Adaptations for Constants

System Boundary (B):

In the case of π and ϕ, the "System Boundary" could be defined as the mathematical system in which these constants are defined or the physical systems where they manifest. For example, for π, this could range from simple geometrical shapes to complex phenomena like waveforms.

Quantization Limit (Q):

These constants themselves could be seen as natural "quantization limits" in certain systems. For instance, π serves as a limit in the ratio of a circle's circumference to its diameter, irrespective of the circle's size. Similarly, the golden ratio serves as a fixed point in certain growth processes and aesthetic proportions.

Spillover Effect (S):

The ubiquity of these constants in various systems could be seen as a "spillover effect." Their presence in unrelated domains might indicate some form of informational or structural spillover.

Adjacent or Nested Systems (A):

These constants appear in both nested systems (like Russian dolls of golden rectangles) and adjacent systems (like π appearing in both geometry and quantum mechanics).

Feedback Loop (F):

The recursive nature of these constants (e.g., π in iterative algorithms or ϕ in recursive sequences like the Fibonacci series) could be seen as a natural feedback loop.

Observer's Frame of Reference (O):

The observer effect here could be the mathematical tools we use to calculate these constants. For example, π and ϕ are irrational numbers, and their complete decimal expansion is not possible. The "resolution" with which we understand these constants is limited by computational resources.

Theorem Statement Adaptation:

For any given system with a boundary influenced by constants like π or ϕ, these constants serve as natural quantization limits. Their ubiquity across various systems can be considered a spillover effect, which might initiate new cycles in adjacent or nested systems, facilitated by feedback loops inherent in the mathematical properties of these constants. The perception and computation of these constants are subject to the limitations imposed by the observer's frame of reference.

This adaptation allows the theorem to explore the cyclic nature and ubiquity of constants like π and ϕ across different systems and scales, providing a unified framework that can potentially offer new insights into why these constants appear so universally.