**Appendix C: Detailed Graph Data (Edges)** **C.1: Edges Originating from Classical Mechanics (CM) Nodes:** | Source Node ID | Target Node ID | Type | Rationale | Conf. | Dir. | |:-------------------------- |:-------------------------- |:----- |:-------------------------------------------------------------------------------------------------------------------------------------- |:---- |:--- | | `CM` | `GR` | `F-LIM`| Classical Mechanics (specifically Newtonian gravity) is the weak-field, low-velocity limit of General Relativity. | H | D | | `CM` | `QM` | `F-LIM`| Classical Mechanics emerges as a limit of Quantum Mechanics (e.g., via correspondence principle, decoherence at large scales). | H | D | | `CM::NewtonsLaws` | `CM` | `S-COMP`| Newton’s Laws are core defining components of the paradigm of Classical Mechanics. | H | D | | `CM::Law2_Force` | `CM::Determinism` | `L-ENT`| Given initial positions/momenta and F=ma, future states are mathematically determined. | H | D | | `CM::GravityLaw_Newton` | `CM::InstantActionDistance` | `L-ENT`| The $1/r^2$form with no time delay implies instantaneous propagation of gravitational influence. | H | D | | `CM::GravityLaw_Newton` | `GR::EFE` | `F-LIM`| Newton’s gravity law is the weak-field, static limit approximation of Einstein’s Field Equations. | H | D | | `CM::AbsoluteSpace` | `GR::DynamicSpacetime` | `L-CTR`| The concept of a fixed, absolute background space contradicts GR’s dynamic spacetime that interacts with matter/energy. | H | S | | `CM::AbsoluteTime` | `GR::DynamicSpacetime` | `L-CTR`| The concept of a universal, absolute time contradicts GR’s notion of time being relative and part of a dynamic spacetime manifold. | H | S | | `CM::AbsoluteSpacetime` | `GR::DynamicSpacetime` | `F-REQ`| Empirical evidence against absolute space (e.g., null result of Michelson-Morley related experiments) necessitated GR’s dynamic view. | H | D | | `CM::Determinism` | `QM::IntrinsicIndeterminism`| `L-CTR`| Classical determinism (predictable outcomes) contradicts the fundamentally probabilistic nature of measurement in standard QM. | H | S | | `CM::Determinism` | `Phil::Determinism` | `L-INST`| Classical mechanics provides a prime physical instantiation and historical support for the philosophical thesis of determinism. | H | D | | `CM::ObjectiveProperties` | `QM::MeasurementCollapse` | `L-CTR`| The idea that properties have definite values independent of measurement contradicts the QM postulate where measurement defines the state. | H | S | | `CM::ObjectiveProperties` | `QM::Complementarity` | `F-CHL`| The classical assumption of simultaneous definite values for all properties is challenged by QM’s principle of complementarity. | H | D | | `CM::ObjectiveProperties` | `Phil::Realism` | `L-CON`/`L-INST`| The classical view of objective properties aligns with and instantiates philosophical realism about measured quantities. | H | D | | `CM::InstantActionDistance` | `GR::Locality` | `L-CTR`| Instantaneous gravitational action contradicts the local propagation of influences at speed c inherent in GR. | H | S | | `CM::Locality` | `QM::NonLocality` | `L-CTR`| The implicit assumption of locality in classical mechanics (excluding Newtonian gravity) is contradicted by QM entanglement. | H | S | | `CM` | `Phil::Physicalism` | `L-CON`| The ontology of classical mechanics (matter, forces in spacetime) is generally considered compatible with physicalism/materialism. | M/H | S | | `CM::GravityLaw_Newton` | `Obs::FlatRotationCurves` | `L-CTR`| Applying Newton’s gravity law to visible mass fails to predict observed flat galactic rotation curves. | H | S | | `CM::GravityLaw_Newton` | `Math::Calculus` | `S-FORM`| Newtonian gravity is formulated using standard differential and integral calculus. | H | D | **C.2: Edges Originating from General Relativity (GR) Nodes:** | Source Node ID | Target Node ID | Type | Rationale | Conf. | Dir. | |:------------------------- |:--------------------------- |:---------------- |:-------------------------------------------------------------------------------------------------------------------------------------- |:---- |:--- | | `GR` | `CM` | `F-LIM` (Inverse) | General Relativity contains Classical Mechanics (Newtonian Gravity) as its weak-field, low-velocity limit. | H | D | | `GR` | `LCDM` | `F-BAS` | General Relativity provides the fundamental framework for gravity and spacetime dynamics used in the ΛCDM model. | H | D | | `GR` | `Math::DifferentialGeometry` | `S-FORM` | General Relativity is mathematically formulated using the language and tools of differential geometry (manifolds, tensors). | H | D | | `GR` | `Problem::QuantumGravity` | `F-REQ` | General Relativity’s incompatibility with quantum mechanics necessitates a more fundamental theory of quantum gravity. | H | D | | `GR::EquivalencePrinciple` | `GR` | `S-COMP` | The Equivalence Principle is a core conceptual component and motivation for General Relativity’s geometric interpretation. | H | D | | `GR::EquivalencePrinciple` | `CM::GravityLaw_Newton` | `F-CHL` | The Equivalence Principle motivates viewing gravity as geometry, challenging the Newtonian concept of gravity as a force. | H | D | | `GR::GeneralCovariance` | `GR` | `S-COMP` | The Principle of General Covariance is a key principle guiding the mathematical formulation of General Relativity. | H | D | | `GR::SpacetimeManifold` | `GR` | `S-COMP` | The modeling of spacetime as a pseudo-Riemannian manifold is the central mathematical structure defining GR. | H | D | | `GR::SpacetimeManifold` | `GR::MetricTensor` | `S-COMP` | The metric tensor defines the geometry of the spacetime manifold in GR. | H | D | | `GR::MetricTensor` | `GR::EFE` | `S-COMP` | The metric tensor (and its derivatives forming the Einstein tensor) is a key component of the Einstein Field Equations. | H | D | | `GR::GeodesicMotion` | `GR` | `S-COMP` | The principle that free particles follow geodesics is a core postulate describing motion within GR. | H | D | | `GR::GeodesicMotion` | `CM::Law2_Force` | `F-CHL` | Geodesic motion (path determined by geometry) offers an alternative explanation for gravitational trajectories compared to F=ma. | H | D | | `GR::EFE` | `GR` | `S-COMP` | The Einstein Field Equations are the central dynamical law of General Relativity. | H | D | | `GR::EFE` | `GR::StressEnergyTensor` | `C-NEC` | The Stress-Energy Tensor acts as the source term in the EFE, determining the spacetime curvature. | H | D | | `GR::EFE` | `GR::CosmologicalConstant` | `S-COMP` | The cosmological constant Λ is an optional term within the structure of the EFE. | H | D | | `GR::EFE` | `GR::Determinism` | `L-ENT` | The EFE are deterministic partial differential equations for the metric tensor, given source terms and initial/boundary conditions. | H | D | | `GR::EFE` | `CM::GravityLaw_Newton` | `F-LIM` (Inverse) | Einstein’s Field Equations reduce to Newton’s law of gravitation in the appropriate weak-field, low-velocity limit. | H | D | | `GR::StressEnergyTensor` | `Concept::Energy` | `F-BAS` | The Stress-Energy Tensor generalizes the concept of energy density and includes momentum density and stress. | H | D | | `GR::DynamicSpacetime` | `CM::AbsoluteSpacetime` | `L-CTR` | GR’s dynamic spacetime, which interacts with matter, contradicts CM’s fixed, absolute spacetime background. | H | S | | `GR::DynamicSpacetime` | `Phil::Physicalism` | `F-REQ` | The dynamic nature of spacetime in GR requires physicalism to clarify whether spacetime itself is a fundamental physical entity. | M | D | | `GR::ContinuumAssumption` | `Problem::PlanckScale` | `F-REQ` | GR’s assumption of a smooth spacetime continuum is expected to break down at the Planck scale, requiring a quantum gravity description. | H | D | | `GR::Determinism` | `QM::IntrinsicIndeterminism` | `L-CTR` | The deterministic evolution described by GR conflicts with the probabilistic nature of measurement outcomes in standard QM. | H | S | | `GR::Locality` | `QM::NonLocality` | `F-CHL` | GR’s description of local interactions propagating at speed c is challenged by the non-local correlations of quantum entanglement. | H | D | | `GR::Singularity` | `GR` | `F-CHL` | The prediction of singularities indicates a breakdown or limit of applicability of the GR framework itself. | H | D | | `GR::Singularity` | `Problem::GRSingularity` | `L-INST` | GR’s predictions are the primary instances defining the singularity problem in physics. | H | D | | `GR::Singularity` | `Problem::QuantumGravity` | `F-REQ` | Singularities signal the need for a quantum theory of gravity to describe physics under extreme conditions. | H | D | **C.3: Edges Originating from Quantum Mechanics (QM - Standard Interpretation) Nodes:** | Source Node ID | Target Node ID | Type | Rationale | Conf. | Dir. | |:-------------------------- |:-------------------------- |:----- |:-------------------------------------------------------------------------------------------------------------------------------------- |:---- |:--- | | `QM` | `CM` | `F-LIM` (Inverse) | Quantum Mechanics contains Classical Mechanics as its macroscopic or large quantum number limit (Correspondence Principle). | H | D | | `QM` | `SM` | `F-BAS`| Quantum Mechanics provides the foundational principles upon which the Standard Model (as a QFT) is built. | H | D | | `QM` | `Math::HilbertSpace` | `S-FORM`| The standard mathematical formulation of QM uses Hilbert spaces to represent states and operators. | H | D | | `QM` | `Math::ProbabilityTheory` | `F-BAS`| QM fundamentally relies on probability theory via the Born rule to connect formalism to experimental outcomes. | H | D | | `QM` | `Problem::QuantumGravity` | `F-REQ`| Standard QM does not incorporate gravity dynamically, necessitating a unification with GR (Quantum Gravity). | H | D | | `QM` | `Problem::QMeasurement` | `S-COMP`| The measurement problem is an internal conceptual inconsistency or incompleteness within standard QM interpretations. | H | D | | `QM::StateVector` | `QM` | `S-COMP`| The state vector (wavefunction) is the core representation of a system’s state in QM. | H | D | | `QM::StateVector` | `QM::Superposition` | `L-ENT`| The vector nature of states in Hilbert space directly entails the principle of superposition (linear combinations are valid states). | H | D | | `QM::HilbertSpace` | `QM::StateVector` | `F-BAS`| Hilbert space is the mathematical structure within which state vectors are defined. | H | D | | `QM::ObservableOperator` | `QM` | `S-COMP`| Associating observables with operators is a core postulate of QM. | H | D | | `QM::ObservableOperator` | `QM::Quantization` | `L-ENT`| The postulate that measurement results are eigenvalues of operators directly leads to observed quantization. | H | D | | `QM::SchrodingerEq` | `QM` | `S-COMP`| The Schrödinger equation is the core postulate governing time evolution (between measurements). | H | D | | `QM::SchrodingerEq` | `QM::UnitaryEvolution` | `L-ENT`| The mathematical form of the Schrödinger equation ensures that time evolution is unitary. | H | D | | `QM::UnitaryEvolution` | `QM::MeasurementCollapse` | `L-CTR`| Unitary evolution is deterministic and preserves superposition, contradicting the non-unitary, probabilistic nature of collapse. | H | S | | `QM::BornRule` | `QM` | `S-COMP`| The Born rule is the core postulate linking the formalism (state vector) to experimental probabilities. | H | D | | `QM::BornRule` | `QM::IntrinsicIndeterminism`| `F-BAS`| The Born rule provides probabilities, which are interpreted as fundamental indeterminism in standard QM. | H | D | | `QM::MeasurementCollapse` | `QM` | `S-COMP`| The projection postulate (collapse) is a core component of the standard description of measurement. | H | D | | `QM::MeasurementCollapse` | `CM::ObjectiveProperties` | `L-CTR`| Collapse implies properties are defined/created by measurement, contradicting classical objective reality. | H | S | | `QM::MeasurementCollapse` | `Phil::Realism` | `F-CHL`| The nature of collapse challenges simple forms of scientific realism about pre-measurement properties. | M/H | D | | `QM::MeasurementCollapse` | `Problem::QMeasurement` | `L-ENT`| The collapse postulate is central to the measurement problem, conflicting with unitary evolution. | H | D | | `QM::Superposition` | `QM` | `S-COMP`| Superposition is a fundamental principle arising from the linearity of the Hilbert space structure. | H | D | | `QM::Quantization` | `CM` | `L-CTR`| Observed quantization contradicts the classical assumption that physical quantities can take continuous values. | H | S | | `QM::IntrinsicIndeterminism`| `CM::Determinism` | `L-CTR`| Fundamental indeterminism contradicts classical determinism. | H | S | | `QM::IntrinsicIndeterminism`| `Phil::Determinism` | `L-CTR`| Standard QM interpretation contradicts the philosophical thesis of determinism. | H | S | | `QM::IntrinsicIndeterminism`| `Phil::CausalClosure` | `F-CHL`| If quantum events are truly random, it potentially challenges the idea that all physical events have sufficient *physical* causes. | M | D | | `QM::Complementarity` | `CM::ObjectiveProperties` | `F-CHL`| Complementarity challenges the classical assumption that all properties can possess definite values simultaneously. | H | D | | `QM::UncertaintyPrinciple` | `QM` | `S-COMP`| The Uncertainty Principle is a core principle derived from the QM formalism (non-commuting operators). | H | D | | `QM::UncertaintyPrinciple` | `CM::ObjectiveProperties` | `F-CHL`| The Uncertainty Principle places fundamental limits on the simultaneous definability of properties assumed objective in CM. | H | D | | `QM::Entanglement` | `QM` | `S-COMP`| Entanglement is a key phenomenon predicted and described by the QM formalism for multi-part systems. | H | D | | `QM::Entanglement` | `CM::Locality` | `F-CHL`| Entanglement demonstrates non-local correlations that challenge the implicit locality assumptions of classical mechanics. | H | D | | `QM::Entanglement` | `Phil::LocalRealism` | `L-CTR`| Experimental verification of Bell’s theorem violations using entangled states contradicts local realism. | H | S | | `QM::NonLocality` | `GR::Locality` | `L-CTR` / `F-CHL` | QM non-locality conflicts with the local nature of interactions described by classical GR field equations. | H | S/D | | `QM::ClassicalDescriptionReq`| `QM` | `S-COMP`| Bohr’s requirement for classical descriptions is a component of the Copenhagen interpretation. | H | D | | `QM::ClassicalDescriptionReq`| `Problem::QMeasurement` | `C-INF`| This requirement contributes to the measurement problem by creating an ambiguous quantum-classical divide. | M | D | | `QM::MeasurementProblem` | `QM` | `F-CHL`| The measurement problem indicates a fundamental conceptual incompleteness or inconsistency within standard QM interpretations. | H | D |