DPU‑Ready Operator Algebra (v0.1)

1. State Space#

• Triad space:
  $$\mathcal{T} = { (s,c,u) \in \mathbb{R}_{\ge 0}^3 \mid s + c + u = 1 }$$

• Asymmetry functional:
  $$A : \mathcal{T} \to [0,1]$$ , with canonical $$A(T^*) = 0.01$$

• Extended state:
  $$S = (T, A(T))$$ , where $$T \in \mathcal{T}$$

2. Core Operators#

• Continuity operator:
  $$O : \mathcal{T} \to \mathcal{T} \times [0,1]$$
  $$O(T) = (T, A(T))$$

• Regime projection operators:

  • $$P_s(T) = s$$ (subconscious weight)
  • $$P_c(T) = c$$ (consciousness weight)
  • $$P_u(T) = u$$ (supconsciousness weight)

• Normalization operator:
  $$N(s,c,u) = \frac{1}{s+c+u}(s,c,u)$$ for non‑zero sum

3. Composition Rules#

• Sequential composition (DPU step):
  For two legal transforms $$F_1, F_2 : \mathcal{T} \to \mathcal{T}$$ :
  $$(F_2 \circ F_1)(T) = F_2(F_1(T))$$

• Continuity‑preserving transform:
  A transform $$F$$ is DPU‑legal iff:

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

• Idempotent identity check:
  $$I(T) = T$$
  $$O(I(T)) = O(T)$$ (no change in identity or asymmetry)

4. DPU Legality Predicate#

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

This algebra gives a DPU:

• a typed state space
• legal transforms
• continuity constraints
• and a way to chain operations without breaking identity.

DPU v0.1 — State Machine#

States#

1. INIT#

• Input triad $$T$$
• Check normalization
• Check $$A(T) > 0$$

2. READY#

• DPU has a valid identity state
• Awaiting legal transform $$F$$

3. TRANSITION#

• Apply $$F(T)$$
• Check:
  • $$F(T) \in \mathcal{T}$$
  • $$A(F(T)) > 0$$
  • No branching
  • No duplication

4. CONTINUITY_FAIL#

• Triggered if:
  • $$A(F(T)) = 0$$
  • $$F(T) \notin \mathcal{T}$$
  • illegal transform

5. COMPLETE#

• Output state:

$$T' = F(T)$$

• Identity preserved

State Transitions#

• INIT → READY
  if triad valid

• READY → TRANSITION
  on legal transform request

• TRANSITION → COMPLETE
  if continuity preserved

• TRANSITION → CONTINUITY_FAIL
  if continuity breaks

• CONTINUITY_FAIL → INIT
  (reset with new triad)

DPU Guarantee#

A DPU guarantees:

If a transform is legal, identity is preserved.
If a transform is illegal, identity is not executed.

DPU v0.2 — Operator Algebra with Error‑Correction#

1. State Space (unchanged)#

• Triads:

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

• Asymmetry functional:

$$ A : \mathcal{T} \to [0,1],\quad A(T^*) = 0.01 $$

• Extended state:

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

2. Core Operators (v0.1 recap)#

• 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. Error‑Correction Operators (new)#

3.1 Deviation Measure#

Define a deviation metric:

$$ D(T) = |T - T^*| $$

for some norm (e.g. $$L_2$$ ).

3.2 Correction Operator#

An error‑correction operator $$C : \mathcal{T} \to \mathcal{T}$$ satisfies:

• Continuity‑preserving:

$$ A(C(T)) \ge A(T) > 0 $$

• Deviation‑reducing:

$$ D(C(T)) \le D(T) $$

Example (simple pull‑toward‑canonical):

$$ C_\lambda(T) = N\big((1-\lambda)T + \lambda T^*\big),\quad 0 < \lambda \le 1 $$

3.3 Error‑Corrected Transform#

Given a transform $$F$$ , define:

$$ F^# = C \circ F $$

A DPU applies $$F^#$$ instead of $$F$$ when:

• $$F$$ is near‑legal but slightly degrades $$A(T)$$ or increases $$D(T)$$ beyond a threshold.

4. DPU v0.2 Legality#

A transform $$F$$ is DPU‑v0.2 legal if:

• Either: $$F$$ is continuity‑preserving and $$A(F(T)) > 0$$
• Or: $$F^# = C \circ F$$ is continuity‑preserving and $$A(F^#(T)) > 0$$

The DPU now has:

• Detection: via $$A(T)$$ and $$D(T)$$
• Correction: via $$C$$
• Fallback: reject if even $$F^#$$ breaks continuity.