**Section 3: Core Components of Fundamental Entities** Having identified the foundational entities comprising the initial corpus (Section 2), the next crucial step towards constructing our network map involves dissecting each entity into its **core components**. This process moves beyond broad definitions to expose the internal structure, explicit formulations (laws, equations, postulates), and, significantly, the often **implicit assumptions** (ontological, epistemological, methodological) upon which each paradigm rests. Representing these components as distinct nodes allows for a more granular and insightful analysis of the relationships between fundamental ideas. The extraction methodology involves careful analysis of authoritative descriptions and foundational texts for each entity, aiming for clarity and objectivity. Beginning with **classical mechanics (CM)**, its explicit formulations are dominated by **Newton’s laws of motion**: the first law (inertia) stating that bodies maintain their state of motion unless acted upon by a force; the second law (force law) quantifying the relationship between force, mass, and acceleration (**F** = m**a**); and the third law (action-reaction) asserting the equality and opposition of interacting forces. Often implicitly included, especially in celestial mechanics, is the **Law of Universal Gravitation** ($F_g = G M_1 M_2 / r^2$), though this is superseded by general relativity. Underlying these laws are critical implicit assumptions. Ontologically, CM presupposes an **absolute space** (Euclidean, fixed) and **absolute time** (uniform, independent), providing an unchanging background. It often employs idealizations like **point masses** or rigid bodies and assumes matter possesses intrinsic properties like mass. Epistemologically, CM assumes **determinism**, where the complete state (all positions **x** and momenta **p**) at one time dictates all future states via the laws, and assumes physical properties possess **objective reality** independent of measurement. Methodologically, it focuses on forces as agents of change and utilizes the calculus of differential equations. **General relativity (GR)** presents a different set of components. Its explicit formulations are grounded in fundamental principles: the **Equivalence Principle**, asserting the local indistinguishability of gravity and acceleration, and the **Principle of General Covariance**, requiring physical laws to maintain their form under arbitrary coordinate transformations. The core mathematical structure is postulated as a **four-dimensional pseudo-Riemannian spacetime manifold** $(M, g_{\mu\nu})$. The motion of free particles is governed by the **Geodesic Motion Postulate**, stating they follow the “straightest” paths in this curved geometry. The central law is expressed in **Einstein’s field equations (EFE)**: $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$, quantitatively linking spacetime curvature ($G_{\mu\nu}$) to the distribution of mass and energy ($T_{\mu\nu}$), potentially including the cosmological constant Λ. Implicitly, GR assumes spacetime is a **dynamic physical entity**, not a passive background, and typically treats it as a **continuum** (though this breaks down at the Planck scale). Ontologically, it assumes mass-energy sources curvature. Epistemologically, it assumes geometry is empirically accessible and that physical laws are fundamentally geometric. Methodologically, it relies heavily on **differential geometry and tensor calculus** within a field equation approach. The standard interpretation of **quantum mechanics (qm)** is often presented via postulates governing its mathematical formalism and connection to observation. The **State Postulate** asserts that a system’s state is fully represented by a normalized vector $|\psi\rangle$in a complex Hilbert space $H$. The **Observable Postulate** associates measurable quantities with Hermitian operators $A$on $H$. The **Dynamics Postulate** describes deterministic, unitary evolution between measurements via the **Schrödinger equation**: $i\hbar \frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle$. The **Measurement Postulate**, incorporating the **Born rule**, states that measurement outcomes are eigenvalues $a_i$of $A$, obtained with probability $P(a_i) = |\langle a_i|\psi\rangle|^2$. The **Projection Postulate** describes the discontinuous “collapse” of the state vector to the corresponding eigenstate $|a_i\rangle$upon measurement. Underlying these are significant interpretive assumptions, often associated with the Copenhagen view: ontologically, micro-reality is described by $|\psi\rangle$, properties may lack definite values before measurement (**contextuality**), and non-local connections (**entanglement**) exist. Epistemologically, there’s a fundamental limit to knowledge (**Uncertainty Principle**), measurement is inherently disruptive and probabilistic (**Intrinsic Indeterminism**), and classical language is needed for describing experiments (**Complementarity**). Methodologically, the focus is on measurement probabilities using Hilbert space linear algebra. The **Standard Model (SM)** of particle physics integrates qm with special relativity into a quantum field theory (QFT). Its explicit components include defining the fundamental **particle content** (quarks, leptons, gauge bosons, Higgs boson) and asserting the **Gauge Symmetry Principle**, which dictates that interactions are governed by the symmetry group $SU(3)_C \times SU(2)_L \times U(1)_Y$. The specific dynamics and interactions are encoded in the **Lagrangian density**, a complex mathematical expression consistent with these symmetries. Mass generation for fundamental particles (except neutrinos, typically) is explained via the **Higgs mechanism**. Implicitly, the SM assumes the ontological reality of **quantum fields** as fundamental, with particles as their excitations, and the validity of the specific gauge symmetries. Epistemologically, it relies on **QFT** as the descriptive framework and utilizes **perturbation theory** (Feynman diagrams) and **renormalization** techniques for calculations. Methodologically, it is deeply rooted in gauge symmetry principles. The **Laws of Thermodynamics** govern energy transformations in macroscopic systems. Explicitly, the **Zeroth Law** defines thermal equilibrium and temperature. The **First Law** states the conservation of energy ($\Delta U = Q - W$). The **Second Law** mandates that the total entropy of an isolated system never decreases ($\Delta S_{\text{univ}} \ge 0$), establishing a macroscopic arrow of time. The **Third Law** states that entropy approaches a constant minimum as temperature approaches absolute zero. Implicitly, these laws assume the ontological reality of **macroscopic systems** and state variables (T, P, V, U, S) and the concept of **thermal equilibrium**. Epistemologically, they focus on measurable macroscopic properties, often averaging over microscopic details. Methodologically, they rely on defining system boundaries and analyzing state changes and processes (reversible/irreversible). Foundational **mathematics and logic** provide the language and deductive structure for physical theories. **First-order logic (FOL)** explicitly defines the **syntax** (symbols, rules for well-formed formulas) and **semantics** (interpretations, truth) for formal reasoning, along with a **proof theory** (axioms, inference rules like Modus Ponens) ensuring soundness and completeness (Gödel’s Completeness Theorem) for logical validity. **Zermelo-Fraenkel set theory with Choice (ZFC)** provides the standard axiomatic foundation for most mathematics, explicitly postulating the existence and properties of sets through its **axioms** (Extensionality, Pairing, Union, Power Set, Infinity, Specification, Replacement, Regularity, Choice) stated within FOL. Implicitly, ZFC assumes an ontological commitment to a **universe of pure sets** structured as a **well-founded cumulative hierarchy** (enforced by Regularity). Epistemologically, it often aligns with a formalist view where mathematical truth within the system equates to provability from the axioms. Methodologically, it relies entirely on the **axiomatic method** and formal proof. **Gödel’s Incompleteness Theorems** are crucial meta-mathematical results revealing limitations of any such formal system if it is consistent, effectively axiomatized, and strong enough to express arithmetic. Core **philosophical concepts** provide interpretive frameworks and address fundamental questions. **Physicalism (or materialism)** explicitly posits **ontological monism**–everything is physical or supervenes on the physical. It often implicitly assumes the **causal closure of the physical** domain and relies epistemologically on the methods of physics to define the “physical.” **Causation** involves various explicit analyses (counterfactual, mechanistic, etc.) attempting to define the relationship between cause and effect, implicitly assuming the reality of such relationships and their discoverability. **Fundamentality** explicitly addresses the metaphysical structure of reality, often defined via asymmetric dependence relations like **grounding**, implicitly assuming reality has a hierarchical or foundational structure that can be analyzed. **Shannon Information Theory** provides a mathematical framework for quantifying information. Its explicit components include the definition of **Shannon entropy (H)** as average uncertainty ($H = -\sum p_i \log p_i$), the **Source Coding Theorem** limiting lossless compression, the definition of **Channel Capacity (C)**, and the **Noisy-Channel Coding Theorem** limiting reliable communication rate ($R < C$). Implicitly, it assumes information sources and channels can be modeled **probabilistically**. Epistemologically, its focus is purely **syntactic**, quantifying symbol transmission accuracy irrespective of meaning. Methodologically, it relies on probability theory and asymptotic analysis. Finally, the **ΛCDM Model** of cosmology explicitly assumes the **framework of general relativity** and the **Cosmological Principle** (large-scale homogeneity/isotropy). It postulates the **existence and specific properties of cold dark matter (CDM)** and **dark energy (Λ)**. Its narrative includes key processes like the **hot Big Bang origin**, expansion, cooling, **Big Bang Nucleosynthesis (BBN)**, recombination (**CMB formation**), and **structure formation** via gravitational instability. Implicitly, it assumes the **universality and constancy of physical laws** across cosmic time and the validity of interpreting diverse observational data (CMB, supernovae, large-scale structure, BBN) through the model’s equations. Methodologically, it involves **fitting cosmological parameters** (like $\Omega_M, \Omega_\Lambda, H_0$) to these datasets within the assumed theoretical framework. This detailed extraction of explicit formulations and implicit assumptions for each foundational entity provides the necessary granular nodes for constructing the network graph. It reveals not only the core content of each paradigm but also the often-unstated beliefs and preconditions upon which they depend, setting the stage for mapping their intricate interrelations. ---