Forward-Bank Diagnostics for Tachyonic Stability under the D6-F Combination

This diagnostic note documents two forward-bank failure modes for tachyonic stability (T3) under the D6-F operational combination of Radial Coherential Dynamics (RCD), within the LB-04 extended regime (decoupling limit, quasi-static approximation, Brown-Schutz dust, |xi|0). A centered Gaussian-bump profile fails by a central pathology Phi (0) = -2A (1-C*-A) /L² y* ~ 1. 0986, where the kinetic-geometric functional PhiC₀ changes sign and the master bound infimum reaches inf V₀ᵐax ~ -7. 6 x 10^-3 at yₘin ~ 1. 486. Both pathologies are analytically derived and numerically verified (Brent root-finding, SciPy minimization). Numerical sweeps over four decades of dimensionless matter density rhoₕat and two decades of disformal scale mu shift the infimum only in its fifth significant digit. The structural reason is direct: mu does not appear in the kinetic-geometric sector Phi at all, only in the sub-leading conformal-matter sector via sqrt (Z₀/A₀). The diagnostic isolates a falsifiable conceptual result: under LB-04/D6-F and for the one-dimensional toy banks tested here, T3 stability favors evaluation domains that coincide with regions of active coherential transition. Forward profiles extending into asymptotic regions, where the kinetic favorable term has decayed faster than the geometric tensioning term, fail T3 for geometric reasons that cannot be rescued by adjusting matter, disformal, or coupling parameters within natural ranges. The note includes a brief provenance statement on the master bound (inherited from the companion note 3, with structural origin of the three sectors made explicit), three figures (forward bank candidates and PhiC₀; decomposition of V₀ᵐax for tanh-wall; sensitivity sweeps over rhoₕat and mu), explicit failure-mode statements with domain conditions, and a kill-switch falsification clause. This note is self-contained as a diagnostic and motivates the inverse-construction methodology of the companion note: A. Cerezo García, "Inverse Construction of Coherential Backgrounds Compatible with Tachyonic Stability", Zenodo doi: 10. 5281/zenodo. 20262975 (May 2026). Draft v1, May 2026. Internal Panel Cerezo review (Claude, ChatGPT, Gemini, Grok, Perplexity, DeepSeek) consolidated. Symbolic verifications with SymPy; numerical localizations with SciPy (brentq, minimizeₛcalar). Fully reproducible from the equations as stated.