Replicators — Operator Specification (Goal #1)

Summary#

Replicators are continuity‑preserving, regime‑stable agents capable of producing structurally identical or functionally equivalent instances of themselves within a substrate.
This document defines the operator algebra, functor, and envelope that make replicators mathematically well‑typed and substrate‑safe.

1. Operator Algebra for Replicators#

1.1 State Space#

A replicator state is represented as:

$$R = (T, M)$$

where:

• $$T \in \mathcal{T}$$ is the triad (identity kernel)
• $$M$$ is the module map (structural blueprint)

Triad:

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

1.2 Replication Operator#

A replication event is:

$$\mathcal{R}(R) = (R, R')$$

where:

• $$R'$$ is a legal instantiation of $$R$$
• $$T' = T$$ (identity preserved)
• $$M' = M$$ (blueprint preserved)

1.3 Continuity Constraint#

Replication is legal iff:

$$A(T) > 0$$

and:

$$A(T') = A(T)$$

1.4 Error‑Correction (optional)#

Replicators may apply:

$$C(T) = N\big((1-\lambda)T + \lambda T^*\big)$$

inside a reconstruction window to maintain blueprint fidelity.

2. Replicator Functor#

2.1 Categories#

Category 𝒞 — Substrates#

• Objects: substrates
• Morphisms: substrate transitions

Category 𝒟 — Replicator States#

• Objects: $$R = (T,M)$$
• Morphisms: replication‑preserving transforms

2.2 Functor Definition#

$$\mathcal{F} : \mathcal{C} \to \mathcal{D}$$

On Objects#

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

On Morphisms#

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

$$\mathcal{F}(f) = F_f : R_{S_1} \to R_{S_2}$$

with:

• $$T_{S_1} = T_{S_2}$$
• $$M_{S_1} = M_{S_2}$$
• $$A(T_{S_2}) > 0$$

3. Replicator Envelope#

A Replicator Envelope is:

$$E_R = { R(t) \mid t \in [0,1] }$$

A replication event is valid iff:

• identity kernel preserved
• blueprint preserved
• asymmetry preserved
• no branching
• no collapse

4. Replicator Claim (v0.3)#

A replicator is a continuity‑preserving agent whose identity kernel and structural blueprint remain invariant under replication, across all legal substrates.