To determine whether the minimal path sum $A(N)$ for the given matrix $M_{i,j} = a_i + b_j$ indeed always passes through the points $M_{x,y}$ such that $a_x$ is the minimum of all $a_i$ and $b_y$ is the minimum of all $b_j$, let’s analyze the situation mathematically.

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1. Restating the Hypothesis

The hypothesis is that the minimal path sum will traverse through a cell $M_{x,y}$, where:

• $x$ is the index for which $a_x = \min(a_1, a_2, \ldots, a_N)$,
• $y$ is the index for which $b_y = \min(b_1, b_2, \ldots, b_N)$.

2. Expression for Path Sum

For a path in the $M$-matrix from $M_{1,1}$ to $M_{N,N}$:

• The total sum of a path is the sum of the entries encountered along the path.
• At each step, the sum includes one $a_i$ and one $b_j$.

The goal is to minimize the total sum over all valid paths.

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3. Properties of $M$-Matrix

The matrix $M_{i,j} = a_i + b_j$ is formed by summing elements of two independent sequences:

• $\{a_i\}$ depends only on $s_{2n-1}$,
• $\{b_j\}$ depends only on $s_{2n}$.

Each $M_{i,j}$ is composed of $a_i$ from the row index and $b_j$ from the column index, which are independent.

4. Key Insight for Minimal Path

For any step (either right or down):

• Moving right adds $b_j$ for the next column, where $b_j \geq \min(b_1, \ldots, b_N)$,
• Moving down adds $a_i$ for the next row, where $a_i \geq \min(a_1, \ldots, a_N)$.

The sum of a path can be thought of as: $\text{Total Sum} = \text{(constant part)} + \text{(cost of steps)},$ where the cost of steps is determined by which rows and columns are traversed.

• To minimize this cost, choosing $x$ and $y$ corresponding to the minimal $a_x$ and $b_y$ will ensure that the smallest possible contributions are added along the path.

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5. Proof by Contradiction

Suppose the minimal path $A(N)$ does not pass through $M_{x,y}$, where $a_x = \min(a_1, \ldots, a_N)$ and $b_y = \min(b_1, \ldots, b_N)$.

Then there exists another path $P$ that has a smaller sum. However:

• The path $P$ must include $a_i + b_j$ for some $i \neq x$ and $j \neq y$,
• By construction, $a_i \geq a_x$ and $b_j \geq b_y$,
• Hence, $a_i + b_j \geq a_x + b_y$, which means $P$ cannot have a smaller sum.

This contradiction implies that the minimal path must include $M_{x,y}$.

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6. Conclusion

The minimal path sum $A(N)$ indeed always passes through $M_{x,y}$, where $a_x$ is the minimum of all $a_i$ and $b_y$ is the minimum of all $b_j$. This result holds because the independent nature of $a_i$ and $b_j$ ensures that any deviation from the minimal values will increase the path sum.

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Would you like a detailed example or computational validation?
Here are 5 related questions to expand your understanding:

1. How does the modular operation affect the sequences $\{a_i\}$ and $\{b_j\}$?
2. What would the path look like in a smaller $N \times N$ case (e.g., $N = 3$)?
3. Could there be cases where $a_x = a_y$ or $b_x = b_y$? How would this impact the path?
4. How does the symmetry (or lack thereof) in $a_i$ and $b_j$ affect the minimal path?
5. Can this principle be generalized to non-additive matrices or other dimensions?

Tip: For large $N$, consider simulating small $M$-matrices to verify edge cases and assumptions numerically.