Move to reciprocal coordinates
Transform the crossing equation so the relevant dependence can be interpolated more cleanly.
← All researchMF-PRISM-MATH-2026-16 / interactive guide
THE IDEA, MADE VISIBLE
Watch the best marginal move.
Change the model parameter and inspect the actual balanced-partition polynomials from the paper.
Candidate resolutionOrdered Polynomial Crossings in Balanced Partition Models
Balance an integer population
Split Z into k parts that differ in size by at most one. Quotient q and remainder r specify the partition exactly.
- Population
- 12
- Maximizing index
- 3
- Maximum marginal
- 1.2370
At Z = 12 and p = 0.65, the largest evaluated marginal is f3 = 1.2370. Ties, if any, are resolved here by first index.
Rₖ = (k−r)qpᑫ + r(q+1)pᑫ⁺¹; fₖ = Rₖ − Rₖ₋₁
TRY THIS
Move p from 0.2 to 0.9 and follow the highlighted maximizing marginal. Then switch to crossing curves.
What this experiment represents. Direct floating-point evaluation of the manuscript’s displayed polynomials. Plotted intersections are visual aids, not exact root certificates or empirical educational claims.
Mathematics · Preprint
Ordered Polynomial Crossings in Balanced Partition Models
Reciprocal interpolation and Foregger’s root-ordering conjecture
Candidate resolutionMF-PRISM-MATH-2026-16 / v1.1
Uses reciprocal interpolation and unimodality to order the crossing roots of balanced-partition marginal polynomials. The construction also identifies identical profiles and an exact selection rule within the stated model.
- Identifier
- MF-PRISM-MATH-2026-16
- Series
- Mathematics
- Edition
- Version 1.1
- Length
- 14 pages
- Reserved DOI
- 10.5281/zenodo.22728342 (record reserved)
01
Plain-language guide
3 minute explanation
What this paper is really saying.
The paper orders the parameter values at which balanced-partition marginal polynomials cross, using reciprocal interpolation and unimodality rather than a numerical plot.
polynomial rootsbalanced partitionsinterpolation
When two polynomial profiles are compared, the point where they cross marks a change in which profile is larger. Foregger’s conjecture predicts a particular order for those crossing points in a balanced-partition model.
The paper transforms the roots reciprocally, interpolates across the model’s piecewise structure, and uses unimodality to prove uniqueness and ordering, while treating identical profiles as explicit exceptions.
02
How it works
The argument, without the notation.
The paper’s technical details matter, but the basic route can be understood in three moves.
Control the piecewise model
Handle floor-function discontinuities through interval averaging and prove the required unimodality.
Order the crossings
Establish uniqueness for simple crossings, identify identical-polynomial exceptions, and derive the exact selection rule.
03
What it adds
The useful takeaways.
The stated root-ordering conjecture receives a candidate proof within the mathematical model.
Simple crossings are ordered through structural inequalities rather than empirical curve fitting.
Identical profiles and discontinuity cases are separated explicitly instead of being hidden as numerical edge cases.
04
Scope & limits
What this does—and does not—establish.
Current status
Candidate proof of the stated root-ordering conjecture; no empirical education-outcome claim.
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 makes no claim about real educational outcomes or policy effectiveness.
Its conclusion is limited to the stated balanced-partition polynomial model.
Floor-function discontinuities and uniqueness of crossings are central review points.
05
Why it matters
Why anyone should care.
A proof would settle the stated structural ordering question inside the mathematical model while avoiding unsupported claims about real educational outcomes.
06
Key terms
The vocabulary, decoded.
These are the terms needed to understand the claim. The full paper uses them more precisely.
Polynomial root
A value of the variable at which a polynomial equals zero.
Crossing
A parameter value where two profiles are equal and may exchange order.
Interpolation
Constructing or analyzing a function from its values or structure between known points.
Unimodal
Increasing up to one peak and then decreasing, or the reverse.
07
Independent review
Where scrutiny should concentrate.
The strongest review is not a general reaction. It tests the steps most capable of changing the conclusion.
- 01Reciprocal interpolation across floor-function discontinuities.
- 02Unimodality under interval averaging and uniqueness of simple crossings.
- 03Identical-polynomial exceptions, betweenness inequalities, and the original Foregger conjecture.
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.
- Document
- Preprint
- Review status
- Open for independent review
- Published
- 2026-09-12
- DOI status
- Reserved · 10.5281/zenodo.22728342
- Canonical record
- MF-PRISM-MATH-2026-16
Metriq PRISM Laboratory, Ordered Polynomial Crossings in Balanced Partition Models, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-16, Version 1.1, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728342.