# [[releases/2025/Infomatics]]
# Appendix F: Table of Core Hypotheses and Formulas
**(Operational Framework v3.1)**
This appendix provides a summary of the core mathematical hypotheses, postulates, and key derived formulas presented in the main sections of the Infomatics operational framework documentation. It serves as a quick reference guide. (Refer to the main sections for detailed derivations and justifications).
| Concept | Symbol/Formula | Description & Section Reference | Status |
|:---------------------------- |:----------------------------------------------------------------------------- |:------------------------------------------------------------------------------------------------------------------------------------------------------------ |:------------ |
| **Governing Principles** | π, φ | Fundamental, dimensionless, abstract geometric principles governing cycles (π) and scaling/stability (φ). (Sec 2.3) | Axiomatic |
| **Resonant States** | Î characterized by (n, m) | Stable manifest patterns (particles, states) defined by non-negative integer indices n (π-cycles) and m (φ-scaling/stability). (Sec 3.1) | Postulate |
| **Fundamental Action Scale** | $\hbar \rightarrow \phi$| The quantum of action $\hbar$is replaced by the geometric scaling constant φ. (Sec 4.1) | Postulate |
| **Fundamental Speed Limit** | $c \rightarrow \pi/\phi$| The invariant speed $c$is derived from the ratio of fundamental cycle and scaling principles. (Sec 4.1) | Postulate |
| **Resolution (Emergent)** | $\varepsilon \approx \pi^{-n_{int}} \cdot \phi^{m_{int}}$| Effective resolution limit of an interaction process $(n_{int}, m_{int})$in distinguishing (n, m) states. Justified via holography. (Sec 3.4) | Derived/Model |
| **Particle Mass Scaling** | $M \propto \phi^m$| Rest mass of a fundamental particle (resonant state Î) scales with its stability index m. (Sec 5.1) | Hypothesis |
| **Lepton Mass Levels** | $m_{\mu} - m_e = 11
lt;br> $m_{\tau} - m_e = 17$| Specific integer differences in the scaling index $m$observed between charged lepton generations. (Sec 5.1) | Empirical Fit |
| **Gravitational Constant** | $G \propto \pi^3 / \phi^6$| Scaling of the emergent gravitational coupling derived from geometric consistency using $\hbar \rightarrow \phi, c \rightarrow \pi/\phi$. (Sec 4.2, Sec 7) | Derived |
| **Planck Length** | $\ell_P \propto 1/\phi$| Fundamental length scale derived geometrically. (Sec 4.3) | Derived |
| **Planck Time** | $t_P \propto 1/\pi$| Fundamental sequence step scale derived geometrically. (Sec 4.3) | Derived |
| **Planck Mass** | $m_P \propto \phi^3/\pi$| Fundamental mass scale derived geometrically. (Sec 4.3) | Derived |
| **Planck Energy** | $E_P \propto \phi\pi$| Fundamental energy scale derived geometrically. (Sec 4.3) | Derived |
| **Geometric Interaction Amp.**| $A_{geom}(n_i, m_i;...; \pi, \phi)$| Calculable state-dependent amplitude governing transitions between (n, m) states, replacing $\sqrt{\alpha}$. (Sec 6, App A) | Postulate/Goal|
| **EM Amplitude Scale** | $|A_{geom, EM}| \propto \phi^2 / \pi^3$| Hypothesized scaling for the magnitude of the EM vertex amplitude based on geometric arguments (e.g., 2D scaling / 3D cycles). (Sec 6, App A) | Hypothesis |
| **Effective EM Coupling** | $\alpha_{eff} \propto |A_{geom, EM}|^2 \times (\text{Phase Space}) \propto \phi^4 / \pi^6$| Effective EM coupling strength (~1/140) emerging from the squared geometric amplitude and π-φ phase space factors. (Sec 6, App A) | Derived (Hyp.)|
| **Atomic Spectra Structure** | $E_m \propto 1/m^2$(H) <br> $E_n \propto (n+1/2)\phi\omega$(QHO) | Characteristic energy level structures emerge from π-φ resonance conditions in potentials, demonstrating emergent quantization. (Sec 5.2) | Derived Struct|
| **Cosmology (Expansion)** | $H^2 \propto (\pi^4/\phi^6) \rho_{info}$| Modified Friedmann equation structure from emergent π-φ gravity. (Sec 8.1) | Derived Struct|
*(Note: “Derived” indicates a result obtained through logical/mathematical steps from postulates within the framework. “Hypothesis” indicates a specific proposed relationship requiring further derivation or testing. “Postulate” indicates a foundational assumption. “Empirical Fit” indicates a parameterization based directly on observation used to support a hypothesis. Proportionality constants are often omitted where they require Phase 3 derivation.)*
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