Why Spacetime Is Four-Dimensional and Matter Is Dirac: Making Probability Geometric Leaves No Other Choice

1. Introduction

1.1. Stance (Motivation Only)

Physical laws are conclusions drawn from observation, not axioms the world is handed. What we measure are probabilities; the laws are our most economical summaries of them. This paper takes that stance literally for two structural facts—the dimension of the spacetime metric and the form of the relativistic wave equation—by treating a probability, rather than a law, as the primitive object, and asking what its mere existence and conservation require. The philosophy ends in this paragraph; everything that follows is algebra and computation.

1.2. The Primitive Object

We posit one object: a geometric probability, an element $\psi$ of the even subalgebra of the Clifford algebra $C\ell \left(V\right)$ of the tangent space V of spacetime, equipped with

(P1) 

a positive-definite scalar density $\rho ={\langle \tilde{\psi }{\gamma }_0\psi \rangle }_0\ge 0$, and

(P2) 

a conserved grade-1 current $J=\psi {\gamma }_0\tilde{\psi }$ that is a pure vector (no higher-grade parts).

(P1) is the Born requirement: a probability density must be real and non-negative. (P2) is the requirement that probability flow be described by a conserved four-vector current admitting a continuity equation—the minimal condition for a relativistic probability balance.

1.3. Results (Summary)

Theorem 1 

(Kinematic ceiling). (P1) and (P2) hold if and only if the metric is Lorentzian with exactly one time dimension and total dimension $n\le 4$. The admissible signatures are $(1,0)/(0,1)$, $(1,1)$, $(2,1)/(1,2)$, $(3,1)/(1,3)$; all Euclidean $(n,0)$, all split (e.g., $2+2$), all multi-time, and all $n\ge 5$ signatures are excluded.

Theorem 2 

(Dynamical form). On an admissible arena, the unique information-preserving evolution of ψ compatible with the relativistic mass-shell is the Dirac equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$. Information preservation—vanishing entropy production—fixes the flux·force functional to be linear in the gradient, so the resulting flow is first-order without further assumption.

Lemma 1 

(The weld). $J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$ is simultaneously the geometric current and the Onsager entropy flux; the kinetic bilinear $\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$ splits into a real part equal to $\frac{1}{2}{\partial }_{\mu }{J}^{\mu }$ (continuity) and an imaginary part equal to the flux·force entropy-production density.

Together, the dimension of the spacetime metric and the Dirac equation are the kinematic and dynamical consequences of one object. To our knowledge no prior framework derives both jointly from a common origin.

1.4. Relation to Prior Work

Dimensionality. Ehrenfest [1] and Tangherlini [2] argued for three spatial dimensions from the stability of orbits and atoms. Tegmark [3] argued for $(3+1)$ anthropically: alternative signatures yield “dead worlds” in which the partial differential equations of physics are unstable or non-predictive. Our bound is neither anthropic nor stability-based; it is an a priori algebraic consequence of representing a probability current. It precedes both dynamics and observers.

The Dirac equation from information. Frieden’s Extreme Physical Information [4] derives relativistic equations by extremizing Fisher information; being gradient-squared, that functional yields second-order equations, and Dirac is reached by factorizing Klein–Gordon. D’Ariano and Perinotti [5] derive the Dirac equation from the large-scale limit of a minimal quantum cellular automaton constrained by unitarity, locality, homogeneity, and isotropy—“Dirac without relativity.” We differ on the functional (entropy production, linear-in-gradient, hence first-order directly) and on the central claim (an identity—the weld—rather than a derivation).

2. The Geometric Probability Object

Let V be the tangent space with metric g of signature $(p,q)$, $n=p+q$, and let $C\ell \left(V\right)$ be its Clifford algebra, defined by $e_{\mu }{e}_{\nu }+e_{\nu }{e}_{\mu }=2g_{\mu \nu }$. The even subalgebra$C{\ell }^{+}$, spanned by products of an even number of basis vectors, is closed under the geometric product (even · even = even), and hence is closed under both superposition (addition) and rotor action (multiplication by elements R with $R\tilde{R}=1$, acting as $\psi \mapsto R\psi$). It has dimension $2^{n-1}$. Reversion, written $\tilde{\psi }$, reverses the order of vector factors in each term. We single out a unit timelike basis vector $e_0\equiv {\gamma }_0$.

The even subalgebra is the largest subspace of $C\ell \left(V\right)$ closed under both operations a quantum amplitude requires—superposition and rotation. It is in this precise sense the most general admissible amplitude space, and we adopt it as the home of $\psi$.

3. Theorem 1: The Kinematic Ceiling

3.1. Why a Geometric Probability—and Why It Is More Demanding Than a Spinor

Conventional relativistic quantum mechanics represents the state as a column spinor: an element of an abstract complex vector space ${\mathbb{C}}^k$ carrying a representation of the spin group. Such a representation exists in every dimension and every signature—Clifford algebras admit spinor representations whatever the metric, and one may write a Dirac-type equation on a seven-dimensional or a $(2,3)$-signature spacetime without obstruction. The column spinor is a bookkeeping device: its components are complex numbers carrying no intrinsic geometric meaning, and the matrices ${\gamma }^{\mu }$ act on them from outside. Nothing in that construction cares how many dimensions there are, or how many are timelike. The abstract spinor is dimensionally promiscuous because it lives outside spacetime.

We make a stronger demand, in the spirit of Hestenes’ geometric algebra [6,7]: that the probability amplitude be a geometric object livinginspacetime, not an abstract column attached to it. Concretely, $\psi$ is an element of the Clifford algebra of the tangent space itself—a multivector, a sum of a scalar, an oriented plane (bivector), an oriented volume (pseudoscalar), and so on, each piece a genuine geometric quantity in the spacetime where the physics happens. The “internal” spin degrees of freedom are not indices on an external column; they are the higher-grade parts of a single spacetime multivector. The ${\gamma }^{\mu }$ are not external matrices but the basis vectors of spacetime, acting by the geometric product.

This identification—amplitude = spacetime multivector—is what costs us our generality, and that is exactly the point. Once the probability is required to be a geometric object built from the spacetime metric, two quantities that were free in the abstract-spinor picture become fixed by the metric and dimension themselves:

• its density must be a genuine non-negative scalar of that geometry (P1), and
• its current must be a genuine grade-1 vector of that geometry (P2),

because density and current are now computed from the multivector by the geometric product, not freely posited bilinears. In the abstract picture one may declare ${\psi }^{†}\psi$ positive by choosing the inner product; here $\rho ={\langle \tilde{\psi }{\gamma }_0\psi \rangle }_0$ is fixed by the algebra and either is or is not positive depending on the signature. In the abstract picture the current is a vector because one contracts with ${\gamma }^{\mu }$ by hand; here $J=\psi {\gamma }_0\tilde{\psi }$ is whatever grade the geometric product makes it, and only in the right dimensions is it purely grade-1.

So the question Theorem 1 answers is precisely: in which spacetimes can a probability be a geometric object at all—possessing a real positive density and a conserved vector current as intrinsic features of the spacetime geometry, rather than as conventions imposed from outside? The abstract spinor evades the question by living outside spacetime; the geometric probability cannot. The answer, computed below, is: only in Lorentzian signature, with exactly one time dimension, and total dimension $\le 4$. The very rigidity the abstract spinor lacks is what turns a representability requirement into a prediction about the dimension of the spacetime metric.

3.2. Positive-Definiteness Implies Exactly One Time Dimension

For $\psi ={\sum }_k{\langle \psi \rangle }_k$, the scalar density ${\langle \tilde{\psi }{\gamma }_0\psi {\gamma }_0\rangle }_0$ is a signed sum of squares of the component coefficients. The sign attached to each grade-k blade is the product of the reversion sign of grade k, the sign from commuting ${\gamma }_0$ through the blade, and the metric factors $g_{\mu \mu }$ of the basis vectors composing the blade.

Lemma 2. 

The density form is definite (a single overall sign) if and only if at most one basis vector has signature opposite to the rest—i.e., signature $(1,n-1)$ or $(n-1,1)$. This is established exhaustively for $n\le 4$ by the explicit computation tabulated in Section 3.4; dimensions $n\ge 5$ are excluded independently by Lemma 3.

Worked example, $n=4$, signature $(1,3)$ with ${\gamma }_0^2=+1$, writing $\psi =a+F+bI$ (scalar, bivector with six components $F_{ij}$, pseudoscalar):

$${\langle \tilde{\psi }{\gamma }_0\psi {\gamma }_0\rangle }_0=a^2+b^2+\sum _{i<j}{F}_{ij}^2\left(\mathbf{positive}\mathbf{definite}\right).$$

(1)

By contrast, the Euclidean $4+0$ case yields

$${\langle \tilde{\psi }{\gamma }_0\psi {\gamma }_0\rangle }_0=a^2-b^2-F_{01}^2-F_{02}^2-F_{03}^2+F_{12}^2+F_{13}^2+F_{23}^2,$$

(2)

which contains terms of both signs and is therefore genuinely indefinite. The discriminating property at every dimension is the existence of exactly one timelike direction; the full case-by-case computation is collected in Section 3.4.

3.3. Pure-Vector Current Implies $n\le 4$

The even subalgebra carries grades $\{0,2,4,\dots \}$. For $n=4$ it is exactly $\{\mathrm{scalar},\mathrm{bivector},\mathrm{pseudoscalar}\}=1+6+1=8$, so $\psi =a+F+bI$ exhausts it, and the bilinear $J=\psi {\gamma }_0\tilde{\psi }$ is a pure grade-1 vector—the spacetime-algebra Dirac current, satisfying $J^2={\rho }^2\ge 0$ and future-pointing timelike.

Lemma 3. 

For $n\ge 5$ the even subalgebra acquires an independent grade-4 sector. At $n=5$, for instance, $C{\ell }^{+}$ is $1+10+5=16$-dimensional, so $\psi =a+F+Q$ with Q a five-component grade-4 object (notthe pseudoscalar, which at $n=5$ is the odd grade-5 element and lies outside $C{\ell }^{+}$). The bilinear $J=\psi {\gamma }_0\tilde{\psi }$ then acquires grade-3 and grade-5 parts; the current is no longer a pure vector, and (P2) fails. The failure is monotonic in n—higher even grades accumulate as n grows—so the obstruction is permanent for all $n\ge 5$. (Explicit grade-3/grade-5 contamination at $n=5$: Appendix A.)

3.4. The Admissible Family

Two independent criteria act together: the density must be definite (Lemma 2) and the current must be a pure grade-1 vector (Lemma 3). Dimensions $n\ge 5$ are excluded outright by the current test, so the density need only be checked for $n\le 4$—a finite list, which we settle by directly computing ${\langle \tilde{\psi }{\gamma }_0\psi {\gamma }_0\rangle }_0$ in every signature:

n	Signature	${\langle \tilde{\psi }{\gamma }_0\psi {\gamma }_0\rangle }_0$	definite?
1	$1+0$	$a^2$	yes
1	$0+1$	$-a^2$	yes (global sign)
2	$1+1$	$a^2+b^2$	yes
2	$2+0$	$a^2-b^2$	no
2	$0+2$	$-a^2+b^2$	no
3	$2+1,1+2$	$a^2+F_{01}^2+F_{02}^2+F_{12}^2$	yes
3	$3+0$	$a^2-F_{01}^2-F_{02}^2-F_{12}^2$	no
3	$0+3$	$-a^2+F_{01}^2+F_{02}^2+F_{12}^2$	no
4	$3+1,1+3$	$a^2+b^2+{\sum }_{i<j}{F}_{ij}^2$	yes
4	$2+2$	$a^2-b^2$, bivectors split $3+/3-$	no
4	$4+0,0+4$	see Equation (2); terms of both signs	no

The table is exhaustive for $n\le 4$, and $n\ge 5$ is closed by Lemma 3. In every case the density is definite if and only if exactly one basis vector carries a signature opposite to the others—i.e., the signature is $(1,n-1)$ or $(n-1,1)$. (In the indefinite rows the explicit per-term signs depend on basis ordering and conjugation convention; the convention-independent fact, and all that the theorem requires, is that terms of both signs are present.) Imposing both criteria leaves exactly the family

n	Signature	density	current	verdict	fails by
1	$1+0,0+1$	definite	grade-1	✓	—
2	$1+1$	definite	grade-1	✓	—
2	$2+0,0+2$	indefinite	—	×	density indefinite
3	$2+1,1+2$	definite	grade-1	✓	—
3	$3+0,0+3$	indefinite	—	×	density indefinite
4	$3+1,1+3$	definite	grade-1	✓ (max)	—
4	$2+2$	indefinite	—	×	density indefinite
4	$4+0,0+4$	indefinite	—	×	density indefinite
$\ge 5$	any	—	grades 3,5,...	×	current not a vector

The surviving family is exactly $0+1$, $1+1$, $2+1$, $3+1$: every dimension up to four, each with a single time direction, and nothing beyond four. Each of these arenas is in fact realized in nature—a particle on a line $(0+1)$, one-dimensional quantum wires and chains $(1+1)$, planar electron systems and anyons $(2+1)$, and the ambient world $(3+1)$—so admitting all of them is the framework being correct, not a defect to be repaired.

3.5. The Character of the Amplitude Across the Ladder

The even-subalgebra dimension is $2^{n-1}=1,2,4,8$ for $n=1,2,3,4$, and the amplitude type climbs accordingly:

• $0+1$: a single real number. Density $a^2$; no phase, no rotation, no spin. This is classical probability—one weight, no interference.
• $1+1$: two real components. A two-component object with left- and right-moving sectors; the genuine $1+1$ Dirac structure.
• $2+1$: a complex two-spinor. Here a true $\mathbb{C}$ phase and an $SU\left(2\right)$ rotor group first appear—a qubit—the arena of planar Dirac physics.
• $3+1$: a four-component Dirac spinor; full spin-$\frac{1}{2}$, the saturating maximal case.

Quantum phase is therefore not assumed; it switches on at a definite rung of the ladder and saturates at the ceiling. The complex structure that makes quantum mechanics quantum appears as a consequence of climbing toward the maximal admissible dimension, not as an input.

4. Theorem 2: The Dirac Equation as Information-Preserving Flow

4.1. Physical Setup: What Evolves, and What Costs Information

We now have the arena and the object: a geometric probability $\psi$ on an admissible ($n\le 4$, one time) spacetime, with a conserved vector current $J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$. The constructive stance permits exactly one question about its dynamics: how can this probability evolve while committing to nothing beyond what conservation requires?

To make “committing to nothing” precise we need a measure of how much a candidate evolution injects into, or extracts from, the information content of the probability. That measure is the rate of entropy change along the flow. An evolution that increases entropy is forgetting—coarse-graining, mixing, losing information about the state. One that decreases entropy is manufacturing information from nowhere—asserting more than the data warrant. The maximally non-committal evolution does neither: it transports the probability while creating and destroying no information. This is the dynamical analogue of the maximum-entropy principle of Jaynes [8]: where maximum entropy fixes a state by adding nothing beyond the constraints, information preservation fixes a flow by adding nothing beyond conservation of the probability.

So before writing any equation we must ask: what does a local rate of entropy change look like for a flowing probability? We do not answer this by fiat—irreversible thermodynamics already fixes the form, and we are obliged to use it.

4.2. The General Form of Local Entropy Production

In the thermodynamics of irreversible processes, the local rate of entropy production has a single universal structure, fixed not by any particular system but by the requirement that it be the rate at which a conserved quantity is redistributed against a gradient. It is always a sum of flux times force [9]:

$$\sigma =\sum _a{J}^a·X_a,$$

(3)

where each thermodynamic force $X_a$ is the gradient of an intensive variable (a temperature gradient, a chemical-potential gradient, a phase gradient) and each flux$J^a$ is the conjugate current it drives. This is the Onsager form [10], and three of its features are not optional—they are what make it entropy production:

1.

It is bilinear—flux times force. A rate of entropy change is the contraction of what is flowing with what drives the flow; neither alone produces entropy.

2.

The force is a first gradient. Entropy is produced by spatial variation of an intensive quantity; a uniform field produces nothing. The force therefore carries exactly one derivative. This single fact makes the resulting dynamics first-order—in sharp contrast to a Fisher-information functional, which is gradient-squared and so yields second-order equations.

3.

It is a scalar formed by contracting a vector flux with a vector force.

We apply this universal form to our object. The conserved current is already in hand: it is the probability current $J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$—the only vector the geometric probability supplies, and (by Theorem 1) a pure vector precisely because $n\le 4$. The conjugate force is the gradient of the amplitude, ${\partial }_{\mu }\psi$. There is no remaining freedom: given that we are constructing entropy production for this probability, the most general local flux·force density expressible from the object is

$$\sigma -\mathrm{density}\sim \overline{\psi }{\Gamma }^{\mu }{\partial }_{\mu }\psi +\left(\mathrm{conjugate}\right),$$

(4)

where ${\Gamma }^{\mu }$ is the still-undetermined coupling between the flux (carrying the index $\mu$) and the force ${\partial }_{\mu }\psi$—the Onsager coefficient on the internal (spinor) space. We have not assumed ${\Gamma }^{\mu }$ to be the Dirac matrices; at this stage it is an arbitrary set of coupling matrices, and everything physical about the Dirac equation will come from determining it.

4.3. The Reversible Sector: Vanishing Entropy Production

A general coupling ${\Gamma }^{\mu }$ decomposes into a symmetric (Hermitian) and an antisymmetric (anti-Hermitian) part. The symmetric part is the genuinely dissipative sector: it yields $\sigma >0$, the irreversible relaxation of a system toward equilibrium—heat flowing down a temperature gradient, probability diffusing toward uniformity. The antisymmetric part yields $\sigma =0$: it transports the probability without net entropy production.

Our criterion—evolve while neither creating nor destroying information—is exactly the demand to sit in the $\sigma =0$ sector. We are not maximizing entropy production (that gives dissipative, parabolic relaxation); we demand it vanish identically along the flow. This selects the antisymmetric coupling, making the functional the reversible flux·force action

$$\Sigma [\psi ,\overline{\psi }]=\int d^{4}x\left[\overline{\psi }{\Gamma }^{\mu }{\partial }_{\mu }\psi -\left({\partial }_{\mu }\overline{\psi }\right){\Gamma }^{\mu }\psi -2\kappa \overline{\psi }\psi \right].$$

(5)

The antisymmetrization of the first two terms is the reversibility condition made explicit. The final term, with Lagrange multiplier $\kappa$, is the single constraint we impose beyond conservation: that the probability carry the relativistic mass-shell—the only place a scale enters.

4.4. The Stationary Statement

Treating $\psi$ and $\overline{\psi }$ as independent and varying $\overline{\psi }$:

$${\Gamma }^{\mu }{\partial }_{\mu }\psi =\kappa \psi .$$

(6)

This is first-order, and that is the entire payoff of the flux·force structure being linear in the gradient. We are careful about what this step does and does not do. The variation does not generate the operator ${\Gamma }^{\mu }{\partial }_{\mu }$; it returns the operator already present in the flux·force density, stripped of its conjugate field. The content of Theorem 2 is therefore not that variation manufactures the Dirac equation, but that the Dirac operator is the stationary characterization of the most general reversible entropy-production functional—the configuration along which the probability flows at constant (vanishing) entropy production. The substantive work is done not by the variation but by the constraint that fixes ${\Gamma }^{\mu }$.

4.5. The Mass-Shell Determines the Coupling: Clifford, and Dirac

The coupling ${\Gamma }^{\mu }$ remains arbitrary. We fix it by the one physical constraint imposed—compatibility with the relativistic mass-shell. Requiring that iterating the first-order flow reproduce the relativistic dispersion relation, $\left({\Gamma }^{\nu }{\partial }_{\nu }\right)\left({\Gamma }^{\mu }{\partial }_{\mu }\right)\psi ={\partial }^{\mu }{\partial }_{\mu }\psi$, forces the symmetric part of ${\Gamma }^{\nu }{\Gamma }^{\mu }$ to be the metric:

$$\frac{1}{2}\{{\Gamma }^{\mu },{\Gamma }^{\nu }\}=g^{\mu \nu }\mathbf{1}.$$

(7)

This is the Clifford algebra—derived here as a compatibility condition, exactly as it was forced in Dirac’s original “square root of Klein–Gordon” argument [11], not assumed. Its solution is ${\Gamma }^{\mu }={\gamma }^{\mu }$ (the gamma matrices, in the minimal four-dimensional representation—the same Dirac–Hestenes object of Section 3), with $\kappa =m$. The reversible (anti-Hermitian) root ${\Gamma }^{\mu }=i{\gamma }^{\mu }$, selected above, gives at last

$$\begin{array}{|c|}\hline (i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0.\\ \hline\end{array}$$

(8)

The Dirac equation is the statement that the geometric probability flows along directions of constant entropy, subject to the relativistic mass-shell.

4.6. The Weld (Lemma 1)

We show the kinetic bilinear splits into the continuity equation and the entropy-production density, and that the geometric current is identically the entropy flux.

Write the kinetic density $\mathcal{K}=\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$, with Dirac adjoint $\overline{\psi }={\psi }^{†}{\gamma }^0$. Its Hermitian (real) part is

$$Re\mathcal{K}=\frac{1}{2}\left(\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi +\left({\partial }_{\mu }\overline{\psi }\right){\gamma }^{\mu }\psi \right)=\frac{1}{2}{\partial }_{\mu }\left(\overline{\psi }{\gamma }^{\mu }\psi \right)=\frac{1}{2}{\partial }_{\mu }{J}^{\mu },$$

(9)

using $\overline{\overline{\psi }{\gamma }^{\mu }\chi }=\overline{\chi }{\gamma }^{\mu }\psi$ and the adjoint Hermiticity ${\gamma }^0{\gamma }^{\mu †}{\gamma }^0={\gamma }^{\mu }$. On any solution of (8) this vanishes, giving the continuity equation${\partial }_{\mu }{J}^{\mu }=0$.

For the anti-Hermitian (imaginary) part we use the Gordon decomposition, which separates the Dirac current into a convective piece driven by the phase gradient and a spin piece:

$$J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi ={\displaystyle \frac{1}{m}}\left[\underset{\mathrm{convective}}{\underbrace{\rho {\partial }^{\mu }S}}+\underset{\mathrm{spin}}{\underbrace{{\partial }_{\nu }\left(\overline{\psi }{\sigma }^{\mu \nu }\psi \right)}}\right],$$

(10)

with ${\sigma }^{\mu \nu }=\frac{i}{2}[{\gamma }^{\mu },{\gamma }^{\nu }]$, $\rho =\overline{\psi }\psi$, and S the Takabayasi phase. Substituting into the anti-Hermitian part of $\mathcal{K}$, the convective term contracts the flux with the phase-gradient force $X_{\mu }={\partial }_{\mu }S$ and constitutes the entropy-production density. The spin term ${\partial }_{\nu }\left(\overline{\psi }{\sigma }^{\mu \nu }\psi \right)$ is a separate, identically conserved Gordon current: because ${\sigma }^{\mu \nu }$ is antisymmetric, ${\partial }_{\mu }{\partial }_{\nu }\left(\overline{\psi }{\sigma }^{\mu \nu }\psi \right)=0$ identically, so it does not enter the flux·force balance. Hence

$$Im\mathcal{K}=J^{\mu }{X}_{\mu },X_{\mu }={\partial }_{\mu }S,$$

(11)

which is exactly the flux·force entropy-production density of (5): the flux is the current $J^{\mu }$, the force is the phase gradient.

Collecting both parts,

$$\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi =\frac{1}{2}{\partial }_{\mu }{J}^{\mu }+i{J}^{\mu }{X}_{\mu },$$

(12)

the real part being probability conservation and the imaginary part the entropy production. The geometric current $J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$ is therefore identically the Onsager entropy flux. This closes the circle between the two theorems: the same current whose grade-1 purity (P2) bounds the dimension in Section 3 is the flux whose flux·force contraction is the entropy production in Section 4. Theorem 1 may accordingly be restated thermodynamically:

The requirement that entropy production be expressible as a vector flux·force caps the spacetime metric at $n\le 4$.

The kinematic ceiling and the dynamical law are two readings—geometric and thermodynamic—of one object.

5. Uniqueness and What Is Explained

Both results are uniqueness theorems conditional on the geometric-probability premise (P1)+(P2), as every physical uniqueness theorem is conditional on its axioms (uniqueness of a gauge group is conditional on anomaly freedom; uniqueness of a vacuum on a stability criterion; and so on). Under that premise:

• The Lorentzian signature, the single time dimension, and the four-dimensional ceiling of the spacetime metric are the unique admissible solution; every alternative is excluded by an explicit obstruction, computed term by term. The dimensionality of the spacetime metric is thereby explained rather than postulated.
• The Dirac equation is the unique information-preserving, mass-shell-compatible flow; it is explained as the dynamics that neither creates nor destroys information. First order is not an additional stipulation: vanishing entropy production makes the flux·force functional linear in the gradient, and a linear-in-gradient action is first-order.

We state the scope honestly:

(a) The dimensional result is a ceiling: it admits the entire family $0+1$, $1+1$, $2+1$, $3+1$, with one time dimension throughout. This is not an incompleteness to be repaired by singling out $3+1$; it is the correct prediction. Each admissible arena is in fact realized in nature—a particle on a line $(0+1)$, quantum wires and chains $(1+1)$, planar electron systems and anyons $(2+1)$, and the ambient world $(3+1)$—and a fundamental structure that hosts a geometric probability must therefore admit all of them, not select one. The framework reproduces exactly the observed family: every dimension up to four, each with a single time direction, and nothing beyond four. The ceiling is the content; there is no further “floor” to impose.

(b) The Dirac uniqueness is relative to two named inputs: the mass-shell constraint, and the choice of the information-preserving (reversible) root over the dissipative one. Both are explicit; neither is hidden.

(c) The premise itself—that there is a geometric probability carried by a metric—is assumed, not derived. We do not eliminate postulates; we replace dynamical postulates about the world (laws the universe obeys) with a single representational postulate about probability (an object describing what we measure), and show that two pillars of physics follow.

6. Discussion

The construction realizes, for two structural facts, a single discipline: add no information beyond what representing and conserving a probability requires. The arena $n\le 4$ is the condition of possibility for a geometric probability—logically prior to any dynamics, demanding even less than a dynamical inference principle needs, since it concerns only whether the object can exist as a positive vector current at all. The Dirac equation is then that same probability’s information-preserving evolution. Representability and information preservation are two faces of one discipline, and this work rests on both.

What is gained is an answer to two “why” questions usually met with a postulate. Why is the spacetime metric Lorentzian, with one time dimension and at most four dimensions? Because exactly these metrics can host a geometric probability with a positive density and a vector current; no other signature can, and none of dimension five or higher can. Why does matter obey the Dirac equation? Because that is the unique evolution of such a probability that conserves information. That the same object answers both—the admissible dimensions through its representability, the dynamics through its conservation—is the result we wish to record.

What is not gained, and we do not claim it, is the inevitability of the probability object itself. Every reconstruction bottoms out in some primitive; ours bottoms out in a geometric probability. The next “why”—why a probability, why positivity, why a metric to carry it—recurses beyond this paper. But the primitive we rest on is an epistemic object, a structure describing measurement, rather than a dynamical postulate about the world; and from it the dimension of spacetime and the law of relativistic matter are not assumed but forced.