K-Matrix — Intra-Agent Gain Kernel¶

Jean-Paul Niko · RTSG v8

The K-matrix encodes how intelligence dimensions interact within a single agent.

Definition¶

\(K\) is an \(8 \times 8\) symmetric matrix with:

\(K_{ss} = 1\) always (self-gain is baseline, enforced by Hebbian \(s \neq t\) constraint)
\(K_{st} > 1\) = synergy (dimensions amplify each other)
\(K_{st} < 1\) = interference (dimensions suppress each other)
Entries \(\geq 0\), but K is NOT positive semi-definite

Key Properties¶

Suppression spectrum: By Perron-Frobenius, \(\lambda_1 > 0\) with positive eigenvector. But if \(\lambda_k < 0\), there exists a direction \(v_k\) in intelligence space where the gain kernel is a net suppressor — the harder the agent pushes in that direction, the worse the outcome.

Observable consequences:

Gifted-but-struggling: High raw capacity but K suppresses certain type-combinations
Savant profiles: Extreme \(\lambda_1\) (dominant mode) with deep negative \(\lambda_8\)
Therapeutic target: Pathological K (PTSD, addiction) creates deep negative eigenvalue. Therapy goal: modify K until \(\lambda_{\min} \geq 0\)
Invisible to psychometrics: \(K^2\) is always PSD — suppression info lost in the K \(\to\) R bridge

K \(\to\) R Bridge (Bifactor Generative Model)¶

\[\Sigma_I = \rho(K\mathbf{1})(K\mathbf{1})^T + (1-\rho)K^2\]

Population correlations (R) are generated by K acting on shared resources (\(\rho\)). First term = rank-1 g-factor explaining ~40-60% of variance. IQ works not because intelligence IS one thing, but because cross-type synergies create correlations.

J-Matrix (Inter-Agent Coupling)¶

\[J_{st} = 1 + \eta(K_{st} - 1)\]

\(\eta\) = communication bandwidth \([0,1]\). \(\eta=0 \to\) isolated. \(\eta=1 \to\) full collaboration.

Surplus Adjacency¶

\[A_{st} = \max\{0, K_{st} - 1\}, \quad A_{ss} = 0\]

Used in all graph/routing formulations. Communicability \(= e^{\beta A} C\).

Hebbian Consolidation¶

\[dK_{st}/dt = \eta \cdot \alpha_s \cdot \alpha_t \quad (s \neq t)\]

Co-activation strengthens coupling. Off-diagonal only.

Addiction = Hebbian runaway. PTSD = K-matrix scarring from extreme single event.