In The Incidence Reduction, we showed that ODE steady-state values recover exact integers from the net topology — incidence degrees to terminal transitions. In Tropical Petri Nets, we showed that timed event graphs are max-plus linear systems, and that the tropical eigenvalue compresses an entire reachability graph to a single throughput scalar. Both analyses start from the same object: the incidence matrix C.
But they're doing something deeper than analysis. They're doing compression — and they're compressing the same structure in compatible ways. When we add a third analysis — zero-knowledge proofs — all three independently discover the same structural boundary inside the net.
That convergence is what this post is about.
Every Petri net we've built — tic-tac-toe, coffee shop, Texas Hold'em — has a natural split. There's a part that does things (moves pieces, brews coffee, deals cards) and a part that watches (detects wins, checks completion, validates hands). We call the first part the core and the second part the observer.
The split isn't just organizational. It has a precise structural definition: observer transitions consume from core places but never produce into them. This is the sink property. The observer reads the core's state destructively — it takes tokens to make a verdict — but it never feeds tokens back. If no observer transition ever fires, the core evolves identically whether or not the observer exists.
For tic-tac-toe, the decomposition is:
Core: 18 play transitions (9 cells × 2 players), 33 places. Each play transition consumes an empty-cell token and a turn token, produces a marked-cell token and the opposite turn token. Every token consumed is produced elsewhere — the core is reversible in the invariant-theoretic sense.
Observer: 17 transitions (8 win lines × 2 players + 1 draw). Each win transition consumes 3 cell tokens + the game_active flag + a turn token, and produces a single verdict token. Five tokens in, one token out. The board state is destroyed. Only the verdict survives.
The compression ratio tells the story. Core transitions have ρ = 1: every token consumed is produced elsewhere. Observer transitions have ρ = 5 (wins) or ρ = 10 (draw): more is consumed than produced, and the difference is the earned compression — the exact amount of state that the verdict makes irrelevant.
Not all compression is equal. JPEG is lossy — it throws away information and hopes you won't notice. Gzip is lossless — it exploits redundancy but preserves everything. Earned compression is something else: it throws away information and proves that what it threw away doesn't matter for the target property.
Formally, a map κ: S → S' is an earned compression with respect to a property π if:
The tightness condition is what distinguishes this from approximation. We're not claiming "close enough." We're claiming: this is the shortest description from which the target property can be recovered, and we can prove it.
Here is the central result. Three independent analyses — each with different mathematical foundations, different computational models, and different target properties — all discover the same partition of the net's transitions into core and observer.
The mass-action ODE system ẋ = C · v(x) evolves the core to a stable equilibrium. This is what we explored in integer reduction: set all rates to 1.0, let the topology speak, and the steady state reveals the incidence degrees.
The observer doesn't participate in the continuous dynamics. Its transitions are sinks — they pull tokens out of the system but don't feed any back. The ODE reaches equilibrium independently of whether the observer exists; the observer is evaluated at that equilibrium, not during the evolution.
The core-observer boundary, from the ODE's perspective: core evolves continuously, observer evaluates at rest.
The tropical eigenvalue compresses the core's entire reachability dynamics to a single scalar λ — the asymptotic throughput. For tic-tac-toe's core, λ = 1: one X-play then one O-play, alternating, at unit rate. The critical circuit has length 2 (an X-play followed by an O-play), total weight 2, mean = 1.
This works because the core is a timed event graph: every place has exactly one producer and one consumer. (The turn places are a subtle exception — they have 9:9 fan structurally, but mutual exclusion via cell tokens makes them behave like 1:1. More on this below.)
The observer breaks it. Place x11 (X's mark on the center cell) has four consumers — the four win transitions for each line through the center. With multiple consumers per place, the tropical semiring can't express the dynamics. The observer is inexpressible in max-plus algebra.
The core-observer boundary, from tropical analysis: core is a timed event graph, observer violates the event-graph property.
The incidence matrix compiles directly into an R1CS constraint system for ZK proofs. A Groth16 proof of a transition says: "a valid move occurred from marking m to marking m'" — in 128 bytes, verifiable in constant time, revealing nothing about which move or what the board looks like.
Core transitions have uniform delta structure: m' = m + Δ_t, where a boolean multiplexer selects the right column of C. This compiles to a clean, uniform R1CS circuit. Observer transitions break the uniformity — they consume tokens from places shared with other observer transitions, requiring auxiliary witness variables to prove non-interference.
The core-observer boundary, from ZK encoding: core compiles to uniform R1CS, observer requires non-uniform witnesses.
No single formalism can prove the boundary is structural. The ODE analysis might be an artifact of mass-action kinetics. The tropical analysis might be an artifact of event-graph restrictions. The ZK encoding might be an artifact of R1CS structure.
But three formalisms sharing no mathematical machinery — real-valued differential equations, the max-plus semiring, and finite-field arithmetic — converge on the same partition. The same transitions are classified as core by all three. The same transitions are classified as observer by all three.
The convergence is the proof. The boundary is in the net, not in the analysis.
The incidence matrix C is the central object. Each analysis is a compression — a map that discards information irrelevant to a specific target property while preserving that property exactly.
| Stage | Input | Output | ρ | Property preserved |
|---|---|---|---|---|
| Domain → Net | Game rules | C ∈ ℤ^{35×33} | — | All firing semantics |
| Net → Eigenvalue | C (1155 integers) | λ ∈ ℝ (1 scalar) | 1155 | Throughput |
| Net → Invariants | C | ker(C), ker(C^T) | P | |
| Net → Observation | m, t_obs | verdict ∈ {w,l,d} | 5–10 | Game outcome |
| Net → ZK proof | C, m, t | π ∈ {0,1}^128 | ~10⁴ | Transition validity |
Each row is earned: the output is strictly smaller, the target property is exactly recoverable, and no smaller output suffices.
The eigenvalue compression is striking. A 35×33 matrix — 1155 integers — compresses to a single real number λ = 1. That one number tells us the maximum throughput of the core: one complete turn cycle per unit time. The other 1154 integers were irrelevant to throughput. The tropical eigenvalue earned its compression by proving they don't matter for that property.
The ZK compression is the most extreme. The entire statement "transition t is valid from marking m to marking m'" — which involves the full incidence matrix, the current state, the enabledness conditions — compresses to 128 bytes. The verifier learns only that the transition was valid, and nothing else. Compression ratio ~10⁴.
The core-observer decomposition is thermodynamic.
The core is reversible: P-invariants guarantee that every token consumed by a core transition is produced elsewhere. The cell invariant p_ij + x_ij + o_ij = 1 says each cell is always exactly one of empty, X-marked, or O-marked. Total tokens are conserved. No information is destroyed.
The observer is irreversible. A win transition consumes 5 tokens and produces 1. The board state — which cells formed the winning line — is destroyed. Only the verdict survives. The game cannot be "un-won."
This is the same pattern as measurement in physics — a measurement apparatus couples irreversibly to a reversible system, producing a classical outcome that cannot be undone. The mechanisms differ (no Born rule, no non-commutativity here), but the structural pattern is shared: observation that claims to be passive is actually irreversible, and the irreversibility is what produces a definite outcome.
The compression ratio ρ quantifies the irreversibility. Core transitions: ρ = 1 (nothing destroyed). Win transitions: ρ = 5 (four tokens of information destroyed per verdict bit). Draw: ρ = 10 (nine moves of history compressed to "nobody won"). The destroyed information is exactly the information irrelevant to the verdict — the specific cells that formed the winning line don't matter once you know who won.
| Observer transition | Consumed | Produced | ρ | Information discarded |
|---|---|---|---|---|
| x_win_* (8 transitions) | 5 | 1 | 5 | Which 3 cells formed the line |
| o_win_* (8 transitions) | 5 | 1 | 5 | Which 3 cells formed the line |
| draw | 10 | 1 | 10 | Full move history (9 moves) |
| Core transitions (18) | 3 | 3 | 1 | None (reversible) |
There's a subtlety that exposes the limits of structural analysis. The turn-control places (x_turn, o_turn) have 9 producers and 9 consumers — every play transition for one player produces into the other's turn place. Structurally, this violates the timed event graph property.
But behaviorally, they're fine. A single token occupies each turn place, and mutual exclusion via the cell places ensures exactly one transition fires per turn. The token ping-pongs 1-to-1 through reachable markings, even though the structure has 9:9 fan.
The three formalisms see the turn places differently:
ODE: The turn token is a real-valued concentration. Mass-action kinetics naturally enforces mutual exclusion — when any x_play fires, the x_turn concentration depletes, suppressing the others. The ODE doesn't distinguish 9:9 from 1:1.
Tropical: The structural analysis sees all 9 producers and 9 consumers and reports causal paths between transitions that can never actually co-fire. It overapproximates.
Discrete/RNN: The firing rule enforces mutual exclusion exactly. An RNN trained on discrete trajectories never observes non-alternating play — the net prevents it.
The turn places sit at the exact boundary where continuous and discrete analysis agree but structural analysis diverges. The incidence matrix C — which all three formalisms share as input — is insufficient to classify tropicality at the turn-control layer. Reachability information is required, and reachability is inherently discrete.
Whether there exists a useful formal weakening — "tropical under mutual exclusion" — that captures this behavioral property without full state-space enumeration is an open question.
The core-observer pattern isn't specific to games. It appears anywhere a computational core is monitored by an observation layer.
Consider a payment settlement network with three parties: Alice → Bob → Carol → Alice. Each channel has a send transition (debit sender, create pending) and a settle transition (clear pending, credit receiver).
The circular structure is visible in the net: tokens flow from alice_bal through send_ab to pending_ab, then settle_ab credits bob_bal, and so on around the cycle. Each place has exactly one producer and one consumer — a timed event graph. The core incidence matrix is:
a b c p_ab p_bc p_ca
send_ab -1 0 0 1 0 0
settle_ab 0 1 0 -1 0 0
send_bc 0 -1 0 0 1 0
settle_bc 0 0 1 0 -1 0
send_ca 0 0 -1 0 0 1
settle_ca 1 0 0 0 0 -1
The core is a timed event graph — every place has one producer and one consumer. The tropical eigenvalue is λ = 1: the system is balanced, no channel is a bottleneck. The P-invariant a + b + c + p_ab + p_bc + p_ca = const is the accounting identity: total money in the system is conserved. Double-entry bookkeeping is a theorem of the incidence matrix, not a policy imposed on it.
The observer checks business rules: overdraft guards read balance places (catalytically — without consuming), and if the guard fails, the candidate payment is deleted. Not rolled back, not compensated — it never becomes a valid morphism. Invalid payments don't exist in the system because they were never admitted.
The same formalism serves both domains, but the tropical eigenvalue transforms from a structural curiosity (tic-tac-toe: λ = 1 confirms alternation) to a design parameter (payments: λ determines maximum sustainable settlement rate).
This three-party cycle is just one instance. Settlement networks form a category — The Category Settle — where channels compose along shared boundary places, and the structural invariants (conservation, throughput, ZK circuits) compose with them.
A single 35×33 integer matrix encodes the full tic-tac-toe game:
No other representation achieves all five from the same data. The incidence matrix is the maximally compressed description of the domain — the shortest description from which all behavioral, structural, and cryptographic properties can be recovered without loss.
This is the Kolmogorov-flavored claim at the heart of the paper: a domain is understood when its shortest description is also its most useful description. The incidence matrix is that description for Petri-net-structured domains.
The strongest result isn't any single compression. It's that the boundary between what compresses and what doesn't is itself the most informative structural feature.
The core compresses because it has clean algebraic structure: event-graph topology, conservation laws, uniform constraint patterns. The observer doesn't compress — at least not in the same ways — because it has irreducible complexity: cells participate in multiple win lines, and there's no cheaper way to check all of them.
But that irreducible complexity is what makes the observer useful. The observer's non-tropical structure is not a failure of analysis. It's a feature of the domain. Win detection in tic-tac-toe is irreducibly non-tropical because a single cell participates in multiple win lines. The fan-out is the point — it's what makes the measurement informative.
The generalization: in any system with a computational core and an observation layer, the core admits algebraic compression (throughput, conservation, proofs) and the observer admits only discrete compression (reachability, enabledness). Three formalisms predict that this boundary will be discovered independently by any sufficiently expressive analysis framework.
In each case, the incidence matrix is both the specification and the analysis input. The boundary tells you where the algebraic tools work and where you need discrete reasoning. And the convergence of three independent formalisms on the same boundary is the evidence that this isn't a modeling choice — it's structure.
For the tropical algebra that makes this possible: Tropical Petri Nets
For the ODE analysis that started it: The Incidence Reduction
For how settlement networks compose categorically: The Category Settle
For the categorical structure underneath: Symmetric Monoidal Categories