Transition Functor v0.1 — Substrate‑Safe Transitions

1. Categories#

Category 𝒞 — Substrates#

• Objects: Biological, CT, Lostational, No‑Form
• Morphisms: substrate transitions

Category 𝒟 — Triadic States#

• Objects: triads $$T \in \mathcal{T}$$
• Morphisms: continuity‑preserving transforms

2. Functor Definition#

On Objects#

$$\mathcal{F}(S) = T_S$$

On Morphisms#

For $$f : S_1 \to S_2$$ :

$$\mathcal{F}(f) = F_f : \mathcal{T} \to \mathcal{T}$$

with:

• $$F_f(T_{S_1}) = T_{S_2}$$
• $$A(T_{S_1}) > 0 \Rightarrow A(T_{S_2}) > 0$$

3. Functoriality#

Identity#

$$\mathcal{F}(\text{id}S) = \text{id}{T_S}$$

Composition#

$$\mathcal{F}(g \circ f) = \mathcal{F}(g) \circ \mathcal{F}(f)$$

4. Transporter as Functor‑Legal Path#

A transporter event is a morphism $$f : S_1 \to S_2$$ such that:

• $$\mathcal{F}(f)$$ is continuity‑preserving
• $$A(T_{S_1}) > 0 \Rightarrow A(T_{S_2}) > 0$$

This makes transporters:

Functor‑legal, continuity‑preserving substrate transitions with a stable triadic identity.