# The state-window gate: does a temporal WINDOW of wave-relative states beat a SINGLE state?

**The design under test (the user's words).** *"We decide first what makes an
'environment state' ... then we feed as input a long list of these states, long
enough to have one or more patterns of movement inside."* The premise is that SBC
is a coincidence detector and that a **temporal list** of states, fed instead of a
single tick, turns "coincidence" into "movement pattern".

**Verdict: NO.** On held-out battles a window of wave-relative states **does not**
predict DrussGT's lateral position at the bullet's arrival better than the single
state at the **same decision tick**. On the pre-fire frame the window is strictly
and substantially *worse* at every window length, every coarseness and every
smoothing strength. On the during-flight frame the apparent gain is entirely the
**later decision tick**, not the window — and where the window finally edges the
single state (K=32, Q=4) it is by 0.045 bits on a subset where the 32-window
**never repeats at all**. The design, as stated, is **dead**. The *state
definition* survives and is the useful result.

This is a gate, not a gun. **No live win is claimed and none is measured here.**

---

## 1. The state (design artifact)

The dodge is a response to **our bullet**, so the state is **wave-relative**, never
absolute world coordinates. One state at absolute tick `t`, in the frame of the
bullet fired at `t0` along `u = (cos dir, sin dir)` (with `n = (-u_y, u_x)` the
lateral unit vector):

| field | definition | why |
|---|---|---|
| `lat` | `(D(t) − P0) × u` — DrussGT's signed perpendicular offset from the bullet line, px | the quantity that decides the hit; signed so "which side" is visible |
| `vlat` | `lat(t) − lat(t−1)` — signed lateral velocity, px/tick | crossing vs returning |
| `toa` | `(t0 − t) + karr` — ticks until the bullet reaches its arrival plane | how much time is left to move |
| `room` | ray distance from `D(t)` along `sign(vlat)·n` until the arena wall (18 px bot radius) | room to keep running vs being cornered — the only *directional* wall field, unlike plain "distance to nearest wall" |
| `turn` | `wrap180(eh(t) − eh(t−1))` — signed turn rate, deg/tick | surfers slow to turn; turning reveals a reversal |

**Refinements to the starting proposal, and why.** (a) was kept signed, not
absolutised, because the target is signed. (b) was made a *difference of the
lateral offset*, so it is exactly the lateral component of velocity in the bullet
frame rather than a speed. (d) "distance to the wall" was made *directional*
(room to run in the current lateral-motion direction); a plain nearest-wall
distance is not wave-relative and is nearly constant at long range. (e)
"heading/turn rate" was reduced to the **turn rate** alone, because heading
relative to the bullet is already implied by the sign of `vlat`, and the turn rate
is the thing that distinguishes a surf from a corner.

**Quantisation (the numerosity dial).** Each field is binned into `Q ∈ {2, 3, 4}`
symbols. A state is therefore:

| Q | bits / state | distinct states |
|---|---|---|
| 2 | **5.0** | 32 |
| 3 | **7.9** | 243 |
| 4 | **10.0** | 1024 |

Bin edges are **quantiles of the TRAIN side only** (no test leakage); a coarse
state makes two similar situations look the same, which is what lets a
"coincidence" repeat. The sweep below is exactly the question of whether coarser
buys more recurrence than it loses in resolution.

### The two windows tested, and why both

A window is `K` consecutive states. Crucially, **the baseline is always the single
state at the same decision tick `D`** — otherwise a longer window would "win" only
because its last state is closer to the answer.

* **`pre`** — `D = t0` (the fire tick); the window is the `K` pre-fire states
  ending at `D`. This is what a bot can compute *before* pulling the trigger. All
  `K ≤ 48` are feasible (max recorded flight is 43 ticks).
* **`fly`** — `D = t0+K−1`; the window is the first `K` states of the flight, and
  the baseline is the single state at `D`. Only shots with `karr > 31` are used,
  so the whole window is strictly before arrival (8156 of 54939 shots).

The task's field `toa` only exists once a bullet is in the air, so `fly` is the
"reaction" reading; `pre` is the "am I about to be dodged?" reading. Both are
reported.

### The target

`perp_arr` — DrussGT's **signed** perpendicular offset from the bullet line at the
tick the bullet reaches its along-track plane. This is the miss offset that
decides the hit; unlike "required lead" it does not depend on the bullet speed. It
is quantised into 7 bins with edges `±120, ±60, ±18` px; the central `±18` px bin
is the hit window (bot radius 18 px).

**Bin width in degrees-at-450px.** The central hit bin is 36 px wide = one bot
diameter. `atan(18/range)` is the hit half-window: **2.29°** at 450 px, 4.23° at
the median fire range (487 px), and 20.4° at 100 px. So the 36 px central bin is
**4.58° at 450 px**. The three inner bins are 36/42/60 px wide (4.58/5.34/7.63° at
450 px) — the target resolution is *finer* than the ~2.29° hit half-window, so a
model that predicts the bin is not being asked an impossible question.

---

## 2. Data and split

* **Data (MEASURED present before use):** `/tmp/tfil_ab2/out/<A..E>/runN.jsonl` +
  `.events.jsonl` + `.rounds.json` — **70 real live battles vs the unmodified
  DrussGT, 490 rounds, 54 939 shots** by ModularBot, recorded by
  `tools/robocode_shim/run_bridge_battle.sh`. Per-shot geometry is re-derived by
  the validated instrument `common_libs/tests/analyze_drussgt_dodge_vs_power.py`
  (which this extractor imports). Attribution (`e*`=DrussGT, `s*`=us) is the one
  already documented in `docs/drussgt_dodge_vs_power.md`.
* **Split (stated explicitly): BY BATTLE, never by tick.** All rounds of a battle
  go to one side. A 70 %/30 % battle split (**49 train / 21 test battles**),
  repeated over **3 seeds**. Bin edges are derived from the train side only. Two
  ticks inside one round never straddle the split.

---

## 3. Model (deliberately dull)

The question is about **information**, not modelling, so the model is a plain
**interpolated (Jelinek–Mercer) suffix-backoff table** — the direct analogue of
SBC "counting coincidences":

```
P = global target-bin histogram
for j = 1..K:   P <- (count(suffix_j) + A*P) / (total(suffix_j) + A)
```

It is order-sensitive, it can never do much worse than the shorter context, and it
degrades gracefully when a suffix is unseen. Context depth is capped at 12 (orders
above that are never repeated often enough to matter — see §5). `A ∈ {1,5,20}` is
swept; **A = 5 is primary**. A majority/no-window predictor is the floor. The
model predicts the 7-bin displacement distribution; we report held-out **accuracy**
(argmax), **log-loss** in bits, and the **implied hit probability** `P(central
bin)`.

---

## 4. Result — does the window beat the single state? **No** *(MEASURED)*

70 battles, 3 battle-split seeds, mean held-out log-loss (bits). `delta = window −
single@D`; **negative would mean the window wins**.

### 4.1 `pre` frame — window ends at the fire tick, looks back *(54 939 shots)*

`single@D` is the fire-tick state; it does **not** change with K. A = 5:

| K | Q=2 single → window (Δ) | Q=3 single → window (Δ) | Q=4 single → window (Δ) |
|---|---|---|---|
| 1 | 2.4524 → 2.4524 (—) | 2.3782 → 2.3782 (—) | 2.3460 → 2.3460 (—) |
| 4 | 2.4524 → 2.5107 (**+0.058**) | 2.3782 → 2.5461 (**+0.168**) | 2.3460 → 2.5656 (**+0.220**) |
| 8 | 2.4524 → 2.8015 (**+0.349**) | 2.3782 → 2.8336 (**+0.455**) | 2.3460 → 2.7265 (**+0.381**) |
| 16 | 2.4524 → 3.1310 (**+0.679**) | 2.3782 → 2.9473 (**+0.569**) | 2.3460 → 2.7450 (**+0.399**) |
| 32 | same as K=16 (depth cap) | same | same |
| 48 | same as K=16 (depth cap) | same | same |

Accuracy tells the same story (Q=4: single `0.409` → window `0.383`/`0.364`/
`0.362`). The per-seed deltas are essentially identical (e.g. Q=2/K=4:
`+0.0572, +0.0534, +0.0644`) — there is no seed where the window helps. The
**heavy-smoothing** arm is the fairest to the window and it still loses:
Q=4, A=20: K=4 `2.3928` vs single `2.3557` (+0.037); Q=2, A=20: K=4 `2.4643` vs
`2.4524` (+0.012).

**The extra history is not just useless, it is actively harmful**, and the reason
is visible in the shuffle control below: the long ordered contexts are sparse, and
the interpolation spends held-out probability on them.

The baseline is not trivial: the single state halves the majority floor
(majority log-loss **2.6983**, acc **0.2348**; best single Q=4 **2.3460**, acc
**0.4094**).

### 4.2 `fly` frame — window starts at the fire tick, ends at `t0+K−1` *(8 156 shots)*

Here the **single@D itself** improves steeply as K grows, because the decision tick
is later and therefore closer to the answer. A = 5:

| K | Q=2 single@D → window (Δ) | Q=3 single@D → window (Δ) | Q=4 single@D → window (Δ) |
|---|---|---|---|
| 1 | 2.6645 → 2.6645 (—) | 2.6741 → 2.6741 (—) | 2.7020 → 2.7020 (—) |
| 4 | 2.6376 → 2.7695 (**+0.132**) | 2.6262 → 2.8858 (**+0.260**) | 2.6397 → 2.9108 (**+0.271**) |
| 8 | 2.5659 → 3.0331 (**+0.467**) | 2.5302 → 3.0207 (**+0.490**) | 2.6101 → 2.9084 (**+0.298**) |
| 16 | 2.4047 → 3.1643 (**+0.760**) | 2.3103 → 2.6362 (**+0.326**) | 2.3853 → 2.5918 (**+0.207**) |
| 32 | 1.8365 → 1.9970 (**+0.161**) | 1.4355 → 1.4625 (**+0.027**) | **1.3121 → 1.2668 (−0.045)** |

Read this as: **the gain is the later tick, not the window.** `single@D` alone drops
from `2.70` at K=1 to `1.31` at K=32 (Q=4) — the single state at `t0+31` is 2
ticks from arrival and already knows the answer. The window adds nothing on top;
the only cell where it "wins" is K=32/Q=4 by **0.045 bits**, and that subset's
32-state window **never repeats** (§5), so the model is running on the single
state plus a trace of recent acceleration. Q=3 at the same K is *worse*, and
K=4–16 are worse by 5–20× that margin.

### 4.3 The mandatory shuffle-order control *(MEASURED)*

The order of the K states inside each window is randomly permuted, preserving the
multiset. In the `pre` frame the shuffled window is **worse** than the temporal
window at K=4 (e.g. Q=4: `2.5672` vs `2.5656`… within noise) but **better** at
K≥8 (Q=4/K=16: shuffled `2.5613` vs temporal `2.7450`). In the `fly` frame at
K=32 the shuffled window is much worse (Δ = `+0.35/ +0.80/ +1.12`).

**How to read that, honestly.** The shuffle penalty at `fly` K=32 is **not**
evidence the window works: the 32-state window never repeats, so the only order
the model can actually use is the *recency* of the last state. Shuffling destroys
*which state is current*, and the model degrades because recency matters — a
statement about the **single state**, not the window. Where the shuffled window is
*better* than the temporal one (pre frame, K≥8), it is because random order
disables the sparse long contexts that the temporal order feeds to the model.
Either way, the window itself contributes nothing.

A deterministic **reverse** control (reverse time, preserves recurrence) is worse
than temporal everywhere, confirming the model is genuinely order-sensitive and
that the last state is the informative one.

---

## 5. Recurrence — the "we can find coincidences" premise *(MEASURED)*

Distinct **ordered** window tuples over the whole corpus (54 939 shots; 8 156 in
the `fly` subset). A window is *learnable by exact coincidence* only if its count
is well above 1.

| K | Q=2 distinct (mean count) | Q=3 | Q=4 |
|---|---|---|---|
| 1 | 16 (3433) | 82 (670) | 372 (148) |
| 4 | 1 747 (31.4) | 11 018 (5.0) | 24 223 (2.3) |
| 8 | 13 474 (4.1) | 37 337 (1.5) | 49 891 (**1.1**) |
| 16 | 41 447 (1.3) | 54 163 (1.01) | 54 707 (1.00) |
| 32 | 54 454 (**1.01**) | 54 754 (1.00) | 54 779 (1.00) |
| 48 | 54 744 (1.00) | 54 772 (1.00) | 54 789 (1.00) |

(In the `fly` subset the same collapse happens earlier and harder: Q=4, K=8 already
1.02 mean count, and **K=32 is 8 156 distinct of 8 156 — literally every window is
unique**.)

So the "long list with patterns inside" premise fails as a **combinatorial**
matter, not merely a modelling one:

* At the coarsest state worth using (Q=2, 5 bits), only **K ≤ 8** has meaningful
  recurrence — and even there the window does not beat the single state.
* At the finest (Q=4, 10 bits), K=4 already averages 2.3 observations; K≥8 is
  essentially all singletons. **Learning from a window that never recurs is
  impossible**, and the sweep shows it does not happen.
* The single state is the only object with a real coincidence structure: 16–372
  distinct values, every one repeated hundreds of times.

---

## 6. Direct answer

**Does a window of wave-relative states predict DrussGT's future lateral position
better than a single state on held-out battles? NO.**

* `pre` frame: the window is worse at **every** K, Q and A; at K=8 it costs
  **+0.35 to +0.46 bits** over the single state.
* `fly` frame: the whole apparent improvement is the later decision tick
  (`single@D` 2.70 → 1.31 bits as K goes 1 → 32). The window adds nothing except
  a 0.045-bit sliver at K=32/Q=4 on a subset whose 32-windows never repeat.
* The recurrence statistics say the same thing independently: re-usable
  "patterns" of movement do not recur at any window length that carries extra
  information.

**The design — feed SBC a long temporal list of states — is dead.** Do not build
it. *(MEASURED)*

**What survives (and is worth keeping).** The **state definition itself is a
strong single-state predictor**: the fire-tick state with 4 symbols/field
(10 bits) halves the majority log-loss (2.6983 → **2.3460** bits) and triples the
chance the predicted displacement bin is right (0.2348 → **0.4094**) on battles
the model has never seen. If anything from this line is taken forward, take the
**single** wave-relative state at Q=4 and use it as a context for a *different*
decision problem — but note the implied hit probability it produces is still the
base rate (~0.09), so it is informative about **which side** the miss falls on, not
about **whether** the shot hits. *(MEASURED/INFERRED)*

---

## 7. MEASURED vs INFERRED

**MEASURED** (exact commands in §8; numbers above)

* 70 battles / 490 rounds / 54 939 shots; 8 156 with `karr ≥ 33`.
* Every held-out log-loss / accuracy / implied-hit number in §4, over 3
  battle-split seeds, with bin edges from train only.
* The shuffle-order and reverse-order controls.
* The distinct-window and repeat counts in §5.

**INFERRED**

* That the residual fly-frame "win" at K=32/Q=4 is a trace of recent acceleration
  rather than a movement pattern — it is consistent with the never-repeating
  32-window and with the single state already containing `vlat`.
* That a *different* (e.g. TSetlin/SBC) learner would not reverse the answer. This
  is not proved. What **is** proved is the recurrence floor: a coincidence-based
  method cannot learn from windows that never recur, and by Q=4/K=8 they already
  essentially never do.
* "Design is dead" applies to the **long temporal window**. It does not say the
  state is useless, and it does not say a *short* (K=2) context is worthless — K=2
  was not separately swept because at Q≥3 it is already 1.5–5 observations per
  context, i.e. below the point where it could help.

**Caveats.** (1) The `fly` frame is a biased long-range subset (all shots with
`flight ≥ 33`, i.e. the longest-range quarter). (2) The model's context depth is
capped at 12; K=16/32/48 therefore share a row, but §5 shows orders above ~4 are
never repeated often enough to matter, so the cap is not the binding constraint.
(3) The target is the miss offset, not the hit; the implied hit probability is the
base rate for every model, so nothing here is a hit-rate claim.

---

## 8. Reproducing

```bash
# corpus (live captures; not in the repo)
ls /tmp/tfil_ab2/out            # 70 battles: <A..E>/runN.jsonl{,.events.jsonl,.rounds.json}

python3 common_libs/tests/state_window_gate.py \
    --tfil /tmp/tfil_ab2/out --seeds 3 --frames pre,fly \
    --json common_libs/tests/fixtures/state_window_gate.json \
    > common_libs/tests/fixtures/state_window_gate_report.txt
```

Runtime ≈ **5 minutes**, single-threaded pure Python (no numpy required). The
script imports `common_libs/tests/analyze_drussgt_dodge_vs_power.py` for the
per-shot geometry; `common_libs/tests/fixtures/state_window_gate_report.txt` is
the verbatim captured output and `…state_window_gate.json` is the same numbers as
JSON (including the per-seed values). No offline fixture replay, no simulator:
these are the recorded real battles.