The fair-discrepancy propagation conjecture for fully balanced matrices

About 8 years old · traced to 2018

Let $A\in\mathbb{M}_{n\times m}(\mathbb{R})$ be a fully balanced matrix. For each row $i$ , let

M_i=\frac{1}{m}\sum_{j=1}^{m}a_{ij}

be its average discrepancy. If, for a fixed $1\leq i\leq n$ , one has

|M_i-a_{ij}|<\epsilon

for the relevant entries $j$ , then $|M_i-a_{ij}|<\epsilon$ for all $1\leq i\leq n$ .

Fair-discrepancy propagation conjecture. If one fixed row has discrepancy less than $\epsilon$ , then every row has discrepancy less than $\epsilon$ .

The claim concerns propagation of a fair discrepancy from one row to all rows of a fully balanced matrix. The supplied text gives no resolution or additional context.

References

Primary source

Theophilus Agama and Gael Kibiti, “Balanced matrices”, arXiv:1810.07542 (2026).

balanced-matrices matrix-theory

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A reader-submitted example appears to disprove the conjecture, but it has not been independently checked, so the general question remains open.

The statement is recorded as Conjecture 6.6 in Balanced matrices: fair discrepancy in one row of a fully balanced matrix should propagate to every row. No published proof or disproof was found.

Known results

For positive fully balanced matrices of size $2\times2$ , fair discrepancy along rows is equivalent to fair discrepancy along columns (Theorem 6.4).
In the same $2\times2$ setting, fair discrepancy in one row implies fair discrepancy in all rows (Proposition 6.5).

Community submission (unverified)

The August 22, 2026 submission argues that $A=\begin{pmatrix}5&5&5\\1&5&7\\7&5&1\end{pmatrix}$ has equal row and column squared energies, while its first row has zero discrepancy and its second row violates the condition for $\varepsilon=1$ ; scaling is claimed to extend this to every $\varepsilon>0$ .

Current status (as of September 2026): The positive $2\times2$ case is settled, while the general conjecture remains open and the submitted counterexample is unverified.

Sources

CounterexampleThis solution needs a summarySee full solutionHide full solution

Counterexample to discrepancy propagation, even for strictly positive, exactly balanced matrices.

For a matrix $A=(a_{ij})\in\mathbb{R}^{n\times m}$ , write

R_i=\sum_{j=1}^{m}a_{ij}^{2},\qquad C_j=\sum_{i=1}^{n}a_{ij}^{2},\qquad M_i=\frac{1}{m}\sum_{j=1}^{m}a_{ij}.

The conjecture asserts that, for a fully balanced matrix and a fixed $\varepsilon>0$ , the inequalities

\lvert M_r-a_{rj}\rvert<\varepsilon \qquad (1\le j\le m)

for one row $r$ imply the corresponding inequalities in every row. Consider the strictly positive integer matrix

A= \begin{pmatrix} 5&5&5\\ 1&5&7\\ 7&5&1 \end{pmatrix}.

Its row energies are

\begin{aligned} R_1&=5^2+5^2+5^2=75,\\ R_2&=1^2+5^2+7^2=75,\\ R_3&=7^2+5^2+1^2=75, \end{aligned}

and its column energies are

\begin{aligned} C_1&=5^2+1^2+7^2=75,\\ C_2&=5^2+5^2+5^2=75,\\ C_3&=5^2+7^2+1^2=75. \end{aligned}

Consequently, $A$ is simultaneously horizontally and vertically balanced with exact equality, which is stronger than the approximate balance required in the original definition. Set $\varepsilon=1$ . The first row has mean

M_1=\frac{5+5+5}{3}=5,

and therefore

\lvert M_1-a_{1j}\rvert=0<1 \qquad (j=1,2,3).

However, the second row has mean

M_2=\frac{1+5+7}{3}=\frac{13}{3},

so its first entry satisfies

\lvert M_2-a_{21}\rvert =\left\lvert\frac{13}{3}-1\right\rvert =\frac{10}{3}>1.

Thus the first row has fair discrepancy while the second does not, disproving the conjecture.

In fact, the obstruction persists for every prescribed tolerance: for any $\varepsilon>0$ , choose $t>3\varepsilon/10$ and replace $A$ by $tA$ . Every row and column then has squared energy $75t^2$ ; the first-row deviations remain zero, whereas

\left\lvert M_2(tA)-(tA)_{21}\right\rvert =\frac{10t}{3}>\varepsilon.

If desired, taking $t\ge1$ also preserves the stronger entrywise condition $(tA)_{ij}\ge1$ . Hence neither exact energy balance nor strict positivity restores discrepancy propagation.

Source: T. Agama and G. Kibiti, Balanced matrices, arXiv:1810.07542v3, Definition 3.1, Conjecture 6.6, and Remark 6.7.