A.419 FALSE — standing two hundred seventy-six

What the thesis claimed

A.419 prints (a = algebraic connectivity / Fiedler value, g = girth):

with the lower bound attained by graphs made of a cycle on ⌊n/2⌋ vertices and the remaining vertices forming a path attached to the cycle — the tadpole T(⌊n/2⌋, ⌈n/2⌉).

Why it is false

The true minimiser is the tadpole whose cycle has ⌈n/2⌉ vertices. The claim fails at every odd order. Smallest counterexample n=7: captioned T(3,4) has a/g ≈ 0.075126 while T(4,3) has ≈ 0.069130 — lower by 8.6731%. Exhaustive census of all 842 connected order-7 graphs containing a cycle shows T(4,3) is the unique minimiser. Same pattern at order 9 (T(5,4) among 261,033 graphs behind AGX_A419_FULL=1). For every odd n from 7 to 41 the strict inequality is proved in exact rational arithmetic by LDLᵀ inertia counts against an explicit separating rational; it persists in the tadpole family to n=61. At n=4 and n=5 the caption prescribes a 2-cycle and is not well-defined.

Defect: a pure floor/ceiling parity slip. ⌊n/2⌋ = ⌈n/2⌉ when n is even, so the caption is correct at every even order; replacing floor by ceil repairs it everywhere.

Self-refuting same-page control

A.418 (a+g) and A.420 (a·g) on the same page name the triangle-plus-path family T(3,n−3) and are verified correct by exhaustive census. Only the ratio member of the quadruple has a non-trivial extremal cycle length (interior optimum ⌈n/2⌉); only it is mis-stated. The printed upper bound a/g ≤ n/3 is true and attained uniquely by complete graphs. The printed lower bound itself is “?????” — no formula — so nothing but the extremal claim is refuted. Inequalities of neighbouring conjectures untouched.