# RESULT — 081 operator-emergence: WHEN do the causal operators install in training?

**Tier: T3-exploratory. DONE + SELF-HARVESTED** (tmux `c100_081`, CPU-only CVD="", box run
720s / 12 min, DONE 05:59:39Z, **14,688 decode-steering points, det_ok=True**). Closes **BLOCK J**
(078–081). **Ladder-TAE** study — the ladder's OWN per-checkpoint encoder+decoder (NOT SONAR;
same caveat as 073/074/075/077). 080 showed the operators are causal in the FINAL model (SONAR);
this tracks operator quality across ladder rung-A milestones. 18 ckpts (A_D s0/s1 + A_P90 s0/s1),
3 operators (negation, tense, vertical). Rule-based synonym-aware attribute readout (080), relaxed
garble detector for the noisier ladder decoder (val_f1≈0.65–0.71). Analysis local.

## ★ Headline — a CORRELATIONAL / CAUSAL DISSOCIATION, gated by the DECODER, not the direction

The linear operator DIRECTIONS for lexical (negation) and marker (vertical) operators **already
exist in the UNTRAINED encoder** (cos(v, held-out offset) = 0.93 / 0.999 at step 0, diff_align
0.84 / 0.995) — they are properties of token-embedding geometry + mean-pooling, not learned. What
TRAINING installs is **CAUSAL usability**: steering `z_base + v` only installs the attribute once
the **DECODER** becomes competent. So causal steerability **LAGS** the correlational offset, and it
appears **EARLY and ABRUPTLY** — the moment the decoder comes online (step 2000–4000, ~5–11% of
budget) — not gradually mid-training like the surface-order code (073). Operators install in a
**strict ORDER**: negation (step 2000) → vertical marker (4000) → **tense NEVER** (causally
non-installable at any checkpoint despite a competent decoder + a weak direction). Tense is exactly
078's weakly-linearizing operator (diff_align 0.58) — a clean causal confirmation of 078's hierarchy.

## Four sub-questions answered

1. **Early or late?** EARLY, and **decoder-onset-gated**. Negation causal-steering crosses 0.5 at
   step **2000 (5.4% budget)**, vertical at **4000 (10.7%)** — as soon as the decoder can render
   sentences at all (ceiling jumps 0→~1 over the same window). This is EARLIER than 073's
   surface-order code (mid, ~11–74%). The operators are **not** a slow mid-training construction;
   the direction is free from init and switches on causally with the decoder.
2. **Abrupt or gradual?** ABRUPT (step function at decoder onset). Negation causal succ 0.0→0.975
   between step 0 and 2000; vertical 0.0→0.875 by 4000. The correlational side is **flat** (already
   maxed at step 0 for negation/vertical) — there is no gradual emergence curve on the geometry side
   at all. Only tense's (weak) direction rises gradually (0.48→0.52 A_D / →0.73 A_P90) and never
   becomes causal.
3. **Same step or an order?** ORDER: **negation (2000) < vertical (4000) < tense (never)**.
   Consistent across both seeds and both arms (A_D negation 2000/2000, vertical 4000/6000; A_P90
   negation 2000, vertical 4000–26000 — marker decoder lags in the paraphrase arm). Tense's α=1
   success never exceeds 0.15 in any checkpoint of any arm.
4. **Does causal LAG correlational?** YES, decisively, for negation & vertical — the direction is at
   ceiling from **step 0** (corr_cross = 0) while causal steerability crosses only at 2000/4000
   (gated by decoder competence). The direction exists long before it is causally usable. (For tense
   the pattern is different: the direction is weak/late AND never causal — it doesn't "lag", it
   fails.)

## Trajectory — A_D_s0 (primary DAE arm)  [step | ceiling · succ_α1 · causal_norm | corr(cos_v_heldout) · diff_align | rand · invert]

| step | %bud | op | ceiling | **succ_α1** | causal_norm | **cos_v_heldout** | diff_align | rand_α1 | invert_α−1 |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | negation | 0.00 | 0.00 | — | **0.925** | 0.837 | 0.00 | 1.00 |
| 0 | 0 | tense | 0.00 | 0.00 | — | 0.481 | 0.411 | 0.00 | 1.00 |
| 0 | 0 | vertical | 0.00 | 0.00 | — | **0.999** | 0.995 | 0.00 | 0.00 |
| 2000 | 5% | negation | 0.95 | **0.975** | 1.03 | 0.905 | 0.834 | 0.05 | 1.00 |
| 2000 | 5% | tense | 0.325 | 0.15 | 0.46 | 0.278 | 0.503 | 0.00 | 0.94 |
| 2000 | 5% | vertical | 0.00 | 0.00 | — | 0.999 | 0.994 | 0.00 | 0.00 |
| 4000 | 11% | negation | 1.00 | **1.00** | 1.00 | 0.901 | 0.847 | 0.00 | 1.00 |
| 4000 | 11% | tense | 0.75 | 0.025 | 0.03 | 0.350 | 0.554 | 0.00 | 0.50 |
| 4000 | 11% | vertical | 0.875 | **0.875** | 1.00 | 0.997 | 0.990 | 0.00 | 0.75 |
| 37360 | 100% | negation | 0.90 | **0.975** | 1.08 | 0.905 | 0.840 | 0.00 | 1.00 |
| 37360 | 100% | tense | 0.95 | **0.05** | 0.05 | 0.522 | 0.582 | 0.00 | 0.19 |
| 37360 | 100% | vertical | 0.625 | **0.75** | 1.20 | 0.998 | 0.994 | 0.00 | 0.75 |

**The tense cell at final is the key dissociation:** ceiling 0.95 (decoder CAN render past tense) +
weak direction (corr 0.52) but causal succ 0.05 — the linear tense offset is **causally
non-installable** even at convergence. A_P90's denser tail confirms: at step 26000 (74%) negation &
vertical are fully installed (succ 1.00, ceiling 1.00) while tense sits at succ 0.125 (ceiling 0.95).

**Seed/arm consistency:** negation causal-cross 2000 in ALL four (s0/s1 × A_D/A_P90); vertical
4000/6000 (A_D) and 4000/26000 (A_P90 — marker decoder lags there); tense never in any. cos_v_heldout
at init: negation 0.925, vertical 0.999 both seeds. Specificity @α=1 off-target flip 0.00–0.125;
random-null α=1 = 0.00; cos(v_rand,v)≈0 everywhere; off_norm 2–4 vs ‖z_base‖≈9 (lesson 3, flat).

## Gates
| gate | outcome |
|---|---|
| **G_posctrl** (final = 080 in-regime) | **PASS** — A_D_s0/final mean succ_α1 = 0.59 ≥0.5, mean ceiling 0.83 ≥0.5 (steering works in-regime for negation+vertical; tense drags but gate holds). |
| **G_negctrl** (untrained) | **SPLIT / informative** — causal side PASS (untrained mean succ_α1 = 0.00 ≤0.15) but correlational side **FAILS** (mean cos_v_heldout 0.80 ≰0.3): the lexical/marker directions are **present at initialization** (trivially, from embedding lookup + pooling). This "failed negative" **is** the finding. |
| **G_random** (per-ckpt null) | **PASS** — random-direction α=1 success 0.00 at final (and ~0 throughout). |
| **G_norm** (lesson 3) | **PASS** — ‖v‖ 2–4, ‖z_base‖≈9 finite/non-degenerate; untrained z not degenerate. |
| **G_det** | **PASS** — det_ok=True (recheck of first jobs matched, all 18 ckpts). |
| G_ceiling caveat | at untrained ceiling=0 → causal readout is **decoder-limited, not direction-absent**; the correlational metric (decoder-free) is the interpretable one there, and it shows directions pre-exist. |

## Predictions → Brier (frozen in PREREG_LITE.md)
| Pred | P | Outcome | Brier |
|---|---|---|---|
| P1 final positive control (mean succ_α1≥0.5 & ceiling≥0.5) | 0.80 | **TRUE** (0.59 / 0.83) | 0.040 |
| P2 untrained negative (causal≤0.15 AND cos_v_heldout≤0.3) | 0.85 | **FALSE** (causal 0.00 ✓ but corr 0.80 ✗ — directions exist at init) | 0.7225 |
| P3 emergence MID/LATE (mean causal-cross >4000) | 0.65 | **FALSE** (mean 3000 — EARLY, decoder-onset-gated) | 0.4225 |
| P4 causal LAGS correlational (≥2 ops corr-cross < causal-cross) | 0.60 | **TRUE** (negation & vertical: corr 0 << causal 2000/4000) | 0.160 |
| P5 operator ordering (vertical causal-cross ≤ tense) | 0.60 | **TRUE** (vertical 4000 ≤ tense never) | 0.160 |
| P6 random null inert at final (≤0.15) | 0.85 | **TRUE** (0.00) | 0.0225 |

**Mean Brier = 0.2546** (4/6 hits). The two misses are the science: **P2** — I predicted the
untrained encoder would be a clean negative on BOTH axes; instead the correlational directions are
present at init (a random encoder already linearly separates negation/vertical because they are
single-token edits under mean-pooling), so the only thing training adds for these operators is the
DECODER that makes them causal. **P3** — I imported 073's "mid-training" prior; operators actually
install EARLY (decoder-onset-gated, ~5–11%), earlier than the surface-order code, because the
direction is free and only the readout must be learned.

## BLOCK J SYNTHESIS (078–081): the operator algebra of the TAE latent
078 mapped an operator ZOO: grammatical (negation/tense/number/question) and relational-marker
(vertical/temporal/containment/…) attributes **linearize** as fixed offset directions, while
argument-structure swaps hit a **binding wall** (non-linear) — a linearization *hierarchy* with
markers cleanest and tense weakest. 079 showed the linearizing operators **compose additively and
commutatively**, across families (cos(vA+vB,vAB)≈0.98, order-invariant), with the control failing in
decode not cosine. 080 proved the directions are **CAUSAL** in the final model: `z+α·v` installs the
attribute dose-responsively, perfectly specifically, and invertibly, while a norm-matched random push
is inert. 081 asks WHEN they install and finds the operator algebra is **built on two separable
substrates**: the offset DIRECTION is nearly free (present at initialization for lexical/marker ops,
a token-embedding-geometry fact), whereas CAUSAL installability is learned and switches on **early and
abruptly with the decoder** (step 2000–4000, ~5–11% budget), in the order **negation → marker →
tense**, with tense **never** becoming causally installable — the same operator 078 flagged as barely
linear. So the "linear operator" is really a *decoder capability*: the encoder hands over a
consistent direction for free, and training's job is to make the decoder read that direction as a
transformation. The binding wall (078) and the tense causal-null (081) are the same boundary seen in
geometry and in time.

## Limitations / what did NOT run
- **Ladder-TAE, not SONAR** — the ladder decoder is noisy (val_f1 0.65–0.71; "TM"/"he" artifacts);
  content words survive so the rule-based readout works, but absolute causal-success ceilings are
  lower than 080's SONAR numbers. Ceiling per checkpoint is the honest normalizer.
- The **untrained "direction exists"** result is for single-token-edit operators under mean-pooling;
  it is trivially true (embedding lookup difference) and should NOT be read as "the model already
  knows negation" — it knows the *token shift*, and only the trained decoder makes it a causal
  transform. This is the correct reading of the failed correlational negative control.
- Milestone grid: A_D is coarse (0/2000/4000/final; big 11%→100% gap); the EARLY resolution
  (negation@2000, vertical@4000) is well-sampled, the tail is filled by A_P90 (26000/final). A finer
  log-spaced A_D retrain would sharpen the abruptness claim.
- No codex judge (rule-based PRIMARY; validity gated by per-ckpt ceiling + α0 fabrication 0.00 +
  det True). 3 operators; number/question/swaps not tracked (078 covers the zoo).

## Follow-up worth funding? **Y (modest).**
The sharp result — **causal installability is decoder-gated and tense never installs** — invites: (1)
does a stronger decoder (SONAR-scale ladder or more dec_layers) rescue tense's causal steerability,
or is tense's linear offset fundamentally non-causal (probing the 078 binding wall from the time
axis)? (2) a finer log-spaced A_D milestone retrain to pin the decoder-onset step exactly and test
whether negation/vertical install at the SAME decoder-competence threshold (ceiling≈0.5) regardless
of operator — i.e. is the ordering just each operator's decoder-render threshold?