Counterexample to naive hit-on-cycle / least-entry conjectures

Theorem

Take the Mordell-hard prime

Status: exact finite counterexample

Date: 2026-08-15

Depends on: EXTERNAL-NR-FACTOR-CYCLE.md, fixed-k signed-divisor criterion

Claim boundary: this does not weaken the universal factor-cycle theorem. It falsifies stronger conjectures that a cycle itself, or the component entered from the least external nonresidue, must contain a successful fixed-k FAB placement.

1. Prime

Take the Mordell-hard prime

\boxed{p=5569.}

Its least external quadratic-nonresidue prime is

\boxed{q_1=13.}

Using the canonical shift multiplier

\sigma(q)=\begin{cases}1,&q\equiv3\pmod4,\\3,&q\equiv1\pmod4,\end{cases}

the natural least-external-factor map gives the exact cycle

\boxed{13\to701\to137\to13.}

Every vertex is 1 mod 4, so the fixed-k auxiliary is k=3q at each stage.

2. Vertex 13

k=39, \qquad C=\frac{5569+39}{4}=1402=2\cdot701.

The signed divisor box modulo 39 is

\{1,2,19,20,37,38\}.

The exact target is

-p^{-1}\equiv5\pmod{39},

which is absent. Thus the vertex fails.

The external factor 701|C supplies the edge

13\to701.

3. Vertex 701

k=2103, \qquad C=\frac{5569+2103}{4}=1918=2\cdot7\cdot137.

The signed divisor box has 27 residues modulo 2103; its exact target is

-p^{-1}\equiv719\pmod{2103},

which is absent.

The external factor 137|C gives

701\to137.

4. Vertex 137

k=411, \qquad C=\frac{5569+411}{4}=1495=5\cdot13\cdot23.

The signed divisor box has 23 residues modulo 411; its exact target is

-p^{-1}\equiv20\pmod{411},

which is absent.

The external factor 13|C closes the cycle:

137\to13.

5. Consequence

Therefore both statements are false:

every external-nonresidue factor cycle contains a fixed-k FAB hit;
the component entered from the least external nonresidue must contain a hit before returning to a cycle.

The factor-cycle theorem remains valuable as a defect transport mechanism, but the universal ES argument must use additional information, such as the Kneser full-stabilizer defect attached to failed vertices, or a global argument over multiple components / external primes.

In particular, a proof cannot rest on a purely graph-theoretic no-cycle or extremal-entry claim.