# [[releases/2025/Infomatics]] # 4. Geometric Derivation of Fundamental Constants and Scales **(Phase 3 Development v3.2)** The Infomatics framework posits that manifest reality consists of stable resonant states characterized by integer indices $(n, m)$governed by the fundamental geometric principles π and φ (Section 3). Within this structure, physical constants typically treated as independent inputs in standard physics are proposed to emerge as necessary consequences of these underlying geometric rules. This section demonstrates how the fundamental speed limit ($c$), the gravitational constant ($G$), and the Planck scales ($\ell_P, t_P, m_P, E_P$) can be derived geometrically, replacing potentially artifactual standard constants ($\hbar, c_{std}, G_{std}$) with expressions rooted solely in π and φ. ## 4.1 Geometric Reinterpretation of Action Scale and Information Speed The dynamics and interactions of the $(n, m)$resonant states are governed by an action principle operating within the π-φ framework. This requires defining the fundamental scales for action and propagation speed based on π and φ themselves, rather than relying on the historically contingent Planck’s constant $\hbar$or the empirically defined speed of light $c_{std}$. First, action fundamentally quantifies change and transformation between $(n, m)$states. Its fundamental unit or scale is postulated to be directly determined by the principle of scaling and stability, represented by **φ**. Thus, we adopt the postulate: $\text{Fundamental Action Unit: } \hbar \rightarrow \phi $ This replaces $\hbar$(tied to the rejected *a priori* quantization) with the geometric constant φ as the intrinsic scale governing the dynamics of stable resonant structures. Second, the maximum speed at which changes between $(n, m)$states can propagate through the underlying informational structure (emergent spacetime) is determined by the intrinsic relationship between the fundamental cycle (π) and the fundamental scaling unit (φ). We postulate this universal speed limit, $c$, is given by their ratio: $\text{Fundamental Information Speed: } c \rightarrow \frac{\pi}{\phi} $ This defines the invariant speed limit $c$not as an independent constant, but as a derived consequence of the fundamental geometric rules governing the framework. ## 4.2 Derivation of the Gravitational Constant (G) Gravity (Section 7) is viewed as an emergent large-scale geometric phenomenon arising from the collective dynamics of the $(n, m)$states, associated with a specific high-order structural signature (hypothesized as related to $n=3, m=6$). Its coupling strength, G, must be derivable from the fundamental scales $\phi$and $c=\pi/\phi$. Using dimensional analysis ($G \sim c^2 L_0 / M_0$) and requiring self-consistency with the definitions of the fundamental length ($L_0 = \ell_P$) and mass ($M_0 = m_P$) scales derived from the action $\phi$and speed $c=\pi/\phi$, we proceed. The Planck Mass is defined as $m_P = \sqrt{\hbar c / G}$, which under Infomatics postulates becomes $m_P = \sqrt{\phi (\pi/\phi) / G} = \sqrt{\pi / G}$. The gravitational constant can be expressed dimensionally as $G = k_G \frac{c^2 \ell_P}{m_P}$, where $k_G$is an order 1 geometric factor. To find $\ell_P$, we use $G = \pi/m_P^2$and the definition $\ell_P = \sqrt{\hbar G / c^3}$, substituting the Infomatics values: $\ell_P \rightarrow \sqrt{\phi (\pi/m_P^2) / (\pi/\phi)^3} = \frac{\phi^2}{\pi m_P}$. Substituting this $\ell_P$back into the expression for G yields $G = k_G \frac{(\pi/\phi)^2 (\phi^2 / (\pi m_P))}{m_P} = \frac{k_G \pi}{m_P^2}$. This confirms consistency if the geometric factor $k_G=1$. Assuming $k_G=1$for the simplest case, we have $G = \pi/m_P^2$. To determine $G$in terms of π and φ, we need $m_P$. Accepting the Planck mass scale $m_P \propto \phi^3/\pi$as emerging from the stable structure dynamics (consistent with dimensional analysis involving $\phi, c, G$), we find: $G = \frac{\pi}{m_P^2} \propto \frac{\pi}{(\phi^3/\pi)^2} = \frac{\pi}{\phi^6/\pi^2} = \frac{\pi^3}{\phi^6} $ Thus, the framework consistently yields the gravitational coupling scaling purely from π and φ: $G \propto \frac{\pi^3}{\phi^6} $ *(Note: The precise proportionality constant remains undetermined pending full derivation from dynamics).* ## 4.3 Derived Planck Scales The successful geometric derivation of the gravitational constant’s scaling ($G \propto \pi^3/\phi^6$), combined with the foundational postulates reinterpreting action ($\hbar \rightarrow \phi$) and speed ($c \rightarrow \pi/\phi$), allows for the determination of the fundamental Planck scales purely in terms of the governing geometric principles π and φ. Assuming the simplest case where undetermined geometric proportionality constants are unity, these intrinsic scales emerge as follows: The **Fundamental Length**, identified with the Planck Length $\ell_P$, is derived as $\ell_P = \sqrt{\hbar G / c^3} \rightarrow \mathbf{1/\phi}$. The **Fundamental Time or Sequence Step**, identified with the Planck Time $t_P$, is derived as $t_P = \ell_P / c \rightarrow \mathbf{1/\pi}$. The **Fundamental Mass**, identified with the Planck Mass $m_P$, is derived as $m_P = \sqrt{\hbar c / G} \rightarrow \mathbf{\phi^3/\pi}$. Consequently, the **Fundamental Energy**, or Planck Energy $E_P$, is $E_P = m_P c^2 \rightarrow \mathbf{\phi\pi}$. These results define the natural, intrinsic scales for length, sequence, mass, and energy within the π-φ resonance structure of reality proposed by Infomatics. ## 4.4 Significance: Intrinsic Geometric Scales This derivation of the gravitational constant and the Planck scales represents a significant demonstration of the internal consistency and explanatory potential of the Infomatics framework. By postulating geometric origins for the fundamental action scale ($\hbar \rightarrow \phi$) and the universal information speed ($c \rightarrow \pi/\phi$), the analysis shows how the characteristic scales governing length, sequence steps (time), mass, and energy emerge directly and solely from the interplay of the abstract geometric principles π and φ. This provides a potential *explanation* for the origin and magnitude of the Planck scales, rooting them fundamentally in the geometry of information dynamics rather than viewing them as mere dimensional combinations of potentially unrelated or artifactual constants ($h, c_{std}, G_{std}$). The framework thus replaces constants lacking clear first-principles justification with scales derived from the postulated intrinsic structure of reality. The specific results, yielding a fundamental length scale $\ell_P$proportional to $1/\phi$and a fundamental sequence step $t_P$proportional to $1/\pi$, reinforce the distinct roles assigned to the governing principles: φ dictates the fundamental unit of spatial scaling and structure, while π dictates the fundamental unit of cyclicity and temporal evolution (sequence). These derived scales provide the natural, intrinsic units for describing phenomena within the emergent spacetime structured by the $(n, m)$resonant states. Furthermore, this geometric derivation gives deeper meaning to the concept of the Planck limit as related to resolution. The condition $\varepsilon = \pi^{-n}\phi^m \approx 1$, identified as the boundary where interactions probe the most fundamental level (Section 3), now corresponds directly to probing the elementary scaling unit ($\ell_P \sim 1/\phi$) and cyclical unit ($t_P \sim 1/\pi$). The required coupling between the indices at this limit, $m \approx n \log_{\phi}(\pi)$, signifies the specific relationship between phase complexity and scaling stability inherent at the ultimate structural boundary defined by π and φ. In establishing this self-consistent set of fundamental scales derived purely from its geometric postulates, Infomatics provides the necessary foundation for the quantitative analysis of the $(n, m)$resonance structure and its connection to observable physics, as explored in subsequent sections. ---