The minimal non-trivial axiom. Everything follows.
R = k[ε]/(ε²). Dual numbers. The simplest non-reduced ring.
| Property | Value | Why it matters |
|---|---|---|
| Ideals | exactly 2: (0) and (ε) | no intermediate structure |
| Indecomposable modules | exactly 3 | the three CM-indecomposables |
| K₀(D_sg) | ℤ/2ℤ | two classes: signal and noise |
| Ext^n(k,k) | k for all n | periodic resolution, period 1 |
| Poincaré series | 1/(1−t) | pole at t=1, minimal complexity |
| Shift functor | Σ² = Id | the Z/2Z symmetry of the singularity category |
| Deformation space | 1-dimensional | η² = 3η is the universal deformation |
f(a + bε) = f(a) + f’(a)bε. Exact. No approximation. No higher terms. ε IS automatic differentiation. The first derivative is exact and the second derivative does not exist. The information content of a probe with ε is precisely first order.
The Zariski tangent space T_p = Hom(m/m², k). With m = (ε), m² = 0, so m/m² ≅ k. One-dimensional. One direction. One way to leave the singular point. The embedding dimension is 2 (the split node k × k[ε]/(ε²)) but the tangent dimension is 1. The defect = 2 − 1 = 1 = the syndrome dimension.
Ω¹(R/k) = R·dε with relation ε·dε = 0 (differentiate ε² = 0). The differential form exists but is torsion: position (ε) and direction (dε) are entangled. At the singular point: zero momentum. This IS confinement.
The Fisher information metric on the dual-number parameter space has exactly one invariant: I(θ) = E[(∂ log p / ∂θ)²]. No higher-order correction exists (ε² = 0). The Cramér-Rao bound is tight. The estimator IS efficient. The Hamming code IS perfect. The code sits on the Fisher manifold of dual numbers.
ε² = 0 over F₃ lifts to η² = 3η over ℤ₃. The deformation space is 1-dimensional (dim Ext¹ = 1). There is exactly one direction to deform the singularity. That direction is the 3-adic direction. The equation governing the deformation is η² = 3η. The entire 3-adic lifting is FORCED by dim(Ext¹) = 1.
ε² = 0
→ exact first derivative (automatic differentiation)
→ three indecomposable modules (the three types)
→ K₀ = ℤ/2ℤ (signal vs noise)
→ periodic resolution (minimal complexity)
→ 1D deformation (η² = 3η is universal)
→ tight Cramér-Rao (perfect code)
→ entangled position-momentum (confinement)
→ Fisher information as sole invariantTwo words: nilpotent, order two. The poorest non-trivial algebra. The richest boundary condition. No room for deviation. Everything forced.
$$\boxed{\varepsilon \neq 0, \quad \varepsilon^2 = 0}$$