We've written up the integer reduction technique as a formal paper.
The paper proves what the blog posts demonstrated experimentally: when you build an analysis net with catalytic source places and drain transitions, then run mass-action ODE to equilibrium, the steady-state concentrations are exact reciprocals of drain counts. No iterative eigenvector computation, no coupled fixed-point convergence — each accumulator obeys an independent ODE with a closed-form solution.
The decoupling lemma. Each accumulator place obeys \dot{x}_i = 1 - n_i x_i, where n_i is the number of drain transitions. The ODEs are fully decoupled — source places hold constant concentration because the play transitions are catalytic. Equilibrium is x_i^* = 1/n_i, available in closed form.
The integer reduction theorem. After normalizing by the minimum concentration, strategic values are V_i = n_\text{max} / n_i — exact positive rationals determined entirely by net topology. When drain counts divide evenly, the values are integers.
Four experimental validations:
| Game | Topology | Distinct Levels | Max:Min |
|---|---|---|---|
| Tic-tac-toe (3×3, 5×5, 7×7) | Square grid, full-line wins | 3 | 2:1 |
| Poker hand rankings | Frequency-weighted drains | 9 | 32:1 |
| Connect Four (7×6) | Rectangular grid, 4-in-a-row | 9 | 4.33:1 |
| Hex (5×5) | Hexagonal grid, shortest paths | 7 | 16:1 |
ZK integration. The same incidence matrix that defines the ODE system provides the constraint structure for Groth16 zero-knowledge proofs of game-state transitions — a unified pipeline from strategic analysis to on-chain verification.
The earlier blog posts covered tic-tac-toe and poker. The paper adds two games with richer topology.
A 7×6 grid with 69 win lines (all 4-in-a-row segments: horizontal, vertical, both diagonals). The drain count matrix:
3 4 5 7 5 4 3
4 6 8 10 8 6 4
5 8 11 13 11 8 5
5 8 11 13 11 8 5
4 6 8 10 8 6 4
3 4 5 7 5 4 3
The center column dominates at every row — matching the well-known heuristic that column 4 is the strongest opening move. The peak cells (rows 2–3, column 3) have drain count 13 vs. 3 for corners, giving a 4.33:1 value ratio. Nine distinct drain-count levels, compared to tic-tac-toe's three.
A 5×5 hexagonal board with path-based win conditions. Instead of fixed-length lines, a player wins by forming any path of adjacent cells connecting opposite edges. We enumerate all 96 shortest winning paths (48 top-to-bottom + 48 left-to-right) and create drains from path membership. The drain count matrix:
2 7 15 23 32
7 16 26 32 23
15 26 32 26 15
23 32 26 16 7
32 23 15 7 2
The striking result: the five cells on the anti-diagonal all share the maximum value of 16.0. This anti-diagonal is the "bridge" connecting both pairs of opposite edges — the most strategically contested territory in Hex. The board exhibits exact 180° rotation symmetry (V_{r,c} = V_{4-r,4-c}), and the 16:1 value ratio is the largest among all tested games.
Exactness without search. The values aren't heuristic approximations — they're provably exact rationals determined by topology alone. No game tree, no Monte Carlo sampling, no neural network.
Generality. The technique works across square grids (tic-tac-toe), non-square grids (Connect Four), frequency-weighted domains (poker), and non-Cartesian topology (Hex). The constraint type can be fixed-length lines, variable-length segments, or shortest paths.
Dual use. The incidence matrix serves both the continuous analysis (ODE equilibrium for strategic values) and the discrete proof system (Groth16 ZK proofs for game-state transitions). One mathematical object, two applications.
The paper, tests, and ODE solver are all in pflow-rs.