We introduce and validate the lottery codec hypothesis, which states that untrained subnetworks within randomly initialized networks can serve as synthesis networks for overfitted image compression, achieving rate-distortion (RD) performance comparable to trained networks. This hypothesis leads to a new paradigm for image compression by encoding image statistics into the network substructure. Building on this hypothesis, we propose LotteryCodec, which overfits a binary mask to an individual image, leveraging an over-parameterized and randomly initialized network shared by the encoder and the decoder. To address over-parameterization challenges and streamline subnetwork search, we develop a rewind modulation mechanism that improves the RD performance. LotteryCodec outperforms VTM and sets a new state-of-the-art in single-image compression. LotteryCodec also enables adaptive decoding complexity through adjustable mask ratios, offering flexible compression solutions for diverse device constraints and application requirements.