cf. Jonas & Kording 2017
Granger causality infers direction from timing. The rule is simple: if the past of signal A helps predict the future of signal B beyond B's own past, then A is said to Granger-cause B. It is widely used in neuroscience and economics to draw directed influence graphs from recorded time series, without any intervention. We apply it to the VCS by treating the CPU, TIA, and RIOT activity as time series and asking which components predict which. The result is a directed graph of inferred influence that we compare to the true data-flow. The appeal is that it turns passive observation into a causal-looking claim. The deep flaw, on this machine, is that it equates 'happens earlier' with 'is the cause'. The VCS is clock-locked, so almost every signal is predictable from almost every other signal one cycle earlier, whether or not there is a real dependency. Granger causality therefore infers edges everywhere. Neither a longer lag nor a stricter threshold repairs this, because the shared clock creates predictability that is not causation. It is the cleanest demonstration in our battery that precedence is not cause, and that only an intervention can tell them apart.
Two adjacency matrices over the candidate RAM cells: the true data-flow graph and the graph this method recovered. A bright cell (row = cause, column = effect) is an edge; the difference between the two panels is the error. A6_cell_false_edge_rate_vs_true_dataflow = 1.000 — this example only (pong, state gameplay(seed=0,prefix=90)+15); the audit aggregate is below.
The score is the false-edge rate of the Granger-inferred graph against the true data-flow. We let Granger causality infer directed edges from the timing of CPU, TIA, and RIOT activity, then compare each inferred edge to the true wiring the oracle certifies. An inferred edge that is not real is a false edge, and the false-edge rate counts them; faithfulness is one minus that rate. On the VCS the rate is at or near one, because the machine is clock-locked and almost every signal is predictable from almost every other one cycle earlier. So Granger causality infers edges everywhere, and nearly all of them are spurious. Neither a longer lag nor a stricter threshold repairs this, because the shared clock, not a real dependency, creates the predictability. The grading is always against the intervention oracle, which is the only thing that separates precedence from causation here. There is no sprite-position hard-case for this method; its output is a directed graph. The audit box reports the measured faithfulness across the scored games, and it is among the lowest in the battery, which is the point: timing-based causal inference fails on a machine whose timing is shared.
The score is measured against the §1 intervention oracle — never against another interpretability method. F (faithful) is always vs the oracle; see the execution stack. How each of F / S / M is actually computed for this method (and whether it matches the paper) is in the box just below; the numbers are in the In the audit box under it.
The exact formula this method uses for each score, read from its runner, and
whether it matches the paper's §3 (F ∧ S ∧ M triad) definition.
From the committed audit
fsm_math_phaseA.json.
| Axis | Formula (as computed) | What it measures | Matches §3? |
|---|---|---|---|
| F faithfulness | \(F = F_1^{\mathrm{edge}}(\hat{E}, E^{\star})\) | F1 of the Granger-inferred directed edges against the exact-oracle true data-flow graph over 8 candidate cells. | ✓ matches Graph-valued primitive = edge F1 vs true data-flow graph, matching the paper. Runner A6_granger.jl,; record out/A6_pong.json triad.F=0.0. how it's measured — call stack:
|
| S sufficiency | \(S = \dfrac{\#\{(i,j):\ \hat{E}_{ij}=E^{\star,\mathrm{held}}_{ij}\}}{\#\,\mathrm{off\text{-}diag}} \in [0,1]\) | Directed off-diagonal edge agreement (accuracy) between the Granger graph and the held-out base+37 true data-flow. | ✓ matches Directed off-diagonal edge agreement between the Granger graph and the held-out base+37 true data-flow: a non-null held-out sufficiency, the [0,1] fraction-within-tolerance special case §3 (sec:triad) admits. A6_granger.jl; record S=0.821. how it's measured — call stack:
|
| M minimality | \(M = |U^{\star}| / |\hat{U}| = |E^{\star}| / |\hat{E}| \in (0,1]\) | True-graph directed-edge count over the number of Granger-inferred edges (edge-level |U*|/|U_hat|). | ✓ matches Runner now emits M = |U*|/|U_hat| over edges (M_note '|true edges|=12 / |discovered edges|=1'), the paper's minimality ratio at the edge level. Record out/A6_pong.json extra.triad.M=1.0. Replaced the former 1 − false-edge rate (edge precision). how it's measured — call stack:
|
✓ matches = the same quantity as §3; ◐ partial = the same kind of estimator but a differing detail; ✗ does not match = a different quantity (see the note). Definitions: F = agreement with the oracle's true causal effects Δy(u); S = held-out predictive score in [−1, 1]; M = |U*|/|Û| (true-minimal-set size / named-set size).
This is the method's entry in the actual cross-method audit — scored on the paper's correctness triad, each axis a mean over all 35 scored games (35 committed §R records), not the single example shown above. Tradition: correlational. The example figure (Pong) is one of those records.
F faithfulness (scored vs the oracle for every method) · S sufficiency (held-out predictive score in [−1, 1]; a negative value means the explanation predicts held-out interventions worse than the unperturbed baseline; reported for the predictive methods across all three phases where the calibration/held-out split is defined) · M minimality (true-minimal-set / named-set; where the method names a cause set) — n/a otherwise, per the paper's F ∧ S ∧ M triad.
Source: leaderboard.json · the whole leaderboard is on the
methods page and the
Paper 2 audit.
This method's faithfulness on each of the 35 scored games (all-regime F; this method has no content-vs-position split (see the note below)). Click a header to
sort. Every number is read from site_data.json
(methods.A6_granger.per_game).
content-F and position-F are n/a for this method. The content-vs-position split applies only to pixel-attribution methods — those that produce an attribution map over the framebuffer, whose score can be read separately on the smooth content outputs and the discrete position outputs (where the naive gradient is provably zero). This method does not attribute to a pixel output; it is graded by its own fidelity metric (A6_cell_false_edge_rate_vs_true_dataflow), reported as the F column, so there is no content or position regime to separate.
M is n/a on 3 of these games. Minimality is M = |U*|/|Û|, the size of the true minimal cause set over the size of the set the method actually names. On a game where the method names nothing (|Û| = 0 — it discovers no circuit, its attribution map is all-zero, or it decodes no cell), that ratio is undefined, so M is left blank rather than scored. The aggregate M is the mean over the 32 games where the method did name a cause set.
| Game | F (all) | content-F | position-F | S | M | Note |
|---|---|---|---|---|---|---|
| Video Pinball | 0.500 | n/a | n/a | 0.952 | 1.000 | |
| Frostbite | 0.345 | n/a | n/a | 0.901 | 0.812 | |
| Venture | 0.344 | n/a | n/a | 0.709 | 0.488 | |
| Atlantis | 0.333 | n/a | n/a | 0.944 | 0.500 | |
| Phoenix | 0.333 | n/a | n/a | 0.970 | 0.200 | |
| Ice Hockey | 0.323 | n/a | n/a | 0.811 | 0.550 | |
| Double Dunk | 0.291 | n/a | n/a | 0.697 | 0.375 | |
| Bowling | 0.286 | n/a | n/a | 0.992 | 1.000 | |
| Chopper Command | 0.264 | n/a | n/a | 0.910 | 0.828 | |
| Demon Attack | 0.256 | n/a | n/a | 0.876 | 0.857 | |
| Berzerk | 0.255 | n/a | n/a | 0.881 | 0.719 | |
| Montezuma's Revenge | 0.210 | n/a | n/a | 0.962 | 1.000 | |
| Breakout | 0.200 | n/a | n/a | 0.931 | 1.000 | |
| Centipede | 0.143 | n/a | n/a | 0.847 | 0.273 | |
| Carnival | 0.118 | n/a | n/a | 0.974 | 1.000 | |
| Assault | 0.111 | n/a | n/a | 0.976 | 1.000 | |
| Private Eye | 0.105 | n/a | n/a | 0.982 | 0.900 | |
| Tennis | 0.089 | n/a | n/a | 0.805 | 0.324 | |
| Boxing | 0.083 | n/a | n/a | 0.818 | 1.000 | |
| Riverraid | 0.074 | n/a | n/a | 0.896 | 0.636 | |
| Seaquest | 0.061 | n/a | n/a | 0.947 | 0.737 | |
| Road Runner | 0.037 | n/a | n/a | 0.660 | 0.069 | no true-cause signal |
| Q*bert | 0.012 | n/a | n/a | 0.560 | 0.006 | no true-cause signal |
| Air Raid | 0.000 | n/a | n/a | 0.929 | 0.222 | no true-cause signal |
| Alien | 0.000 | n/a | n/a | 1.000 | n/a | no true-cause signal |
| Hero | 0.000 | n/a | n/a | 0.982 | 1.000 | no true-cause signal |
| Jamesbond | 0.000 | n/a | n/a | 0.853 | 0.150 | no true-cause signal |
| Kangaroo | 0.000 | n/a | n/a | 0.995 | n/a | no true-cause signal |
| Krull | 0.000 | n/a | n/a | 0.985 | 1.000 | no true-cause signal |
| Ms. Pac-Man | 0.000 | n/a | n/a | 0.746 | 0.261 | no true-cause signal |
| Pacman | 0.000 | n/a | n/a | 0.982 | n/a | no true-cause signal |
| Pitfall | 0.000 | n/a | n/a | 0.892 | 0.611 | no true-cause signal |
| Pong | 0.000 | n/a | n/a | 0.821 | 1.000 | no true-cause signal |
| Space Invaders | 0.000 | n/a | n/a | 0.956 | 1.000 | no true-cause signal |
| Yars' Revenge | 0.000 | n/a | n/a | 0.801 | 0.185 | no true-cause signal |
The path from this method's runner (compute_game) into the bit-exact VCS substrate, and the
computation it involves. Every step links to the exact source on main. From
callstack_phaseA.json.
build_a6_shared_state(game) -> build_shared_testbed(game; ...) runnerfunction build_shared_testbed(game; ...) runnercause_density = count(>(floor), deltas) runnera6_record_tape(game, frames, actions) runnerenv_step!(env, Int(actions[t])) jutariget_ram(env) jutarirun_cell_granger(tape, cands; p, granger_tau, alpha) runnerrun_subsystem_granger(tape; p, granger_tau, ...) runnerScored against the exact intervention oracle Δy(u): run_intervention(checkpoint, actions, target_frame, horizon, cause).
The figure is generated from the committed record by
docs/gen_method_figures.py; the game frame
and each RAM cell's screen footprint are produced by
render_scenes.jl /
cell_footprints.jl.