Hunt, not a result. Nothing here is evidence for or against RH (docs/08).
Verdict: the arm is settled, and it is a rediscovery. The matrix $M_{ij} = a_i + a_j$ with rank $\le 2$ and $S$-unit off-diagonal entries is exactly the object of Alon and Solymosi, "Rank of matrices with entries from a multiplicative group", Corollary 10, whose proof is four lines long and constructs the same matrix. It gives $|A| \le f(\mathrm{rank}\,\Gamma)$, which answers the qualitative question and is catastrophically worse than $2^k$: reading their explicit form, $e^{256k}$ against $3\cdot 2^{k-1}$.
Rank 2 is however special, and the general machinery is not needed for it. Two rows of $M$ collapse the whole problem to a single two-variable $S$-unit equation, giving the explicit and much better
$$g(k) \;\le\; 2 + 3\cdot 7^{2k+3},$$
base $49$ per prime. Still worse than base $2$. The arm does not improve $3\cdot 2^{k-1}$ and, with the best counting bounds now known for $S$-unit equations, it cannot.
Landing note (2026-10-21). Landed on main as Hunt #97 from a local-only branch, support/99fc686a, that was never pushed; PR #144 counted this arm among the seven that "produced no branch at all". The run's files are as it wrote them apart from punctuation and the dated notes marked as added on landing.
Grading (added on landing, 2026-10-21). Alon and Solymosi were read from their PDF, so §4's account of Theorems 1 and 9 and Corollary 10 is primary. The other two constants the verdict uses were not: the run took Evertse's $3\cdot7^{[K:\mathbb{Q}]+2|S|}$ and Beukers–Schlickewei's $2^{8(r+2)}$ from search summaries and rates them medium-low in its own HANDBACK.json. The Evertse form agrees with Győry–Stewart–Tijdeman 1986's quotation of it, $3\cdot7^{2w+3}$, which the sibling literature arm read in primary text (Hunt #143, support_fcffa3b9 T4). The Beukers–Schlickewei form differs from the $2^{8r+8}$ quoted on main (Hunt #104, and Hunt #143's T6, itself secondary), and neither is checked against the paper here, so the §2 table is hedged in place (its base per prime, $2^{16}$, is the same under either form). The two-row collapse (§1) is this lab's ordinary argument, unreviewed and pending external verification, and so are the rectangle-condition expansion and the nondegeneracy analysis of §3. The §6 table is exact arithmetic on the seven r_186989 witnesses, a check of the mechanics and not of any bound. In the terms of ALIGNMENT.md §5 the verdict is restricted to the two-row $S$-unit route and to the counting bounds now known: with them it cannot beat base 2. That is not an obstruction to the route, whose improvement is exactly the open counting problem of §5.
Overlap with arms already on main. The two-row collapse is the reduction of Hunts #96 (support_d5d5ccae) and #104 (support_f3ab3e34); #104 notes that it is how the $c\log|A|$ bounds are proved. It is also Győry–Stewart–Tijdeman 1986 §3, quoted verbatim by Hunt #143 (landed with this arm), which answers §7's first question. Two sibling arms landed with this one derive it again (Hunts #99 and #106). §3's four-element route is the reduction of Hunts #93, #95 and #105, which name the same missing fibre count; Hunt #98 (support_d0f0aefb Theorem 4, landed with this arm) makes a three-anchor version injective. This arm's own contributions are the prior-art identification of the rank-2 matrix with Alon–Solymosi's Corollary 10, the reduction of their rectangle condition to the distinctness of $A$, the observation that their Theorem 1 does not apply (diagonal $S$-units in all seven witnesses), and the pricing of the three-term route at about $10^{1341}$ per prime.
Reproduce with .venv/bin/python hunts/support_99fc686a/probe.py (stdlib only, about a second). It writes probe_out.json (its docstring says results.json); results.json is the run's hand-written summary. Re-run on landing, probe_out.json reproduced byte for byte.
1. The reduction, stated and proved
Let $S$ be $k$ primes, $U \subset \mathbb{Q}^*$ the group of $S$-units ($U \cong \{\pm 1\} \times \mathbb{Z}^k$, rank $k$). Let $A \subset \mathbb{Z}_{>0}$ be finite with $a+b \in U$ for all distinct $a,b \in A$.
Lemma (two-row collapse). Fix distinct $a,b \in A$ and put $c = a-b \ne 0$. For every $d \in A \setminus \{a,b\}$ set $x_d = a+d$, $y_d = b+d$. Then $x_d, y_d \in U$ and
$$x_d - y_d = c \quad\text{for every } d,$$
i.e. $(x_d,y_d)$ solves $\tfrac{1}{c}X - \tfrac{1}{c}Y = 1$ in $(X,Y) \in U^2$. The map $d \mapsto (x_d,y_d)$ is injective, since $x_d = a+d$ determines $d$. Hence
$$|A| - 2 \;\le\; N_{\mathbb{Q}}(k) := \max_{\alpha,\beta \in \mathbb{Q}^*} \#\{(X,Y) \in U^2 : \alpha X + \beta Y = 1\}. \qquad\square$$
The step that matters is that $x_d - y_d$ does not depend on $d$, and that is precisely the rank-$2$ statement: every column of $M$ lies in $\mathrm{span}\{(a_i)_i, (1)_i\}$, so any two rows of $M$ differ by a constant vector. The low rank is used, and nothing else is. No minors, no determinant identity, no matrix rigidity theorem.
The uniformity that makes this work. $c = a - b$ is in general not an $S$-unit, so the equation has arbitrary rational coefficients. What is needed is a solution count depending only on $k$ and not on the coefficients, and that is exactly the form Evertse's theorem takes.
2. Explicit bounds, and the comparison the brief asked for
| route | bound on $g(k)$ | base per prime | source |
|---|---|---|---|
| Erdős–Turán 1934 | $3\cdot 2^{k-1} - 1$ | 2 | the incumbent |
| two-row collapse + Evertse 1984 | $2 + 3\cdot 7^{2k+3}$ | 49 | §1 above |
| two-row collapse + Beukers–Schlickewei 1996 | $2 + 2^{8(2k+3)}$ | $65\,536$ | rank of $U^2$ is $2k$ |
| four-element identity + subspace theorem, 3 terms | not even a bound on $\lvert A\rvert$ without more work | $\sim 10^{1341}$ | see §3 |
| Alon–Solymosi Cor. 10, as published | $\exp(256\,k)$ for large $k$ | $e^{256}$ | their $\log n \le r d^8$ with $d = 2$, $r = k$ |
Evertse (1984): $\alpha X + \beta Y = 1$ has at most $3 \cdot 7^{[K:\mathbb{Q}] + 2|S|}$ solutions in $S$-units. Over $\mathbb{Q}$ with $S = \{\infty, p_1,\dots,p_k\}$ this is $3\cdot 7^{2k+3}$.
Hedged on landing (2026-10-21). Both upper-bound constants in this table came from search summaries, not from the papers. Evertse's form agrees with Győry–Stewart–Tijdeman 1986's quotation of it, $3\cdot7^{2w+3}$, read in primary text by the sibling literature arm (Hunt #143, T4). The Beukers–Schlickewei row uses the form $2^{8(r+2)}$, while main quotes $2^{8r+8}$ (Hunt #104; Hunt #143's T6, secondary), and neither was checked against the paper here; read that row's exponent as unpinned. No conclusion of this file depends on it, since the Evertse row is the better one either way.
Answer to "does it improve $2^k$": no, by a factor of about $(49/2)^k$ at best. The two-row collapse beats the published Alon–Solymosi constant by a very large margin, and is still an order of magnitude worse in the exponent than a proof from 1934.
3. The nondegeneracy issue, isolated exactly
The brief asked for this precisely, and there are two different answers depending on which reduction is used.
In the two-row collapse there is no nondegeneracy issue at all. A two-term equation $\alpha X + \beta Y = 1$ has proper subsums $\alpha X$ and $\beta Y$, both nonzero because $X, Y \in U$ are units. Evertse's count applies to every solution. Nothing has to be assumed about $A$ beyond $a \ne b$.
In the general-rank argument the issue is real and is exactly the Alon–Solymosi rectangle condition. Their Theorem 1 requires the diagonal entries $m_{jj} \notin \Gamma$. Here $m_{jj} = 2a_j$, and this fails: the probe finds $S$-unit diagonal entries in every one of the seven r_186989 witnesses (1, 1, 2, 5, 4, 7, 9 of them for $k=1..7$). Their Theorem 9 replaces the hypothesis by
$$a_{jj}a_{ik} \ne a_{jk}a_{ij} \qquad (i < j \ne k),$$
which for $M_{ij} = a_i + a_j$ reads $(2a_j)(a_i+a_k) \ne (a_j+a_k)(a_i+a_j)$, and expands to $(a_j - a_i)(a_j - a_k) \ne 0$. So the nondegeneracy condition is exactly the distinctness of the elements of $A$, and is free.
In the four-element route the same thing happens. For distinct $i,j,l,m$, rank $2$ gives $(a_i+a_j) + (a_l+a_m) = (a_i+a_l) + (a_j+a_m)$; dividing by $a_i+a_l$ yields $X + Y - Z = 1$ with $X,Y,Z \in U$. Its vanishing proper subsums would be $X + Y = 0$ (impossible, both positive), $X - Z = 0$ (forces $a_i = a_m$) and $Y - Z = 0$ (forces $a_l = a_j$). Every such solution is nondegenerate, again exactly because the elements are distinct. The three-term route is nonetheless strictly worse: the subspace-theorem count $A(m,r) \le (8m)^{4m^4(m+mr+1)}$ (Amoroso–Viada, as quoted by Alon–Solymosi) gives at $m=3$ a bound $24^{324(3k+4)}$, i.e. a base of about $10^{1341}$ per prime, and the map from $4$-subsets of $A$ to solutions is not injective (fixing $(X,Y,Z)$ leaves a one-parameter family of $4$-tuples), so it does not even bound $|A|$ without further work.
4. What the literature actually contains
Searched and read:
- N. Alon and J. Solymosi, "Rank of matrices with entries from a multiplicative group." Theorem 1: for $\Gamma \le \mathbb{C}^*$ of rank $\le r$ and $M$ with off-diagonal entries in $\Gamma$ and diagonal entries outside it, $\mathrm{rank}(M) \ge (\log n / r)^{1/8}$ for large $n$. Theorem 9 weakens the diagonal hypothesis to the rectangle condition. Corollary 10 is our problem: "if $\mathrm{rank}(\Gamma) \le r$ and $A + A \subset \Gamma$ then $|A| \le f(r)$", proved with $m_{ij} = a_i+a_j$ and the observation that its rank is at most two. Their $A+A$ includes the diagonal $2a$; our hypothesis is weaker (off-diagonal only), and Theorem 9 covers it because the rectangle condition holds regardless.
- O. Roche-Newton and D. Zhelezov, cited there: the same statement for $A - A \subset \Gamma$, also via the subspace theorem.
- Evertse–Schlickewei–Schmidt, with the Amoroso–Viada bound $A(m,r) \le (8m)^{4m^4(m+mr+1)}$, is the engine underneath all of it.
- J.-H. Evertse (1984) and Beukers–Schlickewei (1996) for the two-variable counts used in §2.
- arXiv 2602.07545 (Erdős–Turán theorem and Eulerian integers, 2026) restates the classical bound as $3\cdot 2^{k-1}$ and does not improve it; it extends the lower-bound technique to $a^2 \pm ab + b^2$ via a lemma of Győry, Sárközy and Stewart.
Nothing in this literature improves the base $2$. The subspace theorem is ineffective and its counting bounds are enormous; it buys qualitative finiteness in settings where elementary arguments give none (distance sets, $A-A$, Eulerian integers), and it loses badly wherever an elementary argument already exists. Erdős #126 is such a place.
5. The one thing worth taking forward
The two-row collapse is a clean equivalence-flavoured reduction with the implication direction checked:
If $N_{\mathbb{Q}}(k) = \exp(o(k))$, then $g(k) \le 2 + N_{\mathbb{Q}}(k) = \exp(o(k))$, and Erdős #126 is true.
The converse does not follow: $g(k)$ small says nothing about $N_{\mathbb{Q}}(k)$, since a solution set of a unit equation need not come from a set with all pairwise sums smooth. So this is a one-way reduction of #126 to a problem in Diophantine approximation.
Status of the target, which is what makes this interesting rather than circular:
- best known upper bound $N_{\mathbb{Q}}(k) \le 3\cdot 7^{2k+3}$, exponential;
- best known lower bound, Erdős–Stewart–Tijdeman: $N_{\mathbb{Q}}(k) \ge \exp\!\big((4+o(1))(k/\log k)^{1/2}\big)$, which is $\exp(o(k))$.
The lower bound does not obstruct the hypothesis. The gap between $\exp(\Theta(k))$ and $\exp(k^{1/2+o(1)})$ for two-term $S$-unit equations is open, and closing it downward would settle Erdős #126. Whether this reduction is already recorded somewhere we did not establish; it is elementary enough to be folklore, and we make no novelty claim for it. What we do claim is provenance: it was derived here, and it is checked. (Annotated on landing, 2026-10-21: it is recorded. Győry–Stewart–Tijdeman 1986 §3 uses exactly this step, "each element $a_i$ yields a solution $x=a_i+b_1$, $y=a_i+b_2$, $z=1$ of the equation $x-y=(b_1-b_2)z$", quoted from the primary text by Hunt #143, support_fcffa3b9 §2(b).)
6. Verification
probe.py re-checks all seven r_186989 witnesses from scratch by trial division. All pass:
| check | result |
|---|---|
| every off-diagonal sum is an $S$-unit (the hypothesis) | true, 7/7 |
| $\mathrm{rank}_{\mathbb{Q}} M = 2$ exactly (Fraction Gaussian elimination) | true, 7/7 |
| rectangle condition holds for all $i<j\ne k$ | true, 7/7 |
| some diagonal entry $2a_j$ is an $S$-unit (so Thm 1 does not apply) | true, 7/7 |
| two-row reduction valid: $x_d,y_d \in U$ and $x_d-y_d=a-b$ | true, all ordered pairs |
| $d \mapsto (x_d,y_d)$ injective | true, all ordered pairs |
The $k=7$ witness $\{1,3,5,6,7,11,15,19,21,29,49\}$ with $a=1,b=3$ forces 9 distinct solutions of $X - Y = -2$ in $\{2,3,5,7,11,13,17\}$-units: $(6,8),(7,9),(8,10),(12,14),(16,18),(20,22),(22,24),(30,32),(50,52)$. Evertse's bound permits $3\cdot 7^{17} \approx 7\times10^{14}$. The counting theorem is not close to tight on the instances this problem produces, which is the honest reason the route loses.
7. What we could not settle
- Whether the two-row collapse is in the literature. We found the rank-2-matrix framing (Alon–Solymosi) but not this specialisation. A targeted search of Győry–Sárközy–Stewart and of the Erdős problem list commentary was not run. (Answered on landing, 2026-10-21: the specialisation is Győry–Stewart–Tijdeman 1986 §3; see the note at the end of §5.)
- Whether using more than two rows helps. Fixing $m$ elements gives each $d$ a point in the intersection of $m-1$ unit-equation solution sets. We know of no theorem whose bound decreases with the number of simultaneous equations, and we did not look for one. This is the only place in the arm where an improvement could plausibly live.
- Any bound with base below 2. Not attempted; there is no candidate.
The doors
This run measured a ceiling (which exponential base each route can reach), so it owes the list.
1. Active constraints at the optimum.
| Rank | What binds | Evidence |
|---|---|---|
| 1 | The $S$-unit solution-count bound itself. Everything downstream is a single application of it. | $g(k) \le 2 + N_{\mathbb{Q}}(k)$ is one line; all the loss is in $N_{\mathbb{Q}}(k) \le 3\cdot7^{2k+3}$. |
| 2 | Ineffectivity and looseness of the subspace theorem. | Measured slack at $k=7$: 9 solutions forced against $7\times10^{14}$ permitted. |
| 3 | Using only two rows. The other $n-2$ rows of $M$ are thrown away. | Simultaneous systems are untouched, see §7. |
2. Frozen-constant inventory.
| Frozen | Value | What relaxing it trades | ||||
|---|---|---|---|---|---|---|
| number of fixed elements $(a,b)$ | 2 | The door with genuine trade shape. $m$ fixed elements gives $m-1$ simultaneous unit equations for the same unknown $d$; a bound decaying in $m$ would beat everything here. Costs: needs a theorem that does not currently exist. | ||||
| choice of the pair $(a,b)$ | arbitrary | Nothing. The bound is uniform over coefficients, which is why any pair works. | ||||
| the counting theorem | Evertse 1984 | Beukers–Schlickewei is worse over $\mathbb{Q}$ (base $65536$ vs $49$). Any future improvement in $N_{\mathbb{Q}}(k)$ transfers verbatim. | ||||
| ambient field | $\mathbb{Q}$ | Over a number field $K$ the same proof gives $ | A | \le 2 + 3\cdot7^{[K:\mathbb{Q}]+2 | S | }$. Free generalisation, no extra work. |
| the group $\Gamma$ | $S$-units | The reduction only uses that $\Gamma$ is finitely generated. Works for $A$ in any field with $A+A$ in a rank-$r$ group. |
3. Information class. The two-row collapse stays inside the data the family already reads (integers and the smoothness of their pairwise sums), and its ceiling is exactly the ceiling of two-term $S$-unit counting. Door 1 (more rows, simultaneous unit equations) requires reading more: it needs a Diophantine theorem about systems, not about a single equation, and no such theorem is cited by anyone in this literature. That is where the arm's remaining value sits, and it is a research programme rather than a probe.