Foundations from Minimal Change

Research article · Jinku Guo

A Spectral Construction of Natural-Number Arithmetic

SSRN Zenodo Read manuscript PDF Files and citation

Author: Jinku Guo

Work type: Research article

SSRN: Abstract and download page

Zenodo: 10.5281/zenodo.22512592

Current repository copy: main

Updated: 26 October 2026.

Research overview

Arithmetic with its construction history retained

This paper constructs natural-number arithmetic from a correlation spectrum and then resolves its values into the expressions by which they are generated. Its central achievement is to connect three levels that are usually studied separately: arithmetic values, the histories of arithmetic operations, and an evolution acting on those histories. The resulting system retains both the familiar numerical laws and information that numerical evaluation alone removes.

The source is minimal unbiased compositional change. The least-dimensional faithful irreducible real orthogonal carrier is a rotation plane. Independent composition, exact same-shape self-similarity and unit covariance determine square-root scaling and a Gaussian law. Local information and normalized total correlation give the characteristic local and global lengths. Preservation of constants selects Neumann closure within the stated separated nondegenerate homogeneous local boundary class.

Spectral projections are ordered by their Gaussian response before an integer enumeration is introduced. Orientation selects the positive analytic modes. Pointwise modulation and circle-map composition define their addition and multiplication; modulation by the first positive mode defines successor. The paper proves the Peano recursion structure, unique prime factorization and canonical transport under scale and orientation changes. Winding degree certifies the resulting natural-number arithmetic.

Resolving how a value is generated

The next construction retains ordered, bracketed addition and multiplication expressions. Only the specified zero, unit and absorbing reductions are imposed at this level. Arithmetic evaluation identifies expressions with the same value, while each value has a finite expression fibre. This gives a precise space in which the construction of a number can itself be observed.

The simplest example is six. Adding six unit leaves and multiplying an additive expression for three by an additive expression for two give the same root value and the same root energy. Their leaf counts are six and five, while their accumulated internal quadratic defects are thirty and thirty-one. In both cases leaf energy plus accumulated defect is thirty-six. The equality at the root coexists with a measurable difference in the construction.

A recursive positive law gives equal weight to the admissible root channels and assembles their subhistories independently. This law differs from uniform counting of all complete expressions. Phase invariance determines a finite phase extension. Summing the resolved source over histories and phases recovers the original Gaussian mass of each mode exactly. Expression multiplicity therefore enriches the source without changing its original partition function.

Weighted amplitude readout, probability pushforward and conditional expectation have distinct definitions. Exact variance and entropy identities separate the information in the root value from the information remaining within its fibre. The example of six has strictly positive conditional leaf-count variance, giving a concrete instance of information preserved by the resolved construction.

A complete operator on generation histories

The same source weights determine a parent-child assembly response. The paper constructs a nonnegative self-adjoint trace-class operator on the complete counting history space. Its trace tail has an explicit Gaussian bound independent of the number of expressions in each fibre. Finite root cutoffs converge in trace norm, and the full operator generates a contraction evolution with a dissipation identity.

The decisive result is a nonzero transfer from an initially invisible history difference to a later value response. A difference supported on the six histories of four has zero initial value readout, yet its coupling through the complete infinite assembly operator is strictly nonzero. The result persists under small changes of the relevant conditional weights.

Writing $Q$ for the weighted value readout and $H$ for the assembly operator, the phenomenon is expressed by a vector $h$ satisfying

$$ Qh=0,\qquad QHh\ne0. $$

Arithmetic evaluation consequently discards information that can affect this later response. Eliminating the history-difference component produces an exact integral equation with a memory kernel and an initial-history contribution. This supplies a rigorous mechanism by which generation history enters the dynamics of evaluated quantities.

What distinguishes this construction

Circle degree provides a classical arithmetic certification. The additional results come from constructing the correlation source, its native operations, the resolved positive history law and the assembly response together. Direct circle arithmetic supplies the values and their operations; the present construction also supplies normalized history probabilities, conditional information and an explicit evolution that can return hidden history to an observable value.

Formal differences, fractions and completion extend the arithmetic to scalar fields and Hilbert spaces, with finite rational certificates and spectral approximation estimates. A Lean companion records the indexed arithmetic and history constructions, their assembly interfaces and formal dependencies. The paper's contribution is thus both foundational and dynamical: it constructs the arithmetic carrier and proves that the route to a value can remain mathematically consequential after that value has been obtained.

Research subjects and results

Paper keywords

Construction of natural numbers; Arithmetic expression trees; Conditional expectation; Trace-class operators; Non-Markovian dynamics; Memory kernels.

Primary subjects

Construction of natural numbers; Arithmetic expression trees; Conditional expectation; Trace-class operators; Non-Markovian dynamics; Memory kernels; Natural-number objects; Peano arithmetic; Unique factorization; Arithmetic expressions; Spectral theory; Gaussian semigroups; Formalized mathematics.

Research subjects

Foundations of arithmetic; Ordered semirings; Recursive constructions; Arithmetic expression trees; Expression enumeration; Evaluation maps; Finite fibres; Generation histories; Winding number; Circle endomorphisms; Gaussian distributions; Neumann boundary conditions; Conditional variance; Conditional entropy; Scale covariance.

Methods

Universal algebra; Structural recursion; Spectral decomposition; Heat semigroups; Probability measures; Hilbert spaces; Orthogonal projections; Self-adjoint operators; Trace-class estimates; Contraction semigroups; Integral equations; Projection methods; Gaussian tail bounds; Lean theorem proving; Machine-checked proofs.

Results and research questions

Spectral construction of natural-number arithmetic; Finite evaluation fibres of arithmetic expressions; Unique factorization on generated arithmetic modes; Exact separation of value and expression-history information; Nontrivial coupling between arithmetic values and expression histories; Trace-class operator from arithmetic expression assembly; Exact memory equation for arithmetic readout; Canonical scale and orientation transport of arithmetic.

Chinese subject terms

自然数构造; 自然数对象; 皮亚诺算术; 唯一分解; 算术表达式树; 生成历史; 条件期望; 条件熵; 迹类算子; 非马尔可夫动力学; 记忆方程; 形式化数学.

Public file

This directory provides the public PDF. LaTeX sources and editable research files remain outside this repository.

Citation and archival records

Jinku Guo. A Spectral Construction of Natural-Number Arithmetic. Public manuscript, SSRN, 2026. doi:10.2139/ssrn.7421398.

Public records:

The Zenodo concept DOI identifies the work across its deposited versions. The repository contains the current 91-page manuscript. Git history and the file checksum identify the reading copy supplied here.

Research context

Related research and results · Programme overview · Proofs and calculations