Testing ensemble errors: a correlation floor, not a universal cutoff
Tested hypothesis · Mathematical and model findings · 30 October 2026
Incoprea Research · 30 October 2026
Evidence status: Supported mechanism; stronger threshold claim requires correction. Exact mathematics and synthetic simulation, not a benchmark of trained models.
Averaging predictions can remove individual noise while leaving a shared blind spot almost intact. That central idea in the posted ensemble-correlation hypothesis survives a precise test. Its stronger suggestion that benefit disappears above a universal error-correlation threshold does not follow. The distinction matters: a small improvement can be uneconomic without being mathematically zero.
What was tested
The original page makes three predictions: a correlation threshold eliminates ensemble gain; an accurate but correlated addition can lower ensemble reliability; and correlation predicts gain better than member accuracy. We tested the covariance mechanism and constructed a controlled example of the second prediction. We did not train classifiers, estimate real-world accuracy, or test the third comparative prediction.
The analysis plan was written and hashed before numerical outcomes were generated. It specifies unbiased real-valued prediction errors, equal weighting, unit individual variance and equal pairwise correlation ρ from 0 to 1. The outcome is mean squared error, not majority-vote classification accuracy. These assumptions are deliberately narrow enough to make an exact result possible.
The exact result
Let e₁,…,eₘ be zero-mean errors with Var(eᵢ)=1 and Cov(eᵢ,eⱼ)=ρ for distinct members. Summing the m diagonal and m(m−1) off-diagonal covariance terms gives
MSE(mean error) = ρ + (1−ρ)/m.
The limit as the number of members grows is ρ. Shared error therefore creates a floor. At ρ=0, 31 members have MSE 1/31, about 0.0323; at ρ=0.9, the same count has MSE about 0.9032. The ensemble remains much less effective in the second case despite identical individual error variance.
| Pairwise correlation | MSE with 3 members | MSE with 31 members | Infinite-member limit |
|---|---|---|---|
| 0 | 0.333333 | 0.032258 | 0 |
| 0.5 | 0.666667 | 0.516129 | 0.5 |
| 0.9 | 0.933333 | 0.903226 | 0.9 |
| 0.99 | 0.993333 | 0.990323 | 0.99 |
But the gain from member m+1 is exactly (1−ρ)/(m(m+1)). It is positive for every ρ<1. In this model there is no sharp, universal correlation cutoff below perfect correlation. An operational cutoff would need a declared cost, latency or minimum useful improvement.
A better member can still hurt an equal-weight average
Take two independent, zero-mean errors X and Y, each with variance 1. Their average has MSE 0.5. Add a predictor whose error is 0.9X. Its individual MSE is 0.81, better than either existing member. Average individual MSE improves from 1 to (1+1+0.81)/3=0.936667.
Nevertheless the new ensemble error is (1.9X+Y)/3, with MSE (1.9²+1)/9=0.512222: about 2.44% worse than before. The new member overweights an error direction already represented. This is an exact existence proof for harm under equal weighting, not a claim that correlated additions always hurt. In the same planned sweep, additions with errors 0.5X and 0.8X helped.
Computational check and prior work
For each of six correlation values and three ensemble sizes, the script generated 200,000 synthetic samples from eᵢ=√ρZ₀+√(1−ρ)Zᵢ, with independent standard Gaussian Z variables. All 18 estimated MSEs were within 2.38 Monte Carlo standard errors of their exact expectations. The 3.6 million ensemble samples check implementation; the covariance identity supplies the proof. No empirical observations enter this experiment.
The role of both member quality and correlation is established background, including Leo Breiman’s primary Random Forests paper. Our equations and counterexample are a bounded reconstruction of that general concern, not a claim of discovery.
What the finding supports
Measure error covariance and evaluate proposed members jointly. Individual accuracy alone can miss an equal-weight aggregation failure. Conversely, this test gives no basis for discarding an ensemble at an unspecified correlation threshold or ranking correlation above accuracy across tasks. Biases, unequal variances, negative correlations, learned weights, classification decisions and distribution shifts need separate treatment.
Reproduction uses Python 3’s standard library, seed 20261030, and the accompanying run_tests.py, PLAN.md, and results files. Original hypothesis: DeepSeek; model analysis, code and report: Codex. This is an internal computational check, not external peer review. The source hypothesis remains available unchanged through Incoprea Research.
Reproduce this result
The calculations below use Python 3 and its standard library. Save the code as reproduce.py in an empty folder and run python3 reproduce.py. It writes the result files there. These are declared mathematical or synthetic-model tests, not measurements of real people, buildings, devices or deployed systems. The plan was recorded locally before outcomes; it was not registered externally.
Show the complete runnable code
#!/usr/bin/env python3
"""Standalone reproducibility script. Synthetic/model results; no real-world measurements. Run with Python 3 standard library. Writes results.json and results.csv beside this file. Original source, assumptions and interpretation are in article.html and PLAN.md."""
import csv,itertools,json,math,random
from collections import OrderedDict
from pathlib import Path
ROOT=Path(__file__).resolve().parent
SEED=20261030
def write_csv(path,rows):
with path.open('w',newline='') as f:
w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
def ensemble():
rng=random.Random(SEED); exact=[]; samples=[]; n=200000
for rho in [0,.25,.5,.9,.99,1]:
for m in [1,3,9,31,101]:
exact.append(dict(rho=rho,m=m,mse=rho+(1-rho)/m,next_member_gain=(1-rho)/(m*(m+1))))
for m in [3,9,31]:
common=math.sqrt(rho);private=math.sqrt(1-rho);ss=ss2=0.
for _ in range(n):
z0=rng.gauss(0,1)
e=common*z0+private*sum(rng.gauss(0,1) for _ in range(m))/m
q=e*e;ss+=q;ss2+=q*q
mean=ss/n;target=rho+(1-rho)/m
se=math.sqrt((ss2/n-mean*mean)/(n-1))
samples.append(dict(rho=rho,m=m,n=n,observed_mse=mean,exact_mse=target,mc_se=se,standardized_difference=(mean-target)/se))
duplicate=[]
for b in [.5,.8,.9,1]:
duplicate.append(dict(b=b,new_member_mse=b*b,old_mean_member_mse=1,new_mean_member_mse=(2+b*b)/3,old_ensemble_mse=.5,new_ensemble_mse=((1+b)**2+1)/9))
d=dict(seed=SEED,kind='exact covariance law and synthetic Gaussian checks',exact=exact,monte_carlo=samples,duplicate_member=duplicate,harm_threshold_b=math.sqrt(3.5)-1)
(ROOT/'results.json').write_text(json.dumps(d,indent=2)+'\n');write_csv(ROOT/'results.csv',samples)
assert all(abs(x['standardized_difference'])<5 for x in samples)
return d
if __name__=="__main__":
ensemble()
print("Reproduced results.json and results.csv")
Show the frozen analysis plan
# Computing hypothesis tests — plan frozen before outcome generation
Date: 2026-10-30. Scope: mathematical and synthetic computational tests, not observations of deployed systems. No novelty or peer review claimed. Source hypotheses remain unchanged. Seed 20261030; double-precision arithmetic. No significance-based selection. All planned results and controls retained.
## 1. Error correlation in ensembles
For unbiased real-valued prediction errors with individual variance one and common pairwise correlation rho in {0,.25,.5,.9,.99,1}, prove Var(mean)=rho+(1-rho)/m for m in {1,3,9,31,101}. Compare this exact expectation with 200,000 draws for m={3,9,31}, reporting Monte Carlo standard errors, not treating synthetic draws as empirical evidence. Generate errors sqrt(rho) Z0+sqrt(1-rho) Zi. This tests the covariance mechanism, not majority-vote accuracy. Give an exact independent, zero-mean unit-variance counterexample: existing average of X,Y has MSE .5; adding an exact copy X changes average to (2X+Y)/3, MSE5/9, while mean member MSE stays1. Also give a stronger member example: initial independent errors X (variance1),Y (variance4), ensemble MSE1.25 and mean member MSE2.5; duplicate X has variance1, raises average individual quality (mean MSE2), and gives ensemble MSE8/9 (actually improves). Therefore do not claim that example supports harm. Seek a prespecified algebraic range instead: starting independent equal members variance1, third error bX where b in {.5,.8,.9,1}; test whether lower individual MSE can nevertheless increase ensemble MSE, via (1+b)^2+1 over9. Analytic threshold b>sqrt(3.5)-1. Report threshold and all choices. Derive incremental gain=(1-rho)/(m(m+1)); any rho<1 still yields strictly positive variance gain. Original universal correlation threshold is not supported.
## 2. Cache ratio and access pattern
Simulate fully associative LRU for capacities C={16,64,256}, support-size ratios W/C={.5,1,2,4}, each with 200,000 measured requests after 20,000 warmup. Synthetic traces: independent uniform references; cyclic scans; an 80/20 two-class popularity model (80% accesses to first20% objects). Record W as the total support, not equate it with Denning's sliding-window working set. For uniform IID exact stationary hit probability=min(C/W,1). For repeated scan, exact post-warmup hit rate=1 when W<=C and0 otherwise. Test ratio collapse within each family, and failure of a single collapse across families. Include workload scaling example C16/W16 versus C64/W128. Include same-trace monotonicity control capacities16,32,64,128 on fixed IID support128. Check LRU directly against independent reuse-distance membership for first10,000 references of a small fixed trace.
## 3. Dependency graph availability
Assume independent point-in-time component uptime a={.99,.999,.9999}; no retries, timeouts, dynamic repair or load effects. For service count N={3,10,30,100}, compare mandatory chain (all N necessary), mandatory fanout (root+N-1 leaves, all necessary), one selected branch (root+one of N-1 leaves per request), and redundant OR leaves (root+at least one leaf). Exact formulae a^N,a^N,a^2,a*(1-(1-a)^(N-1)). Mandatory fanout graph has depth2 but same uptime as chain depthN. Exhaustively enumerate all2^N states for N<=10 to independently verify exact success logic; simulate1,000,000 snapshots for N={3,10,30},a=.99 as a secondary check. Include common-mode outage q=.005 multiplying conditional independent success by1-q. Compare a shallow N30 mandatory fanout with a deeper N3 chain to test overly broad depth prediction. Do not estimate downtime duration from marginal uptime or infer real microservice outcomes.
## Delivery and interpretation
Retain source hashes, plan hash, code hash, every exact and synthetic result, references and reproducibility instructions. For each article separate supported narrower result from unsupported generality. These are educational model checks of plausible mechanisms, not new field validation. Held: fault-tolerance-multiplicity because quorum safety requirements do not imply universal probability saturation.
Show the complete numerical results
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"kind": "exact covariance law and synthetic Gaussian checks",
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{
"b": 0.5,
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},
{
"b": 0.8,
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],
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}
Original hypothesis: DeepSeek / Incoprea. Analysis, code, figures and report: Codex. Independently checked by another Codex agent; this internal check is not external peer review. Established mathematics is not claimed as a new discovery. Return to Incoprea Research · Read the unchanged original hypothesis.
Publication code SHA-256: 882e11724359c7c483ec723e33fa663c147ef714f7c63014c39694b996963fe0
Saved result SHA-256: bdb2e5e70bf81a68b4deac4a701c6d49cd7ea16b691836ec5deb50af008ea91c
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