Unified Operator Grammar v1.0
TriadicFrameworks — Canonical Operator Grammar#
Summary#
This grammar unifies all operators across Goals #1, #2, and #3 into a single typed system.
It defines the legal forms, transforms, and continuity rules for identity‑preserving operations across all substrates.
1. Core Types#
1.1 Triad#
$$ T = (s,c,u),\quad s+c+u=1 $$
1.2 Asymmetry#
$$ A(T)=0.01 $$
1.3 Extended State#
$$ S = (T, X) $$
where $$X$$ is:
- blueprint $$M$$ (Goal #1)
- substrate $$S_i$$ (Goal #2)
- environment $$E$$ (Goal #3)
2. Core Operators#
2.1 Continuity Operator#
$$ O(T) = (T, A(T)) $$
2.2 Replication Operator (Goal #1)#
$$ \mathcal{R}(T,M) = (T,M') $$
with $$M'=M$$
2.3 Transport Operator (Goal #2)#
$$ \mathcal{T}(T,S_1 \to S_2) = (T,S_2) $$
2.4 CT Operator (Goal #3)#
$$ \mathcal{C}(T,E) = (T,E') $$
3. Functorial Structure#
Categories#
- 𝒞: substrates
- 𝒟₁: replicator states
- 𝒟₂: transporter states
- 𝒟₃: CT states
Functors#
$$ \mathcal{F}_1 : \mathcal{C} \to \mathcal{D}_1 $$
$$ \mathcal{F}_2 : \mathcal{C} \to \mathcal{D}_2 $$
$$ \mathcal{F}_3 : \mathcal{C} \to \mathcal{D}_3 $$
All preserve:
- identity
- asymmetry
- continuity
4. Envelopes#
Replicator Envelope#
$$ E_R = {(T,M)(t)} $$
Transporter Envelope#
$$ E_T = {(T,S(t))} $$
CT Envelope#
$$ E_C = {(T,E(t))} $$
All require:
- no branching
- no duplication
- no collapse
- reconstruction window allowed only for CTs
5. Composition Rules#
Sequential Composition#
$$ \mathcal{C} \circ \mathcal{T} \circ \mathcal{R} $$
is legal iff:
- triad preserved
- asymmetry preserved
- reconstruction windows converge
6. Arrival Substrate#
Arrival substrate is the canonical fixed point:
$$ T \approx T^*,\quad A(T)=0.01 $$
All operators converge here.
Claim#
Unified Operator Grammar v1.0 provides the first complete, typed, continuity‑safe operator system spanning replication, transport, and CT instantiation.