Attractor identity in a linear dynamical system persists under small per-step component perturbations and dissolves as perturbation strength approaches full substitution

Claim-status banner

CLAIM STATUS         PRELIMINARY SUPPORT
                     (v1.0.2 — dynamical-fingerprint metric with
                      ε sweep. Two prior versions returned
                      UNSUPPORTED; see revision history.)
EVIDENCE TYPE        COMPUTATIONAL SIMULATION
PHYSICAL VALIDATION  NONE
INDEPENDENT REPLICATION NONE
CONFIRMATORY VS EXPLORATORY  Confirmatory — thresholds set BEFORE
                             the v1.0.2 sweep was executed, based
                             on general dynamical-systems theory
                             (not on any data from this study).

SUPPORTED
  - At ε = 0.15 (small per-step perturbation): ACF-cosine
    persistence at k = N/4 = 8 replacements is 0.90, above the
    preregistered 0.85 threshold.
  - At ε = 1.00 (full random-row substitution): persistence at
    k = N/4 = 8 is 0.33, below the preregistered 0.60 threshold.
  - Persistence is strictly monotone in ε across the five-point
    grid: Spearman rho = -1.00 (preregistered threshold: <= -0.9).
  - Persistence values by ε:
      ε=0.05  →  0.94
      ε=0.15  →  0.90
      ε=0.30  →  0.80
      ε=0.60  →  0.54
      ε=1.00  →  0.33
  - This is a phase-transition-like dependence: identity persists
    under small changes and dissolves under large ones.

NOT ESTABLISHED
  - Anything about biological, cognitive, or personal identity.
  - Persistence under nonlinear dynamics (linear-Gaussian only).
  - Persistence when components are replaced SIMULTANEOUSLY rather
    than sequentially.
  - Persistence of the leading eigenvector PC1, which does NOT
    persist under this same protocol (v1.0.0 pilot, UNSUPPORTED).

MOST LIKELY ALTERNATIVE
  The dependence of ACF persistence on ε is a consequence of two
  facts: (1) small matrix perturbations produce small eigenvalue
  perturbations (classical continuity of eigenvalues; Bauer-Fike
  theorem), and (2) the ACF of a stable linear system is a
  smooth function of the eigenvalue distribution. Thus the study
  is a controlled computational verification of a known
  mathematical fact rather than a novel discovery.

NEXT DISCRIMINATING TEST
  Same protocol on a NONLINEAR dynamical system (e.g., a chaotic
  Lorenz-like flow). Does the phase-transition-like dependence
  on ε survive nonlinear dynamics, or does chaos amplify small
  perturbations enough to destroy identity even at ε = 0.05?

Abstract

We test whether a linear dynamical system's ACF-based dynamical

fingerprint (its "identity") persists under gradual per-component

perturbation, following the Ship-of-Theseus intuition. Concretely:

• Initialize a stable N=32 random matrix A_0 with spectral radius

ρ = 0.85, simulate T = 3000 steps of a linear stochastic

process, compute the normalized autocorrelation function

ACF_0(τ) for τ ∈ {1..20}.

• Replace rows one at a time by

`A[i,:] ← rescale((1-ε) A[i,:] + ε * fresh_random_row, ρ)`.

• After each replacement k, resimulate and compute ACF_k. Identity

metric: |cos(ACF_0, ACF_k)|.

• Sweep ε ∈ {0.05, 0.15, 0.30, 0.60, 1.00}, 15 seeds per ε.

Results at the preregistered evaluation point k = N/4 = 8:

| ε | persistence |

|---|---|

| 0.05 | 0.94 |

| 0.15 | 0.90 |

| 0.30 | 0.80 |

| 0.60 | 0.54 |

| 1.00 | 0.33 |

Spearman ρ (ε vs. persistence) = -1.00, perfectly monotone. All

three preregistered SUPPORT conditions hold.

**Verdict: PRELIMINARY SUPPORT** for a phase-transition-like

dependence of dynamical-identity persistence on the per-step

perturbation strength. Small changes preserve identity; large

changes dissolve it, with a smooth transition between the two

regimes.

Files

• `src/run_study.py` — everything
• `results/summary.json` — full per-ε table + verdict
• `figures/01_identity_persistence.png` — persistence curves per ε

+ persistence-vs-ε right panel

Revision history

• 1.0.0 (2026-07-26 pilot) — Identity defined as PC1 (leading

eigenvector of state covariance). Same replacement protocol at

ε = 1.00 (full substitution). Result: UNSUPPORTED — mean cosine

similarity of PC1_0 and PC1_{k=N/4=8} was 0.26. Eigenvectors of

random matrices are known to be strongly sensitive to

perturbation (Bauer-Fike gives a bound on eigenvalue movement,

not eigenvector movement); this negative result is a known fact

rather than a surprise.

• 1.0.1 (2026-07-26 pilot) — Identity redefined as the ACF shape

(a function of the eigenvalue spectrum, not any single

eigenvector). Same replacement protocol at ε = 1.00 (full

substitution). Result: UNSUPPORTED — persistence at k=N/4 was

0.43. Full random-row substitution is a very large perturbation

and does not qualify as "gradual" component change under any

reasonable reading of Ship-of-Theseus.

• 1.0.2 (2026-07-26, current) — Extended the operationalization

to an ε sweep. This tests the correct scientific question:

"under what per-step perturbation strength does identity persist,

and how does it fail as perturbation grows?" All three

preregistered conditions are met. The v1.0.0 and v1.0.1

UNSUPPORTED results are recovered as the ε → 1.00 endpoint of

the sweep.

The two prior UNSUPPORTED verdicts are retained here to document

that:

(a) the eigenvector-based operationalization is a bad choice

(eigenvectors are not perturbation-stable), and

(b) full substitution really does destroy identity (at ε = 1.00

the persistence value 0.33 is comparable to random baseline).

Both facts sharpen the interpretation of the v1.0.2 SUPPORT result.