Lab 6: Dimensional Nested Loops — Equations

Purpose#

To formalize the mathematical architecture behind dimensional nesting and triadic loop resonance, as explored in the lab and supported by Resonant Time Operationalizing Zhang’s Triadic Ontology.

Key Equations#

1. Dimensional Loop Depth#

$$ D_n = \sum_{i=1}^{n} T_i $$ Where ( T_i ) is the time resonance at dimension ( i ), and ( D_n ) is the cumulative loop depth.

2. Triadic Resonance Operator#

$$ R_{ijk} = @\left(D_i, D_j, D_k\right) $$ The @() operator formalizes resonance across three nested dimensions.

3. Nested Loop Feedback#

$$ F_n = R_{n,n-1,n-2} - R_{n-3,n-4,n-5} $$ Used to detect phase shifts and cognitive emergence.

4. Dimensional Echo Function#

$$ E(t) = \int_{0}^{t} R_{ijk} \cdot \sin(\omega t) , dt $$ Models the echo of nested loops over time.

Symbol Legend#