RESULTS: the arithmetic difference graph arm (support run 6da47ca3)

Hunt, not a result. Nothing here is evidence for or against RH (docs/08).

Answer to the parent, in three lines.

1. The bipartite picture is correct but the bipartiteness is not what carries the weight. All the quantitative force sits in one sub-pattern, $K_{2,m}$, and it converts the problem into a two-term $S$-unit equation with a fixed coefficient. That gives an unconditional explicit bound $g(k) \le 3\cdot 7^{2k+3} + 2$, hence $f(n) \ge 0.2569\log n - 1.79$.
2. Erdős #126 follows from the standard (open) conjecture that a two-term $S$-unit equation has $\exp(o(k))$ solutions. (The run calls it the standard conjecture and cites no source; annotated on landing, see §4.) That is the theorem-level reduction asked for, and it is the honest location of the wall: every known uniform upper bound on that count is $\exp(O(k))$, and $\exp(O(k))$ is exactly $\log g(k) = O(k)$, which is the classical statement, not the target.
3. The extra within-part pair-sum structure yields no improvement in the exponent. It buys one real thing, a free nondegeneracy guarantee, and nothing else, because using it forces three-term unit equations whose best uniform bounds are worse in $k$ by eleven orders of magnitude in the slope.

The audit of hunts/r_186989/RESULTS.md found it sound in every checkable claim, reproduced its table independently, and refutes one of its four loose threads with an explicit counterexample.

Landing note (2026-10-21). Landed on main as Hunt #106 from a local-only branch, support/6da47ca3, 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, the dated notes marked as added on landing, and one word: the noun in "nondegeneracy guarantee" is a landing substitution, made here, in probe.py, in HANDBACK.json and in the regenerated results.json, because the run's word was a form of the one reserved to zeta/rigor.py and the Lean arm.

Grading (added on landing, 2026-10-21). Theorem 1, Corollaries 2 and 3 and Lemma 4 are this lab's ordinary arguments, unreviewed and pending external verification; none is kernel-checked. Corollary 2's constant uses Evertse's $3\cdot7^{d+2s}$ "as I recall it", not re-read in the run; it agrees with Győry–Stewart–Tijdeman's quotation of Evertse, which the sibling literature arm read in primary text (Hunt #143, T4), but this arm did not check it. §4's "standard conjecture" and "believed conjecture" cite no source and are annotated in place. §8 rests on the published abstracts, since both PDFs came back undecoded, and the run says so. The §6 table is exhaustive inside bounded boxes, so every entry is a lower bound. Disposition (ALIGNMENT.md §5): the attempt is unresolved. The reduction is sufficient, not necessary; Corollary 2 re-proves the classical $f(n)\gg\log n$ with a tracked constant and moves nothing; the verdict that the within-part structure buys no improvement in the exponent is a comparison with the known three-term bounds, not an obstruction; and r_186989's loose thread 3 is a refuted candidate. probe.py was re-run on landing after the rewording; results.json changed in that one string and nowhere else.

Overlap with arms already on main. Theorem 1 is the reduction of Hunts #96 (support_d5d5ccae) and #104 (support_f3ab3e34), and of Hunts #97 and #99 landed with this arm; Győry–Stewart–Tijdeman 1986 §3 has it too (quoted by Hunt #143). Corollary 2's explicit $3\cdot7^{2k+3}$ is #96's constant. Lemma 4 is the nondegeneracy step of the four-element reductions of Hunts #93, #95 and #105. Konyagin–Soundararajan's lower bound is also cited on main by Hunt #102 (from the abstract). The refutation of loose thread 3 (§7) was on main before this arm landed (Hunts #93, #95, #101, #103), and the OEIS thread was closed by the literature arm (Hunt #143). This arm's own contributions are the exactness of the $K_{2,m}$ correspondence, the measured crown-versus-$K_{2,m}$ loss of §6 ($g_N(5)=8$ against $N_2^{(N)}(5)\ge 141$), and the Győry–Hajdu–Tijdeman specialisation of §8, that their theorems are representability results with no bound of the form $|A|\le F(|S|)$.

Reproduce with .venv/bin/python hunts/support_6da47ca3/probe.py (a few seconds, stdlib only, writes results.json).

0. Notation

$S = \{p_1,\dots,p_k\}$ a set of $k$ primes, $\mathbb{Z}_S$ the rationals with denominators supported on $S$, $\Gamma = \mathbb{Z}_S^*$ the group of $S$-units (rank $k$, torsion $\{\pm 1\}$). $A$ is a set of distinct positive integers with every off-diagonal sum $a+b$ an $S$-unit, and $g(k) = \max |A|$ over all such $A$ and all $S$ with $|S| = k$. Following r_186989, Erdős #126 ($f(n)/\log n \to \infty$) is exactly $\log g(k) = o(k)$.

1. The bipartite pattern, and what is actually in it

Put $B = -A$ and take the vertex set $A \sqcup B \subset \mathbb{Z}$. Then for $a \in A$ and $-b \in B$,

$$a - (-b) = a + b ,$$

so the hypothesis says exactly: in the $S$-unit difference graph $\mathcal{G}S(A \cup -A)$, the crown graph $K{n,n}$ minus its perfect matching is a subgraph. The parent's framing is right. Two riders that matter, and the second is the load-bearing one.

Rider 1: subgraph, not induced subgraph. Nothing constrains $a_i - a_j$ (within-part differences) or $2a_i$ (the removed matching). The Győry–Hajdu–Tijdeman notion of representation is the induced one, so their theorems are about a strictly stronger relation than ours. For an upper bound on $n$ that is the safe direction (a subgraph statement is what we may use); for lower bounds it means their constructions do not transfer.

Rider 2: the crown is triangle-free, and that is normally fatal. The standard route from a difference graph to a unit equation needs a triangle: $x,y,z$ pairwise joined gives $(x-y) + (y-z) = (x-z)$, i.e. $u + v = w$ with $u,v,w \in \Gamma$, i.e. after division the coefficient-free equation $u/w + v/w = 1$. A bipartite graph has no triangle, so that route is closed here. What replaces it is the 4-cycle, and a 4-cycle in a difference graph gives a two-term equation with a fixed inhomogeneous coefficient: $u - v = c$ with $c$ a fixed non-zero rational. This is the whole technical content of the arm.

2. The reduction

Define, for $k \ge 1$,

$$N_2(k) \;=\; \max_{|S| = k} \; \max_{c \in \mathbb{Q}^*} \; \#\{(u,v) \in \Gamma^2 \;:\; u - v = c\} .$$

(The inner maximum is over all non-zero rationals $c$, not just $S$-units.)

Theorem 1 (reduction). $\;g(k) \le N_2(k) + 2$.

Proof. Let $A$ be admissible with $|A| = n \ge 3$. Fix two distinct elements $a_1, a_2 \in A$ and set $c = a_1 - a_2 \ne 0$. For every $x \in A \setminus \{a_1,a_2\}$ both $u_x = x + a_1$ and $v_x = x + a_2$ are $S$-units (they are off-diagonal sums), and $u_x - v_x = c$. The map $x \mapsto (u_x, v_x)$ is injective, since $x = u_x - a_1$. Hence $n - 2 \le N_2(k)$. $\square$

Three things to notice about this proof, because they are the answer to the parent's three sub-questions.

• It uses only the sub-pattern $K_{2,m}$: two vertices of one part joined to $m$ vertices of the other. Neither the bipartiteness, nor the removed matching, nor the antisymmetric labelling $B = -A$, nor positivity is used. Theorem 1 therefore holds verbatim for any $A \subset \mathbb{Q}$ all of whose differences to a fixed 2-element set are $S$-units.
• Conversely the correspondence is exact: given $x_1, x_2$ with $c = x_1 - x_2$, the vertices $y$ joined to both are in bijection with the solutions of $u - v = c$. So $N_2(k)$ is the largest $m$ with $K_{2,m} \subseteq \mathcal{G}S(\cdot)$ for some $|S| = k$. Nothing is thrown away at this step; the loss is entirely in going from the crown down to $K{2,m}$ (measured in §5).
• This is also, I believe, the reason the GHT threshold is cubical (= subgraph of a hypercube): in $Q_d$ two vertices have at most two common neighbours, so $K_{2,3} \not\subseteq Q_d$. The crown $K_{n,n}$ minus a matching has two vertices with $n-2$ common neighbours, so it is cubical exactly for $n \le 4$ (and the $n=4$ crown is $Q_3$). The one place where our pattern leaves the cubical world is precisely the place where the quantitative content lives.

3. The unconditional consequence, with the constant tracked

Corollary 2. $g(k) \le 3\cdot 7^{2k+3} + 2$, hence $\log g(k) \le (2\log 7)\,k + \log 3 + 3\log 7 = 3.8918\,k + 6.936$, hence for $n \ge 3$ $$f(n) \;\ge\; \frac{\log(n-2) - \log 3}{2\log 7} - \frac{3}{2} \;=\; 0.25695\log n - 1.782 + o(1).$$

Input. Evertse (Invent. Math. 75 (1984), 561–584): for a number field $K$ of degree $d$ and a set $S$ of $s$ places of $K$ containing the archimedean ones, and fixed $\alpha,\beta \in K^$, the equation $\alpha x + \beta y = 1$ has at most $3\cdot 7^{d+2s}$ solutions $x,y \in O_S^$. Uniform in $\alpha,\beta$, which is what lets $c$ be arbitrary. Apply with $K = \mathbb{Q}$, $d = 1$, $s = k+1$, and $(\alpha,\beta) = (1/c, -1/c)$ applied to $u - v = c$. So $N_2(k) \le 3\cdot 7^{2k+3}$.

(Beukers–Schlickewei's $2^{8(r+1)}$ for the coefficient-free equation gives $2^{8(k+2)}$ after absorbing $c$ into the group, slope $8\log 2 = 5.545$; Evertse's slope $3.892$ is better here. Both are $\exp(O(k))$, which is the point.)

Corollary 2 is a re-proof of the classical $f(n) \gg \log n$ with an explicit constant, obtained from the bipartite picture rather than from whatever the original argument was. It does not move the problem, and I want to be flat about that: $\log g(k) = O(k)$ is the known statement, and the target is $o(k)$.

4. Where the wall actually is, stated as a conditional

Corollary 3 (the reduction the parent asked for). If $\log N_2(k) = o(k)$, then $\log g(k) = o(k)$, i.e. Erdős #126 holds.

So Erdős #126 is implied by a purely Diophantine counting statement: two-term $S$-unit equations, uniformly in the coefficients, have subexponentially many solutions in $|S|$. That statement is open, and its status is worth stating precisely because it decides whether this arm is fundable:

Every known lower bound is $\exp(k^{\theta})$ with $\theta < 1$. So $\log N_2(k) = o(k)$ is consistent with everything known, and it is what the $S$-unit-equation literature expects to be true. Erdős #126 is therefore a consequence of a believed conjecture in a different field, and the 92-year wall on it is the same wall as the gap between $\exp(O(k))$ and $\exp(k^{0.59})$ in unit-equation counting.

Annotated on landing (2026-10-21). "What the literature expects" and "a believed conjecture" cite no source, and the run's second caveat below says as much. The sibling literature arm looked for exactly this statement and did not find it in a source it read (Hunt #143, support_fcffa3b9 §5, first bullet, which names Evertse–Győry, Unit Equations in Diophantine Number Theory, Ch. 6, as the place to check). Read the two sentences above as the run's reading of the field, not as a citation: what is established is the table (every known lower bound is $\exp(k^{\theta})$ with $\theta<1$), so $\log N_2(k) = o(k)$ is consistent with everything known. Whether it is a stated conjecture of that field is open.

Two honest caveats.

• Sufficient, not necessary. The reduction throws away most of the crown (§5 measures how much), so $\log g(k) = o(k)$ could be provable without $\log N_2(k) = o(k)$.
• The conjecture is needed uniformly in $c$. The many-solution constructions are built for particular equations, and I did not verify in-session whether the expected subexponential truth is asserted uniformly in the coefficients in the literature. Treat "expected truth" as my reading, not as a citation.

5. Does the within-part pair-sum structure buy anything? No.

The parent's specific question. Three attempts, all measured or argued:

(a) More anchor pairs. The crown gives Theorem 1 for every pair $(a_i,a_j)$, so $n - 2 \le \min_{i<j} \#\mathrm{Sol}(a_i - a_j)$. A minimum of quantities each bounded only by the uniform $N_2(k)$ is still bounded only by $N_2(k)$. No gain.

(b) More anchors at once. Using $a_1,\dots,a_j$ as anchors asks for $x$ with $x + a_1,\dots,x+a_j$ all $S$-units: the $S$-integral points on $\mathbb{P}^1$ minus $j+1$ points. Every known bound for that is proved by reducing to two-term unit equations and so inherits $\exp(O(k))$; the gain is in the constant, not the exponent, and the constant is already the min in (a).

(c) Use the within-part sums as unknowns. This is where the extra structure genuinely appears, and it does give one real thing:

Lemma 4 (free nondegeneracy). For distinct $a,b,c,d \in A$, the relation $$(a+b) + (c+d) - (a+c) - (b+d) = 0$$ is a non-degenerate $S$-unit relation: no proper subsum vanishes.

Proof. All four terms are positive, so any vanishing proper subsum contains one $+$ and one $-$ term. The four possibilities force $b=c$, $a=d$, $a=d$, $b=c$ respectively; the two triple subsums are non-zero because $(a+b)+(c+d) > \max\{a+c, b+d\}$. $\square$

Nondegeneracy is normally exactly where these arguments die, so this is worth recording. It is nonetheless useless for the exponent: after dividing by $a+c$, Lemma 4 gives a solution of $X + Y - Z = 1$ in $\Gamma$, and the best uniform bound for non-degenerate solutions of a three-term unit equation in a group of rank $r$ is Evertse–Schlickewei–Schmidt's $\exp\big((6n)^{3n}(r+1)\big)$ with $n = 3$, i.e. $\exp(1.98\times 10^{11}\,k)$. Even after $\binom{n}{4}$ quadruples inject into that solution set, the resulting bound $\log n \lesssim 5\times10^{10}k$ is worse than Corollary 2 by eleven orders of magnitude in the slope. The within-part structure is a qualitative gift and a quantitative loss.

6. How lossy the reduction is, measured

$g$ and $N_2$ measured inside a bounded universe (both are lower bounds on the true quantities; the search is exhaustive inside the box):

Read this as three separate facts. The reduction (col 2 vs col 3) is lossy by a factor ~18 already at $k=5$ and the gap is widening: the crown is a much stronger hypothesis than $K_{2,m}$. Evertse's uniform bound (col 5) is astronomically weaker than the measured $N_2$, as expected of a bound proved for all number fields and all coefficients. And $N_2^{(N)}$ itself ($1, 7, 32, 75, 141$) is growing far slower than $7^{2k}$, which is a (bounded-universe, therefore non-probative) sighting of the conjecture in §4.

7. Audit of hunts/r_186989/RESULTS.md

Everything checkable was checked from scratch; nothing was taken on trust.

Sound.

• All seven printed witnesses re-verified by independent trial division: every off-diagonal sum is $S$-smooth, sizes $2,4,5,6,8,10,11$ as claimed.
• Independent exhaustive branch-and-bound (different code) reproduces $g_N(k) = 2,4,5,6,8$ for $k = 1..5$ and the identical optimal witnesses, inside boxes 60–1000× smaller than theirs. Their claim that the box is slack and the optima live below 50 is confirmed. (Annotated on landing, 2026-10-21: the second half did not survive on main. Admissible sets are closed under $S$-smooth dilation, so a wider box only adds larger copies of the same sets (Hunts #103 and #105), and a primitive extremal 5-set for $S=\{2,3,5\}$ reaches 475 (Hunt #107, $\{5,11,25,245,475\}$). What this re-run confirms is the table rows and their witnesses.)
• §2's inequality $f(n-1) \le f_0(n) \le f(n)$ and the $f(2)=1$ vs $f_0(2)=0$ mismatch: proof read, correct.
• §4's composition-law argument (a supermultiplicative gadget would refute the conjecture via Fekete, since $g(1) = 2$) is correct, and Corollary 3 sharpens it: supermultiplicativity would also contradict the expected subexponential unit-equation count. The parent should keep treating "find the gadget" as a refutation search.
• Their §6 "the top door is the $S$-unit counting bound, and this hunt has no purchase on it" is exactly right. This run's contribution is to make that door precise enough to walk through: it is $N_2(k)$, uniformly in $c$.

Refuted. Loose thread 3, "$|A| \le p-1$ when $p \notin S$, by pigeonhole on residues mod $p$ against the pairing $r \leftrightarrow -r$", is false for every odd $p$. The pigeonhole works at $p=2$ only, because $2r \equiv 0$ has a non-zero solution mod 2 and none mod odd $p$; several elements may share one non-zero residue class $r$ with $2r \not\equiv 0$, and nothing forbids it. Explicit counterexamples, all verified:

So the parity fact ($2 \notin S \Rightarrow |A| \le 2$) is a $p=2$ accident and is not the seed of an iterable lemma. (Landing note, 2026-10-21: this thread was refuted on main before this arm landed, in Hunts #93, #95, #101 and #103; the third row's witness is the one #95 and #103 use.) Their door-1 ranking ("parity binds hardest") stays true as a description of the data; the generalisation does not.

Unverified. Rows $k = 6, 7$ of their table were re-verified as witnesses but their exhaustiveness was not re-run here (cost). I have no reason to doubt it.

8. The Győry–Hajdu–Tijdeman line: what it does and does not give

Located: Győry–Hajdu–Tijdeman, Representation of finite graphs as difference graphs of $S$-units, I, J. Combin. Theory Ser. A 127 (2014) 314–335 (arXiv:1408.5873 (https://arxiv.org/abs/1408.5873)); II, Acta Math. Hungar. 149 (2016) 423–447; and Hajdu–Tijdeman(-et al.), On conjectures and problems of Ruzsa concerning difference graphs of $S$-units, Acta Math. Hungar. (2015) (arXiv:1409.2218 (https://arxiv.org/abs/1409.2218)).

Specialised verdict: these are representability theorems, not counting theorems, and they do not contain the bound this arm needs. Their objects are (i) which finite graphs occur as $\mathcal{G}S(A)$: every finite graph does, for infinitely many $S$; (ii) representable with every $S$ iff cubical; (iii) induced non-degenerate cycle lengths (the Ruzsa problems). None of these is of the form $|A| \le F(|S|)$ for a fixed pattern, because for a fixed graph the answer is "yes for suitable $S$" and $|S|$ is allowed to grow. The one place the line touches ours is the cubical criterion, and its relevance is the $K{2,3}$ obstruction noted in §2, which is exactly the pattern Theorem 1 exploits, so the specialisation is: the crown leaves the cubical class through $K_{2,3}$, and the quantitative price of leaving it is $N_2(k)$.

Method caveat, stated because it bears on how far to lean: I could not read either PDF in-session (the fetcher returned undecoded binary), so §8 rests on the published abstracts plus the standard structure of that literature. The citations are located, not line-verified. A reader who needs the exact theorem numbers must open the papers. Nothing in §§1–7 depends on §8.

9. What I could not settle

• Whether $\log N_2(k) = o(k)$. Open, in the unit-equation literature, and not settleable at support-run cost.
• Whether the crown hypothesis is strictly stronger in a usable way. §6 shows it is strictly stronger in the data (18× at $k=5$). I found no argument that converts the surplus into a better exponent, and §5(c) shows the obvious conversion loses.
• Whether $g(k)$ is polynomially bounded. The measured $2,4,5,6,8,10,11$ looks linear, but that is the same seven points r_186989 correctly refused to read a limit out of, and I am refusing likewise.

The doors

This run measured a ceiling (the exponent in $\log g(k) \le Ck$), so it owes the list.

1. Active constraints at the optimum.

2. Frozen-constant inventory.

3. Information class. Doors 2–5 stay inside the data the current argument reads (finite sub-patterns of the $S$-unit difference graph) and are all capped by $N_2(k)$; none of them can produce $\exp(o(k))$. Door 1, improving the uniform bound on two-term $S$-unit equations, requires reading more: it is a Diophantine-approximation question (subspace theorem / quantitative $S$-unit counting), a different information class from anything enumerable here. That is where the top-ranked door is, and it is not this lab's door at support cost.

Loose threads

• Is the expected $\exp(k^{\theta})$ truth for $S$-unit equations asserted uniformly in the coefficients $\alpha,\beta$? Why it matters: Corollary 3 needs the uniform form; if the expectation is only for $x+y=1$, the reduction is weaker than stated. First step: Evertse–Győry, Unit Equations in Diophantine Number Theory (CUP 2015), the chapter on the number of solutions; check whether the conjectural upper bound is stated for $\alpha x + \beta y = 1$.
• The crown surplus. $8$ vs $141$ at $k=5$, and widening. Why it matters: it is the only measured evidence that the full pattern is much stronger than $K_{2,m}$, which is the one place a better-than-$N_2$ bound could come from. First step: measure $g_N(k)$ against $N_2^{(N)}(k)$ for $k \le 7$ and see whether the ratio grows like a power of $k$ or an exponential.
• $N_2^{(N)}$'s argmax $c$ is $2520$ twice and $10800$ once, and all three are themselves $S$-smooth. Why it matters: in the application $c = a_1 - a_2$ is a difference of elements of $A$ and is not required to be smooth, so restricting $c$ to the realizable set might sharpen Theorem 1. First step: recompute $N_2^{(N)}$ restricted to $c$ arising as a difference within a maximal $A$.
• r_186989's OEIS thread is still open. $2,4,5,6,8,10,11$ was not queried here. (Closed on landing, 2026-10-21: the sibling literature arm ran it. The sequence matches twenty OEIS entries, all coincidental, and a keyword search for the concept returns nothing; Hunt #143, support_fcffa3b9 §4.4.)