Rank-Three Scalar-Field Recurrence, Three Lepton Families, and the Inertial Threshold for Mass-Factor Reports

This paper is archived as a speculative research work. This paper develops an EAS scalar-field account of the charged-lepton-like family hierarchy as a support-owned report result. The construction begins from the vacuum as zero-report unresolved balanced scalar-field potentiality, not as empty absence, primitive charge, primitive energy, or background substance. Defined association resolves this vacuum potentiality into scalar contrast. That contrast is organized by a triangle-like every-cycle-remap carrier, the carrier exposes a closure defect, the closure defect recruits support candidates, and only support-owned records contribute to family readability, inertial threshold passage, and mass-factor reporting. The central structural claim is that the triangle-like carrier to electron-like support transition, the electron-like to muon-like transition, and the muon-like to tau-like transition all use the same universal rank-three exterior remap. The family index is therefore recursive captured-remap depth, not a family-specific remap rule. Independent family readability is controlled by the support-owned phase-return space: for a rank-three support, dim V_ret(S) = 3, so a depth-d captured-layer system can be independently family-readable only when rank(M_d) = d. This excludes d > 3 by return-slot capacity and gives positive nullity at d = 4. The inertial threshold is not identified with scalar load alone, SOO gradient alone, layer count, Θ_d = d, path-facing assignment alone, or P(k) alone. It is identified with sustained projected support-owned, phase-specific, path-selected SOO recurrence burden through the inertial remap-settling interval P(k_d). For the admitted depths d = 1, 2, 3, the cycle indices k_d = 2d - 1 give k = 1, 3, 5 and accepted spans P(1) = 1, P(3) = 111, and P(5) = 1033. After inertial persistence opens the mass-factor report domain, the accepted endpoint K = 5 yields the electron-normalized hierarchy 1, 206.769465875020, and 3477.303549170596. The threshold diagnostics are claim-stratified. Label-free whole-field redressing tests are treated only as diagnostic support that loss-before-span and sustained-through-span outcomes can emerge without giving the scalar-value generator the threshold verdict. The result is a scalar-field report mechanism for exactly three charged-lepton-like family reports, their inertial-persistence threshold, and their mass-factor hierarchy.