Curvature and Kernel Structure in Sum–Integral Comparison: A Higher-Order Theory

This repository contains the complete materials for the paper Curvature and Kernel Structure in Sum–Integral Comparison: A Higher-Order Theory, by Bob Jefferson (2026).

The archive includes the full LaTeX source of the paper, the complete Lean 4 formalisation, and the supporting project files required to verify the formal development.

Overview

This paper develops a higher-order theory of sum–integral comparison based on curvature, Peano kernels, and explicitly computable kernel mass constants.

The central object is a family of even-order corrected functionals D₂m(f; a,b). For each m ≥ 1, the paper proves an exact kernel representation

D₂m(f; a,b) = ∫ₐᵇ K₂m(a,b,t) f⁽²ᵐ⁾(t) dt,

where the even-order kernel is

K₂m(a,b,t) = ((t − a)^m(b − t)^m) / (2m)!.

This kernel is nonnegative on [a,b], vanishes at the endpoints, and has the explicit mass

∫ₐᵇ K₂m(a,b,t) dt = C_m(b − a)²ᵐ⁺¹,

where

C_m = (m!)² / ((2m)!(2m+1)!).

In particular, C₁ = 1/12 and C₂ = 1/720, recovering the familiar second-order and fourth-order constants from a single structural principle.

The paper also explains why the same positivity-based mechanism is intrinsically even-order. The corresponding odd-order kernels change sign and have zero mass, so they do not support the same nonnegative kernel-mass theory.

Main Results

The paper establishes the following results.

• Exact Peano-kernel representations for a hierarchy of even-order corrected functionals

• Explicit nonnegative kernels K₂m

• Closed-form kernel mass constants C_m

• Sharp interval-scale bounds governed by C_m(b − a)²ᵐ⁺¹

• Partition decompositions with canonical local weights C_mΔᵢ²ᵐ⁺¹

• Oscillation-control estimates for higher derivatives

• Asymptotic equivalence of weighted sampled sums

• Ratio asymptotics under mild positivity assumptions

• A structural distinction between the even-order and odd-order cases

Together, these results give a unified higher-order framework for understanding quadrature-type sum–integral errors through kernel positivity, explicit mass, and partition-level stability.

Lean 4 Formalisation

All principal results are formalised in Lean 4 using mathlib.

The formalisation includes definitions of the corrected higher-order functionals, the even kernels and their basic properties, the explicit kernel mass theorem, kernel upper bounds, the kernel representation theorem, partition-theoretic decompositions, oscillation estimates, asymptotic equivalence results, ratio-asymptotic results, and the final structural theorem.

The root file CurvatureTheory.lean serves as the public entry point and imports the full formal development.

The Lean project has been checked with:

lake build CurvatureTheory

and builds successfully.

Repository Structure

CurvatureTheory.lean
Public Lean entry point

CurvatureTheory/
Lean 4 source files

docs/
Paper source and supporting documentation

README.md
Project overview and build instructions

LICENSE
Project licence

CITATION.cff
Citation metadata

Mathematical Context

The paper is self-contained and is intended to be read independently.

It is related to earlier work by the author on curvature-kernel methods for sum–integral comparison.

Curvature and Kernel Structure in Sum–Integral Comparison
DOI: 10.5281/zenodo.19442754

Curvature and Kernel Structure in Sum–Integral Comparison: II. Non-Uniform Partitions and Scaled Bounds
DOI: 10.5281/zenodo.19502252

Curvature and Kernel Structure in Sum–Integral Comparison: III. Asymptotic Equivalence and Ratio Asymptotics
DOI: 10.5281/zenodo.19744168

Those works provide related background and motivation. The present paper develops the higher-order theory as a standalone contribution, with its own definitions, statements, proofs, and Lean formalisation.

Significance

The paper gives a structurally transparent approach to higher-order quadrature error. Rather than treating correction terms primarily through asymptotic expansion or cancellation, it identifies the kernel features that drive the analysis:

• positivity

• explicit mass

• endpoint vanishing

• partition additivity

• oscillation control

• stability of weighted sampled sums

This makes the theory particularly well suited to formal verification. The central constants, kernels, and inequalities are explicit, and the main structural results are mechanically checked in Lean 4.

Author

Bob Jefferson
ORCID: 0009-0003-1460-055X

License

MIT License. See LICENSE.

Citation

Please cite this work using the metadata provided in CITATION.cff.