common mass–charge matching. Module C — constructs common upstream continuation and projection intertwining. Module D — establishes exact certificate-fibre exhaustion and unique lawful role occupancy. Complete Certificate Closure — integrates the full FEP-1.0 gravity–quantum realization certificate. Module B therefore closes the common mass coordinate required by the remaining continuation and certificate stages. Relation to Modules A and C The division of labor is exact: Module A constructs the operational source and attaches it to the metric projection. Module B lifts that source into the conserved generator architecture and proves common mass–charge matching. Module C uses the now-source-locked and mass-locked structures to construct a common upstream continuation whose projections reproduce the required gravitational and quantum developments. Module B cannot precede Module A because its conserved charge requires an admissibly constructed source. Module C, in turn, requires Module B because the two projected histories must preserve one common mass reference rather than merely carry numerically similar parameters. Scope and Nonclaims The manuscript does not claim: an independent derivation of the universal AASC kernel; a new derivation of the Einstein or Schrödinger equations; mass equality without the declared common source and calibration structure; a theorem covering unstable resonances as though they were exact massive sectors; generic mass definition on every curved spacetime; universal local gravitational-energy density; equivalence of arbitrary phenomenological gravitational and inertial parameters; derivation of all Standard Model masses; derivation of the Higgs mechanism or its coupling constants; treatment of massless sectors as massive Casimir sectors; completion of self-force, radiation reaction, or strong-field many-body dynamics; construction of the common upstream continuation; a nonzero interacting quantum-gravity law; uniqueness across different targets, sectors, or calibration packages; completion of the final realization certificate. These matters either lie outside the frozen target or belong to downstream modules. Conceptual Significance Module B replaces postulated equality with common-origin identity. The gravitational and inertial roles are not treated as two primitive masses that happen to share a numerical value. They are two projections of one source-generated, translation-bearing, stable-sector invariant. Their equality follows because the common physical interior does not license two independently standing mass charges where one conserved Casimir already occupies the required role. This establishes a stronger result than numerical agreement alone: one operational source; one translation generator; one stable-sector Casimir; two lawful projections; one shared mass reference. The module thereby closes a central compatibility condition between Einstein and Schrödinger dynamics and prepares the construction of their common upstream continuation.