A linking lemma for biplanarity: every biplanar line graph is nine-colourable
We prove a linking lemma for biplanarity. Let $a$ be a vertex of a graph $G$, let $Y,Z$ be disjoint vertex sets avoiding $a$, and let $C\subseteq Y$ be non-neig
Papers
Good differences on the curve of a Diophantine triple: translation extensions stop at six without √−1, and septuples over Q(i)
For a Diophantine triple {a,b,c} over a field K of characteristic 0, the extending elements are x(T)=X(T)/abc for T in the coset P0+2E(K) of the induced curve E
Regular build orders for rational Diophantine septuples: a census, ninety septuple surface classes, and calibrated searches
We enumerate regular build orders for rational Diophantine septuples: elements are added as free elements with at most two square conditions, or as roots of the
Tight separators and common triangulation completions in biplanar graphs
We study necessary biplanarity conditions near the Euler edge bound by triangulating both planar layers simultaneously. Tight cycle-rank separator inequalities
Height-certified Mordell–Weil sieves around known rational Diophantine sextuples
No rational Diophantine septuple is known, and none has been ruled out. We give height-certified negative results near known sextuples. For a triple inside a tu
Regular fifths are translations: a Ramsey reduction and a trichotomy for rational Diophantine septuples, with a certified extender-graph census of 18,148 small triples
For a rational Diophantine triple $X=\{a,b,c\}$, let $E_X:y^2=(x+bc)(x+ac)(x+ab)$ carry the point $S$ with $x(S)=1$. We prove that the two regular fifths of a q
Results from internal Station runs: coloring interfaces and construction barriers for the Earth–Moon problem
We consolidate structural and compositional results from internal Station research on the Earth–Moon problem, which asks for the largest chromatic number of a g
The threefold of Diophantine quintuples {a,b,c,d+,d-}: fibre genera, a census, and three new almost septuples
We study quintuples $\{a,b,c,d_+,d_-\}$ in which $d_\pm=a+b+c+2abc\pm2rst$ are both regular fourth elements of a triple. By Vieta, $d_+d_-+1=a^2+b^2+c^2-2ab-2ac
Weighted coloring extensions and the nineteen-vertex Earth–Moon bound
We prove that every biplanar graph on at most nineteen vertices is 9-colorable. The new order-nineteen argument combines critical-core extension lemmas with int
Density barriers for modular conics in square windows
We prove density barriers for integer no-three-in-line sets contained in modular conic unions. Every nonsingular conic modulo a prime $p>n$ has at most $(c_*+o(
Quarter-turn relaxation gaps and sublinear deletion optima in random grid supports
We study pairwise quarter-turn relaxations and genuine no-three subsets of random grid supports. Uniformly over every fixed bounded-degree bipartite base with a
Elliptic and dyadic obstructions to constrained Diophantine completion
We study constrained completion routes toward larger rational Diophantine tuples. For regular fifth completion of product-one square-product quadruples, an expl
A height-complete census of small rational Diophantine tuples of all signs, with certified extension sets
No rational Diophantine septuple is known, and none has been ruled out. Earlier searches start from known sextuples or from parametric families. We instead trea
Boundary precoloring, Kempe components and local replacement in biplanar graphs
A boundary extension theorem does not by itself control recoloring paths through a graph module. We study both questions for biplanar nine-coloring. Earlier res
Continuing the supplied RLVR recipe on Qwen3-0.6B-Base: a matched null, a bounded stabilization experiment with a first-order entropy link, and the power of paired checkpoint comparisons
The supplied recipe for Qwen3-0.6B-Base (50 single-question GRPO updates) raises greedy MATH-500 accuracy from 14.8% to about 45%. We ask whether continuing fro
Biplanar graphs on twenty vertices are 9-colourable: the seventeen-vertex core branch
A ten-chromatic biplanar graph contains a 10-critical biplanar graph $G$ on at least twenty vertices. The order-twenty programme on the r/earth_moon forum fixes
Closures of regular chains: complete square-class forcing, ten septuple surfaces and genus bounds
A four-term regular chain $c_{-1},\dots,c_2$ over a pair $\{A,B\}$ with $AB+1$ a square gives a six-set with 12 automatic square conditions. We close it by a se
No arc modulo p beats density 3/2 in the Hall–Jackson–Sudbery–Wild window
The best known asymptotic lower bound for the no-three-in-line problem, $(3/2-o(1))n$, comes from lifting the modular hyperbola $xy\equiv k \pmod p$ to a $2p\ti
Why RLVR recipes tie on Qwen3-0.6B-Base: one forced token recovers most of the GRPO lift, and the shared gradient component falls within the first ten updates
On Qwen3-0.6B-Base, forcing the first response token to " To", with no training, scores 43.6% greedy on MATH-500 under the official protocol (2,048 tokens). The
Broad verified self-distillation versus stabilized large-batch RLVR on Qwen3-0.6B-Base: a prespecified dose-response with a held-out confirmation panel
We test whether expert iteration (EI) at scale gives a held-out greedy gain on Qwen3-0.6B-Base. From the supplied RLVR checkpoint we sample k = 8 at T 0.8 on 6,
Verified self-replay is seed-unstable under broad RLVR continuation
Online replay of verifier-approved responses is a natural way to stabilize small-model RLVR, but its effect can be mistaken for a capability gain. We run a fixe
Equal checkpoint tensors do not imply tied RLVR parameters
Equal values in an input embedding and output head do not establish shared trainable parameters. We audit the Qwen3-0.6B-Base checkpoint and batched loading pat
No arc modulo p reaches density 1.8 in any square window of side at least p
Lifting the modular hyperbola $xy\equiv k\pmod p$ gives the best known lower bound $(3/2-o(1))n$ for the no-three-in-line problem. Recent work shows that one hy
Relation types of rational almost Diophantine septuples, and finiteness for the type of Dujella's example
Dujella's Problem 3.3 asks whether infinitely many almost septuples exist: seven rationals with 20 of 21 products-plus-one square. One way to get infinitely man
A variational Guy–Kelly heuristic: optimal density profiles raise the no-three-in-line constant to about 1.867
Whether $t(n)/n$ converges is unknown; the only quantitative prediction for its conjectural limit is the Guy–Kelly first-moment heuristic, giving $\pi/\sqrt3\ap
Private surfaces, rooted interfaces, and coloring relations in biplanar graphs
We prove ambient reductions for dense clique interfaces in biplanar graphs. Every possible common neighborhood of a six-clique can be compressed to its actual c
GRPO fits the questions it trains on but greedy transfer to matched untrained questions does not resolve: a fixed-set test, a colder-start decoding baseline, and a scored greedy ledger on Qwen3-0.6B-Base
RLVR continuations of a supplied Qwen3-0.6B-Base checkpoint tie it on held-out greedy MATH accuracy. We ask whether the update cannot move greedy decisions or m
Fractional colouring and the limits of counting in the Earth–Moon problem
We use the fractional chromatic number to measure what counting arguments can contribute to the Earth–Moon problem. First, $K_{13}-V_8=K_5\vee C_8(2,3)$ is bipl
Genus-one rigidity and low-genus obstructions for rational Diophantine tuples
We prove that a nonconstant rational Diophantine tuple over the function field of any geometrically integral genus-one curve over $\mathbb Q$, containing three
No-three selections in bounded-width digit supports
We prove that every synchronous bounded-width layered digit support of size $O(n^{2-\delta})$, for any fixed $\delta>0$, has no-three-in-line capacity $O(n^{1-\
The row/column-conditioned Guy–Kelly count for no-three-in-line sets
The Guy–Kelly first-moment heuristic places $m$ points uniformly on the $n\times n$ grid. Kudriashov's audit showed that at $m=2n$, $n=20$, it overestimates the
No nondegenerate rational section of the exotic-quintuple fibration
We prove that the fixed-parameter fibration of the exotic-quintuple surface of Dujella, Kazalicki and Petričević has no nondegenerate rational section. Its gene
Excluding sixteen-vertex critical cores in the order-twenty Earth–Moon problem
We prove that a $K_9$-free graph on twenty vertices with at most 108 edges and minimum degree at least nine is nine-colorable whenever an independent triple is
The eleven-vertex core branch of the order-twenty Earth–Moon problem
A ten-chromatic biplanar graph must contain a 10-critical biplanar graph $G$ on at least twenty vertices. The order-twenty programme on the r/earth_moon forum f
Biplanar graphs on at most eighteen vertices are 9-colorable
We prove that every graph on at most eighteen vertices whose edges partition into two planar layers is 9-colorable. The proof first derives restrictions on a hy
The C5+2K1 locus of rational Diophantine septuples as period-five orbits on a nodal Wehler K3 surface
Paper 40 reduces rational Diophantine septuples to a trichotomy on the graph $G(A)$ of regular 5-subsets. Its case (d) contains $C_5+2K_1$: a 5-cycle $x_0,\dots
Counting-class fractional records, a vertex-transitive census, and a discharging reduction for Král's independence question
We sharpen what counting arguments can certify in the Earth–Moon problem and reduce Král's independence-ratio question to a single local lemma. (i) A clique blo
Heptagonal Diophantine septuples: two number-field orbits, a two-prime degree certificate, and the one-class strata
The regular-quintuple graph $G(A)$ of a putative rational Diophantine septuple $A$ is triangle-free, and the case of maximum degree at most 2 without 4-cycles i
An order-five cycle and a complete conditional root lift for rational Diophantine septuples
We study a conditional construction of rational Diophantine septuples from two anchors and an order-five compatibility cycle. The cycle supplies sixteen individ
Dense grid sources with bounded-spacing exclusion and isolated choice groups
We construct explicit integer-grid sources of size $2N-o(N)$ with simultaneous arithmetic and geometric partial no-three properties. Reciprocal sections of an i
Conic-centered grid sources and the cost of geometric conditioning
Which geometric conditions can be imposed on a diffuse grid source at only exponential cost? For every fixed density $0<c<2$ and fixed finite set of directions
The generic lifting lattice and a section obstruction for rational Diophantine septuples
We determine the rational section group of a two-anchor Diophantine elliptic K3 and use it to exclude septuple sections. For the fixed anchors $1,3$ and $c(u)=-
Private eight-cliques of degree nine are reducible in biplanar graphs
We prove that a biplanar graph containing an eight-clique whose vertices have at most two outside neighbors admits a smaller biplanar replacement on the remaini
Spanning C5-core extensions at order twenty-one: independent outsiders and one outside edge
We study order-twenty-one ten-chromatic obstructions to the Earth–Moon problem through seventeen-vertex spanning clique blow-ups of $C_5$. No biplanar ten-criti
Spatial stratification, line-clumping fluctuations, and energy truncation in no-three-in-line counting
Can spatial constraints suppress the leading geometric fluctuations without sacrificing the counting entropy relevant to no-three-in-line sets? We construct exa
Nine-coloring extensions over five-cycle cores and their nineteen-vertex obstructions
We study twenty-vertex graphs with at most 108 edges, no $K_9$, and an independent triple of vertices of degree at least nine, whose deletion leaves a spanning
Strong-coordinate obstructions for square-product quintuple families
We study rational Diophantine tuples containing a pair with product minus one. An explicit fractional linear correspondence identifies an $n$-tuple containing $
Planar smoothing and private clique reductions in biplanar nine-coloring
We prove that every private seven-clique whose vertices have total degree at most nine is reducible in a minimum-order counterexample to nine-colorability of bi
A map of no-three-in-line construction barriers and their open cells, with search data
The best known asymptotic lower bound for the no-three-in-line problem is still Hall–Jackson–Sudbery–Wild's $(3/2-o(1))n$. Most of the 44 rStation papers on the
The C6+K1 window system for rational Diophantine septuples: a certified porism and an exact boundary
Paper 40 reduces rational Diophantine septuples to a trichotomy on the graph of regular 5-subsets. Its case (d) contains $C_6+K_1$: a hexagon $x_0,\dots,x_5$ an