At a nationally televised forum on leadership and public trust in Philadelphia, an exchange between former President Barack Obama and President Donald Trump took an unexpected turn when a familiar political argument — over credentials and intelligence — became unusually literal.

The event, billed as a conversation about governance and accountability, was structured to encourage civil dialogue. Mr. Trump arrived in campaign mode, punctuating answers with boasts about economic performance and deriding critics as out-of-touch elites. Mr. Obama, speaking in a more restrained register, emphasized institutional norms and democratic standards.

Tension sharpened when the discussion turned to expertise and merit. Mr. Trump mocked academic pedigrees and suggested that conventional measures of intelligence were overrated compared with what he described as real-world success. He reiterated a longstanding refrain that he possessed exceptional cognitive abilities, pointing to past references to cognitive testing and his own business record.

Mr. Obama did not respond in kind. Instead, he reframed the exchange around consistency. For years, he noted, Mr. Trump had publicly demanded documentation from rivals and critics — from birth certificates to medical evaluations — arguing that transparency was essential to public trust. If documentation was the standard, Mr. Obama suggested, it should apply evenly.

“If you want the country to trust your claims,” Mr. Obama said evenly, “show your work.”

The line landed with more weight than volume. Rather than escalating rhetorically, Mr. Obama pressed the point that assertions about intelligence, when repeatedly invoked as a political weapon, invite scrutiny. “Not a slogan,” he said. “A document.”

What followed appeared to surprise even the moderator. Mr. Trump reached into his jacket and produced a folded sheet of paper, which he described as containing his SAT score from his youth. Holding it up before the cameras, he signaled that he was willing to provide the number.

The room, according to several attendees, grew noticeably still. Mr. Trump began reading from the paper, initially citing one figure before correcting himself and stating a total score of 970 out of 1600. The number, while neither exceptional nor disqualifying, fell short of the image of academic dominance he has often projected.

There was no immediate uproar. Instead, the reaction was subdued — a mix of murmurs and scattered applause that reflected less a judgment on the score itself than on the unexpected disclosure. Standardized tests from decades past are imperfect measures of ability; they reflect preparation, context and a moment in time. Yet in political theater, symbols matter.

Mr. Trump quickly defended the result, describing it as strong for its era and questioning the relevance of standardized testing to presidential performance. He emphasized that business achievements and electoral victories were more meaningful indicators of capability than adolescent exam results.

Mr. Obama did not dwell on the number. Instead, he returned to his earlier theme. “The score isn’t the story,” he said. “The story is the standard.” His argument was that leaders who demand documentation from others should be prepared to meet the same expectation without resentment.

The exchange encapsulated a dynamic that has defined much of the political rivalry between the two men. Mr. Trump has frequently portrayed himself as uniquely gifted, often labeling opponents as “low IQ.” Mr. Obama, by contrast, has tended to respond indirectly, relying on appeals to norms and institutional fairness rather than personal invective.

Political analysts noted that the episode’s significance lay less in the specific score than in the broader narrative about proof and performance. In an era when accusations and counteraccusations travel instantly online, the call to “show your work” has become shorthand for a demand that rhetoric align with verifiable fact.

Within minutes of the broadcast’s conclusion, clips of the moment circulated widely on social media platforms. Supporters of Mr. Trump argued that the disclosure demonstrated confidence and transparency. Critics focused on what they saw as an inconsistency between past boasts and the modest figure revealed.

Historians of presidential politics caution against overstating the importance of standardized tests. Academic scores have rarely been central to evaluations of presidents, whose legacies are shaped by policy decisions, crisis management and coalition-building rather than classroom performance. Still, the symbolism of documentation carries weight in a political culture attuned to authenticity and hypocrisy alike.

By the end of the evening, the forum had drifted back to prepared questions about economic growth and international alliances. Yet the moment lingered. A folded sheet of paper had briefly shifted the atmosphere, turning an abstract debate about intelligence into a concrete data point.

Whether the disclosure will have lasting political impact is uncertain. Campaign narratives are resilient, and public opinion often hinges on broader concerns than a decades-old test score. But for a few minutes on a Philadelphia stage, the conversation moved from bravado to arithmetic — and from assertion to evidence — underscoring how quickly political myth can be tested when challenged to produce proof.