Realized Physics Arc — Paper IX Mass, Inertia, and Persistence CompressionThe Deformation-Resistance Role of Particle Identity under Field-Carrier Mediation By Amos Jay Maley This manuscript is the ninth paper in the Realized Physics Arc, deriving the minimal mass/inertia role from the previously established framework of field-carrier mediation, particle persistence classes, gauge-role structure, bounded perturbation transport, and quotient comparison structure. The paper proves that mass and inertia first enter realized physics not as primitive scalar quantities, substances, energy ledgers, or measurement outputs, but as the operative deformation-response profile by which a localized particle persistence class preserves standing identity under admissible deformation and transport-state change. The central result is the Mass-Inertia Role Induction Theorem: (ΦD, ΠD, GD, BoundTransD, CD)∧DefAdmD∧IdRetD⇒MassInertiaRoleD(\Phi_D,\; \Pi_D,\; G_D,\; \mathrm{BoundTrans}_D,\; C_D) \wedge \mathrm{DefAdm}_D \wedge \mathrm{IdRet}_D \Rightarrow \mathrm{MassInertiaRole}_D(ΦD,ΠD,GD,BoundTransD,CD)∧DefAdmD∧IdRetD⇒MassInertiaRoleD with induced profiles: μD(Π):ΔΠ⇀CD\mu_D(\Pi) : \Delta_\Pi \rightharpoonup C_DμD(Π):ΔΠ⇀CD and ιD(Π)=μD(Π)∣ΔΠtr\iota_D(\Pi) = \mu_D(\Pi)\vert_{\Delta_\Pi^{tr}}ιD(Π)=μD(Π)∣ΔΠtr where μD\mu_DμD is the mass-role profile and ιD\iota_DιD is the transport-state restriction corresponding to inertia. The manuscript develops: admissible deformation, standing-preserving deformation, compression response, mass-role profiles, inertia-role profiles, deformation mediation, and mass-class witnesses. A central structural claim of the paper is that mass is not primitive quantity but deformation resistance: the admissible carrier/restoration/transport adjustment required for a particle persistence class to remain the same under standing-threatening deformation. The paper proves that: primitive scalar mass assignments fail, primitive substance amounts are insufficient, energy ledgers alone cannot ground mass, gauge-presentation differences do not constitute mass, hidden mass reservoirs collapse, and measurement/extraction records are downstream rather than foundational. The induced compression-response profile is represented through quotient comparison classes: μD(Π)(δ)=[cmp(δ,Π)]\mu_D(\Pi)(\delta) = [\mathrm{cmp}(\delta,\Pi)]μD(Π)(δ)=[cmp(δ,Π)] where admissible deformations are evaluated relative to preservation of: standing signature, carrier mediation, restoration profile, perturbation burden, quotient comparison, boundary traces, and transport structure. The manuscript further establishes: deformation-response mediation, stable localization under deformation, gauge-compatible response preservation, transport-state deformation structure, and quotient-stable compression-response comparison. A major consequence of the paper is that later mass-generating and inertial frameworks — including: Higgs/Yukawa mechanisms, running and effective masses, QCD mass generation, relativistic mass-energy relations, inertial response laws, stress-energy coupling, and Standard Model mass structure — are interpreted as later mathematical and dynamical specializations of a more primitive deformation-response role already forced by persistence-class preservation under admissible deformation. The analysis remains fully compatible with: quantum field theory, Standard Model mass generation, gauge theory, relativistic dynamics, and stress-energy frameworks, while rejecting reduction of mass or inertia to primitive scalar labels, hidden substances, or purely descriptive energy bookkeeping. As the ninth theorem in the Realized Physics Arc, the paper establishes the earliest admissible mass-facing structure from which later reconstructions of Higgs structure, running masses, inertial laws, stress-energy coupling, relativistic mass-energy relations, and Standard Model mass organization can proceed: mass first enters realized physics as the gauge-compatible, boundary-fixed, quotient-stable deformation-response profile of a localized particle persistence class, while inertia is its transport-state-change manifestation. This paper is downstream of: Minimal Conditions for Admissible Construction The Structure of Admissibility