Simulation
families (Lack Kernel, Spectral Experiment, IPM Protocol, Collective Regimes)
consistently produced three reproducible patterns under the tested conditions
(100 runs, ε=0.15, bins=30, max_lag=20).
R1 —
Lack-Degradation
Higher
perturbation → lower coherence. Coherence drops from 0.97 to 0.55 as noise
increases from 0.02 to 1.2. In the IPM Protocol, Φ* drops approximately 16%
under thermal perturbation before recovering.
1.1 Minimal Model of Lack Dynamics
A
one-dimensional system with slow memory mₜ follows:
xₜ₊₁ = xₜ + λ(mₜ − xₜ) + η,η ~ N(0, σ²)
Parameter
Definition
Value /
Domain
xₜ
System state
at time t
—
mₜ
Memory (slow
reference trajectory)
—
λ
Coupling pull
toward memory
0.15,
illustrative value used in the minimal model (stable domain: 0 < λ ≤ 1)
η
Stochastic
perturbation
Gaussian,
zero mean, variance σ², i.i.d. per step
σ
Perturbation
intensity
Varied across
runs (0.02 – 1.2)
Coherence is
defined as:
cₜ = exp(−|xₜ − mₜ|)
This model is
not proposed as a universal law. It serves as a minimal dynamical illustration
of the Lack–Coupling relationship underlying R1.
Result: as σ
increases, mean coherence decreases monotonically — illustrating R1. The term
(mₜ − xₜ) operationalizes Lack as deviation between current state and memory; λ
governs Coupling intensity. Φ* and 𝒞 are scalar compressions derived
from the statistics of this process.
R2 —
Integration-Persistence
Higher
integration → longer persistence under perturbation. Higher integration yields
longer metastability under moderate perturbation.
R3 —
Observed Clustering Under Specific Projections
Under the
tested observer projections (CCI, DIG-proxy, LMS) and simulation conditions,
three coupling regimes formed separable clusters. Whether this reflects a
property of the systems or an artifact of the chosen projections is not
determined. Generalization not established.
These
are computational regularities, not universal invariants.
This
is one functional form among many that satisfy the same boundary conditions
(monotonicity in ε and h, rigidity penalty, chaos penalty, interior peak).
Other compressions are possible. No claim is made that this specific form
preserves all relevant information or that ε, h, D share a common dimension.
Temporal Compressibility: 𝒞
A complementary
estimator based on inter-event intervals:
𝒞 = E[ log( ψ(τᵢ | Hᵢ₋¹) / ψ(τᵢ) ) ]
τᵢ =
inter-event interval. 𝒞 is scale-dependent (discretization,
resolution). Under specific conditions it reduces to transfer entropy, mutual
information rate, or excess entropy.
3. Falsification (Programmatic)
The framework
is weakened by:
•Systematic non-replication
of R1–R3 in new simulation families or labs.
•Loss of inverted-U pattern
under parameter variation.
•Φ* > 0 in thermodynamic
equilibrium (no gradients).
4. Known Limitations
Limitation
Description
Mitigation
/ Direction
Empirical
base
Four
simulation families only
Independent
validation on real-world data required
Φ* embedding
dependence
Parameters
(dimension, delay, k) affect stability
Sensitivity
analysis required per application
𝒞 and
long-range memory
Fails for
non-stationary long-range memory
Use block
entropy estimators, Lempel-Ziv complexity, or fractal dimension methods —
domain-specific choice
Cross-domain
generalization
Not
established
Remains a
working hypothesis
5. The 𝒞 Reduction
Under specific
stationarity and process conditions, 𝒞 reduces to known
information-theoretic quantities:
Condition
Reduces To
History =
immediate past
Transfer
entropy
Markovian
order k
Mutual
information rate (standard form)
Stationary,
ergodic
Excess
entropy
Otherwise
𝒞 is
a scale-dependent estimator, not an invariant.
References
Bateson, G.
(1972). Steps to an ecology of mind. Chandler.
Friston, K.
(2010). The free-energy principle: A unified brain theory? Nature Reviews
Neuroscience, 11(2), 127–138.
Kondepudi, D.,
& Prigogine, I. (1998). Modern thermodynamics. Wiley.
Maturana, H.
R., & Varela, F. J. (1980). Autopoiesis and cognition. D. Reidel.
Prigogine, I.,
& Stengers, I. (1984). Order out of chaos. Bantam.
Simondon, G.
(2020). Individuation in light of notions of form and information. University
of Minnesota Press.
Tononi, G.
(2004). An information integration theory of consciousness. BMC Neuroscience,
5, 42.
This experiment analyzes how dynamical metrics respond to controlled perturbations in coupled systems. A controlled increase in thermal noise is applied during a specific interval. The experiment measures three projection functionals: Φ* (spectral organization), DIG (temporal autocorrelation), and coherence (self-model alignment). Results show consistent metric changes during perturbation (Φ*: -15.9%, DIG: -25.3%) and correlation (0.594) between Φ* and DIG. The experiment demonstrates co-responsiveness to global noise, not dimensional independence or epistemological validation of metrics.
Note on scope: This is a preliminary sensitivity analysis. The observed correlation may arise from shared dependence on global noise, system energy, or latent field dynamics. This experiment does not distinguish these possibilities.
1. What Is This Experiment?
The experiment analyzes how dynamical metrics respond to controlled perturbations in a coupled system. It addresses a methodological question: do these metrics change consistently when the dynamical regime shifts?
The approach: apply a controlled perturbation and observe whether metrics change consistently. Consistent response demonstrates co-responsiveness to regime shifts.
2. What the Experiment Contains
The code simulates a minimal universe with two types of systems that evolve together:
Component
Description
Continuous spectral field
The collective substrate. Dynamics governed by reaction-diffusion PDE in Fourier space.
Multiple discrete agents (toy model)
Each with internal state, synaptic plasticity (Hebbian learning), and self-model.
The coupling: The two scales interact — field influences agents, agents influence field. Perturbation affects both scales simultaneously.
3. What Is the Perturbation?
The perturbation is a controlled increase in thermal noise applied during a specific interval (turns 700-820). During this period, systems continue to evolve but temporarily lose coherence and predictability.
The hypothesis: If metrics are sensitive to dynamical regime shifts, they must change consistently during perturbation.
4. The Metrics (Projection Functionals)
Metric
Definition
What It Captures
Φ (Phi Asterisk)*
Φ* = Φ - λ·Δ, where Φ = 1 - spectral entropy
Spectral organization (spatial structure)
DIG (Dynamical Independence Gap)
Autocorrelation (lag-1 and lag-2) of agent states
Temporal memory / predictability
Coherence
Alignment between agent state and self-model
Self-model stability
Important note: Φ* and DIG are low-order statistics on correlated states of the same system. Their correlation (~0.59) may arise from:
Shared dependence on the same driver (global noise)
Shared dependence on system energy/entropy
Different filtering of the same latent field
This experiment does not distinguish these possibilities.
5. Limitations of This Experiment (Explicit)
Limitation
Implication
Global noise only
Does not separate thermal sensitivity from structural sensitivity
Correlation by construction
Metrics may correlate because they all depend on system energy/entropy
No structural variation
Topology of coupling is fixed; only noise intensity varies
Low-order statistics
Both metrics are filtered versions of the same underlying dynamics
Preliminary scope
Demonstrates co-responsiveness, not dimensional independence
What this experiment does NOT show:
That metrics capture "integration" rather than just global noise
Separation between thermal and structural regimes
That Φ* and DIG measure independent dimensions of the system
Epistemological adequacy across contexts
6. How to Run It
Run in any Python environment with: numpy, torch, matplotlib.
Step 150 | 🟢 NORMAL | Φ*=0.5167 | DIG=0.8500
Step 300 | 🟢 NORMAL | Φ*=0.5243 | DIG=0.8500
Step 450 | 🟢 NORMAL | Φ*=0.5294 | DIG=0.8204
Step 600 | 🟢 NORMAL | Φ*=0.5338 | DIG=0.8500
Step 750 | 🔴 PERTURBATION | Φ*=0.3669 | DIG=0.2783
Step 900 | 🟢 NORMAL | Φ*=0.3765 | DIG=0.8500
Step 1050| 🟢 NORMAL | Φ*=0.4226 | DIG=0.8500
Step 1200| 🟢 NORMAL | Φ*=0.4454 | DIG=0.8499
Step 1350| 🟢 NORMAL | Φ*=0.4603 | DIG=0.8500
Final analysis:
Metric
Pre-Perturbation
Post-Perturbation
Change
Φ*
0.5335
0.3749
-15.9%
DIG
0.8451
0.5918
-25.3%
Correlations:
Global (all data): 0.594
Post-perturbation: 0.654
9. Interpretation
Observation
Interpretation
Φ and DIG drop during perturbation*
Both metrics show co-responsiveness to increased global noise
Metrics recover after perturbation
System returns to previous regime when noise decreases
Correlation (0.594) between Φ and DIG*
Metrics correlate under global noise — not evidence of dimensional independence
**DIG drops more (-25.3%) than Φ* (-15.9%)**
Temporal autocorrelation is more sensitive to noise than spectral organization
10. What This Experiment Demonstrates
Demonstrated:
Metrics respond consistently to increased global noise
Metrics correlate with each other under perturbation
Metrics recover when perturbation ceases
Not demonstrated (explicit limitations):
That metrics capture "integration" rather than just global noise
Separation between thermal and structural regimes
That Φ* and DIG measure independent dimensions of the system
Epistemological adequacy across contexts
Important note on correlation: The observed correlation (~0.59) may arise from shared dependence on:
The same driver (global noise)
System energy/entropy
Different filtering of the same latent field
This experiment does not distinguish these possibilities.
11. Next Steps for Deeper Analysis
To separate sensitivity to thermal regime from structural regime, the next experiment should:
Condition
What varies
What is fixed
A
Noise intensity
Coupling topology
B
Coupling topology
Noise intensity
This would allow asking:
Do Φ* and DIG respond differently to thermal vs structural changes?
Do they desynchronize under certain conditions?
Which metric is more sensitive to each dimension?
This is the critical experiment to determine whether metrics capture structure or just energy.
12. Conclusion
The IPM Protocol demonstrates that Φ* and DIG show co-responsiveness to controlled perturbation in a coupled dynamical system. The experiment shows:
*Φ changes by -15.9%** during increased noise
DIG changes by -25.3% during increased noise
Φ and DIG correlate at 0.594* under global noise
Metrics recover after perturbation — system returns to previous regime
This is a preliminary sensitivity analysis, not epistemological validation. The experiment demonstrates consistent metric response to global noise, not that metrics capture "ontological integration" or separate thermal from structural regimes.
The observed correlation may arise from shared dependence on global noise, system energy, or latent field dynamics. Distinguishing these possibilities requires the next experiment (varying coupling topology independently of noise).
My Results:
Device: cpu
======================================================================
IPM PROTOCOL - DYNAMIC REGIME SENSITIVITY ANALYSIS
======================================================================
Systems: 4
Perturbation: turns 700 to 820
λ* = 0.25
----------------------------------------------------------------------
Step 150 | 🟢 NORMAL | Φ*=0.5165 | DIG=0.8500
Step 300 | 🟢 NORMAL | Φ*=0.5241 | DIG=0.8500
Step 450 | 🟢 NORMAL | Φ*=0.5293 | DIG=0.8019
Step 600 | 🟢 NORMAL | Φ*=0.5338 | DIG=0.8500
Step 750 | 🔴 PERTURBATION | Φ*=0.3666 | DIG=0.3085
Step 900 | 🟢 NORMAL | Φ*=0.3766 | DIG=0.8500
Step 1050 | 🟢 NORMAL | Φ*=0.4223 | DIG=0.8500
Step 1200 | 🟢 NORMAL | Φ*=0.4449 | DIG=0.8499
Step 1350 | 🟢 NORMAL | Φ*=0.4596 | DIG=0.8500
✅ Simulation complete.
======================================================================
REGIME SHIFT ANALYSIS - METRIC RESPONSE TO PERTURBATION
======================================================================
📊 PRE-PERTURBATION:
Φ* = 0.5334 | DIG = 0.8436
📊 POST-PERTURBATION:
Φ* = 0.3749 | DIG = 0.5951
📈 VARIATIONS:
Φ*: 0.5334 → 0.3749 (-15.9%)
DIG: 0.8436 → 0.5951 (-24.8%)
📊 CORRELATIONS:
Global (all data): 0.596
Post-perturbation: 0.668
──────────────────────────────────────────────────
📋 PRELIMINARY ASSESSMENT:
• DIG change: -24.8%
• Φ* change: -15.9%
• Φ* × DIG correlation: 0.596
======================================================================
LIMITATIONS (EXPLICIT):
• Demonstrates co-responsiveness to global noise only
• Correlation (~0.59) may arise from shared dependence on:
- Same driver (global noise)
- System energy/entropy
- Different filtering of the same latent field
• Does not separate thermal from structural regimes
• Metrics are low-order statistics on correlated states
• Preliminary sensitivity analysis, not dimensional validation
======================================================================
