# Verification report

Finite Likelihood Traps on Connected Boltzmann Trees. Alper Ferudun, Mercury Software GmbH. Version 1.0, 17 October 2026. Source identifier: AIM-PROBABILITY-0007.

## Accepted mathematical scope

For q_rho(x)=(1+rho x_2 x_3)/8, the finite three-leaf RBM point with zero biases and weights (t,0,0) is a nonglobal local maximum precisely when 0<|rho|<sinh(t)^2. Above this threshold it is a strict saddle; at equality a cubic escape proves a nonstrict saddle. The full-support 32-observation example takes rho=1/4 and t=(log 3)/2. Both mixture weights are 1/2 and all conditional probabilities are interior.

Joining k branches at one observed center gives a connected allowed tree with at least 2^k distinct local-maximum visible laws in the full 6k+1 dimensional raw parameter space. The all-good law is the global empirical fit. The other displayed laws lose (k-|S|)Delta(1/4) in normalized log likelihood. The sample size is 2*16^k. The maxima are nonstrict and some allowed weights are zero; effective nonzero connectivity, a positive barrier and arbitrary-data stability are not claimed. The original AIM algorithm-design question remains open.

## Analytic audit

Absorbing Walsh unary coefficients into visible fields is a smooth invertible coordinate change. The uniform-law fiber has two free coordinates. Its five-dimensional normal Hessian is uniformly negative on a neighborhood when the inequality is strict. The integrated Taylor inequality proves local maximality in all seven coordinates, not merely on a symmetric slice. At equality, reducing the first weight while activating the other two yields a positive cubic term.

For the connected family, conditioning on the observed center separates the branch likelihoods. Profiling a smooth finite center bias bounds a bad branch by the single-star theorem. Conditional KL inequalities bound every good branch and the actual center marginal. The sum bounds arbitrary nearby perturbations of all parameters; independently profiled biases are not incorrectly imposed on the shared actual center.

## Supplementary checks

In each normal and optimized run, 18,974 exact predicates and 1,507 floating-point checks pass. Conditional-hidden moments produce 60 raw seven-dimensional Hessian tests. Rational congruence elimination verifies inertia; the canonical characteristic polynomial and degenerate threshold are checked exactly. Separate explicit hidden-state summation with rational exponentiated 0/1 parameters verifies all 30 connected configurations through k=4, their probabilities, finite sample counts and allowed tree graphs.

The floating-point component tests 1,500 seeded nearby points, a finite improvement and six threshold escapes. Those samples are falsification attempts, not a neighborhood proof. The separate probability implementation is in the same checker and is not an independent reviewer.

All seven rendered manuscript pages have been directly inspected. The publication workflow additionally requires successful extracted-source rebuild and normal/optimized replay before readiness is recorded. Earlier environment and Boolean-conversion failures remain in the research dossier; they were corrected before the successful runs.

This is originating-researcher self-audit, not external review, peer review or proof-assistant certification. No gap was identified in the stated analytic scope. AI assistance is disclosed in the article, and absolute priority is not certified.