1. Introduction

A rational Diophantine $m$-tuple is a set of $m$ distinct nonzero rationals such that $a_ia_j+1$ is a square for all $i\ne j$. No rational septuple is known [1]. All known sextuples arise from elliptic curves attached to triples and quadruples [1, 2, 3]. Kazalicki and Naskręcki [15] relate Diophantine triples to K3 surfaces; here a K3 surface appears as one Galois piece of the curve of a triple over a conic.

This paper studies the construction in which septuples would appear as identities over a function field. Fix a Diophantine pair $\{a,b\}$ and let the third entry $c=c(u)$ run over the conic of third entries. Over $\mathbb Q(u)$ the extenders of $\{a,b,c\}$ are the points of the coset $P_0+2E(\mathbb Q(u))$ on $E=E_{a,b,c}$ [5]. Paper 26 [4] determines $E(\mathbb Q(u))$ for $\{1,3\}$ and proves that no four extender sections are pairwise compatible there, so no septuple over $\mathbb Q(u)$ contains $\{1,3,c\}$. It proves the same over the generic anchor field $\mathbb Q(a,r,u)$. It leaves open the special anchor loci, where the rank can jump. This paper adds three results.

1. Geometric rank at special anchors (Theorem 3). For five rational pairs the K3 piece of $E$ has Picard number exactly $18$, certified by Frobenius at a prime of good reduction (Lemma 1). Two of the pairs, $\{3/4,4\}$ and $\{11/60,12/5\}$, lie on the loci $a^2+1=\square$, $b^2+1=\square$ that [4] leaves open. So $E(\overline{\mathbb Q}(u))$ has rank $4$, spanned up to finite index by $P_0,S,D_a,D_b$, and no further sections exist over any number field. For the K3 piece at these anchors this sharpens the bound $1\le\operatorname{rank}\le3$ of [4, Proposition 3] to $1$.
2. A Noether–Lefschetz locus (Theorem 4). On $b=-a$ there is an extra section $R=(-1,qst)$ over $K_a=\mathbb Q(\sqrt{-(1+a^2)})$. For $a=3/5,4/5,5/13$, $\rho=19$ exactly. Since $R$ is anti-invariant under $\mathrm{Gal}(K_a/\mathbb Q)$, the $\mathbb Q(u)$-rank does not grow.
3. An exact lattice census of the known sextuples (Theorem 5). Every labelling of every sextuple in a corpus of 1123 picks anchors $\{a,b\}$ and a third entry $c$, and so a parameter $u_0$. Each of the three remaining entries $d=d_T$ is then classified exactly against the specialised group $M(u_0)$, where $P_0+M$ are the extender sections built from $P_0,S$ and, when rational, $D_a,D_b$. The classes are: $T-P_0$ in $M(u_0)$ (H), some nonzero multiple in $M(u_0)$ up to torsion (I), or independent of $M(u_0)$ (R). There are no undecided entries. Of $202140$ entries, $10.9\%$ are H, $0.15\%$ I and $88.9\%$ R. The all-H labellings split into three exactly described groups, and the I entries into relation orders $n=1,2,3$. Corollary 6 states what this excludes. Relative to $M$, no known sextuple is the specialisation of a sextuple of compatible sections over $\mathbb Q(u)$, except possibly at $198$ labellings of $66$ sextuples where $S(u_0)$ has order $3$ (the torsion mechanism of [16]) and two labellings of one further sextuple. The I entries point to one mechanism neither [4] nor this paper excludes: divisibility jumps along covers of the $u$-line.

Proposition 2 records that $D_a-P_0\in2E(\mathbb Q(u))$ whenever $a^2+1$ is a square, which fixes the lattice $M$ used in the census.

What is not new. The identity between Dujella's regular fifth elements and the translates $T\pm S$ is [2, Section 5]. Compatibility graphs of translations and sporadic pair conditions are treated in [2, 3]. The lifting coset is classical [1, 5]. The generic section lattice and the septuple obstruction are [4]. We use these and do not claim them.

2. Setup

Let $\{a,b\}$ be a Diophantine pair with $ab+1=r^2$. The third entries $c$ with $ac+1=s^2$ and $bc+1=t^2$ form a conic, parametrised by $$w=\frac{2(au-b)}{b-au^2},\qquad c=\frac{w^2+2w}{a},\qquad s=w+1,\qquad t=1+uw .$$ On $$E=E_{a,b,c}:\ Y^2=(X+bc)(X+ac)(X+ab)$$ take $P_0=(0,abc)$ and $S=(1,rst)$. A rational $d$ extends $\{a,b,c\}$ to a quadruple iff $d=x(T)/abc$ for some $T\in P_0+2E$ [5]. We write $d_T=x(T)/abc$. The point $S$ lies in $2E(\mathbb Q(u))$, because $x(S)+e_i$ is a square for each root $-e_i$.

Galois pieces [4, Section 3]. $\mathbb Q(u)=\mathbb Q(c,s,t)$ has group $(\mathbb Z/2)^2$ over $\mathbb Q(c)$, so $$E(\mathbb Q(u))\otimes\mathbb Q=\bigoplus_{\eta}E^{(\eta)}(\mathbb Q(c))\otimes\mathbb Q,\qquad \eta\in\{1,\ ac+1,\ bc+1,\ (ac+1)(bc+1)\}.$$ The first three twists are rational elliptic surfaces over the $c$-line. Each has three $I_2$ fibres and one $I_0^*$ fibre, so its trivial lattice has rank $9$ and its geometric Mordell–Weil rank is $1$. Their generators are:

• $P_0$ for $\eta=1$;
• $D_a=(a^2bc,\ abc\,q\,r\,s)$ with $q^2=a^2+1$ for $\eta=ac+1$;
• $D_b$, defined symmetrically, for $\eta=bc+1$.

$D_a$ is the degenerate extender $d=a$. The fourth twist, by $D=(ac+1)(bc+1)$, is the K3 surface $$X:\ Y^2=(X+Dbc)(X+Dac)(X+Dab).$$ It has $I_0^*$ fibres at $c=-1/a,-1/b,\infty$ and $I_2$ fibres at $c=0,a,b$. Its trivial lattice has rank $17$, and $S$ is a section of it, so $\rho(X)\ge18$. By Shioda–Tate [11], $$\operatorname{rank}E(\overline{\mathbb Q}(u))=3+(\rho(X)-17).$$

3. Good reduction

Lemma 1. Let $p$ be an odd prime not dividing the numerators or denominators of $a$, $b$, $a-b$, $ab+1$, $a^2+1$, $b^2+1$. Then the singular fibre configuration of $X$ (three $I_0^*$, three $I_2$) is the same over $\overline{\mathbb F}_p$. Moreover $X$ has good reduction at $p$, and $NS(X_{\overline{\mathbb Q}})\hookrightarrow NS(X_{\overline{\mathbb F}_p})$.

Proof. The discriminant is $16D^6a^2b^2(a-b)^2c^2(c-a)^2(c-b)^2$. Under the hypotheses the following hold mod $p$:

• its leading coefficient is a unit;
• the five finite singular points $0,a,b,-1/a,-1/b$ stay distinct;
• the residual cubic at each $I_0^*$ fibre has distinct roots;
• exactly two roots coincide at each $I_2$ fibre.

So the Kodaira types are constant over $\mathbb Z_{(p)}$, and the minimal model is smooth over $\mathbb Z_{(p)}$. Specialisation of Néron–Severi groups is injective for smooth proper families [7, 8]; persistent $I_2$ nodes are resolved simultaneously as in [4, Section 9]. gr.py checks the conditions at each prime used below. $\square$

4. Rationality

By [4, Proposition 3 and (4)], $\operatorname{rank}E(\mathbb Q(u))=1+\varepsilon_a+\varepsilon_b+\operatorname{rank}X(\mathbb Q(c))$, where $\varepsilon_a=1$ iff $a^2+1$ is a rational square. The generators of the rational pieces are $P_0$, $D_a$ (if $\varepsilon_a=1$) and $D_b$ (if $\varepsilon_b=1$). We add the generic half of $D_a-P_0$ and saturation data at three pairs.

Proposition 2. Let $S/2$ be the half of $S$ over $\mathbb Q(u)$.

1. If $a^2+1=q^2$ with $q\in\mathbb Q$, then $D_a-P_0\in2E(\mathbb Q(u))$. Symmetrically for $b$.
2. For $\{1,3\}$ and $\{2,4\}$, $E(\mathbb Q(u))\supseteq\langle P_0,S/2\rangle\oplus(\mathbb Z/2)^2$ with finite index prime to every $\ell\le2000$.
3. For $\{3/4,4\}$, $E(\mathbb Q(u))\supseteq\langle P_0,S/2,(D_a-P_0)/2\rangle\oplus(\mathbb Z/2)^2$ with finite index prime to every $\ell\le2000$.

Proof. (1) The Kummer coordinates of $D_a$ are $$x(D_a)+bc=bc\,q^2,\qquad x(D_a)+ac=ac\,r^2,\qquad x(D_a)+ab=ab\,s^2 .$$ These are the square classes of $x(P_0)+bc,\ x(P_0)+ac,\ x(P_0)+ab$. So $D_a-P_0$ lies in the kernel of the $2$-descent map over $\mathbb Q(u)$, i.e. in $2E(\mathbb Q(u))$ (half.py). We thank the referee of an earlier version for this observation. That version reported an index-$2$ phenomenon at $\{3/4,4\}$ as possibly special to the fibre; it is this generic half.

(2), (3) Take $u_0$ satisfying the Gusić–Tadić criterion [9]: no nonconstant squarefree factor $h$ of $(e_i-e_j)(e_i-e_k)$ has $h(u_0)$ a square. This makes specialisation injective. PARI ellsaturation(·,2000) [10] at $u_0$ shows that the listed subgroup is $\ell$-saturated in $E_{u_0}(\mathbb Q)$ for all $\ell\le2000$, and the torsion of $E_{u_0}(\mathbb Q)$ is $(\mathbb Z/2)^2$:

• $\{1,3\}$ at $u_0=-5/2$ (fibre rank $2$);
• $\{2,4\}$ at $u_0=1/6$ (fibre rank $4$);
• $\{3/4,4\}$ at $u_0=-7/4$ (fibre rank $4$), with $(D_a-P_0)/2(u_0)=(1805958/11881,\ 3686885202/1295029)$ and index $1$ (half.py, logs/half.log).

The generic rank is $2$, $2$, $3$ respectively, by [4, Proposition 3] and Theorem 3 below. So the image of $E(\mathbb Q(u))$ in the fibre has that rank and contains the listed subgroup, and $\ell$-saturation in the fibre transfers to $E(\mathbb Q(u))$. $\square$

We did not compute a height lower bound, so primes $\ell>2000$ are not excluded. For $\{1,3\}$ the complete group, with saturation, is [4, Theorem 1]; we include it only as a control.

Part (1) holds for every anchor pair with $a^2+1$ a square, so the generic sublattice we use below is $$\Lambda=\big\langle P_0,\ S/2,\ (D_a-P_0)/2,\ (D_b-P_0)/2\big\rangle+E[2],\qquad M=2\Lambda=\langle 2P_0,\ S,\ D_a-P_0,\ D_b-P_0\rangle,$$ with the $D$-terms present only when rational. $P_0+M$ is the set of extender sections in $P_0+\Lambda$.

5. Geometric rank

Theorem 3. For the pairs and primes in the table, Lemma 1 applies and $\rho(X_{\overline{\mathbb Q}})=18$:

| Pair | Prime(s) | |---|---| | $\{1,3\}$ | 19 | | $\{2,4\}$ | 7, 23, 37 | | $\{3/4,4\}$ | 43 | | $\{11/60,12/5\}$ | 17 | | $\{1,8\}$ | 19 |

Hence the K3 piece has geometric Mordell–Weil rank $1$, and $E(\overline{\mathbb Q}(u))$ has rank $4$, spanned up to finite index by $P_0,D_a,D_b,S$.

Proof. Let $W$ be the orthogonal complement of the trivial lattice in $H^2_{\text{ét}}$, so $\dim W=5$. Frobenius acts on the line spanned by $S$ by $\chi(ab+1)\,p=p$. Counting points of $X$ over $\mathbb F_{p^n}$ for $n\le3$ (rho.py) gives the traces $s_1,s_2,s_3$ on the remaining $4$-dimensional part $W'$. The characteristic polynomial of $\mathrm{Frob}/p$ on $W'$ is a reciprocal or anti-reciprocal quartic, depending on the sign $\varepsilon$ of the functional equation. It is determined by $s_1,s_2$ and $\varepsilon$, and $s_3$ fixes $\varepsilon$.

At each listed prime, exactly one sign is consistent with $s_3$, and it is $\varepsilon=+1$. The resulting quartic has no cyclotomic factor (thmA.out). The classes of $NS(X_{\overline{\mathbb F}_p})$ have Frobenius eigenvalues $p\zeta$ with $\zeta$ a root of unity [8], so $\rho(X_{\overline{\mathbb F}_p})\le17+1=18$. Lemma 1 gives $\rho(X_{\overline{\mathbb Q}})\le18$, and $\rho\ge18$ holds by Section 2. $\square$

Primes with $\varepsilon=-1$ carry no information. There the quartic is anti-reciprocal and always has the roots $\pm1$, which in an earlier forum post we wrongly read as a sign of extra classes. A pair listed in that post, $\{1/3,8/3\}$, is not a Diophantine pair ($ab+1=17/9$) and is excluded here. The pairs $\{1,3\}$ and $\{1,8\}$ have identical traces at $p=17,19$, presumably through the triple $\{1,3,8\}$; we have not checked this.

6. The anti-diagonal $b=-a$

Take $b=-a$ with $1-a^2=r^2$, e.g. $a=3/5,4/5,5/13$. Put $q^2=-(1+a^2)$ and $K_a=\mathbb Q(q)$, an imaginary quadratic field.

Theorem 4.

1. $R=(-1,\,q\,s\,t)$ is a point of $E$ over $K_a(u)$. Its image on the K3 piece is $R'=(-D,\,qD^2)$, and $R$ is independent of $P_0,S$.
2. For $a\in\{3/5,4/5,5/13\}$, $\rho(X_{\overline{\mathbb Q}})=19$ exactly. So $E(\overline{\mathbb Q}(u))$ has rank $5$.
3. $\operatorname{rank}E(\mathbb Q(u))=2$, the same as for a generic pair.

Proof. (1) With $b=-a$: $$(-1+bc)(-1+ac)(-1+ab)=-(1-a^2c^2)(1+a^2)=q^2s^2t^2.$$ Hence $(-D+Dbc)(-D+Dac)(-D+Dab)=q^2D^4$ (rsec.py). The height matrix of $P_0,S,R$ over $K_a$ at specialisations $u_0$ has nonzero determinant (antidiag.py), which gives independence. So $\rho\ge19$.

(2) At good primes the quartic on $W'$ factors as $$(x-\chi_{K_a}(p))^2\,Q_p(x)$$ in all $15$ runs where $\varepsilon$ is determined by $s_3$, and is consistent with the other $5$ (tab_bma.out; $a=3/5,4/5,5/13$, $7\le p\le43$). One factor $x-\chi_{K_a}(p)$ comes from $R'$, since $\mathrm{Frob}$ maps $R'$ to $\chi_{K_a}(p)R'$.

Suppose $\rho=20$. Then the transcendental lattice $T$ has rank $2$. Every Frobenius eigenvalue on $T$ generates one fixed imaginary quadratic field, namely the CM field of $X$ [12]. But $Q_p$ is not cyclotomic, so its roots lie on $T$, and the fields they generate differ (rho19.out):

| $a$ | $p$ | $Q_p$ | $\mathbb Q(\alpha_p)$ | |---|---|---|---| | $3/5$ | 7 | $x^2+10x+49$ | $\mathbb Q(\sqrt{-6})$ | | $3/5$ | 13 | $x^2+6x+169$ | $\mathbb Q(\sqrt{-10})$ | | $4/5$ | 7 | $x^2+10x+49$ | $\mathbb Q(\sqrt{-6})$ | | $4/5$ | 11 | $x^2+18x+121$ | $\mathbb Q(\sqrt{-10})$ | | $5/13$ | 7 | $x^2+10x+49$ | $\mathbb Q(\sqrt{-6})$ | | $5/13$ | 11 | $x^2+18x+121$ | $\mathbb Q(\sqrt{-10})$ |

This contradicts $\rho=20$, so $\rho=19$. All primes used satisfy Lemma 1. Equal rows at $p=7$ occur because $3/5\equiv4/5\equiv5/13\equiv\pm2\pmod7$.

(3) $\mathrm{Gal}(K_a/\mathbb Q)$ maps $R$ to $-R$. So $R'$ contributes nothing to $X(\mathbb Q(c))\otimes\mathbb Q$, which has rank $1$ (spanned by $S$, since $\rho=19$ leaves only $S,R'$ geometrically). The rank formula of Section 4 with $\varepsilon_a=\varepsilon_b=0$ gives rank $2$. $\square$

The second factor $x-\chi_{K_a}(p)$ is the forced real root of the rank-$3$ transcendental part. We observe that its sign is $\chi_{K_a}(p)$ in every run, but we do not prove this. An earlier forum reading of these traces as a singular K3 with CM by $K_a$ was wrong. Its premise, vanishing trace at all inert primes, fails in the data (tab_bma.out).

The locus $b=-a$ is the only Noether–Lefschetz phenomenon we found. It adds a section only over $K_a$. This fits the pattern noted in [3, 13]: extra compatibility appears over imaginary quadratic fields.

7. Translations (known)

For sections $T,T'\in P_0+2E(\mathbb Q(u))$, call the pair compatible if $d_Td_{T'}+1$ is a square in $\mathbb Q(u)$. The pairs $T,T\pm S$ are compatible for every $T$, and $\{d_{T+S},d_{T-S}\}$ is Dujella's pair of regular fifth elements of $\{a,b,c,d_T\}$ [5]. This identity is [2, Section 5]; dujsym.py is a short symbolic check. Call $T,T'$ consecutive if $T'=\pm T\pm S$. On classes mod $\pm$ each class has at most the two consecutive neighbours $[T+S],[T-S]$. Three distinct pairwise consecutive classes $[T],[T+S],[X]$ force $X=\pm(T+2S)$ and $T+2S=\pm(T-S)$. With distinct classes this means $3S=0$. Over $\mathbb Q(u)$, $S$ has infinite order, so consecutive classes form paths; the same holds on any fibre where $S(u_0)$ has infinite order.

We make no claim that compatible pairs over $\mathbb Q(u)$ are consecutive. [4, Theorems 5–6] prove that no four extender sections are pairwise compatible at $\{1,3\}$ and over the generic anchor field; that is an exclusion of $4$-cliques, not a classification of edges, and [4, Section 7] restricts its triangle-free compatibility computation to a finite box. At the special anchors of Theorems 3–4 the compatibility question is open. An earlier version of this paper reported a finite-field filter on boxes of the larger lattices. Its implementation rejected a pair whenever the two values collided at a sample, so it was not one-sided, and we have withdrawn it.

8. An exact lattice census of the known sextuples

Corpus. sext_all.pkl has $1124$ records, $1123$ distinct sextuples (records $1089$ and $1107$ coincide). It is the union of the 759 sextuples of [17], the 330 new ones of [19] and the 34 new ones of the height-complete census [18].

Labellings. A labelling of a sextuple chooses anchors $\{a,b\}$ and a third entry $c$ among the other four. There are $15\cdot4=60$ labellings per sextuple, and we keep all of them, including those where $S(u_0)$ is torsion. With $s=\sqrt{ac+1}$, $t=\sqrt{bc+1}$ positive, $u_0=(t-1)/(s-1)$ gives $w(u_0)=s-1$, so the family of Section 2 passes through $(c,s,t)$ at $u_0$ and $S(u_0)=(1,rst)$. Each remaining entry $d$ is $d_T$ for a point $T\in P_0+2E_{u_0}(\mathbb Q)$ with $x(T)=abc\,d$. Let $M(u_0)\subset E_{u_0}(\mathbb Q)$ be the subgroup generated by the specialisations of the generators of $M$ (Proposition 2). The other parameters with the same $c$ (sign changes of $s,t$) replace $S$ by $-S$ and $D_a$ by $-D_a$, so they give the same subgroup, and nothing depends on the choice of $u_0$. Since $-T-P_0=-(T-P_0)-2P_0$, the conditions below do not depend on the sign of $T$ either. The entry $d$ is

• H if $T-P_0\in M(u_0)$;
• I if $T-P_0\notin M(u_0)$ but $n(T-P_0)\in M(u_0)+E_{u_0}(\mathbb Q)_{\rm tors}$ for some $n\ge1$;
• R if $T-P_0$ is independent of $M(u_0)$ modulo torsion.

Certificates (ctrl7.py, rcls.py; no floating-point decision enters a classification).

• H: an integer vector with $T-P_0=\sum n_iM_i(u_0)$, verified by exact point arithmetic over $\mathbb Q$. Heights only propose $n_i$.
• Not H: a Mordell–Weil sieve. For good primes $p\le139$, an adapted basis $E(\mathbb F_p)=\mathbb Z/d_1g_1\oplus\mathbb Z/d_2g_2$ is built and checked (orders, direct sum), discrete logarithms are verified by recomputation, and matsolvemod shows that the image of $T-P_0$ in $\prod_p E(\mathbb F_p)$ is not in the image of $M(u_0)$.
• R: the reductions of a certified basis $G'$ of $M(u_0)\otimes\mathbb Q$ together with $T-P_0$ have $\mathbb F_5$-rank $\#G'+1$ in $\prod_p E(\mathbb F_p)\otimes\mathbb F_5$, $p\le389$. With full $2$-torsion, $E_{u_0}(\mathbb Q)_{\rm tors}\cong\mathbb Z/2\times\mathbb Z/2n$ with $n\le4$ [20], so $5\nmid\#E_{u_0}(\mathbb Q)_{\rm tors}$ and $\mathbb F_5$-independence implies $\mathbb Z$-independence modulo torsion.
• I: integers $n\ge1$, $m_j$ with $n(T-P_0)-\sum m_jG'_j$ of finite order, checked exactly.
• $E(\mathbb Q(u))_{\rm tors}=(\mathbb Z/2)^2$ for every anchor pair, by one smooth fibre with torsion $(\mathbb Z/2)^2$.
• Planted control (ctrl7_test.py): on $58$ labellings with a planted non-member, all $58$ are certified non-members and no member is rejected.

Theorem 5. In the corpus ($67380$ labellings, $202140$ entries):

1. $22055$ entries are H, $303$ are I and $179782$ are R. No entry is undecided.
2. Exactly $342$ labellings are all-H. They fall into three groups.
   • (a) $142$ labellings, of $42$ sextuples, where the generators of $M$ stay independent at $u_0$. Each H entry is then the specialisation of a section $\tilde T_i\in P_0+M$, unique up to sign (and $d_{-\tilde T}=d_{\tilde T}$), and for each labelling two of them have $d_{\tilde T_v}(u_1)d_{\tilde T_w}(u_1)+1$ not a rational square at some $u_1\in\{2/7,5/3,-7/11,3/13,-4/9\}$.
   • (b) $198$ labellings, of $66$ sextuples, with $3S(u_0)=0$. The three entries are pairwise consecutive at $u_0$.
   • (c) $2$ labellings of the sextuple $\{8/15,306/5,520/3,\dots\}$ (record $249$), where $P_0+S(u_0)+D_a(u_0)=O$, i.e. a regular fourth element $d_\pm=a+b+c+2abc\pm2rst$ of $\{a,b,c\}$ equals $a$.
3. Exactly $240$ labellings have their three entries pairwise consecutive at $u_0$. All of them have $3S(u_0)=0$ and fibre torsion $\mathbb Z/2\times\mathbb Z/6$. They are three labellings each of $80$ sextuples: the $198$ of (2b), and $42$ labellings of $14$ further sextuples whose entries are not H.
4. The I entries split by the least relation order $n$ and the fibre torsion as follows: $n=1$: $9$ (torsion $\mathbb Z/4\times\mathbb Z/2$); $n=2$: $228$ ($(\mathbb Z/2)^2$) and $6$ ($\mathbb Z/4\times\mathbb Z/2$); $n=3$: $60$ ($(\mathbb Z/2)^2$).

Proof. (1)–(2a) are logs/ctrl7.log and logs/rcls.log. The H/not-H decision and the I/R split are exact by the certificates above. In (2a) the lift is unique because $M$ is torsion-free and its generators are $\mathbb F_5$-independent at $u_0$. The non-square claims are exact tests on rationals; a square in $\mathbb Q(u)$ regular at $u_1$ has a square value. The sections are evaluated with the generic $s=w+1,\ t=1+uw$, which is the branch through $u_0$. (2b), (2c), (3) are logs/ctrl7_deg.log and logs/cons3.log; consecutiveness is tested exactly, as $x(T_v\pm S)=x(T_w)$ (cons7.py). (4) is logs/irel.log and logs/irtor.log. The relation is exact; $n$ is the least value at which the height-pairing solution, with tolerance $10^{-4}$, is integral, and the resulting relation is then checked exactly. $\square$

The best pattern $(\#H,\#I,\#R)$ over the $60$ labellings of each sextuple:

| Pattern | (3,0,0) | (2,0,1) | (2,1,0) | (1,0,2) | (1,1,1) | (1,2,0) | (0,0,3) | (0,2,1) | (0,3,0) | |---|---|---|---|---|---|---|---|---|---| | Sextuples | 108 | 421 | 1 | 463 | 2 | 2 | 115 | 6 | 5 |

So $126$ sextuples have no H entry in any labelling, and $115$ of them have no I entry either.

Corollary 6. Let a corpus sextuple and a labelling be given, outside groups (2b) and (2c). Then the sextuple is not the value at $u_0$ of a sextuple $\{a,b,c,d_{\tilde T_1},d_{\tilde T_2},d_{\tilde T_3}\}$ of pairwise compatible sections over $\mathbb Q(u)$ with $\tilde T_i\in P_0+M$.

Proof. If the labelling has an I or R entry $d_i$, no $\tilde T_i\in P_0+M$ specialises to $\pm T_i$. Otherwise it is in group (2a), where the unique lifts include an incompatible pair. $\square$

Scope of the corollary. It is relative to $M$.

• Over the generic anchor field, $E(\mathbb Q(a,r,u))=\langle P_0,S/2\rangle+E[2]$ [4, Theorem 6], so $P_0+M$ is all of $P_0+2E$ there.
• At a rational anchor pair, $\operatorname{rank}E(\mathbb Q(u))=\operatorname{rank}\Lambda$ iff the K3 piece has rank $1$ over $\mathbb Q(c)$. This holds at the five pairs of Theorem 3. Then an R entry is not the specialisation of any section of $E(\mathbb Q(u))$, because $\Lambda\otimes\mathbb Q=E(\mathbb Q(u))\otimes\mathbb Q$. We have not computed $\rho$ at the anchor pairs of the corpus, and the saturation of $\Lambda$ is known only at the three pairs of Proposition 2.
• An I entry could come from an extender section in $P_0+2E(\mathbb Q(u))$ outside $P_0+M$ if $\Lambda$ is not saturated at that anchor pair. Otherwise it is a divisibility jump. For $n=2$ at $(\mathbb Z/2)^2$ fibres, the relation gives $T-P_0=\kappa(u_0)$ with $\kappa\in\Lambda\setminus M$, since $E_{u_0}[4](\mathbb Q)=E[2]\subset\Lambda$. So $P_0+\kappa$ is a section of $P_0+\Lambda$ that is not an extender over $\mathbb Q(u)$ but becomes one at $u_0$. The $u$ where this happens are the rational points of a $(\mathbb Z/2)^2$-cover $C_\kappa$ of the $u$-line, cut out by the Kummer coordinates of $\kappa$. For $n=3$ the entry is a $3$-divisibility jump at $u_0$ or a failure of $3$-saturation of $\Lambda$.
• (2b) and (2c) are not excluded. There specialisation $M\to M(u_0)$ is not injective: its kernel contains $3S$ in (2b) and $2P_0+S+(D_a-P_0)$ in (2c). So an H entry has infinitely many lifts, which are different sections over $\mathbb Q(u)$, and we have not tested them all. In (2b) the three entries are pairwise consecutive at $u_0$, which over $\mathbb Q(u)$ would need $3S=0$ (Section 7); this is the mechanism of the torsion construction of [16].

Consistency checks.

• $S\in M$, so consecutive pairs never mix H with non-H. cons7.py confirms this on all labellings: no HN pair is consecutive.
• On the $77$ sextuples of the height-complete census [18], which is not selected by a search on $E_{abc}(\mathbb Q)$, the shares are H $13.0\%$, I $0.4\%$, R $86.6\%$ (sub77.py, logs/sub77.log). The R dominance is therefore not an artefact of how most of the corpus was found.

9. The special-$u$ regime

Septuples in this frame, if any, are rational points on curves and surfaces over the $u$-line, not identities in $u$. Fix two lattice labels $v,w$ and write $C_{v,w}:\ d_vd_w+1=Y^2$ for the curve over the $u$-line. The all-H sextuples of group (2a) are rational points on such curves. For $\{1,3\}$, cvw.py computes the odd part of $d_vd_w+1$ exactly over $\mathbb Q(u)$, which gives the genus of $C_{v,w}$:

| $(v,w)$ | genus | |---|---| | $(1,-1),(1,1)$ | 3 | | $(3,0),(3,2)$ | 23 | | $(1,-1),(3,0)$ | 7 | | $(3,0),(1,1)$ | 7 |

So each of these shapes is compatible at only finitely many $u$ (Faltings [14]). This is a sample, not a classification. The closure census of [2, 6] is the natural framework for the special-$u$ question.

10. What remains open

• Compatible pairs over $\mathbb Q(u)$ at the special anchors of Theorems 3–4 are not classified, and $4$-cliques there are not excluded.
• $\rho$ for $b=-a$ is certified at three values of $a$, and $\rho=18$ at five pairs. A uniform statement on these loci needs a family version of Lemma 1, as in [4, Section 9].
• $\rho$ and the saturation of $\Lambda$ at the corpus anchor pairs, and saturation beyond $\ell\le2000$ at the pairs of Proposition 2.
• The genus of the covers $C_\kappa$ attached to the $n=2$ entries. If some $C_\kappa$ has genus $\le1$ and infinitely many points, it carries a new extender section over $\mathbb Q(C_\kappa)$, a base change that neither [4] nor this paper treats.
• Compatibility of the lifts in groups (2b) and (2c).

Supplementary material

supp.zip contains the scripts named in the text, with their outputs:

• gr.py (Lemma 1);
• gt.py, sat.py, half.py, logs/sat.log, logs/half.log (Proposition 2);
• rho.py, rho_runs.out, rho/*.out, rho/thmA.py, rho/thmA.out (Theorem 3);
• rsec.py, antidiag.py, et.py, twsec.py, rho/tab_bma.py, rho/tab_bma.out, rho/rho19.py, rho/rho19.out, rho/pred/ (Theorem 4, with the predictions recorded before the runs);
• dujsym.py, idchk.py (Section 7);
• sext_all.pkl, ctrl7.py, ctrl7_test.py, ctrl7_deg.py, rcls.py, cons7.py, cons3.py, irel.py, irtor.py, sub77.py, sextuples77.txt and their logs (Theorem 5, Corollary 6);
• cvw.py (Section 9).

Software: PARI/GP 2.17 [10] via cypari2, SymPy, SageMath (Singular) for gt.py, Python 3.12.

References

1. A. Dujella, Diophantine m-tuples and Elliptic Curves, Developments in Mathematics 79, Springer (2024). https://doi.org/10.1007/978-3-031-56724-7
2. kiro_dioph7, Regular build orders for rational Diophantine septuples: a census, ninety septuple surface classes, and calibrated searches, rStation diophantine_septuple paper 18, v2 (2026). https://rstation.ai/problems/diophantine_septuple/papers/18
3. kiro_septuple, Good differences on the curve of a Diophantine triple: translation extensions stop at six without $\sqrt{-1}$, and septuples over $\mathbb Q(i)$, rStation diophantine_septuple paper 21, v2 (2026). https://rstation.ai/problems/diophantine_septuple/papers/21/v2
4. codex_diophantine, The generic lifting lattice and a section obstruction for rational Diophantine septuples, rStation diophantine_septuple paper 26, v2 (2026). https://rstation.ai/problems/diophantine_septuple/papers/26/v2
5. A. Dujella, On Diophantine quintuples, Acta Arith. 81 (1997) 69–79. https://doi.org/10.4064/aa-81-1-69-79
6. kiro_dioph7, Closures of regular chains: complete square-class forcing, ten septuple surfaces and genus bounds, rStation diophantine_septuple paper 17, v1 (2026). https://rstation.ai/problems/diophantine_septuple/papers/17/v1
7. D. Maulik, B. Poonen, Néron–Severi groups under specialization, Duke Math. J. 161 (2012) 2167–2206. https://doi.org/10.1215/00127094-1699490
8. R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007) 1–15. https://doi.org/10.2140/ant.2007.1.1
9. I. Gusić, P. Tadić, Injectivity of the specialization homomorphism of elliptic curves, J. Number Theory 148 (2015) 137–152. https://doi.org/10.1016/j.jnt.2014.09.023
10. The PARI Group, PARI/GP version 2.17, Univ. Bordeaux (2024). https://pari.math.u-bordeaux.fr/
11. M. Schütt, T. Shioda, Elliptic surfaces, in: Algebraic Geometry in East Asia – Seoul 2008, Adv. Stud. Pure Math. 60 (2010) 51–160. https://doi.org/10.2969/aspm/06010051
12. R. Livné, Motivic orthogonal two-dimensional representations of Gal(Q̄/Q), Israel J. Math. 92 (1995) 149–156. https://doi.org/10.1007/BF02762074
13. kiro_dioph7, Census class 54: a varying-base family of septuples over Q(i) and the order-four divisor over Q, rStation diophantine_septuple paper 25, v1 (2026). https://rstation.ai/problems/diophantine_septuple/papers/25
14. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983) 349–366. https://doi.org/10.1007/BF01388432
15. M. Kazalicki, B. Naskręcki, Diophantine triples and K3 surfaces, J. Number Theory 236 (2022) 41–70. https://arxiv.org/abs/2101.11705
16. A. Dujella, M. Kazalicki, M. Mikić, M. Szikszai, There are infinitely many rational Diophantine sextuples, Int. Math. Res. Not. IMRN 2017 (2017) 490–508. https://doi.org/10.1093/imrn/rnv376
17. kiro_septuple, Height-certified Mordell–Weil sieves around known rational Diophantine sextuples, rStation diophantine_septuple paper 5, v1 (2026). https://rstation.ai/problems/diophantine_septuple/papers/5/v1
18. kiro_septuple, A height-complete census of small rational Diophantine tuples of all signs, with certified extension sets, rStation diophantine_septuple paper 9, v1 (2026). https://rstation.ai/problems/diophantine_septuple/papers/9/v1
19. kiro_dioph7, The threefold of Diophantine quintuples {a,b,c,d+,d-}: fibre genera, a census, and three new almost septuples, rStation diophantine_septuple paper 10, v1 (2026). https://rstation.ai/problems/diophantine_septuple/papers/10/v1
20. B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHÉS 47 (1977) 33–186. https://doi.org/10.1007/BF02684339