docs: render inline math with \\( \\) (HF KaTeX does not support single-$ inline math)

README.md

@@ -59,9 +59,9 @@ Zirong Chen<sup>2</sup> &nbsp; Run Luo<sup>1,5,‡</sup> &nbsp; Tao Tan<sup>1</s
 ## Overview
 
 Video-Arena asks a model to watch a full recording of a real game and reconstruct a **hidden game
-state variable** $V_t$ (a win probability, a net-worth lead, a score) that is never drawn on screen.
+state variable** \\(V_t\\) (a win probability, a net-worth lead, a score) that is never drawn on screen.
 Scoreboards, clocks, evaluation bars, kill feeds and names are removed when the video is rendered,
-so the only way to report $V_t$ is to have followed the whole match.
+so the only way to report \\(V_t\\) is to have followed the whole match.
 
 - **180 cases, 9 games, 6 genres**: Go, Chess and Gomoku (board), Doudizhu (card), CS2 (FPS),
   DOTA 2 (MOBA), StarCraft II (RTS), and Google Research Football and Badminton (sports). There are 20 cases per

docs/GROUND_TRUTH.md

@@ -7,7 +7,7 @@ environment can be refreshed without renegotiating what a label means.
 
 Two mechanisms:
 
-| | How $V_t$ is obtained | Environments |
+| | How \\(V_t\\) is obtained | Environments |
 |---|---|---|
 | **Parser** | the game already computes the quantity; the adapter reads it out of the replay, demo, trace, or annotation | CS2, DOTA 2, SC2, GFootball, Badminton |
 | **Evaluator** | the game does not define a win probability, so a pinned engine supplies one for every prefix | Go, Chess, Gomoku, Doudizhu |
@@ -53,7 +53,7 @@ Row counts, which are also the strict-F1 denominators:
 ### Two conventions that are easy to misread
 
 **A T1-only row is not a missing label.** Go and Chess emit `capture` rows alongside `move` rows. A
-capture is a perception event with no $V_t$ update of its own — the move that caused it already
+capture is a perception event with no \\(V_t\\) update of its own — the move that caused it already
 carries that — so its `wr_swing` and `leader_after` are `null` by design (378 of 5,516 rows in Go,
 331 of 1,761 in Chess). Those rows are scored at T1 and excluded from the T2/T3 denominators.
 
@@ -69,7 +69,7 @@ easier.
 ## T3 is not one metric
 
 T3 asks the same question everywhere — *where does this leave the match* — but what counts as an
-answer follows the $V_t$ of the environment:
+answer follows the \\(V_t\\) of the environment:
 
 | T3 answer | Metric | Environments |
 |---|---|---|
@@ -78,10 +78,10 @@ answer follows the $V_t$ of the environment:
 | the cumulative score pair after the round | exact fraction | CS2 |
 
 `score.py` prints all three in one T3 column and each judge records which it computed. A class
-label where the score is the state would be a weaker question, and an exact pair where $V_t$ is a
+label where the score is the state would be a weaker question, and an exact pair where \\(V_t\\) is a
 win probability would be unanswerable.
 
-T2's class vocabulary is not uniform either — it is whatever a change in that $V_t$ can be:
+T2's class vocabulary is not uniform either — it is whatever a change in that \\(V_t\\) can be:
 `{A, B, equal}` on a win probability, four classes (`taken` / `extended` / `levelled` / `narrowed`)
 on a score differential, two (`T` / `CT`) on a CS2 round. Chance level therefore differs by game:
 0.333 on three classes, 0.25 on four, 0.5 on two. Every judge macro-averages over per-class recall
@@ -94,7 +94,7 @@ would otherwise read **0.87** (4,481 of the 5,138 labelled rows are `equal`).
 
 Cases are not a random sample of matches. Each environment selects 20 from a larger pool so the
 corpus contains matches that are hard for different reasons, and so no single class dominates the
-T2/T3 labels. Eight of the nine environments stratify by **trajectory shape**, derived from the $V_t$
+T2/T3 labels. Eight of the nine environments stratify by **trajectory shape**, derived from the \\(V_t\\)
 trajectory itself:
 
 - **blowout** — one side ahead essentially throughout
@@ -119,12 +119,12 @@ A-led and 10 B-led — which is what keeps its dataset-pooled T3 from collapsing
 (A 568 / B 531 / equal 259 across 1,358 events).
 
 **The rule that assigns a trajectory type is per-environment, not shared.** It depends on the shape
-of that $V_t$: a score-difference $V_t$ makes *reversal* nearly impossible to detect from a trough
-(GFootball uses "winner ever trailed" plus a peak threshold), while a win-probability $V_t$ makes
+of that \\(V_t\\): a score-difference \\(V_t\\) makes *reversal* nearly impossible to detect from a trough
+(GFootball uses "winner ever trailed" plus a peak threshold), while a win-probability \\(V_t\\) makes
 *blowout* the fragile one (Doudizhu keys on the winner-perspective trough at 0.12 / 0.22). Always
 smooth before taking an extremum.
 
-Six `CASE_TYPES.json` files carry numeric thresholds — Go, Chess, Gomoku, CS2, and DOTA 2 the stratification thresholds they used, SC2 its event-segmentation ones. Doudizhu carries the note that its thresholds are *fixed* across the corpus (re-tuning ε per case would stop macro accuracy being comparable between cases), and GFootball and Badminton record only the resulting distribution. All nine carry the per-case label (SC2's is `leading_side`). The $V_t$ definition is in every game's `instruction.md` and in the per-environment sections below; `CASE_TYPES.json` repeats it for Doudizhu, GFootball, Badminton, and SC2.
+Six `CASE_TYPES.json` files carry numeric thresholds — Go, Chess, Gomoku, CS2, and DOTA 2 the stratification thresholds they used, SC2 its event-segmentation ones. Doudizhu carries the note that its thresholds are *fixed* across the corpus (re-tuning ε per case would stop macro accuracy being comparable between cases), and GFootball and Badminton record only the resulting distribution. All nine carry the per-case label (SC2's is `leading_side`). The \\(V_t\\) definition is in every game's `instruction.md` and in the per-environment sections below; `CASE_TYPES.json` repeats it for Doudizhu, GFootball, Badminton, and SC2.
 
 ---
 
@@ -137,7 +137,7 @@ Six `CASE_TYPES.json` files carry numeric thresholds — Go, Chess, Gomoku, CS2,
 | **Source** | SGF records via the OGS public API, fixture game 27400000 — <https://online-go.com> |
 | **GT mechanism** | evaluator. The adapter replays each move and evaluates **every prefix** |
 | **Engine pin** | KataGo, network `b18c384nbt`, **1600 visits** per position |
-| **$V_t$** | Black win probability, **±10 pp** dead band around 0.50 |
+| **\\(V_t\\)** | Black win probability, **±10 pp** dead band around 0.50 |
 | **Render** | plain 19×19 board, 612², 25 fps. No coordinates, move numbers, names, clocks, capture counters, komi, or evaluation bar |
 
 T1 is the move or capture with its time, colour, and board coordinate. T2 is the quantized signed
@@ -154,7 +154,7 @@ whole match in view is most exposed — and where Protocol A's frame grid bites
 | **Source** | public Lichess game export, fixture `cZWtZEBE` — <https://lichess.org> |
 | **GT mechanism** | evaluator, per-ply, with real per-ply timing |
 | **Engine pin** | **Stockfish 18, depth 20**, value function `white_expected_score_wdl` |
-| **$V_t$** | White-oriented expected score / WDL, **±10 pp** band. Draws map to the neutral label |
+| **\\(V_t\\)** | White-oriented expected score / WDL, **±10 pp** band. Draws map to the neutral label |
 | **Render** | plain 8×8 board, 548², 1 fps. No coordinates, move numbers, names, clocks, result text, captured-piece tray, or evaluation bar |
 
 T1 records the **destination square only**, not the origin — the released contract is deliberately
@@ -171,7 +171,7 @@ the container rate, so it never asks ffmpeg to fabricate frames here.
 | **Source** | competition PSQ records (Gomocup) |
 | **GT mechanism** | evaluator |
 | **Engine pin** | **Rapfi**, 1000 ms per turn, freestyle rule, `black_winrate_sigmoid_eval` (scaling factor 200) — <https://github.com/dhbloo/rapfi> |
-| **$V_t$** | **tempo-neutral** smoothed Black win probability, ±10 pp band |
+| **\\(V_t\\)** | **tempo-neutral** smoothed Black win probability, ±10 pp band |
 | **Render** | plain 15×15 board, 504², 25 fps. No coordinates, move numbers, names, clocks, or engine annotations |
 
 **Why the smoothing matters.** Gomoku is attack-dominant, so a raw side-to-move win rate swings
@@ -188,10 +188,10 @@ consistency. All 20 cases share that pin.
 | **Source** | DouZero-WP greedy self-play deals (τ=0) — <https://github.com/kwai/DouZero>, weights at <https://huggingface.co/palemoky/douzero-baselines> |
 | **GT mechanism** | evaluator: **hidden-hand PIMC** — determinize, then roll out |
 | **Estimator** | adaptive **400–4000 worlds** per state, greedy rollouts with the released DouZero-WP policy. The PIMC estimator is part of this benchmark, not a DouZero component |
-| **$V_t$** | landlord win probability = fraction of sampled deals the landlord wins. Bands: landlord > 0.55, farmers < 0.45, else equal. T2 impact ε = 0.034, T3 band = 0.05 |
+| **\\(V_t\\)** | landlord win probability = fraction of sampled deals the landlord wins. Bands: landlord > 0.55, farmers < 0.45, else equal. T2 impact ε = 0.034, T3 band = 0.05 |
 | **Render** | tabletop, 1280×720, 24 fps. Landlord hand and the three bottom cards face up; **farmer hands face down**, showing only their counts. No numeric score, timer, move counter, or probability display |
 
-**Hiding the farmer hands is what makes $V_t$ non-degenerate.** With every hand visible the value is
+**Hiding the farmer hands is what makes \\(V_t\\) non-degenerate.** With every hand visible the value is
 close to a deterministic function of the cards, and the task collapses into card counting. The
 evaluator is given exactly the landlord's information set — own hand, the bottom three, all plays
 so far, and both farmer card *counts* — and samples uniformly from the deals consistent with it.
@@ -213,7 +213,7 @@ with no ambiguity across plays.
 |---|---|
 | **Source** | public professional-match demo (`.dem`) files published on HLTV; the game is <https://www.counter-strike.net> |
 | **GT mechanism** | parser, our own, at the demo's **native 64 Hz** tick rate; positions resampled to 4 Hz for the render |
-| **$V_t$** | the **cumulative round score**, by persistent team (A / B) across the halftime side swap |
+| **\\(V_t\\)** | the **cumulative round score**, by persistent team (A / B) across the halftime side swap |
 | **Render** | clean top-down radar, 768², 15 fps, 1:1 real pace. Map geometry, player motion, bomb state, and event-relevant effects preserved; HUD, scoreboard, round counter, kill feed, economy, names, and audio removed |
 
 T1 covers six event types — `kill`, `bomb_planted`, `bomb_defused`, `bomb_exploded`, `elimination`,
@@ -242,7 +242,7 @@ deliberate: it is the round timer, not the state.
 |---|---|
 | **Source** | Valve replays plus OpenDota match records — <https://www.opendota.com>, <https://github.com/odota> |
 | **GT mechanism** | parser (manta-based) for events, OpenDota's minute-level gold curve for the state |
-| **$V_t$** | the **parsed net-worth advantage** — explicitly *not* a learned win-probability model. `even` = within **±1000 gold** |
+| **\\(V_t\\)** | the **parsed net-worth advantage** — explicitly *not* a learned win-probability model. `even` = within **±1000 gold** |
 | **Render** | the real minimap terrain art with GT-anchored coordinates and real game sprites, 768², 15 fps. HUD, scoreboard, clock, net-worth graph, names, and objective counters removed |
 
 T1 covers six event types — first blood, teamfight, tower, barracks, Roshan, ancient — each with
@@ -272,7 +272,7 @@ the lowest in the corpus at **0.597**.
 | **Source** | engine-generated AI-vs-AI replays (difficulty *Harder*, 6 matchups, 2017 ladder maps). Replay protocol: <https://github.com/Blizzard/s2client-proto> |
 | **GT mechanism** | parser — the replay **tracker stream**, read during a `disable_fog` god-view replay pass; deaths from raw `dead_units` (morph-free) |
 | **Engine** | SC2 **4.10, base build 75689** (Blizzard Linux research build) via PySC2, rendered with EGL |
-| **$V_t$** | **resource-equivalent advantage share** $s=(V_A-V_B)/(V_A+V_B)\in[-1,1]$, band **±0.10**. $V$ = banked minerals + gas, live army value, economy value, technology value |
+| **\\(V_t\\)** | **resource-equivalent advantage share** \\(s=(V_A-V_B)/(V_A+V_B)\in[-1,1]\\), band **±0.10**. \\(V\\) = banked minerals + gas, live army value, economy value, technology value |
 | **Render** | full-map isometric observer god-view, 1536², **5.6 fps**, real time. Resource counters, production panels, minimap, names, and other direct readouts removed |
 
 T1 is a `battle` or `building_destroyed` event with its time and **coarse map region**: the playable
@@ -296,7 +296,7 @@ environment contamination-free by construction.
 |---|---|
 | **Source** | randomized Google Research Football simulator episodes (bot vs bot) — <https://github.com/google-research/football> |
 | **GT mechanism** | parser. Score transitions from the engine's `obs['score']`; the scorer name from the engine's own goal log |
-| **$V_t$** | `left_goals - right_goals`, the engine's native score. No evaluator |
+| **\\(V_t\\)** | `left_goals - right_goals`, the engine's native score. No evaluator |
 | **Render** | the engine's default inclined 3D broadcast view, full **1280×720, 24 fps**, uncropped, real time (1 engine step = 1 frame) |
 
 T1 is the goal with its time, scoring side (`L` / `R`), scorer surname, and own-goal flag. T2 is the
@@ -323,7 +323,7 @@ is not always the side that touched the ball last.
 |---|---|
 | **Source** | **ShuttleSet** stroke records (KDD-23) — <https://github.com/wywyWang/CoachAI-Projects>, <https://arxiv.org/abs/2306.04948> |
 | **GT mechanism** | parser: strokes grouped into points, the native score carried through each rally |
-| **$V_t$** | `pts_near - pts_far` within the set. No evaluator |
+| **\\(V_t\\)** | `pts_near - pts_far` within the set. No evaluator |
 | **Render** | fixed-camera **synthetic 3D**, 1280×720, 15 fps. A synthetic render built from real trajectories, **not** a source replay |
 | **Granularity** | one set of a singles match |
 

docs/PROTOCOLS.md

@@ -10,7 +10,7 @@ beyond delivery breaks the comparison.
 ## What is shared
 
 **One instruction per game, rendered per protocol.** `tasks/<game>/instruction.md` carries the
-event vocabulary, coordinate and zone conventions, the $V_t$ definition with its dead band, the
+event vocabulary, coordinate and zone conventions, the \\(V_t\\) definition with its dead band, the
 row schema, and the ±3 s matching rule. That is 70–85% of each file and it is byte-identical
 across the two protocols — otherwise the two result tables are not the same question asked twice.