Adagio Medical Holdings, Inc.

FOR THE 2026 ANNUAL MEETING OF STOCKHOLDERS

TO BE HELD ON JUNE 16, 2026

QUESTIONS AND ANSWERS ABOUT THESE PROXY MATERIALS AND VOTING

What is Adagio Medical Holdings, Inc?

On July 31, 2024 (the “Closing Date”), ARYA Sciences Acquisition Corp IV, a Cayman Islands exempted company (“ARYA”), Aja Holdco, Inc., a Delaware corporation and wholly owned subsidiary of ARYA (“ListCo”), Aja Merger Sub 1, a Cayman Islands exempted company and wholly owned subsidiary of ListCo (“ARYA Merger Sub”), Aja Merger Sub 2, Inc., a Delaware corporation and wholly owned subsidiary of ListCo (“Company Merger Sub”), and Adagio Medical, Inc., a Delaware corporation (“Legacy Adagio”), consummated the previously announced business combination (the “Business Combination”) pursuant to the terms of the Business Combination Agreement, dated February 13, 2024, by and among the foregoing parties, as amended by the Consent and Amendment No. 1 to Business Combination Agreement, dated as of June 25, 2024 (as so amended, the “Business Combination Agreement”), by and between ARYA and Legacy Adagio. In connection with the closing of the Business Combination (the “Closing”), among other things, (i) ARYA Merger Sub merged with and into ARYA (the “ARYA Merger”) and Company Merger Sub merged with and into Legacy Adagio (together with the ARYA Merger, the “Mergers”), with ARYA and Legacy Adagio surviving the Mergers and, after giving effect to such Mergers, each of ARYA and Legacy Adagio became a wholly owned subsidiary of ListCo, and (ii) ListCo changed its name to “Adagio Medical Holdings, Inc.”

Unless the context indicates otherwise, references in this proxy statement to the “Company,” “Adagio,” “we,” “us,” “our” and similar terms refer to Adagio Medical Holdings, Inc.

Why am I receiving these materials?

We have sent you these proxy materials because the Board of Directors (the “Board”) of Adagio Medical Holdings, Inc. is soliciting your proxy to vote at the 2026 Annual Meeting of Stockholders (the “Annual Meeting”), including at any adjournments or postponements of the meeting. You are invited to attend the annual meeting online to vote on the proposals described in this proxy statement. However, you do not need to attend the meeting to vote your shares. Instead, you may simply complete, sign and return the enclosed proxy card, or follow the instructions below to submit your proxy over the phone or through the internet.

We intend to first mail these proxy materials on or about April 30, 2026 to all stockholders of record entitled to vote at the annual meeting.

How do I attend the Annual Meeting?

This year’s Annual Meeting will be a virtual meeting, which will be conducted entirely online via audio webcast to allow greater participation. You may attend, vote and ask questions at the Annual Meeting by following the instructions provided on the proxy card or voting instruction form to log in to www.virtualshareholdermeeting.com/ADGM2026. If you are a stockholder of record, you will be asked to provide the 16-digit control number from your proxy card. If you are a beneficial owner of shares registered in the name of your broker, bank or other agent, follow the instructions from your broker or bank.

The audio webcast of the Annual Meeting will begin promptly at 10:00 a.m. Pacific Time. We encourage you to access the meeting prior to the start time. Online check-in will begin at 9:45 a.m. Pacific Time, and you should allow reasonable time for the check-in procedures.

You are entitled to attend the Annual Meeting if you were a stockholder of record as of the close of business on April 24, 2026, the record date, or hold a valid proxy for the meeting. To be admitted to the Annual Meeting, you will need to visit www.virtualshareholdermeeting.com/ADGM2026 and enter the Control Number found next to the label

“Control Number” on your proxy card or voting instruction form, or in the email sending you the proxy statement. If you are a beneficial stockholder, you should contact the bank, broker or other institution where you hold your account well in advance of the meeting if you have questions about obtaining your control number/ proxy to vote.

Whether or not you participate in the Annual Meeting, it is important that you vote your shares.

What if I cannot find my Control Number?

Please note that if you do not have your Control Number and you are a registered stockholder, you will be able to login as a guest. To view the meeting webcast visit www.virtualshareholdermeeting.com/ADGM2026 and register as a guest. If you login as a guest, you will not be able to vote your shares or ask questions during the meeting.

If you are a beneficial owner (that is, you hold your shares in an account at a bank, broker or other holder of record), you will need to contact that bank, broker or other holder of record to obtain your Control Number prior to the Annual Meeting.

Where can we get technical assistance if we are having trouble accessing the meeting or during the meeting?

If you have difficulty accessing the meeting or during the meeting, please refer to the technical support telephone number posted on the virtual meeting website login page, where technicians will be available to help you.

For the Annual Meeting, how do we ask questions of management and the Board?

We plan to spend up to 15 minutes answering appropriate stockholder questions at the conclusion of the Annual Meeting and will include as many stockholder questions that comply with the rules of conduct for the Annual Meeting as the allotted time permits. If we receive substantially similar questions, we will group such questions together and provide a single response to avoid repetition. Questions that are not pertinent to the Company and in compliance with the rules of conduct for the Annual Meeting will not be responded to. Questions may be submitted during the Annual Meeting through www.virtualshareholdermeeting.com/ADGM2026.

Who can vote at the annual meeting?

Only stockholders of record at the close of business on April 24, 2026 will be entitled to vote at the Annual Meeting.

If on April 24, 2026 your shares were registered directly in your name with the Adagio’s transfer agent, Continental Stock Transfer & Trust Company, then you are a stockholder of record. As a stockholder of record, you may vote online, at the meeting or vote by proxy. Whether or not you plan to attend the meeting, we urge you to fill out and return the enclosed proxy card, vote by proxy over the telephone, or vote by proxy through the internet to ensure your vote is counted.

Beneficial Owner: Shares Registered in the Name of a Broker or Bank

If on April 24, 2026 your shares were held, not in your name, but rather in an account at a brokerage firm, bank or other similar organization, then you are the beneficial owner of shares held in “street name” and these proxy materials should be forwarded to you by that organization. The organization holding your account is considered to be the stockholder of record for purposes of voting at the Annual Meeting. As a beneficial owner, you have the right to direct your broker, bank or other agent regarding how to vote the shares in your account. You must follow the instructions provided by your brokerage firm, bank, or other similar organization for your bank, broker or other stockholder of record to vote your shares per your instructions. Alternatively, many brokers and banks provide the means to grant proxies or otherwise instruct them to vote your shares by telephone and via the Internet, including by providing you with a 16-digit control number via email or on your proxy card or your voting instruction form. If your shares are held in an account with a broker, bank or other stockholder of record providing such a service, you may instruct them to vote your shares by telephone (by calling the number provided in the proxy materials) or over the Internet as instructed by your broker, bank or other stockholder of record. If you did not receive a 16-digit control number via email or on your proxy card or voting instruction form, and you wish to vote prior to or at the virtual Annual Meeting, you must follow the instructions from your broker, bank or other stockholder of record, including any requirement to obtain a valid legal proxy. Many brokers, banks and other stockholders of record allow a beneficial owner to obtain a valid legal proxy either online or by mail, and we recommend that you contact your broker, bank or other stockholder of record to do so.

What am I voting on?

There are two matters scheduled for a vote:

What if another matter is properly brought before the meeting?

The Board does not know of any other matters to be brought before the meeting. If other matters are presented, the proxy holders have discretionary authority to vote all proxies in accordance with their best judgment. Discretionary authority for them to do so is provided for in the proxy card.

How do I vote?

For Proposal 1, you may either vote “For” all the nominees to the Board or you may “Withhold” your vote for any nominee to the Board that you specify. For Proposal 2, you may vote “For” or “Against” or abstain from voting.

The procedures for voting are fairly simple:

If you are a stockholder of record, you may vote at the Annual Meeting, vote by proxy using the enclosed proxy card, or you may vote by proxy over the telephone or through the internet. Whether or not you plan to attend the Annual Meeting, we urge you to vote by proxy to ensure your vote is counted. You may still attend the Annual Meeting and vote at the Annual Meeting even if you have already voted by proxy. This is only required if you want to change your original vote, since votes will not be double counted.

Internet and telephone voting facilities for stockholders of record will be available for 24 hours a day and will close at 8:59 p.m. Pacific Time on June 15, 2026.

If you are a stockholder of record and do not vote by completing your proxy card, by telephone, through the internet or online at the Annual Meeting, your shares will not be voted.

If you return a signed and dated proxy card or otherwise vote without marking voting selections, your shares will be voted, as applicable, “For” the election of all Class II nominees for director and “For” the ratification of the appointment by the Audit Committee of the Board (the “Audit Committee”) of WithumSmith+Brown, PC as our independent registered public accounting firm for the fiscal year ending December 31, 2026. If any other matter is properly presented at the meeting, your proxy holder (one of the individuals named on your proxy card) will vote your shares using that individual’s best judgment.

Beneficial Owner: Shares Registered in the Name of Broker or Bank

If you are a beneficial owner of shares registered in the name of your broker, bank or other agent, you should have received a voting instruction form with these proxy materials from that organization rather than from Adagio. You must follow these instructions for your bank, broker or other stockholder of record to vote your shares per your instructions. Alternatively, many brokers and banks provide the means to grant proxies or otherwise instruct them to vote your shares by telephone and via the Internet, including by providing you with a 16-digit control number via email or on your proxy card or your voting instruction form. If your shares are held in an account with a broker, bank or other stockholder of record providing such a service, you may instruct them to vote your shares by telephone (by calling the number provided in the proxy materials) or over the Internet as instructed by your broker, bank or other stockholder of record. If you did not receive a 16-digit control number via email or on your proxy card or voting instruction form, and you wish to vote prior to or at the virtual Annual Meeting, you must follow the instructions from your broker, bank or other stockholder of record, including any requirement to obtain your 16-digit control number. Many brokers, banks and other stockholders of record allow a beneficial owner to obtain their 16-digit control number either online or by mail, and we recommend that you contact your broker, bank or other stockholder of record to do so.

Internet proxy voting will be provided to allow you to vote your shares online, with procedures designed to ensure the authenticity and correctness of your proxy vote instructions. However, please be aware that you must bear any costs associated with your internet access, such as usage charges from internet access providers and telephone companies.

How many votes do I have?

Each share of common stock you owned as of April 24, 2026 is entitled to one vote.

If I am a stockholder of record and I do not vote, or if I return a proxy card or otherwise vote without giving specific voting instructions, what happens?

If you are a stockholder of record and do not vote by completing your proxy card, by telephone, through the internet or online at the Annual Meeting, your shares will not be voted.

If you return a signed and dated proxy card or otherwise vote without marking voting selections, your shares will be voted, as applicable, “For” the election of all Class II nominees for director and “For” the ratification of the appointment by the Audit Committee of WithumSmith+Brown, PC as our independent registered public accounting firm for the fiscal year ending December 31, 2026. If any other matter is properly presented at the meeting, your proxy holder (one of the individuals named on your proxy card) will vote your shares using that individual’s best judgment.

If I am a beneficial owner of shares held in street name and I do not provide my broker or bank with voting instructions, what happens?

If you are a beneficial owner of shares held in street name and you do not instruct your broker, bank or other agent how to vote your shares, your broker, bank or other agent may still be able to vote your shares in its discretion. Brokers, banks and other securities intermediaries may use their discretion to vote your “uninstructed” shares with respect to matters considered to be “routine,” but not with respect to “non-routine” matters. In this regard, Proposal 1 is considered to be “non-routine,” meaning that your broker may not vote your shares on this proposal in the absence of your voting instructions. Proposal 2 is considered to be a “routine” matter, meaning that if you do not return voting instructions to your broker by its deadline, your shares may be voted by your broker in its discretion on Proposal 2.

If you are a beneficial owner of shares held in street name, and you do not plan to attend the Annual Meeting, in order to ensure your shares are voted in the way you would prefer, youmust provide voting instructions to your broker, bank or other agent by the deadline provided in the materials you receive from your broker, bank or other agent.

What are “broker non-votes”?

As discussed above, when a beneficial owner of shares held in street name does not give voting instructions to his or her broker, bank or other securities intermediary holding his or her shares as to how to vote on matters deemed to be “non-routine,” the broker, bank or other such agent cannot vote the shares. When there is at least one “routine” matter that the broker, bank or other securities intermediary votes on, the shares that are un-voted on “non-routine” matters are counted as “broker non-votes.” Proposal 2 is a “routine” matter and we therefore expect brokers, banks or other securities intermediaries to vote on that proposal. Proposal 1 is considered to be “non-routine” and we therefore expect broker non-votes to exist in connection with that proposal.

As a reminder, if you are a beneficial owner of shares held in street name, in order to ensure your shares are voted in the way you would prefer, youmust provide voting instructions to your broker, bank or other agent by the deadline provided in the materials you receive from your broker, bank or other agent.

Who is paying for this proxy solicitation?

Adagio will pay for the entire cost of soliciting proxies. In addition to these proxy materials, our directors and employees may also solicit proxies in person, by email, by telephone, or by other means of communication. Directors and employees will not be paid any additional compensation for soliciting proxies. We may also reimburse brokerage firms, banks and other agents for the cost of forwarding proxy materials to beneficial owners.

What does it mean if I receive more than one set of proxy materials?

If you receive more than one set of proxy materials, your shares may be registered in more than one name or in different accounts. Please follow the voting instructions on the proxy cards in the proxy materials to ensure that all of your shares are voted.

Can I change my vote after submitting my proxy?

Yes. You can revoke your proxy at any time before the final vote at the Annual Meeting. If you are the record holder of your shares, you may revoke your proxy in any one of the following ways:

Your most current proxy card or telephone or internet proxy is the one that is counted.

Beneficial Owner: Shares Registered in the Name of Broker or Bank

If your shares are held by your broker, bank or other agent, you should follow the instructions provided by your broker, bank or other agent.

What vote is required for adoption or approval of each proposal and how will votes be counted?

For Proposal 1, the election of directors, the two nominees receiving the most “For” votes from the holders of shares present or represented by proxy and entitled to vote on the election of directors will be elected. Only votes “For” will affect the outcome.

To be approved, Proposal 2, ratification of the selection of WithumSmith+Brown, PC as the Company’s independent registered public accounting firm for the year ending December 31, 2026, must receive “For” votes from the holders of a majority of shares present or represented by proxy and entitled to vote on the matter. If you “Abstain” from voting, it will have the same effect as an “Against” vote. Broker non-votes will have the same effect as “Against” votes. Since brokers have authority to vote on your behalf with respect to Proposal 2, we do not expect broker non-votes on this proposal.

Who will count the vote?

Representatives of Broadridge Financial Solutions, Inc. will tabulate the votes and act as inspectors of election.

What is the quorum requirement?

A quorum of stockholders is necessary to hold a valid meeting. A quorum will be present if stockholders holding at least a majority of the voting power of the outstanding shares entitled to vote at the Annual Meeting are present in person or represented by proxy. On April 24, 2026, there were 22,210,459 shares of common stock, outstanding and entitled to vote.

Abstentions and broker non-votes will be counted towards the quorum requirement. If there is no quorum, either the chairperson of the Annual Meeting or the holders of a majority of the voting power of the shares present at the Annual Meeting or represented by proxy and entitled to vote may adjourn the Annual Meeting to another date.

Will a list of record stockholders as of the record date be available?

For the ten days ending the day prior to the annual meeting, a list of our record stockholders as of the close of business on the record date will be available for examination by any stockholder of record for a legally valid purpose at our corporate headquarters during regular business hours. To access the list of record stockholders beginning May 22, 2026, and until the meeting, stockholders should emailir@adagiomedical.com.

How can I find out the results of the voting at the Annual Meeting?

Preliminary voting results will be announced at the Annual Meeting. In addition, final voting results will be published in a current report on Form 8-K that we expect to file within four business days after the Annual Meeting. If final voting results are not available to us in time to file a Form 8-K within four business days after the Annual Meeting, we intend to file a Form 8-K to publish preliminary results and, within four business days after the final results are known to us, file an additional Form 8-K to publish the final results.

What proxy materials are available on the internet?

The proxy statement, Form 10-K and annual report to stockholders are available at www.proxyvote.com.