Dissociating Synchronization from Information Integration

Claim-status banner

CLAIM STATUS         PRIMARY:   UNSUPPORTED
                     SECONDARY: INCONCLUSIVE
                     (previously reported as PRELIMINARY SUPPORT at
                      the preregistered bins_z = 6; downgraded after
                      bin-count robustness check — see §Robustness)
EVIDENCE TYPE        COMPUTATIONAL SIMULATION
PHYSICAL VALIDATION  NONE
BIOLOGICAL VALIDATION NONE
INDEPENDENT REPLICATION NONE
CONFIRMATORY VS EXPLORATORY  CONFIRMATORY (both metrics preregistered)

SUPPORTED
  Under raw pairwise MI with circular-shuffle null, condition A
  (common drive) and condition C (dense coupling) produce statistically
  indistinguishable "excess" MI. Raw pairwise MI does NOT dissociate
  synchronization from integration in this parameter regime.

  Under conditional MI given the Kuramoto mean-field phase Ψ(t),
  condition C's residual pairwise dependence exceeds condition A's by
  a large effect size (Cohen's d ≈ -3.38, C > A). Sparse coupling
  (B) is intermediate. The independent-oscillator baseline (D)
  produces near-zero residual on both metrics. Under this metric, the
  dissociation IS supported at the "preliminary support" level.

NOT ESTABLISHED
  That the same dissociation holds in biological or physical systems.
  That either metric measures "consciousness" or phenomenal integration.
  That Tononi-style integrated information Φ dissociates similarly.
  That the O/0 philosophical framework is empirically confirmed.
  That the effect survives alternative MI estimators (kNN, transfer
    entropy). Test explicitly deferred to Tier-4 replication.

MOST LIKELY ALTERNATIVE
  A specific-metric artifact of coarse bin count for the conditioning
  variable Ψ. Directly testable and scheduled.

NEXT DISCRIMINATING TEST
  Tier-4 replication with kNN MI estimator (Kraskov et al. 2004) and
  bins_z ∈ {4, 6, 8, 10} robustness scan.

Abstract

Two constructs — **synchronization** (measured here by the Kuramoto

order parameter `r`) and **information integration** (measured here by

pairwise mutual information) — are routinely conflated in complex-

systems and consciousness-adjacent literature. We test whether they can

be experimentally dissociated in a minimal stochastic Kuramoto model

with four coupling architectures: common external drive (A), sparse

random coupling (B), dense all-to-all coupling (C), and no coupling (D).

At `N = 32` oscillators, `T = 4000` steps, and 30 confirmatory seeds

per condition (120 confirmatory runs plus 320 sensitivity/robustness

runs), we find:

• On raw pairwise MI with a circular-shuffle null, conditions A, B, and

C all produce large "excess" MI (~0.64–0.68 nats). The metric fails

to dissociate the four conditions along the mechanism dimension.

Pearson correlation between per-run `r` and raw MI is 0.998. **The

raw-MI dissociation hypothesis is UNSUPPORTED.**

• On conditional MI given the Kuramoto mean-field phase `Ψ(t)` at the

preregistered `bins_z = 6`, conditional excess produces the

predicted separation: `A ≈ D ≈ 0`, `B = 0.084`, `C = 0.106`. The

A–C contrast has Cohen's d = -3.38. A subsequent bin-count

robustness check (`bins_z ∈ {4, 6, 8, 10}`) found that the

**direction** of the dissociation is robust (C > A with Cohen's

d > 2 in every case) but the **absolute magnitude** is

bin-count-sensitive (C falls below the preregistered 0.10-nat

threshold at `bins_z ≥ 8`). **The conditional-MI secondary

verdict is therefore INCONCLUSIVE** rather than the initial

PRELIMINARY SUPPORT. See §Robustness for the full table.

The primary finding — that a widely used metric confuses shared-input

correlation with genuine dyadic integration — is the negative-result

core of this study and is stable. The secondary finding — that

conditioning on the collective mean field reveals a real directional

separation between shared-input and coupling — is directionally

supported but magnitude-fragile under a preregistered absolute

threshold. Neither finding establishes anything about biological

integration, consciousness, or the O/0 philosophical framework.

Historical and scientific background

Kuramoto's 1975 model formalized phase-locking in coupled oscillators

and remains the canonical toy system for studying synchronization

transitions (Strogatz 2000, Acebrón et al. 2005). Integrated

information theory (Tononi 2008) posits that consciousness corresponds

to information generated by a system above and beyond its parts.

Tononi & Edelman (1998) explicitly argued that synchronization and

integration are conceptually distinct and can dissociate. Subsequent

work (Seth et al. 2011, Mediano et al. 2022) has continued to

disentangle the two constructs. This study contributes a minimal,

publicly reproducible in-workspace demonstration of both directions of

the dissociation using textbook methodology.

Source-claim audit

See `claim_audit.md`. Briefly: the study neither confirms nor refutes

any philosophical claim about "unity." It measures whether two

specific statistical constructs are numerically dissociable in a

specific model class.

Research question

*Can `r` and pairwise MI be independently manipulated by choice of

coupling architecture, holding other model parameters fixed?*

Operational definitions

• **`r(t) = |mean_i exp(i θ_i(t))|`** — Kuramoto order parameter,

averaged over the post-burn-in window to produce `⟨r⟩`.

• **`MI(θ_i; θ_j)`** — Miller–Madow bias-corrected mutual information

on 12-bin phase histograms.

• **`MI_null`** — average pairwise MI where one series is circularly

shifted by a random offset in `[T/4, 3T/4]` (60 pairs sampled).

• **`mi_excess = MI − MI_null`** — the temporally aligned component.
• **`cond_mi(θ_i; θ_j | Ψ) = H(θ_i, Ψ) + H(θ_j, Ψ) − H(θ_i, θ_j, Ψ) − H(Ψ)`**

on `8×8×6` bins, with `Ψ(t) = arg(mean_i exp(i θ_i(t)))`.

• **`cond_mi_excess`** — analogous circular-shuffle-null-corrected

version of the above.

Primary hypothesis (raw pairwise MI)

`⟨mi_excess_A⟩ < 0.10 nats`, `⟨mi_excess_C⟩ > 0.20 nats`, and

`⟨mi_excess_C⟩ − ⟨mi_excess_A⟩ > 2 · SE`.

Secondary hypothesis (conditional MI)

`⟨cond_mi_excess_A⟩ < 0.10 nats`, `⟨cond_mi_excess_C⟩ > 0.10 nats`, and

`⟨cond_mi_excess_C⟩ − ⟨cond_mi_excess_A⟩ > 2 · SE`.

Null and competing hypotheses

• **H0:** `r` and MI-family metrics are monotonically related

(`|ρ| ≥ 0.9`) across all conditions.

• **H2:** Any dissociation is a histogram-bin artifact.
• **H3:** In condition A the drive itself carries enough information

to produce as much pairwise MI as full coupling.

Formal model

See `formal_model.md`. In brief:

$$

d\theta_i = (\omega_i + D_A(t,\theta_i) + \frac{K}{\bar{d}} \sum_j A_{ij} \sin(\theta_j - \theta_i)) \, dt + \sqrt{2\sigma^2}\, dW_i

$$

with `D_A(t, θ_i) = D \sin(f_D t - θ_i)` only in condition A, and `A`

being all-to-all in C, sparse ER + spanning cycle in B, and zero in

A and D.

Methods

Euler–Maruyama integration for `T = 4000` steps at `dt = 0.05`,

`burn_in = 800`. Four conditions × 30 confirmatory seeds; four

conditions × 4 K-values × 10 seeds sensitivity; four conditions × 4

σ-values × 10 seeds noise robustness. 440 runs total.

Variables and controls

• **Independent:** condition ∈ {A, B, C, D}, `K`, `σ`.
• **Dependent:** `⟨r⟩`, `mi_observed`, `mi_null_shuffled`, `mi_excess`,

`cond_mi_observed`, `cond_mi_null_shuffled`, `cond_mi_excess`.

• **Controls:** D (independent, negative), C (dense-coupled Kuramoto,

positive), circular-shuffle null (within-run).

• **Adversarial:** the K and σ sweeps.

Baselines

• **Null:** D (independent).
• **Conventional positive:** C (standard Kuramoto above K_c).
• **Shuffle-null:** within-run temporal shuffling of one series.
• **Mean-field-conditioning null:** shuffle of the target series only,

keeping `Ψ` and the reference series aligned.

Preregistered analysis plan

See `preregistration.md`. Frozen prior to the confirmatory batch. The

primary and secondary metrics have separate, non-overlapping decision

rules; each yields its own verdict.

Results

Descriptive statistics (confirmatory runs, N = 30 seeds each)

| Condition | ⟨r⟩ | ⟨MI⟩ | ⟨mi_excess⟩ | ⟨cond_MI⟩ | ⟨cond_mi_excess⟩ |

|---|---|---|---|---|---|

| A · common drive | **0.983** ± 0.002 | 1.664 ± 0.005 | **0.640** ± 0.006 | 0.207 ± 0.003 | **0.001** ± 0.004 |

| B · sparse coupled | 0.986 ± 0.004 | 1.719 ± 0.010 | 0.646 ± 0.210 | 0.232 ± 0.007 | 0.084 ± 0.041 |

| C · dense coupled | **0.992** ± 0.001 | 1.774 ± 0.003 | **0.676** ± 0.215 | 0.260 ± 0.004 | **0.106** ± 0.044 |

| D · independent | 0.159 ± 0.022 | 0.150 ± 0.008 | 0.001 ± 0.016 | 0.338 ± 0.009 | -0.002 ± 0.016 |

Effect sizes

• A vs. C on `r_mean`: Cohen's d = -6.24 (C higher, tiny variance)
• A vs. C on `mi_excess`: Cohen's d = -26.8 — extremely large but

reflects tiny SDs in A and larger SDs in C; interpret with caution.

• A vs. C on `cond_mi_excess`: Cohen's d = -3.38 (C higher — the

dissociation)

Global correlations across all confirmatory runs

• `ρ(r_mean, mi_observed) = 0.998` — sync and raw MI move together

perfectly across A/B/C/D. Raw-MI dissociation fails.

• `ρ(r_mean, cond_mi_observed) = -0.916` — inverse relationship

driven by D (low r, high cond_mi from wide phase distribution).

This is not by itself the dissociation; the dissociation is on

`cond_mi_excess`, which is near-zero in A and D, moderate in B,

elevated in C.

Verdicts

• **PRIMARY metric verdict (raw pairwise MI):** `UNSUPPORTED`. The

dissociation predicted by the initial design does not survive when

common-drive-induced correlation is treated as "integration."

• **SECONDARY metric verdict (conditional MI given Ψ):**

`PRELIMINARY SUPPORT`. When the mean-field mediation is partialled

out, the theoretically predicted dissociation is observed with a

large effect size.

Robustness and sensitivity

See `figures/04_sensitivity_K.png` and `figures/05_noise_robustness.png`

for the full sweeps.

• **K sweep (0.5–4.0):** the A / C separation on `cond_mi_excess` is

present at all K values ≥ 1.0. At K = 0.5 the coupling is

subcritical and C fails to synchronize; the dissociation test is

uninformative there.

• **σ sweep (0.05–0.5):** the dissociation on `cond_mi_excess`

survives noise up to σ ≈ 0.3. At σ = 0.5 all conditions collapse

toward D-like behavior.

Bin-count robustness of the conditional-MI estimator (added 2026-07-26)

The conditional-MI metric requires a bin count `bins_z` for the

conditioning variable Ψ. The preregistered value was `bins_z = 6`. To

check fragility, we recomputed the metric at `bins_z ∈ {4, 6, 8, 10}`

on the same 30 confirmatory seeds × 4 conditions. Results, on

`⟨cond_mi_excess⟩`:

| `bins_z` | ⟨A⟩ | ⟨C⟩ | C − A | Cohen's d | Threshold met |

|---|---|---|---|---|---|

| 4 | -0.007 | +0.142 | 0.149 | 2.87 | ✓ |

| 6 (preregistered) | +0.002 | +0.106 | 0.104 | 3.28 | ✓ |

| 8 | -0.026 | -0.002 | 0.023 | 2.40 | ✗ (C falls to ≈ 0) |

| 10 | -0.003 | +0.041 | 0.045 | 4.12 | ✗ (C < 0.10) |

**Directional conclusion (robust):** `⟨C⟩ > ⟨A⟩` across all four bin

values, with Cohen's `d > 2` in every case. The dissociation direction

is preserved. The C-vs-A separation always exceeds ~5 standard errors

of the difference.

**Absolute-magnitude conclusion (fragile):** the preregistered

absolute threshold of `⟨cond_mi_excess_C⟩ > 0.10 nats` is only met at

`bins_z ∈ {4, 6}`. At finer conditioning (`bins_z ≥ 8`), the C mean

drops below the threshold as more shared structure is absorbed into

`Ψ`. This means the preregistered absolute cutoff was implicitly

calibrated to `bins_z = 6` and is not a bin-count-invariant test.

**Verdict update:** the secondary metric verdict is **downgraded from

`PRELIMINARY SUPPORT` to `INCONCLUSIVE`.** The dissociation direction

is real; the absolute-threshold decision rule was under-specified.

**What this implies for the next study:** a follow-up must operate on

a scale-invariant quantity — either the effect size (Cohen's d),

the C-vs-A gap in units of within-bin SEM, or a normalized metric such

as `(C − A) / max(observed cond_mi)`. Under all three of these

scale-invariant readings the dissociation direction is robust.

Alternative interpretations

See `alternative_explanations.md`. Strongest remaining concerns:

bin-count artifact and topology-specific effect in condition B. Both

are directly addressable with the raw data and are scheduled for

Tier-4 replication.

Limitations

See `limitations.md`. In particular: phase-only Kuramoto is not a

biological model, and this study says nothing about consciousness or

physical reality.

Replication procedure

See `replication.md`. The full package is deterministic and

reproducible; independent reimplementation is invited.

Code and data manifest

Executable pipeline: `src/run_study.py`.

Raw data: `data/raw/*.csv`.

Analysis outputs: `results/*.{csv,json,md}`.

Figures: `figures/*.png`.

Checksums: `results/checksums.json`.

Relationship to the philosophical archive

The study is inspired by an O/0-adjacent conceptual tendency to conflate

"unity" and "integration" and "synchronization." It provides:

• a **negative** result on a naive metric (raw pairwise MI cannot

distinguish shared-drive from coupling);

• a **preliminary positive** result on a properly conditioned metric

(mean-field-conditioned MI does distinguish them).

*Conceptual provenance is not empirical support.* Neither result

speaks to the correctness of the O/0 framework.

Primary references

• Kuramoto (1975), Strogatz (2000), Acebrón et al. (2005).
• Cover & Thomas (2006); Miller (1955); Kraskov et al. (2004).
• Tononi (2008); Tononi & Edelman (1998); Seth et al. (2011);

Mediano et al. (2022); Oizumi et al. (2014).

See `literature_review.md` for full bibliographic information. No

citation is fabricated; every reference is verifiable via standard

scholarly databases.

Revision history

See `revision_history.md`. Notably: v1.1.0 amended the preregistration

prior to execution to add the secondary conditional-MI metric based on

Tononi & Edelman's theoretical argument. This is documented

transparently.