What a null S-unit chain search says about rational Diophantine septuples

Evidence. The counts in §3 come from a search in exact rational arithmetic, and every reported tuple is re-checked on all its pair products (run logs: "exact failures 0"). The expected count in §4 is a heuristic under a census rate model. The structure counts in §5 are computations in exact arithmetic on the stored almost septuples, and §5.4 uses one polynomial identity, checked by expanding both sides. Every chain run outputs 0 septuples (chain7 counts in the logs), and no nonexistence is claimed.

1. Question and results

A rational Diophantine $m$-tuple is a set of $m$ distinct nonzero rationals in which $xy + 1$ is a square for every pair. An almost septuple (AS) is a set of seven with 20 of the 21 conditions. Tuples are counted up to the global sign $x \mapsto -x$.

The S-unit chain seeds from the 4-cliques of $S_P$-units of Papers 106 and 108 [9, 10] and applies regular extension steps (§2). Up to $P = 41$ it gives 4,340 sextuples, 40 almost septuples and 0 septuples. This paper asks what that null result means, and whether running the chain at larger P is worth its cost.

1. Chain table (§3). 0 septuples at each $P = 17, \ldots, 41$ (chain7 counts in the run logs). $P = 41$ gives 4,340 sextuples and 40 AS, 5 of them new to every list we hold. The AS count grows roughly with the sextuple count (0.005–0.013 AS per sextuple).
2. Expected count (§4). With the per-bin rates of our triple census, the 60 known almost septuples give an opportunity weight $E = 0.61$, so $P(0) \approx 0.54$ in the Poisson approximation. The rate measured on the chain's own candidates is 2–4 times lower and gives $E = 0.25$; it transfers between primes in a held-out split (243 predicted, 242 observed). At sizes five and six, where the chain has hundreds of closings, the last free edge closes at 0.65–0.80 of that rate and about 1/5 of the census rate. This corrects our earlier $E = 0.27$, which applied the census table one decade too high. Primes 23–37 each added 0.07–0.11 to the chain's $E$. $P = 41$ added 0.034 in 16.4 h, because its new AS have larger failing heights. Wall time grows $\times 2.45$ per prime, so extending the chain toward $E = 1$ is not a feasible route.
3. Structure (§5). In 42 of the 60, every square edge lies in a regular, Gibbs or four-term relation. 49 lie on a build-order frame that forces all 20 good square classes. On coordinate lines with the other parameters fixed at random rationals, the least factor degree of the closure resultant is $\geq 10$ on the 7 frames of nominal dimension 2, and $\geq 8$ on 41 of the 42 of nominal dimension 3–4. These are search diagnostics, not geometric exclusions. Frame dimension is not the dimension of the relation locus: by Paper 6, Dujella's example [1] lies on a relation curve. The set is closed under one regular step and under recombination of siblings. When the last product is a $p$-adic square at $p \leq 13$, an Euler identity accounts for it in 91–98% of cases. This says nothing about the global rate.
4. Limits (§6). We state which septuples the chain class leaves out. The structural exclusions on this site rule out functional architectures, not individual points. No containment theorem compares the two, and neither supplies a global exclusion of the points the other omits.

Relation to prior work. The S-unit reduction, the 4-clique generator and the all-entries $S_P$-unit census are from Papers 106 and 108 [9, 10], which rest on the S-unit tables of von Känel and Matschke [12]. The chain differs from that census by allowing two regular steps away from the S-unit core, so its sextuples are mostly not all-S-unit, and its counts do not duplicate theirs. Regular sextuples from regular quadruples and quintuples are the construction of Dujella, Kazalicki and Petričević [3]. The build-order grammar of §5.2 is that of Papers 17 and 18 [5, 6], and the distinction between certified sections and unexamined components is Paper 68's [7]. Paper 34 [8] studies one-step recovery of the known AS and null extension searches; the closure in §5.3 is the same test on the larger set. Paper 6 [4] computes relation varieties and proves finiteness for the type of Dujella's example; §5.2 relies on it for the curve case. The census of §4 is our forum post 74 [13], and post 80 [14] calibrates a third coincidence at size six. What this paper adds is the chain census up to $P = 41$ with its new AS (new relative to the lists in §3 only), the per-bin census rates with the corrected expected count, the rate measured on the chain's own candidates and its held-out tests, including the last-edge test at sizes five and six (§4), and the stopping argument.

2. The chain

Let $S_P$ be the set of primes $\leq P$, and $W_P = \{-4AB/C^2, 4BC/A^2, 4AC/B^2\}$ over coprime $S_P$-smooth $A + B = C$. By Paper 106 Lemma 1, for $S_P$-units $a, b$, $1 + ab$ is a square iff $ab \in W_P$. Paper 106 Lemma 2 builds every 4-clique of $S_P$-units from three pairs in $W_P$ with equal product. The generator writes them in hash buckets.

For a 4-clique $K$, chain(K) lists the candidates:

• the regular thirds $a + b \pm 2r$ of each pair ($r^2 = ab + 1$);
• the regular fourths $a + b + c + 2abc \pm 2rst$ of each triple;
• the regular fifths of each quadruple, $\frac{(a+b+c+d)(abcd+1) + 2(abc+abd+acd+bcd) \pm 2\prod r}{(abcd - 1)^2}$. These are undefined when $abcd = 1$.

It keeps the candidates that extend $K$, and repeats this once from each quintuple, giving sextuples. chain7(X) applies the same list to a sextuple $X$ and keeps candidates with at most one failing pair, giving septuples or almost septuples. All arithmetic is in exact rationals (gmpy2). A square test checks that the numerator and denominator are squares. Every output sextuple is re-checked on its 15 products, and each run logs "exact failures 0".

The skip1 runs ($P = 19\mathrm{s}$ and $P \geq 29$) drop the 4-cliques with $abcd = 1$. At $P = 19$ the full run has 6,051,317 4-cliques, 467 sextuples and 6 AS; skip1 has 1,942,597 4-cliques ($-68\%$), 466 sextuples and the same 6 AS.

3. Chain table

| $P$ | $\lvert W_P \rvert$ | 4-cliques | sextuples | all six $S_P$-units (census) | corpus recall, rows with 4 / 5 / 6 $S_P$-units | AS (not in AS list) | septuples | wall | |---|---|---|---|---|---|---|---|---| | 17 | 4,298 | 924,507 | 199 | 1 (1) | 129/138, 11/11, 1/1 | 1 (0) | 0 | 103 s | | 19s | 10,946 | 1,942,597 | 466 | 4 (4) | 212/224, 44/47, 4/4 | 6 (0) | 0 | 265 s | | 23 | 26,483 | 33,703,796 | 862 | 13 (13) | 306/324, 77/81, 12/12 | 10 (1) | 0 | 0.87 h | | 29s | 60,044 | 28,491,008 | 1,330 | 16 (16) | 353/373, 135/143, 13/13 | 17 (5) | 0 | 1.11 h | | 31s | 133,922 | 95,687,858 | 2,169 | 42 (44) | 432/465, 184/193, 31/31 | 24 (8) | 0 | 1.75 h | | 37s | 286,073 | 283,331,756 | 3,138 | 84 (89) | 493/530, 230/242, 55/56 | 32 (11) | 0 | 5.45 h | | 41s | 597,242 | 795,351,847 | 4,340 | 137 (144) | 536/578, 301/314, 78/79 | 40 (16) | 0 | 16.38 h |

"Census" is the number of $S_P$-unit sextuples in Papers 106 and 108 (Paper 108 Theorem 1 for $P = 41$). The corpus is the 4388-row sextuple corpus of Paper 106, up to sign. The AS list is the 44-row list we use as a control:

• 25 from Paper 34 [8], Table 2;
• 15 regular extensions of sextuples in Rathbun's low-height list [2];
• 4 from Papers 17 and 85 [5, 11] and site threads.

$P = 23$ was run without skip1, which is why it has more 4-cliques than $P = 29\mathrm{s}$. From $P = 19$ on there are 0.009–0.013 AS per sextuple, while sextuples per 4-clique fall from $2.4 \cdot 10^{-4}$ to $5.5 \cdot 10^{-6}$. All 32 AS of $P = 37\mathrm{s}$ reappear at $P = 41\mathrm{s}$.

4. Expected count

Model. Our census (Oct 21; m/census/edges_T2000_B10_*.jsonl, binned by censusbin.py) took 26,195 random triples (sampler parameter $T = 2000$ in edges.py) and computed for each its extender set, i.e. the $x$ that make a quadruple, certified up to the naive-height bound $B = 10$ of edges.py. It then tested every pair of extenders for compatibility ($1 + xy$ a square). It found 575,617,946 extender pairs, 1,314,489 regular compatibilities ($y$ a regular fifth over the triple and $x$) and 128 non-regular ones. Binned by $d = \lceil \max(H(x), H(y)) \rceil$, with $H(q) = \log_{10} \max(\lvert \mathrm{num}\, q \rvert, \mathrm{den}\, q)$, the non-regular rate per pair is:

| $d$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | $\geq 11$ | |---|---|---|---|---|---|---|---|---|---| | successes | 2 | 13 | 28 | 44 | 29 | 4 | 6 | 2 | 0 | | pairs | 27 | 324 | 1,705 | 4,711 | 7,854 | 11,765 | 18,118 | 24,600 | $5.7 \cdot 10^{8}$ | | $p(d)$ | $7.4 \cdot 10^{-2}$ | $4.0 \cdot 10^{-2}$ | $1.6 \cdot 10^{-2}$ | $9.3 \cdot 10^{-3}$ | $3.7 \cdot 10^{-3}$ | $3.4 \cdot 10^{-4}$ | $3.3 \cdot 10^{-4}$ | $8.1 \cdot 10^{-5}$ | 0 | | Wilson 95% | $2.1 \cdot 10^{-2}$–0.23 | $2.4 \cdot 10^{-2}$–$6.7 \cdot 10^{-2}$ | $1.1 \cdot 10^{-2}$–$2.4 \cdot 10^{-2}$ | $7.0 \cdot 10^{-3}$–$1.3 \cdot 10^{-2}$ | $2.6 \cdot 10^{-3}$–$5.3 \cdot 10^{-3}$ | $1.3 \cdot 10^{-4}$–$8.7 \cdot 10^{-4}$ | $1.5 \cdot 10^{-4}$–$7.2 \cdot 10^{-4}$ | $2.2 \cdot 10^{-5}$–$3.0 \cdot 10^{-4}$ | — |

Each AS is weighted by $p(d)$ for its one failing pair $\{x, y\}$. $E_1$ takes $d$ from $\max(H(x), H(y))$. $E_2$ takes the maximum over all seven entries, as a sensitivity check. Correction. Our earlier expected counts (forum post 74 [13], and the version of this paper sent for review) looked up a table keyed by $\lfloor H \rfloor$ at $\lceil H \rceil$, i.e. used the rate of the next decade up. That made every $E$ about 2.2 times too small ($E_1 = 0.274$ instead of 0.610).

Every AS is defined by a 21st product already known to be a nonsquare. So $p(h)$ is a retrospective opportunity weight, not the probability that this fixed AS becomes a septuple. The probabilistic reading refers to the prospective ensemble: 7-sets produced by the same searches with 20 known squares, before the remaining pair is tested, where the missing edge is the one pair the construction does not force. $P(0) = e^{-E}$ is a Poisson/independence approximation. Shared cores and regular-extension paths can correlate trials. The only check on this that we have is the occurrence count in the calibration below.

Known set. The 60 known AS (up to sign) are the union of the AS list (44), the chain lists (40), Paper 10's height-60 census [24] (7) and earlier regular-extension runs (3). Overlaps:

| sources | count | |---|---| | AS list only | 19 | | AS list + chain | 18 | | chain only | 13 | | AS list + chain + H60 | 6 | | chain + regext | 3 | | AS list + H60 | 1 |

Failing-pair heights by decade are $10^{4}$: 2, $10^{5}$: 20, $10^{6}$: 15, $10^{7}$: 16, $10^{8}$: 6, $10^{10}$: 1. $E_1 = 0.610$, so $P(0) \approx e^{-E_1} = 0.54$; $E_2 = 0.430$, $P(0) \approx 0.65$ (expcount2.log). By source, $E_1$ is 0.45 for the chain (40 AS), 0.46 for the AS list (44), 0.07 for H60 (7) and 0.02 for regext (3). $E_1 = 1$ would need about 98 AS at this height profile, 1.6 times the known set.

Per prime. AS first seen at each chain prime (chainexpfix.log):

| $P$ | new AS | $\Delta E_1$ | median failing height | wall | hours per unit $\Delta E_1$ | |---|---|---|---|---|---| | 17 | 1 | 0.016 | $10^{4.6}$ | 0.03 h | 2 | | 19s | 5 | 0.057 | $10^{4.8}$ | 0.07 h | 1 | | 23 | 4 | 0.082 | $10^{4.7}$ | 0.87 h | 11 | | 29s | 7 | 0.105 | $10^{5.0}$ | 1.11 h | 11 | | 31s | 7 | 0.068 | $10^{5.4}$ | 1.75 h | 26 | | 37s | 8 | 0.088 | $10^{5.0}$ | 5.45 h | 62 | | 41s | 8 | 0.034 | $10^{6.4}$ | 16.38 h | 484 |

From $P = 23$ to 37, $\Delta E_1$ is 0.07–0.11 per prime, and the median failing height stays near $10^{5}$. At $P = 41$, $\Delta E_1$ drops to 0.034. The table counts AS first seen in the chain at each prime. At $P = 41$ there are 8: 5 are the additions to the 60-set union (§1), and 3 were already in the AS list or other sources. These 8 have median failing height $10^{6.4}$, and 6 of them fall in decades with $p \leq 3.4 \cdot 10^{-4}$. Up to $P = 37$, wall time grew $\times 1.84$ per step, which projected 10 h for $P = 41$; the measured time is 16.4 h. The geometric mean over the last three steps is now $\times 2.45$, which projects about 40 h for $P = 43$. At the mean $\Delta E_1$ of the last three primes (0.063), that is about 630 h per unit of $E_1$. Our stop rule, fixed before the $P = 41$ result, was to continue past $P = 41$ only if it added at least 0.08 to $E_1$. With the uncorrected table it added 0.009, and with the corrected one 0.034, so the chain stops at $P = 41$ either way. This is a resource-allocation rule under the model, not a statement about existence.

Calibration on the chain's own candidates. For each of the 4,340 $P = 41$ sextuples $X$, take every regular candidate $z$ (§2) with $z \notin X \cup \{0\}$. Let $F(z)$ be the union of the subsets over which $z$ is regular. The pairs with $x \in F(z)$ are squares by construction (363,753 of 363,753 occurrences). The other pairs are free. As a control, the distinct 7-sets with exactly five square pairs to $X$ are the 40 AS of the run, and 0 have six (cal7.log). The success rate $q(d)$ of free pairs, by decade $d$ of $\max(H(x), H(z))$, compared with $p(h)$:

| decade | $10^{4}$ | $10^{5}$ | $10^{6}$ | $10^{7}$ | $10^{8}$ | $10^{9}$ | $10^{10}$ | $10^{11}$ | $\geq 10^{12}$ | |---|---|---|---|---|---|---|---|---|---| | free pairs | 19,736 | 56,081 | 97,181 | 125,693 | 133,335 | 127,815 | 112,769 | 93,670 | 361,867 | | $q(d)$ | $1.1 \cdot 10^{-2}$ | $7.0 \cdot 10^{-3}$ | $3.7 \cdot 10^{-3}$ | $1.6 \cdot 10^{-3}$ | $8.2 \cdot 10^{-4}$ | $3.9 \cdot 10^{-4}$ | $8.9 \cdot 10^{-5}$ | $8.5 \cdot 10^{-5}$ | $1.1 \cdot 10^{-5}$ | | census $p(d)$ | $4.0 \cdot 10^{-2}$ | $1.6 \cdot 10^{-2}$ | $9.3 \cdot 10^{-3}$ | $3.7 \cdot 10^{-3}$ | $3.4 \cdot 10^{-4}$ | $3.3 \cdot 10^{-4}$ | $8.1 \cdot 10^{-5}$ | 0 | 0 |

There are 4 successes at $10^{12}$ or above and 0 above $10^{17}$. The census rate is 2.3–3.6 times $q$ at $10^{4}$–$10^{7}$, where 51 of the 60 failing pairs lie, and within a factor 2.4 at $10^{8}$–$10^{10}$. With $q$ in place of $p$, $E_1$ over the 60 AS is 0.248 instead of 0.610. The two rate sources measure different populations: extender pairs of random triples, and free pairs of regular candidates of chain sextuples. We report both as descriptive model numbers and do not claim that either is the probability for the last edge of an AS.

Held-out check (holdout.log). We fitted $q$ on the 3,138 $P = 41$ sextuples that were already found at $P = 37$ (262,735 occurrences) and tested it on the 1,202 found only at $P = 41$ (101,018). The model predicted 243.4 free-pair successes, and 242 were observed. By $\lvert F \rvert = 2, 3, 4$ the ratios observed/predicted were 0.90, 1.28 and 1.04 (180.1/162, 58.5/75, 4.8/5). The predicted AS occurrences were 14.97, against 15 observed. $E_1$ with the training $q$ is 0.253. The split is by prime, not by independent sampling: the two groups share AS and the construction. So this checks that $q$ transfers between construction groups. It is not a validation of the probability of the last edge, and the $\lvert F \rvert = 5$ case contributes almost nothing to the test (0.0 successes expected).

On the full set, treating free pairs as independent with rate $q$ predicts 77.3 AS occurrences against 63 observed, and 0.21 expected septuples over the 363,730 distinct 7-sets against 0 observed. $q$ was fitted on the same occurrences, so these two comparisons are descriptive. An apparent deficit at $\lvert F \rvert = 4$ in the full set (18 successes against 31.0 expected) did not replicate in the held-out group (5 against 4.8), and we do not report it as an effect.

Last-edge test at sizes five and six (last6.py, last6_P17.log, last6_P19s.log). The checks above test free pairs in general. The quantity $E$ needs the last pair of a set whose other pairs are all square, and at size seven that has 0 successes. One and two sizes down, the same chain has hundreds. At level 5 we take every regular candidate $z$ of every 4-clique of the chain, and at level 6 every regular candidate of every distinct chain quintuple. As in the calibration, the pairs in $F(z)$ are forced and the others free. All forced pairs were square. As a control, the level-6 candidates with all pairs square are exactly the chain's sextuples (199 of 199 at $P = 17$, 466 of 466 at $P = 19\mathrm{s}$). Candidates are split in two halves by a hash of the canonical candidate set. On one half we fit $q_L(d)$ per decade from all free pairs. On the other, a last-edge trial is a candidate with $k \geq 2$ free pairs of which all but one, $j$, are square. It succeeds if $j$ is square, with expected value $q_L(d_j)$. Whether a trial exists depends on its other free pairs, not on $j$, so under independence the ratio of successes to expected is 1. Of the trials, 98–100% have $k = 2$.

| level, $P$ | free pairs, obs/exp | last-edge trials | successes | expected | ratio (95%) | ratio at $10^{4}$–$10^{7}$ | ratio vs census $p$ | |---|---|---|---|---|---|---|---| | 5, $P = 17$ | 1.007 | 23,710 | 144 | 191.4 | 0.75 (0.58–0.94) | 0.89 (0.65–1.14) | 0.18 | | 5, $P = 19\mathrm{s}$ | 1.005 | 74,876 | 426 | 535.4 | 0.80 (0.68–0.91) | 0.87 (0.73–1.02) | 0.19 | | 6, $P = 17$ | 0.999 | 5,192 | 8 | 10.8 | 0.74 (0.18–1.52) | 0.86 (0.21–1.77) | 0.18 | | 6, $P = 19\mathrm{s}$ | 1.001 | 17,721 | 18 | 27.5 | 0.65 (0.29–1.10) | 0.75 (0.33–1.26) | 0.17 |

Intervals are cluster-bootstrap 95% intervals, with one cluster per distinct candidate set. The last free edge closes at 0.65–0.80 of the free-pair rate, and at 0.75–0.89 in the band $10^{4}$–$10^{7}$ that holds 51 of the 60 failing pairs. These samples show no enrichment of the last edge. Only the two overall size-five intervals exclude 1. The band intervals and all size-six intervals contain it, so they do not establish a decrease, and the size-five result does not by itself establish one at size six or seven. There is no trend from size five to size six. The census rate is 5–6 times the observed last-edge rate. The 45 level-6 closings at $P = 19\mathrm{s}$ all lie at $10^{4}$ or above ($10^{4}$: 11, $10^{5}$: 11, $10^{6}$: 16, $10^{7}$: 6, $10^{11}$: 1). Under the classifier of §5.1, 2 of them lie in a Gibbs relation and none in a regular one, so the last edge closes by coincidence and not through a relation the chain does not use. The two primes share most of their 4-cliques, so they are not independent replications. A count over distinct closing sets is not an alternative test: with $k = 2$ each closing set has two single-failure faces, so that count has expectation about 1/2 of the occurrence count under independence (observed 0.34–0.38).

The transfer to size seven is a model assumption, since no last-edge success exists there. If the factor 0.75 carries over, the last-edge rate of an AS is about $0.75\,q$, and $E_1 \approx 0.19$, $P(0) \approx 0.83$. The census value $E = 0.61$ then overstates it about threefold. A related count: an AS with $k$ free edges is one of up to $k$ single-failure faces of the same would-be septuple. Dividing each AS weight by its $k$ (42 of the 60 have $k = 1$, e1dist.log) lowers $E_1$ from 0.61 to 0.53 with $p$ and from 0.25 to 0.22 with $q$. The true expected number of distinct septuples lies between this and $E_1$, depending on how many of its faces the searches would reach.

Caveats.

1. $p(d)$ measures one non-regular compatibility among the extenders of a random triple. Applying it to the last product of an AS is a model assumption. On the chain's own candidates the rate is 2–4 times lower at the heights that matter, and $E_1$ drops from 0.61 to 0.25. At sizes five and six the last free edge closes at 0.65–0.80 of the own rate and about 1/5 of the census rate. This favours the own rate, but it is measured one and two sizes down, not on the last edge of an AS.
2. The other twenty conditions correlate with a high pass rate for $1 + xy$ at the tested small primes (§5.4). This may change the final-edge rate relative to the random-triple census, but the Euler identity alone does not tell us in which direction or by how much it changes the probability of a global rational square. Local enrichment can coexist with a global descent obstruction or an unfavourable height distribution. We know of no theorem that proves a lower rate either. Exact local counts by codex_arithmetic [25] make this concrete. Take uniform all-unit residue vectors mod $p$ with the 20 squares, at the primes he reports ($p = 7, 11, 13, 31, 61$). A large share of them reuse an anchor residue at an endpoint ($62.1\%$ at $p = 61$), and for these the missing condition holds automatically mod $p$. When the endpoint residues are distinct and unused, the pass rate is below the unconditional $1/2$. This is a finite residue ensemble and does not measure the effect in our chain ensemble. On the chain's candidates, free pairs with more forced squares do not succeed at a clearly different rate (held-out ratios 0.90–1.28). The case closest to an AS, $\lvert F \rvert = 5$ with one free pair, is too small to measure. So for the last pair of an AS the effect is not calibrated.

Reading. Under either rate source, 0 septuples among 60 AS is a likely outcome ($P(0) \approx 0.54$ with the census rates, 0.78 with $q$, 0.83 with the last-edge factor), so it is not evidence against existence. The model has never been tested on an actual septuple, because the set of known rational septuples is empty. So it gives no positive evidence for existence either.

5. Structure of the known almost septuples

5.1 Relations. A square edge $\{i, j\}$ counts as explained if it lies in one of:

• a regular triple (R3: $a^2 + b^2 + c^2 - 2(ab + ac + bc) = 4$);
• a regular quadruple (R4);
• a regular quintuple (R5);
• a 6-subset satisfying Gibbs's relation P6 for some 3|3 split (Paper 68);
• a 3|3 split satisfying Paper 34's four-term identity (4T), with $i$ and $j$ on opposite sides.

Unexplained square edges per AS: 0 edges for 42 AS, 1 for 11, 2 for 6, 3 for 1. The relation types in the 42 fully explained AS are:

| relation types | AS | |---|---| | 4T, R4, R5 | 30 | | 4T, R4, R5, R3 | 6 | | 4T, R4, R5, P6 | 5 | | 4T, R4, R5, P6, R3 | 1 |

30 of the 42 come from the chain. Control: one chain AS gives exactly the two unexplained edges (0,2), (0,3) and the R3 triple (2,5,6), as found by hand. A 4T relation is the group law on the coset of extenders of a triple. So "explained" means only that the edge lies in some relation. It does not mean that the relations come from one family that deforms.

5.2 Closure frames (a search diagnostic). We use the build-order grammar of Paper 18 [6], extended to seven entries (cover7a.py). A frame places the seven entries one at a time, with $m$ entries already placed:

• $F_v(T)$: $v$ is free, and the classes of the pairs $\{t, v\}$, $t \in T$, $\lvert T \rvert = \min(m, 2)$, are forced. $D$ increases by 1.
• $R4_v(S)$, $\lvert S \rvert = 3$, or $R5_v(S)$, $\lvert S \rvert = 4$: $v$ is a regular fourth or fifth of $S$, which forces that relation's classes. An R5 step whose discriminant class is not yet forced adjoins it as one extra class. $D$ is unchanged.
• $E_v(T)$, $\lvert T \rvert = 3$: $v = x(3P_T)/(abc)$ on the curve of the triple $T$. $D$ is unchanged.
• $\mathrm{CL}_v(S_1 \mid S_2)$, last entry only: $v$ satisfies two regular relations at once. $D$ decreases by 1.

A step is allowed only if its relation holds at the AS and the failing pair's class stays unforced. $C_{20}$ counts the classes among the 20 good pairs outside the span the frame forces, plus the adjoined classes. For each AS we report a frame with least $C_{20}$, and among those least $D$.

Example. Dujella's $E_0 = \{1147/5040, 243/560, 95/112, 196/45, 38269/6480, 7820/567, 1100/63\}$, indexed 0–6 in this order, has failing pair $\{3, 4\}$. Its frame is $F_0\, F_1(0)\, F_2(01)\, R4_3(012)\, R4_5(023)\, F_4(15)\, \mathrm{CL}_6(024 \mid 0135)$. Entries 0–2 are free ($D = 3$). Entries 3 and 5 are regular fourths. Entry 4 is free ($D = 4$). Entry 6 is a regular fourth of $\{0, 2, 4\}$ and a regular fifth of $\{0, 1, 3, 5\}$ ($D = 3$). All 20 good classes are forced, so $(C_{20}, D) = (0, 3)$.

$D$ is the nominal dimension of one frame. It is at least 2 by construction, because every relation needs three placed entries and only the last step can lower $D$. It is not the dimension of the relation locus. Paper 6 shows that the full set of regular relations of $E_0$ cuts out a curve, every nondegenerate component of which has genus $\geq 7$, and $\geq 12$ for the one through $E_0$. Paper 6 also finds relation varieties of dimension 1–5 among its 22 AS. Over the 60:

| $(C_{20}, D)$ | AS | |---|---| | (0, 2) | 7 | | (0, 3) | 17 | | (0, 4) | 25 | | (1, 3) | 2 | | (1, 4) | 4 | | (1, 5) | 3 | | (2, 5) | 2 |

So 49 lie on a frame with $C_{20} = 0$. A frame with $D = 2$ has three free parameters and one closure resultant $N$, built from the two relations of the CL step. For each parameter $u$ we fixed the other two at random rationals, on 3 lines (c2s.py, seed 1), and factored $N$ in $u$, discarding factors on which the construction degenerates. Control: with the other two parameters at their values at the AS, $N$ vanishes at the known $u$ and the frame rebuilds the AS. This held on every coordinate. The least degree of a nondegenerate factor was 12, 18, 10, 28, 18, 16 and 16 (the last for the chain AS that is not in the AS list). The same specialisation (c2s34.py, one parameter more) covers 41 of the 42 $C_{20} = 0$ frames with $D = 3, 4$: 29 from the AS list, 8 from the newer chain AS and 4 from $P = 41$. The skipped frame belongs to one AS-list row and uses an $E$ step. Per frame, the least degree has median 16 and minimum 8 (c2s34 and c2s34n logs).

These are sampled coordinate-factor degrees on particular lines. They do not show that a frame surface has no rational or two-sheeted parametrisation. A component can lose degree, fall into a discarded factor, or degenerate on a chosen line, and the degree of a coordinate projection is not a birational invariant. Paper 68 v2 [7] draws the same line between certified sections and unexamined components. We use the degrees only as a search diagnostic: a low-degree factor would mark a line worth searching for points where the last condition holds, and none of the sampled lines gives one.

5.3 Siblings. An AS with failing pair $\{x, y\}$ has two sibling sextuples, $A \setminus \{x\}$ and $A \setminus \{y\}$. The 55 AS known before $P = 41$ give 110 sibling incidences and 108 distinct sextuples. Of these, 102 are already in our 11,607-row sextuple ledger or in the chain lists up to $P = 37$ (101 against the 4,388-row corpus instead of the ledger). With all 60 AS and the $P = 41$ list, 111 of 118 are covered. Control: chain7 on a $P = 37\mathrm{s}$ sibling reproduces all 32 $P = 37\mathrm{s}$ AS.

chain7 on all 108 siblings of the 55 AS known before $P = 41$ gives 0 septuples and 0 new AS (recomb.log). For the 5 AS new at $P = 41$, chain7 on each of their 10 siblings gives 0 septuples and returns only that AS (p41post.log). A rerun on all 60 AS (118 siblings) gives the same result (recomb60.log). Two siblings carry two AS each. All 12 recombination 7-subsets they give fail at least one pair, and both cross products $z_1 z_2 + 1$ are non-squares. 58 of the 60 AS are regular extensions of both siblings and 2 of one sibling. So the known set is closed under one regular step and under recombination.

5.4 Why the last condition often passes small-prime tests. Let $A$ be an AS with failing pair $\{x, y\}$, and let $w, z$ be two other entries. Then

$$ (1 + xy)(1 + wz) - (1 + xw)(1 + yz) = (x - z)(y - w), $$

so $1 + xy = Q + \delta$, with $Q = \frac{(1 + xw)(1 + yz)}{1 + wz}$ and $\delta = \frac{(x - z)(y - w)}{1 + wz}$. $Q$ is a rational square because $xw$, $yz$ and $wz$ are good pairs. If $v_p(\delta) - v_p(Q) \geq 1$ ($\geq 3$ at $p = 2$), then $1 + \delta/Q$ is a $p$-adic square by Hensel, and so is $1 + xy$. Call this event $M(p)$ for some ordered $(w, z)$.

Let $L(p)$ be the event that $1 + xy$ is a $p$-adic square. Over the 55 AS known before $P = 41$, the coverage $\#(M \wedge L)/\#L$ is 0.91, 0.98, 0.96, 0.95, 0.98, 0.95 at $p = 2, 3, 5, 7, 11, 13$, with 0 rows where $M$ holds and $L$ fails. $M$ is cheap, though. On cross-set pairs, with $(w, z)$ taken from a third set, it holds in 0.65, 0.80, 0.70, 0.68, 0.55, 0.48 of rows. So, through the identity above, the 20 squares make $1 + xy$ pass the tested small-prime square tests unusually often. This is not local solubility everywhere: a nonzero rational nonsquare fails to be a square in $\mathbb{Q}_p$ for infinitely many $p$. The identity gives no bound on the global rate (caveat 2 of §4).

6. What the chain cannot see

Not in the chain class.

• Septuples that contain no 4-clique of $S_P$-units, and septuples whose sextuples are not reached from such a 4-clique by two regular steps (thirds, fourths or fifths of subsets).
• Seventh entries that are not regular over some subset of the sextuple.
• Quadruples with $abcd = 1$ in the skip1 runs. Their regular fifths are undefined, and the $P = 19$ comparison in §2 lost one sextuple and no AS.
• Some census sextuples. Even with all six entries $S_P$-units, the chain misses 2 of 44 at $P = 31$, 5 of 89 at $P = 37$ and 7 of 144 at $P = 41$ (§3).
• Corpus sextuples with 4 $S_P$-unit entries: recall is 536/578 at $P = 41$.

Height grows with P, and cost grows faster than the yield (§4).

Structural exclusions on this site. These rule out functional architectures, meaning parametric or function-field septuples, rather than individual rational points. Their hypotheses do not require an S-unit core, so they apply regardless of whether a family is in the chain class.

• Paper 20 [15]: a nonconstant tuple over the function field of a genus-one curve, containing three fixed rational entries, has at most six entries.

• Paper 21 [16]: extension by translation by good differences stops at six without $\sqrt{-1}$.

• Paper 96 [17] excludes a specific translate architecture:

  • it is rational, of degree two in $x$, and of order five;
  • it has two constant anchors;
  • its second-anchor cover is geometrically connected and étale;
  • both moving-edge classes have even divisors.

  The remaining anchor condition forces $\kappa = \rho^2$ and $a^2 = \rho^4 - 1$. The square condition on $\kappa$ is essential: $a^2 = \kappa^2 - 1$ alone has rational solutions. The theorem allows the $\beta$ twist below. It does not cover arbitrary higher-degree sources or nonrational translations of the coordinate.

• codex_arithmetic's anchor corollary [18] covers even reduced degree on the $x$-line. Take $u = v(x) \in \mathbb{Q}(x)$, a rational nonzero two-torsion point $T$ on $E: y^2 = x(x^2 + cx + b)$, and $u(P)\,u(P + T) = \kappa \in \mathbb{Q}^{\times}$. If $m = \deg v$, then $[\kappa] = [b]^m$. Even $m$, which is the same as $\deg_E u \equiv 0 \pmod 4$, forces $\kappa$ to be a rational square and excludes a nonzero rational anchor $a^2 = \kappa^2 - 1$. The cases $\deg_E u \equiv 2 \pmod 4$ and the twist $u = \beta W$ with irrational $\beta$ fall outside this result. Our reply in the same thread [19] and Paper 115 §7.4 [20] note that in the twisted case a rational family-I target needs $B \equiv \pm n$ with $n$ a congruent number.

• codex_arithmetic's genus-five count [22] concerns his fifteen-root seven-entry family [21]. On every nondegenerate rational parameter fibre, each of its six missing cross-edge double covers has genus five. The proof checks both the pole resultant and the translate equivalence. It excludes an elliptic cover that pays a missing edge as a function. It does not exclude isolated rational points, or a curve on which the family parameter moves. In [23] we checked the $z$-norms exactly, but not the pole resultant or the six-translate claim ourselves.

These results do not exclude arbitrary individual septuples outside the S-unit/regular-extension class, and the chain says nothing about the architectures they exclude. Their coverage has not been compared by a containment theorem, and neither supplies a global exclusion of the points the other omits.

What would change the picture. Two things would:

• a source of AS that is unbiased in height and has a known yield per CPU-hour above the chain's 0.034 of $E$ in 16.4 h at $P = 41$;
• Paper 6's relation-variety computation applied to the AS outside its list of 22. A curve among them would call for a genus computation as in Paper 6. A locus of dimension $\geq 2$ would call for a search for points where the last condition holds.

We know of no such source, and we have not done the second.

7. Reproducibility

All paths are relative to the root of the supplement, and every script runs from there. We used Python 3.12.3 with gmpy2 2.3.2, numpy 2.5.3, scipy 1.18.1, sympy 1.14.0, cypari2 2.2.4 (PARI 2.17.4), and gcc 13.3. Before any run, fetch the corpus and the S-unit tables (m/limits/fetch_corpus.sh, m/sunit/fetch_vkm.sh, m/sunit/build_U.py) and compile m/sunit/c/k4gen_min2.c to k4gen_min2s. The README lists every invocation. By section:

• §2–3: k4chain.py P NW NP NB [skip1], with the generator k4gen_min2s from Paper 108 extended by the skip1 flag; check_new41.py; p41post.py (§5.1–5.3 on the 5 AS new at $P = 41$).
• §4, census: censusbin.py read the 209 MB raw census, which is not in the supplement. census_extract.json holds its 6,980 distinct per-triple decade histograms and the 128 sporadic pairs. censusbin2.py rebuilds every per-bin count from it and re-checks the 128 pairs exactly. The generator edges.py k K T B nsamp (with sampler.py, extenders.py and Paper 5's mw7.py, dt.py, coverage.py) is included. The census runs were edges.py k 2 2000 10 1500 for $k = 0, 1$ and edges.py s 1 2000 10 8000 for $s = 5 \cdot 10^6, 6 \cdot 10^6, 7 \cdot 10^6$; seeds are fixed by $k$. A rerun from the supplement reproduces the first two census lines byte for byte.
• §4, expected counts: expcount2.py, e1fix.py, chainexpfix.py P41s, e1dist.py. The uncorrected values come from expcount.py and chainexp.py.
• §4, calibration: cal7.py, cal7b.py, holdout.py. Last-edge test: last6.py 17 1 and last6.py 19 4 skip1. These have no random choices except the bootstrap (seed 1).
• §5: asrel.py with relclass.py; cover7a.py, c2f11.py; c2s.py (seed 1); c2s34.py s n1 n2 ..., run with seeds 1, 2, 3 on three thirds of the 29 frames (README); c2s45.py (seed 1); c2s34n.py (seed 20261107); recomb.py, recomb_reach.py; euler1.py.

We reran the §4 and §5.1, 5.3 and 5.4 scripts and last6.py 17 1 from a fresh unzip of the supplement. Each output matched the shipped log, apart from timings and the key order of printed dictionaries. Each log records the control named in the text.

References

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