# RESULT — Campaign100 035 operator-curvature

**Tier: T3-exploratory.** tmux `c100_035`, phys GPU0 (CVD=0→phys0), folder `035-operator-curvature`.
Decode 98 s (2400 jobs, det_ok=True), LOCAL codex judge 1200 records, geometry local (numpy, BLAS≤8).
Self-harvested in-session. GPU0 freed to 35 MiB, session exited clean, box dir 22 MB.

## ★ Headline
**The offset operators (033/034) are a GEOMETRICALLY CURVED field — but the curvature does NOT
break decode-application, so a single global offset is "good enough."** Re-fitting each transform's
offset WITHIN manifold regions (4 length levels L0..L3; robustness = k-means k=6) shows the regional
offset DIRECTION rotates **20–25° across regions**, versus a within-region bootstrap noise band of
only **~3°** — a **curvature ratio of 6–8×** (k-means: 4.5–6×). The global offset sits near the
mid-length regions; the SHORT (L0) and LONG (L3) extremes deviate most from it (~15–19°), a clean
U-shaped warp along the length axis (031's strongest local-ID axis). **Yet** on held-out sentences a
region's own offset applies the transform NO better than the global offset (Δ success ≈ +0.8 pts),
because these transforms are near-ceiling and robust: a 15–20° direction error still greedy-decodes to
the correct transformed sentence. So: **operators ARE curved (a global fit is a chord, not the true
field), but the curvature is below the threshold that would change the decoded output at this scale.**

## Design (recap)
Controlled templated (before, after) pairs; all 5 transforms {tense, number, negation, question,
compose=number∘tense} applied to the SAME present-3sg before-set. Manifold spread via a controlled
LENGTH gradient (TAIL L0..L3 = 0..3 locative PPs, tense-neutral, ride along unchanged). Regions =
4 length levels (primary) + k-means k=6 on z_before (secondary). off_T re-fit WITHIN each region on
its TRAIN pairs (disjoint vocab from test). n_train = 120/region/transform (all regions ≥ 25 → none
dropped). Bootstrap (1000×) resamples region-train diffs → within-region offset-DIRECTION band.

## Arm 1 — geometry (CPU)
**Curvature = mean pairwise angle between the 4 regional (length) offsets vs within-region band:**

| transform | cross-region angle | within-region band | ratio | curved? |
|-----------|-------------------:|-------------------:|------:|:-------:|
| tense     | 21.2° | 3.4° | 6.2× | YES |
| number    | 25.2° | 3.7° | 6.7× | YES |
| negation  | 25.4° | 3.0° | 8.5× | YES |
| question  | 23.7° | 2.8° | 8.5× | YES |
| compose   | 23.5° | 3.3° | 7.1× | YES |

k-means (k=6) reproduces it: cross-region 20–25° vs band 3.7–5.0° (ratio 4.5–5.9×). **Robust to
partition scheme.** Angle of each length-region offset to the GLOBAL offset (tense/number/negation/
question/compose): L0 14–20° · L1 10–13° · L2 9–13° · L3 19–20° — **U-shaped: the global offset is a
chord anchored near the middle lengths; both length extremes are mis-served.** Global-offset
`diff_align` (operator coherence) reproduces 033/034: tense 0.81 · number 0.78 · negation 0.83 ·
question 0.85 · compose 0.81 (033 had ~0.84) — G_repro PASS.

## Arm 1 — decode validation (GPU decode + neutral codex judge, 033 rubric verbatim)
Region-own offset vs global offset applied to held-out test pairs, transformation-success
(applied yes=1/partial=.5/no=0), n=120/transform each arm:

| transform | success global | success region-own | Δ |
|-----------|---------------:|-------------------:|--:|
| tense     | 0.996 | 1.000 | +0.004 |
| number    | 0.996 | 1.000 | +0.004 |
| negation  | 0.971 | 1.000 | +0.029 |
| question  | 1.000 | 1.000 | +0.000 |
| compose   | 0.996 | 1.000 | +0.004 |

Mean Δ(region-own − global) = **+0.8 pts** (max negation +2.9 pts) → prediction (b) **FALSE** (< 5 pts).
The global offset already applies transforms at 97–100%; region-own reaches exactly 100% (a real but
tiny gain — the curvature never *breaks* application, it just occasionally shaves the last error). For a
majority of pairs the two offsets greedy-decode to the IDENTICAL sentence (hand-check: 8/8 exact, both
arms correct). 1200 judged, 0 parse-fails.

## Curvature vs local-ID (031 connection)
Per-region local-ID (MLE Levina-Bickel, k=15) increases monotonically with length — L0 8.67 < L1 9.65
< L2 10.65 < L3 11.32 — **freshly reproducing 031's length→ID monotonicity in a controlled set.** But
curvature (angle-to-global) is U-shaped in length while ID is monotone → **r(local-ID, angle-to-global)
= +0.14 (level) / −0.05 (k-means): curvature does NOT correlate linearly with local-ID** (pred c FALSE).
(A level-scheme r(ID, within-region diff_align)=+0.86 is NOT robust — only 4 points, flips to −0.14
under k-means; not claimed.) Curvature magnitude is ~equal across transform types (21–25°); there is
NO clean "simpler operator flatter" ordering — tense is flattest (21.2°) but negation is among the most
curved (25.4°), so pred (d) FALSE.

## Arm 2 — composition ROTATION (Paper C): SKIPPED (logged)
The composer is a ladder/SONAR ridge fit `z_AB≈W_A z_A+W_B z_B+b` (`code/tae_comp_math.py`). A stable
regional 1024-d ridge fit needs ~2×10⁴ pairs PER region (the program's p3 fit used 21,947 pairs
GLOBALLY); k separate large fits is NOT a "light" in-session validation. Composition curvature was
instead probed cheaply via the additive `compose` (number∘tense) offset in Arm 1 — it curves 23.5°
(ratio 7.1×), like the atomic operators, i.e. **composition inherits, not amplifies, the curvature.**

## Predictions (frozen in PREREG_LITE) → outcomes
- **(a)** P=0.55 curved beyond band → **TRUE** (ratio 6–8×, robust to scheme). Under-confident.
- **(b)** P=0.25 region-own beats global >5pts → **FALSE** (Δ=+0.8 pts).
- **(c)** P=0.45 curvature~local-ID |r|>0.3 → **FALSE** (r=+0.14; U-shape vs monotone ID).
- **(d)** P=0.50 negation flatter than tense → **FALSE** (neg 25.4° > tense 21.2°; ~equal curvature all types).
- **Brier = 0.1794** (contribs a .2025 / b .0625 / c .2025 / d .25 — under-confident on (a): smoke + 033/034
  priors + the 6–8× ratio justified >0.55; (d) was a coin-flip on a small ~4° gap).

## Gates
- **G_repro** PASS (global diff_align 0.78–0.85 ≈ 033/034 ~0.84; decode success near-ceiling).
- **G_power** PASS (n_train=120/region ≥ 25; 0 regions dropped).
- **G_boot** PASS (1000× within-region direction band computed, ~3°).
- **G_parse** PASS (0 parse-fails / 1200; hand-check 8/8).
- **G_det** PASS (decode determinism True).
- **G_norm** PASS (z-norm flat: before/after ~0.19 both, per offsets.json).

## Limitations
Templated short controlled cloud (not full-manifold domain spread); curvature measured along a LENGTH
gradient — semantic-region curvature (k-means) agrees but is within the same register. 4 length regions
= low power for the local-ID correlation. Greedy decode only. Curvature is a DIRECTION rotation of a
small-norm offset; whether it matters is scale-dependent (here it does not change decode output).

## Follow-up worth funding? **Y (narrow).**
The operator field is provably curved (a global offset mis-serves length extremes by ~18°) yet
application-robust at this scale — so curvature matters for *precision* steering / larger edits, not
for coarse transforms. Worth: (1) test whether curvature breaks application at LOWER cosine budgets or
for the HALF-operators (voice arg-swap, 033) where headroom exists; (2) a length-conditioned (curved)
offset vs global for long-text steering (feeds 080); (3) parallel-transport the offset along the
manifold instead of a single global mean.