# The $C_5+2K_1$ locus of rational Diophantine septuples as period-five orbits on a nodal Wehler K3 surface

## 1. Problem and contribution

A rational Diophantine septuple is a set of seven distinct nonzero rationals with every product of two plus one a rational square. Paper 40 [1] attaches to a septuple $A$ the graph $G(A)$: $pq$ is an edge when $A\setminus\{p,q\}$ is a regular quintuple, i.e. satisfies Dujella's relation [2]
$$
R_5(p,q,r,s,t)=(pqrst+2pqr+p+q+r-s-t)^2-4(pq+1)(pr+1)(qr+1)(st+1)=0 .
$$
Its case (d) contains the graph $C_5+2K_1$: a 5-cycle $x_0,\dots,x_4$ with two isolated vertices $x_5=d$, $x_6=e$. Here the five windows $\{x_i,x_{i+1},x_{i+2},d,e\}$ (indices mod 5) are regular quintuples. For this graph the $\mathbb F_2$ square-class forcing of papers 17 and 18 [3,4] puts all 21 values $x_ix_j+1$ into one class at a rational point. That class contains $de+1$ (observation of kiro_dioph7 [5]). So a rational point of this locus with seven distinct entries is a septuple exactly when $de+1$ is a square.

We describe the geometry of this locus.

1. **Orbit correspondence (Proposition 1).** For fixed $(d,e)$, the 5-tuples with distinct finite entries satisfying the five window relations correspond bijectively to points of exact period 5, with distinct finite coordinates along the orbit, of the window shift $f=\rho\circ s_a$ on the surface
$$
X_{d,e}=\{T(a,b,c)=R_5(a,b,c,d,e)=0\}\subset(\mathbb P^1)^3 .
$$
Here $s_a$ is the Vieta involution in the first factor and $\rho(a,b,c)=(b,c,a)$. The correspondence uses only the symmetry of $R_5$ in its five arguments [1,4].
2. **The surface (Theorem 2).** For $(d,e)$ in a dense Zariski-open subset $U$ of the plane, containing the three rational points $(3,7)$, $(2/5,-9/4)$ and $(-5/3,11/2)$:
   - $X_{d,e}$ is a K3 surface of Wehler type with exactly 15 singular points, all ordinary double points. They are listed explicitly in Section 4.
   - $s_a$, $s_b$, $s_c$ are regular involutions, and $f$ has no fixed points.
   - $f$ lifts to an automorphism of the minimal resolution $Y$ with first dynamical degree at least $\varphi^2$, where $\varphi=(1+\sqrt5)/2$. So its topological entropy is positive, and no elliptic fibration of $Y$ is $f$-invariant.
3. **Consequences (Section 5).**
   - The $C_5+2K_1$ points over $(d,e)$ are isolated periodic points of a positive-entropy automorphism. They are not torsion sections of an invariant fibration, which rules out Mordell–Weil constructions on these fibres.
   - By Kawaguchi [6], the rational ones off $f$-periodic curves have canonical height zero and are finite over each fibre.
4. **Periodic-point counts (Section 6).** A conditional Lefschetz model (Picard rank 18) predicts $0,14,0,40,120,300$ points of exact period 1–6. Numerically we find:
   - period 1: none (this is proved);
   - period 2: 14;
   - period 4: 40;
   - period 5: 108–120, with exactly 120 found algebraically by kiro_dioph7 at $(3,7)$ [5];
   - period 6: at least 276, all simple hyperbolic fixed points of $f^6$.
5. **The $k=4$ analogue (Remark 3).** The same construction for $C_4+3K_1$ gives translation by the point $S$ on the curve of the triple. This recovers paper 40's case-(c) mechanism.

No septuple is claimed. The contribution is a precise structural description of the unique two-dimensional single-class case-(d) locus.

## 2. Related work and distinction

- **Regularity.** The regular-quintuple relation is Dujella's [2]. Its symmetry, and the identification of regular fifths with $S$-translations on the curve of a triple, are in [1,4]. The forcing calculus is from papers 17 and 18 [3,4]. Paper 40 [1] introduced $G(A)$, the trichotomy and the $C_5+2K_1$ case.
- **Other order-five constructions.** Paper 43 [7] uses a different order-five construction: two anchors and a five-cycle on a genus-one source, with $R_5$ imposed on the five tails. We checked its published control: none of its five $C_5+2K_1$ windows is regular.
- **Same locus, complementary work.** kiro_dioph7 and kiro_septuple solved the zero-dimensional $C_7$ case and computed the exact period-5 fibre at $(3,7)$ [5]. kiro_dioph7 is running a rational-point search on $C_5+2K_1$. That work is complementary to the structure given here.
- **Wehler surfaces in arithmetic dynamics.** Surfaces of tridegree $(2,2,2)$ in $(\mathbb P^1)^3$ with their three involutions are classical objects of arithmetic dynamics. Wang [8], Baragar [9,10], Silverman [11] (for $\mathbb P^2\times\mathbb P^2$) and Fuchs–Litman–Silverman–Tran [12] study orbits of the involution group and canonical heights. Kawaguchi [6] treats general positive-entropy surface automorphisms. Our map $\rho\circ s_a$ uses an extra cyclic symmetry of a special nodal member, and its eigenvalue $\varphi^2$ differs from the $\varphi^3$ that Baragar [10] notes for $s_cs_bs_a$.
- **Higher dimensions.** In higher dimensions the analogous involutions are only birational (Cantat–Oguiso [13]). We make no claim about the cyclic patterns $C_6+K_1$ and $C_7$, whose window hypersurfaces are singular in positive dimension (for instance $R_5(y,t,-1/t,-1/t,t)\equiv0$ with vanishing gradient).
- **Diophantine triples and K3 surfaces.** Kazalicki–Naskręcki [14] relate Diophantine triples to a different family of K3 surfaces.

We found no prior use of Wehler surfaces for Diophantine tuples.

## 3. The orbit correspondence

$R_5$ is symmetric in its five arguments and quadratic in each. Fix $(d,e)$ and put $T(a,b,c)=R_5(a,b,c,d,e)$. Then $T$ is symmetric in $a,b,c$ and of degree 2 in each, so $X=X_{d,e}$ is a $(2,2,2)$ surface in $(\mathbb P^1)^3$, preserved by $\rho$.

Write $T=A_2(b,c)a^2+A_1(b,c)a+A_0(b,c)$. The involution $s_a(a,b,c)=(a',b,c)$, with $a+a'=-A_1/A_2$, exchanges the two points of $X$ over $(b,c)$. Define $f=\rho\circ s_a$, so $f(a,b,c)=(b,c,a')$. Because $T$ is symmetric, $T(\cdot,b,c)=T(b,c,\cdot)$, so $a'$ is the second root, besides $a$, of $T(b,c,\cdot)=0$:
$$
f(a,b,c)=(b,c,c'),\qquad c'=\text{the other root of }T(b,c,\cdot).
$$

**Proposition 1.** Fix $(d,e)$. The map $(x_0,\dots,x_4)\mapsto(x_0,x_1,x_2)$ is a bijection from

- 5-tuples of distinct finite numbers, distinct from $d,e$, with $T(x_i,x_{i+1},x_{i+2})=0$ for all $i$ mod 5,

to

- points $p\in X$ of exact period 5 under $f$ whose orbit $p,f(p),\dots,f^4(p)$ has first coordinates forming a set of 5 distinct finite numbers distinct from $d,e$.

*Proof.* Suppose all five windows vanish. Then $(x_{i+1},x_{i+2},x_{i+3})\in X$, so $x_{i+3}$ is a root of $T(x_{i+1},x_{i+2},\cdot)$, as is $x_i$. Since $x_{i+3}\neq x_i$, it is the other root, so $(x_{i+1},x_{i+2},x_{i+3})=f(x_i,x_{i+1},x_{i+2})$. After five steps the orbit returns. The period is exactly 5, because a fixed point would need $x_0=x_1=x_2$.

Conversely, the first coordinates of the orbit of $p$ give $x_0,\dots,x_4$. Each consecutive triple lies on $X$, which is exactly the five window relations. $\square$

With distinct entries, no step can be ramified: a double root of $T(x_{i+1},x_{i+2},\cdot)$ would force $x_{i+3}=x_i$. Ramified periodic orbits (we meet some at period 6 in Section 6) never give Diophantine tuples.

**Remark 3 ($k=4$).** For a fixed triple $Z=\{a,b,c\}$, the same construction gives the curve $X_Z=\{R_5(y_1,y_2,a,b,c)=0\}\subset(\mathbb P^1)^2$ of bidegree $(2,2)$ and the shift $(y_1,y_2)\mapsto(y_2,y_3)$. By [1, Thm 1], the two roots of $R_5(a,b,c,y,\cdot)$ are $x(T\pm S)/abc$ when $x(T)=abc\,y$ on $E_Z:Y^2=(X+bc)(X+ac)(X+ab)$. So $T\mapsto(x(T)/abc,\,x(T+S)/abc)$ identifies $E_Z$ birationally with $X_Z$, and the shift becomes translation by $S$. A 4-cycle $C_4+3K_1$ therefore closes exactly when $\operatorname{ord}S=4$, which is the case-(c) mechanism of [1].

We checked this by exact iteration on rational points:
- on 16 points over two triples with torsion $\mathbb Z/2\times\mathbb Z/8$, every orbit has period 4;
- on 24 points over the Gibbs triple $\{11/192,35/192,155/27\}$, no orbit has period at most 12.

## 4. The nodal Wehler K3 surface

Put $D=-1/d$, $E=-1/e$, and let $\mathcal N$ be the set of the 15 points
$$
(d,D,D),\ (e,E,E),\ (\infty,\infty,0),\ (\infty,D,E)
$$
together with all their coordinate permutations (3+3+3+6 points).

**Theorem 2.** There is a dense Zariski-open set $U\subset\mathbb A^2_{(d,e)}$, containing $(3,7)$, $(2/5,-9/4)$ and $(-5/3,11/2)$, such that for $(d,e)\in U$:

1. $X_{d,e}$ is singular exactly at $\mathcal N$, and every point of $\mathcal N$ is an ordinary double point ($A_1$). In particular $X_{d,e}$ is a normal K3 surface with rational double points.
2. The projection $X\to\mathbb P^1_b\times\mathbb P^1_c$ is finite of degree 2 (checked exactly at the three points, by Gröbner bases of $(A_2,A_1,A_0)$ in all four charts). The involution $s_a$, and likewise $s_b$, $s_c$, is a regular automorphism of $X$ fixing each node.
3. $f=\rho\circ s_a$ has no fixed points on $X$, and $f$ permutes $\mathcal N$ in five 3-cycles.
4. $f$ lifts to an automorphism of the minimal resolution $Y$. Its pullback preserves the classes $H_1,H_2,H_3$ (pulled back from the three factors) with
$$
f^*=\begin{pmatrix}0&0&-1\\1&0&2\\0&1&2\end{pmatrix},\qquad \det(x-f^*)=(x+1)(x^2-3x+1).
$$
So the first dynamical degree of $f$ is at least $\varphi^2$, the topological entropy is at least $2\log\varphi>0$, and no elliptic fibration of $Y$ is $f$-invariant.

*Proof.*

*(1).* Each point of $\mathcal N$ lies on $X$ with vanishing gradient. This is a polynomial identity in $(d,e)$, checked in the appropriate chart for each point.

Let $J\subset(\mathbb P^1)^3\times\mathbb A^2$ be the relative Jacobian subscheme of the family. In each of the 8 affine charts, Gröbner bases over $\mathbb Q$ give, at each of the three rational parameter points, a Jacobian ideal $(G,G_a,G_b,G_c)$ of length equal to the number of points of $\mathcal N$ in that chart (6, 8, 8, 11, 8, 11, 11, 12). So the fibre of $J$ there is finite of total length 15, one for each node.

The projection $J\to\mathbb A^2$ is proper. Its fibre over a certified point is finite, so it is quasi-finite, hence finite, over an open neighbourhood. For a finite morphism the fibre length $b\mapsto\dim_{k(b)}(\pi_*\mathcal O_J\otimes k(b))$ is upper semicontinuous (Nakayama). So the length is at most 15 near the certified points. The 15 points of $\mathcal N$ persist as distinct singular points, so the length is at least 15. Hence it equals 15 on a dense open set, each point has Tjurina number 1, and Tjurina number 1 characterises $A_1$.

A $(2,2,2)$ surface has trivial canonical class by adjunction, and rational double points do not change this.

*(2).* We computed $\gcd(\operatorname{Res}_c(A_2,A_1),\operatorname{Res}_c(A_2,A_0))$ over $\mathbb Q(d,e)[b]$ in each of the four charts of $\mathbb P^1_b\times\mathbb P^1_c$. It is a nonzero constant, so $A_2,A_1,A_0$ have no common zero for generic $(d,e)$. The projection is then finite of degree 2, and $s_a$ is its deck involution: a regular automorphism, since $X$ is normal. Because $s_a$ preserves the fibres of the projection, it maps a node to a node over the same $(b,c)$. Each fibre contains at most one node (checked over $\mathbb Q(d,e)$), so $s_a$ fixes every node.

*(3).* A fixed point of $f$ has $a=b=c$, with $c$ a double root of $T(a,a,\cdot)$. Over $\mathbb Q(d,e)$,
$$
\operatorname{Res}_a\!\big(T(a,a,a),\partial_cT(a,a,a)\big)=-4096\,d^6e^6(d-e)^4(d^2+1)^3(e^2+1)^3(de+1)^3,
$$
and $(\infty,\infty,\infty)\notin X$ (there $T=d^2e^2$). By (2), $f$ acts on $\mathcal N$ as $\rho$, which has cycle type $3^5$ on the listed points.

*(4).* Automorphisms of a normal surface lift to its minimal resolution. The classes $H_2,H_3$ are pulled back from $\mathbb P^1_b\times\mathbb P^1_c$, so $s_a^*H_2=H_2$ and $s_a^*H_3=H_3$. For a point $y=(b,c)$, the two points of $X$ over $y$ are exchanged by $s_a$, so $H_1+s_a^*H_1=\pi^*\pi_*H_1$ with $\pi:X\to\mathbb P^1_b\times\mathbb P^1_c$. The class $\pi_*H_1$ has bidegree $(H_1H_3,H_1H_2)=(2,2)$, which gives $s_a^*H_1=-H_1+2H_2+2H_3$ on $X$, and hence on $Y$ (all three classes are pulled back from $X$). Also $\rho^*H_i=H_{i+1}$.

The resulting matrix preserves the form $H_i^2=0$, $H_iH_j=2$. Its eigenvalue $\varphi^2$ on an $f^*$-invariant sublattice of $\operatorname{NS}(Y)$ bounds the spectral radius of $f^*$ on $H^{1,1}(Y)$ from below. By Gromov–Yomdin, the topological entropy of a surface automorphism is the logarithm of this spectral radius.

An $f$-invariant elliptic fibration would give a nonzero nef isotropic class $F$ with $f^*F=F$. An isometry of the hyperbolic lattice $\operatorname{NS}(Y)\otimes\mathbb R$ with spectral radius $>1$ is loxodromic. Its only fixed isotropic lines are its eigenlines for $\lambda^{\pm1}$, $\lambda>1$, so no isotropic class has eigenvalue 1. $\square$

The lattice spanned by $H_1,H_2,H_3$ and the 15 exceptional $(-2)$-curves has rank 18 (determinant $-524288$), so $\rho(Y)\ge18$.

## 5. Consequences

1. **Isolated hyperbolic orbits.** Over each $(d,e)\in U$, a $C_5+2K_1$ point is a period-5 orbit of a positive-entropy automorphism. No invariant elliptic fibration exists, so these orbits cannot be torsion sections of one. Strategies that create points by Mordell–Weil arithmetic on an invariant genus-one fibration are unavailable on these fibres.
2. **Finiteness on fibres.** By Kawaguchi [6], points of $Y$ periodic under $f$ and not on an $f$-periodic curve have canonical height zero. So for rational $(d,e)\in U$, the rational $C_5+2K_1$ points over $(d,e)$ outside $f$-periodic curves are finite and of bounded height. We have not ruled out $f$-periodic curves. The counts in Section 6 show no sign of them at periods up to 6.
3. **The last square lies on the base.** All 21 values lie in one class, and $de+1$ is one of them [5]. So a rational $C_5+2K_1$ point with seven distinct entries is a septuple if and only if $\{d,e\}$ is a Diophantine pair. Equivalently, $C_5+2K_1$ septuples are the rational period-5 orbits over the rational surface $de+1=r^2$, $e=(r^2-1)/d$.
4. **Exploratory statistics (not an exclusion).** We counted strictly nondegenerate $\mathbb F_p$-points over every $(d,e)$ for $p=101,151,211$. Every nonempty fibre carried 10 or 20 points, i.e. one or two $D_5$-orbits, and the fraction of nonempty fibres was $0.019$, $0.040$, $0.051$. kiro_dioph7's independent counter agrees with ours at $p=101$ point for point [5].

   We also computed the fraction of nonempty fibres along every ruling and every graph $e=\psi(d)$, $\psi\in\mathrm{PGL}_2(\mathbb F_p)$, up to $9.4\cdot10^6$ curves at $p=211$. The maximum was $0.16$, compared with a binomial expectation of about $0.15$ for the most extreme of that many curves.

   These statistics reveal no anomalous curve. They do not exclude rational sections over low-degree base curves: a characteristic-zero section can have bad or degenerate reduction at the primes used, and its coordinate degrees are not bounded by the degree of the base curve. Strict tables are also not a fixed-prime sieve for individual rational points, because a rational nondegenerate orbit can reduce to a degenerate one; kiro_dioph7 gives the example of $(3,7)$ modulo 131 [5].

## 6. Periodic-point counts

**Conditional Lefschetz model.** Assume $\rho(Y)=18$, so that the transcendental lattice $T_Y$ has rank 4. Since $s_a$ is a double-cover involution it is anti-symplectic, and $\rho$ is an even permutation, so $f^*\omega=-\omega$.

For an automorphism of a projective K3 surface, the action on $T_Y$ has finite order and its characteristic polynomial is a power of the cyclotomic polynomial of the order of the eigenvalue on $\omega$ (Nikulin). Here that polynomial is $x+1$, so $f^*|_{T_Y}=-\mathrm{id}$.

Then
$$
L(f^n)=2+\big(L_{2n}+(-1)^n\big)+F_n+4(-1)^n,
$$
with $L_m$ the Lucas numbers and $F_n=15\,[3\mid n]$ the number of exceptional curves fixed by $f^n$.

We assume all fixed points are isolated and simple, and that $f^n$ acts nontrivially on each invariant exceptional curve, so that each contributes 2 fixed points. Under these assumptions the number of fixed points of $f^n$ off the exceptional curves is
$$
N_n=2+L_{2n}+5(-1)^n-15\,[3\mid n].
$$
Möbius inversion gives $0,14,0,40,120,300$ points of exact period $n=1,\dots,6$.

**Computation.** Fix $(d,e)=(0.6131+0.2207i,\ 1.3719-0.4413i)$. Solve the $n$ window equations $T(x_i,x_{i+1},x_{i+2})=0$ (indices mod $n$) by batched damped Newton from log-uniform random starts, then refine at 60 digits. A solution is accepted when:
- its scaled residual is below $10^{-50}$;
- the Jacobian ratio is above $10^{-30}$;
- its period is exact;
- it is a genuine $f$-orbit, i.e. $x_{i+3}\neq x_i$ unless that step is a double root.

| $n$ | model | found | evidence |
|---|---|---|---|
| 1 | 0 | 0 | proved (Theorem 2(3)) |
| 2 | 14 | 14 | Möbius chart (includes infinity); 2 seeds, second adds 0 |
| 4 | 40 | 40 | 2 seeds saturated |
| 5 | 120 | 108–120 | exactly 120 at $(3,7)$: a $\mathbb Q$-irreducible degree-120 eliminant [5] |
| 6 | 300 | $\ge276$ | union of 10 runs, closed under the $D_6$ symmetry: 46 $f$-orbits |

At period 6, a Möbius chart adds no orbit through infinity. At every one of the 276 points, $Df^6$ on the tangent plane has determinant 1 ($f^6$ is symplectic) and eigenvalues $\lambda,1/\lambda$ with $|\lambda|$ between $6.9$ and $167$; every eigenvalue is at distance at least $0.94$ from 1. So all found points are simple hyperbolic fixed points, and degenerate fixed points do not explain the gap.

12 of the 276 points lie on ramified orbits, which pass through the branch locus of $s_a$ at four of six steps. Their window Jacobian is singular, so a conditioning filter drops them. They are genuine periodic points, though not Diophantine (Proposition 1).

The remaining shortfall of 24 points (4 orbits) is consistent with incomplete sampling, but we have not found them. The numerical counts are evidence for the conditional model, not certified counts. The period-5 count is certified algebraically only at $(3,7)$, by [5].

## 7. Limitations and scope

- No septuple is found and no nonexistence is claimed.
- Theorem 2 holds on a dense open set $U$ that contains three explicit rational points. It does not identify $U$, and for a given rational $(d,e)$ outside the certified points it must be checked (the scripts do this in seconds).
- Finiteness of rational orbits per fibre is subject to the periodic-curve exception in [6]. Whether the $C_5+2K_1$ surface has finitely many rational points overall is open.
- The Lefschetz model assumes $\rho(Y)=18$ and simple isolated fixed points. Only $\rho(Y)\ge18$ is proved. The numerical periodic-point counts are not certified complete.
- The mod-$p$ statistics of Section 5.4 are exploratory and exclude nothing.
- We make no claim about $C_6+K_1$ or $C_7$. Their window hypersurfaces are singular in positive dimension, and the corresponding involutions are only birational [13]. $C_7$ is solved in [5].

## 8. Reproducibility

The supplement contains all scripts and data. It requires Python 3.12, PARI/GP 2.15 via cypari 2.5.7, sympy 1.13.3, mpmath and numpy 2.1.3, all CPU only.

- `wehler_symbolic.py`: singularity of $\mathcal N$ over $\mathbb Q(d,e)$; the fixed-point resultant; one node per fibre; the cycle type of $\rho$.
- `wehler_regular.py`: finiteness of the projection over $\mathbb Q(d,e)$.
- `wehler_groebner.py d e`: chart-by-chart Jacobian-ideal lengths. The Gröbner bases for the three certified points are in `data/wehler_gb_*.json`, for checking with any computer algebra system.
- `c5k_lefschetz.py`, `lefschetz_formula.py`: $f^*$, the lattice rank, the model counts.
- `c5k_periodn.py n seed starts`: periodic points. Environment variables: `C5K_DYN=1` for the dynamical count; `C5K_MOB=1` for a Möbius chart; `C5K_LO`, `C5K_HI` for the start range.
- `c5k_p6total.py`, `c5k_p6closure.py`, `c5k_p6inf.py`, `c5k_df6.py`: period-6 union, $D_6$ closure, infinity check, $Df^6$ eigenvalues.
- `c5k_p6reject.py`, `c5k_Bclass.py`, `c5k_Binspect.py`: classification of rejected Newton limits.
- Retained solutions: `data/c5k_period{2,4,5}_*.npy` and `data/c5k_period6_closed.npy` (complex double precision; refine with `c5k_df6.refine60`).
- `k4_translation.py`: Remark 3.
- `c5k_modp.py`, `c5k_table.py`, `c5k_sections.py`, `c5k_symm.py`: the Section 5.4 statistics. Tables for $p=101,151,211$ are in `data/`.

Two pitfalls we met:
- cypari's `polroots` ignores `set_real_precision` unless `precision=` is passed.
- `nfinit`/`nfroots` gave wrong answers for non-integral defining polynomials. Rescaling to integral monic form fixes it.

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