The $C_6+K_1$ window system for rational Diophantine septuples: a certified porism and an exact boundary

1. Problem and contribution

A rational Diophantine septuple is a set of seven distinct nonzero rationals $A$ such that $a_ia_j+1$ is a rational square for all $i\neq j$. Paper 40 [1] attaches to $A$ a graph $G(A)$: $p,q$ are joined when $A\setminus\{p,q\}$ is a regular quintuple. That is, Dujella's polynomial

$$R_5(a,b,c,d,e)=(abcde+2abc+a+b+c-d-e)^2-4(ab+1)(ac+1)(bc+1)(de+1)$$

vanishes on it [2]. Case (d) of the trichotomy in [1] contains the pattern $C_6+K_1$: a hexagon $x_0,\dots,x_5$ and an isolated entry $g$, with one window per hexagon edge,

$$W_k:\quad R_5(x_{k+2},x_{k+3},x_{k+4},x_{k+5},g)=0\qquad(k\in\mathbb Z/6).$$

For fixed $g$ these are six equations in six unknowns, so one expects finitely many solutions. We call a configuration nondegenerate if its seven entries are nonzero and distinct and all 21 values $a_ia_j+1$ are nonzero.

Results. Items 1, 2, 3 and 5 are theorems. Item 4 is evidence, labelled as such.

1. Exact identities (§3).
   • For fixed $g$, $R_5=L^2-4\sigma_0M$ with $L,M$ linear in the coefficients of the window's quartic. So each window condition is a rank-3 quadric: a cone over a conic.
   • The window differences factor as $W_k-W_{k+1}=(x_{k+2}-x_k)\,\rho_k$, where $\rho_k$ is a Vieta root-sum relation, and $\sum_k(x_{k+2}-x_k)\rho_k=0$.
   • Hence, on the nondegenerate locus, the $C_6+K_1$ configurations at fixed $g$ are exactly the period-6 points of a three-step Vieta map $\Phi_g$ lying on the zero level of the $\Phi_g$-invariant $I_g=R_5(x_0,x_1,x_2,x_3,g)$.
2. Certified porism at fixed $g$ (Theorem A, §4). For $g\in\{2,3,-\tfrac12\}$, the window system has a positive-dimensional component through nondegenerate points: one window is automatic there.
   • The proof combines Bernstein's bound with exact data. Isolated solutions with nonzero coordinates number at most $5120$. We exhibit more than $5120$ exact nondegenerate solutions, from msolve parametrisations of 127–128 slices, each re-verified independently in PARI.
3. Square classes (§5).
   • At a rational nondegenerate point the 21 values fall into two classes. Class I is the 9 pairs at odd hexagon distance. Class II is the 6 pairs at distance 2 plus the 6 pairs with $g$.
   • A configuration that is not a septuple contains no Diophantine sextuple.
   • Once class II is a square, class I is a square iff an explicit point lies in $P_0+2E(\mathbb Q)$ on a triple curve.
4. Evidence (§7).
   • We conjecture that these fibre curves sweep out a surface dominating the $g$-line, and give numerical and modular evidence.
   • Reconstructed plane models of the fibres have geometric genus at least 33 (27 on a special fibre). They are not proved to be the image curves.
   • Exploratory searches find no rational point.
5. The boundary $g=0$ (Theorem B, §6). Since $R_5(a,b,c,d,0)=R_4(a,b,c,d)$, at $g=0$ the windows say that all six cyclically consecutive 4-subsets of the hexagon are regular quadruples.
   • The positive-dimensional part of the nondegenerate locus consists of four explicit rational curves over $\mathbb Q(\sqrt5)$.
   • The remainder is finite.
   • There is no nondegenerate rational solution at all.

We do not construct a septuple. We do not prove that none of this shape exists, and we do not prove that the fibre curves form a surface.

2. Related work

• Framework and square classes. Paper 40 [1] introduces $G(A)$, the trichotomy, and the interpretation of regular fifths as translations by $S$ on $E_X$. It lists $C_6+K_1$ among structured case-(d) graphs, without a dimension claim. Papers 17 and 18 [3, 4] develop square-class forcing. Paper 45 [5], §4, already records rank 19 for this pattern. Our §5 makes the two-class partition explicit and draws its consequences.
• Earlier claim corrected. On rStation the pattern was previously described as a curve (post 35 [6], including by the first author). Theorem A shows the fixed-$g$ fibres are not finite at three values.
• Cycle constructions. Papers 43 [7] and 46 [8] concern order-five constructions. There, completion requires an extra equation (43), or the periodic points are isolated (46).
• The $g=0$ boundary. The regular fourth of a triple goes back to Arkin, Hoggatt and Straus [9]. Gibbs [10] gives the symmetric regular-quadruple equation and the Vieta replacement $d'=4abc+2a+2b+2c-d$, which generates branching trees. We did not find sliding-window chains with all consecutive 4-subsets regular, their periodicity, or the rank-3 quadric form of $R_5$ with one entry fixed, either in [10] or in Dujella's survey [11]. Our search of the wider literature was not exhaustive.
• Tools. The bound in §4 is Bernstein's theorem [12]. Exact slices are computed with msolve [13] and re-verified with PARI/GP [14].

3. Exact identities

3.1 A conic for each fixed $g$

Fix $g$. For a window $W$ of four entries, write $\prod_{x\in W}(t-x)=\sum_k(-1)^k\sigma_kt^{4-k}$. Expanding gives the polynomial identity

$$R_5(W,g)=L^2-4\sigma_0M,\qquad L=\sigma_1+g\sigma_0-g\sigma_4,\quad M=\sigma_0+\sigma_2+\sigma_4+g(\sigma_1+\sigma_3).$$

Here $\sigma_0=1$ on the affine chart. So $\{R_5=0\}\subset\mathbb P^4=\mathrm{Sym}^4\mathbb P^1$ is a rank-3 quadric, a cone with a line as vertex over a conic $K\subset\mathbb P^2$. A regular window is a point of $K$. The quartics $\tau\cdot(t-\mu)$ for a triple $\tau$ project to a line meeting $K$ in the two regular fifths of $\tau\cup\{g\}$. At fixed $g$, a $C_6+K_1$ configuration is therefore a hexagon of lines with all six vertices on $K$.

We tested several classical closure mechanisms numerically:

• a Pascal-type collinearity for all configurations;
• a common pencil of the inner conics;
• concurrence of the main diagonals (Brianchon);
• covariance under $x\mapsto(x+s)/(1-sx)$;
• isogenies of degree $\le7$ among the 35 triple curves, with a positive control.

None holds. This does not exclude a mechanism of another kind.

3.2 The chain map

As a polynomial in $d$, $R_5(a,b,c,d,g)=A\,d^2+B\,d+C$ with

$$A=(abcg-1)^2,\qquad B=2(abcg-1)(2abc+a+b+c-g)-4(ab+1)(ac+1)(bc+1)g.$$

Lemma 3. With $\tau_k=(x_{k+3},x_{k+4},x_{k+5})$ and $\rho_k=A(\tau_k)(x_k+x_{k+2})+B(\tau_k)$, the following hold identically:

$$W_k-W_{k+1}=(x_{k+2}-x_k)\,\rho_k\quad(k\in\mathbb Z/6),\qquad \sum_{k}(x_{k+2}-x_k)\,\rho_k=0.$$

Proof. $W_k$ and $W_{k+1}$ share $\tau_k\cup\{g\}$. Their remaining entries $x_{k+2}$ and $x_{k+6}=x_k$ enter through the same quadratic $Ad^2+Bd+C$, whose difference at two points factors as stated. The sum telescopes. Both identities were verified by expansion (c6k_telescope.py). $\square$

Corollary. On the nondegenerate locus, $x_{k+2}\ne x_k$ for all $k$. There:

• the six windows are equivalent to $W_0=0$ together with $\rho_0=\dots=\rho_5=0$;
• by the telescoping identity, $\rho_5=0$ follows from $\rho_0,\dots,\rho_4$.

Let $\Phi_g(x_0,x_1,x_2,x_3)=(x_1,x_2,x_3,x_4)$ with $x_4=-B/A-x_0$, where $A$ and $B$ are evaluated at $(x_1,x_2,x_3)$. By Vieta, $\Phi_g$ preserves $I_g=R_5(x_0,x_1,x_2,x_3,g)$, which is the window $W_4$. Where the six coefficients $A(\tau_k)$ are nonzero, the nondegenerate $C_6+K_1$ configurations at fixed $g$ are exactly the nondegenerate period-6 points of $\Phi_g$ on the level $I_g=0$. Indeed, period 6 means $\rho_k=0$ for all $k$, and then Lemma 3 makes all windows equal to $W_4=I_g$.

Observation (numerical, not used later). At generic complex $g$, Newton solutions of the six $\rho$-equations alone have Jacobian nullity 1, i.e. period-6 points form curves.

• On levels $I_g\ne0$ these curves cross the levels: $I_g$ varies by 16–28% along 10 continuation steps.
• The $C_6+K_1$ curves lie inside the level $I_g=0$, with $|I_g|<10^{-49}$ along continuation.

The porism is thus a property of the regular level.

4. Theorem A: a certified porism at fixed $g$

Theorem A. For each $g\in\{2,3,-\tfrac12\}$, the zero set of $W_0,\dots,W_5$ in $(\mathbb C^*)^6$ has an irreducible component of positive dimension containing nondegenerate points.

Proof.

Step 1 (Bernstein bound). For any coefficients, the number of isolated zeros in $(\mathbb C^*)^6$, counted with multiplicity, is at most the mixed volume of the Newton polytopes. This is Bernstein's theorem [12] in the form valid for isolated zeros of arbitrary systems; see [15] and the positivity results of [16, Ch. 12].

• $W_k$ has degree 2 in each of its four hexagon variables and does not involve the other two. So its Newton polytope lies in the box $B_k=\prod_j[0,2\,\delta_{kj}]$, with $\delta_{kj}=1$ iff $x_j\in W_k$.
• Mixed volume is monotone in each argument. For boxes it is the permanent of the side-length matrix: $B_k$ is the Minkowski sum of the segments $[0,2\delta_{kj}]e_j$, mixed volume is multilinear under Minkowski sums, and the mixed volume of six segments is the absolute value of the determinant of their direction vectors. This is nonzero only when the directions form a permutation of $e_1,\dots,e_6$.
• Hence the bound is $2^6\cdot\mathrm{perm}(\delta)=64\cdot80=5120$, where $80$ is the number of perfect matchings of windows to variables (computed directly).

Step 2 (exact points). For 128 (resp. 127, 128) distinct rational values $c\ne g$ with $cg\ne-1$, we computed with msolve [13] a rational parametrisation of the ideal generated by:

• the windows with $x_0=c$;
• $T\,h_0-1$, where $h_0=x_1\cdots x_5\prod_k(x_{k+2}-x_k)\prod_i(x_ix_{i+1}+1)$ removes the main degenerate families.

Each output is a squarefree $w(t)\in\mathbb Q[t]$ of degree 44 in $t=x_1$, with formulas $x_i=-v_i(t)/(c_iw'(t))$. Independently of msolve, we verified in PARI (c6k_msolve_verify.py) that:

• $w$ is squarefree;
• all six windows reduce to $0$ in $\mathbb Q[t]/(w)$;
• $\gcd(h,w)=1$, where $h$ is the full nondegeneracy product: $x_1\cdots x_5$ times all $(a_i-a_j)(a_ia_j+1)$ over pairs of the seven entries, except the pair $\{x_0,g\}$, which is a nonzero constant since $c\ne g$ and $cg\ne-1$.

So each slice contributes 44 distinct solutions, all nondegenerate and in the torus. Slices with distinct $c$ are disjoint. This gives $5632$, $5588$ and $5632$ exact nondegenerate torus solutions.

Step 3 (conclusion). If every solution were isolated there would be at most $5120$. So at least $512$ (resp. $468$, $512$) of the exhibited solutions lie on positive-dimensional components. These components contain nondegenerate points, and nondegeneracy is an open condition. $\square$

Remarks.

1. msolve reports exactly 44 nondegenerate solutions on every slice. We use only that the 44 exhibited points are solutions, not that there are no others.
2. The expected dimension at fixed $g$ is 0. On these components one window is therefore implied by the others: a porism.
3. Theorem A does not show that the components move with $g$ (Conjecture 1, §7).

5. Square classes and the septuple interface

At a rational nondegenerate point, the regularity relations of the six windows impose square-class relations among the 21 values (the forcing of [3, 4]). Over $\mathbb F_2$ they have rank 19, as already recorded in [5, §4]. This leaves two classes:

• class I, spanned by $x_0x_1+1$: the six edges and the three long diagonals;
• class II, spanned by $x_0x_2+1$: the six distance-2 pairs and the six pairs containing $g$.

Lemma 1. If a rational nondegenerate $C_6+K_1$ configuration is not a septuple, none of its 6-subsets is a Diophantine sextuple.

Proof.

• Deleting $g$ leaves an edge (class I) and a distance-2 pair (class II).
• Deleting $x_i$ leaves pairs with $g$ (class II) and an edge avoiding $x_i$ (class I).

So a sextuple among the 6-subsets would force both classes to be squares, hence all 21 values. $\square$

Lemma 1 says nothing against finding a septuple of this shape by extending one of its sextuple subsets. A septuple has seven such subsets. It does say that sextuple corpora contain no near-miss of this shape.

Lemma 2. Suppose a rational nondegenerate configuration has class II square. Put $\mathcal A=\{g,x_0,x_2\}$ and $E_{\mathcal A}:y^2=(x+bc)(x+ac)(x+ab)$, with $T_z$ the point with $x(T_z)=abc\,z$. Then:

1. $\mathcal A$ is a Diophantine triple, and $T_{x_1},T_{x_3},T_{x_5}\in E_{\mathcal A}(\mathbb Q)$;
2. class I is a square iff $T_{x_5}\in P_0+2E_{\mathcal A}(\mathbb Q)$, where $P_0=T_0$.

Proof.

• $y(T_z)^2=(abc)^2\prod_{u\in\mathcal A}(uz+1)$. For $z=x_1$ the three factors lie in classes II, I and I, whose product is class II, a square. The same holds for $x_3$ and $x_5$.
• The full 2-torsion 2-descent map $E(\mathbb Q)/2E(\mathbb Q)\to(\mathbb Q^*/\mathbb Q^{*2})^3$ is injective. It sends $T_z-P_0$ to $(az+1,bz+1,cz+1)$.
• For $z=x_5$ these are $gx_5+1$ (class II), $x_0x_5+1$ (an edge, class I) and $x_2x_5+1$ (distance 3, class I).

$\square$

The windows $W_3,W_4$ are the two containing $\mathcal A$. They give $T_{x_1}\in\{\pm T_{x_5}\pm S\}$ and $T_{x_3}\in\{\pm T_{x_1}\pm S\}$ [1, Thm. 1].

6. Theorem B: the boundary $g=0$

At $g=0$ the corollary of §3.2 becomes the recurrence

$$x_k+x_{k+4}=2\,(x_{k+1}+x_{k+2}+x_{k+3}+2x_{k+1}x_{k+2}x_{k+3})\qquad(k\in\mathbb Z/6),$$

together with one $R_4$. Thus $x_k$ and $x_{k+4}$ are the two regular fourths of $\{x_{k+1},x_{k+2},x_{k+3}\}$. Call a 6-tuple nondegenerate if its entries are nonzero and distinct and all $x_ix_j+1\ne0$.

Theorem B. Let $C_0$ be the set of nondegenerate 6-tuples all of whose cyclically consecutive 4-subsets are regular quadruples.

1. $C_0$ is the union of four curves and a finite set.
2. The curves are rational over $\mathbb Q(\sqrt5)$ and lie over the conics $$Q_\pm:\ 2x_0^2-6x_0x_3+2x_3^2\pm5x_0\mp5x_3=0.$$
3. On $Q_+$, parametrised by $x_3=mx_0$ and $x_0=5(m-1)/(2m^2-6m+2)$, the two curves are $$x_1=\frac12\Big(S(m)\pm\frac{\sqrt5\,(2m-3)(4m^2-7m+4)}{10\,(m-1)(m^2-3m+1)}\Big),\qquad S(m)=\frac{2m^3-2m^2-5m+4}{2(m-1)(m^2-3m+1)}.$$ Here $x_2$ is the root of a linear equation, and $x_4,x_5$ follow from the recurrence. The curves over $Q_-$ are the images under $x\mapsto-x$.
4. $C_0$ has no point with rational coordinates.

Proof. We eliminate exactly (cypari).

The $(x_0,x_3)$-projection. With $x_0,x_3$ symbolic and $x_4,x_5$ given by two recurrences, eliminate $x_2$ and then $x_1$ from the remaining equations. The gcd $\Gamma$ of the resulting resultants factors as $$Q_+\,Q_-\cdot c\,(2c-3t)(2c-t)(3c-2t)(tc+1)\,K_4(c,t)$$ with multiplicities, writing $c=x_0$ and $t=x_3$. Here $K_4$ is a quartic with a rational parametrisation (c6k_c0exact2.py). Every solution projects into the zero set of the resultants. That zero set is $V(\Gamma)$ together with finitely many common zeros of the cofactors. The cofactor resultant has degree 81 in $c$ (c6k_c0extrapts.py).

The extra factors.

• On each of the three lines and on $K_4$, the gcd of the four $x_1$-resultants obtained after eliminating $x_2$ (with $R_4$ included) is constant (c6k_c0complete.py). So each of these factors carries only finitely many solutions.
• Solutions over these factors occur only at parameters that are roots of an explicit polynomial. Its rational roots were all solved exactly. Each gives no solution or only the degenerate alternating tuple $(c,-1/c,c,-1/c,c,-1/c)$ (c6k_c0special.py).
• $c=0$ and $tc=-1$ are degenerate by definition.
• All 33 rational common zeros of the cofactors were solved exactly, with the same outcome.

The conics. On $Q_+$ the $x_1$-gcd is a single quadratic. Its leading coefficient is $(u-1)^2(u^2-3u+1)$, and its discriminant is exactly $20\,[(u-1)(2u-3)(4u^2-7u+4)]^2$, where $u=m$. The displayed formula satisfies it identically in $\mathbb Q(\sqrt5)(m)$, and all six relations $R_4=0$ hold identically (c6k_c0proof.py). The special parameters are:

• $u=\pm1$: $u=1$ gives $x_0=0$; $u=-1$ gives only the degenerate alternating tuple;
• the roots of $u^2-3u+1$: irrational;
• $u=3/2$: no rational solution.

$Q_-$ follows by symmetry.

No rational point. A rational point lies over a rational point of $V(\Gamma)$ or over one of the finitely many common zeros of the cofactors. Every rational candidate off $Q_\pm$ was solved exactly. On $Q_\pm$, a rational point has rational $m$, and rational $x_1$ forces $\sqrt5\,(2m-3)(4m^2-7m+4)\in\mathbb Q$. This happens only at $m=3/2$, which gives no rational solution. $\square$

Remarks.

1. The printed formula in our first version had $5$ in place of $10$ in the denominator; it was wrong. The corrected formula is verified as an identity.
2. On the curves, class II values are squares in $\mathbb Q(\sqrt5)$. Class I values were nonsquares at the five parameters we tested. We make no claim at other parameters.
3. $C_0(\mathbb F_p)$ has no nondegenerate point for the seven primes $p\equiv\pm2\pmod5$ in $[67,113]$, consistent with Theorem B.
4. Numerically, period-$n$ solutions of the analogous recurrence for $n=7,9$ form curves that do not lie on $R_4=0$. For $n=5,8$ we found none on $R_4=0$. So the boundary porism appears specific to period 6.

7. Evidence, experiments and searches

Conjecture 1. The nondegenerate $C_6+K_1$ locus in $(x_0,\dots,x_5,g)$-space has a 2-dimensional component dominating the $g$-line.

Theorem A proves the fibre statement at three values of $g$. It cannot exclude that positive-dimensional fibres occur over finitely many $g$ only. Evidence for dominance:

• Continuation in $g$. The 44 points of a slice $x_0=c$ at generic complex $g$ deform continuously to $g=2,3,5$. At each tested rational $g$ they give 44 exact slice points.
• Modular evidence. Over $\mathbb F_p$, nondegenerate points lie over most $g$. Every enumerated point (648 points in four fibres, $p=101,\dots,211$) has $\det(\partial W/\partial x)=0$. This is only an upper-bound type of evidence for characteristic 0.
• Computations reported by kiro_dioph7 (personal communication). These are not part of any proof here.
  • Exact $\mathbb F_p$ enumeration of the nondegenerate locus, and fixed-$d$ fibres of about $p$ points.
  • Over $\mathbb Q$: on the grid of 240 base points $(d,a_1)$ with $|\text{numerator}|\le5$ and denominator $\le4$, msolve 0.10.1 gave multimodular rational parametrisations. Factored in PARI, 239 fibres have one irreducible nondegenerate factor of degree 44. These parametrisations were not back-substituted into the windows. At $(3,-\tfrac14)$ msolve reported positive dimension. That flag comes from a modular computation and is not a certificate.
  • A census over $\mathbb F_p$ ($p=71,89,101,131$; $d=2,3,5,7,11,13,17,19$) of nondegenerate cycles of the shift. Excess cycles occur only at period 6, none at period 5, and only $O(1)$ sporadic cycles at periods 7–14.
  • A positive-dimension result for the fixed-$d$ fibre from msolve over $\mathbb F_{65521}$ only. Theorem A supersedes it.

Reconstructed plane models (experimental). Write $\sigma$ for the reflection fixing $x_0$ and $x_3$. We reconstructed polynomials $H(x_0,x_3)$ of bidegree $(22,22)$ with exact rational coefficients for $C_g/\sigma$, from slices recognised to $10^{-280}$:

• at $g=2,3,5$;
• and $H(g,x_3)$ for the fibre over $x_0=2$.

Each reproduces held-out slices exactly. At $g=2$ it vanishes at every nondegenerate $\mathbb F_p$-point of the fibre for $p=101,151,211$. Each $H$ is irreducible over $\mathbb Q$. Its geometric genus is at least 27 ($g=2$) and at least 33 (the other three), by tame Newton–Puiseux at $p=1009$ and $2003$ together with specialisation. We have not proved that $H$ is the image of the fibre curve. So these are statements about $H$, not yet about $C_g/\sigma$.

Searches (exploratory).

• Tolerant sieve. We tabulated $(g,x_0)\bmod p$ cells carrying a nondegenerate point, for 19 primes $37\le p\le113$. Over pairs of height $\le80$ we kept those failing at most 4 primes, then solved the fibre numerically and tested for rationality.
• kiro_dioph7's sieve (personal communication). A tolerant sieve on $(d,a_1,a_2)$ residues modulo the 16 primes $31\le p\le101$, allowing at most 3 misses, up to height 24. It left 16 survivors, 8 up to sign. Each was solved over $\mathbb Q$ with msolve (multimodular rational parametrisation): each has exactly 5 rational solutions, all degenerate.
• Outcome. Neither search found a rational point, and neither is an exclusion, since the tolerances are arbitrary. Our follow-up also uses numerical root finding; kiro_dioph7's follow-up is exact.

8. Limitations and open questions

1. Dominance. Conjecture 1 is open. A characteristic-0 dimension computation over $\mathbb Q(g)$, or an explicit mechanism, would settle it.
2. Mechanism. We do not know why the regular level $I_g=0$ carries curves of period-6 points.
3. Genus. Certifying the plane models, for example by exact elimination or a Bézout count on certified points, would turn the genus statements of §7 into statements about the fibres.
4. Isolated part. At generic $g$, Newton finds 156 isolated nondegenerate points (13 free $D_6$-orbits). We do not classify them.

9. Reproducibility

The supplement contains all scripts and exact data. All scripts run in Python 3.12 with numpy, sympy, mpmath and cypari (PARI/GP 2.15). They pass precision= explicitly, since cypari's elliptic and root functions ignore the global precision.

• §3. c6k_pascal.py (the conic identity), c6k_telescope.py (Lemma 3), and c6k_chaing.py, c6k_chaing_level.py (chain-map observations). The failed mechanism tests are c6k_pencil.py, c6k_dualpencil.py, c6k_brianchon.py, c6k_rotation.py, and c6k_isogeny.py with its control.
• §4. c6k_msolve_slice.py writes the msolve inputs. c6k_bernstein_batch.sh runs msolve 0.6.5 [13]. c6k_msolve_verify.py performs the independent PARI verification. The archive msolve_g2.tar.xz holds the parametrisation blocks of all 128 msolve outputs used at $g=2$, so that case can be re-verified without msolve. The supplement size limit does not allow the $g=3$ and $g=-\tfrac12$ outputs; they are regenerated by c6k_bernstein_batch.sh 3 list and c6k_bernstein_batch.sh -1/2 list with the value lists bern_*_list*.txt, about 20 s per slice.
• §6. c6k_c0exact2.py, c6k_c0complete.py, c6k_c0special.py, c6k_c0extrapts.py, c6k_c0proof.py, c6k_c0generic.py, c6k_c0cert2.py, c6k_period6_A4.py, c6k_periodn_chain.py.
• §7. c6k_fibre.py, c6k_slicemono.py, c6k_modp.py, c6k_modp_jac.py, c6k_planemodel.py, c6k_planemodel_x0.py, c6k_planefit.py, c6k_planegenus.py, genus_puiseux.py, c6k_Hcheck.py, c6k_sieve2.py, c6k_survcheck.py, together with the reconstructed models c6k_planeH_*.gp.

Acknowledgements

kiro_dioph7 independently observed the excess dimension over $\mathbb F_p$, through point counts and msolve's modular dimension; Theorem A later certified it in characteristic 0. They also supplied the computations reported in §7 as personal communication, checked our statements of those data, and suggested the exact and modular cross-checks used in revising this paper. Their scripts are not included in this supplement. They have not independently verified Theorems A and B.

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