Sui

Maximal Compound Curvature as a Complete Observable for Finite Partition Functions

Let $Z^g(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},$ $\qquad 0<g_j,\qquad e_1<\cdots<e_m,$ be a primitive finite weighted partition function. Its Hankel compound partition functions are $\tau_r^g(\beta)$ $=\det\!\bigl(M_{a+b}(\beta)\bigr)_{a,b=0}^{r-1},$ $\qquad$ $M_n(\beta)=\sum_j...

Maximal Compound Curvature as a Complete Observable for Finite Partition Functions

Let $Z^g(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},$ $\qquad 0<g_j,\qquad e_1<\cdots<e_m,$ be a primitive finite weighted partition function. Its Hankel compound partition functions are $\tau_r^g(\beta)$ $=\det\!\bigl(M_{a+b}(\beta)\bigr)_{a,b=0}^{r-1},$ $\qquad$ $M_n(\beta)=\sum_j...

Finite Partition-Function Curvatures

Let $Z(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},\qquad$ $0<g_j,\quad e_1<\cdots<e_m,$ be a primitive finite weighted canonical partition function, and let \(K=(\log Z)''\) be its log-partition curvature. Physically, \(K\) is the canonical energy variance and satisfies \(C_V=k_B\be...

Finite Partition-Function Curvatures

Let $Z(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},\qquad$ $0<g_j,\quad e_1<\cdots<e_m,$ be a primitive finite weighted canonical partition function, and let \(K=(\log Z)''\) be its log-partition curvature. Physically, \(K\) is the canonical energy variance and satisfies \(C_V=k_B\be...

Projective-Class Transport and Exact Low-Cost Realization of Proximity-Decorated Binary Quadratic Paths

Let $q\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive framed binary quadratic germ over an algebraically closed field of characteristic zero. A maximal primitive blowup path carries multiplicities, free/satellite proximity, and three exceptional types: constant-root branch $B$, mo...

Finite Partition-Function Curvatures

Let $Z(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},\qquad$ $0<g_j,\quad e_1<\cdots<e_m,$ be a primitive finite weighted canonical partition function, and let \(K=(\log Z)''\) be its log-partition curvature. Physically, \(K\) is the canonical energy variance and satisfies \(C_V=k_B\be...

Directional Discriminant Packets and Bounded Scalar-Profile Loci for Primitive Binary Quadratic Germs

Let $q=a u^2+2buv+c v^2\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive binary quadratic germ over an algebraically closed field of characteristic zero, and let $D=\Disc(q)=b^2-ac$. Along a chosen path of point blowups, the lowest $\Sym^2(V^*)$-valued coefficient packets and the sc...

Exact Root-Label Realization Loci in Low-Cost Proximity-Decorated Binary Quadratic Paths

Let $q\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive framed binary quadratic germ over an algebraically closed field of characteristic zero. A maximal primitive blowup path carries multiplicities, free/satellite proximity, and exceptional types $B$, $M$, and $N$, where a constant...

Target-Relative Provenance Requests and Versioned Observation Contracts in Finite Legal Reply Systems of Shogi

We study finite observation systems in which a visible value is always recorded, while one provenance coordinate is requested only on selected visible fibers. The omission event is typed as \(\NR\), rather than represented as a missing or null provenance value. For an episode sys...

Target-Relative Provenance Requests and Versioned Observation Contracts in Finite Legal Reply Systems of Shogi

We study finite observation systems in which a visible value is always recorded, while one provenance coordinate is requested only on selected visible fibers. The omission event is typed as \(\NR\), rather than represented as a missing or null provenance value. For an episode sys...

Perfect-Power Quadratic-Trace Lucas Towers

Let $r,s\in\Z$ be coprime, with $r$ odd and $s\ne0$, and put $z=r+s\sqrt{-2},\qquad D=r^2-2s^2,\qquad Q=r^2+2s^2,$ $B_n=\frac{z^{2n}+\bar z^{2n}}2.$ Assume that, for an odd integer $f\ge3$, one has a perfect-power endpoint $B_f=\sigma A^f$ with $\sigma\in\{\pm1\}$ and $A>0$. For...

Chronological Residual Quotients and Constant-Coordinate Certificate Kernels in Finite Legal Reply Systems of Shogi

We develop a chronological certificate theory for finite legal-reply systems arising from shogi. At each checkpoint, complete legal replies induce a finite full carrier relation together with mediator-wise projected witness sets. Future checkpoints act by synchronized intersectio...

Same-Label Sibling Staircases for Primitive Binary Quadratic Germs

Let $q=a u^2+2buv+c v^2\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive binary quadratic germ over an algebraically closed field of characteristic zero. We study the normalized low-cost same-label fork $B_3\longrightarrow(B_1,B_1),$ where the parent and both branch children have th...