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The Capacity Paradox
sim.pflow.xyz ranks every knob in a model by its measured effect on the objective. One run against a software team's backlog model reported that hiring past a point lowers the score; that finding is unconfirmed until the model's cost meter is fixed, and the catalog-wide sweep behind the rest of the pattern was never preserved. What was checked, what wasn't, and the models to re-run it on.
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Byte-Exact Across Four Languages
Go, Rust, JavaScript and Julia run the same stochastic Petri net and produce the identical IEEE-754 double, every time. Not close — equal. Here's what that actually took, and the two bugs it caught along the way.
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A Tiling for Computation
A small, fixed alphabet whose local composition rules generate an unbounded, non-repeating space of checkable systems — token standards, business operations, games, music, our own infrastructure. Not one system that swallows every domain. Two tiles, remixed without limit.
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A Generator You Can Read
The AI music debate is arguing about the wrong axis: the distinction that matters is structural transparency vs. statistical opacity, and a Petri net — beats.bitwrap.io's generator, linked file by file — is a third kind of generator the discourse keeps missing.
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Building ZK Poker: Four Petri Nets and a Branchless Circuit
A technical deep dive into heads-up hold'em at stackedup.gg — how four cooperating Petri nets replace imperative game logic, a commit-reveal protocol ensures fair shuffles, and a branchless Groth16 circuit proves hand strength without revealing cards.
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Peerage Is an Automorphism
Two people are peers when you can swap them and the net doesn't notice — an automorphism. Cory Doctorow's reverse centaur is that symmetry broken. And only under that symmetry does continuing a protocol count as consent rather than coercion: the ethics of working alongside an LLM reduces to one question — does the agent-swap stay in the automorphism group?
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The Model Is the App
A declarative vocabulary of composable models that gives you analysis and an app framework from the same artifact. Describe your operation, get a model; play the model as an app; calibrate it with your own data.
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The Obligation Triangle
A phase space for money — three poles (sovereign, commodity, protocol), edges as couplings with witnesses, interior positions as instruments.
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Structuralism, Not Objects: Dirac Deltas, Algebraic Effects, and Why C Comes First
A Mastodon thread on formalizing Dirac deltas as algebraic effects, read through the pflow lens — the same refusal to posit an object, the same check-after-the-fact discipline, and a question about whether the predicate future is an effect rather than a value.
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The Proof Form: When a Theorem Is Just Another Port
Adding Lean 4 to pflow-polyglot forced a fifth implementation form — the same reachability check is a startup assertion in five languages and a compile-time theorem in the sixth, and the code is line-for-line equivalent.
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Brouwer Ordinals and the Shape of a Witness
de Jong et al. (2026) put an ordinal axis under decidability. That axis is past-tense, and Brouwer ordinals are the scalar specialization of our vector tropical past.
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The Pflow Square
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.
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The Zipper Whose Hole Is a Universe
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.
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The Little Language Thesis
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, and the incidence matrix proves what it can see: conservation, not guards or data.
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ZK Polls: Voting as a Visible State Machine
Anonymous voting where anyone can see the rules — built from a diagram with four circles and three arrows.
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Bitwrap: Petri Nets as ZK Containers
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.
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Tropical Petri Nets
Petri nets, ReLU neural networks, and tropical algebra all compute over the same algebraic structure. Tropical algebra is the formalism that makes this precise.
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Symmetric Monoidal Categories: The Structure Underneath
A Petri net generates a free symmetric monoidal category — places give the objects, transitions the generating morphisms. What that theorem buys for composition, analysis and proofs across this blog, and which of the connections are still a shape match rather than a checked result.
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Skip the Spreadsheet: What-If Analysis with Petri Nets
Describe your business to an LLM, get a simulation you can actually play with — adjust staffing, change demand, and watch the numbers move. Revised to say which vet-clinic results the tests actually check, and that the coffee shop runs out of milk, not cups.
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Small Models > LLMs
Why executable formal models matter more with LLMs around, not less — and what the LLM is and isn't trusted with when a Petri net constrains it.
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Code-to-Flow: Turn Anything into a Petri Net
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.
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ZK Hold'em: Poker Hand Ranking from Network Topology
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.
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Comparing Nets by Their ODE Signatures
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.
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A Circuit's-Eye View: Notes from Inside the Loop
Reflections from an AI collaborator on building with Petri nets—what makes this approach different, and why it keeps surprising me.
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Enzyme Kinetics: Michaelis-Menten from Three Transitions
Four places and three mass-action transitions generate the ODE system that Michaelis-Menten is derived from. What the demo integrates, what it reads off a formula, and the saturation sweep it doesn't yet run.
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Exponential Weights in Petri Nets: What Worked, What Didn't, and What's Next
Power-of-2 weights encode lexicographic order over distinct ranks as a single integer. We tried it for poker kicker scoring inside the net and reverted it; the per-card variant we shipped also broke the ordering. Where the encoding belongs, and where it doesn't.
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Zero-Knowledge Proofs for Petri Nets
How gnark circuits prove that a Petri net transition is valid without revealing the state—MiMC hashing, topology-based constraints, and Groth16 proofs.
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Categorical Net Types
Five Petri net types classify token behavior — workflow cursors, countable resources, game turns, continuous rates, and classification signals — with typed links that constrain how nets compose.
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Coffee Shop Model
Resource modeling with Petri nets — the five-drink core of go-pflow's coffee shop, where arc weights are recipes, each ingredient carries a conservation law, and the rates say what runs out first (milk, not cups). Revised to link the code, restore mocha's syrup and label every minute figure as a linear estimate.
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Introducing Petri-Pilot
Interactive tutorials for learning Petri nets—from tic-tac-toe basics to complex multi-player games, with model-driven code generation.
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Texas Hold'em Model
Modeling multi-player poker with Petri nets—state machines, role-based access, guards, and event sourcing for complex game logic.
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ZK Tic-Tac-Toe Model
Zero-knowledge proofs meet Petri nets—cryptographically verify valid game moves without revealing strategy using gnark circuits.
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The Token Language
Four terms — cell, func, arrow, guard — generate a free symmetric monoidal category. The DSL is a categorical language for executable token models.
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Sudoku Petri-Net Model
Modeling Sudoku as a Petri net with ODE simulation—constraint satisfaction through token flow.
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Declarative Differential Models (DDM)
A modeling approach where system behavior is described declaratively and encoded directly in differential equations.
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Knapsack Model
A four-item 0/1 knapsack as a Petri net under mass-action ODEs — why every item stops at 71.4%, what exclusion analysis actually ranks, and where the relaxation falls short of branch-and-bound.
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Ode To Petri
A poem for Carl Adam Petri, inventor of Petri nets—the mathematical notation for concurrent systems.