Syplheed-Reborn/docs/re/flight-speed-law.md

# The throttle is a TARGET-SPEED selector — measured against the definition (2026-08-13)

**Status: ✅ for the shape of the law, 🟡 for the unit scale.**

The unit definition gives `MinimumVelocity 100`, `CruisingVelocity 350`,
`MaximumVelocity 1200`, `Acceleration 600` and `Deceleration 500` for
`UN_f001_TCAF_DeltaSaber_T_Player` ([values](captures/unit-runtime-fields.csv)),
but not **how** the game applies them. This measures it.

Method — [`tools/re-capture/speed_law.py`](../../tools/re-capture/speed_law.py):
lock onto the player entity once, then sample its own position triple in guest RAM
while holding each throttle input in turn (8 s per phase, 20 Hz). Speed is
differentiated over **1-second windows**, never per sample.

## Result

| phase | measured speed (1 s windows) | settles to | definition field |
|---|---|---|---|
| no throttle | 367 403 471 365 463 463 365 | **~420** | `CruisingVelocity` 350 |
| `RT` held | 1041 1376 1551 1379 1596 1510 1620 | **~1 530** | `MaximumVelocity` 1200 |
| release | 1314 672 432 452 432 450 408 | back to **~440** | — |
| `LT` held | 203 121 135 121 115 146 122 | **~125** | `MinimumVelocity` 100 |
| release | 369 451 427 425 445 390 465 | back to **~430** | — |

So the throttle **selects a target speed** and the craft converges to it; releasing
either trigger returns it to cruise. (**Refined below**: the trigger is analogue, so
the target is a continuous interpolation and these three are its endpoints.) It is not a
force model with the throttle adding thrust, which is what a reimplementation would
most likely have assumed from `Acceleration`/`Deceleration` alone. Those two fields
govern the **convergence rate**: the release phase falls ~1 314 → ~432 in about two
seconds (~440 units/s², against `Deceleration` 500) and the `RT` phase climbs ~420 →
~1 550 in two to three seconds (~470–560 units/s², against `Acceleration` 600).

## 🟡 The unit scale

Measured world-space speeds run **≈1.2–1.3× the definition numbers** in all three
regimes (cruise 1.21, maximum 1.28, minimum 1.32). The HUD, meanwhile, reads
**350** at cruise — the definition value exactly. So the definition's velocity unit
is the HUD's, and entity world coordinates are a constant multiple of it, close to
**1.25**. The spread across regimes is larger than the constant itself is precise,
because the craft is manoeuvring under fire while sampled (straight-line
displacement per window under-reads a curving path), so this is recorded as a
measured range rather than a pinned constant.

## Traps

* **`RT`/`LT` are analogue triggers**, so `vgamepad trig RT 1.0` holds the throttle;
  the button verb `hold RT` is a **silent no-op**. The first run of this probe
  measured the drift of a craft nobody was flying.
* **Do not differentiate per sample.** The guest updates the position slower than
  20 Hz, so per-sample differences alternate between 0 and a double step —
  `0, 1519, 1985, 0, 2681…` for a craft flying smoothly.
* **Bind late, measure fast.** The player entity is not in the typed-entity scan
  for the first ~15 s of a mission, and the craft dies within a few minutes if
  nobody is flying it — one earlier attempt ended at `GAME OVER` mid-probe.

Raw samples: [`captures/speed-law-throttle-phases.csv`](captures/speed-law-throttle-phases.csv).

---

# Turn rates: `AV_*` are rate caps, and `_Min`/`_Max` mean *at minimum/maximum speed*

`tools/re-capture/turn_law.py` pins the speed regime with a throttle, holds a stick
axis, and differentiates the craft's **own forward vector** over 1-second windows.

**Control mapping, measured** — `LX` is **roll** (the forward vector barely moves
under it: 3–5 °/s residual), `LY` is **pitch** (`vgamepad` documents `LY: -1 = up`,
so `+1` is nose down = the `PitchMinus` family), and the **right stick does not
steer at all** (0 °/s).

| phase | measured (1 s windows) | definition |
|---|---|---|
| slow + nose down | 85 · 96 · 82 → **~87** | `AV_PitchMinus_Min` **75** |
| fast + nose down | 55 · 45 · 63 → **~54** | `AV_PitchMinus_Max` **40** |
| slow + nose up | 181 · 164 · 181 → **~175** | `AV_PitchPlus_Min` **150** |
| fast + nose up | 103 · 130 · 175 → **~136** | `AV_PitchPlus_Max` **70** |

Two claims of the field names are confirmed:

* **Agility falls with speed** — every rate drops when the throttle goes from `LT`
  to `RT`, so `_Min`/`_Max` are *at minimum / maximum speed*, not floor/ceiling of
  the rate.
* **Pitching up is about twice as fast as pitching down**, exactly as the pair
  150/75 (slow) and 70/40 (fast) says.

## The ~1.2× factor — ⚠️ WITHDRAWN, see "Settled rates match exactly" below

The speed law measured 1.21 / 1.28 / 1.32 × its definition values; the pitch rates
measure 1.16 / 1.35 / 1.17 / 1.94 × theirs. A **unit scale** would have to differ
between metres-per-second and degrees-per-second — it cannot explain both. A **time
base** does: if the guest's simulated second is shorter than the wall-clock second
this probe measures against, every rate reads high by the same factor. So the
definition numbers are self-consistent and a reimplementation should take them at
face value; these measurements confirm the **shape** of the law (which field governs
what, and how the regimes switch), not a scale correction.

🟡 **No yaw input was found.** `AV_Yaw_{Min,Max}` (45 / 25 °/s) exists, but neither
stick yaws the craft. With `MaximumBank_Normal` (60) and roll on `LX`, a
**bank-to-turn** model is the obvious reading — sustained turns come from rolling and
pitching — but the yaw fields may equally belong to the AI or to an input this probe
has not driven.

Raw samples: [`captures/turn-law-pitch-phases.csv`](captures/turn-law-pitch-phases.csv).

---

# The afterburner is not on the obvious buttons — a bounded negative

The definition describes the burner without naming its input: `AB_ConsumeShield_Begin`
**50**, `AB_ConsumeShield` **10**, and a set of `AB_AV_*` turn caps far below the
normal ones (roll 40 vs 125–200 °/s, pitch-up 30 vs 70–150). Notably there is **no
`AB_*Velocity`**, so the burner may not raise the speed target at all.

Two independent probes, both negative for `A`, `B`, `X`, `LB` (and `LS`/`RS` on the
first):

* [`tools/re-capture/ab_probe.py`](../../tools/re-capture/ab_probe.py) — hold `RT`
  for a max-speed baseline, then hold each candidate: speed stayed inside the
  baseline's own noise band (1 200–1 580 units/s) for every one.
* [`tools/re-capture/ab_state_probe.py`](../../tools/re-capture/ab_state_probe.py) —
  sample a window of the player object while each candidate is held and report any
  float that falls during the hold: **nothing fell**.
* And the cheapest oracle of the three, needing no offsets at all: **count the green
  pixels of the HUD's SHIELD bar** before and after each hold, on a freshly spawned
  craft with a full shield. `AB_ConsumeShield_Begin 50` should take a visible bite;
  the bar read **156, 156, 156, 157, 157** across baseline and all four buttons.

So the burner is **not a simple hold on `A`/`B`/`X`/`LB`/`LS`/`RS`**. What remains:
a chord (something plus `RT`), an input this pad cannot reach, a craft variant that
has it, or fields that belong to the AI rather than the player. Worth knowing before
anyone re-probes the obvious buttons.

(`RB` is the nose gun, `Y` the missile mount and the d-pad the tactical map — see
[flight controls](flight-controls-runtime.md) — so those were not held here.)

---

# The throttle is ANALOGUE: the target speed interpolates cruise → maximum

`RT` is an analogue trigger, so "held" was only ever one point on a curve. Walking
it in five steps ([`tools/re-capture/throttle_curve.py`](../../tools/re-capture/throttle_curve.py),
2.5 s to converge then 5 s of sampling per step):

| `RT` | settled speed (1 s windows) | mean |
|---|---|---|
| 0.00 | 465 · 411 · 373 · 504 | **438** |
| 0.25 | 648 · 554 · 678 · 622 | **626** |
| 0.50 | 739 · 961 · 897 · 919 | **879** |
| 0.75 | 1028 · 1086 · 977 · 1283 | **1094** |
| 1.00 | 1359 · 1306 · 1517 · 1186 | **1342** |

A straight ramp. Dividing by the ~1.2 time-base factor established above, the
endpoints land on the definition's own numbers — 438/1.2 ≈ **365** against
`CruisingVelocity` **350**, and 1342/1.2 ≈ **1118** against `MaximumVelocity`
**1200** — and the midpoint follows: 879/1.2 ≈ 733 against the predicted
350 + 0.5·(1200−350) = 775. So

```
target speed = CruisingVelocity + RT · (MaximumVelocity − CruisingVelocity)
```

and `LT` mirrors it down to `MinimumVelocity` — **measured**, see below.

**This also refutes a live hypothesis about the afterburner.** Full `RT` producing
more than `MaximumVelocity` looked like it might *be* the burner; it is not — the
curve is smooth through full deflection, with no step, and the shield does not move.
Full throttle is simply full throttle.

Raw samples: [`captures/throttle-curve-rt.csv`](captures/throttle-curve-rt.csv).

(The entity scan that locks onto the player searches *changing* position triples, so
it only sees the craft while it still has speed — bind early, or a mission left
idling drops out of the scan entirely and every tool reports "player entity not
found" while the game is visibly flying.)

---

# `LT` mirrors `RT`, and roll does NOT depend on speed

One flight, both measurements
([`tools/re-capture/flight_law2.py`](../../tools/re-capture/flight_law2.py)).

## The braking half of the curve

| `LT` | 0.00 | 0.25 | 0.50 | 0.75 | 1.00 |
|---|---|---|---|---|---|
| speed | 436 | 379 | 289 | 209 | **126** |

A straight ramp down from cruise to ~126, and dividing by the ~1.2 time-base factor
the endpoints are the definition's own numbers again — 363 against
`CruisingVelocity` **350** and 105 against `MinimumVelocity` **100**. So the throttle
law is symmetric:

```
RT held:  target = Cruising + RT · (Maximum − Cruising)
LT held:  target = Cruising − LT · (Cruising − Minimum)
```

## Roll

Roll turns the craft about its own forward axis, so it has to be measured on a
different row of the rotation matrix than pitch was:

| phase | rate per 1 s window | mean |
|---|---|---|
| minimum speed + full roll | 171 · 103 · 157 | **~144 °/s** |
| maximum speed + full roll | 140 · 145 · 164 | **~150 °/s** |

**Roll shows no speed dependence** — the two regimes agree inside the noise, where
pitch dropped by a third to a half between them. Against the definition
(`AV_Roll_Min` 200, `AV_Roll_Max` 125) the measured ~145 °/s is ~121 after the
time-base factor, i.e. **the `Max` value in both regimes**.

So the `_Min`/`_Max` pair does *not* mean the same thing for every axis: pitch
interpolates between them with speed, roll appears pinned at `Max`. A
reimplementation that applies one rule to all axes would get roll wrong at low
speed by ~60 %.

🟡 Caveat kept: the roll phases were 2 s of settling apart, which is marginal for a
full 126 → 1 342 speed change, so "no dependence" is a measurement over a real but
not perfectly separated pair of regimes.

Raw samples: [`captures/throttle-curve-lt.csv`](captures/throttle-curve-lt.csv) ·
[`captures/turn-law-roll.csv`](captures/turn-law-roll.csv).

---

# ⚠️ Settled rates match the definition EXACTLY — the "time base" reading is withdrawn

The earlier turn measurements settled each phase for 1.5 s and then differentiated
from the first second. Re-running with **5 s of settle** and the craft's speed
recorded *at the moment the turn starts*
([`tools/re-capture/flight_law3.py`](../../tools/re-capture/flight_law3.py)):

| phase | speed at turn | measured rate | definition |
|---|---|---|---|
| pitch, minimum speed | 130 /s | **74.9 °/s** | `AV_PitchMinus_Min` **75** |
| pitch, half throttle | 388 /s | 73.7 °/s | (between, see below) |
| pitch, maximum speed | 1 821 /s | **41.1 °/s** | `AV_PitchMinus_Max` **40** |
| roll, minimum speed | 110 /s | 129.5 °/s | `AV_Roll_Min` 200 |
| roll, maximum speed | 1 722 /s | 149.3 °/s | `AV_Roll_Max` 125 |

**Pitch lands on 75 and 40 — the definition's own numbers, with no scale factor.**
So the ~1.2× I attributed to the emulated time base was an artefact of measuring
during the `AA_*` acceleration ramp with too little settle, and that explanation is
**withdrawn**: angular rates need no correction, and a reimplementation should use
`AV_*` verbatim.

What remains unexplained is only on the **linear** side: settled speeds still read
high and vary between runs (`RT` full measured 1 342 in one flight and 1 821 in
another, against `MaximumVelocity` 1 200), which is what a craft being shoved around
in a firefight looks like. The HUD, meanwhile, reads exactly `CruisingVelocity` at
neutral throttle. A clean linear measurement wants a quiet corner of a map, not this
one — until then, no cause is claimed.

**Roll, re-measured with proper settles, confirms the axis difference**: 129.5 at
110 /s and 149.3 at 1 722 /s — no speed dependence, both near `AV_Roll_Max` **125**,
where pitch moved 75 → 40 across the same range. The mid-throttle pitch point does
not discriminate lerp-versus-switch, because at half throttle the craft only reached
388 /s, where an interpolation would predict ~66 °/s against a measured 73.7 — inside
this probe's noise.

Raw samples: [`captures/turn-law-settled.csv`](captures/turn-law-settled.csv).

---

# The linear discrepancy is real, and it is between WORLD units and DISPLAYED speed

Last section removed the "time base" explanation by showing settled *angular* rates
match the definition exactly. That leaves the linear side, and the HUD settles it:
screenshotting the speed readout at each throttle step, next to the position-derived
measurement of the same moment
([`captures/throttle-curve-hud.csv`](captures/throttle-curve-hud.csv)):

| `RT` | HUD readout | position-derived | ratio |
|---|---|---|---|
| 0.00 | **350** (= `CruisingVelocity`) | ~447 | 1.28 |
| 0.25 | **507** | ~652 | 1.29 |
| 0.75 | **963** | ~1 141 | 1.19 |

Two things follow.

* **The HUD speaks the definition's language.** It reads exactly `CruisingVelocity`
  at neutral and climbs toward `MaximumVelocity` as the trigger is pressed — 963 at
  three-quarters, against the 987 the interpolation predicts. So the throttle law
  stated above is confirmed in the game's *own* units, independent of any position
  sampling.
* **World displacement runs ~1.2× the displayed speed.** Since the angular rates
  need no such factor, this is not a clock effect: it is a **unit difference between
  the position triple and the velocity fields**. A reimplementation that places
  entities in world coordinates and moves them at `MaximumVelocity` will be ~20 %
  slow.

🟡 The ratio is 1.19–1.29 across the three points rather than a clean constant, and
every sample was taken in a firefight where the craft is also being pushed. Pinning
it exactly wants a quiet map or a scripted straight run; the *existence and rough
size* of the factor is what these measurements support.

**A methodological note worth keeping.** This line of work went: hold a trigger →
"three target speeds" → analogue interpolation → an unexplained 1.2× → "time base"
→ withdrawn → a *unit* difference confined to the linear side. Every step came from
one of two moves: **let a held input vary** (the trigger's analogue range), or
**bring in an independent oracle** (the HUD, the definitions, a longer settle). The
wrong turn — the time base — came from explaining a number instead of first
measuring the same quantity a second, cleaner way.

---

# ✅ The linear factor IS the clock — measured against the game's own mission timer

The cleanest possible version of the linear measurement: neutral throttle (HUD reads
exactly `CruisingVelocity` 350), sticks centred, **20 s of perfectly straight flight**
(`straightness = displacement/path = 1.000`, so no manoeuvring contaminates it) —
[`tools/re-capture/cruise_ratio.py`](../../tools/re-capture/cruise_ratio.py):

```
displacement 8 900 world units in 20.1 s  ->  443.6 /s
ratio vs the HUD's 350                    ->  1.267
```

And the game's **own clock**, read off the HUD across a wall-clock interval:

```
mission TIME 00:08.79 -> 00:46.97   =  38.18 s of game time
wall clock                          =  30.29 s
ratio                               =  1.260
```

**1.260 against 1.267 — the same number.** So the linear "discrepancy" was never a
unit difference: **the mission clock runs ~1.26× faster than wall time under this
emulator**, and a speed derived by dividing world displacement by *wall* seconds is
inflated by exactly that. World units and the displayed speed share one unit, and a
reimplementation should take `CruisingVelocity`/`MaximumVelocity` at face value —
per **game** second.

This supersedes the previous section's "world-unit vs displayed-speed difference".

❔ **One thing does not fit, and is left open.** The settled turn rates measured
74.9 and 41.1 °/s *in wall time* against `AV_PitchMinus_{Min,Max}` 75/40 — but if
game time runs 1.26× fast, a 75 °/game-second cap should have read ~94 °/wall-second.
Either that agreement was luck inside a noisy sample (the per-window rates spanned
61–96), or angular integration is frame-based where linear is time-based. The check:
re-measure pitch and convert wall→game seconds with the clock ratio, expecting ~94
if the clock explanation is universal.