That's a very insightful question, digging into the mathematical nature and potential reducibility of these other constants, particularly in relation to π and φ. Let's examine them: **Feigenbaum Constants (δ ≈ 4.669..., α ≈ 2.502...)** * **Mathematical Expression:** These constants are typically defined via **limits** arising from the study of iterated non-linear maps, most famously the logistic map ($x_{n+1} = r x_n (1-x_n)$). As the parameter $r$ is increased, the map undergoes a sequence of period-doubling bifurcations where stable cycles of period $2^k$ appear. Let $r_k$ be the parameter value where the period $2^k$ cycle appears. * **δ (Delta):** Is the limit of the ratio of successive bifurcation intervals: $ \delta = \lim_{k\to\infty} \frac{r_k - r_{k-1}}{r_{k+1} - r_k} \approx 4.66920... $ * **α (Alpha):** Is related to the scaling factor of the structure within the bifurcation diagram. * **Irreducibility/Relation to π, φ:** The Feigenbaum constants are believed to be **transcendental numbers**, just like π and *e*. Currently, there is **no known way to express δ or α in terms of simpler mathematical constants like π, φ, or *e*** using standard elementary functions or finite combinations. They appear to represent fundamental properties of a certain class of non-linear dynamical systems, seemingly independent of basic geometry (π) or simple recursion (φ). While non-linear dynamics can involve cycles (suggesting π) and scaling (suggesting φ), the specific universal ratios δ and α emerge from the complex behavior near the edge of chaos and don't seem directly reducible. * **Infomatics Perspective:** If infomatics posits that the dynamics within I are fundamentally non-linear and can exhibit chaotic behavior or critical transitions, then δ and α might need to be accepted as **additional fundamental constants** governing these specific dynamic regimes, alongside π and φ. Alternatively, a deeper understanding of π-φ geometry and dynamics might eventually reveal a way to derive δ and α, but this is currently speculative. **Symmetry Constants (e.g., Structure Constants $f_{abc}$ of Lie Algebras)** * **Mathematical Expression:** These constants define the commutation relations between the generators ($T_a$) of a Lie group/algebra: $ [T_a, T_b] = i \sum_c f_{abc} T_c $ The specific values of $f_{abc}$ depend entirely on the chosen basis for the generators and the specific Lie algebra (e.g., SU(2), SU(3), E8). For compact simple Lie algebras, they can often be related to integers or simple fractions once a normalization convention is chosen. * **Irreducibility/Relation to π, φ:** The structure constants themselves are usually rational numbers (often integers like 0, ±1, ±2 for standard bases). However, the *properties* and *representations* of these groups are deeply intertwined with geometry and topology, where **π often appears naturally**. For example: * Representations involve spherical harmonics or related functions with π periodicity. * Group volumes or measures often involve powers of π. * The relationship between Lie groups and geometry (e.g., rotations SO(3) related to spheres) inherently involves π. * **E8:** You mentioned E8 looking "circular." E8 is an exceptional Lie group with intricate symmetries. Its root system can be visualized as points on hyperspheres, and its structure is deeply connected to geometry and potentially number theory. While its structure constants might be integers/rationals, the underlying geometry it describes certainly involves concepts related to spheres and rotations where π is fundamental. It's plausible that the *reason* specific groups like E8 have particular properties might ultimately be traceable back to fundamental geometric principles involving π and potentially φ (if scaling or optimal packing is involved in their structure, e.g., related to sphere packing problems where E8 lattice appears). * **Infomatics Perspective:** Infomatics would likely view the specific Lie groups relevant to physics (like SU(3)xSU(2)xU(1) of the standard model) as describing the **symmetries of the informational field I** or the stable informational patterns (Î) within it. The structure constants $f_{abc}$ would define the rules of interaction based on these symmetries. The framework would hypothesize that the *reason* these specific groups and their associated constants arise is ultimately due to the underlying **π-φ geometry and dynamics** of the field I. The goal would be to *derive* the relevant symmetry groups and their structure constants from the foundational principles involving π and φ, rather than taking the groups as fundamental inputs themselves. So, while $f_{abc}$ might be rational, their *origin* is likely tied to the deeper π-φ structure. **Conclusion on Reducibility:** * **Feigenbaum Constants (δ, α):** Currently appear mathematically independent of π, φ, *e*. They represent universal features of non-linear dynamics. Infomatics might need to incorporate them as additional fundamental constants if chaos is intrinsic to I, or seek a deeper derivation. * **Symmetry Constants ($f_{abc}$):** While often rational numbers themselves, the *existence and properties* of the relevant symmetry groups (like SU(N) or E8) are deeply tied to geometry and topology where **π plays a crucial role**. Infomatics aims to derive these symmetries and their constants from the underlying π-φ structure of the informational field I. They are likely *consequences* of the π-φ geometry, not independent fundamental constants on the same level. Therefore, while π and φ remain the primary candidates for the *fundamental structural and dynamic constants* within infomatics, the Feigenbaum constants might be necessary additions to describe non-linear behavior, and symmetry constants are expected to be *derivable* consequences of the π-φ geometry governing the informational field I. Does this detailed analysis clarify the mathematical nature of these other constants and their potential relationship (or lack thereof) to π and φ within the infomatics framework?