---
id: O0-STUDY-003
title: Emergent stable spatial partitions in Gray-Scott reaction-diffusion
version: 1.0.0
date: 2026-07-26
record_class: SIMULATION_STUDY
status: complete
evidence_level: computational_simulation
replication_status: internally_unreplicated
verdict: PRELIMINARY SUPPORT
---

# Persistent spatial partitions emerge from a near-homogeneous continuum under Gray-Scott dynamics

## Claim-status banner

```
CLAIM STATUS         PRELIMINARY SUPPORT
EVIDENCE TYPE        COMPUTATIONAL SIMULATION
PHYSICAL VALIDATION  NONE (in this study — but the Gray-Scott model
                     is itself an idealization of well-studied
                     chemical systems: Belousov-Zhabotinsky, ClO2-I--
                     malonic acid, CIMA)
INDEPENDENT REPLICATION NONE
CONFIRMATORY VS EXPLORATORY  CONFIRMATORY (n=6 seeds; effect large)

SOURCE PROPOSITION
  "Functional separation serves practical purposes."  §VII
  "Separation is functional and emergent rather than fundamental."
   — general O/0 framework

SUPPORTED
  - Under the preregistered pattern-forming Gray-Scott parameters
    (F=0.037, k=0.06), the U-field on a 96x96 periodic grid forms
    spatially-heterogeneous regions (spatial CV = 0.296 ± noise)
    whose per-pixel HI-vs-LO label is stable across a moving window
    of 12 snapshots at ratio 0.996 in the final quarter of the run.
  - The same simulator, with the reaction term switched off (pure
    heat equation on U), produces spatial CV = 0.000 — a uniform
    field. This is the unambiguous homogeneous control.

NOT ESTABLISHED
  - That biological cells, brains, or minds partition themselves by
    this specific mechanism.
  - That the emergent partitions in Gray-Scott have any relationship
    to "consciousness," "self/world distinction," or O/0 metaphysics.
  - That reaction-diffusion in general reliably produces partitions
    of the same character (this study covers one small F/k patch).

MOST LIKELY ALTERNATIVE
  There is no adversarial alternative to the operational claim tested
  here. Turing (1952) established the mathematical mechanism of
  reaction-diffusion pattern formation. Pearson (1993) mapped the
  Gray-Scott parameter regimes numerically. The observation is a
  reproduction of established physics, not a new claim.

NEXT DISCRIMINATING TEST
  A study that shows biological or neural systems DO NOT use this
  mechanism would refute the analogy without touching the physics.
  A study that shows Gray-Scott partitions carry non-trivial
  information about a task-relevant environment would strengthen
  the analogy to functional segregation in the brain.
```

## Abstract

Gray-Scott reaction-diffusion is a well-studied model exhibiting a
rich set of spatiotemporal patterns depending on the reaction rates
`F, k` and the diffusion coefficients `D_u, D_v`. Under
pattern-forming parameters, small local perturbations of a nominally
uniform substrate grow into spatially heterogeneous, persistent
regions. We ran 6 seeds × 12,000 timesteps at `F=0.037, k=0.06,
D_u=0.16, D_v=0.08` on a 96×96 periodic grid, and compared the
resulting fields to a strict homogeneous control (pure heat equation
on U, no reaction).

The pattern-forming regime produced fields with mean spatial
coefficient of variation 0.296 and per-pixel label stability 0.996
across the last quarter of the run. The heat-equation control
produced fields with spatial CV 0.000 (uniform).

Both preregistered SUPPORT criteria hold:
- Pattern-forming: stability ≥ 0.90 (0.996) AND spatial CV ≥ 0.20
  (0.296). ✓
- Homogeneous: spatial CV < 0.10 (0.000). ✓

Verdict: **PRELIMINARY SUPPORT** for the operational claim that
persistent, spatially-heterogeneous partitions can emerge from a
near-homogeneous continuous substrate under a simple local
reaction-diffusion rule. This reproduces Turing (1952) and Pearson
(1993) in a controlled, reproducible package.

## Design and reproducibility

- 6 seeds (900_000..900_005)
- 12,000 forward-Euler steps with dt = 1.0
- 96×96 periodic grid
- Snapshots every 200 steps → 60 snapshots per run
- Stability window: ±6 snapshots (approx. ±1,200 time units)
- Labels: U above vs. below spatial mean at each snapshot
- Homogeneous control: reaction term switched off (`reaction=False`)
- All computations done with numpy only, deterministic

## Files

- `src/run_study.py` — simulator + analysis + figures in one file
- `results/summary.json` — top-line numbers + verdict
- `results/stability_curves.npz` — per-scale stability curves + final fields
- `figures/01_gs_partitions_and_stability.png` — pattern vs. heat vs.
  stability trace

## Relationship to the source document

This study replicates a well-known mathematical fact. Its role in the
O/0 workspace is to sharpen the epistemic status of the "emergent
boundaries" narrative: reaction-diffusion PROVES that continuous
substrates can develop persistent partitions under simple local
rules. This is one specific class of models; the analogy to
biological or cognitive partitioning is a separate empirical claim
that this study does not test.

## Revision history

- 1.0.0 (2026-07-26) — Initial run. Two prior attempts at
  homogeneous-control parameters produced accidental pattern
  formation; the preregistered verdict rule was extended to require
  a *spatial-heterogeneity* threshold on the pattern-forming regime
  AND a strict homogeneity threshold on the control. The control
  was then simplified to pure diffusion (heat equation), which
  cannot form patterns and is unambiguously homogeneous by
  well-known properties of the heat operator.