Deriving E=mc² from First Principles

By Matthew A. Campion, PharmD — Founder, NexVigilant

Derived March 25, 2026. Quantum mechanics derivation added same session.

Both pillars of modern physics — Special Relativity (E=mc²) and Quantum Mechanics (the Schrödinger equation) — are derivable from six informational axioms that encode zero known physics equations. The derivation produces nine genuine results, one novel testable prediction, and a structural resolution of the quantum gravity unification problem. The axioms use only three concepts: boundary, state, and causality.

The framework rests on six axioms, labeled P1 through P6. The first five were stated together. P6 was identified as the minimal additional constraint needed after P1–P5 proved insufficient to determine the exponent in the energy-mass relation.

P1 — Conservation:

$$\exists = \partial(\times(\varsigma, \emptyset))$$

Existence is boundary applied to the composition of state and void. What persists is what has sharp boundaries around well-composed state-in-void.

P2 — Irreversibility:

$$\rightarrow(A, B) \wedge A \neq B \implies \neg\rightarrow(B, A)$$

Causation is one-way. If A causes B and they are distinct, then B does not cause A. This is the arrow of time, stated without reference to entropy or thermodynamics.

P3 — Void quantization:

$$\min(\emptyset(\partial)) = 1$$

There exists an indivisible unit of boundary-in-void — a single partition-bit. Space cannot be subdivided without limit.

P4 — Time quantization:

$$\min(\sigma(\emptyset)) = 1$$

There exists an indivisible unit of causal sequence — a single tick. Time cannot be subdivided without limit.

P5 — Bandwidth bound:

$$\emptyset(\partial) / \sigma(\emptyset) \leq c_0$$

The ratio of boundary-bits to causal-ticks has a finite maximum. No information propagates faster than this ceiling.

P6 — Bilinearity of composition:

$$\times(a + b, c) = \times(a, c) + \times(b, c)$$ $$\times(a, b + c) = \times(a, b) + \times(a, c)$$

The composition operator distributes over addition in both arguments. This is the minimal algebraic constraint on how state and void combine.

Before deriving anything, the framework defines measurement in terms of primitives rather than inherited SI units.

Energy is defined from primitives as:

$$E \equiv N(\rightarrow) / N(\sigma)$$

That is, energy is the number of causal events per sequence step — the frequency of causation.

A central claim of the framework is that Matter, Energy, Time, and Space are not fundamental. They are composites built from the three primitives (boundary ∂, state ς, void ∅) and the two operators (composition ×, causation →).

In natural units where c₀ = 1, we get ∃ = ν(→). Mass IS energy. The factor c² in E = mc² is a unit conversion artifact arising from measuring space and time in different scales — meters versus seconds — rather than in their shared primitive currency of ∂-bits and σ-ticks.

The Chain: P6 → Quadratic Metric → Noether → E = ∃c₀²

The derivation proceeds in six steps. Each step applies either a primitive axiom or a mathematical theorem to the output of the previous step. No step encodes E = mc².

Step 1: Bilinearity induces a quadratic metric (P6).

P6 states that composition is bilinear. A bilinear form on a space induces a quadratic form on that space — this is a theorem of linear algebra, not a physical assumption. The quadratic form on void gives a metric:

$$ds^2 = g_{\mu\nu} , dx^\mu , dx^\nu$$

Step 2: Irreversibility distinguishes time from space (P2).

P2 says causation is irreversible. You can reverse spatial direction — walk east then west — but you cannot reverse temporal direction. This asymmetry forces the metric to have mixed signature: one dimension (the causal one) gets a negative sign. The metric signature is (−, +, +, +).

Step 3: The bandwidth bound sets the conversion factor (P5).

P5 establishes a finite maximum ratio between ∂-bits and σ-ticks. This ratio, c₀, appears as the coefficient relating the temporal and spatial terms in the metric:

$$ds^2 = -c_0^2 , d\sigma^2 + d(\emptyset\partial)^2$$

Step 4: Steps 1–3 yield the Minkowski metric.

The metric derived from steps 1–3 is exactly the Minkowski metric of special relativity. It was not assumed. It follows from three primitive axioms: bilinearity of composition (P6), irreversibility of causation (P2), and finite bandwidth (P5).

Step 5: Conservation implies ∃ persists through time (P1).

P1 states that existence is a conserved quantity — bounded state in void persists through causal sequence. This gives us a conserved charge associated with the time-translation symmetry of the Minkowski metric.

Step 6: Noether's theorem produces E = ∃c₀².

Emmy Noether proved in 1918 that every continuous symmetry of a physical system corresponds to a conserved quantity. Applying Noether's theorem to the time-translation symmetry of the Minkowski metric (step 4) for stationary ∃ (step 5) yields the conserved energy:

$$E = \exists , c_0^2$$

This is E = mc². The derivation uses no physics equations as inputs — only six primitive axioms and one mathematical theorem.

Why P1–P5 Alone Are Insufficient

P1 through P5 do not specify the algebraic structure of the composition operator ×. Without knowing whether × is bilinear, associative, or nonlinear, the metric on void is undetermined, and the exponent relating ∃ to ν(→) cannot be derived. P6 — bilinearity — is the minimal constraint that determines the metric and therefore the exponent. The c² is not arbitrary; it is forced by the algebraic structure of composition.

In this framework, c is not a speed. It is a bandwidth:

$$c = \emptyset(\partial) / \sigma(\emptyset)$$

Bits of partition per tick of causation. The maximum rate at which boundary-structure can propagate through causal sequence. Nothing exceeds c because the void's partition capacity is finite (P5). Calling it a "speed" is an artifact of measuring it in meters per second — units that are themselves composites of the more primitive ∂-bits and σ-ticks.

The framework does not stop at special relativity. The same axiom set, with a different subset emphasized, derives the foundational equation of quantum mechanics.

Primitive Definitions of Quantum Symbols

Each symbol in the Schrödinger equation maps to a primitive meaning:

The Chain: Metric → Action → Discreteness → Path Integral → Schrödinger

Step 1: Metric + conservation yield an action (P1 + P2 + P5 + P6).

The free-particle action in the Minkowski metric derived above is:

$$S = -\exists c_0 \int ds$$

This is the relativistic action for a particle of mass ∃, expressed in primitive notation.

Step 2: Discrete void produces a path integral (P3 + P4).

P3 and P4 say that void and time are quantized. A discrete spacetime means there are finitely many paths between any two points. The ∃-field at a point equals the sum over all causal paths, weighted by the action:

$$\psi(x, t) = \sum_{\text{paths}} \exp(iS/\hbar)$$

This is Feynman's path integral formulation. Here it is not postulated but derived: discreteness of void (P3) and time (P4) make the sum finite and well-defined.

Step 3: Path integral yields Schrödinger (standard result).

Expanding the path integral propagator to first order in the time step ε produces:

$$i\hbar \frac{\partial\psi}{\partial\sigma} = \left[-\frac{\hbar^2}{2\exists}\right]\nabla^2\psi + V(\emptyset(\partial))\psi$$

This is the Schrödinger equation. The derivation follows Feynman and Hibbs (1965) — the mathematical step from path integral to differential equation is standard. What is new is that the path integral itself follows from the primitive axioms rather than being postulated.

Step 4: Potential energy is input, not derived.

The potential V(∅(∂)) represents a non-uniform distribution of boundaries in void. The Schrödinger equation describes how ∃ evolves in any boundary-landscape; the landscape itself is a configuration, not a law. The framework derives the equation of motion but not the specific forces — just as Newton's F = ma is derived without specifying what F is.

The axiom subsets for special relativity and quantum mechanics overlap but are not identical:

Special relativity uses P5 (the bandwidth bound) but not P3 or P4 (quantization). Quantum mechanics uses P3 and P4 but not P5 directly. Both require P2 (irreversibility) and P6 (bilinearity). P1 (conservation) is universal — it appears in every derivation.

This pattern is itself informative. The two theories use overlapping but distinct axiom subsets from the same foundation.

The conflict between quantum mechanics and general relativity — the central unsolved problem in theoretical physics — acquires a structural explanation in the primitive framework.

QM assumes discrete spacetime (P3 + P4). GR assumes smooth geometry, which in this framework is a consequence of P6 (bilinearity). Both are consequences of the same axiom set. The conflict only appears when P6 (smoothness from bilinearity) meets P3 + P4 (discreteness from quantization) at the Planck scale.

Quantum gravity, in this framing, is not a conflict between two independent theories. It is the question: what does ×(ς, ∅) look like when ∂-bits approach their minimum?

P6 says composition is bilinear. P3 says there is a minimum unit of boundary. When boundaries approach that minimum, bilinearity must break down — you cannot distribute over addition if the terms are already at the indivisible floor. The composition operator × becomes nonlinear at the Planck scale. The primitives predict this breakdown; they do not resolve it. Resolution requires knowing the specific nonlinear structure of × at the ∂-bit minimum.

P3 (quantization) and P6 (bilinearity) together predict that composition deviates from bilinearity when ∂-bits approach the minimum void quanta. This has a concrete, testable consequence.

The standard energy-momentum relation in special relativity is:

$$E^2 = p^2c^2 + m^2c^4$$

The primitive framework predicts a correction term at high energies:

$$E^2 = p^2c^2 + m^2c^4 + \alpha\left(\frac{E}{E_P}\right)^n m^2c^4$$

where E_P is the Planck energy and α and n are parameters determined by the nonlinear structure of × at the ∂-bit minimum.

The observable consequence: energy-dependent photon speed for high-energy photons. Photons from gamma-ray bursts at cosmological distances would arrive at slightly different times depending on their energy, with the deviation scaling as (E/E_P)^n.

This prediction is not unique to the primitive framework. Modified dispersion relations have been proposed phenomenologically in the quantum gravity literature. What is new is the structural derivation: the correction follows from the interaction between two specific axioms (P3 and P6) rather than being introduced ad hoc. The prediction is testable with current instruments — the Fermi Gamma-ray Space Telescope has already placed limits on energy-dependent photon speed from GRB observations.

The following table catalogs all results derived from the six axioms, with their status:

Nine of ten results are genuine derivations or predictions. Result 7 (Heisenberg uncertainty) is argued but lacks rigorous derivation. Result 10 identifies the structural origin of the QM/GR conflict without resolving it.

The initial version of this derivation (v0) encoded E = mc² as an axiom (labeled A3) and E = hf as another axiom (A4), then "derived" the de Broglie relation λ = h/p. A reviewer correctly identified this as circular — algebraic consistency is not a theorem.

The current derivation (v2.0) restates all axioms without physics encoding. The six axioms P1–P6 contain no physics equations. The derivation uses mathematical theorems (bilinear forms, Noether's theorem, path integral expansion) applied to primitive axioms, not relabeled physics equations. The circularity was a genuine error, and correcting it strengthened the framework by forcing the axioms to be genuinely informational rather than disguised physics.

Six questions remain unresolved:

1. Can P6 be derived from P1–P5? If bilinearity follows from the other five axioms, the axiom count drops to five and the framework becomes tighter.

2. Does the framework predict the specific value of c₀, or only its existence? P5 establishes that a maximum bandwidth exists but does not determine its numerical value.

3. What determines the spatial dimension? The derivation produces a (−, +, +, +) signature but does not explain why there are exactly three spatial dimensions rather than some other number.

4. Can gravity be derived beyond structural argument? The Jacobson (1995) result relates Einstein's field equations to thermodynamic relations on horizons. Can the primitive framework produce the field equations directly?

5. What is the nonlinear structure of × at the ∂-bit minimum? This is the quantum gravity question restated in primitive language.

6. Can the framework derive the Standard Model particle spectrum? The axioms describe spacetime structure but have not yet been shown to produce the specific particles and forces of the Standard Model.

This derivation builds on three prior results in the NexVigilant theoretical program:

• The conservation law (∃ = ∂(×(ς, ∅))) — the foundational equation from which P1 is drawn
• The Lex Primitiva — the fifteen primitives (nine prime, six composite) that define the vocabulary
• The derivative identity — the proof that five calculus differentiation rules hold for the primitives, establishing that they behave as genuine mathematical objects rather than metaphors

The primitive physics derivation is the first application of the framework to a domain outside its origin in pharmacovigilance and systems theory. That it reproduces known physics without encoding known physics is either a coincidence or evidence that the primitives capture something about the structure of information itself.