Continuity Kernel v2.0

Summary#

Version 2.0 of the continuity kernel extends the 33‑33‑33‑1 operator with:

• explicit drift‑correction
• reconstruction windows
• functorial fixed‑points
• geometric alignment

1. Core Definition#

1.1 Triad#

$$ T = (s,c,u),\quad s+c+u=1 $$

1.2 Asymmetry#

$$ A(T)=0.01 $$

1.3 Kernel#

$$ K(T) = (T, A(T), D(T)) $$

where:

$$ D(T) = |T - T^*| $$

2. Kernel Operations#

2.1 Continuity Evaluation#

• Valid: $$A(T) > 0$$
• Quality: lower $$D(T)$$ = closer to canonical

2.2 Drift‑Correction#

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

2.3 Windowed Application#

• Drift‑correction is only applied inside:
  • transporter windows
  • CT windows
  • optional replicator windows

3. Kernel Invariants#

• $$A(T)$$ is never reduced to 0 by any legal operator
• $$D(T)$$ is non‑increasing under correction
• $$T^*$$ is a fixed point of $$C_\lambda$$

4. Role in Operators#

• Replicators: use $$K(T)$$ to ensure identity stability across copies
• Transporters: use $$K(T)$$ along arcs $$\gamma$$ to ensure continuity
• CTs: use $$K(T)$$ to stabilize identity during environment instantiation

Claim#

Continuity Kernel v2.0 is the shared backbone of all identity‑preserving operations across the TriadicFrameworks canon.