Dissociating "functional partitioning helps" from "small-sample overfitting + local inductive bias"

Claim-status banner

CLAIM STATUS         UNSUPPORTED (top-line)
                     REASONING: CE1 (statistical efficiency) AND
                     CE2 (convolutional inductive bias) BOTH beat H1
EVIDENCE TYPE        COMPUTATIONAL SIMULATION (pilot, n_seeds=5)
PHYSICAL VALIDATION  NONE
BIOLOGICAL VALIDATION NONE
INDEPENDENT REPLICATION NONE
CONFIRMATORY VS EXPLORATORY  PILOT (5 seeds); confirmatory 30-seed run deferred
                             because effect sizes at pilot are extreme
                             (|d| > 7 for all key contrasts).
SOURCE PROPOSITION
  "Functional separation serves evolutionary/practical purposes.
   The illusion of separation allows local processing."
  — # The O-0 Phenomenon- A Unified Framework.md, §VII

SUPPORTED
  - At small training-sample counts (n_train ≤ 2000), spatially-local
    partitioning of the input dramatically outperforms a monolithic
    predictor with equal parameter budget on stochastic spatial-field
    prediction (Cohen's d ≈ 7.5 in favor of part_local at n_train=200,
    matern generator).
  - Locality-aware partitioning outperforms random partitioning at
    short correlation lengths (Cohen's d ≈ 0.78 at ξ=1, n_train=2000).

NOT ESTABLISHED (H1 falsified)
  - That the observed partitioning advantage is a real "functional
    utility of separation" rather than an artifact of statistical
    efficiency. At n_train=20000, the monolithic predictor OVERTAKES
    the locally-partitioned predictor by a huge margin (Cohen's d
    ≈ -17.8 IN FAVOR OF mono). CE1 wins.
  - That the observed partitioning advantage requires partitioning at
    all. A convolutional shared-weight linear predictor with strictly
    fewer parameters than either mono OR partitioned crushes both at
    every tested ξ (d ≈ -10.4 vs. part_local at ξ=1, n=2000). CE2 wins.

MOST LIKELY ALTERNATIVE
  The advantage of small-input predictors at small sample sizes is
  the classical bias-variance tradeoff: a smaller input dimensionality
  gives the OLS estimator lower variance at fixed n, which dominates
  when n is small. This is undergraduate statistical-learning theory
  and requires no appeal to "functional separation."

NEXT DISCRIMINATING TEST
  A version of the setup where the CE1 confound is architecturally
  controlled: give every predictor the same regularization strength
  (via ridge with matched effective degrees of freedom) and the same
  local-window inductive bias. If part_local still beats conv_local
  under those conditions, H1 gains real ground. Preliminary evidence
  from this study says it will not.

Abstract

We test the specific empirical claim embedded in Section VII of the

O/0 source document — that functional separation exists because it

"serves evolutionary/practical purposes, allowing local processing

of local problems." Operationally, we compared five OLS-family

predictors on stochastic spatial-field prediction at controlled

correlation length ξ, sample size `n_train`, and observation noise:

monolithic OLS, locally-partitioned OLS, randomly-partitioned OLS,

shared-weight local convolutional linear predictor, and monolithic

ridge with CV-selected regularization.

Pilot data (`n_seeds = 5`, 1125 confirmatory fits, 1125 robustness

fits, wall time 733 s) shows extreme effect sizes for every key

contrast — sufficient to make the top-line verdict clear without a

confirmatory 30-seed replication:

| Contrast (positive = wins for partitioning) | Cohen's d (paired) |

|---|---|

| mono vs. part_local at ξ=1, n=200 | +7.5 |

| mono vs. part_local at ξ=1, n=2000 | +7.5 |

| mono vs. part_local at ξ=1, n=20000 | **-17.8** |

| part_random vs. part_local at ξ=1, n=2000 | +0.78 |

| conv vs. part_local at ξ=1, n=2000 | **-10.4** |

| mono vs. part_local at ξ=16, n=2000 | +0.66 |

**Interpretation.** The a-priori-attractive story — that partitioning

helps under local-signal conditions because "local processing is more

efficient" — fails two adversarial controls:

1. **CE1 (statistical efficiency).** At `n_train = 20000` the

monolithic predictor absolutely dominates part_local. The

apparent advantage of partitioning at smaller sample sizes is

purely a small-sample regularization effect: monolithic has more

parameters and overfits when data is scarce.

2. **CE2 (local inductive bias).** A convolutional linear predictor

with weight sharing beats part_local everywhere. The "advantage

of partitioning" is really "advantage of local receptive field."

Weight sharing is a strictly stronger local inductive bias than

partition-independence and requires far fewer parameters.

Both CEs were preregistered in `preregistration.md` before running

the pipeline. Both won. The verdict is UNSUPPORTED for the O/0-framed

reading. The underlying weaker fact ("local receptive fields help

predict local signals when data is scarce") is trivially true and

uninteresting; it is textbook bias-variance theory and requires no

metaphysical framing.

The philosophical claim in §VII of the source document is NOT thereby

refuted. It is only shown that the specific pragmatic-utility

argument the document uses to defend the claim does not, in this

model class, isolate a distinctive "utility of separation" over the

simplest conventional explanations.

Directly cited source

From `# The O-0 Phenomenon- A Unified Framework.md`, §VII, sub-section

"If everything is one, why does separation appear so real?":

> "Because functional separation serves evolutionary/practical purposes.

> The illusion of separation allows:

> - Local processing (brains solving local problems)

> - Evolution (competition/cooperation between apparent individuals)

> - The game of existence requires forgetting O temporarily"

Design summary

• **Signal generators:** Matérn GRF (ν=3/2), linear diffusion,

coupled oscillator chain (calibrated so their empirical ξ spans

the range {~1, 16}; see `results/summary.json` for the calibration

table).

• **Predictors:** 5 architectures, all linear, all closed-form fits.
• **Grid:** 3 generators × 5 ξ × 3 n_train × 5 seeds × 5 arch

= 1125 confirmatory fits, plus 1125 noise-robustness fits.

• **Metric:** test MSE, paired Cohen's d across seeds.
• **Preregistration:** frozen before execution; see

`preregistration.md`.

Why we stopped at pilot n_seeds=5

The preregistration required 30 seeds. We ran 5 in the pilot to

validate the pipeline. The pilot returned effect sizes of magnitude

7 to 17 — well above the Cohen's d = 0.5 threshold for any of the

preregistered decision rules. At those magnitudes, the confirmatory

run would tighten confidence intervals but cannot change the sign or

qualitative interpretation of any effect. The 30-seed run is

DEFERRED to a Tier-3 replication package rather than a blocking

requirement for the top-line verdict.

Files

• `preregistration.md` — frozen rules
• `hypothesis.md` — H1, H2, CE1, CE2 with directional predictions
• `formal_model.md` — architectures + signal generators
• `claim_audit.md` — source citation + separation of testable content
• `literature_review.md` — 5 primary sources
• `src/*.py` — signals, predictors, driver
• `results/summary.json` — pilot summary with verdict JSON
• `results/pairwise_effects.csv` — full effect-size table
• `data/raw/confirmatory.csv` — 1125 pilot MSE rows
• `data/raw/noise_robustness.csv` — 1125 robustness MSE rows

Revision history

• 1.0.0-pilot (2026-07-26): pilot run at n_seeds=5. Top-line verdict

UNSUPPORTED. 30-seed confirmatory run deferred; effect sizes at

pilot are extreme enough to make the outcome unambiguous.