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A Condorcet winner defeats every other alternative by a strict majority in pairwise voting. It is unique when it exists. For an odd electorate with single-peaked preferences, the median peak is a Condorcet winner: alternatives on either side lose to the majority whose peaks lie at or beyond the median in the other direction. With an even electorate ties may prevent a strict winner.

On a fixed common axis, select the median of reported peaks. For odd $n$ this is the unique Condorcet winner. For even $n$, a fixed lower-median or upper-median convention selects a weak Condorcet winner. Holding other peaks fixed, the attainable outcomes form an interval between two order statistics. Truthful reporting obtains the true peak if it is inside that interval, or its nearest interval endpoint otherwise, so no allowed report improves the outcome under single-peaked preferences.

There is a parity qualification in the printed claim. With the usual strict-majority definition, an even electorate need not have a Condorcet winner. On the axis $a<b<c$, the two orders $a\succ b\succ c$ and $c\succ b\succ a$ are both single-peaked preferences, but every pairwise contest is tied. Thus no alternative strictly defeats every other one. We first prove the intended strict-winner result for odd $n$.

Let $n=2k+1$ and order the peaks along the common axis. Their median $w$ has at least $k+1$ peaks at or to each side. For any $y<w$, the $k+1$ voters whose peaks are at or to the right of $w$ prefer $w$ to $y$ by single-peakedness. For any $y>w$, the corresponding $k+1$ voters on the left prefer $w$. Thus the median peak is the unique Condorcet winner.

The corresponding median voter rule is strategyproof. Fix the other $2k$ peaks $q_1\le \cdots \le q_{2k}$. As one voter reports a peak $p$, the selected median iswhere minimum and maximum refer to the common axis. All attainable outcomes lie between $q_k$ and $q_{k+1}$. If the true peak lies inside this interval, truthful reporting obtains the voter's top. If it lies left of the interval, truthful reporting obtains its left endpoint, which the voter's single-peaked order prefers to every larger attainable outcome. The case right of the interval is symmetric. No report improves the outcome. For one voter the rule simply selects its peak.

There is also a proof that does not depend on knowing the axis. If a voter truly prefers a proposed new winner $z$ to the current strict Condorcet winner $w$, that voter already opposes $w$ in the contest $w$ versus $z$. The strict majority supporting $w$ in that contest therefore consists of other voters and is unaffected by its report. Thus $z$ cannot become a strict Condorcet winner after a profitable misreport. This proves strategyproofness on the domain of profiles for which the selected strict winner exists, including all admissible odd-electorate single-peaked profiles.

For even $n$, a weak Condorcet winner is guaranteed: every alternative between the two middle peaks weakly defeats each other alternative, allowing ties. With a fixed common axis, consistently choosing the lower median or consistently choosing the upper median gives a single-valued strategyproof rule, by the same interval-clamping argument for an order statistic. Arbitrary tie selection is not asserted to have this property. The corrected conclusion is therefore a unique strict winner and its strategyproof selection for odd electorates; a weak winner with a specified median rule for even electorates.

A weak Condorcet winner is never defeated by a strict pairwise majority, allowing ties. For even electorates with single-peaked preferences, all alternatives between the two middle peaks have this property. A single-valued selection requires a tie convention; arbitrary tie selection is not automatically strategyproof.