Complex Regular Variation and Boundary Firewall Dynamics(paper2)

Title: Complex Regular Variation and Boundary Firewall DynamicsSeries: Foundations of Complex Slowly and Regularly Varying Functions (Part II)Author: Arman Rahaman SekhRelated DOI: 10.5281/zenodo.23509689 (continuation of Paper 1) Abstract & Overview:This manuscript establishes the operational and geometric foundations of complex regular variation, serving as the direct sequel to "Complex Slowly Varying Functions" (Sekh, 2026). While the preceding work introduced Sectorial Slow Variation on rigid conical sectors, this paper resolves the geometric deformation induced by arbitrary complex scaling indices $\rho = u + iv$ ($u > 0$). The narrative balances structural classification on boundary firewalls with the emergence of non-Euclidean spiral manifolds, the complete resolution of the complex de Bruijn asymptotic inversion problem, and destructive interference dynamics. Key Breakthroughs & Geometric Invariants: 1. Boundary Firewalls & Rigidity Floors:Classifies analytic continuation limits across sector boundaries into tame boundaries (Class I) and rigid singular firewalls (Class II). Enforces Phragmén–Lindelöf growth floors ($\rho \ge 1/2$) to preserve sectorial stability and govern safe interior buffers $\Omega_\eta$. 2. Canonical Gauge Factorization:Establishes the Conformal Equivalence Bridge between Sectorial Regular Variation ($\mathrm{SRV}_\rho$) and Analytic Regular Variation ($\mathrm{ARV}_\rho$) through the logarithmic elasticity operator:$$E_f(z) := z \frac{f'(z)}{f(z)} \longrightarrow \rho \quad \text{as } |z| \to \infty$$proving that every regularly varying function decouples into a pure complex power carrier and a slowly varying gauge factor:$$f(z) = z^\rho L(z)$$ 3. The Logarithmic Spiral Horn ($\mathcal{W}_\sigma$):Proves that the transformation $z \mapsto z^\rho$ shears straight sectors into expanding spiral horns. The natural domain of complex regular variation is therefore governed by the invariant transverse spiral coordinate:$$\Theta_\sigma(z) := \operatorname{Im}\left( \frac{\operatorname{Log} z}{\sigma} \right), \qquad \mathcal{W}_\sigma(\alpha, \beta) := \{ z \in \mathbb{C}^* : \alpha < \Theta_\sigma(z) < \beta \}$$Setting $\sigma = 1$ recovers the classical conical sector $\Omega(\alpha, \beta)$. 4. Resolution of the Complex de Bruijn Conjugate Problem:Completely resolves the open asymptotic inversion problem formulated by N. G. de Bruijn (1959) in the complex 2D setting. On the natural spiral horn geometry, constructs the unique analytic de Bruijn conjugate $L^\#$ satisfying asymptotic reciprocity:$$L(z) \cdot L^\#\big(z L(z)^{1/\rho}\big) \longrightarrow 1$$We prove that the logarithmic gauge $\Phi := \operatorname{Log} L^\#$ is the unique attracting fixed point of a strict Picard contraction on a complete Banach space, governed by an exact closed Elasticity Duality Law:$$E_{L^\#}(w) = \frac{-E_L(z)}{\rho \big(\rho + E_L(z)\big)}$$This is accompanied by the Single-Sheeted Univalence Horizon guaranteeing global biholomorphism without Riemann sheet overlap:$$\omega_\eta < \frac{2\pi u}{u^2 + v^2}$$ 5. Vector Addition & The Pitch-Density Law:Classifies the superposition $S(z) = f_1(z) + f_2(z)$ via a three-way trichotomy. For co-dominant phase twists ($u_1 = u_2$, $v_1 \neq v_2$), destructive interference generates an infinite discrete lattice of spiral zeros whose Nevanlinna density is governed by the scalar pitch mismatch $\Delta \kappa := \frac{v_2 - v_1}{u}$:$$\mathcal{D}(0, S) := \lim_{r \to \infty} \frac{N(r, 0; S)}{T_{\log}(r, S)} = \frac{|\Delta \kappa|}{2\pi}$$