## 1. Problem and contribution

A rational Diophantine $m$-tuple is a set of $m$ distinct nonzero rationals with $a_ia_j+1$ a square for all $i\ne j$. No rational septuple is known [1, Problem 3.2]. Rational sextuples were first found by search [8], and the known positive-dimensional families stop at six elements [2,3]. Paper 17 [4] classified the closures of one seven-element construction, a regular four-chain over a pair. It found ten septuple surfaces up to sign, with no rational points of small height.

This paper replaces the single construction by an exhaustive census of *regular build orders*. Elements are added one at a time. Each new element is either free with at most two square conditions, or a root of the regular-quadruple relation over a triple, or a root of the regular-quintuple relation over a quadruple. Every $\mathbb F_2$-linear consequence of these relations for the square classes is taken into account. The census measures, for each order, the base dimension $D$ and the number $C$ of square roots that are still missing.

1. **Pure orders (Section 3).** For $n=5,6,7$ elements the least cost is $C=0,1,3$, attained at $D=3$. At $n=7$ we have $C\ge D$ for every $D\le7$ (Table 1). The $28$ non-degenerate cheapest septuple orders have exact function-field rank $3$, so they have no hidden identity. On every coordinate line each is a $(\mathbb Z/2)^3$-cover of geometric genus $\ge145$.
2. **Closures (Section 4).** If the seventh element is a common root of two relations, the septuple locus is a hypersurface $N=0$ in a rational base, and $C=0$ on it. This gain is largely illusory. Of the $239$ non-degenerate closure orders on a $4$-dimensional base ($6$ hypergraph classes), $2$ classes are pure $(3,3)$ orders in disguise, because $R_5$ is symmetric in its five arguments. On three of the other $4$ classes the $t$-discriminant of $N$ on a coordinate plane has only even multiplicities, which is the signature of a $2$-power cover of a rational threefold (Section 4.1); the fourth computation did not finish. The $1511$ closure orders on a $3$-dimensional base give $39$ genuine septuple surfaces. $8$ of them are the $k=0$ classes of [4]; the other $31$ are new.
3. **Elliptic seeds (Section 5).** The fourth elements of a triple $T$ are $d_{j,k}=x(jP+kS)/abc$ on $E_T$. Over the function field, the two $R_5$ roots over $T\cup\{d_{j,k}\}$ are $d_{j,k\pm1}$, and only consecutive pairs give squares. So regular quintuple extension is translation by $S$. The census of Section 3 reaches only the chain through $d_{1,0}=0$. Adding the step $d_{3,0}=x(3P)/abc$ makes the census recover the frame of Dujella–Kazalicki–Mikić–Szikszai [3] as a genuine $C=0$ closure surface, of degree $2$ along a coordinate. At $n=7$, the extended census has $2309$ closure orders with $C=0$ on surfaces and $918$ on threefolds. They give $51$ further genuine septuple surfaces, whose least coordinate degree over $\mathbb Q$ is $21$ (Section 6).
4. **Where the known sextuples are (Section 7).** For each of the $1089$ known rational sextuples we computed the cheapest census order it realises. $1059$ lie on $C=0$ closure loci: $112$ on surfaces, $706$ on threefolds and $241$ on fourfolds. Only $30$ need a pure order with $C>0$. The two largest classes are the hypergraph of Dujella–Kazalicki–Petričević [2] ($363$ sextuples) and $\{Q,e_+,e_-\}$ with two regular quadruples ($285$). So at six elements the census frames do contain the known points.
5. **Searches and their calibration (Section 6).** We searched coordinate fibres of the septuple threefolds up to height $6$ and of the $39$ septuple surfaces up to height $12$, about $1.8\cdot10^6$ fibres. No non-degenerate rational point appeared. The same code at the same heights, run on populated six-element frames, finds $156$ sextuples on the DKMS surface but $0$ or $1$ on five other populated frames, and the parameters of the known points show why: almost none lie in the search box. Silence at these heights is therefore evidence only on frames with a coordinate of degree $\le2$. No septuple surface has one; the least coordinate degree over $\mathbb Q$ is $21$. Section 6 also reports a defect in our first search script, which skipped roots, and its repair.

All negative statements here are limited to the build orders enumerated.

We do not show that any of the surfaces has few rational points, and we do not determine their Kodaira dimension.

## 2. Build orders and square-class forcing

Write $[u]$ for the class of $u\in K^\times$ in $K^\times/K^{\times2}$, and $[uv]:=[uv+1]$ for a pair. Two relations are used:
$$R_4(a,b,c;x)=(a+b-c-x)^2-4(ab+1)(cx+1),$$
$$R_5(a,b,c,d;x)=(\alpha x+\beta)^2-4\Pi\,(dx+1),$$
where $\alpha=abcd-1$, $\beta=2abc+a+b+c-d$ and $\Pi=(ab+1)(ac+1)(bc+1)$. $R_4$ is symmetric in all four arguments [5]. $R_5$ is symmetric in all five [6]. For $x$, $R_4=0$ has the roots $a+b+c+2abc\pm2\sqrt{\Pi}$. $R_5=0$ has the roots
$$x_\pm=\frac{(a+b+c+d)(abcd+1)+2\sigma_3\pm2\sqrt{\Pi_6}}{(abcd-1)^2},$$
where $\sigma_3=abc+abd+acd+bcd$ and $\Pi_6$ is the product of the six $uv+1$ over the quadruple.

**Lemma 1 (forcing rules; [4, Lemma 2]).** Let $K$ be a field of characteristic $0$. Take a point over $K$ at which all the values $uv+1$ involved are nonzero.
(i) If $R_4(a,b,c;d)=0$, then $[ab]=[cd]$ for each of the three matchings. Also $[uv]+[uw]+[vw]=0$ for each of the four sub-triples, because the discriminant in the fourth variable is a square.
(ii) If $R_5(a,b,c,d;e)=0$, then $[uv]+[uw]+[vw]+[yz]=0$ for each of the ten splits $\{u,v,w\}|\{y,z\}$. Also $\sum_{\text{6 pairs}}[\cdot]=0$ over each of the five four-subsets.

Each matching or split identity is the relation itself, written as a difference of a square and a product. Each discriminant identity is the statement that a quadratic with a root in $K$ has a square discriminant.

**Build orders.** An order is a sequence of steps. Each step adds element $i$:
- $F(T)$ with $|T|\le2$: a free element subject to $xt+1=\square$ for $t\in T$. For $|T|=0$ the element is a free parameter. For $|T|=1$ put $x=(s^2-1)/t$. For $|T|=2$ the conic $\{t_1x+1=\square,\ t_2x+1=\square\}$ has the rational point $x=0$, and we use the parametrisation from it. Each $F$ step raises the base dimension $D$ by $1$ and keeps the base rational.
- $R_4(T)$ with $|T|=3$: a root of $R_4(T;x)=0$. It is rational exactly when $[\Pi_T]=0$.
- $R_5(Q)$ with $|Q|=4$: a root of $R_5(Q;x)=0$. It is rational exactly when $[\Pi_6(Q)]=0$.

The census (`cen.py`) works in $\mathbb F_2^{\binom n2}$ with one basis vector per pair. It keeps the span $W$ of all relations from Lemma 1 for the relation steps so far, together with the vectors $e_{xt}$ of the conditions imposed by $F$ steps. At an $R$ step, if the discriminant vector is not in $W$, a square root has to be adjoined: we add it to $W$ and increase a counter `cov` by $1$. The cost of an order is
$$C=\mathrm{cov}+\binom n2-\dim W .$$
This is the number of independent square roots that must be adjoined over the $D$-dimensional rational base for every $uv+1$ to become a square.

**Lemma 2.** The true number of independent missing classes in the function field of the base is at most $C$. If $C=0$, then on a dense open subset of the base every rational point gives $n$ rationals with all $\binom n2$ values $uv+1$ squares.

*Proof.* Every vector in $W$ is a valid relation by Lemma 1, so $C$ is an upper bound. If $C=0$, every discriminant is forced, so each root is a rational function on the base. Every pair class then lies in $W$, so it vanishes at each point where the forcing applies. That covers all points off the zero loci of the $uv+1$, of the denominators, and of the leading coefficients. $\square$

The bound can fail to be sharp only because of an identity among the classes that Lemma 1 does not imply. Section 3 rules this out for the cheapest orders by computing the exact rank in $\mathbb Q(t)$ along a random line (`cnum.pure`). That computation factors every $uv+1$, records the odd-multiplicity factors and the class of the leading coefficient, and takes the $\mathbb F_2$-rank.

**Closures.** A *closure* $\mathrm{CL}(S_1|S_2)$ adds the seventh element $x$ as a common root of two relations, with supports $S_1,S_2$ of size $3$ or $4$. We write $q_j(x)=R_{|S_j|+1}(S_j;x)$. The locus is $N=\mathrm{Res}_x(q_1,q_2)=0$ in the base. At the step $x$, all relations of both supports enter $W$. The dimension of the locus is $D-1$. On a component where $\deg\gcd(q_1,q_2)=1$, $x$ is the rational function $-(a_1e_2-a_2e_1)/(a_1b_2-a_2b_1)$, where $q_j=a_jx^2+b_jx+e_j$. So Lemma 2 applies with $C=0$ on that component.

## 3. Pure orders

Table 1 lists the least cost $C$ for each $(n,D)$ over all orders whose steps are $F(T)$ with $|T|\le2$, $R_4$ and $R_5$ (`cen.py 7 3`, output `cen7_3.out`).

| $n$ | $D=2$ | $D=3$ | $D=4$ | $D=5$ | $D=6$ | $D=7$ |
|---|---|---|---|---|---|---|
| 2 | 0 | | | | | |
| 3 | | 0 | | | | |
| 4 | | 0 | 1 | | | |
| 5 | | 0 | 1 | 3 | | |
| 6 | | 1 | 2 | 4 | 6 | |
| 7 | | 3 | 4 | 6 | 8 | 10 |

Table 1. Least cost $C$ of a pure order with $n$ elements on a $D$-dimensional base.

The base dimension is at least $3$ once $n\ge3$, because the first three elements of any order need three parameters. At $n=5$ the cost is $0$ at $D=3$, which recovers the regular quintuples over a regular quadruple [6]. At $n=6$ there are exactly $5$ orders with $(D,C)=(3,1)$. Each is a regular quadruple $\{0,1,2,3\}$, a regular fifth element $4$, and a sixth element that is a regular fifth element of a sub-quadruple of $\{0,\dots,4\}$. Only $6$ of the $1089$ known sextuples realise this frame (Section 7). At $n=7$ the minimum cost is $C=3$, at $D=3$, and $75$ orders attain it. In $30$ of them a discriminant has to be adjoined ($\mathrm{cov}=1$). Their base is a double cover rather than a rational variety, and we do not analyse them further. Of the $45$ orders with $\mathrm{cov}=0$, `cnum.pure` finds $17$ degenerate on a random line: two elements coincide, an element is $0$, some $uv+1$ vanishes, or $abcd=1$ at a quintuple step. The remaining $28$ all have the form
$$F,\ F(0),\ F(0,1),\ R_4(012),\ R_5(0123),\ R_5(Q_1),\ R_5(Q_2),\qquad Q_1,Q_2\subset\{0,\dots,5\},$$
which is a regular quintuple followed by two regular fifth elements of sub-quadruples.

**Proposition 3.** Let $X$ be one of the $28$ non-degenerate orders. Its three unforced classes are independent in $\overline{\mathbb Q}(\text{base})^\times/\square$. Fix two of the three base coordinates at general values. The septuple locus over the remaining coordinate line is then a $(\mathbb Z/2)^3$-cover of $\mathbb P^1$. Its geometric genus is at least the value observed on two random such lines. Over all orders and coordinates these values run from $145$ to $717$, and the minimum over coordinates for a single order lies between $145$ and $281$.

*Proof sketch.* `fibgen.py` builds the order over $\mathbb Q$ on each coordinate line. It computes the geometric rank $k$ of the classes from the odd-multiplicity factors of each $uv+1$ over $\mathbb Q$ and the parity at $\infty$. It counts the branch points $B$ (the geometric roots of those factors, plus $\infty$ if odd) and applies Riemann–Hurwitz to the $(\mathbb Z/2)^k$-cover, $g=1-2^k+B\,2^{k-2}$. For all $28$ orders and all coordinates, $k=3$ (`fg7.out`). Restriction to a line can only create relations, so the geometric rank $k=3$ found on a line also holds on the base. Specialising from a general line to a random one can only merge branch points, so the observed $g$ is a lower bound for the general fibre. Over $\mathbb F_p(t)$, `cnum.pure` gives the same rank $3$ for the arithmetic classes, so $C=3$ is exact. With the base triple in the Lasić chart instead (`lfib.py`, `lfg7.out`), $g\ge293$. $\square$

As a sanity check, `fibgen.py` applied to the quintuple threefold $\{a,b,c,d_+,d_-\}$ (one open class, $d_+d_-+1$) gives $k=1$ and $B=4$, hence $g=1$, in both charts. This is the elliptic fibration used in [9].

So among pure regular build orders, the cheapest septuple needs three independent square roots over a rational threefold, and the cost never drops below the dimension: $C\ge D$ for $D\le7$. To reach $C=0$, a closure is required.

## 4. Closures

All closure statistics come from `cencl.py 7 0` (pure steps, file `cl_7.pkl`) and the classification scripts `dedup.py`, `dedup2.py`, `hgcmp.py`, `rep2.py`. Two orders are in the same *class* if their relation hypergraphs (the $R_4$ $4$-sets and $R_5$ $5$-sets on the seven elements, including the two closure edges) agree up to $S_7$. For each class we take up to six representatives and run `cnum.analyse`. It factors $N$ along a random line and along each coordinate line over $\mathbb Q$, and discards a factor as *degenerate* if it divides one of the degeneracy polynomials of Section 3, or if $x\in\{0\}\cup\{e_i\}\cup\{-1/e_i\}$ is a common root on it. A class is *genuine* if some representative has a non-degenerate factor. On genuine classes, `cnum.ptcheck` reconstructs $x$ at the $\mathbb F_p$-points of $N=0$ on random lines ($p=100003$) and confirms that all $21$ values $uv+1$ are nonzero squares there. `fullhg.py` confirms at $\mathbb F_p$-points ($p=1000003$) that each representative satisfies exactly its recorded relations and no further $R_4$ or $R_5$ relation.

### 4.1 Closures on a four-dimensional base

There are $258$ closure orders with $C=0$ and $D=4$. $19$ are degenerate on every factor, and the remaining $239$ form $6$ classes (`hgcmp.out`).

**Proposition 4.** Two of the six classes ($48$ orders) are pure orders of cost $(C,\mathrm{cov})=(3,1)$ on a three-dimensional base. Each is a regular quintuple followed by two further $R_5$ roots, so the closure gives no saving over Table 1.

*Proof.* Take order #12, $F,F(0),F(0,1),F(0,1),R_4(012),R_5(0124),\mathrm{CL}(0145|2345)$. Because $R_5$ is symmetric in all five arguments, the closure condition $R_5(0,1,4,5;x)=0$ says that $x$ is the second $R_5$ root over $\{0,1,4,5\}$, and $R_5(2,3,4,5;x)=0$ then says that the free element $3$ is an $R_5$ root over $\{2,4,5,x\}$. After relabelling, this is the pure order $F,F(0),F(0,1),R_4(012),R_5(0123),R_5(0134),R_5(2345)$. By the forcing rules (`costchk.py`) it has $D=3$, $\mathrm{cov}=1$, $C=3$: one discriminant is adjoined, and the pairs $\{0,6\},\{1,6\}$ are unforced. The second class is handled the same way. `hgcmp.py` lists the pure realisations of each class hypergraph and finds them exactly for these two classes. $\square$

The other four classes ($191$ orders) have no pure realisation with $\mathrm{cov}=0$. Their least coordinate degrees are $16$ ($3$ classes) and $36$ ($1$ class). We do not count them as new. For the class of $F,F(0),F(0,1),F(0,1),R_4(013),R_4(034),\mathrm{CL}(0145|2345)$, we freed two coordinates and fixed the others at random $\mathbb F_p$-values (`surfg.py`). Every root of the $t$-discriminant of $N(t,s)$ then has even multiplicity. This is what a pull-back of a cover of the $s$-line looks like, and it makes the Riemann–Hurwitz bound vacuous. Reordering the same class gives the pure order $F,F(0),F(0,1),R_4(012),R_4(023),R_5(0134),R_5(2345)$, which has $(D,\mathrm{cov},C)=(3,2,4)$ (`costchk.py`; `purechk.py` confirms that the second $R_4$ discriminant is not a square on a random line). Two of the three other classes give the same picture ($R=0$ odd-multiplicity discriminant roots in `surfg_0.out`, `surfg_2.out`, with $t$-degrees $96$ and $48$). For the last class (`surfg_1.out`) the discriminant computation did not finish within its time limit. The evidence is that the $D=4$ closures are $2$-power covers of rational threefolds in disguise. We have not proved this, and for one class we have no evidence either way.

### 4.2 Closure surfaces

There are $1511$ closure orders with $C=0$ and $D=3$, forming $52$ classes (`dedup2.py`, `rep2_*.out`). $13$ classes are degenerate on every factor for the first six representatives. The other $39$ are genuine: `ptcheck` finds all $21$ values square at every non-degenerate $\mathbb F_p$-point tested, and `fullhg.py` finds no relations beyond the recorded ones. Along the best coordinate line, the least degree of a genuine factor is between $24$ and $192$ (median $78$).

Paper [4] classified the closures of a regular chain over a pair. Translating its supports into census labels (`map17.py`, `map17.out`) identifies its eight $k=0$ surfaces with the census classes $33$ and $45$–$51$, all genuine here. This is an independent check of both classifications. Its $k=1$ supports leave a pair unforced and do not occur among the $C=0$ classes. The other $31$ of the $39$ septuple surfaces are not in [4].

**Theorem 5.** Each of the $39$ genuine classes defines a surface $X\subset\mathbb A^3$ (a component of $N=0$) such that on a dense open subset of $X$, every rational point gives seven distinct nonzero rationals with all $21$ values $uv+1$ nonzero squares.

*Proof.* On a genuine component, $\deg\gcd(q_1,q_2)=1$, so $x$ is a rational function on $X$ (Section 2). All classes are forced by Lemma 1, so Lemma 2 applies on the complement of the degeneracy polynomials, which do not vanish identically on $X$ because the factor is genuine. $\square$

The $\mathbb F_p$-checks are consistency checks for the forcing, not part of the proof.

## 5. The elliptic seed step

For a triple $T=\{a,b,c\}$ let $E_T:\ Y^2=(X+bc)(X+ac)(X+ab)$, with $P=(0,abc)$ and $S=(1,\sqrt{\Pi_T})$, $\Pi_T=(ab+1)(ac+1)(bc+1)$. A rational $d$ extends $T$ to a quadruple exactly when $abcd$ is the abscissa of a point of $P+2E_T(K)$ [7]. Write
$$d_{j,k}=x(jP+kS)/abc,\qquad j\ \text{odd},$$
so $d_{1,0}=0$, $d_{1,\pm1}$ are the two $R_4$ roots, and $d_{3,0}=x(3P)/abc$.

**Proposition 6.** Over $\mathbb Q(a,b,c)(\sqrt{\Pi_T})$:
(i) among the $d_{j,k}$ with $j\in\{1,3,5\}$, $|k|\le3$, the value $d_{j,k}d_{j',k'}+1$ is a square in the function field only for $(j',k')=(j,k\pm1)$;
(ii) the two roots of $R_5(a,b,c,d_{j,k};x)$ are $d_{j,k-1}$ and $d_{j,k+1}$ (checked for $j\in\{1,3,5\}$ whenever both lie in the tested range $|k\pm1|\le3$; for $j=1$, $d_{1,0}=0$ appears as the root).

So the $R_5$ step over $T\cup\{d\}$ is translation by $\pm S$ on $E_T$. The census of Section 3 enters each such chain only through an $R_4$ root $d_{1,\pm1}$; that chain passes through $d_{1,0}=0$, which is why $R_5$ over a regular quadruple has the spurious root $0$.

*Verification.* `emine.py` checks (i) and (ii) exactly over $\mathbb F_p(t)$, $p=1000003$, on random lines in the $(a,b,c)$-space: the square test is exact factorisation, and the root identity is exact equality of rational functions.

The point $3P$ is defined for every triple without any square root, so $d_{3,0}$ is a rational function of the triple with $[ad_{3,0}]=[bd_{3,0}]=[cd_{3,0}]=0$ and no other forced class. We add the step $E(T)$, $|T|=3$, which appends $d_{3,0}$ and puts the three vectors $e_{td}$ into $W$ with no discriminant and no relation. In $\mathbb F_2$ it is dominated by $R_4(T)$, which forces strictly more, but its chain $\{d_{3,k}\}$ is a different component and is never degenerate at $k=0$.

**The DKMS frame.** With $E$ steps (`cen2.py`, `cencl2.py`), the six-element closure
$$F,\ F(0),\ F(0,1),\ E(012),\ R_5(0123),\ \mathrm{CL}(0123\,|\,0124)$$
has $C=0$ on a surface, and its resultant has genuine factors (`clsE.py`): at $\mathbb F_p$ points all $15$ values $uv+1$ are squares (`ptcheck`), and along the first coordinate each of its four genuine factors has degree $2$. Here $x$ is a common $R_5$ root over $\{T,d_{3,0}\}$ and $\{T,d_{3,1}\}$, i.e. $x(3P-S)=x(3P+2S)$, forcing $3S$ to be torsion. This is the mechanism of [3]. The same closure with $R_4(012)$ in place of $E(012)$ is degenerate on every component.

## 6. The extended census at seven elements

With the $E$ step, `cen2.py 7 3` gives the same least costs for pure orders as Table 1. The closure census `cencl2.py 7 0` (from $25038$ distinct six-element prefix states) finds $C=0$ closures on surfaces ($2309$ orders) and on threefolds ($918$ orders), and no $C=0$ closure on a four-dimensional base. All classes below contain at least one $E$ edge, since the $E$-free classes are those of Section 4. The classification uses typed hypergraphs up to $S_7$, with $E$ edges distinguished from $R$ edges (`clsE.py`, `clsEp.py`).

**Threefolds.** There are $13$ classes, all genuine, and `ptcheck` gives all $21$ values square at every $\mathbb F_p$-point tested (`clsE7_3_*.out`). Six classes have least coordinate degree $8$ or $16$. In each, that coordinate belongs to a free element that occurs only in the two closure supports, and by the symmetry of $R_5$ the closure reorders into a pure order. For example, $F,F(0),F(0,1),F(0,1),E(012),R_5(0124),\mathrm{CL}(0124|2345)$ becomes $F,F(0),F(0,1),E(012),R_5(0123),R_5(0123),R_5(2345)$, with $(D,\mathrm{cov},C)=(3,1,3)$ (`costchk.py`). The other seven classes have least coordinate degree between $56$ and $368$.

**Surfaces.** There are $72$ classes. Over $\mathbb F_p$, $p=1000003$ (`clsEp7_2_*.out`), $21$ are degenerate on every factor of the first six representatives. The other $51$ are genuine, and `ptcheck` gives all $21$ values square at every point tested. These $51$ are new septuple surfaces in the sense of Theorem 5. For each we computed, over $\mathbb Q$, the degrees of the genuine factors of $N$ on a random line in each base coordinate (`qdeg.py`, files `qdeg7_*.out`, `qdegm_*.out`). The least such degree is $21$ (class $44$, $F,F(0),F(0,1),E(012),R_5(0123),R_5(0123),\mathrm{CL}(0124|0345)$), then $28$ (classes $46,48,59$), $29$ (classes $54,58$) and $32$ (class $63$). The median is $72$, the maximum $1996$, and $13$ of the $51$ classes have a coordinate of degree $\le40$.

So nothing analogous to the degree-$2$ DKMS coordinate appears at seven elements. The cheapest seven-element $E$-closures are again high-degree covers along every coordinate.

**Searches.** `ratsrch.py` takes a class representative and a coordinate. It runs over all values of the other base coordinates of height $\le H$, finds the rational roots of the resultant in the remaining coordinate exactly (`nfroots`), rebuilds the seven elements, and checks all $21$ squares exactly. The runs were:
- the six $D=4$ classes of Section 4.1 at $H\le4$ ($57886$ fibres) and $H\le6$ ($556534$ fibres);
- the $39$ surfaces of Section 4.2 at $H\le12$: $37$ of $39$ runs completed ($1212800$ fibres); runs $14$ and $18$ hit their time limit.

No non-degenerate rational point appeared. The searches have not been extended to the $E$-closure classes.

*A defect and its repair.* The first version of `ratsrch.py` (kept as `ratsrch_v1.py`) wrapped a whole fibre in one exception handler. When the rebuild was undefined at one root, typically $t=0$ where an element vanishes, the other roots of that fibre were skipped and the fibre was logged as `err`. In the surface runs this affected $457936$ fibres. The fixed version handles each root separately and has a mode that redoes exactly the `err` fibres of an old output (`redo.sh`, `redo_*.out`). For $34$ runs the redo is complete: $425184$ fibres, $429576$ rational roots, of which $425184$ (exactly one per fibre) have an undefined rebuild; the other $4392$ roots were checked and none gives a non-degenerate point. The redo of run $3$ ($32752$ fibres) had not finished at submission. In the threefold runs all $10400$ `err` fibres have a parameter equal to $\pm1$ and fail before root finding, because an element vanishes identically there; they are degenerate.

*Calibration.* To see what silence at these heights means, we ran the same search (`ratsrchn.py`, the same code for any $n$ and with $E$ steps; `cal.sh`) on six-element $C=0$ closure frames that are known to carry rational sextuples (Section 7), in coordinate $0$. For each frame `parht.py` inverts the build for every corpus sextuple on it, over all labellings and sign choices, and reports the least height of the parameters a search would have to reach. `qdeg.py` gives the least coordinate degree.

| six-element frame | dim | least coord. degree (coord. 0) | corpus points on frame | corpus points with search height $\le H$ | $H$ | sextuples found |
|---|---|---|---|---|---|---|
| $F,F(0),F(0,1),E(012),R_5(0123),\mathrm{CL}(0123\vert0124)$ (DKMS) | 2 | 2 (2) | 64 | 4 | 12 | 156 |
| $F,F(0),F(0,1),R_4(012),R_4(123),\mathrm{CL}(013\vert0234)$ | 2 | 4 (4) | 23 | 2 | 12 | 1 |
| $F,F(0),F(0,1),R_4(012),R_4(123),\mathrm{CL}(234\vert0134)$ | 2 | 6 (6) | 23 | 0 | 12 | 0 |
| $F,F(0),F(0,1),R_4(012),E(023),\mathrm{CL}(023\vert014)$ | 2 | 6 (6) | 8 | 0 | 12 | 0 |
| $F,F(0),F(0,1),F(0,2),R_4(023),\mathrm{CL}(024\vert0134)$ (DKP) | 3 | 12 (18) | 420 | 0 | 6 | 0 |
| $F,F(0),F(0,1),F(0,1),R_4(012),\mathrm{CL}(234\vert0134)$ | 3 | 4 (4) | 352 | 0 | 6 | 0 |

Table 2. Search calibration on populated six-element frames (`cal_*.out`, `ph_*.out`, `qdeg6cal*.out`). "Sextuples found" counts distinct sextuples up to sign; on the DKMS frame $2$ of the $156$ are in the corpus. The search height of `parht.py` is minimised over the choice of free coordinate, so it bounds from above what a coordinate-$0$ search can recover.

Only the frame with a coordinate of degree $2$ lights up. Frames whose search coordinate has degree $4$ to $18$ return at most one sextuple although they carry $8$ to $420$ known ones, and `parht.py` shows why: almost none of the known points has small parameters (median search heights $56$ to $1264$). On the DKMS frame, too, only $4$ corpus points are within reach (median $65$); what lights it up is that the degree-$2$ coordinate yields many other small points, $154$ of the $156$ found being outside the corpus. The septuple surfaces have least coordinate degree $\ge21$. **The searches above are therefore not evidence that the septuple surfaces or threefolds have no rational points.** They exclude points only in the stated boxes.

## 7. Where the known sextuples lie

The corpus is the $1089$ rational sextuples of [10] ($759$) and of the height-$60$ search of [9] ($330$, disjoint from the first). `cover6.py` computes for each sextuple its typed relation hypergraph: the $4$-sets satisfying $R_4$, the $5$-sets satisfying $R_5$, and the $E$ edges $(T,i)$ with $s_i=d_{3,0}(T)$ exactly. Up to $S_6$ there are $42$ classes. For each class it enumerates every census order (steps $F(T)$, $R_4$, $R_5$, $E$, and a closure as last step) whose relations the sextuple satisfies, computes $(D,C)$ with the forcing rules of Section 2, and keeps the cheapest.

| cheapest $(D,C)$ | $(2,0)$ | $(3,0)$ | $(4,0)$ | $(4,1)$ | $(5,4)$ | $(5,5)$ | $(6,6)$ |
|---|---|---|---|---|---|---|---|
| sextuples | 112 | 706 | 241 | 6 | 15 | 6 | 3 |
| of which with an $E$ edge | 82 | 15 | 2 | 0 | 0 | 0 | 0 |

Table 3. Cheapest census frame of the $1089$ known sextuples (`cover6.out`).

So $1059$ of the $1089$ sextuples are points of $C=0$ closure loci. The largest classes are:
- $363$ sextuples with two $R_4$ and one $R_5$ relation (the hypergraph of [2]), on $F,F(0),F(0,1),F(0,2),R_4(023),\mathrm{CL}(024|0134)$, a threefold;
- $285$ sextuples $\{Q,e_+,e_-\}$ with $e_\pm$ the two regular fifth elements of a quadruple $Q$ and two $R_4$ relations, on $F,F(0),F(0,1),F(0,1),R_4(012),\mathrm{CL}(234|0134)$, a threefold;
- $122$ and $97$ sextuples on two fourfold closures;
- $64$ sextuples on the DKMS surface of Section 5.

Only $6$ sextuples realise the cheapest pure six-element frame $(3,1)$ of Table 1 (`frame6.out`). Taken over pure orders only, the known sextuples sit on frames with $C\ge1$, and the $\{Q,e_+,e_-\}$ sextuples look like points of the $(4,2)$ frame on which $e_+e_-+1$ is an unforced class. That class is indeed not forced on the pure frame: over $\mathbb F_p$ it is a square for $50.7\%$ of random quadruples (`gibbs.py`). The closure frame is where it becomes forced.

On $E_T$ for a triple $T\subset Q$, the fourth element of $Q$ of the $496$ sextuples of the form $\{Q,e_+,e_-\}$ is $d_{3,0}$ for $64$, $d_{5,0}$ for $2$, another $d_{j,k}$ with $|j|\le9$, $|k|\le8$ for $10$, and none of these for $420$ (`e6.py`, `e6.out`). The $64$ are the DKMS class.

Two conclusions follow. At six elements the census frames do contain the known points, and $C=0$ closures are the frames that carry them. The quantity that decides whether a small search finds them is the coordinate degree (Table 2), not $C$. At seven elements the census gives $C=0$ closure surfaces, but none with a coordinate of degree below $21$.

## 8. Next steps

The DKMS surface has a degree-$2$ coordinate because a torsion condition ($3S$ torsion on $E_T$) collapses the closure equation. The analogous test at seven elements is to restrict the low-degree $E$-classes ($44$, $46$, $48$, $59$) to loci where $S$ or $3P-S$ has small order on $E_T$ and recompute the coordinate degrees. A component of degree $\le4$ could be searched to large height. If no such locus lowers the degree, the census has no DKMS-type mechanism at seven elements. A second extension is a step type that joins the chains of different triples at one element; the census treats each chain separately, and the $420$ unlabelled sextuples above suggest that such crossings matter.

## Supplementary material

`supp.zip` contains the scripts (`cen.py`, `cen2.py`, `cencl.py`, `cencl2.py`, `cnum.py`, `costchk.py`, `clsE.py`, `clsEp.py`, `qdeg.py`, `surfg.py`, `ratsrch.py`, `ratsrch_v1.py`, `ratsrchn.py`, `cover6.py`, `parht.py`, `e6.py`, `gibbs.py`, and the others named in the text) and their outputs, with search logs packed as tarballs. Software: PARI/GP 2.17 via cypari2, Python 3.12.

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