spectralGapCert
plain-language theorem explainer
The spectral gap certificate assembles the nonnegativity of variance for functions on the boolean hypercube together with zero J-cost edge weights for the empty formula into a single structure. Researchers analyzing convergence rates of gradient descent on J-cost SAT landscapes cite it when building the master certificate for P versus NP separation. The definition is a direct record construction that applies the variance nonnegativity theorem and the empty formula flat landscape theorem to each required field.
Claim. The spectral gap certificate is the structure whose first field states that the variance of every real-valued function on the $n$-dimensional boolean hypercube is nonnegative and whose second field states that the empty conjunctive normal form formula induces zero J-cost edge weights for every assignment and variable index.
background
The module develops the spectral gap of the J-cost Laplacian, which governs the rate at which gradient descent on the J-cost landscape converges to a minimum for SAT problems. Variance of a function on the boolean cube is defined as the sum of squared deviations from the mean, and its nonnegativity is immediate from the sum-of-squares expression. The empty formula flat landscape theorem shows that the trivial empty CNF formula produces a completely flat J-cost graph with all edge weights zero, obtained by direct simplification of the edge weight definition.
proof idea
The definition is a one-line wrapper per field that applies the variance nonnegativity theorem to the variance field and the empty formula flat landscape theorem to the flatness field of the SpectralGapCert structure.
why it matters
This certificate supplies the spectral component to the master certificate in the P versus NP assembly. It completes the required nonnegativity and flatness properties for the J-cost model in the Recognition Science framework, supporting convergence analysis via the spectral gap. The module status records an open Cheeger-type inequality for positive gaps on unsatisfiable formulas that remains to be discharged.
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