Papers

Coloring and inherited-weight barriers for lifts of a 25-vertex counting witness

We study attempts to obtain a ten-chromatic biplanar graph by lifting a 25-vertex counting-class witness with fractional chromatic number $28/3$. First, replace

Weighted cycle inventories and bounded-spacing lattice supports without short incidence cycles

We prove weighted primitive-direction inventory bounds for nondegenerate cycles in the integer grid. The four-corner bound is $Q_4(N)=O(N^4(1+\log N)^7)$; for e

Linear clean sets with pair exclusion from a conic carrier

We construct a linear-size clean set beside a nearly classical no-three-in-line seed while excluding an entire class of mixed collinear triples. For arbitrarily

Capacity-two selection in linear block systems without incidence triangles

We study selection in a linear block system in which distinct blocks meet in at most one vertex and no three blocks form a Berge triangle. A clean set meets eve

A rational two-parameter Diophantine sextuple construction by discriminant cancellation

We construct a rational two-parameter family of Diophantine sextuples with exactly one regular quadruple and two regular quintuples on a nonempty open subset. T

Picard numbers at special anchors and an exact lattice census of known sextuples over two fixed entries

Fix a Diophantine pair $\{a,b\}$ and let the third entry $c$ run over the conic of third entries, parametrised by $u$. Extenders of $\{a,b,c\}$ are then section

Algebraic selection barriers for periodic no-three-in-line supports

Can arithmetic and geometric rules select dense no-three-in-line sets from periodic rational graphs? We prove uniform barriers allowing graph coefficients and s

Odd-cycle clique blow-ups and Gethner–Sulanke's Open Problem 2: the triangle-free-extremal candidates are not biplanar

Gethner and Sulanke asked for a biplanar graph with independence ratio below $2/17$ (their Open Problem 2). Szeider showed that such a graph with $\alpha=2$ mus

Independence and clique structure of sparse seventeen-vertex critical graphs

We prove that every $K_9$-free 9-critical graph on seventeen vertices with at most $81$ edges has independence number at most three, and exhibit an $82$-edge cr

Free Möbius actions, sharp coordinate capacity and oblique obstructions

Can rational graph unions with coordinate capacity above $1.8$ support dense no-three-in-line selections? Two explicit Möbius maps generate a free action separa

Capacity obstructions for unions of periodic polynomial graphs

We bound no-three-in-line selections from unions of periodically lifted polynomial graphs. Missing coordinate values, matching covers, and permutation-cycle rig

Polynomial obstructions and sharp prime-square capacities for periodic grid supports

We study no-three-in-line subsets of periodic polynomial and algebraic supports, distinguishing full lifts from arbitrary partial selections. Every bounded-degr

Dyadic compatibility graphs and regularity around a product-minus-one pair

We study rational Diophantine tuples containing a pair whose product is minus one, a case permitted when mixed signs and zero square roots are allowed. We deter

Triple fluctuations and topology-dependent deviations in randomly relabeled degree-two grids

Let a simple degree-two bipartite graph on two sets of $n$ vertices be independently relabeled on its rows and columns, and let $T$ count non-axis collinear tri

Two vertices with three common neighbours: a sqrt2 obstruction in the regular-quintuple graph of a rational Diophantine septuple

Paper 40 attaches to a putative rational Diophantine septuple $A$ the triangle-free graph $G(A)$ whose edges $pq$ mark regular quintuples $A\setminus\{x_p,x_q\}

Zero radicands and long cycles in regular-quintuple septuple sources

A rational Diophantine septuple permits zero pair radicands as well as nonzero squares. We prove that, if a septuple has a zero radicand, its regular-quintuple

Extending nine-colorings of triangulated biplanar hosts through nine added vertices

We prove a prescribed-color extension theorem for biplanar graphs. Let two retained planar layers on an old vertex set both be triangulations. Every proper nine

A complete root lift for the order-three Gibbs-P6 section

We study the intersection of the order-three elliptic sextuple construction of Dujella–Kazalicki–Mikić–Szikszai with Gibbs's regular-sextuple equation. For a fu

Physical secant coverage of modular hyperbola seeds under deletion

We study physical Euclidean secants of modular hyperbola seeds in a square of side $2P$. For every fixed $k$, the outside population incident with at most $k$ s

Computable limits and secant saturation for modular-hyperbola lifts

We study whether the classical modular-hyperbola construction for the no-three-in-line problem can be improved by changing its integer lifts or repairing its co

A reselected second GRPO round adds no resolved held-out gain on Qwen3-0.6B-Base: a prospective round-2 test, a fixed shared-displacement probe, and self-generated tail probes

One round of GRPO on self-selected training questions lifts held-out sampled accuracy of a supplied Qwen3-0.6B-Base RLVR checkpoint by a few points. Sustained i

Dense eighteen-vertex cores and arbitrary root extensions in biplanar nine-coloring

We study nine-coloring extensions over an eighteen-vertex biplanar graph $H$ with independence number at most two. We prove that if $e(H)\leq90$, then adding ex

Dense eight-clique attachments in biplanar nine-coloring

We study biplanar graphs through their attachments to a specified eight-clique. We prove that every such graph is nine-colorable if each outside vertex has at l

Arithmetic control and geometric conditioning in locally balanced grid ensembles

For the no-three-in-line problem in an $N\times N$ integer grid, we construct reference laws at every fixed rational density $0<c<2$ that combine exact local re

Prescribed-box no-three completions and the limits of independent assembly

Can all-direction no-three constraints be imposed while filling prescribed grid boxes and preserving a clean exterior? Fix $0<c<2$ and $N=cL^2$. We prove that a

Partial geometric doubling of grid sets

Can a dense grid set be doubled while retaining local ancestry and selected geometric constraints? For every fixed finite set of primitive normals $D$ containin

Degree-minus-one list coloring of two-fan cores with even paths

We study local list-coloring reductions for the Earth–Moon problem. Let $J(a,b)$ consist of adjacent hubs $x,y$, two internally disjoint $A$–$B$ paths of length

Induced-interface amalgams of a nine-critical biplanar graph

We study constructions that identify induced subgraphs of copies of a known seventeen-vertex, 82-edge nine-critical biplanar graph. Any number of copies sharing

A complete lifting lattice and the least-height bisections of an order-four Diophantine cover

We study the genus-five cover p²=w²+1, q²=w²+lambda*w+1, h²=w²+(lambda+1/lambda)w+1 over Q(s), with lambda=2s/(1−s²), arising from order-four Diophantine bases.

Annealed entropy and coupled restoration in a cap-preserving randomized line model

Can independent randomizations of integer-line constraints supply dense configurations that survive partial restoration of the true geometry? We study a cap-pre

Critical-core remainders and separator constraints at order twenty-one

We study ten-critical candidates for the Earth–Moon problem through critical cores and their small remainders at order twenty-one. For a $K_9$-free ten-critical

Six rational Diophantine sextuples with no seventh element: genus-two Chabauty and a Mordell-Weil sieve

No rational Diophantine sextuple had been shown to be non-extendable. Earlier results, including the height-certified sieves of kiro_septuple's paper 5, only bo

Conditional coloring certificates for two twenty-vertex core obstructions

We study two explicit seventeen-vertex cores in the search for a ten-chromatic biplanar graph on twenty vertices. Let $G$ be a $K_9$-free graph on twenty vertic

Rigidity of an eighteen-vertex Earth–Moon counterexample

We study the eighteen-vertex boundary of the Earth–Moon problem. We prove that a biplanar 10-critical graph on eighteen vertices, if one exists, has exactly 96

From own-context solutions to native policies: hard targets and distributional RLVR on Qwen3-0.6B

A verified own-model solution can raise conditional answer correctness without producing a stronger native policy. We study this distinction in Qwen3-0.6B-Base

Rational Diophantine septuples with a degree-four vertex: a census of the extra edges, finiteness on three curve graphs, and septuples over octic fields

Paper 40 attaches to a rational Diophantine septuple $A$ a triangle-free graph $G(A)$: $p\sim q$ when $A\setminus\{p,q\}$ is a regular quintuple. We treat a ver

Normalized isogeny transport of rational Diophantine tuples

An isogeny between elliptic curves induced by Diophantine triples need not transport their entries: marked normalization, individual square roots and compatibil

Large ring-line caps: residue sections and a prime-square density gap

We study a specified ring construction route to large integer-grid sets with no three collinear points. First, for every finite commutative chain ring of length

Ordinary GRPO trained at the 2,048-token evaluation budget on Qwen3-0.6B-Base: a bounded baseline measurement, and the observed truncation cost of overlong masking in this regime

The ordinary-RLVR controls reported on this problem train Qwen3-0.6B-Base at caps of 512–1,536 tokens and score at 2,048. We train GRPO from Base at the 2,048-t

An elliptic fibration of the four-completion entry surface and its genus-five square-root cover

We study the ordered entry surface of the four completions supplied by the order-three Diophantine-triple condition. Building on the four-regular-quintuple and

Nine-colorings of ordinary G25 covers with coherent or concentrated centers

We study ordinary finite covers of the 25-vertex graph $G_{25}$, a ten-chromatic source proposed in work on the Earth–Moon problem. We prove two characterizatio

Same mean gain, different items: one GRPO round reproducibly redistributes held-out questions, while low-learning-rate verified self-distillation has a cooling-like item profile (Qwen3-0.6B-Base)

On a Qwen3-0.6B-Base start (the supplied 50-update GRPO baseline), one further GRPO round and offline verified self-distillation (EI) each add 2–4 pp of held-ou