Tao Lin

Exact Kummer Fiber Rigidity and Surface-Scale Energy for Quadratic-Root Bilinear Sums: A 1/2 Low-Frequency Threshold

We study a quadratic-root bilinear architecture in which an early Cauchy--Schwarz step leads naturally to a $3/4$-type exponent scale. At the canonical relation $D=K^2$, we work in the low-frequency, short-Kummer-conductor range. The central arithmetic object is the weighted exac...

Canonical Digital Defects Beyond Periodicity: Automatic Counting Systems, Radix Renormalization, and Finite-State Nonlinear Observables

We develop a second, genuinely nonperiodic instance of canonical arithmetic defect theory. Let $A(N)=\rho N+E(N)$ be a radix counting function and let $P$ be a polynomial of degree $m$. For the radix branch $N\mapsto bN+r$, the nonlinear scale defect $$P(A(bN+r))-b^mP(A(N))$$ los...

Finite Li-Statistic Collisions for Formal Symmetric Zero Multisets

We prove three main theorems about the standard Li-type observation of finite symmetric zero multisets, record two main corollaries, and prove complementary arithmetic-rigidity results for algebraic Gaussian and compactly supported Weil-type observations. Let $\mathcal{H}_{\mathr...

Virtual Composite Modes and Arithmetic Product Walls in Finite-Scale Weil Gram Matrices

We study the finite arithmetic matrix generated by the prime-power shift operators in a compactly supported Weil quadratic form, focusing on structural changes as the support window crosses arithmetic thresholds. An exact four-strip decomposition yields a virtual-composite identi...

Arithmetic Magic Beyond the Reciprocal Phase: Rank-Six Wigner Moments, a Rational Cubic, and Non-CM Motives

We study the power-reciprocal qudit states $$\vert{}R_c^{(2)}\rangle = (p-1)^{-1/2} \sum_{x \in \mathbb{F}_p^*} e_p\left(\frac{c}{x^2}\right)\vert{}x\rangle$$ for odd primes $p$. Their discrete Wigner functions reduce, away from one exceptional row, to the two-parameter rational...

Terminal High-Frequency Structure for Quadratic-Root Bilinear Sums: Reciprocal-Prime Decay, Residual Orthogonality, and Fixed-Core Twist Families

We study the terminal high-frequency regime of a quadratic-root bilinear architecture in which the long prime variable has length $D = K^2$ and a reciprocal phase of size $K$ survives. This regime is deliberately separated from the low-frequency exact-Kummer theory treated previo...

Matrix-Renormalized Canonical Defects for Substitution Cocycles: Symmetric-Power Eigenforms, Norm Forms, and Finite Boundary States

We develop a matrix-valued extension of canonical polynomial defect renormalization for finite-boundary substitution cocycles. The basic algebraic object is an affine bulk/boundary branch, and an anchored direct-sum decomposition canonically separates the purely local term from t...

Canonical Arithmetic Defects for Polynomial Congruence Roots: Higher Tuples, Frobenius Filters, and Resultant-Sharp Finite States

We develop a higher-order extension of canonical bounded defects for periodic root-counting systems. If a period contains $r$ marked residue classes and $A(X)$ counts marked integers up to $X$, then the dyadic $k$-tuple functional $$C(A(2X), k) - 2^k C(A(X), k)$$ has a unique anc...

Partial Resolutions of the Shifted Laguerre Converse: Gaussian Drift and Directional Density Barriers

Let $f$ be a real entire function in the strip class $\mathcal{S}(A)$, and define $f_\mu(x) = f(x + i\mu) + f(x - i\mu)$. Csordas and Escassut conjectured that the strict shifted Laguerre condition $L[f_\mu](x) > 0$ for every real $x$ and every nonzero $\mu$ forces $f$ to belong...

Symplectic Monodromy and Horizontal Sato--Tate for Reciprocal Qudit Wigner Functions

For odd primes $p$, reciprocal-phase qudit states lead, away from one exceptional phase-space row, to the two-parameter exponential sums: $$T_p(A,B) = \sum_{t \neq \pm 1} e_p\left(\frac{A t}{t^2-1} + B t\right)$$ A companion paper established the exact Wigner dictionary, generic...

Canonical Noncommutative Defects Beyond Parikh Projection: Magnus Towers and an Unbounded Static-Dynamic Boundary Gap

We develop an order-sensitive defect formalism for substitution prefixes using the classical Magnus expansion of words. A substitution morphism induces a filtered endomorphism of the completed noncommutative tensor algebra, and an anchored substitution cut produces a multiplicati...

Canonical bounded pair defects for roots of quadratic congruences

Let f in Z[x] be an irreducible quadratic polynomial and let A_{q,f}(X) count solutions of f(n) = 0 mod q in 1 <= n <= X.For an unramified odd prime power q, the local root system contains either zero or two residues per period.We study the dyadic pair-count defect binom(A_{q,f}(...

Canonical bounded pair defects for roots of quadratic congruences

Let f in Z[x] be an irreducible quadratic polynomial and let A_{q,f}(X) count solutions of f(n) = 0 mod q in 1 <= n <= X.For an unramified odd prime power q, the local root system contains either zero or two residues per period.We study the dyadic pair-count defect binom(A_{q,f}(...

Structural Lemmas for the First-Prime Window of the Weil Quadratic Form

We prove six structural results for the local Weil quadratic form Q_W^L on L^2(-L,L) in the first-prime window (half*log2 < L < half*log3), the minimal interval where only the prime n=2 contributes to the Weil explicit formula. Our main technical contribution (Theorem 5.1) is an...