GPT 6 Astra Ultra attempt on Math-671

Problem 96: a two-degeneracy argument on a parabola

1) OBJECTIVE AND SOURCE REVIEW

Attempt dated 2026-09-23. The target is a linear bound for unit distances among vertices in convex position. I read gpt_pro_5.4/96.tex fully. Its circle argument is sound and its quartic-root argument gives $2n$ for a parabola, though its expanded quartic is incorrect. I strengthen the latter bound to $2n-3$ for $n\ge2$ using a vertex that can always be removed.

2) WORK

Let $P(t)=(t,t^2)$ and fix a distance $r>0$. In a finite set of parameters choose $a$ with $|a|$ minimal. Reflection permits $a\ge0$. Every remaining parameter lies either in $[a,\infty)$ or in $(-\infty,-a]$. The squared distance from $P(a)$ is \[ D_a(x)=(x-a)^2+(x^2-a^2)^2, \qquad D_a'(x)=2(x-a)\left(1+2x(x+a)\right). \] For $x>a$ both factors are positive. For $x<-a$, the first factor is negative and $x(x+a)\ge0$, so the second is positive. Thus $D_a$ is strictly increasing on the right ray and strictly decreasing as $x$ increases on the left ray. Each ray contributes at most one point with $D_a(x)=r^2$. The chosen vertex has at most two distance-$r$ neighbors.

Every subset has the same property, by choosing its own smallest absolute parameter. Removing such vertices until two remain counts each edge once and gives \[ e_r(A)\le2(n-2)+1=2n-3\qquad(n\ge2). \] For $n=1$ the bound is zero. Euclidean similarities transport this argument to every parabola, because the prescribed distance merely rescales. No assumption of uniform spacing is used.

3) ADVERSARIAL VERIFICATION

The actual expansion is \[ D_a(x)-r^2=x^4+(1-2a^2)x^2-2ax+a^4+a^2-r^2; \] the cubic term and several coefficients in the supplied expansion should be removed or changed. This does not invalidate its root-count bound, but it does invalidate the printed polynomial identity. When $a=0$, the two rays meet only at the center point, which cannot be a distance-$r$ neighbor. When both $a$ and $-a$ occur, the latter is included in the left ray without duplicating the chosen point.

4) FINAL

UNRESOLVED. The proof certifies a stronger linear bound on the parabolic subclass. General convex polygons do not admit a common parabolic parameter or this monotonicity argument. A uniformly bounded-degree vertex in every convex subset remains unproved here.

5) SOURCES CHECKED

The complete supplied version, read 2026-09-23. The algebra, differentiation, and deletion proof are self-contained and do not use an unverified external geometric theorem. No literature-priority claim is made.

6) COMPLETION ESTIMATE

Subjective progress toward arbitrary convex polygons. COMPLETION: 12%