Publications

In this collaboration with Quantum Gravity Research, we take the Tri-Quarter Framework—originally a 2D method for folding a flat triangular grid in on itself via circular inversion with perfect, lossless precision—and launch it up into the higher dimensions with hyperspherical inversion, all the way to the legendary E₈ lattice in 8D, one of the most elegant and symmetric structures in all of mathematics. The core idea is a kind of mathematical origami: a reversible “inversion” that folds far-away points in these vast lattices back into a small, tidy region and unfolds them again without ever losing a speck of information—no rounding, no approximations, just exact arithmetic. Because the boundary sphere is a perfect mirror, every one of the infinitely many points in the sprawling outer region gets a unique “twin” inside the finite inner zone—so an endless lattice is faithfully encoded into a small, bounded space, with nothing duplicated and nothing lost. That exactness is the whole point, because E₈ and its 8D cousins sit at the heart of cutting-edge physics models for quasicrystals and quantum gravity, where the sheer number of points to track explodes so fast that direct computation becomes hopeless. By folding that explosion down to size, our method runs these calculations 83,000 to 260,000 times faster than the standard exact approach—and the bigger the problem, the bigger the advantage. As a bonus, the same fold neatly reveals a famous 4D shape (the 600-cell) nested inside E₈ and scaled by the golden ratio, which is a satisfying hint that this structure was hiding there all along. Onwards to unification! 🚀 😎

This is the original Tri-Quarter Framework in its full 2D form: a way of taking a flat, endlessly repeating triangular grid and giving it a rich set of perfectly reversible symmetries— rotations, mirror reflections, and a circular “inside-out” inversion that swaps the region inside a circle with the region outside it—all computed with exact arithmetic, no approximations or floating-point hassles. Every one of these moves can be undone without losing a shred of information, which is what lets the framework fold, mirror, and reorganize the grid with total precision. The engine driving it all is a 24-element collection of symmetries I call 𝕋₂₄, and here's the part I find genuinely delightful: this hand-built collection turns out to be a famous structure that crystallographers and mathematicians have studied for over a century—the hexagonal symmetry group that describes how honeycombs, snowflakes, and many real crystals are organized. So rather than inventing something exotic and isolated, the framework plugs straight into a deep, well-understood body of classical mathematics, with everything ever proven about that structure coming along for free. And the payoff is practical, because the grid splits cleanly into six interlocking color groups that never interfere with each other, so the work can be spread across modern parallel hardware like a graphics card, and on an ordinary gaming laptop the symmetry-aware version runs the calculations a median of roughly 8× faster than a standard single-core approach—an advantage that only grows as the problem gets bigger. It's a foundation for fast, exact, symmetry-aware computing, with applications spanning geometry, graph algorithms, tiling, clustering, and parallel computation. BOOM SHAKALAKA! 🏀

Every time your phone, a satellite, or a smart device sends data, it's really sending a stream of 1s and 0s encoded as a wave—and the receiver has to correctly read each bit even when the signal is buried in noise. This case study takes the Tri-Quarter Framework and puts it to work on exactly that problem, for a common encoding scheme called BPSK. The clever part is that our approach is “model-free”: it doesn't need to be told in advance what kind of noise it's dealing with. Most standard decoders are tuned for one specific flavor of noise—the gentle, bell-curve “static” that engineers usually assume—and they perform beautifully there but fall apart when the interference gets spiky and unpredictable, the kind you actually get in a crowded city, a factory floor, or out at sea. Our method uses simple sign-and-distance rules to decode signals on the fly, and in 100,000 simulated trials it holds its own against the textbook techniques in calm conditions while dramatically outperforming them when the noise turns nasty—all while staying cheap enough to run in just a handful of CPU cycles. The upshot: a versatile, robust decoder for the messy real-world environments—urban cellular, industrial sensors, ocean buoys, naval comms—where the usual assumptions break down, with a clear path to extend it to richer encodings down the road.

This is the founding paper of the Tri-Quarter Framework, where the core idea first took shape. That idea is deceptively simple: take an ordinary circle drawn on a flat plane and give it a job. Normally a circle is just a passive boundary, but here it becomes an active divider that cleanly separates “inside” from “outside” and keeps track of which direction every point is facing. To do this, we attach extra directional labels (called phase pairs) to every point and unify the three coordinate systems mathematicians normally juggle separately—rectangular, polar, and complex—into one coherent picture. We then prove two elegant duality theorems: one showing the circle perfectly partitions space into matched inner and outer regions, and a second—named after the artist M.C. Escher—showing a mirror-like “inside-out” flip that swaps those regions while preserving all the directional information. Beyond its geometric elegance, the framework is practical: in a worked example, it cuts the number of decision-checks a computer must perform from up to seven down to four, a tidy efficiency win. It's a fresh lens for analyzing anything with circular symmetry—from black hole physics to signal processing—and it laid the conceptual groundwork that every later Tri-Quarter Framework paper builds upon.

In 2015, humanity directly detected gravitational waves for the first time—ripples in the fabric of spacetime itself, set off by two black holes colliding over a billion light-years away. This essay, written for the FQXi “Wandering Towards a Goal” contest, steps back to ask a bigger question: how did we get there? The detection wasn't just a triumph of incredible engineering—it rested on a full century of theoretical work by scientists who, in their own time, were wandering through abstract mathematics with no guarantee any of it would ever connect to reality. We explore that wandering and argue that this seemingly aimless roaming through “mindless mathematical laws” is in fact how real progress toward the grand goal of unifying physics tends to happen—a reflection on the strange, winding path from pure idea to world-changing discovery.

A Latin square is the mathematical idea hiding inside every Sudoku puzzle: a grid where each symbol appears exactly once in every row and every column. They look like a game, but they quietly encode the structure of deep algebraic objects—and that makes them valuable raw material for building cryptographic systems that keep data secure. This thesis digs into a hard, still-unsolved question about special diagonal patterns within these grids (called transversals), surveys what's known, and builds software tools to generate Latin squares and count those patterns. The centerpiece is a new algorithm I designed that snaps small prime-sized Latin squares together like building blocks to construct larger “super-symmetric” ones, letting me predict their transversal counts with accuracy. As a capstone, I use one of these structures to build a simplified version of Grøstl, a real cryptographic hash function—showing how this puzzle-like mathematics reaches all the way to modern data security.

Science advances not only by proposing new ideas but by carefully checking and correcting claims that don't hold up—and that's exactly what this paper does. A group of researchers had published a series of bold assertions: that Einstein's general relativity is wrong and that the Universe isn't actually expanding. Those are extraordinary claims, and we examine them rigorously. We show that they stem from misunderstandings of basic concepts in gravitation and cosmology, that neither claim is actually demonstrated, and that the alternative “iso-gravitation” theory being proposed conflicts with well-established physics like the Equivalence Principle. Even the one idea with some surface appeal—an alternative explanation for the cosmic redshift—falls apart on the numbers, coming out about a million times too weak to account for what we observe. It's a careful defense of well-tested physics against overreaching claims.

This short note proposes a way to sharpen a model of an exotic kind of chemical bond. Beyond the familiar bonds that hold ordinary molecules together, some theories describe “magnecules”—structures held together by magnetic effects among their electrons. The idea here is to upgrade that model by equipping it with extra mathematical bookkeeping (called order parameters) that more precisely capture how the electrons swirl and align, borrowing tools from the physics of superfluids—those strange, frictionless states of matter. The payoff of adding this machinery is an extra degree of freedom : a richer way to describe a magnecule's possible states and how it shifts between them, which could reveal more about the underlying physics of these unusual bonds and how they might be put to practical, industrial use. It's an early-stage hypothesis flagged for further investigation.

Nature is full of fractals—the endlessly repeating, self-similar patterns you see in coastlines, snowflakes, ferns, and lightning—and as the saying goes, fractal geometry is the language of chaos. This short note launches a brand-new twist on them, which we call iso-fractals. The key idea is to take a classic like the famous Mandelbrot set and give it an adjustable extra dial (an “iso-unit”) that can be turned continuously or in discrete steps—even driven by famous number sequences like Fibonacci's—to generate whole families of related fractals that morph and rescale while preserving their essential structure. Adding this extra degree of freedom makes the resulting toolkit far more general and flexible than traditional fractals for tackling problems in chaos. It's an early sketch of an emerging sub-discipline, closing with open questions to chart where it might go next.

Protons and neutrons are built from quarks, which come in three “colors”—and counting a particle and its antiparticle together gives six. This paper builds on an earlier quark-confinement proof and reworks it using Santilli's iso-mathematics, an alternative mathematical framework. The setup places a matched particle-antiparticle pair on a six-colored honeycomb-like lattice and upgrades the equations that describe them (their wavefunctions and matrices) into “iso” versions with an extra adjustable ingredient. The early results suggest this richer machinery might offer a cleaner way to describe how these particles jump between energy states. It's an exploratory proposal flagged for further scrutiny.

This paper sketches an early, speculative version of a string theory—the idea that the universe's most fundamental objects are tiny vibrating strings rather than point particles. The twist is to build it using Santilli's iso-mathematics, with a number system designed so that space and time mirror each other as duals. In the proposal, open strings encode matter particles (fermions) while closed strings encode force particles (bosons), and the whole construction is meant to capture the twisting, winding, and spiraling of information structures while respecting the holographic principle. It's explicitly a hypothesis and preliminary construction—an opening move, offered up for collaboration and refinement.

Many objects in physics are spherical and change size—expanding, contracting, pulsing. This paper proposes a generalized mathematical way to track the states of such growing-and-shrinking spheres using Santilli's iso-mathematics. The central new idea is an “effective iso-radius” that lets these spheres smoothly transition between states, building toward what the paper calls “effective dynamic iso-sphere holographic rings.” It poses a direct question—could these structures find real use in physics and chemistry?—and answers, hypothetically, yes, while explicitly calling for the rigorous follow-up work needed to test the conjecture.

Protium is the simplest atom—ordinary hydrogen—and antiprotium is its antimatter mirror image. This brief paper applies a particular “dual” model of spacetime, paired with a six-colored structure borrowed from quark physics, to describe both of them together. The encouraging early result is that the same approach might extend to every chemical element, not just hydrogen. It's a preliminary proposal, offered up for the scrutiny and refinement that would be needed to develop it further.

One of the deepest clues toward a theory of quantum gravity is the idea that a black hole's surface area isn't smooth but comes in tiny discrete steps—like rungs on a ladder. The evidence increasingly supports this, but physicists don't yet agree on the spacing between the rungs. This paper examines the key work on that spacing, in both its simpler “strictly thermal” form and a more refined “non-strictly thermal” version that includes entropy (a measure of a black hole's hidden information). By showing how a series of corrections connects these previously separate results, it unifies several earlier advances into one picture, where the black hole's natural “ringing” vibrations—its quasi-normal modes—mark the transitions between energy levels.

Mathematicians have long used complex numbers to describe the 2D plane; this paper works with their 3D cousin, the triplex number, and combines it with Santilli's iso-mathematics and a particular spherical “holographic ring” structure. The goal is foundational: to carefully define new iso-triplex numbers for doing fractal geometry in this enriched 3D setting, and to state a series of supporting lemmas (small proven building-block results) that pin the new structures down. It's an opening assessment, suggesting this encoding might be a versatile tool for attacking a broad range of problems—while calling for the rigorous collaborative work needed to test and refine it.

This paper takes a fixed mathematical sphere and, using Santilli's iso-mathematics, splits it into a linked pair: one version that magnifies outward and a mirror version that shrinks inward, with the two locked together as “inverse duals.” A neat feature falls out: a measurement on one (its size) exactly matches a complementary measurement on the other (its curvature), and vice versa. The significance is that the inside and outside regions become two faces of the same object— each can be inferred from the other—hinting at a new mode of geometry worth exploring further. Notably, this inside-outside duality theme is an early ancestor of ideas that later matured in the Tri-Quarter Framework.

When two hydrogen atoms bond into a molecule, one model describes the shared electrons as a combined object called iso-electronium. This paper proposes a step-by-step upgrade to that model, adding extra mathematical bookkeeping (order parameters borrowed from the physics of superfluids) to more precisely capture how the paired electrons swirl, align, and form their bond. As with the related magnecule work, the payoff is an extra degree of freedom—a richer description of the molecule's states that could shed light on the underlying physics. It's an early hypothesis, explicitly offered up for rigorous testing and refinement.

Black holes slowly evaporate by emitting Hawking radiation, and they also “ring” like a struck bell with characteristic vibrations called quasi-normal modes. A deep insight is that these two phenomena are linked, with the ringing marking the black hole's quantum energy levels. Earlier work established this for plain and spinning black holes; this paper extends it to electrically charged ones (the Reissner-Nordström case). That step matters because a charged black hole can absorb or emit a particle carrying both mass and charge, not just mass—and the analysis shows the ringing modes still cleanly correspond to quantum levels for both emission and absorption.

A tesseract is the four-dimensional analogue of a cube—a famously hard shape to picture. This paper offers an unusual route to building one: using Santilli's iso-mathematics, it assembles a tesseract from two interlocking mirror-image cubes following a set of simple, shape-preserving instructions. The elegant result is that the whole 4D structure can be inferred from a single starting cube rather than needing two independent ones, revealing a new geometric link between “inside” and “outside” dynamical systems—another early echo of the inside-outside duality themes that run throughout this body of work.

This paper introduces dynamic iso-spaces—mathematical spaces, built with Santilli's iso-mathematics, that can stretch and rescale according to a changing dial rather than staying fixed. To make the idea concrete, it works two famous examples: it drives the dial using the Fibonacci sequence to build a “Fibonacci dynamic iso-space,” and using the Mandelbrot set to build a “Mandelbrot dynamic iso-space.” Taken together, these constructions suggest the outlines of a possible new branch of mathematics—an exploratory first step.

Real black holes spin, and a rotating one is described by the Kerr solution. This paper extends a key insight to that spinning case: that a black hole's slow Hawking radiation and its bell-like quasi-normal mode vibrations are naturally linked. We show that for a Kerr black hole, these ringing modes can be read as quantum energy levels, so that emitting or absorbing a particle becomes a jump between two levels—and we generalize the notion of the black hole's “effective state” to match. It's part of a broader program connecting black hole radiation to a discrete, quantum picture of gravity.

Predicting how many objects move under their mutual gravity—the famous n-body problem—gets brutally hard as the number of bodies grows; even three is notoriously difficult. This paper proposes a preliminary algorithm that tries to tame the mess by collapsing many bodies down toward a simpler two-body picture. It works by planting one mass at the origin as a reference point and then following a recipe to consolidate all the remaining bodies onto a spherical ring structure. Through a worked step-by-step example, it shows the method yields net “effective” quantities—an effective energy and motion for the whole system. It's an early, alternative viewpoint aimed at sparking simpler solutions down the road.

The Mandelbrot set is the iconic, infinitely intricate fractal generated by repeating one simple equation. This paper explores what happens when you modify that equation using Santilli's iso-mathematics—swapping in a tweaked form of multiplication controlled by an adjustable “iso-unit” dial. Turning that dial produces a whole family of Mandelbrot iso-sets, each a variation on the original that stays connected in one piece but bends and rescales in two distinct ways (reshaping its overall scale and its boundary). The takeaway is a more general toolkit than the single classic Mandelbrot set—an early probe into the emerging realm of iso-fractals.

Complex numbers are a powerful tool for describing the 2D plane; this paper builds their 3D counterpart, the triplex number, to extend that power into higher dimensions. The aim is a clean, easy-to-visualize vector framework for particle and astrophysics that respects the holographic principle. Using complex and triplex numbers as building blocks, it systematically encodes a “dual” model of spacetime—where space and time are interconnected—in both 3D and 4D, and equips that spacetime with extra fields describing how it can deform. It closes with three example applications to show the framework in action. This triplex-number work is a direct conceptual ancestor of the coordinate unification that later anchors the Tri-Quarter Framework.

One of the great puzzles of particle physics is quark confinement: why quarks are never found alone, always locked inside particles like protons. This paper—my collaboration with Andrej Inopin—sets out a proof of confinement for a matched particle-antiparticle pair, built on a custom, deliberately easy-to-visualize model of spacetime grounded in experimental data. The construction assembles a “dual” 4D spacetime, equips it with fields that spontaneously break several symmetries at once, and uses those to build the particle's quantum description, connecting it to elegant structures like skyrmions and Higgs-like excitations. The paper argues the framework sharpens the predictions of standard physics and even nods to M.C. Escher's art as a visual metaphor for its core duality. This is the ambitious early work from which much of my later research—including the inside-outside duality at the heart of the Tri-Quarter Framework—ultimately grew.

DNA and proteins are long sequences spelled out in a small alphabet, and one way to study them is to count how often every short “word” of a given length (a k-mer) appears—paying special attention to nullomers, the words that never show up at all. This thesis builds software to extract and statistically analyze these word-frequency patterns across the genomes and proteomes of ten organisms plus several bacteria and viruses. It finds that naturally evolved sequences look clearly different from randomly generated ones. As a second experiment, it trains machine-learning models on 18 months of genome snapshots to see whether future patterns can be forecast—and finds that, to a modest degree, they can. It's a blend of bioinformatics, statistics, and early machine learning.

This early paper—from my undergraduate days—is about using robot simulations as a teaching tool. The goal was to get undergraduates excited about real scientific research by having them design and program virtual mobile robots. The novel ingredient is “Physicomimetics”: a control approach where simulated robots coordinate by acting as if they're governed by artificial physics forces, like particles attracting and repelling each other. The paper describes the hands-on experience of building such a simulated robot environment and reflects on how effective it was for drawing students into research.