Newton to Einstein: Axiom-Based Discovery via Game Design

(Axiom 1) covers how rays move within a single uniform medium

Straight-Line Propagation. (Axiom 1) covers how rays move within a single uniform medium

reflection (Axiom 2) and refraction (Axiom 3) ensure we can describe all standard interactions at interfaces

The Laws at Boundaries. reflection (Axiom 2) and refraction (Axiom 3) ensure we can describe all standard interactions at interfaces. From these three, one can construct ray diagrams for mirrors, lenses, prisms, fiber optics, and more. They suffice to describe image formation,...

cannot be deduced from reflection or refraction laws (which involve boundaries)

Rectilinear Propagation. cannot be deduced from reflection or refraction laws (which involve boundaries)

cannot be derived from rectilinear propagation, or the refraction law, It specifically concerns a boundary interaction where the angle of incidence equals angle of reflection

Reflection Law. cannot be derived from rectilinear propagation, or the refraction law, It specifically concerns a boundary interaction where the angle of incidence equals angle of reflection

is not derivable from reflection or rectilinear propagation

Refraction Law (Snell’s). is not derivable from reflection or rectilinear propagation. It specifically relates sines of angles to the refractive indices of distinct media. 31 E.2 Wave Optics Wave optics describes light as an electromagnetic wave. In a non-conducting, linear me...

Wave Equation. In a linear, isotropic, source-free region with refractive index n and speed of light in vacuum c, the electric and magnetic fields, E and B, satisfy the wave equations: ∇2E − n2 c2 ∂2E ∂t2 = 0, ∇2B − n2 c2 ∂2B ∂t2 = 0

Boundary Conditions. At an interface between two linear media (with permittivities ϵ1, ϵ2 and refractive indices n1, n2), the tangential components of E and B are continuous, while the normal components obey: ϵ1 E⊥ 1 − ϵ2 E⊥ 2 = σ, B⊥ is continuous. Here σ is any free surface ...

In source-free, non-conducting regions, electromagnetic waves are transverse: k · E = 0, k · B = 0, E ⊥ B, where k is the wave vector (direction of propagation)

Transverse Nature of Light. In source-free, non-conducting regions, electromagnetic waves are transverse: k · E = 0, k · B = 0, E ⊥ B, where k is the wave vector (direction of propagation). E.2.2 Completeness These three axioms suffice to explain all major wave-optical phenomena:

Governs propagation, superposition, interference, diffraction, and lays the foundation for polarization and dispersion when combined with a frequency-dependent index

Wave Equation. Governs propagation, superposition, interference, diffraction, and lays the foundation for polarization and dispersion when combined with a frequency-dependent index

Essential for describing behavior at interfaces (reflection, refraction, transmis- sion)

Boundary Conditions. Essential for describing behavior at interfaces (reflection, refraction, transmis- sion)

Ensures electromagnetic waves in a non-conducting medium have no longitudinal electric (or magnetic) components, thus capturing polarization

Transverse Nature. Ensures electromagnetic waves in a non-conducting medium have no longitudinal electric (or magnetic) components, thus capturing polarization. E.2.3 Independence

cannot be deduced from boundary conditions or transversality alone

Wave Equation. cannot be deduced from boundary conditions or transversality alone. It is the fundamental statement of how fields evolve in free space or uniform media

are not derivable from the free-space wave equation; they define behavior at interfaces and thus must be stated separately

Boundary Conditions. are not derivable from the free-space wave equation; they define behavior at interfaces and thus must be stated separately

does not automatically follow from the wave equation in a general scenario

Transverse Nature. does not automatically follow from the wave equation in a general scenario. It requires that charges/currents are absent in the medium and that we are dealing with electromagnetic waves, specifically. Hence, none of the three axioms can be derived from the o...

Since A′ 1 implies a plane-wave solution moving at speed c/n, and A′ 3 fixes that speed inside M, we deduce the wave equation for electromagnetic fields E and B

− →W1. Since A′ 1 implies a plane-wave solution moving at speed c/n, and A′ 3 fixes that speed inside M, we deduce the wave equation for electromagnetic fields E and B. Formally, plane-wave solutions to ∇2Ψ − n2 c2 ∂2Ψ ∂t2 = 0 33 are consistent with constant-index propagation....

Rewrite Reflection (A2) as a tangential-k continuity condition (B2)

Rewrite Refraction (A3) as a tangential-k continuity condition for the transmitted wave (B3)

wave-based reasoning

Combine B2 and B3 into full electromagnetic field boundary conditions ( W2). The tangential continuity of k maps onto tangential continuity of E and B. Gauss’s law for E and B gives the normal-component relations (ϵ1E⊥ − ϵ2E⊥ = σ and B⊥ continuous). 35 E.3.3 Axiom 3 Goal: Show...

There is a wave field F (which later we may identify with E or B),

the phase fronts for the reflected wave have the same in-plane wavevector as the incident wave

That wave obeys reflection ( A2) and refraction (A3) at boundaries between media. Rewrite Step 1: Replace Reflection and Refraction by Tangential Wave-Vector Continuity. { A2, A3} − → {B2, B3}. • B2 (Reflection as a Wave Condition): k∥(inc) = k∥(ref), meaning the tangential co...

ˆE(r, t), ˆB(r, t) − →ˆak, ˆa† k with [ˆak, ˆa† k′ ] = δkk′ , The electromagnetic field is promoted to an operator in a Hilbert space

Field Quantization. ˆE(r, t), ˆB(r, t) − →ˆak, ˆa† k with [ˆak, ˆa† k′ ] = δkk′ , The electromagnetic field is promoted to an operator in a Hilbert space. Each mode k is described by annihilation ˆak and creation ˆa† k operators satisfying canonical commutation relations. This...

iℏ d dt |Ψ(t)⟩ = ˆH |Ψ(t)⟩ , In the Schrödinger picture, the quantum state |Ψ(t)⟩ of the field evolves according to the Hamiltonian ˆH

Unitary Evolution (Schrödinger or Heisenberg Picture). iℏ d dt |Ψ(t)⟩ = ˆH |Ψ(t)⟩ , In the Schrödinger picture, the quantum state |Ψ(t)⟩ of the field evolves according to the Hamiltonian ˆH. In the Heisenberg picture, field operators evolve instead, while states remain fixed. ...

ˆO = ˆO†, outcomes = eigenvalues of ˆO, p (outcome = λ) = ∥ ˆPλ |Ψ⟩ ∥2, Physical observables (e.g., photon number, field quadratures) correspond to Hermitian operators

Quantum Measurement Postulate. ˆO = ˆO†, outcomes = eigenvalues of ˆO, p (outcome = λ) = ∥ ˆPλ |Ψ⟩ ∥2, Physical observables (e.g., photon number, field quadratures) correspond to Hermitian operators. Measurement outcomes are given by their eigenvalues, with probabilities deter...

Field Quantization allows us to represent light in terms of discrete or continuous quantum states (Fock, coherent, squeezed, etc.), capturing inherently quantum behaviors like vacuum fluctuations and photon antibunching

Unitary Evolution governs the time evolution and interactions of these quantum states, enabling analyses of multi-photon interference, beam-splitter transformations, and quantum entanglement generation

collapses

Quantum Measurement accounts for how photodetection, homodyne detection, or other measurement schemes yield observable outcomes and how the act of measurement influences or “collapses” the quantum state. Together, they suffice to derive virtually all standard quantum-optical p...

Field Quantization is not inferable from the unitary evolution or measurement principles alone: one may have quantum evolution and measurement in other (non-photonic) systems without imposing photon creation/annihilation operators or the discrete energy levels of the electroma...

Unitary Evolution cannot be deduced from quantization and measurement alone, as it prescribes how states evolve in time under a Hamiltonian and is not captured by just asserting that a quantum field exists with certain commutation relations

Hence, none of the three axioms can be derived from the others, ensuring their logical independence in describing the quantum nature of light

Quantum Measurement Postulate does not automatically follow from field quantization and unitary dynamics; it explicitly states how measurement outcomes arise and how they probabilistically update the state of the system. Hence, none of the three axioms can be derived from the ...

By recasting the classical fields as operator-valued mode expansions, we impose canonical commuta- tion and obtain creation/annihilation operators

Wave Equation → Field Quantization: W1 =⇒ Q1. By recasting the classical fields as operator-valued mode expansions, we impose canonical commuta- tion and obtain creation/annihilation operators

The requirement of consistent field solutions across interfaces leads to a single global quantum state and the Hamiltonian-based time evolution

Boundary Conditions → Unitary Evolution: W2 =⇒ Q2. The requirement of consistent field solutions across interfaces leads to a single global quantum state and the Hamiltonian-based time evolution

Transverse Nature → Quantum Measurement: W3 =⇒ Q3. The restriction to transverse degrees of freedom becomes the quantized operator framework, requiring a measurement postulate (Hermitian observables and Born rule) to predict experimental outcomes. Hence, each of the three Wave...

(KCL) X n In(t) = 0 , (KVL) X n Vn(t) = 0

Kirchhoff’s Laws. (KCL) X n In(t) = 0 , (KVL) X n Vn(t) = 0 . Statement. At any node, the algebraic sum of currents is zero (Kirchhoff’s Current Law, KCL); around any loop, the algebraic sum of voltages is zero (Kirchhoff’s V oltage Law, KVL)

Resistor: V (t) = R I(t), Capacitor: I(t) = C dV dt , Inductor: V (t) = L dI dt ,

Linear Constitutive Relations. Resistor: V (t) = R I(t), Capacitor: I(t) = C dV dt , Inductor: V (t) = L dI dt , . . . Statement. Each (passive) two-terminal component relates its voltage and current via a linear, time- invariant (LTI) differential equation. In the simplest ca...

Provide global constraints on the voltages and currents, ensuring that charges and energy flows are conserved throughout the circuit

Kirchhoff’s Laws. Provide global constraints on the voltages and currents, ensuring that charges and energy flows are conserved throughout the circuit

Define how each circuit element (resistor, capacitor, inductor, etc.) responds to voltage and current, laying the foundation for time-domain and frequency-domain analyses

Linear Constitutive Relations. Define how each circuit element (resistor, capacitor, inductor, etc.) responds to voltage and current, laying the foundation for time-domain and frequency-domain analyses. 42 G.1.3 Independence

Kirchhoff’s Laws cannot be derived from the constitutive relations or superposition alone; they express fundamental conservation principles that apply regardless of how individual circuit elements behave

Linear Constitutive Relations are not deducible from Kirchhoff’s Laws or superposition. Kirchhoff’s Laws and superposition only constrain how voltages/currents combine or sum at nodes and loops; they do not dictate the specific linear equations linking voltage and current of a...

Each signal X(t) can take on only two distinct values, { 0, 1}, representing logical ‘false’ or ‘true.’ Statement

Binary Signal States. Each signal X(t) can take on only two distinct values, { 0, 1}, representing logical ‘false’ or ‘true.’ Statement. All digital signals are confined to two nominal voltage levels (or two symbols). Intermediate values or analog considerations are assumed to...

∀ (AND, OR, NOT, etc.) : output = fgate(inputs), Statement

Boolean Gate Operations. ∀ (AND, OR, NOT, etc.) : output = fgate(inputs), Statement. Each combinational logic element (gate) transforms its input bit(s) according to a Boolean function (truth table). For instance, an AND gate outputs ‘1’ if and only if all inputs are ‘1’; a NO...

Flip-flops, registers: Q(t + ∆t) = F Q(t), X(t) on clock edge

Synchronous State Update (Clocked Memory). Flip-flops, registers: Q(t + ∆t) = F Q(t), X(t) on clock edge. Statement. Sequential circuits store and update states at discrete time intervals (clock cycles). Flip- flops or registers sample input signals and produce outputs synchro...

Establishes the discrete, two-level (0/1) nature of signals, enabling logical (rather than continuous) interpretation of circuit behavior

Binary Signal States. Establishes the discrete, two-level (0/1) nature of signals, enabling logical (rather than continuous) interpretation of circuit behavior

Captures how combinational logic processes input bits via gates (AND, OR, NOT, NAND, NOR, etc.)

Boolean Gate Operations. Captures how combinational logic processes input bits via gates (AND, OR, NOT, NAND, NOR, etc.). Permits constructing arbitrary Boolean functions

Introduces memory elements and timing, forming sequential circuits

Synchronous State Update. Introduces memory elements and timing, forming sequential circuits. This encompasses registers, counters, finite-state machines, and synchronous system design. G.2.3 Independence

Gates and flip-flops could, in principle, operate on multi-valued logic; the two-level nature of digital signals must be stated independently

Binary Signal States is not derivable from Boolean gates or clocked memory alone. Gates and flip-flops could, in principle, operate on multi-valued logic; the two-level nature of digital signals must be stated independently

Even with two-level signals and timing, one must specifically define how the inputs map to outputs for each gate

Boolean Gate Operations cannot be inferred from the binary state axiom or from the presence of a clock or memory. Even with two-level signals and timing, one must specifically define how the inputs map to outputs for each gate

decomposed signals

Synchronous State Update does not follow from binary signals or Boolean operations. One can have purely combinational (stateless) logic or asynchronous designs unless the notion of clocked memory is explicitly introduced. Hence, none of these three axioms can be derived from t...

Hence: S2 = { A′ 1, A′ 2} − → S3 = { D1}, where D1 : All signals are 0 or 1 (two distinct logical levels)

− →D1 : {Binary states only}. Hence: S2 = { A′ 1, A′ 2} − → S3 = { D1}, where D1 : All signals are 0 or 1 (two distinct logical levels). Final Set (Digital Axiom 1, Binary Signal States): S3 = n D1 : {0, 1} logic levels o I.e. each output node saturates to LOW or HIGH, represe...

For instance, a saturating amplifier with certain resistor networks might implement AND, OR, NOT, etc

− →B3 : fgate : (0 , 0) 7→ X, . . . For instance, a saturating amplifier with certain resistor networks might implement AND, OR, NOT, etc. once we interpret VLOW → 0, VHIGH → 1. Hence, S2 − →S3 = { B3}, where B3 : Each linear sub-block + saturations =⇒ Boolean function table. ...

By adding a threshold-based saturation stage, continuous voltages are forced into LOW or HIGH, yielding discrete {0, 1}

From Kirchhoff + Linear Relations to Binary Signal States: {A1, A2} =⇒ D1. By adding a threshold-based saturation stage, continuous voltages are forced into LOW or HIGH, yielding discrete {0, 1}

Once inputs/outputs are binary, linear sub-blocks with saturations become truth tables, i.e

From Linear Devices to Boolean Gate Operations: {A2, A3} =⇒ D2. Once inputs/outputs are binary, linear sub-blocks with saturations become truth tables, i.e. Boolean gates (AND, OR, NOT, etc.)

Incorporating active components and clock signals transforms a latch into a flip-flop or register, enforcing synchronous updates of the internal state

From Bistable Circuits to Synchronous State Updates: {A1, A2} =⇒ D3. Incorporating active components and clock signals transforms a latch into a flip-flop or register, enforcing synchronous updates of the internal state. Hence, each of the three Analog Electronics Axioms can b...

A1 : All information is encoded using a finite alphabet Σ (e.g

Discrete Symbol Representation. A1 : All information is encoded using a finite alphabet Σ (e.g. bits). Statement. At the lowest level, any digital system uses a finite set of distinct symbols Σ (e.g. {0, 1} in binary) to represent data and instructions. Intermediate analog eff...

A2 : S δ − →S′, S, S ′ ∈ S, S finite, δ : S ×Σ → S

Finite Control and State Transitions. A2 : S δ − →S′, S, S ′ ∈ S, S finite, δ : S ×Σ → S. Statement. A digital computer at any instant is in one of a finite number of internal states S. Input symbols (from Σ), along with the current state S, determine the next state S′ via a w...

A3 : ∃ U: ∀ f ∈ C, ∃ p ∈ Σ∗, ∀ x ∈ Σ∗ : U(p, x) = f (x)

Algorithmic Universality (Effective Computability). A3 : ∃ U: ∀ f ∈ C, ∃ p ∈ Σ∗, ∀ x ∈ Σ∗ : U(p, x) = f (x). Statement. There exists a universal mechanism U (e.g. a programmable processor) such that for every effectively definable function f over the alphabet Σ, there is a fin...

Discrete Symbol Representation guarantees that all data and instructions are handled as symbolic tokens from a finite set (bits, characters, etc.), enabling unambiguous, reproducible manipulation

This underpins everything from simple finite-state controllers to CPU instruction cycles

Finite Control and State Transitions encodes how a digital system moves from one configuration to another in discrete time steps (or events). This underpins everything from simple finite-state controllers to CPU instruction cycles

This is the essence of universal computability

Algorithmic Universality confirms that by combining discrete symbols and a finite control mechanism, one can implement every computable function (assuming sufficient memory/time resources). This is the essence of universal computability. Together, they suffice to describe how ...

One could imagine a continuous control system without explicitly stipulating discrete symbols

Discrete Symbol Representation is not derivable from finite-state control or universality alone. One could imagine a continuous control system without explicitly stipulating discrete symbols

One must explicitly stipulate how the machine progresses from one moment to the next in a finite-state manner

Finite Control and State Transitions cannot be inferred merely from having a discrete alphabet and the notion of a universal mechanism. One must explicitly stipulate how the machine progresses from one moment to the next in a finite-state manner

Without an explicit statement about the generality of these transitions, the system might be limited to a small class of tasks (not fully universal)

Algorithmic Universality does not follow automatically from having discrete symbols and finite transitions. Without an explicit statement about the generality of these transitions, the system might be limited to a small class of tasks (not fully universal). Hence, none of thes...

A1 : |ψ⟩ ∈ H, dim(H) < ∞, ∥ |ψ⟩ ∥= 1

Quantum State Space (Hilbert Space). A1 : |ψ⟩ ∈ H, dim(H) < ∞, ∥ |ψ⟩ ∥= 1. Statement. Information is represented by vectors in a finite-dimensional complex Hilbert space H. Each qubit (or collection of qubits) is described by a normalized state |ψ⟩. Superposition and entanglem...

A2 : |ψ(t + ∆t)⟩ = ˆU |ψ(t)⟩ , ˆU † ˆU = ˆI

Unitary Evolution and Quantum Gates. A2 : |ψ(t + ∆t)⟩ = ˆU |ψ(t)⟩ , ˆU † ˆU = ˆI. Statement. In the absence of measurement, the evolution of quantum states is governed byunitary transformations ˆU. These can be discrete gates (e.g., single- or multi-qubit logic gates) or conti...

collapses

Quantum Measurement Postulate. A3 : ˆO = ˆO†, Outcomes = Eigvals( ˆO), p (outcome) = ∥ ˆPoutcome |ψ⟩ ∥2. Statement. Physical observables are represented by Hermitian operators ˆO. Upon measurement, the system “collapses” (or is updated) to an eigenstate of ˆO with probability ...

Formalizes the representation of quantum information, including superposition and entanglement

Quantum State Space (Hilbert Space). Formalizes the representation of quantum information, including superposition and entanglement

Describes how quantum information is processed or transformed, via discrete gates or continuous Hamiltonian evolution

Unitary Evolution and Quantum Gates. Describes how quantum information is processed or transformed, via discrete gates or continuous Hamiltonian evolution

Explains how classical outcomes emerge upon measurement and how the quantum state updates according to the measurement result (via projection or more general maps)

Measurement Postulate. Explains how classical outcomes emerge upon measurement and how the quantum state updates according to the measurement result (via projection or more general maps). Together, these axioms enable universal quantum computation: one can construct quantum ci...

One could posit abstract transformations and observations without specifying a complex Hilbert-space structure

Quantum State Space is not inferable from unitarity or measurement alone. One could posit abstract transformations and observations without specifying a complex Hilbert-space structure

It must be explicitly stated that closed-system dynamics are governed by a norm-preserving (unitary) operation

Unitary Evolution cannot be deduced purely from the existence of a Hilbert space and a measurement postulate. It must be explicitly stated that closed-system dynamics are governed by a norm-preserving (unitary) operation

internal state + symbol

Measurement Postulate does not follow from the first two axioms. Even with a Hilbert space and unitary gates, one needs an additional rule specifying how measurement outcomes (eigenvalues) arise and how the post-measurement state is determined. Hence, these three axioms are mu...

We lift the finite alphabet Σ into a finite-dimensional Hilbert space H, permitting superpositions of basis states

From Discrete Symbols (Axiom 1) to Quantum State Space (Q1). We lift the finite alphabet Σ into a finite-dimensional Hilbert space H, permitting superpositions of basis states

The classical transition function δ generalizes to a linear, norm-preserving operator ˆU, ensuring reversibility of quantum evolution

From Finite Control (Axiom 2) to Unitary Evolution (Q2). The classical transition function δ generalizes to a linear, norm-preserving operator ˆU, ensuring reversibility of quantum evolution

Classical outputs must be read out from quantum states, so we formalize observables via Hermitian operators and outcome probabilities via the Born rule

From Algorithmic Universality (Axiom 3) to Measurement Postulate (Q3). Classical outputs must be read out from quantum states, so we formalize observables via Hermitian operators and outcome probabilities via the Born rule. Together, these steps illustrate how each digital axi...

In other words, space is three-dimensional, Euclidean, and absolute; time flows uniformly and is the same for all observers

Absolute Space and Time ∀ Frames 1, 2 : dt1 = dt2, ri = (xi, yi, zi), and metric ds2 i = dx2 i + dy2 i + dz2 i , ds2 1 = ds2 2 Where dti and dxi, dyi, dzi denote the time separation of two events, as measured in frame i. In other words, space is three-dimensional, Euclidean, a...

Existence of Inertial Frames ∃ Frame in which ⃗F = ⃗0 ⇒ d⃗ v dt = ⃗0, v i = dxi dt There exist reference frames (inertial frames) in which Newton’s laws of motion hold true without modification

Newton’s Second Law (Law of Acceleration) ⃗F = m⃗ a= m d⃗ v dt , v i = dxi dt In an inertial frame, the net external force acting on a body equals the rate of change of its momentum

Newton’s Third Law (Law of Action and Reaction) ⃗F = − ⃗R For every action ⃗F , there is an equal and opposite reaction ⃗R; forces between two bodies are equal in magnitude and opposite in direction. 54

Principle of Superposition ⃗F = X i ⃗Fi When multiple forces ⃗Fi act on a body, the net force ⃗F is the vector sum of the individual forces acting on it

I.1.2 Completeness

Law of Universal Gravitation ⃗Fg = Gm1m2 r2 ˆr Every point mass 2 attracts every other point mass 1 with a force along the unit vector ˆr pointing from 1 to 2, proportional to the product of their masses m1, m2, and inversely proportional to the square of the r distance betwee...

Absolute Space and Time (Axiom 1) shows the mathematical structure of system parameters and how they can be transformed between reference frames

Existence of Inertial Frames (Axiom 2) shows that Newton’s laws can be applied to our universe

Newton’s Second Law (Axiom 3), Newton’s Third Law (Axiom 4), and Principle of Superposition (Axiom 5) give a way of predicting a body’s motion, provided forces acting on it

Law of Universal Gravitation (Axiom 6) describes the fundamental force of gravity, which attracts any two objects. From these axioms, one can predict the motion of celestial objects, transform between reference frames, draw force diagrams, and model the behavior of arbitrary s...

Absolute Space and Time sets the mathematical framework to which other other axioms should be applied

Existence of Inertial Frames can’t be derived from other axioms, which explicitly assume being in an inertial reference frame

Newton’s Second Law is the only axiom that quantitatively describes the effect of an arbitrary force on a body’s motion

Newton’s Third Law is the only axiom that refers to an arbitrary interaction between two bodies

Principle of Superposition is the only action that demonstrates the effects of multiple forces on a body

I.2 General relativity The fundamental difference between general relativity and Newtonian mechanics is the assumption that the speed of light is the same in all reference frames

Law of Universal Gravitation refers to gravity, a concept not introduced in previous axioms. I.2 General relativity The fundamental difference between general relativity and Newtonian mechanics is the assumption that the speed of light is the same in all reference frames. This...

Principle of Equivalence (Einstein’s Equivalence Principle) Gravity indistinguishable from acceleration Locally, the effects of gravity are indistinguishable from those of acceleration. This means that in a small enough region of spacetime, the laws of physics reduce to those ...

gµν and g′ µν are the corresponding metrics, and L(xµ, gµν ) = L′(x′µ, g′ µν ) corresponds to the Lagrangian density in different reference frames

Principle of General Covariance g′ µν (x′) = ∂xα ∂x′µ ∂xβ ∂x′ν gαβ(x), L(xµ, gµν ) = L′(x′µ, g′ µν ) Where primed x′µ and unprimed xµ coordinates correspond to different reference frames. gµν and g′ µν are the corresponding metrics, and L(xµ, gµν ) = L′(x′µ, g′ µν ) correspond...

The metric tensor defines distances and angles in spacetime and determines its geometric and causal structure

Spacetime as a Four-Dimensional Pseudo-Riemannian Manifold (M4, gµν ) with signature (−, +, +, +) Spacetime is modeled as a smooth, four-dimensional manifold M4 equipped with a metric tensor gµν of Lorentzian signature (−, +, +, +). The metric tensor defines distances and angl...

The curvature of spacetime is directly related to the energy and momentum content within it

Einstein’s Field Equations Rµν − 1 2 gµν R + Λgµν = 8πG c4 Tµν Where Rµν is the Ricci curvature tensor, gµν is the metric tensor, R is the Ricci scalar (trace of the Ricci tensor), Λ is the cosmological constant, G is the gravitational constant, and c is the speed of light. Th...

In the absence of non-gravitational forces ⃗F , free-falling test particles move along timelike geodesics of spacetime

Geodesic Principle (Motion of Free Particles) ∀ ⃗F = 0 ⇒ ∇τ v = 0, v µ = dxµ dτ Where ∇τ denotes the covariant derivative, and τ the proper time. In the absence of non-gravitational forces ⃗F , free-falling test particles move along timelike geodesics of spacetime

I.2.2 Completeness

Metric Compatibility and Torsion-Free Connection (Levi-Civita Connection) ∆λgµν = 0, Γλ µν = Γλ νµ The connection on the spacetime manifold is the unique torsion-free, metric-compatible Levi-Civita connection. I.2.2 Completeness

Principle of Equivalence (Axiom 1) allows one to connect general relativity to special relativity to derive the geodesics equation and predict the nature of spacetime

Principle of General Covariance(Axiom 2), Spacetime as a Four-Dimensional Pseudo-Riemannian Manifold (Axiom 3), and Metric Compatibility and Torsion-Free Connection (Levi-Civita Con- nection) (Axiom 6) lay the basis of a mathematical framework that should describe reality

It introduces the empirical gravitational constant, which lets one describe the expansion of the universe

Einstein’s Field Equations (Axiom 4) quantitatively relate the curvature of spacetime to present matter. It introduces the empirical gravitational constant, which lets one describe the expansion of the universe

The geodesic principle (Axiom 5) describes the motion of free particles. Together, these axioms let one describe the universe as curved spacetime and allow for a quantitative description of various phenomena, such as black holes, gravitational waves, and gravitational lensing....

Principle of Equivalence cannot be derived from other axioms and establishes the nature of gravity

Principle of General Covariance is the only universal statement about physical laws in the general relativity framework that is crucial for deriving them

Spacetime as a Four-Dimensional Pseudo-Riemannian Manifoldis essential to introduce the notion of a metric and curvature to describe spacetime. 56

Einstein’s Field Equations contain the empirical cosmological constant, which can’t be evaluated from other axioms

The geodesic principle is the only equivalent of Newton’s laws that lets one predict the motion of a free particle

Metric Compatibility and Torsion-Free Connection is an assumption about the Riemannian mani- fold that needs to be imposed and is not explicitly introduced by other axioms. I.3 Transformation I.3.1 Axiom 1 Goal: Transform a subset of Newtonian axioms A3, A5 A3 : Newton’s secon...

This establishes the concept of temperature as a fundamental property and justifies consistent tempera- ture measurement

Zeroth Law of Thermodynamics ∀ systems A, B, C: A in thermal equilibrium with C ∧ B in thermal equilibrium with C =⇒ A in thermal equilibrium with B . This establishes the concept of temperature as a fundamental property and justifies consistent tempera- ture measurement

There exists an extensive state function U called the internal energy

First Law of Thermodynamics dU = δQ − δW. There exists an extensive state function U called the internal energy. Its change in any process is equal to the heat supplied to the system minus the work done by the system

Entropy S is thus a state function, given by the integral of the reversible heat exchange divided by the temperature

Second Law of Thermodynamics ∆S = Z δQrev T (for a reversible process), and for an isolated system, ∆Sisolated ≥ 0. Entropy S is thus a state function, given by the integral of the reversible heat exchange divided by the temperature. In any spontaneous (irreversible) process, ...

As temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum value

Third Law of Thermodynamics lim T →0 S = S0, where S0 is a universal constant (often taken as zero for a perfectly ordered system). As temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum value. 68 J.1.2 Completeness

(Axiom 1) underpins the concept of temperature, enabling consistent thermo- metric definitions and comparisons

Thermal Equilibrium. (Axiom 1) underpins the concept of temperature, enabling consistent thermo- metric definitions and comparisons

(Axiom 2) ensures that all forms of energy transfer (heat, work) can be accounted for through the internal energy of the system

Energy Conservation. (Axiom 2) ensures that all forms of energy transfer (heat, work) can be accounted for through the internal energy of the system

Definition and Directionality via Entropy. (Axiom 3) explains how entropy is defined ( ∆S =R δQrev/T in reversible processes) and clarifies that in real (irreversible) processes, the entropy of an isolated system can never decrease

(Axiom 4) characterizes the limiting behavior of systems at the lowest temperatures, completing the description of equilibrium thermodynamics

Behavior at Absolute Zero. (Axiom 4) characterizes the limiting behavior of systems at the lowest temperatures, completing the description of equilibrium thermodynamics. From these four, one can analyze heat engines, refrigerators, phase transitions, and a wide range of thermo...

cannot be deduced from energy conservation or entropy statements; it establishes the transitive property of thermal equilibrium and the concept of temperature

Zeroth Law. cannot be deduced from energy conservation or entropy statements; it establishes the transitive property of thermal equilibrium and the concept of temperature

is not derivable from the existence of thermal equilibrium or entropy principles alone; it adds a distinct statement about energy being a state function and conserved in processes

First Law. is not derivable from the existence of thermal equilibrium or entropy principles alone; it adds a distinct statement about energy being a state function and conserved in processes

Second Law. introduces the definition of entropy (through δQrev/T ) and its non-decreasing property in isolated systems; this does not follow from thermal equilibrium (zeroth law) or energy conservation (first law)

focuses on the entropy’s limiting value at zero temperature

Third Law. focuses on the entropy’s limiting value at zero temperature. It cannot be derived from the other axioms, which govern equilibrium, energy conservation, and irreversibility but do not dictate the absolute value of entropy as T → 0. J.2 Statistical Mechanics Statistic...

Each microscopic configuration (microstate) of the system is specified by generalized coordinates and momenta

State Space (Phase Space) Postulate A system with f degrees of freedom is represented by a point in a 2f-dimensional phase space. Each microscopic configuration (microstate) of the system is specified by generalized coordinates and momenta

All microstates are assigned a nonnegative probability density ρ(Γ), normalized to unity

Probabilistic Description Postulate ρ(Γ) ≥ 0, Z ρ(Γ) dΓ = 1 . All microstates are assigned a nonnegative probability density ρ(Γ), normalized to unity. Any measurable macroscopic quantity (observable) is given by ⟨A⟩ = Z A(Γ) ρ(Γ) dΓ

In an equilibrium regime, the probability distribution remains constant in time when viewed in the appropriate phase-space representation

Stationary Distribution for Equilibrium dρ dt = 0 = ⇒ ρ(Γ) is time-invariant along Hamiltonian trajectories. In an equilibrium regime, the probability distribution remains constant in time when viewed in the appropriate phase-space representation. 69

The time average of an observable along a single long trajectory equals the ensemble average over the phase-space distribution

Ergodic Hypothesis lim τ →∞ 1 τ Z τ 0 A Γ(t) dt = Z A(Γ) ρ(Γ) dΓ, for any phase function A(Γ). The time average of an observable along a single long trajectory equals the ensemble average over the phase-space distribution. In particular, when A is the Hamiltonian, we get the i...

This defines the entropy as a functional of the probability distribution ρ(Γ)

Statistical Entropy Postulate S[ρ] = − kB Z ρ(Γ) ln ρ(Γ) dΓ. This defines the entropy as a functional of the probability distribution ρ(Γ). In equilibrium ensembles, S[ρ] reduces to the thermodynamic entropy

The entropy functional S[ρ] cannot decrease over time when the system is isolated (no exchange of energy or particles), reflecting the irreversible increase in macroscopic entropy

Statistical Second Law ∆S[ρ] ≥ 0 for an isolated system. The entropy functional S[ρ] cannot decrease over time when the system is isolated (no exchange of energy or particles), reflecting the irreversible increase in macroscopic entropy

As temperature T → 0, the equilibrium distribution collapses to a minimal-entropy ground state

Statistical Third Law lim T →0 S[ρeq] = constant (often taken as zero). As temperature T → 0, the equilibrium distribution collapses to a minimal-entropy ground state. The statistical entropy thus approaches a constant, mirroring the thermodynamic third law. J.2.2 Completeness

(Axioms 1, 2) define the phase-space setting and the probabilistic nature of macroscopic quantities

Microscopic Foundations. (Axioms 1, 2) define the phase-space setting and the probabilistic nature of macroscopic quantities

(Axioms 3, 4) ensure that we can identify equilibrium states (stationary distributions) and relate time averages to ensemble averages (ergodic hypothesis)

Equilibrium and Dynamics. (Axioms 3, 4) ensure that we can identify equilibrium states (stationary distributions) and relate time averages to ensemble averages (ergodic hypothesis)

(Axiom 5) completes the link between the probability distribution and the thermodynamic concept of entropy

Entropy. (Axiom 5) completes the link between the probability distribution and the thermodynamic concept of entropy

(Axioms 6, 7 ) guarantee non-decreasing entropy (second law), and specify the limiting entropy at zero temperature (third law)

Statistical Analogs of Thermodynamic Laws. (Axioms 6, 7 ) guarantee non-decreasing entropy (second law), and specify the limiting entropy at zero temperature (third law). These axioms together allow one to construct the canonical, microcanonical, and grand canonical ensembles ...

It establishes the fundamental structure of microstates

Phase Space Representation (Axiom 1) is not derivable from probabilistic or equilibrium arguments. It establishes the fundamental structure of microstates

Probability Postulate (Axiom 2) does not follow merely from the existence of a phase space; it specifies how probabilities are assigned to microstates

Equilibrium Stationarity (Axiom 3) must be separately stated; it cannot be inferred from the shape of ρ(Γ) alone without the notion of time invariance in Hamiltonian systems

Ergodic Hypothesis (Axiom 4) is not implied by any combination of the first three axioms; it specifically identifies long-time behavior with ensemble statistics

Statistical Entropy (Axiom 5) introduces a functional S[ρ], which is not deducible solely from Axioms 1–4

Statistical Second Law (Axiom 6) cannot be proved from earlier axioms alone; it asserts the non- decreasing nature of the entropy functional in isolated systems

A system with f degrees of freedom is represented by a point in a 2f-dimensional phase space

Statistical Third Law (Axiom 7) specifically addresses the behavior of S[ρ] as T → 0, thus requiring a separate statement not implied by the others. J.3 Transformation J.3.1 Axiom 1 Goal: Transform a subset of the classical thermodynamics axioms T2, T3 T2 : First Law: dU = δQ ...

A system with f degrees of freedom is represented by a point in a 2f-dimensional phase space

− →(M1) We let: Γ = q1, . . . , qf ; p1, . . . , pf . Hence each microstate is a specification of generalized coordinates (qi) and momenta (pi) for the f degrees of freedom in the system. Symbolically: M1 : Γ = ( qi, pi), i = 1, . . . , f. Thus U (Γ) and S(Γ) become functions ...

number of microstates,

− →(M2) To quantify “number of microstates,” assign a nonnegative weightw(Γ) ≥ 0 to each microstate Γ. In equilibrium ensemble theory, w(Γ) might later be related to Boltzmann factors, but we do not assume that form yet. We only assert: M2 : w(Γ) ≥ 0 for all Γ, so that the tot...

U, P, T, S) stop changing with time: M3 : d(Macrostate) dt = 0 ( time-invariance of macroscopic variables)

− →(M3) At equilibrium, observable thermodynamic quantities (e.g. U, P, T, S) stop changing with time: M3 : d(Macrostate) dt = 0 ( time-invariance of macroscopic variables). Symbolically, M3 expresses that once equilibrium is established, the system’s measured properties are s...

S is a measure of microstates (non-decreasing in isolation)

− →(M4) From statistical mechanics, a single system with temperature T > 0 explores many microstates Γ over time. Denote its phase-space trajectory by Γ(t). Symbolically: M4 : Γ( t) is the system’s microscopic path under Hamiltonian dynamics. Although T ′ 4 indicates that at T...

Entropy S must be expressible as a functional of ρ(Γ) reflecting how widely ρ spreads

− →(M5) Recall from previous axioms that each microstate Γ has a probability density ρ(Γ) ≥ 0, normalized to 1. Thus, S1 suggests that: ∆S ≥ 0 ⇐ ⇒the distribution ρ(Γ) spreads or remains the same, 78 never shrinking onto fewer states for an isolated system. Similarly, S → S0 a...

spread out

Increases when ρ becomes more “spread out” (no microstates are removed; partial mixing or spreading always raises entropy)

Goes to a minimal value when ρ is concentrated on as few microstates as possible (consistent with T → 0)

microstates do not decrease for an isolated system

Is additive over independent subsystems (extensivity). These conditions identify the Gibbs/Shannon form: SM5 : S[ρ] = − kB Z ρ(Γ) ln ρ(Γ) dΓ, where kB is Boltzmann’s constant (fixing units). This formula reproduces the classical thermodynamic entropy under equilibrium ensemble...

All possible states correspond to points in this space

Phase Space (Axiom 1) A system is described by a 2n-dimensional phase space (qi, pi). All possible states correspond to points in this space

Hamiltonian Function (Axiom 2) H = H(qi, pi), which encodes the total energy (often T + V ) in terms of positions and momenta

These specify how the system evolves in time

Hamilton’s Equations (Axiom 3) ˙qi = ∂H ∂pi , ˙pi = − ∂H ∂qi . These specify how the system evolves in time. K.1.2 Completeness

Phase Space (Axiom 1) provides the mathematical arena: each point (qi, pi) specifies a complete state

From it, one derives the equations of motion for both integrable and more general systems

Hamiltonian Function (Axiom 2) supplies a single scalar that contains all the energetic information. From it, one derives the equations of motion for both integrable and more general systems

Together with the Hamiltonian, they enable predictions of trajectories and conserved quantities in phase space

Hamilton’s Equations (Axiom 3) determine the unique time evolution of (qi, pi). Together with the Hamiltonian, they enable predictions of trajectories and conserved quantities in phase space. Hence, these three axioms suffice to describe classical mechanics in Hamilton’s forma...

One must explicitly posit a 2n-dimensional space of (qi, pi)

Phase Space cannot be inferred from the existence of a Hamiltonian or the dynamical equations alone. One must explicitly posit a 2n-dimensional space of (qi, pi)

It is the unique scalar whose partial derivatives produce the system’s time evolution

Hamiltonian Function does not automatically follow from the notion of phase space or from the form of Hamilton’s equations. It is the unique scalar whose partial derivatives produce the system’s time evolution

The scalar function H(q, p) is the primary energy for the system

Hamilton’s Equations themselves are not implied by merely specifying (qi, pi) and an energy function. They must be taken as the prescription for how states in phase space evolve with time. None of the three can be derived from the others, ensuring their mutual independence. K....

All possible physical states of a system are represented by (equivalence classes of) vectors in H

State Space (Hilbert Space) |ψ⟩ ∈ H, ⟨ψ|ψ⟩ = 1, where |ψ⟩ is a (normalized) state vector in a complex Hilbert space H. All possible physical states of a system are represented by (equivalence classes of) vectors in H

Observables as Self-Adjoint Operators ˆA = ˆA†, Physical observables (e.g., position, momentum, energy) are represented by self-adjoint (Hermitian) operators ˆA on H. 85

Measurement Postulate (Born Rule) P (αi) = ⟨ϕi | ψ⟩ 2 , When measuring an observable ˆA with eigenvalues αi and corresponding eigenstates |ϕi⟩, the proba- bility of obtaining outcome αi is given by the squared amplitude of the state projected onto |ϕi⟩

collapses

State Collapse |ψ⟩ measure ˆA − − − − − → |ϕi⟩ , Immediately after obtaining a specific measurement outcome αi, the system’s state “collapses” to the corresponding eigenstate |ϕi⟩

The Schrödinger equation governs how the state vector changes in time

Time Evolution (Schrödinger Equation) iℏ d dt |ψ(t)⟩ = ˆH |ψ(t)⟩ , Between measurements, a closed quantum system evolves unitarily under the Hamiltonian operator ˆH. The Schrödinger equation governs how the state vector changes in time

The spin quantum number s can be integer or half-integer (0, 1 2 , 1, 3 2 , 2,

Intrinsic Spin ˆS2 |ψ⟩ = ℏ2s(s + 1) |ψ⟩ , S z |ψ⟩ = ℏms |ψ⟩ , In addition to orbital degrees of freedom, particles possess an intrinsic spin described by the spin operator ˆS, which satisfies the standard angular-momentum commutation relations. The spin quantum number s can be...

,ri, σi,

Exchange Symmetry and Pauli Principle Ψ(. . . ,ri, σi, . . . ,rj, σj, . . .) = (−1)2s Ψ(. . . ,rj, σj, . . . ,ri, σi, . . .), Identical particles with spin s have a many-particle wavefunction Ψ that gains a factor of (−1)2s upon exchanging any two such particles (including the...

(Axiom 1) ensures each physical configuration of the system corresponds to a vector in a complex Hilbert space

State Space. (Axiom 1) ensures each physical configuration of the system corresponds to a vector in a complex Hilbert space

(Axiom 2) stipulates that physical quantities are represented mathematically by self- adjoint operators, allowing us to predict their possible eigenvalues

Observables. (Axiom 2) stipulates that physical quantities are represented mathematically by self- adjoint operators, allowing us to predict their possible eigenvalues

(Axiom 3) gives the probabilistic interpretation needed to connect the mathematical formalism to measurable outcomes

Measurement Postulate. (Axiom 3) gives the probabilistic interpretation needed to connect the mathematical formalism to measurable outcomes

(Axiom 4) specifies how the state changes upon measurement, ensuring consistency with observed discrete measurement results

State Collapse. (Axiom 4) specifies how the state changes upon measurement, ensuring consistency with observed discrete measurement results

(Axiom 5) describes the deterministic (unitary) evolution of the system in the absence of measurement

Time Evolution. (Axiom 5) describes the deterministic (unitary) evolution of the system in the absence of measurement

(Axiom 6) incorporates intrinsic angular momentum, crucial for understanding fine structures, magnetic phenomena, and multi-particle spin states

Intrinsic Spin. (Axiom 6) incorporates intrinsic angular momentum, crucial for understanding fine structures, magnetic phenomena, and multi-particle spin states

(Axiom 7) governs the symmetry or antisymmetry of the wavefunction for identical particles, explaining why fermions cannot share a quantum state while bosons can

Exchange Symmetry and Pauli Principle. (Axiom 7) governs the symmetry or antisymmetry of the wavefunction for identical particles, explaining why fermions cannot share a quantum state while bosons can. From these seven axioms, one can predict the evolution of quantum systems, ...

(Axiom 1) cannot be derived from assumptions about measurements or time evolution; it is the foundational structure that underlies the rest

State Space. (Axiom 1) cannot be derived from assumptions about measurements or time evolution; it is the foundational structure that underlies the rest

Observables. (Axiom 2) cannot be inferred solely from the state space; specifying that observables correspond to self-adjoint operators is an independent assertion about how physical quantities map to operators. 86

(Axiom 3) introduces the probabilistic interpretation (Born Rule)

Measurement Postulate. (Axiom 3) introduces the probabilistic interpretation (Born Rule). It is not implied by the structure of the state space or the form of the Hamiltonian

(Axiom 4) is not derivable from continuous time evolution or the existence of observables; it uniquely addresses how the state changes upon a measurement

State Collapse. (Axiom 4) is not derivable from continuous time evolution or the existence of observables; it uniquely addresses how the state changes upon a measurement

(Axiom 5) is independent of the measurement process; knowing how measurements yield probabilities does not itself define the unitary dynamics between those measurements

Time Evolution. (Axiom 5) is independent of the measurement process; knowing how measurements yield probabilities does not itself define the unitary dynamics between those measurements

Intrinsic Spin. (Axiom 6) is not implied by the previous axioms on its own; it is an additional degree of freedom that arises from the fundamental quantum nature of particles and cannot be derived from orbital degrees of freedom alone

Relativistic

Exchange Symmetry and Pauli Principle. (Axiom 7) does not follow from the preceding ax- ioms; it makes a unique statement about the symmetry properties of many-particle wavefunctions, distinguishing bosons from fermions and dictating the Pauli exclusion principle for half-inte...

Law of Segregation ∀ O ∈ Diploid, ∀ G = (A1, A2) : A1 and A2 separate during gamete formation, each gamete inherits exactly one Ai. 94

Law of Independent Assortment ∀ (G1, G2) on separate chromosomes : ( segregation of G1) ⊥ (segregation of G2)

Law of Dominance ∀ (genotype Aa) : ( phenotype of A) > (phenotype of a)

M.1.2 Completeness

Chromosomal Inheritance ∀ G : G ⊆ (chromosomes) ∧ chromosomes segregate in meiosis . M.1.2 Completeness

(Axiom 1) covers how alleles for a single gene partition into gametes within a diploid organism

Law of Segregation. (Axiom 1) covers how alleles for a single gene partition into gametes within a diploid organism

Independent Assortment & Dominance. Axiom 2 and Axiom 3 ensure that multiple genes (on separate chromosomes) follow independent segregation patterns, and also define how phenotypes manifest when one allele is dominant over another

(Axiom 4) encapsulates the physical basis of these laws by placing genes on chromosomes, unifying the segregation and assortment mechanisms with observable meiotic behavior

Chromosomal Inheritance. (Axiom 4) encapsulates the physical basis of these laws by placing genes on chromosomes, unifying the segregation and assortment mechanisms with observable meiotic behavior. From these four, one can construct Punnett squares and describe monohybrid or ...

cannot be deduced from independent assortment or dominance alone, nor from chromosomal inheritance without explicitly stating that each gamete receives exactly one allele per gene

Law of Segregation. cannot be deduced from independent assortment or dominance alone, nor from chromosomal inheritance without explicitly stating that each gamete receives exactly one allele per gene

cannot be derived from segregation, dominance, or chromoso- mal inheritance alone, since it specifically addresses the independence of allele pairs on separate chromosomes

Law of Independent Assortment. cannot be derived from segregation, dominance, or chromoso- mal inheritance alone, since it specifically addresses the independence of allele pairs on separate chromosomes

is not derivable from segregation or independent assortment, which only address allele partitioning, not how one allele’s expression can mask another’s

Law of Dominance. is not derivable from segregation or independent assortment, which only address allele partitioning, not how one allele’s expression can mask another’s

does not follow from any combination of segregation, independent assortment, or dominance

Chromosomal Inheritance. does not follow from any combination of segregation, independent assortment, or dominance. These three axioms describe the distribution of alleles and their expression, but do not guarantee a physical location (chromosomes) for genes without stating th...

(DNA is formed by two complementary strands S1 and S2

Double Helix & Replication ∀ D : D = (S1, S2) - complementary strands ∧ h (∀ i) Si → (template for new S′ i) i . (DNA is formed by two complementary strands S1 and S2. Each strand serves as a template in replication.)

(A gene G in DNA is transcribed into messenger RNA, which is then translated into a polypeptide chain.)

Central Dogma ∀ G ⊆ D : G → (transcription → mRNA) → (translation → polypeptide). (A gene G in DNA is transcribed into messenger RNA, which is then translated into a polypeptide chain.)

Genetic Code Universality ∀ (codon c) : c 7→ amino acid a, (with minor exceptions, the same codons specify identical amino acids across most organisms.) 95

(Each gene’s expression level depends on transcription factors, promoter sequences, enhancer elements, and epigenetic modifications.) M.2.2 Completeness

Regulation of Gene Expression ∀ (gene g) : expression(g) = f (TFs, promoters, enhancers, epigenetics). (Each gene’s expression level depends on transcription factors, promoter sequences, enhancer elements, and epigenetic modifications.) M.2.2 Completeness

(Axiom 1) describes the physical structure of DNA and its templated replication mechanism

Double Helix & Replication. (Axiom 1) describes the physical structure of DNA and its templated replication mechanism

Axiom 2 and Axiom 3 capture the flow of genetic information (DNA to RNA to protein) and the universal mapping from codons to amino acids

Central Dogma & Code. Axiom 2 and Axiom 3 capture the flow of genetic information (DNA to RNA to protein) and the universal mapping from codons to amino acids

(Axiom 4) incorporates the control systems that modulate gene expression, making the informational framework responsive to cellular and environmental states

Regulation. (Axiom 4) incorporates the control systems that modulate gene expression, making the informational framework responsive to cellular and environmental states. These four axioms provide a foundation for describing DNA replication, transcription, translation, and regu...

cannot be inferred from the Central Dogma or the Genetic Code alone; they address the flow of information but not the physical structure enabling replication

Double Helix & Replication. cannot be inferred from the Central Dogma or the Genetic Code alone; they address the flow of information but not the physical structure enabling replication

is not implied by the double-helix structure or the universality of codons; it specifically prescribes transcription and translation as core processes

Central Dogma. is not implied by the double-helix structure or the universality of codons; it specifically prescribes transcription and translation as core processes

does not follow from the Central Dogma or replication; it concerns the specific mappings from codons to amino acids

Genetic Code Universality. does not follow from the Central Dogma or replication; it concerns the specific mappings from codons to amino acids

Segregation

Regulation of Gene Expression. cannot be deduced from any combination of the preceding axioms alone, since it describes how the basic processes are modulated and controlled, rather than how they fundamentally operate in sequence. M.3 Transformations M.3.1 Axiom 1 Goal: Transfo...

We assume each considered function f is 2π-periodic and integrable (e.g

Periodicity & Integrability ∀ f : f is 2π-periodic ∧ f ∈ L1[−π, π] . We assume each considered function f is 2π-periodic and integrable (e.g. piecewise continuous or in L1[−π, π]). This ensures integrals of f over one period are well-defined

This states that the set {e i n x| n ∈ Z} forms an orthogonal system on [−π, π]

Orthogonality of Exponential Basis ∀ m, n∈ Z, Z π −π e i m xe− i n xdx = 2 π δm,n, where δm,n is the Kronecker delta (= 1 if m = n, 0 otherwise). This states that the set {e i n x| n ∈ Z} forms an orthogonal system on [−π, π]

Each f admits coefficients (cn) given by the above integral

Fourier Coefficient Formula ∀ f, c n = 1 2π Z π −π f (x) e− i n xdx (∀ n ∈ Z). Each f admits coefficients (cn) given by the above integral. In some formulations, n ranges over integers for exponential expansions, or includes separate sine/cosine expansions

Under suitable conditions (e.g

Fourier Series Representation f (x) ∼ ∞X n=−∞ cn e i n x, meaning f (x) is represented (pointwise, or in some norm sense) by the infinite series whose nth term is cn e i n x. Under suitable conditions (e.g. Dirichlet, Jordan, or L2 criteria), this series converges to f in the ...

(Axiom 1) sets the domain and guarantees well-defined integrals over one period

Periodicity & Integrability. (Axiom 1) sets the domain and guarantees well-defined integrals over one period

Axiom 2 and Axiom 3 provide a systematic way to compute coefficients from an orthogonal exponential basis, ensuring each f has a unique set {cn}

Orthogonality & Coefficients. Axiom 2 and Axiom 3 provide a systematic way to compute coefficients from an orthogonal exponential basis, ensuring each f has a unique set {cn}

(Axiom 4) states that f can be expressed as a (possibly infinite) sum of the exponential basis functions weighted by these coefficients, completing the idea of a Fourier expansion

Series Representation. (Axiom 4) states that f can be expressed as a (possibly infinite) sum of the exponential basis functions weighted by these coefficients, completing the idea of a Fourier expansion. From these four, one can derive the fundamental properties of Fourier ser...

Periodicity & Integrability. cannot be deduced from orthogonality, coefficient formulas, or the final series statement; it specifically restricts the function’s domain and integrable property, laying the groundwork for meaningful integrals

cannot be inferred from the other axioms

Orthogonality of Exponential Basis. cannot be inferred from the other axioms. It must be stated that {e i n x} is orthogonal under the integral inner product, as this is not implied by the mere existence of a periodic integrable function or a coefficient formula

is not derivable from periodicity, orthogonality alone, or the final series representation

Fourier Coefficient Formula. is not derivable from periodicity, orthogonality alone, or the final series representation. One must explicitly define how cn are computed from f

mother wavelet

Fourier Series Representation. does not follow from any combination of periodic integrability, orthogonality, or the definition of coefficients. It is an additional statement that such a series does represent (or approximate) f in a valid sense, and thus must be stated as a se...

mother wavelet

Mother Wavelet & Integrability ∃ ψ : Z ∞ −∞ ψ(x) dx = 0, Z ∞ −∞ |ψ(x)| dx <∞. There is a single function (the “mother wavelet”) ψ(x) with zero mean and finite integral norm, ensuring local oscillation and admissibility

A scaled and shifted family is constructed from ψ, parameterized by integer j (scale) and k (position)

Dilation & Translation ∀ j ∈ Z, ∀ k ∈ Z, ψ j,k(x) = 2 j 2 ψ 2j x − k . A scaled and shifted family is constructed from ψ, parameterized by integer j (scale) and k (position). This defines the wavelet system {ψj,k}

For anorthonormal wavelet, distinct wavelets in {ψj,k} are pairwise orthogonal in L2(R)

Orthogonality (or Tight Frame) Z ∞ −∞ ψj,k(x) ψj′,k′ (x) dx = δj,j′ δk,k′ . For anorthonormal wavelet, distinct wavelets in {ψj,k} are pairwise orthogonal in L2(R). (In a more general frame setting, one replaces this with a tight-frame condition.)

Every square- integrable function f can be reconstructed from its wavelet coefficients ⟨f, ψj,k⟩ via the orthonormal (or tight) wavelet basis

Wavelet Expansion ∀ f ∈ L2(R) : f (x) = X j,k∈Z f, ψj,k ψj,k(x), where the series converges in L2 norm (or pointwise under additional regularity). Every square- integrable function f can be reconstructed from its wavelet coefficients ⟨f, ψj,k⟩ via the orthonormal (or tight) wa...

(Axiom 1) imposes the zero-mean and finite-energy properties necessary for localization in both time (or space) and frequency domains

Mother Wavelet & Integrability. (Axiom 1) imposes the zero-mean and finite-energy properties necessary for localization in both time (or space) and frequency domains

(Axiom 2) organizes the wavelet system by scaling and shifting ψ, creating a multi-scale family of analyzing functions

Dilation & Translation. (Axiom 2) organizes the wavelet system by scaling and shifting ψ, creating a multi-scale family of analyzing functions

(Axiom 3) ensures each ψj,k is orthonormal to all others, a key property for simplifica- tion of expansions and coefficient computations

Orthogonality. (Axiom 3) ensures each ψj,k is orthonormal to all others, a key property for simplifica- tion of expansions and coefficient computations

(Axiom 4) completes the theory by stating that any function (in L2) can be expressed as a linear combination of these wavelets, ensuring a full basis for signal reconstruction

Wavelet Expansion. (Axiom 4) completes the theory by stating that any function (in L2) can be expressed as a linear combination of these wavelets, ensuring a full basis for signal reconstruction. From these four, one obtains the standard results of wavelet theory: multi-resolu...

cannot be deduced from dilation/translation structure, orthogonality, or the final expansion statement

Mother Wavelet & Integrability. cannot be deduced from dilation/translation structure, orthogonality, or the final expansion statement. It sets the fundamental shape and integrable nature of the wavelet, a separate requirement

is not derivable from zero-mean integrability, orthogonality conditions, or the statement of expansions

Dilation & Translation. is not derivable from zero-mean integrability, orthogonality conditions, or the statement of expansions. It must be explicitly stipulated that ψ generates a family via scale and shift

does not follow from mother wavelet assumptions, nor from simply dilating/translating ψ

Orthogonality. does not follow from mother wavelet assumptions, nor from simply dilating/translating ψ. It is a special structural property (or in general, a tight-frame condition) that must be imposed

mean-free

Wavelet Expansion. cannot be concluded from integrability, the existence of a dilated/translated family, or orthogonality alone. It is an additional global claim that every L2 function can be represented in this wavelet basis. O.3 Transformations O.3.1 Axiom 1 Goal: Transform ...

Re-index from n ∈ Z to (j, k) ∈ Z2 to allow a two-dimensional (scale, shift) parameter space

Replace exponentials by a localized function ϕ subject to dilation and translation

Inner Products

Introduce the 2j/2 factor to preserve L2-norm consistency across scales, yielding the standard wavelet family. Hence, we arrive at the wavelet system {ψj,k} from the original concept of a discrete orthogonal set {e i n x}. O.3.3 Axiom 3 Goal: Transform a subset of Classical Fo...

Rewrite the coefficient formula in terms of an inner product with a basis function

Generalize from exponentials to any orthonormal family {ψα}

Hence, the uniqueness of Fourier coefficients morphs into the wavelet requirement that each basis elementψj,k is orthogonal to all the others in the wavelet system

Specialize to wavelets ψj,k, asserting orthogonality across scales and shifts, giving the wavelet orthogonality axiom. Hence, the uniqueness of Fourier coefficients morphs into the wavelet requirement that each basis elementψj,k is orthogonal to all the others in the wavelet s...

Rewrite Fourier sum as an orthonormal expansion P n⟨f, ϕn⟩ ϕn

Extend to a general orthonormal basis {ψα} in L2(R), removing the 2π-periodic domain restriction

Thus, the final statement W4 proclaims a full wavelet expansion for any square-integrable function, analogous to a Fourier series but in time-frequency localized wavelets

Restrict to wavelet basis {ψj,k}, yielding a discrete sum over scales and shifts that covers L2(R). Thus, the final statement W4 proclaims a full wavelet expansion for any square-integrable function, analogous to a Fourier series but in time-frequency localized wavelets. P Fro...

Matter consists of fundamental, indivisible units called atoms, each associated with a specific element and a unique atomic mass

Atomic Theory ∃ fundamental units (atoms) ∀ elements, each element has unique atomic mass mA. Matter consists of fundamental, indivisible units called atoms, each associated with a specific element and a unique atomic mass

No net mass is lost or gained in an isolated chemical reaction

Conservation of Mass ∀ Reaction R : X i∈Reactants mi = X j∈Products mj, where mi is the total mass of reactant i and mj is the total mass of product j. No net mass is lost or gained in an isolated chemical reaction

, Ek) such that m(E1) m(E2) = constant, for a fixed ratio of masses of any two elements E1, E2 in C

Definite Composition ∀ Compound C : ∃ (elements E1, . . . , Ek) such that m(E1) m(E2) = constant, for a fixed ratio of masses of any two elements E1, E2 in C. A pure chemical compound always contains the same elements in the same proportions by mass

m(B)C2 m(A)C2 = p q , where p, q∈ Z+ are small integers

Multiple Proportions ∀ (elements A, B), ∀ (compounds C1, C2 of A, B) : m(B)C1 m(A)C1 . m(B)C2 m(A)C2 = p q , where p, q∈ Z+ are small integers. If two elements form more than one compound, then the ratio of the masses of one element that combine with a fixed mass of the other ...

molecular particles

Avogadro’s Law (Classical Form) ∀ (gases G1, G2) ∀ V > 0, (same T, P) : N (G1, V) N (G2, V) = 1, where N (Gi, V) is the number of “molecular particles” of gas Gi occupying volume V at the same temperature T and pressure P . Equal volumes of any ideal gases, under the same cond...

( Axiom 1) establishes that matter is composed of fundamental, indivisible atoms, which serve as the basis for all chemical reactions and mass relationships

Atomic Theory. ( Axiom 1) establishes that matter is composed of fundamental, indivisible atoms, which serve as the basis for all chemical reactions and mass relationships

( Axiom 2) ensures that in a closed system, mass is neither created nor destroyed during chemical reactions

Conservation of Mass. ( Axiom 2) ensures that in a closed system, mass is neither created nor destroyed during chemical reactions

Definite Composition & Multiple Proportions. Axiom 3 and Axiom 4 describe how elements combine in fixed, well-defined ratios by mass, and how multiple distinct compounds from the same elements exhibit integer mass-ratio relationships

Avogadro’s Law. ( Axiom 5) extends these foundational mass relationships to the realm of gases, providing a direct link between macroscopic volumes and microscopic particle counts under uniform conditions. These five axioms together allow prediction of reaction stoichiometry, ...

cannot be deduced from mass conservation, definite composition, multiple proportions, or Avogadro’s law

Atomic Theory. cannot be deduced from mass conservation, definite composition, multiple proportions, or Avogadro’s law. It asserts the discrete nature of matter, which is an independent foundational concept

Conservation of Mass. cannot be deduced from atomic theory, definite composition, multiple propor- tions, or Avogadro’s law, which all presume mass-based or volumetric combinations of elements but do not guarantee total mass constancy in reactions

is not derivable from mass conservation, atomic theory, multiple proportions, or Avogadro’s law alone

Definite Composition. is not derivable from mass conservation, atomic theory, multiple proportions, or Avogadro’s law alone. It specifically states each compound has a fixed mass ratio of elements, a separate assertion from the invariance or ratio rules of other axioms

Multiple Proportions. cannot be obtained from atomic theory, mass conservation, or definite composi- tion alone (each compound’s internal ratio is fixed, but this does not imply theinteger ratio pattern across multiple compounds) nor from Avogadro’s law (gas volume relationshi...

does not follow from any combination of the other four axioms

Avogadro’s Law. does not follow from any combination of the other four axioms. Conservation of mass and elemental ratio axioms describe solid and solution-phase compound formation but do not dictate how gas volumes relate to particle numbers. P.2 Bohr Atomic Theory Bohr’s atom...

Stable Orbits ∀ (electron e in atom A) : e moves in circular orbits about A’s nucleus without radiating electromagnetic energy, whenever in an allowed orbit. Electrons orbit the nucleus under electrostatic attraction, and do not continuously emit energy (as classical electrody...

Only integral multiples of h/(2π) are permitted

Angular Momentum Quantization Ln = n h 2π , n ∈ Z+, where Ln is the orbital angular momentum of an electron in the nth allowed orbit, and h is Planck’s constant. Only integral multiples of h/(2π) are permitted. 115

Each allowed orbit corresponds to a specific quantized energy En

Energy Level Postulate En = − Z 2 e2 8πϵ0 a0 1 n2 , n ∈ Z+, for a hydrogen-like atom of nuclear chargeZe. Each allowed orbit corresponds to a specific quantized energy En. (In the simplest hydrogen case, Z = 1 and a0 is the Bohr radius.)

An electron transitioning between orbit m and n releases or absorbs a photon of energy h νequal to the difference in orbital energies

Spectral Transition Condition ∆E = Em − En = h ν, where ν is the frequency of the emitted (or absorbed) photon. An electron transitioning between orbit m and n releases or absorbs a photon of energy h νequal to the difference in orbital energies. P.2.2 Completeness

(Axiom 1) exempts electrons from classical radiation loss in these special orbits, preventing them from spiraling into the nucleus

Stable Orbits. (Axiom 1) exempts electrons from classical radiation loss in these special orbits, preventing them from spiraling into the nucleus

Axiom 2 and Axiom 3 specify how electron orbits are discretized in angular momentum and corresponding energies, addressing why only certain atomic radii/energies are observed

Quantization (Angular Momentum & Energy Levels). Axiom 2 and Axiom 3 specify how electron orbits are discretized in angular momentum and corresponding energies, addressing why only certain atomic radii/energies are observed

(Axiom 4) connects these discrete energy levels to observed emission/absorption lines

Spectral Transitions. (Axiom 4) connects these discrete energy levels to observed emission/absorption lines. It explains how atomic spectra arise from electrons jumping between quantized states, each jump releasing or absorbing a photon of specific energy. Together, these four...

cannot be deduced from angular-momentum quantization, energy-level discretization, or spectral transitions alone

Stable Orbits. cannot be deduced from angular-momentum quantization, energy-level discretization, or spectral transitions alone. It specifically states that electrons do not radiate in those allowed orbits

does not follow from stable orbits, energy postulates, or photon emission rules

Angular Momentum Quantization. does not follow from stable orbits, energy postulates, or photon emission rules. One must explicitly declare that orbital angular momentum comes in multiples of h/(2π)

is not derivable from stable orbits, angular-momentum quantization, or transitions alone

Energy Level Postulate. is not derivable from stable orbits, angular-momentum quantization, or transitions alone. Assigning a particular En ∝ 1/n2 is a separate assumption tied to specific electrostatic coupling and boundary conditions in the hydrogenic system

solar system

Spectral Transition Condition. does not follow from stable orbits, angular-momentum quantization, or the specific energy formula alone. One must posit that electrons emit/absorb single photons with energies equal to differences of orbit energies. P.3 Transformations P.3.1 Axio...

All celestial bodies X have their primary orbital centers at Earth (which is fixed and unmoving)

Central Earth ∀ X ̸= Earth : CenterOrbit( X) = Earth. All celestial bodies X have their primary orbital centers at Earth (which is fixed and unmoving)

Uniform Circular Motion ∀ X : Orbit( X) = X (UniformCircles), i.e., each orbit is either a single circle or a finite sum of epicyclic circles, all traversed with constant angular velocity

Sphere of Fixed Stars ∃ S⋆ : ( all stars are on S⋆) ∧ (dist(S⋆, Earth) = constant), where S⋆ is a single spherical shell (or near-sphere) centered on Earth, rotating daily to carry the stars around

Q.1.2 Completeness

Epicycles for Planets ∀ P ∈ {planets} : Orbit( P ) = (deferent circle) + (epicycle circle), where each planet P moves on a small circle (epicycle) whose center itself moves along a larger circle (deferent) around Earth, explaining retrograde motion and varying brightness. Q.1....

(Axiom 1) fixes Earth as the motionless center for all celestial bodies

Central Earth. (Axiom 1) fixes Earth as the motionless center for all celestial bodies

Uniform Circular Motion & Fixed-Star Sphere. Axioms 2 and 3 assert that celestial paths are composed of one or more uniform circles and that stars reside on a single rotating sphere, accounting for daily rotation and consistent stellar distances

Epicycles. (Axiom 4) extends the simple circular notion to capture observed planetary anomalies (retrograde, brightness changes) by introducing a small circle (epicycle) superimposed on a main orbit (deferent). From these four, the classical Ptolemaic geocentric cosmology can ...

cannot be deduced from uniform circular motion, the star sphere, or epicyclic constructs

Central Earth. cannot be deduced from uniform circular motion, the star sphere, or epicyclic constructs. One must explicitly posit that orbits are centered on Earth

is not derivable from Earth-centrism, the star sphere, or epicycles alone; it establishes a geometric ideal for celestial movements (constant angular velocities on perfect circles)

Uniform Circular Motion. is not derivable from Earth-centrism, the star sphere, or epicycles alone; it establishes a geometric ideal for celestial movements (constant angular velocities on perfect circles)

does not follow from Earth-centrism, uniform circles, or planetary epicycles

Sphere of Fixed Stars. does not follow from Earth-centrism, uniform circles, or planetary epicycles. It specifically asserts that all stars are affixed to a single spherical shell around Earth

cannot be inferred from the other three axioms

Epicycles. cannot be inferred from the other three axioms. Even with Earth-centrism and uniform circles for some bodies plus the star sphere, explicit epicycles are required to handle planetary retrograde motions and brightness variations. Q.2 Heliocentric Model The heliocentr...

All planets P (including Earth) move around the Sun as the primary orbital center

Central Sun ∀ P ̸= Sun : CenterOrbit( P ) = Sun. All planets P (including Earth) move around the Sun as the primary orbital center

Earth’s Dual Motion Rotation(Earth, 24h) ∧ Revolution(Earth, T⊕) Earth both rotates on its own axis once per day (producing daily cycles) and orbits the Sun with period T⊕ ≈ 1 year

Uniform Circular Orbits (Copernican Ideal) ∀ (planet P ) : Orbit( P ) = X (UniformCircles), each planet’s path around the Sun is modeled as one or more perfect circles (possibly epicycles) traversed with constant angular speed, accounting for observed phenomena such as retrogr...

Distant Fixed Stars & Minimal Parallax ∃ S⋆ : (dist( S⋆, Sun) ≫ dist(planet, Sun)) ∧ (stellar parallax ≈ 0). Stars lie on a vastly larger sphere (or shell) far beyond planetary distances, so any annual stellar parallax is negligible (explaining why no parallax was detected wit...

(Axiom 1) places the Sun at the orbital focus for all planets, including Earth

Central Sun. (Axiom 1) places the Sun at the orbital focus for all planets, including Earth

(Axiom 2) explains daily cycles (Earth’s rotation) and seasonal phenomena (Earth’s annual revolution)

Earth’s Dual Motion. (Axiom 2) explains daily cycles (Earth’s rotation) and seasonal phenomena (Earth’s annual revolution)

(Axiom 3) maintains the classical ideal of circular (or epicyclic) paths at uniform angular velocities, reproducing retrograde loops from the new perspective of Earth’s own motion

Uniform Circular Orbits. (Axiom 3) maintains the classical ideal of circular (or epicyclic) paths at uniform angular velocities, reproducing retrograde loops from the new perspective of Earth’s own motion

(Axiom 4) accounts for the absence of observed stellar parallax, attributing it to the immense distance of the fixed stars relative to planetary orbits

Distant Stars. (Axiom 4) accounts for the absence of observed stellar parallax, attributing it to the immense distance of the fixed stars relative to planetary orbits. From these four axioms, one reconstructs a Copernican-style heliocentric cosmology, explaining the same pheno...

cannot be derived from Earth’s dual motion, uniform circular orbits, or distant stars alone

Central Sun. cannot be derived from Earth’s dual motion, uniform circular orbits, or distant stars alone. It must be stated that the Sun (not Earth) is the center of planetary paths

is not inferable from the Sun-centric arrangement, purely circular orbits, or the distant star sphere

Earth’s Dual Motion. is not inferable from the Sun-centric arrangement, purely circular orbits, or the distant star sphere. One must explicitly claim that Earth rotates once per day and revolves annually

does not follow from any combination of a Sun-centered system, Earth’s rotation/revolution, or distant stars

Uniform Circular Orbits. does not follow from any combination of a Sun-centered system, Earth’s rotation/revolution, or distant stars. It specifically asserts the classical geometric principle that planetary paths are formed by perfect circles (or sums thereof)

All orbits are centered on Earth

Distant Fixed Stars. is not implied by the other three axioms. One must introduce the idea that the stars are so far away that no measurable parallax is observed, reconciling the lack of star shift throughout Earth’s orbit. Q.3 Transformations Q.3.1 Axiom 1 Goal: Transform the...

Excess epicycles in a purely Earth-centered system suggest an overly complex framework

Earth’s behavior is akin to other planets, motivating the notion that Earth, too, orbits a central luminary

Hence, Geocentric Axiom 1 (Earth at center) transforms into Heliocentric Axiom 1 (Sun at center) once one reinterprets Earth as simply another planet

Sun at center simplifies orbits (especially retrograde explanation), yielding Copernicus’s heliocentric principle. Hence, Geocentric Axiom 1 (Earth at center) transforms into Heliocentric Axiom 1 (Sun at center) once one reinterprets Earth as simply another planet. 124 Q.3.2 A...

Uniform Circular Motion (geocentrism) includes a daily star-sphere plus epicycles for planets

Assign daily rotation to Earth instead of the entire star-sphere, simplifying the model drastically

sphere of stars

Add Earth’s annual revolution around the Sun, forming the twofold motion central to heliocentrism. Hence, Geocentric Axiom 2 (all bodies follow sums of uniform circles around Earth, including a daily star-sphere) is replaced by Heliocentric Axiom 2 (Earth rotates daily and rev...

Sphere of Fixed Stars is replaced by the idea that star motion is apparent (due to Earth’s spin), so no actual rotating sphere is needed near Earth

Focus on planetary orbits around the Sun, relegating stars to a distant background

overtakes

Assert uniform circular orbits for planets in a Copernican sense, retaining the classical ideal of perfect circles (or sums of them) for each planet’s path. Thus, Geocentric Axiom 3 (a rotating sphere of stars around Earth) becomes Heliocentric Axiom 3 (each planet’s orbit aro...

Epicycles are largely unnecessary once Earth’s orbital motion explains retrograde loops

No observed stellar parallax suggests stars are extremely distant

R From Heliocentric to Cosmological Model R.1 Heliocentric Model This section is the same as Section Q.2

Hence, we adopt H_4, stating stars lie at vast distances, producing negligible parallax. R From Heliocentric to Cosmological Model R.1 Heliocentric Model This section is the same as Section Q.2. R.2 Cosmological Model A modern cosmological model (as in the Friedmann–Lemaître–R...

No position in the universe is preferred on sufficiently large (cosmological) scales

Homogeneity ∀ (comoving observers O1, O2), ρ O1, t = ρ O2, t , where ρ is the average matter-energy density in a given large-scale region at cosmic time t. No position in the universe is preferred on sufficiently large (cosmological) scales

No direction is special for a comoving observer on large scales; all directions appear statistically the same (e.g

Isotropy ∀ (comoving observer O), ∀ (directions d1, d2) : Properties( d1, O) = Properties(d2, O). No direction is special for a comoving observer on large scales; all directions appear statistically the same (e.g. same average temperature, same large-scale structure distribution)

All cosmologically relevant distances scale uniformly by a(τ ) over time

Expansion (Scale Factor) Metric(τ ) = a(τ )2 dΣ2, where a(τ ) is the scale factor (a function of cosmic time τ) and dΣ2 is a (fixed) 3D spatial metric of constant curvature. All cosmologically relevant distances scale uniformly by a(τ ) over time

This governs how the scale factor a(τ ) evolves with cosmic time according to matter, radiation, and other energy contents

Relativistic Dynamics (Einstein Field Equations) Gµν + Λ gµν = 8π G c4 Tµν , where Gµν is the Einstein tensor, Λ is the cosmological constant, gµν is the metric, Tµν is the stress- energy tensor of matter/energy, and G is Newton’s gravitational constant. This governs how the s...

(Axioms 1, 2) specify the large-scale distribution of matter/energy in the universe, ensuring no preferred locations or directions at cosmic scales

Homogeneity & Isotropy. (Axioms 1, 2) specify the large-scale distribution of matter/energy in the universe, ensuring no preferred locations or directions at cosmic scales