Math — Fractional Memory
Fractional Memory with Shock Bounds
We couple Caputo fractional memory with a shock‑aware outer loop so
history isn’t diluted during extremes. Instead, stability matters more and
shocks are encoded as bound events that trigger guardrails and mode switches.
Version: 0.18.018
Built: 2025-08-13T20:14:08.142255Z
Core Update
For a signal \(x(t)\) with baseline operator \(\mathcal{B}\), inner fractional memory:
\[
\mathcal{M}_\alphax = {}^C D_t^\alpha x(t), \quad \alpha\in(0,1)
\]
Outer shock loop detects excursions when \(|x(t) - L^\pm(t)| > \tau\), emitting a Bound Event:
\[
\mathcal{S}x = \mathbf{1}_{\{|x - L^\pm|>\tau\}} \cdot \mathrm{Return}(\cdot)
\]
During shocks, weighting increases on stable references; history gets more influence, not less.
Guardrails (dynamic) depend on memory depth and slope of change within that depth.