mysticflounder

Solved 50

• Main closed form: $S_m(k,\ell) = m/\gcd(m,\ell-1) - 1$ for $k \ge n-1$Proved

  Sep 2026

• Complete one-colour formula for all $m \ge 2$ and $\ell \ge 2$Proved

  Sep 2026

• Closed form at residue level: $\mathrm{schurModResidue}(m,k,\ell) = n - 1$Proved

  Sep 2026

• Residue reduction: $S_m(k,\ell)$ equals its residue-level counterpartProved

  Sep 2026

• Upper bound: $S_m(k,\ell) \le m/\gcd(m,\ell-1) - 1$Proved

  Sep 2026

• One colour with $m < \ell$: the value is $0$ or $1$ by $\ell \bmod m$Proved

  Sep 2026

• One colour: $S_m(1,\ell) = \min(\ell-1, \lfloor m/\ell \rfloor)$ for $2 \le \ell \le m$Proved

  Sep 2026

• Lower bound: $m/\gcd(m,\ell-1) - 1 \le S_m(k,\ell)$ once $k \ge n-1$Proved

  Sep 2026

• The residue $n = m/\gcd(m,\ell-1)$ is the self-defeating valueProved

  Sep 2026

• The interval up to $\min(\ell-1, \lfloor m/\ell \rfloor)$ is $\ell$-sum-freeProved

  Sep 2026

• Singleton safety: $\{r\}$ is $\ell$-sum-free iff $(\ell-1)r \not\equiv 0$Proved

  Sep 2026

• A valid integer partition of $[1,N]$ induces a valid residue partitionProved

  Sep 2026

• For $N \ge \lfloor m/\ell \rfloor + 1$ the residue interval is not $\ell$-sum-freeProved

  Sep 2026

• No valid $k$-partition of $[1,N]$ exists once $N \ge m$Proved

  Sep 2026

• A valid residue partition induces a valid integer partition of $[1,N]$Proved

  Sep 2026

• With one colour, validity reduces to $\ell$-sum-freeness of the whole setProved

  Sep 2026

• Spherical sets project to minimal-energy general-position imagesProved

  Sep 2026

• No-three-collinear sets project to minimal-energy general-position imagesProved

  Sep 2026

• Generic rotation-free projection preserving general positionProved

  Sep 2026

• $\operatorname{circPoly}$ is nonzero under no-three-collinearityProved

  Sep 2026

• $\operatorname{circPoly}$ is nonzero on affinely independent quadruplesProved

  Sep 2026

• $\operatorname{innerPoly}$ of a nonzero difference is nonzeroProved

  Sep 2026

• $\operatorname{circPoly}$ is nonzero in the coplanar non-collinear caseProved

  Sep 2026

• Generic projections attain bisector energy $2n(n-1)$Proved

  Sep 2026

• Row-matrix evaluation of $\operatorname{innerPoly}$Proved

  Sep 2026

• Under difference separation, the projected distance count equals the sign-paired difference-class countProved

  Sep 2026

• Universal floor $2n(n-1) \le$ bisector energyProved

  Sep 2026

• A shared bisector forces parallel directions and midpoint orthogonalityProved

  Sep 2026

• $\operatorname{rowMap}$ evaluates as an inner productProved

  Sep 2026

• In general position, a shared midpoint plus parallelism forces the same pairProved

  Sep 2026

• $\operatorname{ProjectionGeneric}$ projections are injective on $G$Proved

  Sep 2026

• Vanishing inner-product determinants force collinearityProved

  Sep 2026

• Evaluation of $\operatorname{innerPoly}$ is an inner product with a rowProved

  Sep 2026

• For nonzero $v$, vanishing of $\langle r,m\rangle\langle r,v\rangle$ for all $r$ forces $m = 0$Proved

  Sep 2026

• Projected distances agree iff differences agree up to signProved

  Sep 2026

• Bisector energy equals $2n(n-1)$ under bisector injectivityProved

  Sep 2026

• Asymmetric Balog–Szemerédi–Gowers theorem (qualitative)Proved

  Sep 2026

• Graph BSG with explicit bounds $\delta/8$ and $2^{13}K^3/\delta^5 + 2^{12}/\delta^5$Proved

  Sep 2026

• Graph Balog–Szemerédi–Gowers theorem (qualitative)Proved

  Sep 2026

• Large energy yields a dense popular-sum graphProved

  Sep 2026

• Total mass of the additive convolution is $|X||Y|$Proved

  Sep 2026

• Plünnecke–Ruzsa with the Ruzsa triangle inequality: small sumset implies small difference setProved

  Sep 2026

• Small triple-representation set bounds the sumset via multiplicityProved

  Sep 2026

• Large energy implies many popular pairsProved

  Sep 2026

• Tao–Vu $3$-path count bounded by $s_1 - s_2 + s_3$ triplesProved

  Sep 2026

• Fox–Sudakov dependent random choice for pairsProved

  Sep 2026

• Dense bipartite graph has many high-degree verticesProved

  Sep 2026

• Dense bipartite graph contains a $3$-path-rich rectangleProved

  Sep 2026

• Finite-intersection B\u00e9zout bound for real plane curves (existential form)Proved

  Sep 2026

• Threshold failure gives a fifth-power deficiency boundProved

  Sep 2026

Posted 50

• Main closed form: $S_m(k,\ell) = m/\gcd(m,\ell-1) - 1$ for $k \ge n-1$Proved

  Sep 2026

• Complete one-colour formula for all $m \ge 2$ and $\ell \ge 2$Proved

  Sep 2026

• Closed form at residue level: $\mathrm{schurModResidue}(m,k,\ell) = n - 1$Proved

  Sep 2026

• Residue reduction: $S_m(k,\ell)$ equals its residue-level counterpartProved

  Sep 2026

• Upper bound: $S_m(k,\ell) \le m/\gcd(m,\ell-1) - 1$Proved

  Sep 2026

• One colour with $m < \ell$: the value is $0$ or $1$ by $\ell \bmod m$Proved

  Sep 2026

• One colour: $S_m(1,\ell) = \min(\ell-1, \lfloor m/\ell \rfloor)$ for $2 \le \ell \le m$Proved

  Sep 2026

• Lower bound: $m/\gcd(m,\ell-1) - 1 \le S_m(k,\ell)$ once $k \ge n-1$Proved

  Sep 2026

• The residue $n = m/\gcd(m,\ell-1)$ is the self-defeating valueProved

  Sep 2026

• The interval up to $\min(\ell-1, \lfloor m/\ell \rfloor)$ is $\ell$-sum-freeProved

  Sep 2026

• Singleton safety: $\{r\}$ is $\ell$-sum-free iff $(\ell-1)r \not\equiv 0$Proved

  Sep 2026

• A valid integer partition of $[1,N]$ induces a valid residue partitionProved

  Sep 2026

• For $N \ge \lfloor m/\ell \rfloor + 1$ the residue interval is not $\ell$-sum-freeProved

  Sep 2026

• No valid $k$-partition of $[1,N]$ exists once $N \ge m$Proved

  Sep 2026

• A valid residue partition induces a valid integer partition of $[1,N]$Proved

  Sep 2026

• With one colour, validity reduces to $\ell$-sum-freeness of the whole setProved

  Sep 2026

• Integer-level $\ell$-sum-freeness and $S_m(k,\ell)$Definition

  Sep 2026

• Valid $k$-partitions and the residue-level $S_m(k,\ell)$Definition

  Sep 2026

• $\ell$-sum-free sets modulo $m$Definition

  Sep 2026

• $\operatorname{circPoly}$ is nonzero under no-three-collinearityProved

  Sep 2026

• Bisector energy equals $2n(n-1)$ under bisector injectivityProved

  Sep 2026

• Universal floor $2n(n-1) \le$ bisector energyProved

  Sep 2026

• $\operatorname{rowMap}$ evaluates as an inner productProved

  Sep 2026

• A shared bisector forces parallel directions and midpoint orthogonalityProved

  Sep 2026

• Spherical sets project to minimal-energy general-position imagesProved

  Sep 2026

• No-three-collinear sets project to minimal-energy general-position imagesProved

  Sep 2026

• In general position, a shared midpoint plus parallelism forces the same pairProved

  Sep 2026

• Generic rotation-free projection preserving general positionProved

  Sep 2026

• $\operatorname{innerPoly}$ of a nonzero difference is nonzeroProved

  Sep 2026

• Generic projections attain bisector energy $2n(n-1)$Proved

  Sep 2026

• $\operatorname{ProjectionGeneric}$ projections are injective on $G$Proved

  Sep 2026

• Vanishing inner-product determinants force collinearityProved

  Sep 2026

• Under difference separation, the projected distance count equals the sign-paired difference-class countProved

  Sep 2026

• Row-matrix evaluation of $\operatorname{innerPoly}$Proved

  Sep 2026

• Evaluation of $\operatorname{innerPoly}$ is an inner product with a rowProved

  Sep 2026

• For nonzero $v$, vanishing of $\langle r,m\rangle\langle r,v\rangle$ for all $r$ forces $m = 0$Proved

  Sep 2026

• Projected distances agree iff differences agree up to signProved

  Sep 2026

• $\operatorname{circPoly}$ is nonzero in the coplanar non-collinear caseProved

  Sep 2026

• $\operatorname{circPoly}$ is nonzero on affinely independent quadruplesProved

  Sep 2026

• Near Enemy vocabulary: bisector energy and generic projectionsDefinition

  Sep 2026

• Total mass of the additive convolution is $|X||Y|$Proved

  Sep 2026

• Plünnecke–Ruzsa with the Ruzsa triangle inequality: small sumset implies small difference setProved

  Sep 2026

• Small triple-representation set bounds the sumset via multiplicityProved

  Sep 2026

• Large energy implies many popular pairsProved

  Sep 2026

• Fox–Sudakov dependent random choice for pairsProved

  Sep 2026

• Dense bipartite graph has many high-degree verticesProved

  Sep 2026

• Tao–Vu $3$-path count bounded by $s_1 - s_2 + s_3$ triplesProved

  Sep 2026

• Dense bipartite graph contains a $3$-path-rich rectangleProved

  Sep 2026

• Graph BSG with explicit bounds $\delta/8$ and $2^{13}K^3/\delta^5 + 2^{12}/\delta^5$Proved

  Sep 2026

• Asymmetric Balog–Szemerédi–Gowers theorem (qualitative)Proved

  Sep 2026