# Verification of the connected likelihood model

The written proof establishes a compact connected rational three-segment model with all four probabilities at least \(1/32\), a smooth distinguished point \(q\), and a local entropy boundary for its logarithmic Voronoi cell. It also proves nonsemialgebraicity over the reals and the sharp four-outcome lower bound under the stated smooth positive-dimensional hypothesis.

## Main proof checks

- The connector joins exactly the specified endpoints. Its first-pair mass decreases strictly between the masses of the two other segments, ruling out unintended intersections.
- The distinguished point is interior to the second segment and is neither a corner nor an isolated point.
- The two segment maxima are obtained by strict concavity. Their likelihood difference is \(\log4-H(r)-z\log(3/2)\).
- At the central tie point, exact completed-square identities bound the exponential of twice the connector score difference by \(75/128<1\). Compactness extends this strict gap to an open data neighborhood. The boundary is therefore a global-MLE boundary of the whole connected model.
- Nonsemialgebraicity follows from the logarithmic derivative of the entropy boundary. The argument permits real coefficients and does not confuse an arithmetically transcendental endpoint with a nonsemialgebraic curve.
- Smooth-point stationarity gives \(d\) independent constraints in a simplex of dimension \(m-1\). Convexity of cells makes every cell of dimension at most one semialgebraic, yielding the stated lower bound when \(m\leq3\).
- The contrasting convex-model proposition uses a sufficient first-order concavity inequality, with only positive denominators cleared.

## Reproducible checks

The supplied checker passed 12,019 exact rational checks and 9,000 floating-point diagnostic checks in both ordinary and optimized Python modes. The two reports agree. Exact checks cover normalization, the lower probability bound, connector endpoints, product identities, completed squares and separating masses. Floating diagnostics independently maximize along the connector by concave derivative bisection and test both sides of the proposed boundary.

These counts are regression tests, not exhaustive proofs of continuum claims. The analytic neighborhood and nonsemialgebraicity arguments are established in the text, not inferred from the diagnostics.

The five-page PDF compiled successfully with the desktop LaTeX compiler and Tectonic, without undefined references, overfull or underfull boxes, missing glyphs or compilation warnings. All five rendered pages were directly inspected and were readable without clipped mathematics.

## Scope and attribution

The AIM source is Question 26 on printed page 7 of its computational algebraic statistics notes. The following Seth Sullivant entry is not part of that question. The source asks broadly about implications of semialgebraic geometry for maximum likelihood and is not completely closed by this note.

Alexandr and Heaton's theory and two-disjoint-line computation precede this construction. Alexandr's 19 August 2026 preprint already proves nonalgebraicity in a cuspidal example and arithmetic transcendence in a smooth one. The present contribution is the explicit connected uniformly positive example and outcome bound. The bounded source review does not certify absolute priority.

This is the originating researcher's self-audit, with AI assistance. No independent review or formal verification is represented as having occurred.