Operator‑Spectral Marketing II: Formal χ‑Operators, χ‑Spectra, χ‑Geodesics and χ‑Flow Derived from Raw Transactions, Event Logs, UTM Markup and Cohort Tables

This article presents a fully formal, operator‑spectral reconstruction of customer lifetime value (CLV) and multi‑channel attribution directly from raw transactional data, event logs, UTM markup, and cohort tables. Modern marketing systems generate high‑dimensional, heterogeneous, and temporally irregular behavioral traces, yet existing CLV models rely on heuristic discounted sums, probabilistic retention assumptions, and inconsistent attribution rules. They lack operator structure, spectral decomposition, geometric interpretation, boundary constraints, and falsifiability. We resolve these foundational limitations by deriving χ‑operators, χ‑spectra, χ‑geodesics, and χ‑flow explicitly from real data sources. Raw transactions define χ‑state vectors; event logs generate χ‑transition kernels; UTM markup induces χ‑channel projections; cohort tables calibrate χ‑boundary operators; and RFM aggregates anchor χ‑geometric coordinates. The resulting χ‑operator family produces a complete χ‑spectral expansion, where λ₀ encodes retention stability, λ₁ captures churn‑instability, λ₂ represents latent response, and higher modes correspond to noise. Customer journeys are reconstructed as χ‑geodesics on the χ‑manifold, with curvature identifying switching shocks and geodesic length quantifying behavioral friction. Retention and churn emerge as fixed points or collapses of χ‑flow, governed by the balance between χ‑operator excitation and χ‑boundary suppression. Attribution is defined as χ‑residue at χ‑infinity, yielding a unique, globally consistent decomposition of channel influence. All constructs satisfy χ‑falsifiability criteria, enabling structural validation of CLV models. Numerical recipes for χ‑CLV, χ‑attribution, χ‑geodesics, χ‑curvature, and χ‑flow are reproducible and data‑driven. This establishes operator‑spectral marketing as a rigorous, falsifiable scientific discipline grounded directly in observable behavioral data.