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The Obligation Triangle
A sketched phase space for money — three poles (sovereign, commodity, protocol), instruments placed inside as barycentric mixes, two edges labelled with witnesses, and a proposed F ⊣ U between the sovereign and protocol poles.
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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 Pflow Square
One square that collects six earlier posts — the free SMC on a net, the zipper W(M) = 𝓛 × M × 𝓡, and three analyses that read the ρ boundary off the incidence matrix — with what has been checked and what is still only drawn.
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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 Category Settle
Settlement networks form a sub-SMC of open Petri nets. The core-observer split is really two boundaries — an algebraic one (ρ) inside the incidence matrix and a categorical one (contextual arcs) outside it — and a ZK circuit dissolves only the second.
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Earned Compression
ODE simulation, tropical analysis and zero-knowledge proofs each draw a line between a Petri net's core and its observer. Where the three agree on tic-tac-toe, why the agreement follows from one definition, and where the code does not yet match the theorem.
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Tropical Petri Nets
Timed event graphs are max-plus linear and ReLU networks are tropical rational, so the two meet over the tropical semiring. This post covers what those two theorems give us, what pflow-tropical checks in code, and how far the Petri-net/neural-net analogy has actually been earned.
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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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JSON-LD as Declarative Infrastructure
Why JSON-LD's purely declarative semantics and monotonic schema expansion make it reliable infrastructure for composable systems.
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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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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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Revisiting the Flows
Reflecting on old ideas with new tools — from Petri nets to proofs, from Bash scripts to composable universes.