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emo-v36-paper(ENG).txt

@@ -80,7 +80,57 @@ Efficiency improvements for EmoNavi, EmoFact, and EmoLynx:
 
 Supplementary Material (2): Formal Proof of emoDrive Boundedness
 
-(Sections 1-3: Mathematical evaluation of Normal, Acceleration, and Emergency zones)
+1. Objective
+
+We prove that emoDrive, which applies a dynamic correction to the learning rate in the EmoNAVI update rule, possesses both upper and lower bounds at any step t. This guarantees that the update magnitude \Delta w_t does not explode and that the convergence conditions are satisfied.
+
+2. Lemma: Boundedness of the Emotional Scalar \sigma _t
+
+The emotional scalar in EmoNAVI takes the form
+\sigma _t=\tanh (x).
+From the properties of the hyperbolic tangent function, the following holds for any input x\in \mathbb{R}:
+-1<\sigma _t<1.
+Therefore, the absolute value |\sigma _t| always lies within the interval [0,1).
+
+3. Theorem: Proof of the Boundedness of emoDrive
+
+Based on the implementation code (v3.6.1), the definition of emoDrive is evaluated by dividing it into the following three regions:
+
+(A) Normal Zone (No Intervention Zone):
+|\sigma _t|\leq 0.25\quad \mathrm{or}\quad 0.5<|\sigma _t|\leq 0.75
+In this region, according to the implementation, the value is:
+\mathrm{emoDrive}=1.0.
+
+(B) Acceleration Zone (emoDrive Active Region):
+0.25<|\sigma _t|<0.5
+In this region, emoDrive is defined as:
+\mathrm{emoDrive}=\mathrm{emoDpt}\times (1.0+0.1\cdot \mathrm{trust}),
+where
+\mathrm{emoDpt}=8.0\times |\mathrm{trust}|,
+and trust is the signed value of (1.0-|\sigma _t|).
+- Evaluation of |\mathrm{trust}|:
+For |\sigma _t|\in (0.25,0.5), we have
+|\mathrm{trust}|\in (0.5,0.75).- Range of emoDpt:
+8.0\times 0.5<\mathrm{emoDpt}<8.0\times 0.75- hence
+4.0<\mathrm{emoDpt}<6.0.- Overall evaluation:
+The factor 1.0+0.1\cdot \mathrm{trust} lies within the range 0.9 to 1.1 regardless of the sign of trust.
+Therefore, the maximum value B_{\mathrm{up}} in the acceleration zone satisfies:
+B_{\mathrm{up}}<6.0\times 1.1=6.6.
+
+(C) Emergency Zone (Rapid Braking Zone):
+|\sigma _t|>0.75
+In this region,
+\mathrm{emoDrive}=\mathrm{coeff},
+where
+\mathrm{coeff}=1.0-|\mathrm{scalar}|.
+Since |\sigma _t|\in (0.75,1.0), the minimum value B_{\mathrm{low}} satisfies:
+0<B_{\mathrm{low}}\leq 0.25.
+
+4. Conclusion
+From the above evaluations, we have proven that emoDrive satisfies the following boundedness condition in all regions:
+0<(1-|\sigma _{\max }|)\leq \mathrm{emoDrive}\leq 6.6.
+(Even when |\sigma _t| approaches 1, implementation details such as eps ensure that a small positive value is maintained.)
+The existence of this bounded multiplicative coefficient provides the mathematical foundation that allows EmoNAVI to retain the Adam‑type convergence rate O(1/T) while achieving constant‑factor acceleration.
 
 In summary, EmoNAVI encapsulates three forms of "intelligence" within a single update loop:
 
@@ -91,7 +141,7 @@ In summary, EmoNAVI encapsulates three forms of "intelligence" within a single u
     Action Intelligence (emoDrive): Autonomously decides the "step-size" based on the judgment, similar to COCOB or D-adaptation.
 
 
-6. References
+References
 
     Kingma, D. P., & Ba, J. (2015). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.  
 

emo-v36-paper(JPN).txt

@@ -128,7 +128,7 @@ emoDrive=1.0
   行動の知能 (emoDrive): 判定に基づき、COCOB や D-adapt のように「歩幅（Step-size）」を自律的に決定する。
 
 
-6. 参考文献 (References)
+参考文献 (References)
 
 Kingma, D. P., & Ba, J. (2014). Adam: A Method for Stochastic Optimization.