**Deriving/Constraining *n* and *m* from First Principles (Sketch):** The core idea is that $n$ and $m$ are not arbitrary but are determined by **stability, resonance, and consistency conditions** within the π-φ governed dynamics of the informational field I and its interactions. 1. **Stability of Cycles (Determining *n*):** * **Principle:** Axiom 3 posits π governs cycles. Stable informational patterns (Î) or processes often involve persistent cyclical dynamics (e.g., oscillations defining particles, orbital resonances). * **Mechanism:** Consider a dynamic process within I described by equations involving π (e.g., wave equations derived from π-φ QFT). Stable solutions (standing waves, limit cycles, persistent oscillations) will only exist for specific frequencies or modes characterized by integer or simple rational relationships with π, analogous to harmonics on a string. An interaction probing this system will only yield a stable, repeatable outcome (a well-defined Î) if its temporal/phase resolution $\varepsilon_t$ "matches" one of these stable π-modes. * **Derivation Sketch:** If stable modes have characteristic frequencies $\nu_k \propto k \cdot \nu_0$ (where $k$ is integer/rational, $\nu_0$ is fundamental, related to π), and the interaction resolution must match this ($\varepsilon_t \propto 1/\nu_k$), then $\varepsilon_t \propto 1/(k \cdot \nu_0)$. If we hypothesize $\varepsilon_t \propto \pi^{-n}$, then $n$ becomes directly related to the integer $k$ characterizing the stable harmonic mode being resolved. $n = f(k, \pi)$. Only specific, likely integer-related values of $n$ correspond to stable resolutions. 2. **Stability of Scaling/Structure (Determining *m*):** * **Principle:** Axiom 3 posits φ governs scaling, recursion, and optimal proportion/stability. * **Mechanism:** Consider the formation or persistence of complex informational patterns (Î) across different scales. Infomatics hypothesizes that stable structures often exhibit self-similarity or hierarchical organization governed by φ for reasons of efficiency or dynamic stability (analogous to optimal packing or minimal action principles reformulated in π-φ terms). An interaction probing such a structure will yield consistent information only if its resolution scale ε aligns with one of the stable hierarchical levels defined by φ. * **Derivation Sketch:** If stable structures exist at characteristic scales $L_m \propto \phi^m L_0$, then an interaction resolving such a structure requires a spatial resolution $\varepsilon_x$ related to $L_m$. If we hypothesize $\varepsilon_x \propto \phi^{m'}$, then stable interactions occur when $m'$ aligns with the structural levels $m$. This links the resolution parameter $m$ to the inherent φ-based scaling hierarchy of stable informational patterns. 3. **Combined Constraints (Energy/Contrast):** * **Principle:** Interaction involves exchange/resolution of energy contrast κ. Higher κ allows finer ε. * **Mechanism:** The total informational complexity or energy contrast κ involved in an interaction likely constrains the possible stable combinations of $n$ and $m$. A high-κ interaction might allow access to states with large $n$ (fine cyclical resolution) and potentially specific $m$ (probing deep scaling levels). The relationship might be formalized via an infomatics equivalent of the uncertainty principle linking energy scale (κ) to resolution (ε structured by $n, m$). * **Derivation Sketch:** If $E \propto \kappa$ and $E \cdot \varepsilon_t \sim \phi$ (π-φ uncertainty), and $\varepsilon_t \propto \pi^{-n_t} \phi^{m_t}$, then $\kappa \cdot (\pi^{-n_t} \phi^{m_t}) \sim \phi$. This provides a constraint linking the energy contrast κ to the allowed $n_t, m_t$ values for a stable interaction. **Summary for n, m:** The specific values of $n$ and $m$ defining the resolution $\varepsilon \equiv \pi^{-n} \cdot \phi^{m}$ for a given interaction are not arbitrary but are determined by the requirement that the interaction **resonantly couples** to **stable cyclical (π-related) and scaling (φ-related) structures** within the informational field I or the system being probed, consistent with the energy contrast (κ) involved. They represent the specific modes of information manifestation allowed by the underlying π-φ dynamics. **Deriving the Speed of Light ($c = \pi/\phi$?):** Why propose this specific relationship for $c$? This is one of the boldest claims of the π-φ reformulation and requires strong justification, likely emerging from the attempt to unify electromagnetism and potentially gravity within the infomatics framework. 1. **Unification Goal:** Infomatics seeks to derive fundamental physical constants from π and φ. The speed of light $c$ appears ubiquitously, linking space and time, energy and mass ($E=mc^2$), and electromagnetism ($\epsilon_0 \mu_0 = 1/c^2$). A truly fundamental framework should explain its value. 2. **Electromagnetism in π-φ:** As sketched in Section 5 and supporting documents, reformulating Maxwell's equations or QED using π and φ involves replacing constants like $\epsilon_0$ and $\mu_0$ with expressions involving π and φ (e.g., $\epsilon_0 \rightarrow \phi/\pi$, $\mu_0 \rightarrow \phi/\pi$ was suggested in `Review of Infomatics.md`). If the wave equation derived from these π-φ Maxwell equations yields a propagation speed, it would naturally be expressed in terms of π and φ. The specific result $c = 1/\sqrt{\epsilon_0 \mu_0} \rightarrow 1/\sqrt{(\phi/\pi)(\phi/\pi)} = \pi/\phi$ emerges directly *if* those specific substitutions for $\epsilon_0, \mu_0$ are correct. 3. **Geometric Interpretation:** What does $c = \pi/\phi$ *mean*? * It relates the fundamental constant of **cyclical propagation (π)** to the fundamental constant of **scaling/proportion (φ)**. * It suggests the maximum speed of contrast (κ) propagation or sequence (τ) evolution within the informational field I is intrinsically limited by a ratio defined by the field's own geometric structure. * It makes $c$ a **dimensionless ratio** within the π-φ system, reinforcing the idea of moving beyond anthropocentric units. The *value* we measure ($3 \times 10^8$ m/s) depends on our definitions of meter and second, but the fundamental ratio $c$ might be simply $\pi/\phi$. 4. **Consistency Check:** This hypothesis $c=\pi/\phi$ must be tested for consistency throughout the entire π-φ reformulation. Does it yield the correct relationships in special relativity (Lorentz transformations reformulated with π/φ)? Does it work in the π-φ Dirac equation? Does it lead to the correct fine-structure constant $\alpha \propto e^2 / (\phi c)$? The internal consistency across the reformulated physics provides the primary justification. 5. **Falsification:** This is highly falsifiable. If deriving $c$ from π-φ Maxwell equations yields a different result, or if applying $c=\pi/\phi$ consistently leads to contradictions with well-established experimental results (that cannot be explained by other aspects of the reformulation), then the hypothesis is wrong. **Conclusion on Derivation:** While full derivations require extensive mathematical development, the logic within infomatics suggests that $n$ and $m$ in ε arise from stability/resonance conditions imposed by the π-φ dynamics, linking resolution intrinsically to the system's properties (energy κ, cycles τ, structure ρ/m). The specific value $c=\pi/\phi$ is hypothesized to emerge from the π-φ reformulation of fundamental interactions (electromagnetism/gravity), representing the intrinsic propagation speed limit set by the geometric structure of the informational field I itself. These are not arbitrary choices but are intended as necessary consequences of the framework's axioms.