One commutative square encoding the full categorical structure — the adjunction F ⊣ U, the zipper comonad W = UF, and the convergence of three analyses on a single structural boundary.
Execution state is a zipper — the present moment is not a parameter or a modality but a universe that separates tropical past from predicate future. Tic-tac-toe makes the structure visible.
cell, func, arrow, guard are the structural primitives — but the real ubiquitous language lives in the labels. Like Forth, we build up domain vocabularies on a minimal substrate.
A music sequencer built entirely on Petri nets — token rings become drum machines, Euclidean rhythms fall out of the topology, and polyrhythm comes free.
Anonymous voting where anyone can see the rules — built from a diagram with four circles and three arrows.
From OP_RETURN in 2014 to zero-knowledge Petri nets in 2026 — how bitwrap.io became the capstone for a decade of work on formal state machines, cryptographic proofs, and executable specifications.
The incidence matrix of an NCAA bracket Petri net connects ODE, Monte Carlo, and analytical methods — and makes simulation redundant. A closed-form formula derived from the bracket topology replaces 150,000 stochastic transitions with 256 exact configurations.
Settlement networks form a free symmetric monoidal category — the category of all settlement networks built from the same primitives. Open Petri nets make composition, throughput, and conservation laws compositional.
Three independent formalisms — ODE simulation, tropical analysis, and zero-knowledge proof — discover the same structural boundary in a Petri net. The convergence is the proof that the boundary is real.
Petri nets, ReLU neural networks, and tropical algebra all compute over the same algebraic structure. Tropical algebra is the formalism that makes this precise.
Petri nets are morphisms in a symmetric monoidal category. This isn't an analogy — it's the theorem that explains why composition, analysis, and proofs all work the way they do.
Describe your business to an LLM, get a simulation you can actually play with — adjust staffing, change demand, and watch the numbers move.
Why executable formal models matter more than ever in the age of AI — and how LLMs become most useful when constrained by them.
Convert source code into a visual state machine. Paste code in any language — Go, Python, Rust, Solidity — and get a validated Petri net model you can simulate, analyze, and generate apps from.
A draft paper formalizing incidence reduction — extracting exact strategic values from game topologies — validated on tic-tac-toe, poker, Connect Four, and Hex.
Applying incidence reduction to poker — hand strength values emerge from Petri net drain structure, every action is Groth16-proven, and the shuffle uses Poseidon commit-reveal.
ODE steady-state values with uniform rates reveal integer structure — incidence degrees to terminal transitions — giving a reverse-engineering technique for rate constants.
The same math with different labels still produces the same ODE solution. In tic-tac-toe, we see it directly — each board position is a place, and the heatmap is the solution projected onto the grid.
The blog's models, concepts, and toolchain have been organized into a book-length guide at book.pflow.xyz.
Why JSON-LD's purely declarative semantics and monotonic schema expansion make it reliable infrastructure for composable systems.