Inverse Construction of Coherential Backgrounds Compatible with Tachyonic Stability

This work is a toy inverse-construction benchmark, not an astrophysical background. We introduce an inverse methodology for generating one-dimensional coherential profiles C₀ (y) compatible with tachyonic stability (T3) under the D6-F operational combination of Radial Coherential Dynamics (RCD), within a previously ghost/gradient-controlled extended regime (LB-04 with decoupling limit, quasi-static approximation, Brown-Schutz dust, |xi|0). Forward attempts with Gaussian-bump and tanh-wall profiles produce structural failures driven by the sign of the kinetic-geometric functional PhiC₀ = 2 (C₀') ² + (1-C₀) C₀'', not by matter or coupling parameters. Imposing PhiC₀ >= 0 as a design constraint and integrating, we obtain a closed-form first integral (u') ² = epsilon₀/2 + C₁ u⁴ with u = 1-C₀. The saturation branch (Phi = 0) yields a non-viable hyperbolic profile; the simplest strict-positivity branch (Phi = epsilon₀ > 0, C₁ = 0) yields a linear bank with a closed-form upper bound rhoₘax (m, alpha, mu) = 4m² / alpha * sqrt (1 + (m²/mu⁴) / (1+alpha) ) on the dimensionless matter density. For representative parameters (m, alpha, mu) = (0. 4, 1e-4, 1) this gives rhoₘax ~ 5. 94 x 10³, a four-order margin against natural values. The linear bank is shown to be optimal within its family: the non-linear branch C₁ > 0 preserves Phi exactly but strictly degrades the matter margin and contracts the domain length. The result is offered as a methodological contribution: a viable inverse-construction route exists for T3 in D6-F, with the linear bank as its first closed example, not its final destination. Scope is explicitly bounded: the result is a toy bank, not an astrophysical background; T1 and T2 are inherited from previous D6-F analysis and not rederived here; no claim is made about global RCD stability or about deriving the program's parameter alpha. The note includes an appendix with a self-contained derivation of the master bound V₀ᵐax from the D6-F action in the LB-04 extended regime, two figures (forward vs. inverse methodology; profile and kinetic-geometric functional across three banks), and explicit references to the EFT-for-dark-energy framework of Frusciante et al. (2016, 2017, 2019) and to the Brown-Schutz matter action (Brown 1993; Sorkin 1997). Draft v1, May 2026. Internal Panel Cerezo review applied. Symbolic verifications with SymPy and numerical evaluations with SciPy fully reproducible from the equations as stated.