Fatou–Julia Decomposition in Grothendieck Topoi via Frame-Theoretic Equicontinuity

Version 2 (May 2026). The mathematical results and scope of the originalsubmission are preserved. This version improves the exposition (introductionreorganized around observed dynamical systems and a legitimacy-testsarchitecture), refines hypothesis annotations through a stratification ofadmissibility (observable, backward, forward, fully admissible), correctsinternal references, and includes minor terminological precision (notationφ_∀ for the right adjoint, replacing φ_!). Section 8 (étale frame) has beencalibrated to register open questions without tacit predictions. --- We construct a systematic point-free Fatou–Julia theory within thesubobject/frame structure of Grothendieck topoi, formalized through observeddynamical systems (X, φ, 𝒪), where 𝒪 ⊆ Sub(X) is an admissible observablesubframe. For such systems we define a frame-theoretic Fatou object F_𝒪(φ)by coverwise equicontinuity, together with observable and ambient Juliaobjects J_𝒪 = ¬_𝒪 F in 𝒪 and J_L = ¬_L F in L = Sub(X), with J_𝒪 ≤ J_L ingeneral. The construction recovers the classical Fatou locus on ℙ¹(ℂ) and theBerkovich Fatou locus on ℙ¹,ᵃⁿ under explicit representability hypotheses.Over ℂ_p it yields the equilibrium-measure identity J_L(*) = supp(μ_eq) forrational maps of degree at least two; over general complete algebraicallyclosed non-Archimedean fields of characteristic zero, the inclusion J_L(*) ⊇supp(μ_eq) holds. In compact representable settings, the construction isidentified with a condensed Montel–Fatou object ℱ^M = F_𝒪(φ) defined throughquasi-compactness of orbit closures in the condensed mapping space, byArzelà–Ascoli and the compact-Hausdorff comparison inside Cond. Beyond the standard regimes, the framework records structure visiblespecifically at the level of observed systems: the gap J_𝒪 ≤ J_L, strict instandard Zariski and Berkovich examples; an internal boundary condition F ∨J_𝒪 < 1 realized for φ(z) = z² on ℙ¹,ᵃⁿℂ_p; and, in the Zariski regimeover F̄_q, a coverwise arithmetic signal. For separable polynomialendomorphisms of 𝔸¹ of degree at least two over F̄_q, the Zariski Fatouobjects F_Zar and Fb,Zar vanish, while the backward coverwise ambientenvelope Fb,Zarcw,L remains nontrivial and detects the totallyinvariant, equivalently superattracting, maps. The resulting framework iscalibrated by recovery in the standard regimes and becomes genuinelyinformative through new observable, logical, and arithmetic phenomena.