Independent mathematics research
The same structure keeps
returning, wearing different names.
Avadari studies the parallels that surface again and again across algebraic, differential, and advanced mathematics — the recurring architecture behind Euler's identities, the zeta function, and the machinery that connects them. We think these aren't three subjects. They're one grammar, spoken in three dialects.
The Avadari Method
Three threads, one grammar
At Avadari, we work quietly, test results against a small circle of collaborators before they leave the building, and publish only when a proof is load-bearing — when it holds weight for more than one field at once.
The working hypothesis underneath everything here: algebraic identity, differential behavior, and the deeper machinery of primes and zeta functions are not separate territories that occasionally intersect. They are the same underlying shape, observed from three different instruments.
A working analogy
Three scripts, one slab, one decree — that's the whole idea.
In 1799, a single decree was carved into the Rosetta Stone three times: once in hieroglyphic, once in Demotic, once in Greek. Not three different messages — one message, in three notations. Once scholars could read the Greek, the stone became a machine for teaching them the other two.
The Avadari Method treats algebraic, differential, and advanced mathematics the same way: three notations for one underlying text. A result that's stubborn in one language often turns out to already be solved in another — we just have not learned to read the translation yet. Finding the inscription that survives the trip from one script to the next, and using it to carry a proof across, is the method itself.
Structure & symmetry
Where identity is exact. We look at what a relation preserves when everything around it is allowed to change — the invariants that survive transformation.
Change & flow
Where identity is a rate. The same structures reappear as behavior over time — as the rule a system obeys while it's in motion, not just at rest.
ζ, primes & Euler
Where the two meet. The zeta function and its relatives sit at the seam — analytic objects that encode strictly algebraic information about primes.
Publications
Research Preprints
Full-length formal manuscripts detailing extended proofs, modular parity decompositions, and foundational geometric-algebraic structures.
A Modular Parity Decomposition of the Riemann Zeta Function and Its Asymptotic Floor Boundary
We present a modular parity decomposition of the Riemann Zeta function partitioned modulo 4 into its constituent Hurwitz sub-series. By analyzing parity sums, alternating series differences, and cross-parity ratios, we isolate the transcendental complexity into a single parameter-free coordinate remainder k(n) contracting at (2/3)n, and analyze the floor boundary conjecture ⌊ζ(n, 3/4)/ζ(n)⌋ = ⌊(4/3)n⌋ connecting discrete Diophantine boundaries to continuous modular L-functions.
The Geometric-Algebraic Structure of the Partitioned Riemann Zeta Function
We present a geometric-algebraic partitioning of the Riemann zeta function to consider a previously unreported structural relationship between its symmetric components. A geometric interpretation of beta(n) and eta(n) is offered, exposing a coordinate ratio K(n) with a remainder k(n) that links discrete and known transcendental values of zeta(n).
Reference Benchmarks
Mathematical Pearls
Concise, single-topic working notes and coordinate reference tables establishing verified numerical benchmarks with permanent DOIs.
Modulo-4 Analytic Partitions: Bridging Euler's Exponential Decompositions to Dirichlet Series and the Coordinate Ratio K(s)
We present a direct structural correspondence between Euler's modulo-4 partition of the exponential series and the modulo-4 arithmetic partition of Dirichlet series. By evaluating the symmetric sums and alternating differences of both systems, we demonstrate that the rational factor 2s − 1 emerges as the Dirichlet analogue of the hyperbolic partition ratio, while the Coordinate Ratio K(s) emerges as the Dirichlet analogue of the circular cotangent partition ratio, defined by the quotient of the Dirichlet beta function and the scaled Dirichlet eta function.
Cayley Transform and Unitary Coordinate Representation of the Dirichlet Beta-Eta Parity Ratio
We define the normalized parity remainder k(n) between the Dirichlet Beta and Dirichlet Eta series across integer dimensions n ≥ 2, and establish its Cayley transform C(n) and orthogonal complement W(n) mapping the parity ratio onto the unit circle C(n)2 + W(n)2 = 1.
The Role of k(n) in the β − η Bridge at the s = 1 Pole Boundary
We define the ratio coordinate K(s) = 2s β(s)/η(s) and its remainder k(s) = K(s) − 2s as the continuous bridge between the Dirichlet Beta function β(s) and the Dirichlet Eta function η(s). At the s = 1 pole boundary, the evaluation β(1) = π/4 emerges as an algebraic balance point where the ln(2) magnitude of the alternating harmonic series η(1) cancels with the inverse logarithmic scale of the scaffold derivative.
The ζ Coordinate Framework: Normalized-to-Analytic Conversion and β − η Parity Decomposition
We unify the primary ζ coordinate framework with the β − η parity difference decomposition. Table 1 defines the conversion between normalized integer coordinates and continuous analytic L-function evaluations across the ζ Parent Series, the Even Parity Sector, and the Odd Parity Sector via the scale 4n/ζ, yielding the Coordinate Ratio K(n). Table 2 isolates the parity difference sector to extract the Coordinate Remainder k(n) = K(n) − 2n.
Algebraic Compendium
Identities & Bridges
Under Construction
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