Knowing how much a causal predictor could improve need not reveal the gain of the repair actually learned. We quantify this gap in a scalar Gaussian causal experiment with known intervention geometry: auxiliary data identify effect magnitude up to bounded contamination, while diagnostics identify direction. The target is the squared-loss gain of the realized trained repair relative to a fitted reference. Jointly optimizing the learner and assessor under uniform learning MSE η avoids the trivial solution of making no repair. At the usual 1/k learning scale, every feasible learner incurs a k−2 assessment floor, even when oracle potential is estimable at a faster rate. In the magnitude-rich regime, we characterize a sharp leading-log frontier: the assessment exponent is minℓk,2kηk/U to first relative order, where ℓk=log(1/(k2Ek)) and Ek is auxiliary precision. A diagnostic-abstention rule attains this exponent with unknown nuisance parameters. We also bound the critical allowance window and transfer the frontier to adaptive sampling by exact Gaussian simulation. Finite-grid experiments distinguish sign-tail suppression from total MSE and expose conservative finite-budget behavior. The result isolates how the assessment target changes information requirements in this experiment; it is not a general causal identifiability claim.
Figures & tables
Figure 1: Leading-log frontier, normalized by ℓk . The admissible assessment exponent saturates at auxiliary precision Ek . The curve depicts Theorem 2 , not an exact finite-sample risk. Theorem 3 shifts the first-order center Uℓk/2 to zc=Uℓk/(2−ℓk/k) . That correction can exceed the stated allowance window even when ℓk=o(k) ; it is not discarded in the refined result.
d(Q)=k8V∗logϵη8Q,T(Q)=max{0,Q−d(Q)}.
Algorithm 1: The same declared rule attains the leading exponent and the critical-window upper bound, with eventual uniform learning feasibility in their respective regimes. No unknown effect, variance, or sign is an algorithm input.
Budgets (m,n,k)
Population H
Fitted oracle A
Learned gain Ga
Decision class
Target only
Target only
Fixed clipped mean
(t,t2,t2)
t−1
t−2
t−2
(t3,t3,t)
t−3
t−3
t−2
Table 1: Same data, explicitly different decision classes. Clean data ( Δ=0 ); entries are assessment-MSE orders. The first two columns optimize only assessment (and policy if adaptive); the last fixes both the direct design and the clipped-mean learner. Joint learner selection is the separate problem in Theorem 1 .
Figure 2: Controlled mechanism checks. Left: identical report Q , different targets, at m=n=k3 , h=.3 ; bars are ±2 Monte Carlo SEs. Right: integrated wrong-sign MSE at k=224 , h2=1.1η>0 , and z=kη . The fixed-slack pair isolates the diagnostic gate; the fast schedule is Algorithm 1 . Tail suppression need not materially reduce total MSE.
Suite
Comparison
Cells
P90
Max.
Target scale
Corrected / Q
16/16
0.677
0.681
Diagnostic gate
Diagnostic / magnitude †
90/120
1
1
Robustness
Projected / raw fallback
16/24
1
1
Residual strength
Joint / mean
27/27
1.14
1.9
Fitted oracle
Selected / clipped unbiased
24/24
3.67
39.4
Pooled budgets
Joint / mean
18/18
1.59
1.83
Table 2: All seven original suites: cellwise total-MSE ratios (first report divided by second). P90 and maximum summarize the stated grid, not a minimax supremum. The Cells column gives positive-denominator cells / all cells; omitted cells have both observed MSEs zero. † Changes the learner and its target; every other row compares assessors of the same target.
Learner + report
Clean, rich
Stress
Lmax
R90
Lmax
R90
Clipped mean + Stein
0.174
6.12×103
0.178
8.69
Clipped mean + Q−V/k
0.174
17.6
0.178
8.27
Magnitude gate + IQ
0.5
7.6
10.7
16.5
Diagnostic gate + IQ
0.5
7.6
3.27
14.5
Fallback + projected report
0.122
21.2
0.291
8.37
Table 3: Finite-budget pipelines at kη=4zcclean : 192 clean, magnitude-rich cells and 768 stress cells (balanced, no-cheap, contamination-bound-only, and contaminated). Lmax is the grid maximum of learning MSE divided by η ; R90 is P90 of assessment MSE divided by EN+k−2 . Each pipeline assesses its own realized repair; only the first two rows share a learner. Values are estimates, not uniform feasibility or dominance certificates.
Appendix figures & tables10 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Analytic center-shift/window ratio for ℓ=k0.8 and U=2.5 . The condition ℓ=o(k) does not make the correction smaller than the allowance window.
k
h
k3RA(Q)
k2RG(Q)
k2RG(corr.)
k2RG(Stein)
16
.3
15.4(0.0893)
14.2(0.195)
9.67(0.149)
46.8(0.755)
16
.25/k
8.08(0.081)
13.9(0.176)
8.44(0.132)
32.4(0.505)
32
.3
17.4(0.11)
13.6(0.19)
9.21(0.145)
58.1(0.866)
32
.25/k
8.08(0.0804)
13.4(0.183)
8.36(0.141)
33(0.544)
64
.3
17.4(0.11)
13.3(0.191)
8.92(0.146)
83.1(1.18)
64
.25/k
8.15(0.0802)
12.9(0.184)
8.18(0.142)
32.7(0.566)
Appendix
Table 4: Full target-scale suite. Entries are scaled MSE (one Monte Carlo SE). The variance report is Q ; the corrected report is Q−V/k .
z
Gate
−log(k2RW)/z
E(a−θ)2/η
E(g−G)2/Ek
64
Magnitude
0.1413
6.60×10−9(7.54×10−13)
998(0.114)
64
Diagnostic, fixed
0.1413
6.60×10−9(7.54×10−13)
998(0.114)
64
Diagnostic, fast
0.1413
6.60×10−9(7.54×10−13)
998(0.114)
256
Magnitude
0.2334
3.28×10−12(4.59×10−14)
7.93(0.111)
256
Diagnostic, fixed
0.4146
3.28×10−12(4.59×10−14)
7.93(0.111)
256
Diagnostic, fast
0.4373
3.28×10−12(4.59×10−14)
7.93(0.111)
Appendix
Table 5: Conditional integration at k=224 , h2=1.1η , positive h . The exponent uses the log-domain mean of the wrong-sign MSE. Remaining columns use nuisance Monte Carlo means (one SE).
k
h
Δ
ζ
Joint
Mean
Variance
16
0.0
0.0
0.0
0.02(2.44×10−4)
0.072(0.00102)
0.0201(2.55×10−4)
16
0.0
0.1
0.0
0.0212(2.21×10−4)
0.0724(0.00104)
0.0206(2.57×10−4)
16
0.0
0.1
0.1
0.046(3.63×10−4)
0.0764(0.00107)
0.0526(4.33×10−4)
16
0.3
0.0
0.0
0.0172(1.47×10−4)
0.0811(0.0011)
0.0198(1.51×10−4)
16
0.3
0.1
0.0
0.0149(1.44×10−4)
0.0812(0.0011)
0.0197(1.54×10−4)
16
0.3
0.1
0.1
0.0297(2.57×10−4)
0.0876(0.00117)
0.0354(2.82×10−4)
Appendix
Table 6: Residual-strength suite, m=512 , n=4096 , p=1.5 . MSE (one SE); all three procedures share the same target and data. The mean method clips the corrected coefficient before applying the power.
m
k
h
Δ
Selected
Unbiased, clipped
Unbiased, raw
128
32
0.0
0.0
0.0282(4.18×10−4)
0.00968(3.00×10−4)
0.012(2.97×10−4)
128
32
0.0
0.1
0.0527(6.27×10−4)
0.011(3.41×10−4)
0.0135(3.39×10−4)
128
256
0.0
0.0
6.99×10−4(1.59×10−5)
5.95×10−4(1.41×10−5)
6.36×10−4(1.41×10−5)
128
256
0.0
0.1
7.24×10−4(1.58×10−5)
6.09×10−4(1.39×10−5)
6.55×10−4(1.39×10−5)
128
4096
0.0
0.0
3.14×10−5(6.88×10−7)
3.11×10−5(6.79×10−7)
3.11×10−5(6.79×10−7)
128
4096
0.0
0.1
3.25×10−5(6.77×10−7)
3.22×10−5(6.66×10−7)
3.23×10−5(6.66×10−7)
Appendix
Table 7: Fitted oracle-potential suite, n=4096 . MSE (one SE); Δ=ζ . Count selection chooses between the variance and mean plug-ins using EN≤1/k . The competing unbiased estimator is shown both raw and clipped.
n
k
h
Δ
Pooled joint
Mean
0
64
0.0
0.0
0.03(5.55×10−4)
0.0164(2.59×10−4)
0
64
0.0
0.1
0.0291(5.04×10−4)
0.0171(2.75×10−4)
0
64
0.3
0.0
0.0401(6.02×10−4)
0.031(4.42×10−4)
0
64
0.3
0.1
0.0427(6.05×10−4)
0.0325(4.67×10−4)
0
64
0.7
0.0
0.0615(4.57×10−4)
0.0653(4.84×10−4)
0
64
0.7
0.1
0.0615(4.58×10−4)
0.0671(4.90×10−4)
Appendix
Table 8: Pooled-budget suite, m=128 , p=1.5 , Δ=ζ . MSE (one SE). The pooled joint estimator uses diagnostic residuals even at n=0 .
c
h
k
Gain plug-in
Stein
Stein bias
Prn(G<0)
0.3
0.0
16
0.193(0.00311)
0.133(0.00237)
−8.83×10−4(0.00182)
0.783(0.00206)
0.3
0.0
128
0.004(6.77×10−5)
0.003(5.39×10−5)
4.59×10−4(2.74×10−4)
0.506(0.0025)
0.3
0.0
512
4.39×10−4(7.47×10−6)
3.76×10−4(6.39×10−6)
9.63×10−5(9.70×10−5)
0.294(0.00228)
0.3
0.3
16
0.264(0.00447)
0.197(0.00357)
0.00208(0.00222)
0.43(0.00248)
0.3
0.3
128
0.00986(1.48×10−4)
0.00888(1.26×10−4)
−0.00116(4.71×10−4)
0.0883(0.00142)
0.3
0.3
512
0.00192(2.29×10−5)
0.00186(2.10×10−5)
−1.14×10−4(2.16×10−4)
0.0265(8.02×10−4)
Appendix
Table 9: Clipping-boundary suite, m=128 , n=64 . Both reports assess the same clipped-mean repair. MSE and mean bias (one SE); the last column is the observed negative-gain fraction, not a confidence guarantee.
k
h
ζ
Base learning
Fallback learning
Base gain
Fallback gain
65536
0
0.0
6.99×10−15(7.06×10−17)
6.99×10−15(7.06×10−17)
0(0)
0(0)
65536
.95t
0.0
0.0195(1.66×10−10)
0.0195(1.66×10−10)
0(0)
0(0)
65536
1.05t
0.0
6.13×10−13(6.09×10−15)
6.13×10−13(6.09×10−15)
0.0239(1.84×10−10)
0.0239(1.84×10−10)
65536
.3
0.0
1.71×10−13(1.72×10−15)
1.71×10−13(1.72×10−15)
0.09(3.56×10−10)
0.09(3.56×10−10)
65536
0
0.05
0.05(1.73×10−9)
0.05(1.73×10−9)
−0.05(1.73×10−9)
−0.05(1.73×10−9)
65536
0
0.2
0.2(1.86×10−9)
0.2(1.86×10−9)
−0.2(1.86×10−9)
−0.2(1.86×10−9)
Appendix
Table 10: Robustness suite: learning MSE and realized mean gain (one SE). Here t2=η/64 , m=n=k3 , and η=8192log(k)/k . The base learner is magnitude-only; fallback changes the learner.
k
h
ζ
Base report/ learner
Raw / fallback
Projected / fallback
Stein / mean
65536
0
0.0
0(0)
0(0)
0(0)
7.39×10−9(1.88×10−10)
65536
.95t
0.0
0(0)
0(0)
0(0)
2.43×10−6(2.51×10−8)
65536
1.05t
0.0
5.85×10−14(5.82×10−16)
5.85×10−14(5.82×10−16)
5.85×10−14(5.82×10−16)
2.89×10−6(2.90×10−8)
65536
.3
0.0
6.17×10−14(6.21×10−16)
6.17×10−14(6.21×10−16)
6.17×10−14(6.21×10−16)
1.15×10−5(1.14×10−7)
65536
0
0.05
0.01(6.87×10−10)
0.01(6.87×10−10)
0.00179(9.18×10−7)
7.68×10−9(1.99×10−10)
65536
0
0.2
0.16(2.87×10−9)
0.16(2.87×10−9)
0.00719(3.81×10−6)
9.18×10−9(2.56×10−10)
Appendix
Table 11: Robustness suite: MSE (one SE). Raw and projected reports share the same fallback-learner target. The Stein column assesses a different direct-mean learner and is a pipeline comparator, not a same-target dominance comparison.
z
ρ
P(T>0,IQ=1)
−log(k2RW)/z
R/Ek
R/Rmag
16
1
0
−0.126
6.13×104
1
16
2
0
−0.126
6.13×104
1
16
4
0
−0.126
6.13×104
1
16
16
0
−0.126
6.13×104
1
64
1
0
0.141
8.99
1
64
2
0
0.141
8.99
1
Appendix
Table 12: Variance-bound ablation at k=16384 , m=n=k3 , h2=1.1η , c=.6 , τ=2 , and fast slack. The rule uses ρV∗ with V∗=4 ; the data and allowance are unchanged within a cell. P(T>0,IQ=1) is the nuisance-draw fraction with a positive eligible threshold. The last column compares total MSE with the matched magnitude-only rule. Negative normalized exponents at small z are permitted.