RESULT 38: 2D update-field analysis — localized curl blob at beta=1e-2 (conservative near basins), EP/BP different-basin equal-depth (barrier 8.25); fieldviz.py

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_014FAPDWQ49M5Ye3NpTndTpn

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diff --git a/docs/campaign/CASCADE_ABLATION_PLAN.md b/docs/campaign/CASCADE_ABLATION_PLAN.md
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@@ -568,6 +568,36 @@ direction), not training-under-fault; wave-2 = co-training with faults injected
 
 
 
+### RESULT 38 (2026-07-17): 2D UPDATE-FIELD ANALYSIS (professor's suggestion) — the finite-beta field IS non-conservative, but only where training doesn't live; EP and BP reach DIFFERENT, equally deep basins.
+Method: fieldviz.py — full-model PCA plane over 27 snapshots of 5 runs (49% trajectory variance);
+13x13 grid; loss contours + three update fields (BP, EP@3e-4, EP@1e-2; 2-batch averaged;
+block-subspace projection); finite-difference curl; ride<->BP linear interpolation.
+| field | curl RMS |
+|---|---|
+| BP (conservative reference / noise floor) | 4.38e-6 |
+| EP @ 3e-4 (operating beta) | 4.41e-6  (= floor) |
+| EP @ 1e-2 | 1.37e-5  (3.1x floor) |
+Findings:
+1. NON-CONSERVATIVITY IS REAL AND LOCALIZED: at beta=1e-2 a coherent positive-curl blob appears
+   in the steep high-loss region (large states/gradients -> large bias field c(theta)); BP and
+   EP@3e-4 show pure noise. NEAR THE BASINS — where training actually operates — the EP field
+   stays conservative-within-noise even at 10x operating beta. Same-batch BP control shows no
+   blob => EP-specific, not data noise.
+2. Two-level echo for the dynamics paper: parameter-space curl concentrates exactly where the
+   loop-gain hazard lives (big sigma*||J||) — the state-space non-conservativity story reappears
+   one level up, scaled by beta. Training under finite-beta EP = a mildly non-gradient flow that
+   is gradient-like precisely in the region the governor keeps it in.
+3. DIFFERENT BASINS, EQUAL DEPTH: ride(s50k) 1.4325 vs BP(s55k) 1.4156 with an 8.25 barrier
+   (~random-level) between them — same seed/init, methods branch at step 0 and land in linearly
+   disconnected but equally good minima. The EP family (plain/ride/cent) clusters in one plane
+   region, BP in another.
+   CONTROL PENDING: BP-s1 <-> BP-s2 interpolation (does BP-vs-BP also barrier? decides whether
+   different-basin is EP-specific or generic) — runs when the seed chain delivers bp_s2.
+   Cheap follow-ups queued: EP<->EP interp (expected flat); curl-vs-beta scaling curve at a
+   near-basin point and a steep point (the quantitative bridge figure for aep-dynamics).
+Caveats: 2D slice (49% variance); curl from 2-batch-averaged fields on a 13x13 grid; barrier
+statement is linear-path only (no permutation alignment attempted).
+
 ### RESULT 37 (2026-07-17): RIDE FAILS AT 72M — weight poisoning via permissive guards during beta surfing; crown-3 relaunched as PLAIN from the clean ckpt.
 Timeline (fw72m_ride): healthy surf to ~48k; at 72M the governor never found the 42M-style smooth
 hover — beta banged between 1.5e-4 and 1.85e-2 (rho-hat is noisy/laggy at this scale). ~54k: