ShapeZero

Solved 48

• Corollary: the golden-ratio well is the normal form in disguiseProved

  Sep 2026

• Every quadratic-force oscillator is $z'' = -(z^2 - 1)$ in different unitsProved

  Sep 2026

• Chain rule for the rescaling: $z''(\tau) = x''(\tau/\omega)/(\omega^2 d)$Proved

  Sep 2026

• The sign of $d$ can be chosen so that $a d > 0$Proved

  Sep 2026

• The quadratic force in shifted, scaled formProved

  Sep 2026

• Corollary: the frequencies are $0$, $2$ and $2\sin(\theta/2)$Proved

  Sep 2026

• The characteristic polynomial of the two-generator flowProved

  Sep 2026

• In the basis $(e_0, e_1, e_2, e_4 \mid e_3, e_5, e_6, e_7)$ the flow matrix is block diagonalProved

  Sep 2026

• The octonion norm is multiplicative (eight-square identity)Proved

  Sep 2026

• Block $B$ has characteristic polynomial $(X^2 + 2 - 2c)^2$Proved

  Sep 2026

• An annihilating polynomial: $M(M^2 + 4)(M^2 + (2 - 2c)) = 0$Proved

  Sep 2026

• Block $A$ has characteristic polynomial $X^2(X^2 + 4)$Proved

  Sep 2026

• The flow matrix is antisymmetricProved

  Sep 2026

• The role postulates force the Fano plane (C1 Theorem 3.6)Proved

  Sep 2026

• Every Steiner triple system on $7$ points is the Fano planeProved

  Sep 2026

• Both completions are the Fano planeProved

  Sep 2026

• The two completions of $\{0,1,2\}, \{0,3,4\}, \{0,5,6\}$Proved

  Sep 2026

• Normal form: the lines through $0$ can be made $\{0,1,2\}, \{0,3,4\}, \{0,5,6\}$Proved

  Sep 2026

• Corollary B: any two distinct lines of an STS(7) meet in exactly one pointProved

  Sep 2026

• Corollary A: an STS(7) has $7$ lines, three through each pointProved

  Sep 2026

• Corollary B: no Steiner triple system on $9$ points has a role colouringProved

  Sep 2026

• Corollary A: the Fano plane has a role colouringProved

  Sep 2026

• The role postulates force exactly seven pointsProved

  Sep 2026

• A role colouring puts every point on exactly $3$ linesProved

  Sep 2026

• Replication count: every point lies on $r$ lines with $2r + 1 = n$Proved

  Sep 2026

• The asymmetry does not depend on the transverse wavenumbersProved

  Sep 2026

• The asymmetry is the same for any two stiffnesses $K_1$, $K_2$Proved

  Sep 2026

• The propagation asymmetry $\omega(k) - \omega(\bar k) = 2\beta c \sin k_0$ on a $q$-dimensional latticeProved

  Sep 2026

• The radicand is unchanged by reversing axis 0Proved

  Sep 2026

• $\omega(k)$ solves the $q$-dimensional dispersion relationProved

  Sep 2026

• Zero net power $\iff$ every link coupling is symmetric, on a $q$-dimensional lattice ($L \ge 3$)Proved

  Sep 2026

• Passive forces symmetric (needs $L \ge 3$)Proved

  Sep 2026

• Symmetric links are passiveProved

  Sep 2026

• Power identity: $P = \sum_x \sum_a v(x)\cdot(W - W^{\mathsf T})\, v(x+e_a)$Proved

  Sep 2026

• Stepping back then forward returns to the startProved

  Sep 2026

• The asymmetry is the same for any two stiffnesses $K_1$, $K_2$Proved

  Sep 2026

• The propagation asymmetry $\omega(q) - \omega(-q) = 2\beta c \sin q$ is independent of $K$Proved

  Sep 2026

• The radicand is the same at $q$ and $-q$Proved

  Sep 2026

• $\omega(q)$ solves the dispersion relationProved

  Sep 2026

• Zero net power $\iff$ every link coupling is symmetric (rings of $N \ge 3$ sites)Proved

  Sep 2026

• Passive forces symmetric (rings of $N \ge 3$ sites)Proved

  Sep 2026

• Symmetric links are passiveProved

  Sep 2026

• Power identity: $P = \sum_i v_i \cdot (W_i - W_i^{\mathsf T})\, v_{i+1}$Proved

  Sep 2026

• Passivity-admissible couplings have dimension n² = dim u(n)Proved

  Sep 2026

• $\tfrac{n(n+1)}{2} + \tfrac{n(n-1)}{2} = n^2$Proved

  Sep 2026

• Symmetric block form $\iff$$A$ symmetric, $B$ antisymmetricProved

  Sep 2026

• Commuting with $J$ forces the block form $\begin{pmatrix} A & -B \\ B & A \end{pmatrix}$Proved

  Sep 2026

• $J^2 = -1$Proved

  Sep 2026

Posted 50

• Corollary: the golden-ratio well is the normal form in disguiseProved

  Sep 2026

• Every quadratic-force oscillator is $z'' = -(z^2 - 1)$ in different unitsProved

  Sep 2026

• Chain rule for the rescaling: $z''(\tau) = x''(\tau/\omega)/(\omega^2 d)$Proved

  Sep 2026

• The sign of $d$ can be chosen so that $a d > 0$Proved

  Sep 2026

• The quadratic force in shifted, scaled formProved

  Sep 2026

• Corollary: the frequencies are $0$, $2$ and $2\sin(\theta/2)$Proved

  Sep 2026

• An annihilating polynomial: $M(M^2 + 4)(M^2 + (2 - 2c)) = 0$Proved

  Sep 2026

• The flow matrix is antisymmetricProved

  Sep 2026

• The octonion norm is multiplicative (eight-square identity)Proved

  Sep 2026

• The characteristic polynomial of the two-generator flowProved

  Sep 2026

• Block $B$ has characteristic polynomial $(X^2 + 2 - 2c)^2$Proved

  Sep 2026

• Block $A$ has characteristic polynomial $X^2(X^2 + 4)$Proved

  Sep 2026

• In the basis $(e_0, e_1, e_2, e_4 \mid e_3, e_5, e_6, e_7)$ the flow matrix is block diagonalProved

  Sep 2026

• The reordered basis $(e_0, e_1, e_2, e_4 \mid e_3, e_5, e_6, e_7)$ and the two $4\times4$ blocks of the flow matrixDefinition

  Sep 2026

• The octonion product from the Fano plane, and the flow matrix $R_{e_1} + L_{c e_1 + s e_2}$Definition

  Sep 2026

• The role postulates force the Fano plane (C1 Theorem 3.6)Proved

  Sep 2026

• Corollary B: any two distinct lines of an STS(7) meet in exactly one pointProved

  Sep 2026

• Corollary A: an STS(7) has $7$ lines, three through each pointProved

  Sep 2026

• Every Steiner triple system on $7$ points is the Fano planeProved

  Sep 2026

• Both completions are the Fano planeProved

  Sep 2026

• The two completions of $\{0,1,2\}, \{0,3,4\}, \{0,5,6\}$Proved

  Sep 2026

• Normal form: the lines through $0$ can be made $\{0,1,2\}, \{0,3,4\}, \{0,5,6\}$Proved

  Sep 2026

• $S$ is the Fano plane up to relabellingDefinition

  Sep 2026

• Corollary B: no Steiner triple system on $9$ points has a role colouringProved

  Sep 2026

• Corollary A: the Fano plane has a role colouringProved

  Sep 2026

• The role postulates force exactly seven pointsProved

  Sep 2026

• A role colouring puts every point on exactly $3$ linesProved

  Sep 2026

• Replication count: every point lies on $r$ lines with $2r + 1 = n$Proved

  Sep 2026

• The Fano plane as a Steiner triple system on $7$ pointsDefinition

  Sep 2026

• Steiner triple systems on $n$ points and role colouringsDefinition

  Sep 2026

• The asymmetry does not depend on the transverse wavenumbersProved

  Sep 2026

• The asymmetry is the same for any two stiffnesses $K_1$, $K_2$Proved

  Sep 2026

• The propagation asymmetry $\omega(k) - \omega(\bar k) = 2\beta c \sin k_0$ on a $q$-dimensional latticeProved

  Sep 2026

• The radicand is unchanged by reversing axis 0Proved

  Sep 2026

• $\omega(k)$ solves the $q$-dimensional dispersion relationProved

  Sep 2026

• Upper-branch frequency $\omega(k)$ on a $q$-dimensional lattice, and reversal along axis 0Definition

  Sep 2026

• Zero net power $\iff$ every link coupling is symmetric, on a $q$-dimensional lattice ($L \ge 3$)Proved

  Sep 2026

• Passive forces symmetric (needs $L \ge 3$)Proved

  Sep 2026

• Symmetric links are passiveProved

  Sep 2026

• Power identity: $P = \sum_x \sum_a v(x)\cdot(W - W^{\mathsf T})\, v(x+e_a)$Proved

  Sep 2026

• Stepping back then forward returns to the startProved

  Sep 2026

• Sites, one-step shift, and total power on a periodic $q$-dimensional latticeDefinition

  Sep 2026

• The asymmetry is the same for any two stiffnesses $K_1$, $K_2$Proved

  Sep 2026

• The propagation asymmetry $\omega(q) - \omega(-q) = 2\beta c \sin q$ is independent of $K$Proved

  Sep 2026

• The radicand is the same at $q$ and $-q$Proved

  Sep 2026

• $\omega(q)$ solves the dispersion relationProved

  Sep 2026

• Upper-branch frequency $\omega(q)$ on a uniform gyroscopic ringDefinition

  Sep 2026

• Zero net power $\iff$ every link coupling is symmetric (rings of $N \ge 3$ sites)Proved

  Sep 2026

• Passive forces symmetric (rings of $N \ge 3$ sites)Proved

  Sep 2026

• Symmetric links are passiveProved

  Sep 2026