Chetansing K. Rajput

A Symmetric Gaussian Cubic Form on the Pythagorean Angle Lattice: Unit Triples, a ℤ[√3] Composition Law and Pell Symmetries

A symmetric Gaussian cubic form on the Pythagorean Angle Lattice admits a complete composition law in ℤ[√3], classifies all unit triples via Pell shells, and yields Pell symmetries with explicit spectral and arithmetic dynamics. The Pythagorean angle lattice is PAL = {cot(α/D) :...

Covering Laws for Powers of the Pythagorean Angle Lattice: Pell Twins and a Ternary Invariant

Sharp covering laws for powers of the Pythagorean Angle Lattice are governed by Pell twins and a ternary invariant of the symmetric Gaussian cubic form, yielding the cubic law ρ₃(Q) ≍ Q^{-3/4} and a general lattice-number conjecture. The Pythagorean angle lattice is PAL = {cot(α/...

Dyadic Division Towers of the Angle Difference in the Pythagorean Angle Lattice: Cyclotomic Descent, Exact Decoding, Power Resonance and Arithmetic Moduli

Dyadic division towers of the angle difference in the Pythagorean Angle Lattice are completely classified by cyclotomic descent, exact address decoding, power resonance classes, and arithmetic moduli of Gaussian exponent vectors. Let θ be the smaller acute angle of a primitive Py...

Cyclotomic Square-Return Rigidity in the Pythagorean Angle Lattice: Double Packet Descent, Terminal Prime Support, and the Even-Class-Number Frontier

Square return of a primitive Pythagorean angle at an odd prime p is completely rigid when the class number of ℚ(ζ_p) is odd, and the remaining even-class-number cases are reduced to explicit 2-class data of real cyclotomic fields. Let W = u + iv be a primitive Gaussian integer wi...

Pythagorean Angle Lattice: Rational Division, Degree Rigidity, Source–Division Duality, and Global Reconstruction

Degree–return rigidity on the Pythagorean Angle Lattice identifies algebraic degree with minimal return depth in the quotient T_PAL, yielding a source–depth–branch normal form and a complete global reconstruction of the lattice and its profinite division boundary. The Pythagorean...

Radioactive Highway: Ionizing Radiation on a National Highway — Field detection of unshielded industrial gamma radiography at Padgha, Tal. Bhiwandi, Dist. Thane, Maharashtra, India

Radioactive Highway: Ionizing Radiation on a National Highway — Field detection of unshielded industrial gamma radiography at Padgha, Tal. Bhiwandi, Dist. Thane, Maharashtra, India Unshielded industrial gamma radiography on the Mumbai–Agra National Highway at Padgha Toll Plaza, B...

Pythagorean Angle Lattice: Dyadic Collision Rings, Hasse Depth, and a Carry–Boundary Comparison for Circular Units

In the Pythagorean Angle Lattice, the real carry word of a circular unit and its first dyadic boundary differ only by a reciprocal-unitary Gaussian-period automorphism and share the same Hasse depth on every compatible repeated-root primary block : We study a binary repeated-root...

Pythagorean Angle Lattice: Division Content, Oriented Kummer Structure, Dyadic Descent, and Arithmetic Height Spectrum

Oriented Kummer structure and dyadic descent on the Pythagorean Angle Lattice produce a canonical division content whose exact-content height spectrum has poles at s = 1/C and maximal depth asymptotic to log X / (2 log log X) : The Pythagorean Angle Lattice (PAL) is an arithmetic...

Pythagorean Angle Lattice: Division Content, Cyclotomic Decimation, Lucas Reduction, and Selmer Chamber Rigidity

The Pythagorean Angle Lattice yields an unconditional Lucas reduction of prime square events together with a dyadic Selmer chamber mechanism that excludes all such events for primes p ≡ 1 (mod 24) below 3200 and for p = 5209. We develop the arithmetic of the Pythagorean Angle Lat...

Pythagorean Angle Lattice: The Division-Ramification-Criticality Theorem (DRC theorem)

Proves the Division-Ramification-Criticality (DRC) theorem, establishing that principal dyadic division, Gaussian ramification, and arithmetic criticality govern the exact arithmetic survival of Pythagorean angles under repeated halving : A primitive Pythagorean triple determines...

Pythagorean Angle Lattice: The Division-Ramification-Criticality Theorem (DRC theorem)

Proves the Division-Ramification-Criticality (DRC) theorem, establishing that principal dyadic division, Gaussian ramification, and arithmetic criticality govern the exact arithmetic survival of Pythagorean angles under repeated halving : A primitive Pythagorean triple determines...

The Pythagorean Angle Lattice: Arithmetic Spectrum Barnes-Cone and Gaussian Geometry A Complete Synthesis of Geometry, Spectral Theory, Pell Dynamics, and Gaussian Markoff Geometry

A single arithmetic kernel extracted from metallic means and primitive Pythagorean triples is revealed to be the Barnes double zeta function, generating a complete spectral theory, heat-trace asymptotics, and a geometric lattice whose boundary constant is exactly 4. We present a...

The Pythagorean Angle Lattice: A Unified Framework Connecting Geometry, Number Theory, and the Langlands Program

A single lattice built from primitive Pythagorean triples and Gaussian integers unifies metallic means, an Omni-Metallic Euler constant, explicit S₃ and holomorph Galois extensions of ℚ(i), closed geodesics on the modular surface, and Maass/Bianchi forms in one geometric-arithmet...

The Pythagorean Angle Lattice: Geometry, Algebra, and Hyperbolic Correspondence

A lattice built from the angles of primitive Pythagorean triples systematically generates algebraic, logarithmic and transcendental constants while revealing an S₃-Galois extension and a precise correspondence with closed geodesics on the modular surface : We study the Pythagorea...

The Pythagorean Angle Lattice: Arithmetic Structures and Spectral Geometry A Synthesis of the Omni-Metallic Framework, Non-Abelian Character Invariants, and the Geometry of Integer Triangles

A self-contained arithmetic–geometric framework linking primitive Pythagorean triples to zeta values, modular geodesics, and a conjectural cancellation of non-abelian Artin characters. We present a complete, self-contained unification of the Pythagorean Angle Lattice L and the Om...

Pi from the Pythagorean Angle Lattice: A Direct Geometric Sieve from Integer Right Triangles

π is recovered as the ratio of two full rotations (720°) to a universal constant that emerges from the asymptotic angles of primitive Pythagorean triples. We present a rigorous derivation of the constant π strictly from the geometry of primitive Pythagorean triples. By ordering t...

Pi from the Pythagorean Angle Lattice: A Direct Geometric Sieve from Integer Right Triangles

π is recovered as the ratio of two full rotations (720°) to a universal constant that emerges from the asymptotic angles of primitive Pythagorean triples. We present a rigorous derivation of the constant π strictly from the geometry of primitive Pythagorean triples. By ordering t...

The Pythagorean Angle Lattice: Arithmetic Structures and Spectral Geometry A Synthesis of the Omni-Metallic Framework, Non-Abelian Character Invariants, and the Geometry of Integer Triangles

A self-contained arithmetic–geometric framework linking primitive Pythagorean triples to zeta values, modular geodesics, and a conjectural cancellation of non-abelian Artin characters. We present a complete, self-contained unification of the Pythagorean Angle Lattice L and the Om...

The Pythagorean Angle Lattice: Angle Hierarchy and Spectral Structure

Every primitive Pythagorean triple generates an infinite angle-halving hierarchy that recovers the metallic means precisely when c−b equals 1, 2 or 8, and naturally produces a convergent spectral zeta function. We study the Pythagorean Angle Lattice, a countable set of real numbe...

The Pythagorean Angle Lattice: Angle Hierarchy and Spectral Structure

Every primitive Pythagorean triple generates an infinite angle-halving hierarchy that recovers the metallic means precisely when c−b equals 1, 2 or 8, and naturally produces a convergent spectral zeta function. We study the Pythagorean Angle Lattice, a countable set of real numbe...