# Verification report Manuscript: Critical Points and Local Maxima of Sparse Binary Restricted Boltzmann Machines. Alper Ferudun, Mercury Software GmbH. Version 1.0, 5 October 2026. ## Accepted mathematical scope For a finite binary RBM with at least one visible unit, independent finite real biases and allowed weights, and every hidden degree at most two, the proof classifies all finite critical points of exact unregularized empirical log likelihood. Arbitrary empirical laws, repeated hidden neighborhoods, visible cycles, leaves and isolates are included. Activity is defined by nonzero hidden weight pairs, not by nonzero aggregate interactions. The main theorem gives Hessian inertia (R, n+|A|+R, P-n-|A|-2R), where A is the active pair set and R counts the hidden vertices on inactive pairs with nonzero score. All finite local maxima are global and all finite nonglobal critical points are strict saddles. Additional results describe finite attainment, critical visible laws by effective submodels, and Morse-Bott versus singular-cone maximizing fibers. Two six-observation examples distinguish finite attainment with the same support counts and pairwise support patterns. The three-input example is a nonstrict saddle, not a local maximum. The four-input example is a nonglobal nonstrict local maximum for a positive parity-biased law. The local inequality holds in a full parameter neighborhood. An explicit finite mixture-model point has higher likelihood, so the nonglobality assertion does not rely on an unattained boundary optimum. At parity bias 1/2, the sample has size 32 and the explicit improvement has likelihood gain at least 3/270400. ## Analytic self audit The detailed audit addresses the singular parametrization, pair cancellation, inactive block decoupling, raw-coordinate Hessian congruence, boundary samples, realization of every active submodel, singular fibers, both sample examples, the cubic saddle expansion, and the degree-four whole-neighborhood and finite-improvement inequalities. No mathematical gap was identified within this scope. The originating researcher accepted the written proof; no independent human or external-model review is claimed or required. ## Exact checks The bundled Python checker uses SymPy 1.14.0 and passes 65 exact rational and symbolic assertions. Finite raw RBM Hessians are computed from hidden conditional Bernoulli moments separately from the predicted pair-map inertia. Symmetric congruence elimination and characteristic-polynomial sign counting agree; the latter applies because the matrices are real symmetric. Cases include visible-only models, leaves and isolated units, active zero interactions, positive-negative parallel cancellation, inactive parallel units, mixed triangles and cycles, the two six-state samples, and the parity constructions. Rational mixture parameters and all constants in the degree-four bound are checked. Finite tests do not establish the universal theorems, exhaustive novelty or proof-assistant formalization. An initial checker run stopped after ten passes because it attempted to convert a SymPy Boolean with int. That software error was corrected; subsequent runs passed 53 and then 65 assertions. The failed source/output and later outputs are retained in the research dossier and bundled evidence. No mathematical counterexample was suppressed or failed output overwritten. ## Source coverage and prior work The canonical source is AIM Boltzmann Machines Problem 5.2, represented by AIM-PROBABILITY-0013 in frozen UnsolvedMath v1.6.0. It asks a broader architecture-dependent question. These theorems do not classify all architectures and do not close that source record. The degree-one special case is credited to the dataset's upstream partial attempt. Softplus interaction representation and finite exponential-family MLE criteria are standard background. Montufar and Rauh, Fienberg and Rinaldo, Seigal and Montufar, and Montufar's review are credited. Bad RBM local maxima and parity constructions are already known phenomena. A bounded primary-source search did not establish novelty or absolute priority for the exact statements here; neither is claimed. AI assistance was used in research, derivations, verification code and manuscript preparation. The author remains responsible for the claims. This is a self-audited, unrefereed preprint, not a formally verified or independently peer-reviewed solution.