The Zero Boundary: A Two-Sorted Mathematical Framework Eliminating Singularities in Physics

Abstract 

Overview

This paper introduces a new foundational mathematical framework in which the number “zero” is not treated as a numeric element of the real number system. Instead, zero is redefined as a boundary-state operator: something that marks the limit of the numeric domain rather than acting as a value within it. Using a two-sorted first-order language, the traditional real number line is reconstructed without zero. All nonzero real numbers form the numeric domain, and zero is represented by a separate boundary object that acts as a topological limit. This reformulation removes the contradictions that arise when zero is used simultaneously as a number, a placeholder, an identity, a coordinate, and a boundary point. The framework, referred to as “Zero Boundary Theory,” is formally defined and shown to be mathematically consistent by explicit model construction.

Motivation

Modern mathematics forces zero to behave in many contradictory ways at once:

• as a number,

• as a placeholder,

• as the additive identity,

• as the multiplicative annihilator,

• as a topological boundary point, and

• as a coordinate at which physical fields are evaluated.

This mixture of roles creates every major singularity in physics.
Black hole curvature diverges at “r = 0,” the Big Bang density becomes infinite at “t = 0,” quantum field theory diverges at zero separation, and renormalisation must cancel infinities introduced by evaluating expressions at zero.

The central claim of this paper is that these pathologies are not physical phenomena — they are mathematical category errors created by treating zero-as-boundary as if it were zero-as-number.

Framework and Methods

The paper develops a two-sorted first-order theory consisting of:

1. A numeric sort containing all nonzero real numbers.

2. A null sort containing a single boundary-state object representing what was historically called zero.

3. A null operator that acts on numeric values but performs no change, reflecting the idea that “zero does nothing.”

4. Axioms that define arithmetic, multiplication, inversion, and a strict separation between numeric values and boundary states.

A standard model is built explicitly to show that the theory is internally consistent. Calculus is reformulated using only nonzero values, meaning all limits, derivatives, and integrals operate on arbitrarily small nonzero quantities approaching the boundary, but never on an actual zero. The paper then examines the consequences for topology, algebra, and category theory.

Applications to Physics

The framework is applied to several branches of physics, demonstrating that many long-standing singularities disappear when zero is removed from the domain of physical variables.

General Relativity:
The Schwarzschild singularity at “r = 0” is reinterpreted as a boundary point outside the spacetime manifold. Curvature remains finite everywhere within the domain.

Quantum Field Theory:
Zero-distance divergences do not occur because field quantities are never evaluated at exactly zero separation. Propagators remain finite for all physically allowed separations.

Renormalisation:
Ultraviolet divergences such as the integral of 1/k from k = 0 to infinity do not arise, because k = 0 is not an admissible value. Loop integrals become naturally finite without counterterms.

Cosmology:
The Big Bang singularity is removed because “time zero” is not a valid interior time coordinate. The origin of the universe is reframed as a geometric boundary rather than a point of infinite density.

Conservation Laws and Vacuum Structure:
No physical field or observable quantity ever reaches a literal zero value. Boundary states replace the assumption of perfect vanishing, which resolves several conceptual inconsistencies in energy conservation and vacuum fluctuations.

Mathematical Results

The paper proves:

• A formal consistency theorem for the zero-free numeric universe.

• Syntactic conservativity: all theorems about nonzero values remain valid.

• Semantic conservativity: all statements that do not rely on zero have the same truth value as in classical mathematics.

• The Zero-Error Elimination Theorem: expressions such as division by zero, zero in denominators, multiplying by zero, and zero-to-the-zero all become impossible to form.

• Finite curvature at the centre of black holes and finite density at the Big Bang.

• Finite propagators and loop integrals in quantum field theory.

These results together show that many long-standing infinities and singularities in physics arise solely from misclassifying zero as a number rather than a boundary.

Significance and Contribution

The work presents a rigorous alternative foundation for mathematics, analysis, differential geometry, and physical theory that:

• preserves all classical theorems that do not rely on zero,

• removes all known zero-based physical singularities,

• unifies the behaviour of boundaries in mathematics and physics,

• redefines zero as structural absence instead of a numeric entity,

• creates a coherent cosmological and quantum framework without renormalisation infinities.

This provides a new conceptual lens for understanding spacetime, fields, curvature, and cosmological origins. It suggests that the deepest inconsistencies in physics originate from the assumption that zero is a number rather than a boundary-state.

Keywords
zero-free analysis, boundary operator, mathematical foundations, singularity elimination, general relativity, quantum field theory, renormalisation, cosmology, Schwarzschild geometry, Big Bang, topology, first-order logic, model theory, algebraic structures.