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Here is the single compact formula for the example: \boxed{ f_{n+1}(x)= \tanh\!\left[ \left(3.7+0.3r_n(x)\right) \sin\!\left( 2^{n+1}\pi x+f_n(x) \right) \right] } with the regularity feedback \boxed{ r_n(x)= \tanh\!\left( \alpha\, \frac{|f_n'(x)|}{1+|f_n''(x)|} \right). } Start with, for example, \boxed{f_0(x)=\sin(2\pi x),\qquad 0\le x\le1.} So the complete system is \boxed{ \begin{cases} f_0(x)=\sin(2\pi x),\\[2mm] r_n(x)= \tanh\!\left( \alpha\dfrac{|f_n'(x)|}{1+|f_n''(x)|} \right),\\[3mm] f_{n+1}(x)= \tanh\!\left[ (3.7+0.3r_n(x)) \sin(2^{n+1}\pi x+f_n(x)) \right]. \end{cases}} This gives boundedness + nonlinear feedback + rapidly increasing scales + potentially chaotic dynamics in one compact construction.
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When a Function Watches Itself: Self-Modifying Regularity

  When a Function Watches Itself: Self-Modifying Regularity What if a mathematical function could examine its own behavior and then use that information to change itself? This leads to a strange idea: a self-modifying function whose regularity becomes part of its definition . Instead of simply calculating a value f(x), the system first asks a deeper question: “What kind of function am I at this point?” The answer then determines what happens next. The Basic Idea Start with a function f:\mathbb Q\rightarrow\mathbb Q. Define a new function by \boxed{ g(x)= \begin{cases} 2f(x), & \text{if }f\text{ is continuous at }x,\\[2mm] f(x), & \text{if }f\text{ is not continuous at }x. \end{cases}} The function is effectively examining its own regularity. If it is continuous at x, it doubles its value. If it is pathological at x, it leaves the value unchanged. This can be written more compactly as g(x)=C_f(x)f(x), where C_f(x)= \begin{cases} 2,&f\text{ is cont...
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