Technical Whitepaper
The Double Sigmoid Mencius Function (DSMF)
Recursive Fractal Contextualization in High-Velocity Snap Engines
1. Abstract
This paper introduces the Double Sigmoid Mencius Function (DSMF), a novel mathematical architecture that utilizes recursive nesting—specifically micro-sigmoids embedded within larger sigmoid curves—to capture multi-scale data relationships. By integrating fractional derivatives and quantum state vectors (|ψ⟩), the DSMF enables a classical-to-quantum bridge that maps simultaneous correlational and causal relationships within a single O(1) operation.
2. The Primary Hypothesis: Fractal Recursion
Traditional activation functions (like the standard sigmoid or ReLU) flatten data into a singular, piecewise transition. The DSMF rejects this flattening, proposing that each point within a sigmoid curve is itself composed of a micro-sigmoid.
The Formalism:
γ (Recursion Level)
Represents Causation (sequential dependencies).
α (Fractional Derivative Order)
Represents Correlation (simultaneous movements).
β (Aperture Method)
Acts as a "zoom lens," scaling input to reveal finer micro-sigmoid details.
3. The Hidden Context Layer
The innovation of the "micro-sigmoid within a sigmoid" is that it provides a container for hidden context.
The Macro Curve:
Handles the primary state transition (e.g., from Noise to Signal).
The Micro-Sigmoid:
Captures the "Refractive Jitter"—the high-dimensional context that occurs during the transition—which classical models ignore as error.
4. The Snapdragon Integration
Project SNAPDRAGON uses the DSMF to handle the "Judgment" phase of the data snap.
Bit-Shift (Snapdragon):
The O(1) hardware shift (0x5f) collapses the 4D manifold.
Contextual Hold (DSMF):
The recursive sigmoid layer holds the "hidden" metadata (the correlation strength α and aperture β).
The Result:
A deterministic binary output that is "Quantum-Informed"—it knows why it snapped because it contained the contextual sub-states within its own fractal structure.