The Immune System as a Maintenance Engine: A dm3 Geometric Framework for Innate Immunity, Aging, and the Mathematics of Cellular Upkeep

The Immune System as a Maintenance Engine: A dm3 Geometric Framework for Innate Immunity, Aging, and the Mathematics of Cellular Upkeep The innate immune system sustains tissue homeostasis, organ performance, and the integrity of aging not through reactive mobilisation alone but through continuous, cycle-by-cycle maintenance work. This monograph proposes that this maintenance function is governed by the dm³ operator chain — a five-operator contact-geometric sequence $G_5 = U \circ F \circ K \circ C \circ \mathcal{E}$ in which compression ($C$), commitment ($K$), fold ($F$), unfolding ($U$), and the temporal operator ($\mathcal{E}$) execute in mandatory, irreversible order across every scale of immune function, from single-cell autophagy to whole-organ restoration. The chain is intrinsically non-commutative: $[F,K] \neq 0$, $[K,C] \neq 0$, and $[U,F] \neq 0$. Each commutation error corresponds to a defined clinical category — fold without commitment ($[F,K] \neq 0$) produces autoinflammatory syndromes including Familial Mediterranean Fever, CAPS, and NOMID; commitment without compression ($[K,C] \neq 0$) drives alloreactive rejection; premature unfolding ($[U,F] \neq 0$) is the structural basis of tumour immune evasion. No linear ODE framework can encode these distinctions. The temporal operator $\mathcal{E}$ advances the contact-manifold phase coordinate according to the system's intrinsic oscillation frequency and external zeitgeber field, providing the circadian gating and thermodynamic irreversibility absent from four-operator reductions of the chain. We map recent findings from Hidalgo and colleagues at Yale School of Medicine — the NeuMap transcriptional atlas of neutrophils (Cerezo-Wallis et al., Nature 2025) and the role of cardiac macrophages in mitochondrial clearance (Nicolás-Ávila et al., Cell 2020) — onto this framework, identifying each biological transition as an instance of the G-chain. Aging is formalised as monotonic erosion of the Lyapunov contraction rate, $\mu_{\max}(a) = \mu_{\max}(0)\cdot e^{-\lambda_{\mathrm{age}},a}$, with immune chain parameters $\mu_{\max} = -0.44\ \mathrm{s}^{-1}$, $\beta = 2.0$, and $\kappa^* \in [0.11, 0.19]$. We further map autophagy (Ohsumi, Nobel 2016) onto the fold operator $F$, demonstrating that the mTOR/AMPK decision surface is a Whitney $A_1$ singularity in contact coordinates that fires irreversibly when the normalised nutrient coordinate $r \leq r^*(\lambda) = \sqrt{K_{\mathrm{aa}}/\lambda}$. The post-fold Lyapunov stability ($\mu = -2$) is established by machine proof in Lean 4 (AXLE repository, 0 sorry). Three falsifiable quantitative predictions are derived, each with an explicit experimental protocol and falsification condition. Part of the Principia Orthogona Series