A geometric completion of fluid phase space. The catalog of Benjamin Frohman — research, formalization, fusion control, and consulting — rendered as a tethered field that cannot blow up.
For more than two centuries the mathematical physics community has lived with an incomplete sentence about the Navier–Stokes equations. They describe the air over a wing and the swirl in a cup. They do not yet, in three dimensions, confess whether they remain smooth for all time.
The fear has a name: blow-up. Vorticity intensifies at a point until velocity becomes infinite in finite time, and the model ceases to represent the world. The Clay Millennium Prize asked the question as a detective story. Does every smooth, divergence-free initial field stay smooth forever — or can a perfect fluid spontaneously become a singularity?
The analytic school tried to fight the nonlinearity with estimates. In 3D the vortex-stretching term outruns the bounds. The geometric school placed the fluid on Arnold’s coadjoint orbits, then stopped at ideal flow. Viscosity and stretching were left outside the picture.
Frohmanian Symplectohydrodynamics begins where those two paths fail to meet: a geometric completion in which the equations are given the missing 2-form demanded by their own stretching, and thereby refuse their own destruction.
Sobolev spaces, Fourier analysis, localized energy. A century of attempting to keep the stretching term from outrunning the mathematician.
Arnold and Khesin placed ideal flow on coadjoint orbits. The Frohmanian tether finishes the picture for viscosity and 3D stretching — a leash strictly degenerate to kinetic energy.
The roadmap of how spinning parts of the fluid are carried by the flow itself.
The dangerous growth. A spinning band pulled at both ends spins faster as it thins. This is the engine of possible blow-up.
The specific track the fluid is allowed to run on inside the abstract world of all possible motions.
A geometric leash. Strictly degenerate to kinetic energy. It does not rewrite the physics; it completes the geometry that keeps rotation finite.
The Tether is not an artificial force. It is the minimal metric correction derived directly from the stretching term of the original PDE. Coefficient matching fixes κ uniquely at 1. Linear stretching is cancelled; what remains is a negative quadratic brake.
The equations follow a conserved geometric path on the coadjoint orbit. Route A projects the correction orthogonal to kinetic energy, so reversible Euler flow remains exactly classical.
κ = 1 matching against stretching produces a quadratic braking force. As the fluid tries to spike, the Tether answers with −M².
Maximum rotation is prevented from reaching infinity. Beale–Kato–Majda then guarantees the solution remains C∞ for all time.
Isolate the vorticity equation from Navier–Stokes.
Identify (ω · ∇)u — the engine of potential blow-up.
The stretching’s own energy produces the Riccati bound on the maximum rotation M.
The −M² term overwhelms growth. The explicit solution is a tanh, always bounded by √C.
A bounded vorticity integral forbids finite-time singularity. Parabolic regularity upgrades the solution to global smoothness.
The engine. Core dynamics and the Riccati bound. Fluid motion as a stable geometric system on a completed coadjoint orbit.
Heat, friction, viscosity. Entropy production and geometric stability work in tandem. The 4-bracket restores thermodynamic consistency without violating the bound.
AdS/CFT. Fluid regularity as the boundary projection of cosmic censorship. The tether is dual to horizon area. Naked singularities are geometrically forbidden.
This observatory is the public surface of a research program, not a static CV. Dates below are provenance markers. Source lives on GitHub: some repositories are reusable formalization, others are personal logs of builds and process.
Consulting and research studio for AI, digital assets, and infrastructure, with a focus on mathematical application using novel techniques at frontier pace.
Full-cycle ownership of problem selection, geometric method, architecture, formal documentation, and stakeholder-ready technical proposals.
Early tether constructions in a living LaTeX document. The missing Poisson structure begins to take geometric form.
Major manuscript with novel conjectures on turbulence singularity. Priority surface for the program.
ForMathlib.NS.Tether: first-principles derivations, degeneracy lemmas, five-step uniqueness of the minimal correction. GitHub: BenFrohman.
Earliest public content record of the Frohmanian Symplectic Tether program, preserved for provenance.
Zero-sorry Lean 4 combinatorial engine — well-founded recursion as practice for larger verified arguments. Zenodo packaging 2026-08-19.
Hierarchical Latent Exchange & Coordination Fabric — a five-layer geometry-aware fleet coordination architecture for frontier training infrastructure.
Public instrument of Symplectori Labs: FSymHD catalog hosted at symplectorilabs.com.
Public Lean 4 formalization of the Frohmanian Symplectic Tether: geometric + analytic layers, ForMathlib lemmas intended for Mathlib, CI build. Priority snapshot — not a completed Clay proof.
Public landing page for the tether program. Points the open-source community at the Lean corpus. It is an index, not a reusable library.
Earliest public content record (main.tex + figures). Personal priority log, read-only — not a maintained community package.
Zero-sorry combinatorial Lean 4 exercise — well-founded recursion as practice for larger verified arguments. Personal working log, packaged on Zenodo, not a standalone GitHub library.
Containment and integrity architecture for agentic and infrastructure surfaces. Not a public repository. Scope, threat model, and internals are not published here.
Multi-tenant AI job-hunting SaaS: tenant sandboxes, scanners, trial-then-lifetime unlock. Source is a private GitHub repo — a commercial product, not an open-source contribution.
Modular robotic arms with rotating ball joints driving rolling excitation disks — a kinematic architecture for distributed mechanical excitation. Personal design notes, not a public repo.
Preserving Boolean / discrete logical structure inside numerical integrators and continuous functionals — plasma MHD and FSymHD tethered dynamics. Working notes, not a standalone repository.
Reactor control, in this architecture, is conditional on active geometric forcing. To sustain fusion gain Q > 10, stretching must be suppressed without rewriting conservation laws. The Tether is the control geometry; the physics remains classical.
Metriplectic extension restores viscosity and entropy so the plasma can relax into high-beta equilibria. A tethered Lyapunov functional watches enstrophy precursors. RF injection and edge micro-actuators become synthetic metric corrections — transport barriers on demand.
The holographic dual is a diagnostic: boundary turbulence as a window onto bulk horizon shifts, disruptions visible before they arrive as surface instabilities.
A parallel research thread asks how Boolean and other discrete logical structure can be preserved — not smeared away — when it is computed into numerical integrators and continuous functionals of plasma MHD and FSymHD.
Through A2Zweb 3 (founded 2 February 2022) Benjamin Frohman works as independent research lead: problem selection, geometric method, architecture, formal documentation, and artifacts that a downstream team can execute without keeping the author in the critical path. Open to remote research staff, advisor, and project-lead engagements. Undergraduate study at UT Dallas completed 2022 (business, economics, finance), with concurrent study at St. Edward’s University, Austin.
Tethered phase-space completions, structure-preserving models, and geometric bounds for fluids, MHD, and related nonlinear PDEs.
Metriplectic control logic, ITG/TEM suppression programs, diagnostic duals, and reactor-facing roadmaps from first principles.
HLEX-class coordination fabrics, registries, phased roadmaps, risk registers, and acceptance criteria for frontier labs.
Lean 4 pipelines that check, solve, and compile formal verification into handwritten, legible proofs — explicit and high-level — on frontier open problems.