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THE IDEA, MADE VISIBLE

The coefficient ring changes the answer.

Explore a small arithmetic obstruction before the manuscript’s Schur-matrix construction.

Candidate resolutionPrime Obstructions to Integral Smith Forms of Schur Matrices

Calculated illustration · change the inputs to inspect the mechanism.

01 / 04Guided chapter

An ideal records combinations

The ideal (p,x) consists of polynomial combinations of p and x. Among linear polynomials a+bx over the integers, membership requires p to divide a.

Prime p

Coefficient ring

Constant coefficient a

Prime: 3
Selected a: 4
a + bx in (p,x)?: No

Over Z[x], a linear polynomial belongs to (p,x) exactly when 3 divides its constant coefficient.

a + bx ∈ (p,x) over Z[x] ⇔ p divides a

TRY THIS

Keep p = 3 and switch the coefficient ring. Points excluded over the integers become reachable over the rationals.

What this experiment represents. Teaching analogue (p,x), not the manuscript’s Schur matrix or its full determinantal-ideal calculation. No stable-equivalence certificate is generated here.

Source manuscript & release ↗Read the full explanation ↓

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1. An ideal records combinations

The ideal (p,x) consists of polynomial combinations of p and x. Among linear polynomials a+bx over the integers, membership requires p to divide a.

2. Find an integer obstruction

The grid shows linear polynomials by constant coefficient a and x-coefficient b. Integer combinations cannot produce constant 1 from p and x.

3. Allow rational coefficients

Over the rational polynomial ring, 1/p is allowed. Multiplying p by 1/p produces 1, so the same ideal becomes the whole ring.

4. Connect to the matrix question

The paper uses nonprincipal determinantal ideals to obstruct integral Smith forms, including stabilization. This toy ideal explains why changing the coefficient ring can erase an obstruction.

Mathematics · Preprint

Prime Obstructions to Integral Smith Forms of Schur Matrices

An infinite family of arithmetic counterexamples

Candidate resolutionMF-PRISM-MATH-2026-18 / v0.2

Constructs an infinite family of Schur-related matrices whose entry ideals are nonprincipal over the integral Laurent ring. The obstruction persists under identity stabilization and targets the general Schur-function clause of Kuperberg’s conjecture.

Read the paper Release & files

Cover of Prime Obstructions to Integral Smith Forms of Schur Matrices — Current public editionv0.2 · 2026-09-12

Identifier: MF-PRISM-MATH-2026-18
Series: Mathematics
Edition: Version 0.2
Length: 10 pages
Reserved DOI: 10.5281/zenodo.22728346 (record reserved)

Plain-language guide

3 minute explanation

What this paper is really saying.

The paper constructs an infinite family in which the entry ideal is nonprincipal, blocking an integral Smith form even after identity stabilization and thereby targeting the broad Schur-function clause of Kuperberg’s conjecture.

smith normal formschur functionscommutative algebra

A Smith normal form is a diagonal simplification of a matrix that exposes its arithmetic structure. Over integers and Laurent polynomials, such a form requires certain determinantal ideals to be generated by a single element.

The paper builds Schur-related matrices whose smallest relevant ideal needs two generators. Adding identity blocks does not repair that defect, so the obstruction survives the stabilization allowed by the conjectural statement.

How it works

The argument, without the notation.

The paper’s technical details matter, but the basic route can be understood in three moves.

Choose a Schur family

Construct matrices tied to Schur functions for which the entry ideal can be calculated explicitly.

Prove the ideal is nonprincipal

Work over the integral Laurent ring—not merely over rational coefficients—and show two generators cannot collapse to one.

Preserve the obstruction

Show identity stabilization leaves the obstructing determinantal ideal unchanged and transfer that fact to the conjecture’s stable-equivalence clause.

What it adds

The useful takeaways.

An infinite family of candidate counterexamples is given for the general Schur-function clause.
The obstruction is arithmetic: the relevant ideal is nonprincipal over the integral Laurent ring.
Identity stabilization does not remove the obstruction.

Scope & limits

What this does—and does not—establish.

Current status

Candidate counterexamples to the general Schur-function clause of Kuperberg’s Conjecture 11; other clauses are not resolved.

This manuscript presents a candidate resolution. Publication is not verification: independent specialist review remains necessary before the result should be treated as established.

The paper targets the general Schur-function clause, not every part of Kuperberg’s Conjecture 11.
Special rectangular or otherwise restricted cases may remain unaffected.
The integral calculation is essential; a rational-coefficient simplification would not prove the stated obstruction.

Why it matters

Why anyone should care.

The proposed infinite family would falsify the broad Schur-function clause while leaving unrelated rectangular or specialized clauses untouched.

Key terms

The vocabulary, decoded.

These are the terms needed to understand the claim. The full paper uses them more precisely.

Smith normal form

A diagonal matrix form that records divisibility invariants of a matrix.

Schur function

A symmetric polynomial central to representation theory and algebraic combinatorics.

Laurent ring

A polynomial ring that also allows negative powers of its variables.

Principal ideal

An ideal generated by one element.

Independent review

Where scrutiny should concentrate.

The strongest review is not a general reaction. It tests the steps most capable of changing the conclusion.

01
The exact two-generator ideal calculation over integers rather than rational coefficients.
02
Preservation of the nonprincipal determinantal ideal under every identity stabilization.
03
The stable-equivalence transfer and the precise general Schur clause, excluding unrelated rectangular cases.

Publication record

Go from explanation to evidence.

This page is a reading guide, not a substitute for the manuscript. The public record links the explanation to the paper, source package, review materials, and persistent identifier.

GitHub release Source folder

Document: Preprint
Review status: Open for independent review
Published: 2026-09-12
DOI status: Reserved · 10.5281/zenodo.22728346
Canonical record: MF-PRISM-MATH-2026-18

Metriq PRISM Laboratory, Prime Obstructions to Integral Smith Forms of Schur Matrices, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-18, Version 0.2, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728346.