Checking Is Cheaper Than Finding: The Energy Asymmetry of Proof and Search

For standard problem classes, a presented solution is cheap to check and expensive to find. This paper prices both directions in joules, from operation counts and a stated per-switch energy of 2.8125e-16 J (representative loaded silicon gate, 1/2 C V^2 with C = 1 fF and V = 0.75 V). The algorithmic asymmetry ratio is 10^76.76 for a 256-bit hash preimage, 10^30.67 for a 2048-bit factorisation (general number field sieve, leading term), 10^33.74 for subset-sum with 120 items under exhaustive search, and 10^59.90 for 3-SAT with 200 variables. Against this, the minimum ratio thermodynamics permits, priced by the Landauer cost of the bits a reversibly organised search must erase (its answer and its enumeration counter), is at most a factor of two in every case computed, so the gap between the algorithmic ratio and the permitted floor ratio reaches 76.46 orders of magnitude: the asymmetry of proof and search is algorithmic, not thermodynamic. The paper also prints the collapse cases: a polynomial control class (sorting) has a ratio of only 10^1.12 to 10^1.30, meet-in-the-middle search halves the subset-sum exponent, and at small sizes the exponential classes are absolutely cheap. Model calculations are labelled SIMULATED, NOT MEASURED. The deposit includes the paper PDF and Markdown, the computation script (compute.py) that produces every number in the paper, and its outputs (results.json, results.txt). Computations and drafting were performed with Dave, an AI research system.