MOAT-MATH-ABLATION - Math-only Projector Ablation

Weight-geometry replay over the EXP-57 fp32 trace (338 matrices). Two cadence regimes: S = per-step (global steps, PRIMARY, paper-comparable) and T = per-tick (optimizer ticks, extended-Δ sweep). Metric contract: weight-proj-metrics-v1. Ratio < 1 ⇒ projection beats holding the stale weights.

regime T / per-tick: FAST — sampled estimate (frac=0.001, seed=42, strips=1024; small tensors exact; 1727408 of 1543714304 scalars); verdict-grade = --fidelity full

1 — Ranked OOS projection accuracy (DELIVERABLE A)

All base methods + #49 self-correcting arms ranked by out-of-sample projection accuracy (pred_evr_pooled, higher is better) at the per-tick operating point Δ=5, h=10 (GLOBAL median row). Best-fitting method: armD_W2_ramp. Ratio < 1 ⇒ beats holding the stale weights; state = # scalars ANCHOR carries.

rank	method / arm	knobs	pred_evr_pooled ↑	op ratio ↓	h_safe	state
1	armD_W2_ramp (arm) ★	rolling_ls_k K=3 W=2 gain=linear_ramp	0.0000	0.9999	10	1
2	hold_stale	fixed	0.0000	1.0000	0	—
3	armB_rlsK3 (arm)	rolling_ls_k K=3	-0.0000	1.0000	10	2
4	armA_rlsK3 (arm)	rolling_ls_k K=3	-0.0002	1.0000	10	1
5	adaptive_second_order	rolling_ls	-0.0003	1.0001	5	2
6	full_confirm (arm)	rolling_ls	-0.0004	1.0001	5	1
7	adaptive_linear	rolling_ls	-0.0006	1.0002	5	1
8	armC_mom0.9 (arm)	ewma_momentum β=0.9	-0.0009	1.0003	5	2
9	armC_ewma0.9 (arm)	ewma	-0.0010	1.0004	5	1
10	damped_linear	fixed	-0.0012	1.0000	2	—
11	naive_linear	fixed	-1.9175	1.7273	0	—
12	damped_second_order	fixed	-2.0476	1.7466	0	—
13	naive_second_order	fixed	-16.7523	4.2236	0	—

State-cost vs gain — each arm vs the fixed #47 damped-linear bar (Δratio < 0 AND Δpred_evr > 0 = a real gain that must justify the arm's extra ANCHOR state).

method / arm	knobs	state (#scalars)	pred_evr_pooled	op ratio	Δratio vs bar	Δpred_evr vs bar
damped_linear (#47 fixed bar)	fixed	—	-0.0012	1.0000	0.0000	0.0000
adaptive_linear (expanding incumbent)	rolling_ls	1	-0.0006	1.0002	0.0002	0.0005
armA_rlsK3	rolling_ls_k K=3	1	-0.0002	1.0000	-0.0000	0.0009
armB_rlsK3	rolling_ls_k K=3	2	-0.0000	1.0000	-0.0000	0.0011
armC_ewma0.9	ewma	1	-0.0010	1.0004	0.0004	0.0001
armC_mom0.9	ewma_momentum β=0.9	2	-0.0009	1.0003	0.0003	0.0003
armD_W2_ramp	rolling_ls_k K=3 W=2 gain=linear_ramp	1	0.0000	0.9999	-0.0001	0.0012
full_confirm	rolling_ls	1	-0.0004	1.0001	0.0001	0.0007

2 — Per-scalar linearity R² (the MUST metric; Wang et al. 2026)

Per-individual-scalar OLS R² of the weight value vs the training-step index (constant scalars excluded & counted). Anchors: nearest RL analog R1-Distill-Qwen-1.5B GRPO/DeepScaleR 0.845/0.794 (median/Pr>0.7); SFT Qwen2.5-1.5B+GSM8K 0.426/0.259. Regime S (per-step) is the paper-comparable cadence. A LOW R² is a valid, decisive finding.

regime	R² median	Pr(R²>0.7)	n_excluded_const
T / per-tick	0.485	0.295	0

3 — Projection accuracy vs horizon (both regimes)

4 — Δ-sensitivity (extended to Δ=40, per-tick) & λ-selection

5 — Absolute prediction accuracy (regression view)

Absolute accuracy (pred_evr_pooled, per-scalar pred_r2_scalar_*) is reported first for every arm including hold_stale (evr = 0 by definition). Every projector's pred_evr_pooled is at or below 0 at the primary op-point (per-tick Delta=5, h=10 = 200 global steps); it turns positive ONLY at the very short-Delta / short-h corner. This is the signature of a near-incoherent trajectory: there is little linear structure to fit beyond one 20-step tick.

Regime T / per-tick — operating cell (Δ=5, h=10)

method	pred_evr_pooled	pred R² median (scalar)	frac < 0	n	n_excluded
hold_stale	0.000	-0.935	0.912	1726415	993
naive_linear	-1.917	-0.995	0.978	1726415	993
naive_second_order	-16.752	-0.995	1.000	1726257	1151
adaptive_linear	-0.001	-0.995	0.999	1725329	2079
adaptive_second_order	-0.000	-0.995	0.998	1723986	3422
damped_linear	-0.001	-0.995	0.999	1725329	2079
damped_second_order	-2.048	-0.995	0.997	1724057	3351

EVR = 1 − Σ‖e‖²/Σ‖b‖² (global group); 0 = hold-stale, 1 = perfect; negative values clip to the axis.

6 — Adaptive coefficient trajectories (GLOBAL group, OOS per window)

adaptive_linear (per-tick)

adaptive_second_order (per-tick)

7 — Lane verdict (direction-agnostic)

Linearity R² [regime T / per-tick]: global median = 0.485, Pr(R²>0.7) = 0.295, excluded-constant scalars = 0. vs Wang anchors — RL analog 0.845/0.794, SFT-GSM8K 0.426/0.259.
Per-tick cell (Δ=5,h=10 ticks): OOS-damped ratio = 1.000, naive_linear = 1.727 (vs #45 naive reference 1.158).
Does wider Δ help? (per-tick extended sweep Δ∈[1, 2, 3, 5, 8, 10, 15, 20]): best_delta = 1 (min OOS-damped ratio 1.000 at operating h).
h_safe (per-tick, ticks): 2.
Predictor ladder (per-tick, Δ=5,h=10): hold_stale: ratio 1.000, EVR 0.000, pred R² -0.935; naive_linear: ratio 1.727, EVR -1.917, pred R² -0.995; naive_second_order: ratio 4.224, EVR -16.752, pred R² -0.995; adaptive_linear: ratio 1.000, EVR -0.001, pred R² -0.995, λ*=0.028, warmup 11; adaptive_second_order: ratio 1.000, EVR -0.000, pred R² -0.995, λ*=0.028/λc*=-0.006, warmup 12; damped_linear: ratio 1.000, EVR -0.001, pred R² -0.995, λ*=0.000, warmup 11; damped_second_order: ratio 1.747, EVR -2.048, pred R² -0.995, λ*=0.000, warmup 11. Best OOS arm: damped_linear (1.000).

A decisive negative result (damped never beats hold-stale; wider Δ never helps; ratio > 1 at long h; low R²) is a VALID finding — the lane PASSES by producing a correct, schema-verified, machine-readable answer either way.

Provenance

per-tick: fidelity=fast cadence=per-tick unit=tick n_ticks=50 band=70 n_rows=250096 lam_grid=11pts fingerprint=None panel=509f388a8c4ba855 gates={'hold_stale_identity': True, 'structure_partition': True, 'naive_linear_finite': True, 'bounds_honesty': True, 'oos_bookkeeping': True, 'aggregates_present': True, 'coverage_safety': True, 'operating_points': True, 'r2_well_defined': True, 'pred_evr_bounds': True, 'pred_scalar_present': True, 'fast_sampling_recorded': True}

EXP-61 - MOAT-MATH-ABLATION: Math-only projector ablation