The Urban Consumption Kernel K: Construction, the K-Family, and Structural Diagnosis — Evidence from Greater London, 2015–2025, and a Cross-City Comparison with Greater Manchester

Urban labour demand is usually taken as given. This paper measures the consumption side: the kernel K (Kang, 2026), 𝐾=diag(𝜃)𝑃, with 𝜃𝑖 the share of type i's income retained locally and P row-stochastic. The kernel is built on a three-layer type space — ten income deciles, sixteen sectors, thirty-three boroughs, n = 5,280 — from ONS expenditure, ASHE earnings and Census 2021 commuting flows. Two constructions of P are given: a gravity kernel and a maximum-entropy calibration pinning it to both margins in closed form. Five quantities follow: the spectral radius 𝜌, chain length 𝐶̄ , multiplier 𝑀, composite 𝛷 and non-normality ratio 𝜅𝑉. Three formal results follow: the closed-form solution; a single-axis identity making the spectral radius invariant to the choice of axis; and a scaling invariance making period-by-period recomputation of 𝜃 a discipline. Both constructions are run over eleven periods, two denominator conventions and four settings (88 cells each, two cities); their difference is decomposed with a 2×2 experiment: the rise in 𝜅𝑉 is borne by the landing operator, while the loop that flips the sign of 𝛷 varies with the city. Over eleven fiscal years (FYE2015–FYE2025), a structural low in FYE2021 is common to all four settings under the income denominator yet does not survive a denominator change — and reappears in a second city and under both constructions; the non-normality ratio, not the spectral radius, discriminates the axes; and Greater Manchester reverses the export base. All data and scripts are public.