Tao Lin

Certified Stratified Phase Diagrams for Finite Hopkins Lithography

We study an explicitly declared finite partially coherent Hopkins imaging model and develop a proof-carrying framework for topology changes under focus and threshold variation. Three phase diagrams are separated: bifurcations of the spatial critical set, changes in adjacent criti...

Certified Stratified Phase Diagrams for Finite Hopkins Lithography

We study an explicitly declared finite partially coherent Hopkins imaging model and develop a proof-carrying framework for topology changes under focus and threshold variation. Three phase diagrams are separated: bifurcations of the spatial critical set, changes in adjacent criti...

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 $\binom{A(2X)}{k} - 2^k \binom{A(X)}{k}$ has a un...

Prime-Power Activation Geometry in Finite-Scale Weil Operators: Quadratic Rank, Intrinsic Curvature, and an Arithmetic Weyl Law

We develop a finite-scale structural theory for the quadratic prime-power activation walls associated with Weil-type arithmetic operators. The arithmetic coefficient at a prime power $q = p^m$ is $c_q = \Lambda(q)/\sqrt{q}$, with activation location $L_q = \frac{1}{2}\log q$. Sta...

Prime-Power Activation Geometry in Finite-Scale Weil Operators: Quadratic Rank, Intrinsic Curvature, and an Arithmetic Weyl Law

We develop a finite-scale structural theory for the quadratic prime-power activation walls associated with Weil-type arithmetic operators. The arithmetic coefficient at a prime power $q = p^m$ is $c_q = \Lambda(q)/\sqrt{q}$, with activation location $L_q = \frac{1}{2}\log q$. Sta...

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...

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...

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...

Fourth-Moment Collision K3 Surfaces, Genus-Two Splitting, and a Projective Modular Packet

We study an explicit rational-phase exponential-sum family through the geometry of its mixed fourth moment. After saturating away the ordered pairing components, the residual collision surface is birational to a double plane whose minimal resolution is a $K3$ surface. A symplecti...

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

We study the power-reciprocal qudit states $$\vert{}R_c^{(2)}\rangle=\frac{1}{\sqrt{p-1}} \sum_{x\in\mathbf F_p^\times}e_p(c/x^2)\vert{}x\rangle,$$ for odd primes $p$. Their discrete Wigner functions produce, away from one exceptional row, the two-parameter rational exponential s...

Exact Activation Rank in Multiplicative Excursion Walls: All-Order Boundary Cyclicity, Explicit Finite Caps, and Sharp Low-Order Minors

We close the structural strict-rank problem left open by the boundary-history factorization for prime-power activation walls in finite-scale Weil operators. At an activation q, the order-k wall jet factors through a k-dimensional boundary-history core. We isolate the correct stri...

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...

Rank-Six Symplectic Monodromy, Horizontal USp(6), and Coherent-Spike Renormalization for Reciprocal-Square Qudit Wigner Functions

We study the reciprocal-square qudit state $\psi_c(x) = (p-1)^{-1/2} e_p(c/x^2)$ for $x$ nonzero, with $\psi_c(0)=0$, and the rank-six exponential-sum family arising from its discrete Wigner function. The associated two-parameter $\ell$-adic local system has generic Swan ledger $...

Fourth-Moment Collision K3 Surfaces, Genus-Two Splitting, and a Projective Modular Packet

We study an explicit rational-phase exponential-sum family through the geometry of its mixed fourth moment. After saturating away the ordered pairing components, the residual collision surface is birational to a double plane whose minimal resolution is a $K3$ surface. A symplecti...

Passive Observer Fibers in Two-Level Quantum Dynamics: Local Rigidity, Global Ambiguity, and Controlled Active Completion

We study identifiability of finite-dimensional Hermitian dynamics from complete site-to-site transition probabilities. On the two-level spectral stratum $H = \alpha I + \beta P$, passive probabilities factor through a simple projector-modulus observer. We give the exact static/vi...

Sharp Logarithmic Retained-Score Laws for Local Decoders in Critical Planar Ising Models

We study the loss incurred when a raw observation that is sufficient for the global-spin-even quotient scores considered here is processed only by a spatially local additive decoder. The nearest-neighbor bond field determines the spin configuration modulo the global spin flip, so...

Sharp Logarithmic Retained-Score Laws for Local Decoders in Critical Planar Ising Models

We study the loss incurred when a raw observation that is sufficient for the global-spin-even quotient scores considered here is processed only by a spatially local additive decoder. The nearest-neighbor bond field determines the spin configuration modulo the global spin flip, so...

Passive Observer Fibers in Two-Level Quantum Dynamics: Local Rigidity, Global Ambiguity, and Controlled Active Completion

We study identifiability of finite-dimensional Hermitian dynamics from complete site-to-site transition probabilities. On the two-level spectral stratum $H = \alpha I + \beta P$, passive probabilities factor through a simple projector-modulus observer. We give the exact static/vi...

Fourth-Moment Collision K3 Surfaces, Genus-Two Splitting, and a Projective Modular Packet

We study an explicit rational-phase exponential-sum family through the geometry of its mixed fourth moment. After saturating away the ordered pairing components, the residual collision surface is birational to a double plane whose minimal resolution is a $K3$ surface. A symplecti...

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...