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We are offering 33,290,000 shares of our common stock and, in lieu of common stock to certain investors that so choose,pre-fundedwarrants to purchase 6,710,000shares of common stock in this offering. Our common stock is listed on The Nasdaq Capital Market under the symbol “CMPX.” On August 11, 2025, the last reported sale price for our common stock on TheNasdaq Capital Market was $3.34 per share. There is no established public trading market for thepre-fundedwarrants, and we do not expect a market to develop. We donot intend to list thepre-fundedwarrants on The Nasdaq Capital Market or any other national securities exchange or nationally recognized trading system. The purchase price of eachpre-fundedwarrant to purchase one share of our common stock sold in this offering equals the price at which a share of common stock is soldto the public in this offering, minus $0.0001, and the exercise price of eachpre-fundedwarrant to purchase one share of our common stock will equal $0.0001 per share.This prospectus supplement also relates to the offering of the shares of common stock issuable upon exercise of suchpre-fundedwarrants. We are an “emerging growth company” and a “smaller reporting company” under the federal securities laws and, as such, we have elected to comply with certain reducedpublic company reporting requirements for this prospectus supplement and for future filings. See “Prospectus Supplement Summary—Implications of Being anEmerging Growth Company and Smaller Reporting Company.” (1)See “Underwriting” for a description of the compensation payable to the underwriters. Investing in our common stock andpre-fundedwarrants involves a high degree of risk. See “Risk Factors” beginningon pageS-8of this prospectus supplement, as well as those risks described in our most recent Annual Report on Form10-Kfor the year ended December31, 2024 and in our other filings with the Securities and Exchange Commission thatare incorporated by reference into this prospectus supplement. Neither the Securities and Exchange Commission nor any state securities commission has approved or disapproved of these securities ordetermined if this prospectus supplement or the accompanying prospectus is truthful or complete. Any representation to the contrary is acriminal offense. The underwriters expect to deliver the shares of common stock to purchasers on or about August 14, 2025. Thepre-fundedwarrants are expected to be delivered on orabout August 14, 2025. GuggenheimSecurities Jefferies H.C. Wainwright & Co. Raymond James D. Boral Capital Ladenburg Thalmann Table of Contents TABLE OF CONTENTS Prospectus Supplement About this prospectus supplementSpecial note regarding forward-looking statementsProspectus supplement summaryThe offeringRisk factorsUse of proceedsDilutionDividend policyDescription ofpre-fundedwarrantsUnderwritingLegal mattersExpertsWhere you can find more informationIncorporation of certain information by reference Prospectus About this ProspectusRisk FactorsSpecial Note Regarding Forward-Looking StatementsSummaryUse of ProceedsSecurities We May OfferDescription of Capital StockDescription of Debt Securities Description of Warrants Legal Matters Where You Can Find More Information Incorporation by Reference Table of Contents ABOUT THIS PROSPECTUS SUPPLEMENT This document is in two parts. The first part is this prospectus supplement, which describes the terms of this offering of common stock andpre-fundedwarrants and also adds to and updates information contained in the accompanying prospectus and the documents incorporated by reference into thisprospectus supplement and the accompanying prospectus. The second part, the accompanying prospectus dated September6, 2024, including thedocuments incorporated by reference therein, provides more general information. Generally, when we refer to this prospectus, we are referring to bothparts of this document combined. To the extent there is a conflict between the information contained in this prospectus supplement, on the one hand, andthe information contained in the accompanying prospectus or in any document incorporated by reference that was filed with the Securities and ExchangeCommission (the SEC), before the date of this prospectus supplement, on the other hand, you should rely on the information in this prospectussupplement. If any statement in one of these documents is inconsistent with a statement in another document having a later date—for example, adocument incorporated by reference in the accompanying prospectus—the statement in the document having the later date modifies or supersedes theearlier statement. We have not, and the underwriters have not, authorized anyone to provide any information other than that contained or incorporated by reference in thisprospectus supplement, the accompanying prospectus or in any free writing prospectus that we have authorized for use in connection with this offering.We and the underwriters take no responsib