Charged-Lepton Bounded Supports and the Scalar-Amplitude Koide Structure in EAS

This paper is archived as a speculative research work. This paper develops a conditional scalar-field construction of the charged-lepton Koide structure in EAS. The construction separates the common charged-lepton interface charge readout from the scalar-amplitude readout associated with the full charged-lepton-like scalar-point records. The electron-like, muon-like, and tau-like cases are represented by bounded-support reporting labels L₀, L₁, L₂. They share a charge-facing boundary/dressing presentation but differ in their finite scalar-point edge records. Their full-cycle boundary/dressing presentations coarse-grain to the same unit charge, Q₏₀ₑₓ₈₀₋ ₃ (Delta₀) = Q₏₀ₑₓ₈₀₋ ₃ (Delta₁) = Q₏₀ₑₓ₈₀₋ ₃ (Delta₂) = qₑ, while their scalar-amplitude readouts Aⱼ = A (Pⱼ) remain distinct. Charge equality is therefore an interface coarse-graining of the boundary/dressing channel, not equality of whole-support scalar amplitude. The paper first recalls the scalar-amplitude form of the Koide relation. The triple (A₀, A₁, A₂) is decomposed into a common family component and a zero-sum family component: (A₀, A₁, A₂) = C (1, 1, 1) + (N₀, N₁, N₂), N₀ + N₁ + N₂ = 0. Writing Nⱼ = R cos (theta + 2 pi j / 3), the strict null-boundary orientation of the electron-like branch fixes the scale condition R = sqrt (2) C. This gives the Koide scalar-amplitude form (A₀² + A₁² + A₂²) / (A₀ + A₁ + A₂) ² = 2/3. The main constructive contribution of the paper is a finite harmonic/stiffness backbone for the charged-lepton-like edge records. In the corrected charged-lepton edge skeleton, the electron-like record exposes the dressing triad e0, e2, e4. The muon-like record incorporates that dressing triad into the next support record and exposes e9, e10, e11, while the tau-like record repeats the operation and exposes e12, e13, e14. Thus the records exhibit recursive dressing incorporation. The association paths from the exposed dressing points to the electron-like border/base set 0, 2, 4 have common lengths 1, 3, 5. These lengths are graph-theoretic association indices in the scalar-point record, not spatial distances or geometric directions. The paper then connects this edge-record result with boundary-readable recurrence. SOO, scalar-value imprecision, and positive stiffness can produce oscillatory-like scalar-field presentation in admissible sectors. The charged-lepton boundary does not create recurrence; rather, its rank-3 coherence selects a common support-readable mode from the three boundary-selected path reports. In this common mode, unit path-stage imprecision is read as a cumulative coherent profile cᵣ = r. A positive quadratic stiffness readout therefore gives the addend cᵣ² = r². For one exterior-to-electron path of length n, the finite readout is the sum from r=1 to n of r². Because each charged-lepton-like record has three such path families, the total finite readout is K (n) = 3 * (sum from r=1 to n of r²). For n = 1, 3, 5, this yields Kₑ: Kₘu: Kₜau = 3: 42: 165. This backbone is not claimed to be the empirical mass ratio, and it is not a completed derivation of detector-facing charged-lepton masses. It is a finite positive-stiffness scalar-field readout of the corrected charged-lepton edge records. The construction remains conditional: the completed cyclic rank-3 slot records, full scalar-value/sign assignments, the final positive-stiffness functional, the empirical family angle, the scalar-amplitude functional A, the interface bridge mⱼ propto Aⱼ², and detector-facing calibration remain to be derived. Mass labels and Yukawa coefficients are treated as interface readouts, not scalar-field primitives. The result is therefore a scalar-field route to a Koide-compatible charged-lepton scalar-amplitude structure, with a newly identified finite harmonic/stiffness backbone, not a completed derivation of empirical charged-lepton masses. .