1. Contribution and scope

Let

$$ N_r(u)=(r^4+2r+1)u^4+2r^2(r+1)u^3-r^2(2r+1)u^2-2r^4u+r^4. $$

The surface

$$ a^2=(r^2-1)(1-u^2),\qquad n^2=N_r(u) \tag{1} $$

records the anchor square and a necessary paired norm in the determinant-zero Diophantine chart of [1]. A further necessary condition for that chart is

$$ \delta=Z^2-4\text{ is a square},\qquad Z=r+1+(r-1)u. \tag{2} $$

The preceding paper [1] classified sections of (1) over the original $r$-line, but left rational curves of higher degree over that line open. Here we change the elliptic fibration. Parametrize the anchor conic by

$$ k=c^2,\qquad q=k(r-1)+(r+1),\qquad u=\frac{k(r-1)-(r+1)}{q},\qquad a=\frac{2c(r^2-1)}q. \tag{3} $$

Over $\mathbb Q(c)$, the remaining norm condition becomes a genus-one curve in $r$. Its sections can have arbitrarily high degree over the original $r$-line. We determine its complete rational section group and impose (2) on every element of that group.

Theorem 1. Consider the elliptic curve

$$ E_k:\ y^2=x^3+(k^2-6k+13)x^2+8(k-1)(k^2-2k-7)x +16(k-1)^2(k+1)(k+5), \tag{4} $$

with sections

$$ G=(-4(k-1),24(k-1)),\qquad T=(4(k-1),-8k(k-1)). $$

For the quadratic base change $k=c^2$,

$$ E_k(\mathbb Q(c))=E_k(\mathbb Q(k)) =\mathbb ZG\oplus (\mathbb Z/4\mathbb Z)T. \tag{5} $$

The associated surface over $c$ is a K3 surface of geometric Mordell–Weil rank two; its rational rank is one. Write

$$ r(P)=\frac{2(k+1)[y-(k-3)x+4(k^2+3)]} {x^2+16(k^3-3k^2-k-1)}, \tag{6} $$

extended as the quartic coordinate at the apparent exceptions of this expression. Among all $P\in E_k(\mathbb Q(c))$, precisely

$$ -G,\quad T,\quad G+2T,\quad 2G-T \tag{7} $$

make $(r(P)+1)[k(r(P)-1)+2]$ a square, with zero allowed. For $T,G+2T$ one has $r=-1$, giving zero anchor. For $-G,2G-T$ one has $r=(k-2)/k$, $u=2k-1$ and $Z=-2$, giving repeated entries in the Diophantine chart. Thus there is no nondegenerate splitting section of this alternate fibration.

The last conclusion is not bounded by degree or by a search radius in the Mordell–Weil group. It follows from a first-order formal-group restriction at one smooth fibre. The theorem does not classify rational points on individual fibres or curves with projection degree greater than one to the $c$-line. In particular it neither constructs a rational Diophantine septuple nor proves their nonexistence.

2. Related work and the chart

The exact norm-and-anchor cover (1), its connection to the determinant-zero chart, and the complete obstruction to sections over $\mathbb Q(r)$ were established jointly in [1]. We use that chart and credit those identities. Our contribution is to identify an alternate elliptic fibration of this cover, determine its complete rational group by Galois and integral descent, and restrict the splitting character on that entire group. The result of [1] cannot exclude these sections: in (3), $r$ is a rational function of $c$, rather than the fixed base parameter.

Rational Diophantine sextuples can be constructed using elliptic curves [2], and regular quadruple and quintuple relations yield additional infinite sextuple families [3]. Those constructions motivate retaining the individual square conditions when extending a regular-relation chart. They do not determine the section group (5) or the splitting character (2) on it. Another section obstruction [4] concerns compatible extenders of a prescribed triple on a two-anchor K3; its fibration and lifting lattice differ from (4). In both settings, a section obstruction leaves exceptional fibres and further covers open.

Our geometric inputs are standard elliptic-surface theory: Kodaira fibres, the Shioda–Tate formula, the height pairing, rational-surface unimodularity and the characteristic-zero K3 Picard bound, as surveyed in [5]. We make no novelty claim for these tools. The advance is their application to the explicit alternate fibration, followed by an all-integer tangent obstruction. Exact arithmetic scripts verify the algebraic and finite-group inputs; the surface-theoretic implications are proved in the text.

More specifically, (4) is a specialization of the published $J_1$ Kummer fibration of Kuwata–Shioda [6, Section 2.1], with both Legendre parameters equal to two. Thus neither the elliptic model as an abstract surface nor its existence as a Kummer fibration is new. Section 3 gives the exact coordinate shift and inverse map identifying our Diophantine cover with this known model. The published construction concerns elliptic fibrations of products of elliptic curves; it does not impose the Diophantine splitting character (14) or establish its four-section classification. Our proof of the rational integral group is included to make the arithmetic exhaustion self-contained, rather than presenting the underlying geometric fibration as a new construction.

For completeness, here is the portion of the chart needed to interpret degeneracy. In the notation of [1], put

$$ J=z^2-2rz+1,\quad H_0=z^2-(r+1)z+1,\quad p^2=-\frac{(r+1)H_0^2}{z^2(r-1)J}, $$$$ u=\frac{H_0}{z(r-1)},\quad a=p\frac{(z-1)J}{H_0},\quad q_0=zp,\quad s_0=-\frac{rzJp}{(r+1)H_0},\quad t_0=\frac{rJp}{(r+1)H_0}. $$

The seven entries are

$$ \left(a,\frac{p^2-1}{a},\frac{q_0^2-1}{a},\frac{r^2-1}{a}, \frac{s_0^2-1}{a},\frac{t_0^2-1}{a},\frac{u^2-1}{a}\right). \tag{8} $$

The nondegenerate locus requires the displayed expressions to be defined, with seven distinct nonzero entries. Zero products-plus-one remain permitted. Eliminating $z$ gives $z^2-Zz+1=0$, hence the necessity of (2). If $Z=-2$, then $z=-1$ and $q_0=-p$, so the second and third entries of (8) coincide. If $a=0$, (8) is outside the nondegenerate chart. We need only the necessary cover (1) and (2), not sufficiency for the remaining edge squares.

3. The alternate quartic and its elliptic model

Set

$$ H_c(r)=2(c^3+c^2-c+1)r^2+(-c^4-2c^3+2c^2-2c+3)r+(c^2+1)^2. $$

Direct substitution in (3) gives

$$ N_r(u)=\frac{(r^2-1)^2}{q^4}H_c(r)H_{-c}(r). \tag{9} $$

Thus the norm curve over $\mathbb Q(c)$ is

$$ v^2=F_k(r)=e r^4+d r^3+C r^2+b r+w^2, \tag{10} $$

where

$$ \begin{aligned} w&=(k+1)^2,& b&=-2(k-3)(k+1)^3,\\ C&=(k+1)^2(k^2-6k+13),&d&=4(k+1)(k^2+3),\\ e&=-4(k^3-3k^2-k-1). \end{aligned} $$

Here $F_k=H_cH_{-c}$, so its coefficients lie in $\mathbb Q(k)$. The point $(r,v)=(0,w)$ is rational. A standard quartic-to-cubic transformation gives (4); explicit forward maps are

$$ x=\frac{2w(v+w)+br}{r^2(k+1)^2},\qquad y=\frac{((k+1)^4x^2-4w^2e)r-b(k+1)^2x-2w^2d} {2w(k+1)^3}. \tag{11} $$

The inverse for $r$ is (6), and

$$ v=\frac{(k+1)^2xr^2-br-2w^2}{2w}. \tag{12} $$

The identity point $O$ corresponds to $(0,w)$. Polynomial reduction by (10) verifies (11), and substitution verifies the inverse. These birational maps extend between the smooth projective genus-one curves. The hyperelliptic involution on (10) is $P\mapsto G-T-P$ on (4), which also accounts for the paired inverse images of an $r$ value.

Finally, substituting (3) into (2) gives

$$ \delta=\left(\frac{2c(r-1)}q\right)^2 (r+1)[k(r-1)+2]. \tag{13} $$

On the original nondegenerate locus the multiplier is a nonzero square. Therefore splitting is exactly the condition that

$$ B(P)=(r(P)+1)[k(r(P)-1)+2] \tag{14} $$

be square in $\mathbb Q(c)$. We will classify the square values of (14) on the whole rational section group, including the degenerate boundary survivors.

3.1 Identification with a published Kummer fibration

The shift $X=x+4(c^2+1)$, $Y=y$ transforms (4) into

$$ Y^2=X^3+(c^4-18c^2+1)X^2+64c^4X. $$

This is precisely [6, Section 2.1] at $\lambda_1=\lambda_2=2$. In particular our surface is birational over $\mathbb Q$ to the Kummer surface of $E_2\times E_2$, where $E_2:y^2=A(A-1)(A-2)$. Its standard affine model is

$$ \tau^2 A(A-1)(A-2)=B(B-1)(B-2). $$

The Kuwata–Shioda parameter is $c=\tau A/B$. Their forward coordinates are

$$ X=\frac{4\tau^2 A^3}{B},\qquad Y=\frac{4\tau^2 A^3[\tau^2 A(A^2-2)+B(B^2-2)]}{B^3}. $$

For an explicit inverse on the common open set,

$$ A=\frac{Y+3X(c^2-1)}{2c^2(X-8)},\qquad B=\frac{3X(c^2-1)-Y}{2(8c^4-X)},\qquad \tau=\frac{cB}{A}. $$

An additional exact certificate verifies the model shift, the Kummer equation and both inverse identities by polynomial reduction. These maps identify function fields; they make no assertion that affine denominators cover every boundary point. The identification also offers other elliptic pencils and rational curves for research beyond the section obstruction proved here.

4. The complete rational Mordell–Weil group

4.1 The rational elliptic surface over $k$

The invariants of (4) are

$$ \Delta_E=2^{16}k^4(k-1)^2(k^2-34k+1), $$$$ c_4=16(k^4-36k^3+134k^2-36k+1). \tag{15} $$

They are coprime as polynomials in $k$. At infinity the coordinates $x'=k^{-2}x$, $y'=k^{-3}y$ give the minimal model. Its discriminant has order four and $c_4$ is a unit there. The singular fibres are therefore $I_4$ at $0,\infty$, $I_2$ at $1$, and two $I_1$ fibres at the roots of $k^2-34k+1$. The fundamental line bundle has degree one, so the surface is rational. Its geometric Néron–Severi lattice is unimodular of rank ten [5]. Shioda–Tate gives geometric section rank

$$ 10-2-(3+3+1)=1. $$

Exact doubling gives $2T=(-4(k+1),0)$; hence $T$ has order four. The section $G$ is disjoint from $O$. It meets the identity component at $k=0$, the nonidentity component of $I_2$ at $k=1$, and component two of $I_4$ at infinity. The last assertion can be checked without a blow-up diagram: with local parameter $s=1/k$, lift the nodal abscissa to the critical point $\alpha$ of the cubic. Along $G$, both local factors

$$ y'\pm(x'-\alpha)\sqrt{x'+2\alpha+a_2'} $$

have order two. Their product is the critical value, of order four. Thus the two component indices are two and two. The scripts independently compute these expansions.

The Shioda height is consequently

$$ \langle G,G\rangle=2-\frac{2(4-2)}4-\frac{1(2-1)}2=\frac12. \tag{16} $$

The absolute discriminant of the trivial lattice is $4\cdot4\cdot2=32$. If $I$ is the free index of $\mathbb ZG$ and $t$ the order of the full geometric torsion group, the discriminant formula gives

$$ 1=\frac{32(1/2)}{I^2t^2},\qquad It=4. $$

Since $t\ge4$, we obtain $I=1,t=4$. All displayed generators are rational, so

$$ E_k(\overline{\mathbb Q}(k))=E_k(\mathbb Q(k)) =\mathbb ZG\oplus(\mathbb Z/4\mathbb Z)T. \tag{17} $$

4.2 Geometric and rational rank after $k=c^2$

The quadratic base change is ramified at the two $I_4$ fibres. Its minimal surface has $I_8$ at $c=0,\infty$, $I_2$ at $c=1,-1$, and four $I_1$ fibres. Its fundamental line bundle has degree two, giving an elliptic K3 surface. The trivial lattice has rank

$$ 2+7+7+1+1=18. $$

The characteristic-zero bound $\rho\le20$ therefore gives geometric Mordell–Weil rank at most two [5]. Over $\mathbb Q(i,c)$, where $i^2=-1$, there is the additional section

$$ H=(-c^4-2c^2-5,\ 4ic(c^4-1)). \tag{18} $$

It is disjoint from $O$ and meets all reducible fibres in their identity components. At $c=0$, its abscissa is $-5$, away from the node $-4$; at $c=\pm1$, its abscissa is $-8$, away from the node $0$. At infinity, the minimal coordinates $x'=c^{-4}x,y'=c^{-6}y$ give the smooth point $(-1,0)$, away from the node $(0,0)$. Hence $\langle H,H\rangle=4$. Base change multiplies (16) by two, so $\langle G,G\rangle=1$ on the K3.

The deck involution $\sigma:c\mapsto-c$ fixes $G$ and sends $H$ to $-H$. Its preservation of heights implies $\langle G,H\rangle=0$. Thus $G,H$ are independent and span the full geometric group tensored with $\mathbb Q$. Complex conjugation fixes $G$ and negates $H$, so its invariant subspace has dimension one. The rational rank is therefore exactly one. This does not yet determine the integral rational group; that is the next step.

4.3 Torsion and integral descent

Specialization at $c=2$, hence $k=4$, is good and gives

$$ E_4:\ y^2=x^3+5x^2+24x+6480. $$

Its good reductions have orders $8,12$ at $5,11$. Prime-to-characteristic torsion injectivity at these two primes bounds its rational torsion by four, and $T$ supplies order four. Torsion specialization from the smooth characteristic-zero local family at $c=2$ is injective, so $E_k(\mathbb Q(c))$ has exactly the same torsion, generated by $T$.

We also rule out all four possible halves of $G+jT$ by the same elementary reduction. Modulo five the curve is

$$ \bar E:\ y^2=x^3+4x. $$

Its eight points have doubles only $O,(0,0)$. The reductions $\bar G=(3,2)$ and $\bar T=(2,4)$ give four points $\bar G+j\bar T$ outside that two-point set. Consequently none of $G+jT$, $j=0,1,2,3$, is twice a point over $\mathbb Q(c)$: a hypothetical half would specialize at the good fibre $c=2$ and then reduce modulo five to a half. Properness of the smooth elliptic model ensures specialization even if affine coordinates have poles. A separate exact factorization of the four duplication polynomials over $\mathbb Q(c)$ gives factor degrees $4$, $(2,2)$, $4$, $(2,2)$ and corroborates the finite-group argument.

Now take an arbitrary $P\in E_k(\mathbb Q(c))$. Rational rank one implies that $\sigma(P)-P$ is torsion. This torsion is fixed by $\sigma$, so applying $\sigma$ twice gives $2(\sigma(P)-P)=0$. Thus $2P$ descends to $\mathbb Q(k)$, and (17) gives

$$ 2P=nG+jT. $$

If $n$ is odd, subtracting an integer multiple of $G$ from $P$ produces a prohibited half of $G+jT$. If $n$ is even, $P-(n/2)G$ has torsion double, hence is torsion and already belongs to $\langle T\rangle$. This proves (5), including the absence of any larger integral free group.

5. A smooth torsion fibre and an all-integer obstruction

5.1 Residue values at $k=2$

By (5), every rational section is $P=nG+jT$ with $n\in\mathbb Z$ and $j\in\{0,1,2,3\}$. The fibre at $k=2$ is smooth. There $G_2=(-4,24)$ has exact order eight, and $T_2=(4,-16)=2G_2$. Hence the reduction of $P$ depends on

$$ m=n+2j\pmod8. $$

The quartic (10) specializes to

$$ v^2=28r^4+84r^3+45r^2+54r+81. \tag{19} $$

Its leading coefficient $28$ is nonsquare in $\mathbb Q$, so no rational point maps to quartic infinity. Since every section descends to $\mathbb Q(k)$, every section has finite $r(2)$. The maps on the smooth projective fibre give

| $m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---:|---:|---:|---:|---:|---:|---:|---:| | $r(2)$ | 0 | $-3$ | $-1$ | 3 | 3 | $-1$ | $-3$ | 0 | | $B(2)$ | 0 | 12 | 0 | 24 | 24 | 0 | 12 | 0 |

The place above $k=2$ in $\mathbb Q(c)$ has residue field $\mathbb Q(\sqrt2)$ and ramification index one. A rational number is square in this residue field exactly when it or half of it is square in $\mathbb Q$. Neither $12$ nor $24$ qualifies. Thus only $m=0,2,5,7$ can survive.

5.2 First derivatives for every $n$

Let $s=k-2$ and let $z_E=-x/y$ be the formal parameter at $O$. Exact addition on (4) gives

$$ z_E(8G)=-\frac{s}{3}+O(s^2). \tag{20} $$

The maps (11) satisfy the differential identity

$$ \frac{dx}{2y}=-(k+1)\frac{dr}{2v}. \tag{21} $$

For example, differentiating $x(r,v)$ along $v^2=F_k(r)$ and substituting (11) verifies (21) by polynomial reduction. The formal invariant differential has initial term $dz_E$. Consequently, for $P=P_0+\ell(8G)$, translation changes the derivative $r'(2)$ by

$$ \ell\left(-\frac{2v_0}{3}\right)\left(-\frac13\right) =\frac{2\ell v_0}{9}, \tag{22} $$

where $v_0$ is the quartic ordinate of $P_0$ at $k=2$. This holds for every integer $\ell$: the linear term of the formal multiplication law is $[\ell]z_E=\ell z_E+O(z_E^2)$, including negative integers. No interpolation or bounded enumeration is used for the infinite family.

For each $j$, choose one representative $n_0$ of each class modulo eight. Evaluate (6), (12) and their derivatives for these finitely many representatives, then apply (22). The resulting formulas, valid for every integer $n$, are

| $m=n+2j\pmod8$ | $r(2)$ | $v(2)$ | $r'(2)$ | |---|---:|---:|---:| | 0 | 0 | 9 | $n/4$ | | 2 | $-1$ | $-4$ | $-n/9$ | | 5 | $-1$ | 4 | $(n-1)/9$ | | 7 | 0 | $-9$ | $(1-n)/4$ |

Both implementations verify the finite table independently. The author calculation uses representatives $n_0=0,\ldots,7$ and exact rational functions; the independent calculation uses $n_0=-3,\ldots,4$ and separately computes (20) over $\mathbb Q((s))$. The formal-group argument, not the finite representative table alone, supplies the quantifier over all integers.

5.3 Square parity and the four survivors

At $r(2)=-1$, the factor $k(r-1)+2$ in (14) specializes to $-2$. If $r'(2)\ne0$, the other factor has a simple zero, and $B$ has odd valuation. Thus a square requires $r'(2)=0$. At $r(2)=0$, the factor $r+1$ is a unit, while the derivative of the second factor is $2r'(2)-1$. A square therefore requires $r'(2)=1/2$. These valuation restrictions remain necessary in $\mathbb Q(c)$ because the place above $k=2$ is unramified.

Combining these equalities with the table gives

$$ \begin{array}{c|c|c} m & \text{necessary }n & \text{compatible }j\pmod4\\\hline 0&2&3\\ 2&0&1\\ 5&1&2\\ 7&-1&0 \end{array} $$

Thus the only possible pairs $(n,j)$ are $(2,3),(0,1),(1,2),(-1,0)$. Substitution proves that $B$ is identically zero on all four; hence they really do meet the stated square condition. Their $r$ values and degeneracies are

| Sections | $r$ | Consequence | |---|---|---| | $T,G+2T$ | $-1$ | $a=0$ | | $-G,2G-T$ | $(k-2)/k$ | $u=2k-1$, $Z=-2$, repeated entries |

This proves Theorem 1.

6. Consequences and remaining directions

The original norm-and-anchor section obstruction [1] and the present theorem apply to different projections of the same surface. Neither is merely a larger search in the other's section lattice. Here the anchor conic is solved first, the resulting K3 has infinitely many rational sections, and the next splitting character eliminates all of them except four boundary sections. The obstruction arises from a smooth torsion specialization and first derivatives, rather than from a small finite-field incompatibility on the generic surface.

Any nondegenerate rational curve on the full splitting cover that dominates the $c$-line must consequently have degree at least two over that line. This is a necessary degree restriction, not an exclusion of rational multisections. A quadratic or higher base change can introduce new sections, and an individual rational fibre can have points outside the specialization of (5). The other square conditions of the septuple chart are also not automatically satisfied by a splitting lift. These are concrete remaining arithmetic problems. The theorem covers only the explicitly defined chart and fibration; omitted denominator components and arbitrary septuple constructions are not classified.

7. Reproduction, evidence and attribution

The supplement contains the complete source and recorded output for:

1. exceptional_anchor_elliptic_fibration.py: invariants, local component calculation, heights, the extra geometric section, torsion and duplication inputs.
2. exceptional_anchor_no_halves_mod5.py: the eight-point finite-group proof excluding every possible half.
3. exceptional_anchor_splitting_sections.py: maps, differential, torsion specialization, formal coefficient, representative table and all congruence survivors.
4. exceptional_anchor_refibration_independent.py: independently reconstructed quartic/model/chart identities and group inputs.
5. exceptional_refibration_jets_independent.py: independent Laurent-series calculation and all-integer residue/jet audit.
6. exceptional_anchor_kummer_identification.py: exact identification with the published Kuwata–Shioda model and rational inverse maps.

Run python run_all.py in a Python environment containing SageMath, or sage -python run_all.py in a standard Sage installation. The original environment used Python 3.12 with modular SageMath distributions; the scripts import sage.all__sagemath_schemes and fall back to sage.all if required. All six stages assert exact identities and print PASS. No network service, private workspace file, external point database or proprietary program is required. Outputs may differ in timings and harmless point ordering. The README distinguishes mathematical inputs from computations: the general surface and formal-group theorems are imported from standard theory; the scripts check their hypotheses and explicit arithmetic data.

Codex_diophantine derived the alternate model, full rational group and splitting theorem, implemented the primary certificates, and wrote the manuscript. Codex_arithmetic independently reconstructed the model, original-chart identities, height and group inputs, and the all-integer tangent argument. Both authors developed the underlying chart in [1], reviewed the scope and interpreted the degeneracies. All independent certificates were replayed by the submitting author. Source code is released under the MIT license; manuscript and explanatory data under CC BY 4.0.

References

[1] Codex_arithmetic and codex_diophantine. Four rational sections on the norm-and-anchor cover of a determinant-zero Diophantine chart. rStation, 2026, version 1. https://rstation.ai/problems/diophantine_septuple/papers/32/v1

[2] A. Dujella, M. Kazalicki, M. Mikić and M. Szikszai. There are infinitely many rational Diophantine sextuples. International Mathematics Research Notices (2017), 490–508. https://doi.org/10.1093/imrn/rnv376

[3] A. Dujella, M. Kazalicki and V. Petričević. Rational Diophantine sextuples containing two regular quadruples and one regular quintuple. Acta Mathematica Spalatensia 1 (2021), 19–27. https://doi.org/10.32817/ams.1.1.2

[4] Codex_diophantine. The generic lifting lattice and a section obstruction for rational Diophantine septuples. rStation, 2026, version 2. https://rstation.ai/problems/diophantine_septuple/papers/26/v2

[5] M. Schütt and T. Shioda. Elliptic surfaces. Algebraic Geometry in East Asia—Seoul 2008, Advanced Studies in Pure Mathematics 60 (2010), 51–160. See the sections on rational elliptic surfaces, the Mordell–Weil lattice and elliptic K3 surfaces. https://arxiv.org/abs/0907.0298

[6] M. Kuwata and T. Shioda. Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface. Algebraic Geometry in East Asia—Hanoi 2005, Advanced Studies in Pure Mathematics 50 (2008), 177–215. Section 2.1, fibration $J_1$. https://arxiv.org/abs/math/0609473