demo · nowhere else on the web
The Plane
How far a strategic system sits from equilibrium is not one number. It is two, read from two different mathematical objects, and they do not agree. This is the plane those two coordinates span, with every system anyone has measured so far marked on it — and one quadrant still empty.
Put your game on the plane
Two coordinates, read from two entirely different objects: response asymmetry ℛ is a derivative at one equilibrium, dissipation σ_EP is a flux functional over the whole profile space. Drag your game left and right and watch both move.
The dashed vertical at ℛ = 0.02 is the band an estimate has to clear to count as non-zero; the solid divisions are the exact zeros, where the theorem puts them. A dot is a system with both coordinates read; a dashed line is one where only one coordinate has been read, drawn across everything the other could be. Real-data landmarks are placed by their verdict against their own null and carry their own units — only the in-browser games share the chain's nats-per-step scale. Every landmark is clickable for its artifact and its caveats.
◉ IV. Whirlpool
Edgeworth-cycle territory. Response asymmetry and circulation together; the regime where naive optimisation against a static rival model is worst.
Row player's payoffs — drag a cell up or down
Each cell is what the row player gets when the pair play R, P, S against R, P, S. Change one and the whole equilibrium, the whole chain and both coordinates move.
You are sitting on the faint locus, because every game in this one-parameter family does. Now drag one of the payoff cells on the right: the dot leaves the locus, and a theory with a single axis has nothing left to say about where it went.
Show the maths
ℛ = ‖χ − χᵀ‖F / ‖χ + χᵀ‖F with χ = (I − SB)⁻¹S, S the block-diagonal softmax Jacobian at the logit fixed point and B the block-off-diagonal payoff cross-derivative. σ_EP is Schnakenberg entropy production of the Glauber chain on the nine joint profiles, in nats per step. Both are computed in this page from your dragged payoffs.
The implementation here reproduces the committed calibration exactly. At λ = 1.2 it returns ℛ = 0.6928203 for rock–paper–scissors and ℛ = 0.4346571 for the five-action version, against benchmarks/results/reciprocity_harmonic.json R_rps_3 = 0.6928203230275507, R_rps_5 = 0.434657051228945. On potential games it returns ~5e-17 against reciprocity_potential.json max_R = 8.93e-17.
ℛ's magnitude scales with λ (finding F-0002) — only the zero versus non-zero verdict is λ-free. Every landmark on this plane was read at λ = 1.2, so comparing levels at any other λ is not meaningful, and the figure says so when you move the λ slider.
The current reading is ℛ = 5.415309e-1, σ_EP = 2.190484e-1, solver residual bound met: true, ‖SB‖∞ = 0.693 (values at or above 1 mean the resolvent is near-singular and the level should be read as direction only — the same caveat the phase map carries in benchmarks/results/phase_map.json).
Quadrant III is empty, and that is the experiment
You cannot reach the hatched quadrant from this page, and the reason is a theorem rather than a limitation of the widget. For an exact two-player game under this chain both coordinates vanish together: ℛ = 0 if and only if the game is potential, if and only if σ_EP = 0.
I. Landscape — A potential game.
Comparative statics are trustworthy, pass-through is symmetric, there are no cycles to time, and optimising against a static competitor model is correct.
Sioux Falls road network (5.65 × 10⁻¹⁷) · Dominick's retail panel (0.00112)
II. Driven landscape — Timing matters, structure does not.
Reciprocal structure with circulating dynamics: something exogenous is cycling the system — demand, schedules, cost shocks — rather than the strategic interaction itself.
CAISO SP15 day-ahead (not read)
III. Stalled whirlpool — Structure matters, timing does not.
Asymmetric response with no persistent circulation — one agent structurally leads, but nothing cycles. Pass-through asymmetry is the exploitable object.
no real system measured here yet
IV. Whirlpool — Both.
Edgeworth-cycle territory. Response asymmetry and circulation together; the regime where naive optimisation against a static rival model is worst.
Rock–paper–scissors (0.69 (λ = 1.2)) · Colonel Blotto (budget 3) (0.118)
Quadrant III is reachable only for a real system, where the two coordinates are read by two different instruments from two different kinds of data — pass-through asymmetry on one axis, trajectory irreversibility on the other — so one can sit at its own null while the other does not. Nobody has measured a system there yet.
Show the maths
Asymmetric response with no persistent circulation: one side structurally leads, but nothing cycles. If distance-from-equilibrium were a single scalar this quadrant could not exist. Finding a real system in it is the strongest available confirmation of the two-axis result, which is why it is the programme's open decisive test.
Retail fuel pricing with asymmetric rockets-and-feathers pass-through but no Edgeworth cycling is the named candidate (DIRECTION_v4 section 5, unit R11 science.plane.quadrant_iii).
The two coordinates are established as independent, not merely different. Stratified by the harmonic fraction, the correlation between them collapses as α → 1 and its residual reverses sign: ρ_S(σ_EP, ℛ) runs 0.882, 0.812, 0.856, 0.849, 0.800, 0.870, 0.801, 0.610, 0.323, −0.355 across α = 0.05 … 0.95 (benchmarks/results/chain_comovement.json). The obvious repair — that ℛ misbehaves only because it is a ratio — was pre-registered and refuted by its own test: the numerator alone reaches ρ = −0.368 at high α (benchmarks/results/decoupling_mechanism.json, finding F-0007).
Two scope limits belong on the same page as the claim. The α-stratified collapse is a two-player instrument: at N = 3 and N = 4 the meters do not couple at low α, so the collapse cannot be demonstrated there and the programme declines to certify the claim at N > 2 (F-0023, benchmarks/results/plane_nplayers.json). And the sign of the residual is λ-dependent (F-0010), so the headline is that the plane is two-dimensional — not that the second axis always points the way it points at λ = 1.2.