SIMULATION · O0-CRP-030

Track 2 R3 · Discriminability in Nonlinear Universes — CRP-022 pattern preserved under tanh dynamics

STATUSVerdict: PATTERN_PRESERVED. Substantive reading: under a saturating tanh nonlinearity, δ*(0.7) = 0.20 identical to the linear case. Matched-dual AUC = 0.387 (in chance band). Realization theorem holds bit-identically (IDENTITY and MATCHED_DUAL use identical M and noise). Nonlinearity dampens discriminability slightly at small δ (AUC=0.53 at δ=0.10 vs 0.64 linear) but the boundary at τ=0.7 is unchanged.
EVIDENCE TYPECOMPUTATIONAL · CRP-022 protocol under z_{t+1} = tanh(M z_t) + w_t nonlinear dynamics on 30 fresh world seeds (18000–18029).
REPLICATIONREGISTERED · alternate nonlinearities (ReLU, cubic, sinusoidal), joint nonlinearity × classifier sweep, finer δ grid, chaotic regime.
PHYSICAL VALIDATIONNONE
VERSION1.0.0
DATE

O0-CRP-030 — Scientific Record

**Title:** Discriminability in Nonlinear Universes

**Program:** Track 2 followup R3

**Version:** 1.0.0

**Status:** COMPLETED · Executed 2026-07-28

**Preregistration:** v1.0.1 frozen before final result publication

Claim status

CLAIM STATUS: **PATTERN_PRESERVED · δ*(0.7) = 0.20 identical to linear case under tanh nonlinearity**

EVIDENCE TYPE: COMPUTATIONAL SIMULATION

PHYSICAL VALIDATION: NONE

Key finding

Under `z_{t+1} = tanh(M z_t) + w_t` (nonlinear evolution), the CRP-022 qualitative pattern is preserved:

| Metric | Linear (CRP-022) | Nonlinear (CRP-030) |

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

| δ*(0.7) | 0.20 | **0.20** |

| δ*(0.6) | 0.10 | 0.00 |

| matched-dual AUC | 0.376 ± 0.044 | 0.387 ± 0.025 |

| noise-control AUC | 1.000 | 1.000 |

| realization theorem | PASS | PASS (bit-identical) |

| AUC at δ = 0.10 | 0.643 | 0.529 |

| AUC at δ = 0.20 | 0.896 | 0.818 |

| AUC at δ = 0.30 | 0.969 | 0.952 |

**Verdict: PATTERN_PRESERVED.**

Substantive reading

1. **The CRP-022 result generalizes past linear-Gaussian.** Under a saturating tanh nonlinearity, IDENTITY and MATCHED_DUAL remain statistically indistinguishable (matched-dual AUC = 0.39, well within chance band), and the DUAL(δ) boundary sits at the same δ*(0.7) = 0.20.

2. **Nonlinearity dampens discriminability at small δ but not at the τ=0.7 boundary.** At δ = 0.10, nonlinear AUC (0.53) is much closer to chance than linear AUC (0.64) — the tanh saturation absorbs part of the perturbation. But by δ = 0.20, both cross the 0.7 effective-AUC threshold.

3. **State-space realization holds bit-identically in nonlinear case too.** Since IDENTITY and MATCHED_DUAL use identical M matrices and identical noise, the tanh(M · z) computation is identical bit-for-bit. This is a stronger property than the linear realization theorem — it's a construction-time equality.

Figures

![nonlinear_vs_linear](figures/01_nonlinear_vs_linear.png)

![matched_and_noise](figures/02_matched_and_noise.png)

Adversarial interpretation

  • **Could tanh mask a true DUAL signature that would be visible under a different nonlinearity?** Possibly. tanh has a specific saturation profile; other nonlinearities (ReLU, cubic, sinusoidal) may amplify perturbations differently. Registered followup.
  • **Is δ*(0.7) = 0.20 the same "true" boundary or the same "grid resolution artifact"?** Given CRP-027 finds fresh-worlds get 0.10 under linear dynamics and this study also produced sharper discriminability (0.82 vs 0.90 at δ=0.20), it's possible the *sample-level* boundary is around 0.15 in both cases and we're binning to 0.20. A finer δ grid would resolve.
  • **Could the nonlinearity destroy the realization guarantee at higher perturbation levels?** No — DUAL(δ) explicitly uses a different M matrix (with cross-coupling scaled), so trajectories genuinely differ. What tanh does is bound the divergence, but the difference in dynamics is preserved.

Not established

  • Anything about non-tanh nonlinearities.
  • Anything at higher noise scales in nonlinear regime.
  • Anything about chaos or non-hyperbolic dynamics.
  • Anything about consciousness, metaphysics, or O/0 identity — this is purely a discriminability characterization.

Cross-study implications

  • Combined with **CRP-022** (linear δ* = 0.20), **CRP-026** (SNR-invariant), and **CRP-030** (nonlinear pattern preserved), the qualitative discriminability boundary appears robust across broad dynamical regimes.
  • Combined with **CRP-025** (classifier-dependent) and **CRP-027** (fresh-world shifts to 0.10), the *specific numerical value* of δ*(0.7) has ± 1 grid step uncertainty. True asymptotic boundary likely in [0.10, 0.20].
  • The Track 2 program (deep O/0 identity claim, "observer is source") now has substantial empirical coverage: 6 studies × 3 dynamical regimes × 5 classifiers × multiple seed sets. No study contradicts the qualitative CRP-022 finding.

Replication procedure


cd research/studies/O0-CRP-030/src
python run_study.py    # ~3.5 min
python analyze.py

Reuses O0-CRP-022's `UniverseParams`, `ExperimentConfig`, `build_world_matrix`, `compute_features`, and `discriminate`. Reimplements `simulate_universe` with tanh nonlinearity.

Registered followups

  • R1: Alternate nonlinearities — ReLU, cubic, sinusoidal.
  • R2: Joint (nonlinearity × classifier) sweep to combine CRP-025 and CRP-030.
  • R3: Finer δ grid to pin down whether "true" boundary is 0.10, 0.15, or 0.20.
  • R4: Chaotic dynamics regime (spectral radius > 1).

Revision history

  • v1.0.0 (2026-07-28): initial record after preregistered execution.

Figures

Figure from O0-CRP-030: 01 nonlinear vs linear
Figure from O0-CRP-030: 01 nonlinear vs linear
Figure from O0-CRP-030: 02 matched and noise
Figure from O0-CRP-030: 02 matched and noise

Source proposition

“Followup R3 registered in O0-CRP-022 manifest. Tests whether the linear-Gaussian discriminability boundary extends past the linearity assumption.”

Conceptual provenance is not empirical support.

Related records

O0-CRP-022O0-CRP-024O0-CRP-025O0-CRP-026O0-CRP-027