Tao Lin

Finite-Channel Dirac Tomography on Flat Tori: Nonlinear Recovery, Genuine Blindness, and a Matching Resolution Edge

We study what geometric information about an unknown conformal metric on a marked flat spin torus can be recovered from a fixed low-energy compression of its Dirac operator using a fixed finite collection of operator-valued observation channels. After the nonlinear reparameteriza...

Sparse-Burst Compression and Mesoscopic Screening in the Shifted Laguerre Converse

Let $f$ be a real entire function whose zeros lie in a fixed horizontal strip, and set$$f_\mu(x) = f(x + i\mu) + f(x - i\mu).$$The shifted Laguerre converse asks whether strict positivity of the first Laguerre expression for every nonzero imaginary shift forces $f$ to belong to t...

Quantitative Prolate-Proxy Estimates for Zeta Spectral Models

We isolate and prove the analytic part of a proxy-to-ground-state bridge in recent spectral models for the Riemann zeta function. On $I_\lambda=[\lambda^{-1},\lambda]$ we distinguish the semilocal Weil quadratic form from its associated self-adjoint operator, replace endpoint nor...

Primitive-Quotient Gap Packing in Native-Radix Collision Geometry: Variable-Gap Energy, Residual Blockers, and Capped Hyperbolic Kernels

We develop an arithmetic and combinatorial sequel to the structural native-radix collision theory established previously. Starting from the published primitive-quotient decomposition of genuine augmented cross edges, we prove a mesoscopic gap-or-small-packet dichotomy, radial she...

Critical Effective Rank and Sparse Zero Bursts in the Shifted Laguerre Converse

Let $f$ be a real entire function in the standard horizontal-strip class, with zero Gaussian coefficient, and suppose that every nonzero vertical shift satisfies the strict Laguerre inequality. We study vanishing-height nonreal critical points of $f$, equivalently zeros of the lo...

Minkowski Bounds for Pasten's Arithmetic Derivative Lattices in the Squarefree Subfamily

For each coprime triple $(a,b,c)$ with $a+b=c$, Pasten (2021) constructs a universal derivative lattice $F(a,b)$ of rank $\omega(abc)-1$ from the linear constraint imposed by the weighted arithmetic derivative. We prove that for squarefree coprime triples, the lattice determinant...

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 for $\mu \in \mathbb{R}$ put $f_\mu(x) = f(x+i\mu) + f(x-i\mu)$. Csordas and Escassut asked whether strict shifted-Laguerre positivity $L[f_\mu](x) > 0$ for every real $x$ and every nonzero $\mu$ forces $f...

The Terminal D4 Packet in Rank-Six Arithmetic Magic: A Gaussian Weil Prym Fourfold and a Dihedral Weight-One Form

The rank-six power-reciprocal Wigner family studied previously leaves a terminal rank-four finite-monodromy local system D₄ whose image was only known to lie in Aut(D₄) = W(F₄), and whose intersection-cohomology rank was not identified. We close this packet exactly. Two explicit...

Correlation-Generated Factorability in a Generic Sector of Folded Quadratic Kloosterman Sums

We isolate a factorability mechanism that appears after a $TT^*$ expansion of a folded quadratic-root Kloosterman sum. Starting from a two-variable sum with modular inverse phase and a smooth radial phase, the off-diagonal correlation for two short variables $r_1, r_2$ has a comm...

Exact Finite-Observation Collisions for Symmetric Zero Multisets

We prove an exact non-discrimination theorem for finite Li-type observations of functional-equation symmetric zero multisets. For every prescribed finite order $m$, we construct two finite multisets in the open critical strip, invariant under complex conjugation and $s \mapsto 1-...

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...

Collision Geometry of Finite Radix Value Sets: Restricted Product Regions, Dyadic Divisor Fibers, and Hyperbolic Kernels

We study the finite value sets of two quadratic state forms arising in a binary--ternary native-radix comparison. The negative sectors admit exact factorizations into restricted multiplication-table regions. After removing the repeated binary zero state, the binary and ternary ne...

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...

Pre-Cauchy Barriers for Signed Quadratic Root Packets

We study a signed bilinear model built from quadratic congruence root packets at the critical scales $D = K^2$. The long variable carries a von Mangoldt–Dirichlet coefficient and the short variable a Möbius–Dirichlet coefficient. A first Cauchy–Schwarz or Hilbert-space completion...

Multiplicative Excursion Walls in Finite-Scale Weil Operators: Rational Ghosts, Boundary-Rank Laws, and Dense Arithmetic Curvature

We develop a higher-order extension of the arithmetic wall structure arising from compactly supported finite-scale Weil operators. For products of truncated two-sided shifts, an exact sign-path expansion shows that each overlap cell is controlled by the range of the partial-sum p...

An Arithmetic Resolvent and Heat Kernel for the Riemann Xi Function in the Variable \(s(1-s)\)

We descend the completed Riemann Xi function through the quotient map $s\mapsto s(1-s)$ and obtain an entire function $\mathcal D$ of order $1/2$ and genus zero. Its zeros are the functional-equation orbit values $\rho(1-\rho)$, and the Riemann hypothesis is equivalent to their s...

Collective Screening and Real-Mesh Obstructions in the Shifted Laguerre Converse

1. Background & Foundational Framework 1.1 The Laguerre–Pólya Class $\mathcal{LP}$ A real entire function $f(z)$ belongs to the Laguerre–Pólya class $\mathcal{LP}$ if it can be represented in the Hadamard product form: $$f(z) = C z^m e^{-\alpha z^2 + \beta z} \prod_{k=1}^\infty \...

Finite Spectral Data Do Not Determine a Real-Rooted Entire Function

We prove an explicit non-uniqueness theorem for real-rooted entire functions. Starting from an even genus-zero canonical product with simple positive zeros $\gamma_n$ of exponent of convergence one, and fixing arbitrary integers $k,J$, we perturb the infinite tail while compensat...

Mesoscopic Blindness of Hypersingular Riesz Polarization: Defect Budgets, Tangent Coercivity, and Gap-Moment Thresholds

We study quantitative stability and instability for one-dimensional hypersingular Riesz polarization. For the periodized Riesz kernel of exponent $s>1$, equally spaced points have optimal polarization $P^*_{M,s}=C_sh^{-s}$, where $$C_s=2(2^s-1)\zeta(s),\qquad h=L/M.$$ We construc...

Arithmetic Magic in Reciprocal Qudit States: Exact Wigner Moments, a Singular K3 Surface, and CM Modularity

We study a family of odd-prime-dimensional pure states whose amplitudes carry the reciprocal phase $x\mapsto c/x$. Their discrete Wigner functions reduce, away from one exceptional phase-space row, to the two-parameter finite-field exponential sums $$T_p(A,B)=\sum_{t\ne\pm1} e_p\...