Modern IoT and embedded platforms must start execution from a known trusted state to thwart malware, ensure secure firmware updates, and protect critical infrastructure. Current approaches to establish a root of trust depend on secret keys and/or specialized secure hardware, which drives up costs, may involve third parties, adds operational complexity, and relies on assumptions about an attacker's computational power. In contrast, TRUSTCHECKPOINTS is the first system to establish an unconditional software root of trust based on a formal model without relying on secrets or trusted hardware. Developers capture a full-system checkpoint and later roll back to it and prove this to an external verifier. The verifier issues timing-constrained, randomized k-independent polynomial challenges (via Horner's rule) that repeatedly scan the fast on-chip memory in randomized passes. When malicious code attempts to persist, it must swap into slower, unchecked off-chip storage, causing a detectable timing delay. Our prototype for a commodity ARM Cortex-A53-based platform validates 192 KB of SRAM in approximately 10 s using 500 passes, sufficient to detect single-instruction persistent malware. The prototype then seamlessly extends trust to DRAM. Two modes (fast SRAM-bootstrap and comprehensive full-memory scan) allow trade-offs between speed and coverage, demonstrating reliable malware detection on unmodified hardware.