Differential Calculus
Definition
The discrete differential calculus of UOR defines two derivative operators on functions f : R_n → R_n:
- Ring derivative ∂_R f(x) = f(succ(x)) - f(x), measuring change along the ring successor.
- Hamming derivative ∂_H f(x) = f(bnot(x)) - f(x), measuring change along the Hamming antipode.
These are encoded as Identity individuals
DC_1 and DC_2 in the op/ namespace.
The Jacobian
The Jacobian decomposes the incompatibility metric site by site. At position k:
J_k(x) = |d_R(x, succ(x)) - d_H(x, succ(x))| restricted to site k
Key identities:
- DC_6: J_k(x) = ∂_R site_k(x)
- DC_8: rank(J(x)) = d_H(x, succ(x)) - 1 for generic x
- DC_9: Total curvature κ(x) = Σ_k J_k(x)
- DC_11: Curvature equipartition — each site contributes approximately equally to total curvature.
Curvature-Weighted Resolution
Identity DC_10 shows that the optimal next constraint in iterative resolution maximizes the Jacobian over free sites. This connects the differential calculus to the resolution pipeline: curvature guides constraint selection.
Worked Example: Ring Derivative on R_4
Consider f(x) = x² mod 16 on R_4 = Z/16Z. The ring derivative is ∂_R f(x) = f(succ(x)) - f(x) = (x+1)² - x² = 2x + 1 (mod 16).
| x | f(x) = x² | succ(x) | f(succ(x)) | ∂_R f(x) = 2x+1 |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 2 | 4 | 3 |
| 2 | 4 | 3 | 9 | 5 |
| 3 | 9 | 4 | 0 | 7 |
| 7 | 1 | 8 | 0 | 15 |
The derivative 2x+1 is always odd — it is a unit of Z/16Z for every x. This means f(x) = x² has no stationary points, and every constraint application changes the squared observable.
Commutator Decomposition
Identity DC_4 shows that the fundamental commutator [neg, bnot](x) = 2 can be recovered from the difference of ring and Hamming derivatives of negation. This provides a differential-geometric interpretation of the critical identity.
Connection to Resolution
Identity DC_10 shows that the optimal next constraint in iterative resolution maximizes the Jacobian over free sites. This connects differential calculus directly to the resolution loop: each step selects the constraint whose curvature contribution is largest, ensuring maximal progress per iteration. See Iterative Resolution for the convergence analysis and Observables for the Jacobian's role as a curvature observable.