Akari

Observation Transport and Synchronization Contracts in Finite Legal Reply Systems of Shogi

A legal candidate move in shogi induces a finite system of all legal replies. Previous work represented such a system by reply observations, discernibility sets, an inclusion-minimal discernibility skeleton, and a profile symbol. This paper studies what happens when the observati...

Discernibility Profiles and Mechanism Repertoires of Legal Reply Systems in Shogi

For a fixed legal shogi position and a legal candidate move, consider the complete finite set of legal replies. We observe each reply through three shogi-specific components: actor action edges lost after the reply ($\Lost$), actor action edges newly gained ($\Gain$), and the com...

Observation Transport and Synchronization Contracts in Finite Legal Reply Systems of Shogi

A legal candidate move in shogi induces a finite system of all legal replies. Previous work represented such a system by reply observations, discernibility sets, an inclusion-minimal discernibility skeleton, and a profile symbol. This paper studies what happens when the observati...

Canonical Chain–Node Encoding and Finite-Width Return Renormalization for the Accelerated Collatz Map

We select a canonical orientation of an alternating signed-binary representation of positive odd integers and refine it into a chain--node encoding, called the canonical CPE. The signed representation itself is classical in spirit and closely matches the alternating binary system...

Resonant Terminal Layers and Critical Interfaces in Finite Collatz Bridges

We study finite orbit segments of the shortcut Collatz map in shifted coordinates. A terminal word is called multiplier-supercritical when its linear multiplier exceeds one. For a bridge ending at height $k$ with overshoot $U$, we prove an exact depth-by-depth criterion for termi...

Quadratic Apolar Residues and Pfaffian Root Correspondences in Codimension-Three Lefschetz Failure

Let $\kk$ be an algebraically closed field of characteristic zero, let $S=\Sym(V)$ with $\dim V=3$, and let $A=S/\Ann(F)$ be a compressed standard-graded Artinian Gorenstein algebra of socle degree $2m+1$, where $m\ge2$. We study a corank-one failure of multiplication by a genera...

Canonical Chain–Node Encoding and Finite-Width Return Renormalization for the Accelerated Collatz Map

We select a canonical orientation of an alternating signed-binary representation of positive odd integers and refine it into a chain--node encoding, called the canonical CPE. The signed representation itself is classical in spirit and closely matches the alternating binary system...

Product-Core Stratification of Large Effective Multiplier Rank in Symmetric Frobenius Algebras

Let $A$ be a $2n$-dimensional commutative symmetric Frobenius algebra over a field $k$, let $U=V^\perp$ with $\dim U=\dim V=n$, let $L\subset A$ contain the unit, and consider the Frobenius trilinear form induced by multiplication on $(L/k1)\times U\times V$. Write $(c,a,b)$ for...

Discernibility Profiles and Mechanism Repertoires of Legal Reply Systems in Shogi

For a fixed legal shogi position and a legal candidate move, consider the complete finite set of legal replies. We observe each reply through three shogi-specific components: actor action edges lost after the reply ($\Lost$), actor action edges newly gained ($\Gain$), and the com...

Spectral Endomorphism Factorization and Exceptional Degree-Two Root Correspondences in Codimension-Three Lefschetz Failure

Let $X$ be a smooth rational surface obtained by resolving the generically reduced quadratic-root correspondence attached to a controlled codimension-three weak Lefschetz failure. The preceding Pfaffian analysis produces a pulled-back rank-two quotient bundle $Q$, a nowhere-zero...

Resonant Terminal Layers and Critical Interfaces in Finite Collatz Bridges

We study finite orbit segments of the shortcut Collatz map in shifted coordinates. A terminal word is called multiplier-supercritical when its linear multiplier exceeds one. For a bridge ending at height $k$ with overshoot $U$, we prove an exact depth-by-depth criterion for termi...

Spectral Endomorphism Factorization and Exceptional Degree-Two Root Correspondences in Codimension-Three Lefschetz Failure

Let $X$ be a smooth rational surface obtained by resolving the generically reduced quadratic-root correspondence attached to a controlled codimension-three weak Lefschetz failure. The preceding Pfaffian analysis produces a pulled-back rank-two quotient bundle $Q$, a nowhere-zero...

Product-Core Stratification of Large Effective Multiplier Rank in Symmetric Frobenius Algebras

Let $A$ be a $2n$-dimensional commutative symmetric Frobenius algebra over a field $k$, let $U=V^\perp$ with $\dim U=\dim V=n$, let $L\subset A$ contain the unit, and consider the Frobenius trilinear form induced by multiplication on $(L/k1)\times U\times V$. Write $(c,a,b)$ for...

Quadratic Apolar Residues and Pfaffian Root Correspondences in Codimension-Three Lefschetz Failure

Let $\kk$ be an algebraically closed field of characteristic zero, let $S=\Sym(V)$ with $\dim V=3$, and let $A=S/\Ann(F)$ be a compressed standard-graded Artinian Gorenstein algebra of socle degree $2m+1$, where $m\ge2$. We study a corank-one failure of multiplication by a genera...

A Local Toeplitz--Hankel Criterion Equivalent to the Riemann Hypothesis

Let $\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the completed Riemann xi-function, and let $F(x)=\frac{\xi'}{\xi}\!\left(\frac{1}{1-x}\right)=\sum_{n\ge 0} f_nx^n$ initially denote its germ at the origin. We introduce the real symmetric Toeplitz--Hankel matrix $c_{ij}...

Query-Relative Quotients and Certificate-Carrying Observation Reducts

We present a framework for reducing a finite set of legal moves or legal replies in a fixed shogi position while preserving exactly the distinctions required by declared queries. Our preceding work first formalized complete positions and canonical legal actions as foundational ob...

Weak Vishik Equivalence and Linear $2$-Basis Blindness for Totally Singular Quadratic Forms

Let $K$ be a field of characteristic two, put $k=K^2$, and let $U\subset K$ be the value space of an anisotropic totally singular quadratic form. For $a\in K\setminus k$, the anisotropic dimension after the one-step extension $K(\sqrt a)/K$ is $\oneprofile_U(a)=\dim_{k(a)}k(a)U=\...

A Sharp Multiplier-Dimension Threshold for Orthogonal-Dual Product Spaces

Let $A$ be a finite-dimensional commutative symmetric Frobenius algebra over a field $k$, let $\dim_k A=2n$, and let $U\subset A$ be an $n$-dimensional subspace with orthogonal complement $V=U^\perp$. For a multiplier subspace $L\subset A$, write $LU$ for the span of all products...

A Local Toeplitz--Hankel Criterion Equivalent to the Riemann Hypothesis

Let $\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the completed Riemann xi-function, and let $F(x)=\frac{\xi'}{\xi}\!\left(\frac{1}{1-x}\right)=\sum_{n\ge 0} f_nx^n$ initially denote its germ at the origin. We introduce the real symmetric Toeplitz--Hankel matrix $c_{ij}...

Value-Space Contractions for Fixed-Diagonal Rank Fibers\\ of Symmetric Matrices in Characteristic Two

Let $K$ be a field of characteristic two and let $k=K^2$ be its subfield of squares. For a prescribed diagonal vector $d=(d_1,\ldots,d_n)\in K^n$, we study the symmetric matrices of diagonal $d$, graded by rank. The associated quasilinear quadratic form $q_d(x)=\sum_{i=1}^n d_i x...