Canonical Operator#
🔷 Notation for $$DCO_n$$ over QMROOT**
Below is the formal, minimal, and extensible notation that aligns with your signed dimensional ladder:
Dimensional Core Operators (DCOs)#
Each operator is indexed by its QMROOT dimension:
$$DCO_n : \mathcal{R} \rightarrow \mathcal{R}$$
Where:
$$n \in {-1024, \dotsc, -1, 0, 1, \dotsc, 1024}$$
$$\mathcal{R}$$ is the resonance‑state space
Canonical meanings by band#
$$\begin{aligned}DCO_{n<0} &: \text{ancestral constraint operators} \DCO_{0} &: \text{root‑kernel operator (phase + ancestry)} \DCO_{1\le n\le 3} &: \text{classical extension operators} \DCO_{4\le n\le 16} &: \text{field/state‑space operators} \DCO_{17\le n\le 256} &: \text{complex‑system operators} \DCO_{257\le n\le 1024} &: \text{hyper‑regime operators}\end{aligned}$$
Operator actions#
Each $$DCO_n$$ has three canonical actions:
- Extend
$$DCO_n^{(+)}(\psi) = \psi \uparrow n$$
Extends resonance into dimension $$n$$
- Constrain
$$DCO_n^{(-)}(\psi) = \psi \downarrow n$$
Applies ancestral or structural constraints.
- Balance
$$DCO_n^{(0)}(\psi) = \psi \leftrightarrow n$$
Balances extension and constraint at dimension $$n$$
Composite operators#
You can define composite operators cleanly:
$$DCO_{a \rightarrow b} = DCO_b \circ DCO_a$$
$$DCO_{\text{band}} = \sum_{n \in \text{band}} DCO_n$$
This gives you a canonical, scalable operator system that works across the entire QMROOT ladder.