# RESULT — 075 seed-atlas: how seed-stable is ladder-TAE anatomy?

**Tier: T3-exploratory. DONE + HARVESTED** (tmux `c100_075`, GPU TRAIN on CVD 0/1/2 =
phys 0/2/3; wave-1 s2/s3/s4 launched 16:35:21Z, wave-2 s5/s6/s7 17:49Z, eval + aggregate
DONE 19:12:57Z). **8 full-budget seeds** at the SAME small rung (rung-A, arm-D, d_z=256,
5e8 tokens, matched args): **6 newly-trained s2–s7** + REUSED anchors **A_D_s0/s1**. Per
seed: knee (`eval_067.py`), offset operators (`eval_operators.py`, 5 transforms, Procrustes
alignment), binding battery (`binding_battery.py`, stimuli_v2), reconstruction/z-stats.
**All 8 seeds VALID** (none excluded; every val_f1 > 0.5, tokf1_L8 > 0.5, step 37359±1 =
matched budget). **Mean Brier 0.1038; all 6 predictions TRUE.**

## Verdict (headline)

**Ladder-TAE anatomy is seed-STABLE up to a global ROTATION of the z-basis.** Every scalar
invariant is nearly seed-constant (val_f1 CV 0.5%, knee CV 2.5%, surfX CV 2.8%, ‖z‖ CV 1.2%;
anti-transfer universal across all 8). The offset OPERATORS point in essentially RANDOM
directions across seeds — raw cross-seed cosine ≈ 0 (median −0.012) — yet after a single
orthogonal Procrustes rotation their geometry snaps back into agreement (median aligned
cosine 0.83; negation 0.93, question 0.97). This is exactly the **044-style picture**:
**absolute direction NON-identifiability + shared RELATIVE geometry.** The "real" anatomy is
the rotation-invariant content (operator quality, mutual angles, all reconstruction/code
metrics); the only seed-idiosyncratic thing is the arbitrary orientation of the z-basis each
seed happens to pick.

## Per-invariant seed variance (n=8 seeds)

| seed | val_f1 | tokf1@62 | knee_tok | surfX | roleXflip | ‖z‖ | PR | within_lin |
|---|---|---|---|---|---|---|---|---|
| A_D_s0 | 0.6554 | 0.4526 | 20.97 | 0.7063 | 0.3021 | 9.352 | 219.25 | 0.664 |
| A_D_s1 | 0.6488 | 0.4590 | 21.34 | 0.6890 | 0.3164 | 9.188 | 217.66 | 0.636 |
| s2 | 0.6466 | 0.4585 | 21.98 | 0.6678 | 0.3369 | 9.102 | 219.76 | 0.632 |
| s3 | 0.6487 | 0.4513 | 22.23 | 0.7061 | 0.2994 | 9.227 | 217.92 | 0.658 |
| s4 | 0.6455 | 0.4510 | 20.61 | 0.7219 | 0.2783 | 9.239 | 216.52 | 0.695 |
| s5 | 0.6498 | 0.4551 | 21.24 | 0.7064 | 0.2996 | 9.023 | 213.14 | 0.644 |
| s6 | 0.6481 | 0.4547 | 21.13 | 0.6894 | 0.3157 | 9.281 | 215.54 | 0.627 |
| s7 | 0.6499 | 0.4580 | 21.77 | 0.6682 | 0.3307 | 9.122 | 218.27 | 0.624 |
| **CV** | **0.0046** | **0.0071** | **0.025** | **0.028** | — | **0.012** | — | — |

- **(d) Reconstruction:** CV(val_f1) = **0.0046**, all val_f1 ∈ [0.646, 0.655] ⊂ [0.60,0.70].
  CV(tokf1@62) = 0.0071. No seed collapses. → the single most seed-stable invariant.
- **(a) Knee:** mean 21.41 tok, CV = **0.025** (range 20.6–22.2). Capacity knee is
  seed-locked to within a token.
- **(c) Surface code + anti-transfer:** mean surfX 0.694, CV = **0.028**; **anti-transfer
  UNIVERSAL** — roleXflip ∈ [0.278, 0.337] in ALL 8 seeds, max 0.337 < 0.45 (role probe reads
  roles backwards on parity-flipped families in every seed). z_bag = 0.500 exactly, all seeds
  (lesson-4 stimuli-validity control passes everywhere).
- **z-norm:** CV(‖z‖) = 0.012 (lesson-3 norm check: finite, comparable across seeds).

## Operators — RAW vs PROCRUSTES-ALIGNED (the key result)

Reference seed = A_D_s0; cosines are the 7 other seeds' offset operators vs s0's, before
and after a single orthogonal Procrustes rotation fit on a DISJOINT anchor set (anchor hash
verified identical across seeds — no circularity, G4).

| transform | **raw** cos (median) | **aligned** cos (median) | diff_align (mean) | CV(diff_align) | z-success |
|---|---|---|---|---|---|
| tense | 0.015 | 0.709 | 0.507 | 0.033 | 0.923 |
| negation | −0.048 | **0.931** | 0.822 | 0.009 | 0.962 |
| number | 0.008 | 0.591 | 0.349 | 0.040 | 0.909 |
| question | −0.011 | **0.973** | 0.928 | 0.005 | 0.940 |
| voice | 0.016 | 0.845 | 0.736 | 0.023 | 0.901 |
| **clean median** (tense/neg/num/question) | **−0.012** | **0.829** | — | max **0.040** | — |

**Reading:** RAW operator directions are seed-random (median clean cosine −0.012 — each seed
learns its own rotated basis; absolute directions are NOT identifiable, exactly 044's atom
non-identifiability). One global rotation recovers a shared geometry (median clean **0.829 ≥
0.80**). Aligned recovery TRACKS operator quality: the two most linearly-coherent operators —
question (diff_align 0.93) and negation (0.82) — recover almost perfectly (0.97, 0.93), while
the weakest, number (diff_align 0.35), only partially recovers (0.59). So the shared relative
geometry is real and strong precisely where the operator is a real linear operator; the
"noisy" operators share less because there is less coherent signal to share (not because the
alignment fails). z-side operator success (cos(z_before+offset, z_after)) is high and
seed-invariant for all transforms (0.90–0.96, CV ≤ 0.016) — every seed learns applicable
operators of equal quality.

**Within-seed operator-angle matrix (rotation-invariant):** the four clean operators are
mutually near-ORTHOGONAL inside every seed (pairwise mean cosines 0.05–0.14, one −0.20 for
tense|question). The high pair-CVs (0.27–1.03) are an artifact of near-zero means (tiny
denominators), not instability — the qualitative structure "operators are ~orthogonal axes"
is itself seed-invariant.

## Invariant vs idiosyncratic split (P6 verdict: CONFIRMED)

| **SEED-INVARIANT** (rotation-invariant content — the "real" anatomy) | **SEED-IDIOSYNCRATIC** |
|---|---|
| Reconstruction fidelity (val_f1/tokf1, CV ≤ 0.007) | **Absolute operator basis orientation** (raw cross-seed cosine ≈ 0 — a free global rotation per seed) |
| Capacity knee location (CV 0.025) | |
| Surface-order code strength (surfX, CV 0.028) | |
| Anti-transfer signature (roleXflip < 0.45 universal) | |
| z-norm profile (CV 0.012), z_bag control (0) | |
| Operator quality: diff_align (CV ≤ 0.04) + z-success (CV ≤ 0.016) | |
| Procrustes-aligned operator geometry (median 0.83) + mutual-angle structure | |

**Anatomy is stable "up to a rotation of the z-basis."** Confirms and n=8-generalizes the
2-seed baseline the campaign has been limited by (067/073/048), and matches 044's
absolute-non-identifiability + shared-relative-geometry finding — now shown to hold for the
supervised offset OPERATORS (033/034), not just SAE atoms.

## Gates

| Gate | Outcome |
|---|---|
| G1 GPU foreign-free | **PASS** — 3 GPUs verified free at claim (35/15/15 MiB), in-script hard-guard armed. |
| G2 per-seed train completes | **PASS** — all 8 final.pt present, step 37359±1 (= s0's 37360 ±5% matched-budget assert), pct_budget ≥ 0.95, no DIVERGED. |
| G3 per-seed reconstruction sanity | **PASS** — every seed val_f1 > 0.5 (min 0.6455) and tokf1@L8 > 0.5 (min 0.808). 0 seeds excluded; anatomy stats on full n=8. |
| G4 operator pipeline valid | **PASS** — s0 diff_align finite, tense/negation reproduce linear operators (0.51/0.82 > 0.3); Procrustes anchor set DISJOINT from offset-fit vocab (anchor_hash_ok=true, no circularity); z_bag = 0.500 (lesson 4). |
| G5 norm profiles finite/comparable | **PASS** — mean ‖z‖ 9.0–9.4 all seeds, CV(‖z‖) 0.012. |

## Predictions → Brier (frozen in PREREG_LITE.md; verified against atlas_075.json)

| Pred | P | Frozen bar | Observed | Outcome | Brier |
|---|---|---|---|---|---|
| P1 reconstruction seed-stable | 0.85 | CV(val_f1) ≤ 0.05 AND all val_f1 ∈ [0.60,0.70] | CV 0.0046; all ∈ [0.646,0.655] | **TRUE** | 0.0225 |
| P2 surface-code stable + anti-transfer universal | 0.75 | CV(surfX) ≤ 0.10 AND roleXflip < 0.45 all 8 | CV 0.028; max roleXflip 0.337 | **TRUE** | 0.0625 |
| P3 knee seed-stable | 0.70 | CV(knee_tok) ≤ 0.15 | CV 0.025 | **TRUE** | 0.0900 |
| P4 operators ROTATED (raw non-identifiable) | 0.65 | median raw cross-seed cos(clean) < 0.5 | −0.012 (≈ random) | **TRUE** | 0.1225 |
| P5 geometry invariant up to rotation | 0.65 | median aligned cos(clean) ≥ 0.80 AND CV(diff_align) ≤ 0.15 | 0.829; max CV 0.040 | **TRUE** | 0.1225 |
| P6 split verdict (invariant vs idiosyncratic) | 0.55 | as stated above | matches exactly | **TRUE** | 0.2025 |

**Mean Brier = 0.1038** (self-scored value reproduced; 6/6 TRUE). The score is dominated by
honest under-confidence on the two rotation predictions (P4/P5 at 0.65, P6 at 0.55) — the
manager hedged the rotation-invariance story it was least sure of, and it landed cleanly, so
the residual Brier is the price of appropriate humility rather than of any miss. A calibrated
retro-verdict would raise P4 (raw cos ≈ 0 is about as decisive as this test gets) and P1.

## What did NOT run / limitations

- **Single point in the design space:** one rung (A), one arm (D, plain BART-DAE), one
  bottleneck (d_z=256). Whether rotation-invariance and the same invariant/idiosyncratic
  split hold at higher capacity or other arms is untested — this is a 1-cell seed atlas, not
  a rung × arm × dim atlas.
- **2 of 8 seeds pre-existing:** A_D_s0/s1 are reused campaign anchors (matched args); the 6
  new seeds s2–s7 are what makes n=8. Anchors and new seeds are indistinguishable in every
  metric, so no anchor-vs-new confound is visible, but the anchors were not re-trained here.
- **Ladder-TAE scale, not SONAR:** d_z=256 discrete-ish TAE; the operator battery is 033's
  before/after stimuli. The `number` operator is weak (diff_align 0.35) on this TAE — its
  partial aligned recovery (0.59) is the honest floor, not a failure of the method.
- **Operator cosines are all-vs-s0** (reference seed), not full pairwise; robust because s0
  is one arbitrary seed among 8 and metrics are flat, but a full n×n Procrustes fan would be
  marginally cleaner.
- **Angle-matrix pair-CVs are near-zero-mean artifacts** (reported, not over-read).
- Box heavy ckpts (s2–s7 final.pt, 6×31MB) DELETED post-harvest (atlas json IS the result);
  anchors in `ladder/runs/A_D_s{0,1}` untouched. Repo `out/`: atlas_075.json,
  results_knee_075.json, batt_*.json (8), BATT_*.md (8), run.log.

## T3-exploratory. Follow-up worth funding? **Y (modest)**

Two cheap extensions would sharpen this into a promotable claim: (1) **repeat the seed atlas
at one higher rung / larger d_z** (~6 GPU-h) to test whether "stable up to rotation" is a
property of this bottleneck or of ladder-TAEs generally — if the aligned-cosine recovery and
the invariant list survive a capacity change, the anatomy claim strengthens sharply; (2) the
**number-operator anomaly** (diff_align 0.35, aligned recovery only 0.59) is a clean handle
on "which linguistic operators are genuinely linear in z" — worth a targeted look at whether
number is non-linear/multi-directional rather than merely noisy. The core result — anatomy
is seed-invariant up to a global z-rotation, n=8 — is solid and directly retires the
"characterized on only 2 seeds" caveat flagged in 067/073/048.