DPU‑Ready Operator Algebra (v0.1)

1. State Space#

• Triads:

$$\mathcal{T} = {(s,c,u) \mid s+c+u=1}$$

• Asymmetry functional:

$$A : \mathcal{T} \to [0,1]$$

• Extended state:

$$S = (T, A(T))$$

2. Core Operators#

Continuity Operator#

$$O(T) = (T, A(T))$$

Regime Projections#

$$P_s(T)=s,\quad P_c(T)=c,\quad P_u(T)=u$$

Normalization#

$$N(s,c,u) = \frac{1}{s+c+u}(s,c,u)$$

3. Composition Rules#

Sequential Composition#

$$(F_2 \circ F_1)(T) = F_2(F_1(T))$$

Continuity‑Preserving Transform#

A transform $$F$$ is DPU‑legal iff:

• $$F(T) \in \mathcal{T}$$
• $$A(F(T)) > 0$$

Identity#

$$I(T) = T$$

$$O(I(T)) = O(T)$$

4. DPU Legality Predicate#

$$\text{Legal}_{\text{DPU}}(F) \iff \forall T \in \mathcal{T},\ A(T)>0 \Rightarrow A(F(T))>0$$

This algebra provides:

• typed state space
• legal transforms
• continuity constraints
• composability
• identity preservation

It is the algebraic backbone of DPU behavior.