Log

Tonight I just launched version 4 (V4) of my Tri-Quarter Framework radial dual lattice graph preprint to Zenodo. 📜 In V4 I implemented some technical tightening and Python simulation improvements over version 3 (V3). V4 has all the elegant math from V3 but with even stronger, more refined computations proving it actually pays off in the real world.

The headline: the trihexagonal six-coloring—which carves the lattice graph into six conflict-free independent sets—now runs on a consumer GPU (an RTX 4060 laptop card) to hit a median ~8× speedup over a single CPU thread at ~290K vertices. And the GPU runtime stays nearly flat as the graph grows while the CPU cost climbs linearly, so the compute advantage only widens with scale. The symmetry isn't just pretty on paper—it maps straight to the parallel hardware of an average gaming computer that many folks already own. Yep, this is gaming alright. 🎮 👾 🔥

I also tightened the guarantees. The inversion-based path mirroring now comes in at ~1.6–1.8× with bitwise-exact agreement against full recomputation. And the symmetry-reduced clustering runs in exact rational arithmetic, which reproduces the full computation as the same rational number at every radius to land at a steady 3.5–4.0×. No floating-point madness, no information loss. ⚛

And the best part? V4 makes it explicit that my 𝕋₂₄ symmetry group, as an abstract object, is the classical centrosymmetric hexagonal point group D_6h—a structure mathematicians and crystallographers have studied for over a century. My contribution is its concrete realization as a circle inversion action on the lattice. Discovering that your hand-built-from-scatch symmetry "is" a famous, well-understood group isn't redundancy—it means the whole construction is plugged into deep existing machinery, and everything already proven about D_6h comes along for free as backup for driving the TQF. 🚀 😎

Ok, here's a big one. Teaming up with the folks over at Quantum Gravity Research (QGR), we busted the Tri-Quarter Framework radial dual lattice graph out of it's 2D cage and up into the high dimensions—generalizing it from the Eisenstein integers in 2D to the Hurwitz integers in 4D and the mighty E₈ lattice (e.g., Cayley integers in 8D). ⛓️‍💥 🔥

We built these higher-dimensional radial dual graphs using admissible hyperspherical inversion, with exact arithmetic, coordinate compression, no floating-point approximations, and zero information loss. Forget the drift and hassles of floating-point, this Eisenstein-based beast is integer-exact. The payoff: computational results show speedups of 83,000–260,000× at moderate shell radii. That's not a typo. This could prove seriously useful for modeling quasicrystal and E₈-based physics. BOOM SHAKALAKA! 🏀 🚀 😎

The preprint is here! 📜

I posted some preliminary "mark 1" experimental results for my TQF-ANN model (armed with the mighty ℤ₆ orbit mixing algorithm) squaring off against the big time FC-MLP, CNN-L5, and scaled ResNet-18 models. It's an old skool showdown, son. 🔫 🤠 Data augmentation is not used (and not needed) in the experiment. TQF-ANN smokes 'em in the rotated accuracy and FLOPs. 💀 🔥

I used my Tri-Quarter Framework's Radial Dual Triangular Lattice Graph and 𝕋₂₄ symmetry group to design and implement a new experimental Tri-Quarter Framework Neural Network (TQF-NN) architecture with GPU-driven CUDA benchmark tools in Python. These benchmark tools run configurable simulations where my newly invented TQF-NN competes against some industry-standard big dogs (e.g., FC-MLP, CNN-L5, and ResNet-18) which are all scaled to match ~650K trainable parameters to ensure scientific "apples-to-apples" comparisons. So fire up your GPU and give it a shot! Work hard, train hard! 🤖 🔥

My new preprint titled

"The Tri-Quarter Framework: Radial Dual Triangular Lattice Graphs with Exact Bijective Dualities and Equivariant Encodings via the Inversive Hexagonal Dihedral Symmetry Group 𝕋₂₄"

is now available here, with the Python helper tools and simulation scripts on GitHub! As always, get your game face on and feel free to explore, question, experiment, and have some fun via the Scientific Method! Also, can the interested reader come up with any new additional distance metrics that will be helpful for further applying these lattice graphs to real-world applications?

This paper was a beast. Many late nights this year. Very exciting stuff. Now to see about getting this published in a serious peer reviewed journal. Let's see if it stands the test of time. 🔬 ⚛ 🧪

To summarize this preprint's content in a fun way, I've also posted a 42-minute "podcast-like" Gemini educational discussion, which is available on the videos page! 🚀😎

So the almighty HAL of France—an "open-access" preprint archive—stopped allowing me to post my preprints there based entirely on the fact that I don't have a PhD and I'm not currently affiliated with a university, regardless of the potential merit of my "Tri-Quarter Framework Case Study: BPSK Signal Processing" work. They wrote me such an authentic, polite, "open-access" email:

Dear Nathan Schmidt, Please make the following corrections to your document hal-05145190 (version 1) by login on HAL ( https://hal.science ) To proceed appropriately, we kindly ask for specific clarification regarding your role and relationship with Cold Hammer Research & Development LLC — including whether it represents an official research affiliation and your position within the organization. Additionally, we would like to inform you that, in accordance with our current policies, we are only able to accept submissions from researchers with a formal academic affiliation or a doctoral-level profile. Independent submissions without such credentials unfortunately do not meet our eligibility criteria at this time. We look forward to your response and thank you again for your understanding. Without modifications or answer in the next 5 months, the document will be deleted from the server. You can user the feature "Annotate/Answer" to the modification request. For any exchange with the moderator: view the deposit page, go to the options bar on the right and click on reply to the moderator . Otherwise, contact support at the following address: hal.support@ccsd.cnrs.fr Greetings, -- The HAL/CCSD team CCSD - HAL

So I can't be a part of their cool club. I guess 20+ years of industry and academic experience in cutting-edge software engineering development and scientific research, with dual M.S. degrees in computer science and math, isn't cool enough for them. 😎 Oh well, it's their choice and their loss, not mine. 🤣

Got an attention span of 6 minutes (and 49 seconds)? Check out my new video "The Tri-Quarter Framework: A Brief Gemini Report" on the videos page!

I just finished the preprint for my new paper titled

"Tri-Quarter Framework Case Study: BPSK Signal Processing"

which is available here, with the Python BPSK simulation scripts available for experimentation at GitHub. This is my second paper after taking my 8-year break. For this example application, it was a fun execise to apply the part of the tri-quarter framework to signal processing, and I'm sure there is plenty more that we can do for this in the future as well. For now, let's see if it stands the test of time. 🔬 ⚛ 🧪

Happy 249th birthday USA! 🚀😎

Got an attention span longer than 6 minutes? Get your game face on and check out my new 2.5-hour compsci/math deep-dive video "Tri-Quarter Framework: Upgrading the Complex Plane (Mark 1)" on the videos page!

After taking a break for 8 years, and working late nights as a part-time hobbyist over the last few weeks, I'm back in action with a new paper titled

"The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius"

which is available here and is currently being reviewed by a journal. Oh and there is even a software visualization tool to go with it. And yes, the term "Tri-Quarter" is inspired by the infamous hand-held "tricorder". 🔥⛽🚀😎🤣

Added some initial site content. Reviewing past work of quark confinement and black holes. A bit rusty but back in action. More to come! 🤪

The skeleton site is up and starting to add some content. 💀😎