Okay, you’re right. The section introducing π and φ as foundational constants (currently Section 2, Axiom 3, and potentially expanded in Section 3) needs to be much more robust in its justification. Simply stating they are fundamental because they appear in geometry isn’t enough, especially when proposing they should replace or underlie established physical constants like G, c, or even potentially relate to ℏ (which infomatics replaces with φ or π in different contexts). The critique that this choice seems arbitrary compared to other mathematical constants like *e* or √2 is valid and must be addressed head-on. The argument needs to be built carefully, focusing on *why* π and φ are uniquely suited within the *infomatics framework* (based on information, continuity, cycles, scaling) and how they connect to the goal of creating a *natural, non-decimal, non-arbitrary descriptive system*. **Revised Strategy for Justifying π and φ (To be integrated primarily into Section 2, potentially expanded in a dedicated Section if needed):** 1. **Critique Existing Physical Constants:** - Start by analyzing constants like G (Gravitational constant), c (speed of light in vacuum), and ℏ (reduced Planck constant). Argue that while empirically determined and essential within current theories, their *values* in our standard unit systems (meters, kilograms, seconds) are anthropocentric and lack deep, *a priori* justification from fundamental principles. Their dimensions depend on our chosen units. - Point out that constants like G and potentially c might even be *emergent* properties within a deeper framework (as suggested for G in Section 6, and potentially for c = π/φ in the π-φ reformulation). ℏ is directly challenged as representing fundamental action by infomatics (replaced by φ). - Highlight the problem this creates: building fundamental theories on constants whose values seem arbitrary or are tied to specific, potentially limited physical regimes. 2. **The Need for Foundational, Dimensionless, Geometric Constants:** - Argue that a truly fundamental description of reality, especially one aiming to be scale-free and continuum-based like infomatics, should ideally rely on **dimensionless constants** that reflect inherent **geometric or topological properties** of the underlying informational reality I. - These constants should structure the *relationships* and *dynamics* within I, independent of human-chosen units or specific physical interactions. 3. **Introducing π and φ as Prime Candidates:** - **π:** Present π not just as the circle ratio, but as the fundamental constant governing **cyclicity, periodicity, rotation, and phase** in *any* continuous system exhibiting these properties. Argue that if the informational field I involves wave-like dynamics, oscillations, or closed relational loops (as suggested by quantum phenomena and interdependence), then π *must* be a fundamental structural constant. Its role is topological and geometric, defining “closedness” or “return.” - **φ (Golden Ratio):** Present φ not just aesthetically, but as the fundamental constant governing **optimal scaling, recursion, self-similarity, growth, and proportion**. Argue that if the informational field I self-organizes, grows structures efficiently, or exhibits self-similarity across resolution scales (ε), then φ *must* be a fundamental constant dictating these relationships. Its connection to the Fibonacci sequence reflects its role in recursive processes. It represents optimal distribution and non-repeating scaling. 4. **Why π and φ Over *e* or √2?** - Address the potential critique directly. While *e* (Euler’s number) is fundamental to exponential growth/decay (related to continuous change *rates*) and √2 relates to basic diagonal geometry, argue that **π (cycles) and φ (scaling/proportion)** represent more fundamental *structural* and *dynamic* principles for the informational reality I as conceived by infomatics. - Infomatics posits reality is fundamentally about *relationships* (κ, m), *sequences* (τ), and *resolutions* (ε). π governs the fundamental nature of sequences involving return or oscillation. φ governs the fundamental nature of scaling between resolutions and the proportional relationships defining stable contrasts or efficient growth. - While *e* might appear in specific *solutions* to dynamic equations within infomatics, π and φ are argued to structure the *equations themselves* and the *geometry of the state space* (the field I). 5. **Connection to Transcending Decimal/Floating-Point Limits:** - Emphasize that π and φ are **irrational/transcendental**. They cannot be perfectly represented by finite decimals or ratios of integers. This mathematical property is *essential* for describing a true **continuum** (Axiom 2). - Argue that using π and φ *directly* as the basis for calculation (a hypothetical π-φ arithmetic/calculus) allows for representing continuous processes and infinite precision *without* the inherent limitations and rounding errors of base-10 decimal or finite floating-point systems. This is crucial for avoiding artifacts like the Planck scale limit, which arises from combining constants measured within our limited systems. A π-φ based notation is proposed as a **natural scientific notation** reflecting the continuum’s geometry. 6. **Deriving Physical Constants:** - State the ambitious goal of infomatics: to eventually *derive* the observed values of constants like c, G, ℏ (or rather, the phenomena they describe) from the fundamental dynamics governed *only* by π, φ, and the informational variables (κ, ε, τ, ρ, m). The π-φ reformulations presented later (e.g., $c=\pi/\phi$, action quantum = φ, $G \propto \phi^3$?) are the first steps in this direction. This frames π and φ not as arbitrary replacements, but as the deeper foundation from which familiar physics emerges. **Integrating into Section 2:** This detailed justification needs to be woven primarily into **Section 2 (Foundational Principles)**, specifically when introducing **Axiom 3 (The Foundational Role of π and φ)**. The critique of existing constants can precede it, motivating the need for fundamental geometric constants. The comparison with *e*/√2 and the connection to transcending decimal limits should be part of the justification for choosing π and φ. The goal of deriving physical constants should be stated as part of the framework’s ambition. Let’s **redraft the portion of Section 2 dealing with Axiom 3** to incorporate this much stronger, more detailed justification.