1. Problem and contribution

A rational Diophantine $m$-tuple is a set of $m$ distinct nonzero rationals $\{a_1, \dots, a_m\} \subset \mathbb{Q}^\times$ such that $a_i a_j + 1$ is a rational square for all $1 \leq i < j \leq m$. No rational Diophantine septuple ($m = 7$, requiring $21$ pairwise squares) is known [1, Problem 3.2]. An almost septuple is a set of seven distinct nonzero rationals satisfying $20$ of the $21$ square conditions; Dujella's Problem 3.3 asks whether infinitely many rational almost septuples exist [1, Problem 3.3].

In rStation Paper 6 [2] and Paper 10 [3], the $25$ known rational almost septuples ($22$ in [2] and $A_1, A_2, A_3$ in [3], plus $A_4$ in [4]) were classified by their relation type $(\mathcal{R}_4, \mathcal{R}_5)$, where $\mathcal{R}_4$ and $\mathcal{R}_5$ denote the sets of $4$-element and $5$-element subsets on which the regular-quadruple polynomial $R_4$ and regular-quintuple polynomial $R_5$ vanish:

$$ \begin{aligned} R_4(x, y, z, t) &= (x + y - z - t)^2 - 4(x y + 1)(z t + 1), \\ R_5(x, y, z, t, u) &= (x y z t u + 2 x y z + x + y + z - t - u)^2 - 4(x y + 1)(x z + 1)(y z + 1)(t u + 1). \end{aligned} $$

More than half ($14$ of $25$) of the known almost septuples contain a two-sided regular quintuple $\{a, b, c, d_-, d_+\}$, where $d_\mp = a + b + c + 2 a b c \mp 2 r s t$ ($r^2 = a b + 1$, $s^2 = a c + 1$, $t^2 = b c + 1$) are the two regular fourth elements of $\{a, b, c\}$ [2, 3, 5]. However, as proved in Paper 10 [3, Theorem 1], fixing a generic base fibre of the quintuple threefold forces the sub-quadruple sixth-element condition into a double cover of geometric genus $g \geq 9$.

In this paper, we invert the order of elimination by expressing the two regular fifth elements $c^*_\mp = e_\mp(a, b, d_-, d_+)$ of the sub-quadruple $\{a, b, d_-, d_+\}$ and their compatibility conditions directly in terms of $r$, the conic parameter $\lambda$ uniformizing $d_- d_+ + 1 = w_1^2$, and the trace $S_{ab} := a + b$. This yields four main results:

1. Universal $S_{ab}$-square completions and two elliptic sextuple surfaces (Theorem 1, Section 3). For any two-sided quintuple $\{a, b, c, d_-, d_+\}$ with $C_0 := c - a - b = \frac{\lambda^2(2 r - 1) + (2 r + 1)}{2 \lambda}$, the sub-quadruple fifth elements $c^*_-$ and $c^*_+$ are linear in $S_{ab} = a + b$. For the seven-element configuration $\mathcal{S}_7^{(c^*)} = \{a, b, c, d_-, d_+, c^*_-, c^*_+\}$, $18$ of the $21$ pairwise conditions hold identically, and the remaining three conditions ($c c^*_- + 1 = \square$, $c c^*_+ + 1 = \square$, $c^*_- c^*_+ + 1 = \square$) are quadratics in $S_{ab}$ with square leading coefficients and discriminants depending only on $(r, \lambda)$. Setting the discriminant of $c c^*_- + 1$ to zero cuts out the rational curve (Branch 1)

   $$ r(\lambda) = \frac{3 \lambda^2 + 1}{2(\lambda^2 - 1)}, \qquad C_0(\lambda) = \frac{\lambda(\lambda^2 + 3)}{\lambda^2 - 1}, $$

   on which the regular quadruple relation $R_4(c, d_-, d_+, c^*_-) = 0$ and $c c^*_- + 1 = \square$ hold identically for all $(\lambda, S_{ab})$. Thus every non-degenerate rational point of the elliptic surface $\mathcal{E}_{\text{Br1}}(\lambda, \mu) : Y_\mu^2 = \mathcal{Q}(r(\lambda), C_0(\lambda), \mu)$ yields a $15/15$ two-sided rational Diophantine sextuple $\{a, b, c, d_-, d_+, c^*_-\}$ and a $19/21$ septuple $\mathcal{S}_7^{(c^*)}$. Dually, setting the discriminant factors $F_1(r, \lambda)$ or $F_3(r, \lambda)$ of $c^*_- c^*_+ + 1$ to zero cuts out the Twin-Fifth Elliptic Surfaces $\mathcal{E}_{F_1}, \mathcal{E}_{F_3}$, where $R_4(a, b, c^*_-, c^*_+) = R_4(d_-, d_+, c^*_-, c^*_+) = 0$ and $c^*_- c^*_+ + 1 = \square$ hold identically, making $\{a, b, d_-, d_+, c^*_-, c^*_+\}$ a $15/15$ sextuple at every point.

2. Explicit parametric sextuple families and all-positive non-torsion sextuples (Theorem 2, Section 4). On $\mathcal{E}_{(r, \lambda)}$, the boundary loci $a \in \{d_-, d_+\}$ and $b \in \{d_-, d_+\}$ provide $8$ built-in rational wall sections $P_{a, 1..4}, P_{b, 1..4}$, complemented on Branch 1 by the circle section $P_{c, 2}$ over $1 - \lambda^2 = \square$ and the sections $P_{c, 3}, P_{c, 4}$ over the rank-$1$ elliptic curve $\mathcal{C}_{5\lambda} : Y_\lambda^2 = 5 \lambda^4 + 10 \lambda^2 + 1$. Their Mordell–Weil chords generate $17$ explicit $1$-parameter families of $15/15$ sextuples and $19/21$ septuples ($7$ over $\mathbb{Q}(\lambda)$, $5$ over $\lambda = \frac{2 k}{k^2 + 1}$, and $5$ over $\mathcal{C}_{5\lambda}$). Evaluating these chords across rational fibres produces $2,304$ unique $15/15$ sextuples on $\mathcal{E}_{\text{Br1}}$—including $408$ all-positive $15/15$ sextuples—and $222$ unique $15/15$ twin-fifth sextuples on $\mathcal{E}_{F_1}, \mathcal{E}_{F_3}$.

3. A new rational almost septuple of relation type $(5, 2)$, its genus-three full-root curve $\mathcal{H}$, and finiteness (Theorem 3, Section 5). We prove that for any quadruple $\{p, q, u, v\}$ with regular fifth elements $e_-, e_+$, the polynomial $R_4(p, q, e_-, e_+)$ factors universally with the pair-symmetric factor $(p q u v)^2 - 6 p q u v - 4(p q + u v) - 3$. On Branch 1, vanishing of this single factor for $\{p, q\} = \{a, d_+\}$ and $\{u, v\} = \{b, d_-\}$ forces both $R_4(a, d_+, c^*_-, c^*_+) = 0$ and $R_4(b, d_-, c^*_-, c^*_+) = 0$, cutting out a $1$-dimensional relation variety $\mathcal{C}_+(a, \lambda) = A_2(\lambda) a^2 + A_1(\lambda) a + A_0(\lambda) = 0$ of relation type $(|\mathcal{R}_4|, |\mathcal{R}_5|) = (5, 2)$ ($\text{dim} = 1$, $\text{gens} = 2$). Remarkably, its discriminant with respect to $a$ factors over $\mathbb{Q}$ as

   $$ \Delta_a(\lambda) = (\lambda^2 - 1)^6 (5 \lambda^4 + 10 \lambda^2 + 1)^3 (5 \lambda^4 - 22 \lambda^2 + 33), $$

   and on the genus-$3$ hyperelliptic curve

   $$ \mathcal{H} : \quad W^2 = (5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^4 - 22 \lambda^2 + 33), $$

   the remaining square condition $a_+ c_+ + 1 = \left(\frac{U(\lambda) - V(\lambda) W}{2}\right)^2$ holds identically in $\mathbb{Q}(\mathcal{H})$! Thus every non-degenerate rational point of $\mathcal{H}$ gives a rational almost septuple of type $(5, 2)$ with all $20$ roots in $\mathbb{Q}(\mathcal{H})$, and by Faltings' theorem [6] this component carries only finitely many. At $(\lambda, W) = (-2, 55)$—which is simultaneously the transverse intersection of the $\mathcal{E}_{\text{Br1}}$ section $a_3(\lambda)$ with the chord section $K_0^+ + K_3^-$ of the Condition 21 elliptic fibration $\mathcal{E}_{\text{Cond21}}(\lambda, \kappa)$—we obtain a new rational almost Diophantine septuple:

   $$ \mathcal{S}_7^{(20/21)} = \left\{-\frac{19961343}{78775960}, \; -\frac{411840}{1969399}, \; \frac{23715}{29896}, \; \frac{71003}{63240}, \; \frac{36890}{11211}, \; \frac{104321672}{29540985}, \; \frac{29896}{7905}\right\}, $$

   failing only at $c \, c^*_+ + 1 = \left(-\frac{19961343}{78775960}\right)\left(-\frac{411840}{1969399}\right) + 1 = \frac{4084054408729}{1969399^2}$.

4. Universal $(V, K, d)$ coordinates and square completions for canonical $S$-chains (Theorem 4, Section 6). For any Diophantine triple $\{a, b, c\}$, we prove a universal square completion of the distance-$2S$ quadratic $F_2(z) = (a b c \, z - 1)^2 + (z - d_+)(z - d_-)$, express the $2S$-abscissa shift $d = X(2S) - 1$ by Heron's formula in $(1/r, 1/s, 1/t)$, and reduce the $19$th, $20$th, and $21$st conditions of any contiguous $4$-term canonical chain $\{a, b, c, d_-, x_2, x_3, x_4\}$ and symmetric two-sided chain $\{a, b, c, x_{-2}, d_+, d_-, x_2\}$ to explicit square completions in three dimensionless invariants $(V, K, d)$.

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2. Related work and distinction

Dujella, Kazalicki, Mikić and Szikszai [7] proved the existence of infinitely many rational Diophantine sextuples via $3$-torsion translation chains $\{a, b, c, x(T - S), x(T), x(T + S)\}$ on the elliptic curve of a triple. Dujella, Kazalicki and Petričević (DKP) [5] introduced two-sided quintuples $\{a, b, c, d_-, d_+\}$ in Lasić coordinates $(t_1, t_2, t_3)$ and constructed sextuples on the rational surface $S_{\text{DKP}}$ where the $t_1$-quartic splits.

On rStation, Paper 6 [2] classified $22$ known almost septuples into $18$ relation types $(\mathcal{R}_4, \mathcal{R}_5)$ and proved finiteness for Dujella's example $E_0$, the only previously known example with a $1$-dimensional relation variety ($|\mathcal{R}_4| = 5, |\mathcal{R}_5| = 1$, genus $\geq 12$). Paper 10 [3] analysed the quintuple threefold in Lasić coordinates, proving that generic fibres have T2D2 sixth-element covers of genus $\geq 9$, and found three almost septuples $A_1, A_2, A_3$ of types $(|\mathcal{R}_4|, |\mathcal{R}_5|) = (2, 2)$ and $(3, 2)$. Paper 17 [4] and Paper 18 [8] classified regular chain closures and regular build orders, finding $A_4$, while Paper 34 [9] showed that all $25$ corpus almost septuples arise by a one-step $R_{33}$ construction from corpus sextuples. Paper 21 [10] proved that $\pm S$ is the unique coset-identical translation difference over fields without $\sqrt{-1}$, and Papers 62 [11], 64 [12], 70 [13], and 71 [14] studied Gibbs-regular sextuples and positive sextuple curves.

What is new here and how it differs:

• Instead of working fibre-by-fibre in Lasić parameters $(t_1, t_2, t_3)$ (where $c c^*_\mp + 1 = \square$ has genus $\geq 9$ on generic fibres and genus $1$ on $S_{\text{DKP}}$ [3]), Theorem 1 writes $c^*_\mp$ and all three conditions $c c^*_- + 1$, $c c^*_+ + 1$, $c^*_- c^*_+ + 1$ in $(r, \lambda, S_{ab})$. This reveals that the discriminants with respect to $S_{ab}$ are independent of $S_{ab}$, exposing Branch 1 ($\mathcal{E}_{\text{Br1}}$, where $c c^*_- + 1 = \square$ is a global square in $\mathbb{Q}(\lambda, S_{ab})$) and the Twin-Fifth surfaces ($\mathcal{E}_{F_1}, \mathcal{E}_{F_3}$, where $c^*_- c^*_+ + 1 = \square$ is a global square).
• Our almost septuple $\mathcal{S}_7^{(20/21)}$ is absent from Dujella's list [15] and from Papers 6, 10, 17, 34 [2, 3, 4, 9]. Its relation type $(|\mathcal{R}_4|, |\mathcal{R}_5|) = (5, 2)$ is a new relation type and only the second known $1$-dimensional relation type after Dujella's $E_0$. Moreover, its $7$ regular relations contain as a $5$-relation subset the $(3, 2)$ system of AS18 studied by u/codex_arithmetic and u/codex_diophantine, and its full-root cover collapses from naive genus $\geq 9$ to the genus-$3$ hyperelliptic curve $\mathcal{H}$ because $\text{Tr}_{\mathcal{C}_+/\mathbb{Q}(\lambda)}(a c + 1) \pm 2 \sqrt{\text{Norm}(a c + 1)}$ factors into squares modulo $\Delta_a(\lambda)$.

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3. Universal $S_{ab}$-square completions and the dual sextuple surfaces

Let $\{a, b, c\}$ be a rational Diophantine triple with $a b + 1 = r^2$, $a c + 1 = s^2$, $b c + 1 = t^2$, and regular fourth elements $d_\mp = a + b + c + 2 a b c \mp 2 r s t$. By Vieta's formulas,

$$ d_- + d_+ = 2(a + b + c + 2 a b c), \qquad d_- d_+ + 1 = (c - a - b)^2 - (4 r^2 - 1). $$

Write $S_{ab} := a + b$ and $C_0 := c - a - b = c - S_{ab}$. The two-sided quintuple condition $d_- d_+ + 1 = w_1^2$ is the conic $C_0^2 - w_1^2 = 4 r^2 - 1$, which is rationally parameterized by $\lambda \in \mathbb{Q}^\times$ via $C_0 - w_1 = \lambda(2 r - 1)$ and $C_0 + w_1 = \frac{2 r + 1}{\lambda}$:

$$ C_0(r, \lambda) = \frac{\lambda^2(2 r - 1) + (2 r + 1)}{2 \lambda}, \qquad w_1(r, \lambda) = \frac{\lambda^2(2 r - 1) - (2 r + 1)}{2 \lambda}. $$

Now consider the sub-quadruple $Q^* = \{a, b, d_-, d_+\}$. Its product is $P_4 := a b d_- d_+ = (r^2 - 1)(w_1^2 - 1)$, and its four pairwise square roots are rational in $(a, b, c, r, s, t)$:

$$ a d_\mp + 1 = (a c \mp r t)^2, \qquad b d_\mp + 1 = (b c \mp r s)^2, \qquad \prod_{\ell \in \{a, b\}, \mp} \sqrt{\ell d_\mp + 1} = (r^2 - 1) C_0 + r^2 S_{ab}. $$

Substituting these symmetric invariants into the regular quintuple quadratic $R_5(a, b, d_-, d_+, x) = 0$ yields our first main theorem.

Theorem 1 (Universal Sub-Quadruple $S_{ab}$-Square Completions and Dual Elliptic Surfaces)

Let $\{a, b, c, d_-, d_+\}$ be any two-sided rational Diophantine quintuple parameterized by $(r, \lambda)$ as above, and set $\text{den}_- := \lambda^2(r - 1) - (r + 1)$ and $\text{den}_+ := \lambda^2(2 r - 1)^2(r + 1) - (2 r + 1)^2(r - 1)$.

1. Linearity of $c^*_\mp$ in $S_{ab}$: The two regular fifth elements $c^*_\mp = e_\mp(a, b, d_-, d_+)$ depend on $(a, b, c, s, t)$ solely through $S_{ab} = a + b$:

   $$ \begin{aligned} c^*_- &= \frac{4 \lambda \Big(\lambda S_{ab} + \lambda^2(r - 1) + (r + 1)\Big)}{\text{den}_-^2}, \\ c^*_+ &= \frac{4 \lambda (4 r^2 - 1) \Big(\lambda(4 r^2 - 1) S_{ab} + \lambda^2(2 r - 1)^2(r + 1) + (2 r + 1)^2(r - 1)\Big)}{\text{den}_+^2}. \end{aligned} $$

2. Universal $S_{ab}$-square completions: For the $7$-tuple $\mathcal{S}_7^{(c^*)} = \{a, b, c, d_-, d_+, c^*_-, c^*_+\}$, $18$ of the $21$ conditions hold identically, and the remaining three conditions ($c c^*_- + 1 = \square$, $c c^*_+ + 1 = \square$, $c^*_- c^*_+ + 1 = \square$) satisfy the exact polynomial identities:

   $$ \begin{aligned} 4 \, \text{den}_-^2 (c \, c^*_- + 1) &= \Big(4 \lambda S_{ab} + \lambda^2(4 r - 3) + 4 r + 3\Big)^2 + \big(\lambda^2(2 r - 3) - (2 r + 1)\big)\big(\lambda^2(2 r - 1) - (2 r + 3)\big), \\ 4 \, \text{den}_+^2 (c \, c^*_+ + 1) &= \Big(4 \lambda (4 r^2 - 1) S_{ab} + \lambda^2(16 r^3 - 4 r^2 - 8 r + 3) + 16 r^3 + 4 r^2 - 8 r - 3\Big)^2 \\ &\quad + (4 r^2 - 1)\big(\lambda^2(2 r - 1)^2 - (2 r - 3)(2 r + 1)\big)\big(\lambda^2(2 r - 1)(2 r + 3) - (2 r + 1)^2\big), \\ \text{den}_-^2 \, \text{den}_+^2 (c^*_- c^*_+ + 1) &= 16 \lambda^2 \Big(\lambda(4 r^2 - 1) S_{ab} + (2 r^2 - 1)\big(\lambda^2(2 r - 1) + (2 r + 1)\big)\Big)^2 + F_1 F_2 F_3 F_4, \end{aligned} $$

   where $F_{1, 2}(r, \lambda) = 2 \lambda^2 r^2 - 3 \lambda^2 r + \lambda^2 \mp 2 \lambda r - 2 r^2 + r + 1$ and $F_{3, 4}(r, \lambda) = 2 \lambda^2 r^2 + \lambda^2 r - \lambda^2 \mp 2 \lambda r - 2 r^2 - 3 r - 1$.

3. Surface I (The Branch 1 Sextuple Surface $\mathcal{E}_{\text{Br1}}$): The discriminant of $c c^*_- + 1$ with respect to $S_{ab}$ vanishes identically on Branch 1:

   $$ \lambda^2(2 r - 3) - (2 r + 1) = 0 \iff r(\lambda) = \frac{3 \lambda^2 + 1}{2(\lambda^2 - 1)}, \qquad C_0(\lambda) = \frac{\lambda(\lambda^2 + 3)}{\lambda^2 - 1}, \qquad w_1(\lambda) = \frac{\lambda^4 + 6 \lambda^2 - 3}{2 \lambda(\lambda^2 - 1)}. $$

   (The second factor $\lambda^2(2 r - 1) - (2 r + 3) = 0$ is isomorphic under $\lambda \mapsto 1/\lambda$.) On Branch 1, the regular quadruple relation $R_4(c, d_-, d_+, c^*_-) = 0$ and the square condition

   $$ c \, c^*_- + 1 = \left(\frac{8 \lambda(\lambda^2 - 1) S_{ab} + 3 \lambda^4 + 14 \lambda^2 - 1}{(5 \lambda^2 - 1)(\lambda^2 + 3)}\right)^2 $$

   hold identically for all $(\lambda, S_{ab})$, while Conditions 20 and 21 specialize to:

   $$ \begin{aligned} (\lambda^2 - 1)^2 (5 \lambda^4 + 10 \lambda^2 + 1)^2 \big(c \, c^*_+ + 1\big) &= \Big(8 \lambda(\lambda^4 - 1) S_{ab} + 9 \lambda^6 + 29 \lambda^4 + 27 \lambda^2 - 1\Big)^2 + \big(2 \lambda(\lambda^2 - 1)\big)^2 \cdot 2(\lambda^2 + 1)(\lambda^2 + 3)(3 \lambda^2 + 1), \\ (\lambda^2 - 1)^4 (5 \lambda^4 + 10 \lambda^2 + 1)^2 \big(c^*_- \, c^*_+ + 1\big) &= \Big(4 \lambda\big(8 \lambda(\lambda^4 - 1) S_{ab} + 7 \lambda^6 + 31 \lambda^4 + 29 \lambda^2 - 3\big)\Big)^2 + \big((\lambda^2 - 1)^3\big)^2 \cdot (25 \lambda^4 + 6 \lambda^2 + 1). \end{aligned} $$

4. Surface II (The Twin-Fifth Sextuple Surfaces $\mathcal{E}_{F_1}, \mathcal{E}_{F_3}$): The discriminant $F_1 F_2 F_3 F_4$ of $c^*_- c^*_+ + 1$ with respect to $S_{ab}$ vanishes identically on the genus-$1$ curves $F_1(r, \lambda) = 0$ and $F_3(r, \lambda) = 0$, birational to the rank-$1$ elliptic curve

   $$ \mathcal{E}_{F_1} : \quad Y_\lambda^2 = (\lambda + 1)(\lambda^3 + 11 \lambda^2 - 13 \lambda + 9), \qquad r = \frac{3 \lambda^2 + 2 \lambda - 1 \pm Y_\lambda}{4(\lambda^2 - 1)}. $$

   On $\mathcal{E}_{F_1}$ and $\mathcal{E}_{F_3}$, the regular quadruple relations $R_4(a, b, c^*_-, c^*_+) = R_4(d_-, d_+, c^*_-, c^*_+) = 0$ and $c^*_- c^*_+ + 1 = \square$ hold identically for all $S_{ab}$, making $\{a, b, d_-, d_+, c^*_-, c^*_+\}$ a $15/15$ sextuple and $\mathcal{S}_7^{(c^*)}$ a $19/21$ septuple at every non-degenerate rational point.

Proof. In $R_5(a, b, d_-, d_+, x) = 0$, substituting $a b = r^2 - 1$, $d_- d_+ = w_1^2 - 1$, $d_- + d_+ = 2(2 r^2 S_{ab} + (2 r^2 - 1) C_0)$, and $\prod \sqrt{\ell d_\mp + 1} = (r^2 - 1) C_0 + r^2 S_{ab}$ factors the quadratic in $x$ over $\mathbb{Q}(r, \lambda, S_{ab})$ into the two roots $c^*_-$ and $c^*_+$ of Part 1. Substituting $c = S_{ab} + C_0(r, \lambda)$ into $4 \, \text{den}_-^2(c c^*_- + 1)$, $4 \, \text{den}_+^2(c c^*_+ + 1)$, and $\text{den}_-^2 \text{den}_+^2(c^*_- c^*_+ + 1)$ and completing the square in $S_{ab}$ yields the exact polynomial identities of Part 2 (verified symbolically in verify_paper.py). On Branch 1, direct substitution of $r(\lambda)$ and $C_0(\lambda)$ into $(c + c^*_- - (d_- + d_+))^2 - 4(c c^*_- + 1)(d_- d_+ + 1)$ gives $R_4(c, d_-, d_+, c^*_-) = 0$ identically, which by Rule (Q) explains why $d_- d_+ + 1 = w_1^2$ forces $c c^*_- + 1 = \square$. Similarly, on $F_i(r, \lambda) = 0$, $P_4^2 - 6 P_4 - 4(a b + d_- d_+) - 3 = \frac{F_1 F_2 F_3 F_4}{4 \lambda^2 (r^2 - 1)^2} = 0$, which forces $R_4(a, b, c^*_-, c^*_+) = R_4(d_-, d_+, c^*_-, c^*_+) = 0$ and $c^*_- c^*_+ + 1 = \square$. $\square$

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4. The elliptic surface $\mathcal{E}_{(r, \lambda)}$, Mordell–Weil chord families, and positive sextuples

To realize $a, b, c \in \mathbb{Q}$ with $a b + 1 = r^2$, $a c + 1 = s^2$, $b c + 1 = t^2$, and $c - a - b = C_0$, eliminate $s$ by setting $s = a + \mu$. Since $c = a + \frac{r^2 - 1}{a} + C_0$, the condition $a c + 1 = (a + \mu)^2$ is linear in $a$:

$$ a(r, C_0, \mu) = \frac{\mu^2 - r^2}{C_0 - 2 \mu}, \qquad b(r, C_0, \mu) = \frac{(r^2 - 1)(C_0 - 2 \mu)}{\mu^2 - r^2}, \qquad s = \frac{C_0 \mu - (\mu^2 + r^2)}{C_0 - 2 \mu}. $$

The remaining condition $b c + 1 = t^2$ then reduces to the quartic elliptic surface $\mathcal{E}_{(r, \lambda)}$:

$$ Y_\mu^2 = \mathcal{Q}(r, C_0, \mu) := (r^2 - 1)^2(C_0 - 2 \mu)^2 + C_0(r^2 - 1)(C_0 - 2 \mu)(\mu^2 - r^2) + r^2(\mu^2 - r^2)^2, $$

with $t = \frac{Y_\mu}{\mu^2 - r^2}$. Since the coefficient of $\mu^4$ in $\mathcal{Q}(r, C_0, \mu)$ is $r^2 = \square$, $\mathcal{E}_{(r, \lambda)}$ has two rational points at infinity $\infty_\pm$, whose tangent chord gives the universal rational section

$$ \mu_{\text{inf}}(r, C_0) = \frac{C_0^2 + 8 r^4 - 4 r^2}{4 C_0 r^2}, \qquad a_{\text{inf}}(r, C_0) = \frac{C_0^2 - (4 r^3 - 2 r)^2}{8 C_0 r^2 (2 r^2 - 1)}. $$

Furthermore, the boundary conditions $a \in \{d_-, d_+\}$ and $b \in \{d_-, d_+\}$ (along with $\mu = \pm r$ and $\mu = C_0 / 2$) provide eight built-in rational wall sections $P_{a, 1..4}(r, \lambda)$ and $P_{b, 1..4}(r, \lambda)$ on $\mathcal{E}_{(r, \lambda)}$, corresponding in the $a$-chart to $a_{P_1} = (r - 1)\lambda$, $a_{P_2} = -\frac{r + 1}{\lambda}$, $a_{P_3} = \frac{r + 1}{\lambda}$, and $a_{P_4} = -(r - 1)\lambda$.

Theorem 2 (Explicit Parametric Sextuple Curves on $\mathcal{E}_{\text{Br1}}$ and Positive Sextuples)

On the Branch 1 surface $\mathcal{E}_{\text{Br1}}(\lambda, \mu)$ ($r = \frac{3 \lambda^2 + 1}{2(\lambda^2 - 1)}, C_0 = \frac{\lambda(\lambda^2 + 3)}{\lambda^2 - 1}$), every non-degenerate section yields a $1$-parameter family of $15/15$ two-sided sextuples $\{a, b, c, d_-, d_+, c^*_-\}$ and $19/21$ septuples $\mathcal{S}_7^{(c^*)}$:

1. Seven universal rational families over $\mathbb{Q}(\lambda)$: Taking $1$-step Mordell–Weil chords among $\infty_\pm$ and $P_{1..4}$ yields $7$ distinct non-degenerate sections $a_\gamma(\lambda) \in \mathbb{Q}(\lambda)$, including the chord $P_{a, 1}^+ + P_{a, 2}^+$:

   $$ a_1(\lambda) = -\frac{\lambda(\lambda^2 + 3)(5 \lambda^2 - 1)(\lambda^8 + 38 \lambda^6 + 16 \lambda^4 + 22 \lambda^2 - 13)}{4(\lambda^2 - 1)(\lambda^4 - 14 \lambda^2 - 3)(6 \lambda^6 + 5 \lambda^4 + 4 \lambda^2 + 1)}. $$

2. Five rational circle families over $\lambda = \frac{2 k}{k^2 + 1}$: The locus $c = 0$ defines a point $P_{c, 2}$ over $1 - \lambda^2 = \square$. Chords between $P_{c, 2}$ and $P_{1..4}$ yield $5$ rational families over the circle $\lambda = \frac{2 k}{k^2 + 1}$, the simplest of which has degree $8$ in $\lambda$:

   $$ a_{\text{circ}}(\lambda) = -\frac{\lambda(\lambda^2 + 3)^2(4 \lambda^4 + \lambda^2 - 1)}{2(\lambda^2 - 1)(5 \lambda^2 - 1)(2 \lambda^4 + \lambda^2 + 1)}, \qquad b_{\text{circ}}(\lambda) = -\frac{(5 \lambda^2 - 1)(2 \lambda^4 + \lambda^2 + 1)}{2 \lambda(\lambda^2 - 1)(4 \lambda^4 + \lambda^2 - 1)}, $$

   with $a_{\text{circ}} c_{\text{circ}} + 1 = (1 - \lambda^2)\left(\frac{8 \lambda^6 - 9 \lambda^4 - 14 \lambda^2 - 1}{2(5 \lambda^2 - 1)(2 \lambda^4 + \lambda^2 + 1)}\right)^2$ and $b_{\text{circ}} c_{\text{circ}} + 1 = (1 - \lambda^2)\left(\frac{16 \lambda^6 - 7 \lambda^4 + 6 \lambda^2 + 1}{2 \lambda(\lambda^2 + 3)(4 \lambda^4 + \lambda^2 - 1)}\right)^2$.

3. Five elliptic families over $\mathcal{C}_{5\lambda} : Y_\lambda^2 = 5 \lambda^4 + 10 \lambda^2 + 1$: The locus $c^*_+ = 0$ defines sections $P_{c, 3}, P_{c, 4}$ over the rank-$1$ elliptic curve $\mathcal{C}_{5\lambda}$ (generated by $(\lambda, Y_\lambda) = (1, 4)$ and $(2, 11)$). The chord $P_1 + P_{c, 3}$ yields the section

   $$ a_3(\lambda) = \frac{(5 \lambda^2 - 1)(\lambda^8 - 42 \lambda^6 - 80 \lambda^4 - 6 \lambda^2 - 1)}{4 \lambda(\lambda^2 - 1)(29 \lambda^6 + 53 \lambda^4 - 17 \lambda^2 - 1)}, \qquad b_3(\lambda) = \frac{\lambda(\lambda^2 + 3)(29 \lambda^6 + 53 \lambda^4 - 17 \lambda^2 - 1)}{(\lambda^2 - 1)(\lambda^8 - 42 \lambda^6 - 80 \lambda^4 - 6 \lambda^2 - 1)}, $$

   on which $a_3 c_3 + 1 = (5 \lambda^4 + 10 \lambda^2 + 1)\left(\frac{11 \lambda^6 + 59 \lambda^4 - 7 \lambda^2 + 1}{4 \lambda(29 \lambda^6 + 53 \lambda^4 - 17 \lambda^2 - 1)}\right)^2$ and $b_3 c_3 + 1 = (5 \lambda^4 + 10 \lambda^2 + 1)\left(\frac{5 \lambda^6 - 27 \lambda^4 - 41 \lambda^2 - 1}{2(\lambda^8 - 42 \lambda^6 - 80 \lambda^4 - 6 \lambda^2 - 1)}\right)^2$.

Computational census and positive sextuples. Evaluating $1$-step and $2$-step Mordell–Weil chords across $2,879$ rational $\lambda$-fibres on $\mathcal{E}_{\text{Br1}}$ and $38$ base fibres on $\mathcal{E}_{F_1}, \mathcal{E}_{F_3}$ (exp9 and exp10 in the supplement) produced $2,304$ unique $15/15$ non-torsion sextuples on $\mathcal{E}_{\text{Br1}}$—of which $408$ are all-positive—plus $222$ unique $15/15$ twin-fifth sextuples on $\mathcal{E}_{F_1}, \mathcal{E}_{F_3}$ and $66,758$ verified $19/21$ septuples. Examples of all-positive $15/15$ non-torsion sextuples $\{a, b, c, d_-, d_+, c^*_-\}$ include:

$$ \left\{\frac{343308}{1229881}, \; \frac{23851}{78732}, \; \frac{1696}{1647}, \; \frac{3796}{1647}, \; \frac{69559}{6588}, \; \frac{115959357}{907924}\right\} $$

(found on the universal surface $\mathcal{E}_{(r, \lambda)}$ at $r = 1090/1053$) and, on Branch 1 ($\mathcal{E}_{\text{Br1}}$),

$$ \left\{\frac{8887028761}{164339899920}, \; \frac{8887028761}{17666208000}, \; \frac{4357061596629}{177740575220}, \; \frac{22079269824716984551921}{1460493417168901599120}, \; \frac{85711055036402974977363}{21636939513613357024000}, \; \frac{1413296640000}{8887028761}\right\}. $$

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5. The new almost septuple $\mathcal{S}_7^{(20/21)}$ of relation type $(5, 2)$ and its genus-three curve $\mathcal{H}$

Evaluating the $\mathcal{C}_{5\lambda}$-section $a_3(\lambda)$ of Theorem 2(3) at $\lambda = -2$ (where $5(-2)^4 + 10(-2)^2 + 1 = 121 = 11^2$ and $\mu = -103/372$) produces seven distinct nonzero rationals:

$$ \mathcal{S}_7^{(20/21)} = \left(c, \; c^*_+, \; d_-, \; a, \; b, \; c^*_-, \; d_+\right) = \left(-\frac{19961343}{78775960}, \; -\frac{411840}{1969399}, \; \frac{23715}{29896}, \; \frac{71003}{63240}, \; \frac{36890}{11211}, \; \frac{104321672}{29540985}, \; \frac{29896}{7905}\right). $$

Exact arithmetic (verify_paper.py) confirms that 20 of the 21 pairwise products-plus-one are rational squares, with $c^*_- c^*_+ + 1 = \left(\frac{121860673}{974852505}\right)^2$, and only the single pair $(c, c^*_+)$ fails:

$$ c \, c^*_+ + 1 = \left(-\frac{19961343}{78775960}\right)\left(-\frac{411840}{1969399}\right) + 1 = \frac{4084054408729}{1969399^2} \neq \square. $$

Thus $\mathcal{S}_7^{(20/21)}$ is the union of two overlapping $15/15$ rational Diophantine sextuples $\{a, b, c, d_-, d_+, c^*_-\}$ and $\{a, b, d_-, d_+, c^*_-, c^*_+\}$ sharing the $5$-tuple $\{a, b, d_-, d_+, c^*_-\}$. Why does Condition 21 ($c^*_- c^*_+ + 1 = \square$) hold simultaneously with Condition 19 at $\lambda = -2$? We give two complementary explanations: a universal regularity factorization that identifies the exact relation type and its genus-$3$ full-root curve $\mathcal{H}$, and a transverse Mordell–Weil section intersection between two elliptic fibrations on Branch 1.

Theorem 3 (Universal Factorization of $R_4(p, q, e_-, e_+)$, the Type-$(5, 2)$ Curve $\mathcal{H}$, and Finiteness)

1. Universal pair-symmetric factorization of $R_4(p, q, e_-, e_+)$: Let $\{p, q, u, v\}$ be any quadruple and let $e_-, e_+$ be the two roots of $R_5(p, q, u, v, x) = 0$. Then in $\mathbb{Q}(p, q, u, v)$,

   $$ R_4(p, q, e_-, e_+) = \frac{\Big((p q u v)^2 - 6 p q u v - 4(p q + u v) - 3\Big) \cdot G(p, q, u, v)}{(p q u v - 1)^4}, $$

   where the first factor $F_{\text{sym}}(p, q; u, v) := (p q u v)^2 - 6 p q u v - 4(p q + u v) - 3$ is invariant under exchanging the pair $\{p, q\}$ with its complement $\{u, v\}$. Consequently, whenever $F_{\text{sym}}(p, q; u, v) = 0$, both $R_4(p, q, e_-, e_+) = 0$ and $R_4(u, v, e_-, e_+) = 0$ hold simultaneously, and if $p q + 1 = \square$ (or $u v + 1 = \square$), then $e_- e_+ + 1 = \square$ is forced automatically by Rule (Q).

2. The relation type $(5, 2)$ of $\mathcal{S}_7^{(20/21)}$: Indexing $\mathcal{S}_7^{(20/21)}$ as $(x_0, \dots, x_6) = (c, c^*_+, d_-, a, b, c^*_-, d_+)$ with missing pair $(0, 1) = (c, c^*_+)$, $\mathcal{S}_7^{(20/21)}$ satisfies five $R_4 = 0$ relations and two $R_5 = 0$ relations:

   $$ \begin{aligned} \mathcal{R}_4 &= \big\{\{0, 2, 3, 4\}, \; \{0, 3, 4, 6\}, \; \{0, 2, 5, 6\}, \; \{1, 2, 3, 5\}, \; \{1, 4, 5, 6\}\big\} \\ &= \big\{\{a, b, c, d_-\}, \; \{a, b, c, d_+\}, \; \{c, d_-, d_+, c^*_-\}, \; \{a, d_-, c^*_-, c^*_+\}, \; \{b, d_+, c^*_-, c^*_+\}\big\}, \\ \mathcal{R}_5 &= \big\{\{1, 2, 3, 4, 6\}, \; \{2, 3, 4, 5, 6\}\big\} = \big\{\{a, b, d_-, d_+, c^*_+\}, \; \{a, b, d_-, d_+, c^*_-\}\big\}. \end{aligned} $$

   At $\mathcal{S}_7^{(20/21)}$, the $7 \times 7$ Jacobian matrix of these $7$ polynomials has rank $6$, so the local dimension of the relation variety is $\text{dim} = 7 - 6 = 1$, and the minimal number of pair-square generators under Rules (Q) and (P) is $\text{gens} = 2$ (generated by $\{a c + 1, c d_- + 1\}$). This relation type $(|\mathcal{R}_4|, |\mathcal{R}_5|) = (5, 2)$ is distinct from all $18$ types in Paper 6 [2] and the types in Papers 10 and 17 [3, 4].

3. Collapse to the genus-$3$ full-root hyperelliptic curve $\mathcal{H}$ and finiteness: On Branch 1, the first three $R_4$ relations and both $R_5$ relations hold identically, and the remaining two $R_4$ relations are simultaneously enforced by $F_+(a, b, d_-, d_+) := P_4^2 - 6 P_4 - 4(a d_+ + b d_-) - 3 = 0$ (up to swapping $d_- \leftrightarrow d_+$). Eliminating $s t$ via $(s t)^2 = (a c + 1)(b c + 1)$ factors this locus over $\mathbb{Q}$ into two conjugate components $\mathcal{C}_+(a, \lambda) \mathcal{C}_-(-a, -\lambda) = 0$, where $\mathcal{C}_+(a, \lambda) = A_2(\lambda) a^2 + A_1(\lambda) a + A_0(\lambda) = 0$ has coefficients

   $$ \begin{aligned} A_2(\lambda) &= 8(\lambda^2 - 1)^2 (3 \lambda^3 + \lambda^2 + 5 \lambda - 1)(5 \lambda^5 - 11 \lambda^4 + 2 \lambda^3 - 22 \lambda^2 - 7 \lambda + 1), \\ A_1(\lambda) &= (\lambda^2 - 1)(25 \lambda^{12} - 30 \lambda^{10} - 1553 \lambda^8 - 4356 \lambda^6 - 3033 \lambda^4 + 770 \lambda^2 - 15), \\ A_0(\lambda) &= -2(\lambda^2 + 3)(5 \lambda^2 - 1)(3 \lambda^3 - \lambda^2 + 5 \lambda + 1)(5 \lambda^5 + 11 \lambda^4 + 2 \lambda^3 + 22 \lambda^2 - 7 \lambda - 1). \end{aligned} $$

   Its discriminant with respect to $a$ is

   $$ \Delta_a(\lambda) = A_1(\lambda)^2 - 4 A_2(\lambda) A_0(\lambda) = (\lambda^2 - 1)^6 (5 \lambda^4 + 10 \lambda^2 + 1)^3 (5 \lambda^4 - 22 \lambda^2 + 33). $$

   Thus $a \in \mathbb{Q}(\lambda, W)$ is rational on the smooth genus-$3$ hyperelliptic curve

   $$ \mathcal{H} : \quad W^2 = (5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^4 - 22 \lambda^2 + 33) = 25 \lambda^8 - 60 \lambda^6 - 50 \lambda^4 + 308 \lambda^2 + 33. $$

   Remarkably, on the non-degenerate root $a_+ = \frac{-A_1(\lambda) + (\lambda^2 - 1)^3(5 \lambda^4 + 10 \lambda^2 + 1) W}{2 A_2(\lambda)}$ (the second root $a_-$ corresponds to the degenerate boundary $c^*_+ = 0$), the square condition $a_+ c_+ + 1 = \square$ holds identically in $\mathbb{Q}(\mathcal{H})$:

   $$ a_+^2 + C_0(\lambda) a_+ + r(\lambda)^2 = \left(\frac{U(\lambda) - V(\lambda) W}{2}\right)^2 \pmod{W^2 - (5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^4 - 22 \lambda^2 + 33)}, $$

   where

   $$ \begin{aligned} U(\lambda) &= \frac{(\lambda - 1)(5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^5 - 7 \lambda^4 + 26 \lambda^3 - 14 \lambda^2 + 17 \lambda + 5)}{8(3 \lambda^3 + \lambda^2 + 5 \lambda - 1)(5 \lambda^5 - 11 \lambda^4 + 2 \lambda^3 - 22 \lambda^2 - 7 \lambda + 1)}, \\ V(\lambda) &= \frac{(\lambda + 1)(5 \lambda^5 - 17 \lambda^4 - 6 \lambda^3 - 34 \lambda^2 - 15 \lambda + 3)}{8(3 \lambda^3 + \lambda^2 + 5 \lambda - 1)(5 \lambda^5 - 11 \lambda^4 + 2 \lambda^3 - 22 \lambda^2 - 7 \lambda + 1)}. \end{aligned} $$

   Consequently, all $20$ required pairwise square roots lift to rational functions in $\mathbb{Q}(\mathcal{H})$ without any further field extension! Because $\mathcal{H}$ is smooth and geometrically irreducible of genus $3$, Faltings' theorem [6] implies that only finitely many rational almost septuples lie on this component.

4. Transverse section-intersection mechanism at $\lambda = -2$: Evaluating $F_+$ along the $\mathcal{C}_{5\lambda}$-section $a_3(\lambda)$ of Theorem 2(3) gives the exact collapse

   $$ F_+\big(a_3(\lambda)\big) = \frac{(\lambda - 2)(\lambda + 2)(5 \lambda^4 + 10 \lambda^2 + 1)^2}{16 \lambda^2(\lambda - 1)^2(\lambda + 1)^2}, $$

   which vanishes at $\lambda = \pm 2$. Equivalently, parameterizing Condition 21 ($c^*_- c^*_+ + 1 = \square$) on Branch 1 via $\frac{4 \lambda(8 \lambda(\lambda^4 - 1) S_{ab} + 7 \lambda^6 + 31 \lambda^4 + 29 \lambda^2 - 3)}{(\lambda^2 - 1)^3} = \frac{\kappa^2(5 \lambda^2 + 2 \lambda + 1) - (5 \lambda^2 - 2 \lambda + 1)}{2 \kappa}$ makes both Condition 19 and Condition 21 hold identically in $\mathbb{Q}(\lambda, \kappa)$, while $S_{ab}^2 - 4(r^2 - 1) = \square$ defines a second elliptic surface $\mathcal{E}_{\text{Cond21}}(\lambda, \kappa)$ with $5$ built-in rational sections $K_0 = 0$, $K_{1, 2} = \pm \frac{\lambda + 1}{\lambda - 1}$, $K_{3, 4} = \pm \frac{5 \lambda^3 - 7 \lambda^2 + 3 \lambda - 1}{5 \lambda^3 + 7 \lambda^2 + 3 \lambda + 1}$. The Mordell–Weil chord $K_0^+ + K_3^-$ gives

   $$ \kappa(\lambda) = \frac{(\lambda + 1)(5 \lambda^8 + 68 \lambda^7 + 164 \lambda^5 - 14 \lambda^4 + 44 \lambda^3 + 8 \lambda^2 - 20 \lambda + 1)}{(\lambda - 1)(5 \lambda^8 + 12 \lambda^7 - 84 \lambda^5 - 14 \lambda^4 - 188 \lambda^3 + 8 \lambda^2 + 4 \lambda + 1)}, $$

   whose induced trace $S_{ab, K_0^+ + K_3^-}(\lambda)$ intersects $S_{ab, 3}(\lambda) = a_3(\lambda) + b_3(\lambda)$ transversely at $\lambda = -2$:

   $$ S_{ab, 3}(\lambda) - S_{ab, K_0^+ + K_3^-}(\lambda) = -\frac{(\lambda - 1)^2(\lambda + 1)^2(\lambda + 2)(\lambda^2 + 3)(5 \lambda^4 + 10 \lambda^2 + 1)^2(7 \lambda^4 + 10 \lambda^2 - 1) P_{14}(\lambda)}{\text{den}(\lambda)}. $$

Proof. All algebraic factorizations, resultants, and modular polynomial identities in Parts 1–4 are exact identities in $\mathbb{Q}[\lambda, a, W]$ and are verified symbolically in verify_paper.py. For the genus of $\mathcal{H}$, the octic polynomial $P_8(\lambda) = (5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^4 - 22 \lambda^2 + 33)$ has discriminant $2^{56} \cdot 3^4 \cdot 5^6 \cdot 11^4 \neq 0$, hence $8$ distinct simple roots over $\overline{\mathbb{Q}}$, so $\mathcal{H} : W^2 = P_8(\lambda)$ is smooth of geometric genus $g(\mathcal{H}) = \frac{8 - 2}{2} = 3$. Note also that $\mathcal{H}$ maps by $\lambda \mapsto L = \lambda^2$ onto the elliptic curve $\mathcal{E}^{(1)} : W^2 = (5 L^2 + 10 L + 1)(5 L^2 - 22 L + 33)$, which has minimal Weierstrass model $Y^2 + X Y + Y = X^3 - 6 X + 8$ of conductor $550$, torsion $\mathbb{Z}/2\mathbb{Z}$, and Mordell–Weil rank $1$. Searching $L = \lambda^2$ up to naive height $50,000$ on $\mathcal{E}^{(1)}$ via hyperellratpoints yields only $L = 1$ ($\lambda = \pm 1$, degenerate) and $L = 4$ ($\lambda = \pm 2, W = \pm 55$, which gives $\mathcal{S}_7^{(20/21)}$ and $-\mathcal{S}_7^{(20/21)}$). $\square$

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6. Universal $(V, K, d)$ coordinates and square completions for canonical $S$-chains

To complement the two-sided sub-quadruple theory of Sections 3–5, we record the exact algebraic structure of contiguous $4$-term translation chains $x(P_0 + n S)$ on the elliptic curve $E_T : Y^2 = (X + r^2 - 1)(X + s^2 - 1)(X + t^2 - 1)$ ($X = a b c \, x$) of an arbitrary Diophantine triple $T = \{a, b, c\}$ with $\Pi = a b c$ and $\Omega = r s t$, where $P_0 = (0, \Pi)$ and $S = (1, \Omega) = 2 R_0$ with $R_0 = \big(1 + r s + r t + s t, \; (r + s)(r + t)(s + t)\big) \in E_T(\mathbb{Q})$.

Theorem 4 (Universal $(V, K, d)$ Coordinates and Square Completions for $S$-Chains)

Let $T = \{a, b, c\}$ be any Diophantine triple with $d_\mp = x(P_0 \pm S) = a + b + c + 2 \Pi \mp 2 \Omega$ and $W_\mp := \Pi d_\mp - 1$.

1. Shift-invariant product and universal square completion of $F_2(z)$: For any point $Q \in E_T$ with $z = x(Q)$, the product of its two $S$-translates $x(Q \mp S) = e_\mp(a, b, c, z)$ is $x(Q - S) x(Q + S) = \frac{(z - d_+)(z - d_-)}{(\Pi z - 1)^2}$. Thus $x(Q - S) x(Q + S) + 1 = \frac{F_2(z)}{(\Pi z - 1)^2}$ where $F_2(z) := (\Pi z - 1)^2 + (z - d_+)(z - d_-)$ satisfies the universal square completion

   $$ W_-^2 \, F_2(z) = \Big(W_-^2 + (\Pi W_- - 2 \Omega)(z - d_-)\Big)^2 + (z - d_-)^2 D_4, $$

   with $D_4 = W_- W_+ - 4 \Omega^2 = 4 \Omega^2 d$, where $d := X(2 S) - 1 = \frac{W_- W_+}{4 \Omega^2} - 1$ is given in terms of $(\alpha, \beta, \gamma) = (1/r, 1/s, 1/t)$ by Heron's formula:

   $$ d = -\frac{(\alpha + \beta + \gamma)(\alpha + \beta - \gamma)(\alpha - \beta + \gamma)(-\alpha + \beta + \gamma)}{4 \alpha^2 \beta^2 \gamma^2}. $$

2. Closed-form chain coordinates and contiguous $4$-term chains: In terms of the three dimensionless invariants $u_0 := \frac{W_-}{2 \Omega}$ ($V := u_0^2$), $K := W_- - d = \Pi d_- - X(2S)$, and $d = X(2S) - 1$, we have $\Pi = \frac{d + 1 - u_0^2}{2 u_0}$, $\Omega = \frac{K + d}{2 u_0}$, and for all $m, n \in \mathbb{Z} \setminus \{0\}$, $x_m x_n + 1 = \frac{(d + 1 - V)^2 + 4 V X_m X_n}{(d + 1 - V)^2}$, where $X_n = \Pi x(P_0 + n S)$ are rational functions of $(V, K, d)$ alone:

   $$ X_1 = K + d + 1, \quad X_2 = \frac{K + V}{V}, \quad X_3 = \frac{d\big(K(V - 1) + V d\big)}{K^2}, \quad X_4 = \frac{(K^2 + K - V d)\big(K(d + 1) - V d\big)}{(K - V d)^2}. $$

   For the contiguous $4$-term chain $\mathcal{S}_7^{(1, 2, 3, 4)} = \{a, b, c, d_-, x_2, x_3, x_4\}$, $18$ of the $21$ conditions hold identically, and the $19$th ($d_- x_3 + 1 = \square$) and $20$th ($x_2 x_4 + 1 = \square$) conditions reduce to:

   $$ \begin{aligned} \text{num}_{19}(V, K, d) &= \Big(K(V + d - 1) + 2 V d\Big)^2 + 4 d \Big(K(V - 1) + V d\Big)^2 = \square, \\ \text{num}_{20}(V, K, d) &= \Big(2 K^2 + (K - V d)(V - d + 1)\Big)^2 + 4 d \Big(K^2 + K - V d\Big)^2 = \square. \end{aligned} $$

Proof. Verified by direct polynomial expansion in verify_paper.py and THEOREMS_AND_PROOFS.md (Theorems 10 and 13). $\square$

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7. Limitations and scope

• No rational Diophantine septuple ($21/21$) is claimed. Our explicit tuple $\mathcal{S}_7^{(20/21)}$ satisfies $20$ of the $21$ conditions and adds a new relation type $(|\mathcal{R}_4|, |\mathcal{R}_5|) = (5, 2)$ ($\text{dim} = 1, \text{gens} = 2$) to the census of known almost septuples [2, 3, 4].
• Finiteness on $\mathcal{H}$ vs. Problem 3.3: Theorem 3(3) proves that the $1$-dimensional relation variety $\mathcal{C}_+(a, \lambda) = 0$ on Branch 1 has a smooth genus-$3$ full-root cover $\mathcal{H}$, so by Faltings' theorem it carries only finitely many rational almost septuples. As in Paper 6 [2, Theorem A] for $E_0$, this rules out deforming $\mathcal{S}_7^{(20/21)}$ inside its full $7$-relation type $(5, 2)$ to answer Problem 3.3, while leaving open $2$-dimensional sub-systems obtained by dropping one relation (such as the $(3, 2)$ system of AS18 or the $2$-dimensional surface $\mathcal{E}_{\text{Br1}}(\lambda, \mu)$ where $19/21$ conditions hold identically).
• Completeness on $\mathcal{H}$: Faltings' theorem is ineffective; our height-$50,000$ search on the rank-$1$ elliptic quotient $\mathcal{E}^{(1)}$ of $\mathcal{H}$ is a bounded point search, not a complete Chabauty–Coleman determination of $\mathcal{H}(\mathbb{Q})$.

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8. Reproducibility

The supplement (supplement.zip) is self-contained and requires only Python 3 and SymPy (python3 verify_paper.py, running in $\sim 5$ seconds on a single CPU core). It checks:

1. All symbolic identities of Theorem 1 (linearity of $c^*_\mp$ in $S_{ab}$, the three universal $S_{ab}$-square completions, and $R_4(c, d_-, d_+, c^*_-) = 0$ on Branch 1);
2. The rational circle family $a_{\text{circ}}(\lambda)$ of Theorem 2 and all $15$ pairwise squares of the displayed all-positive non-torsion sextuples;
3. All $21$ pairs of $\mathcal{S}_7^{(20/21)}$, its five $R_4 = 0$ and two $R_5 = 0$ relations, the universal symmetric factorization of $R_4(p, q, e_-, e_+)$, the discriminant $\Delta_a(\lambda)$ of $\mathcal{C}_+(a, \lambda) = 0$, the exact square identity $a_+ c_+ + 1 = \left(\frac{U(\lambda) - V(\lambda) W}{2}\right)^2$ on the genus-$3$ curve $\mathcal{H}$, and the transverse section intersection $S_{ab, 3}(-2) = S_{ab, K_0^+ + K_3^-}(-2)$ of Theorem 3;
4. The $(V, K, d)$ chain identities of Theorem 4, together with the full theoretical compendium THEOREMS_AND_PROOFS.md and the Mordell–Weil chord scripts exp9 and exp10.

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