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 in which compression (\(C\)), commitment (\(K\)), fold (\(F\)), unfolding (\(U\)), and a temporal operator (\(\mathcal{E}\)) execute in a mandatory, irreversible order across every scale of immune function, from single-cell autophagy to whole-organ restoration. Three falsifiable quantitative predictions are derived, each with an explicit experimental protocol and falsification condition.
CONJECTURE / FRAMEWORK The central proposal is that immune maintenance is one turn of the operator cycle:
compression \(C\) (survey / concentrate the signal), commitment \(K\) (curvature toward threshold), fold \(F\) (the irreversible decision), unfolding \(U\) (resolution to a stable state), and the temporal operator \(\mathcal{E}\), which advances the contact-manifold phase according to the system's intrinsic oscillation and external zeitgeber field — supplying circadian gating and thermodynamic irreversibility that a four-operator (time-free) reduction cannot.
The chain is proposed to be intrinsically non-commutative, and each commutation error is mapped to a disease category. These are proposed structural analogies, not established mechanisms:
| Commutator | Proposed failure | Mapped clinical category |
|---|---|---|
| \([F,K]\neq 0\) | fold without commitment | Autoinflammatory syndromes — FMF, CAPS, NOMID |
| \([K,C]\neq 0\) | commitment without compression | Alloreactive transplant rejection |
| \([U,F]\neq 0\) | premature unfolding | Tumour immune evasion |
The framework is mapped onto real experimental results — these are the established science, distinct from the dm³ interpretation:
The monograph identifies each of these biological transitions as an instance of the G-chain. The findings are established; the identification with the operator chain is the bridge (conjecture).
CONJECTURE / BRIDGE Autophagy is mapped onto the fold operator \(F\): the mTOR/AMPK nutrient-decision surface is proposed to be a Whitney \(A_1\) singularity in contact coordinates that fires irreversibly when the normalised nutrient coordinate crosses a threshold,
The post-fold Lyapunov stability, \(\mu = -2\), is established by machine proof in Lean 4 (AXLE repository, reported 0 sorry). This is a statement about the toy-model dynamical system's stability after the fold — it is the mathematics that is verified, not the immunological identification. See AXLE.
CONJECTURE / MODEL Aging is formalised as monotonic decay of the Lyapunov contraction rate with age \(a\):
with immune-chain model parameters \(\mu_{\max} = -0.44\ \mathrm{s}^{-1}\), \(\beta = 2.0\), and \(\kappa^* \in [0.11,\,0.19]\). These are model parameters proposed by the monograph, not independently measured biological constants.
CONJECTURE The monograph derives three quantitative predictions, each with an explicit experimental protocol and falsification condition. Stating predictions this way — with a defined way to be proven wrong — is the honest strength of the framework: it can be tested. The full protocols are in the deposited PDF.
The complete text (173.7 kB PDF), with all derivations, the mapping tables, parameters, and the three prediction protocols, is openly available:
→ dm3_immunology_monograph_v2.pdf (Zenodo, open access)
→ DOI: 10.5281/zenodo.20969152 · Part of the Principia Orthogona series (10.5281/zenodo.19117399)