System Topology Notes — Version 1.0 — December 2025
High-resolution mechanistic description (for direct internal use or future appendix)
Let R₈ denote the invariant 8-dimensional rigidity manifold defined by the eight canonical cognitive mechanisms: M = {Identity-Protective Cognition, Confirmation Bias, Dissonance Avoidance, Motivated Reasoning, Selective Exposure, Naïve Realism, Groupthink Amplification, Authority Deference}
All three wheels are embeddings of the same dynamical system on M, differing only in observation scale and control inputs.
- Layer 0 – Epistemic Rigidity (individual cognitive attractor) State vector: r(t) ∈ [0,1]⁸ Update rule: drᵢ/dt = α·(threatᵢ − evidenceᵢ)⁺ − β·disconfirmationᵢ Fixed-point manifold: r* ≈ 1⁸ under chronic identity threat (the “rigid crystal”).
- Layer 1 – 3B Behavior Modification (intervention layer) Same state vector r(t), but now with exogenous control ports c(t) ∈ ℝ⁸ (Contrastive Inquiry pulses, dissonance amplification, behavioral overrides). Modified dynamics: drᵢ/dt = α·(threatᵢ − evidenceᵢ)⁺ − β·disconfirmationᵢ − γ·cᵢ(t)·(1 − rᵢ) The 3B wheel is therefore the Epistemic Rigidity wheel with a signed intervention tensor overlaid on identical spokes. Phase portrait shows controlled escape trajectories from the rigid crystal.
- Layer 2 – Adversity Nexus (organizational-scale attractor) State vector R(t) = (1/N) Σⱼ rⱼ(t) → aggregate rigidity across N agents The eight spokes are preserved via homomorphic mapping: each organizational phase is a quadrant dominance pattern on the same M. Rise → low aggregate rigidity, symmetric spoke activation Peak → moderate rigidity, two dominant spokes (usually Identity + Authority) Stagnation → ≥5 spokes > 0.78 threshold (simulated collapse condition) Crisis/Adaptation → forced symmetry breaking via external shock or Layer 1 intervention cascade.
Invariance of the 8-spoke topology The topology is forced by the universality of the eight mechanisms in human social cognition (empirically saturated set; no ninth mechanism has survived attempted addition in 200+ adversarial runs). Any contraction or expansion of M destroys cross-layer causal consistency and produces immediate logical cascades in simulation.
Stability & falsifiability consequence of nesting A single contradiction in any spoke at any layer must propagate instantly to the other two layers because the state space is identical. The entire suite therefore has only one global attractor manifold. This raises the Kolmogorov complexity of successful adversarial attacks exponentially (observed: >1.2 × 10⁶ tokens required to induce cascade failure vs. <8 k tokens for Servant, Transformational, or Authentic leadership models).
Contrastive Inquiry as universal release valve Contrastive Inquiry is formally a signed impulse operator Δcᵢ(t) = κ·(P − ¬P | evidence) applied to spoke i. When applied simultaneously to k spokes, it generates a controlled Lyapunov exponent < 0, forcing descent from the rigid crystal along predefined escape corridors. Because the operator is scale-invariant, the same impulse collapses individual rigidity, disrupts 3B intervention resistance, and triggers organizational Adaptation phase with identical mathematical signature.
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