RTSG Equations Reference¶

Fundamental Equations¶

GL Action (Master Equation)¶

\[S[W] = \int \left(|\partial W|^2 + \alpha |W|^2 + \frac{\beta}{2}|W|^4\right) d\mu\]

The Ginzburg–Landau action governing the will field \(W\) on all three co-primordial spaces. \(\alpha\) controls the phase (ordered vs disordered), \(\beta\) controls nonlinear self-interaction.

Intelligence Equation¶

\[U = \frac{V}{E \times T}\]

Understanding = Value / (Energy × Time). The cognitive efficiency metric.

Will Field Equation of Motion¶

\[-\nabla^2 W + \alpha W + \beta |W|^2 W = 0\]

Euler–Lagrange equation from the GL action. Nonlinear Schrödinger-type on CS.

Spectral Equations¶

Graph Laplacian¶

\[L = D - A\]

where \(D\) is the degree matrix and \(A\) is the adjacency matrix. Eigenvalues \(0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n\).

Fiedler Value (Algebraic Connectivity)¶

\[\lambda_2(L) = 0.0653\]

Second-smallest eigenvalue of \(L\). Measures how well-connected the graph is. Related to the Cheeger constant by:

\[\frac{\lambda_2}{2} \leq h(G) \leq \sqrt{2\lambda_2}\]

Spectral Radius¶

\[\mu_1 = \max |\lambda_k(A)| = 1728.6 \approx 12^3\]

Connection to the modular discriminant \(\Delta(\tau)\) of weight-12 cusp forms.

Graph Energy¶

\[E(G) = \sum_k |\lambda_k(A)| = 5{,}466\]

Topological Equations¶

Betti Numbers¶

\[\beta_0 = 765, \quad \beta_1 = 2{,}065\]

\(\beta_0\) = connected components, \(\beta_1\) = independent cycles.

Euler Characteristic¶

\[\chi = \sum_{k} (-1)^k \beta_k = V - E + F = 1\]

Heat Kernel Trace¶

\[K(t) = \text{tr}(e^{-tL}) = \sum_{i=1}^{n} e^{-t\lambda_i}\]

\(t\)	\(K(t)\)	Active Modes
0.1	794.6	Nearly all modes active
1.0	178.3	Fast modes dominate
5.0	10.8	Only persistent structure
\(\infty\)	\(\beta_0\)	Connected components only (Spinoza's God)

Coherence Equations¶

Coherence Matrix¶

\[C_{ij} = \langle W_i | W_j \rangle_{CS}\]

Symmetric positive semi-definite matrix. Eigenvalues \(\{\lambda_k\}\) determine the coherence spectrum.

Coherence Score¶

\[\kappa = \frac{\lambda_{\min}}{\lambda_{\max}}\]

\(\kappa > 0.7\): Integrated
\(0.4 < \kappa \leq 0.7\): Intermediate
\(\kappa \leq 0.4\): Fragmented

Fragmentation Count¶

\[F = |\{k : \lambda_k < \epsilon\}|\]

Number of eigenvalues below threshold \(\epsilon\) (default 0.5). Each represents a dissociated mode.

Dissociation Criterion¶

\[\lambda_k \to 0 \implies \text{block diagonalization of } C \implies \text{DID structure}\]

BRST Equations¶

BRST Charge¶

\[Q^2 = 0, \quad [Q, H] = 0\]

Nilpotent, commutes with the Hamiltonian.

Physical State Condition¶

\[Q|\psi\rangle = 0, \quad |\psi\rangle \not\sim Q|\chi\rangle\]

States are \(Q\)-closed but not \(Q\)-exact.

Gauge-Fixed Action¶

\[S_{\text{gf}} = S[W] + \{Q, \Psi\}\]

where \(\Psi\) is the gauge-fixing fermion (the choice of Grothendieck filter).

Filter Equations¶

Idempotence¶

\[\mathcal{F} \circ \mathcal{F} = \mathcal{F}\]

Every Grothendieck filter is idempotent — applying it twice is the same as applying it once.

Negative Space¶

\[\text{neg}(\mathcal{F}) = \ker(\mathcal{F})\]

The topology of what the filter removes.

Conceptual Irreversibility¶

\[S(\mathcal{F}(X)) \geq S(X)\]

Entropy increases under filter application. Some collapses are thermodynamically irreversible.

Polynomial Invariants¶

F₂ Survival Rate¶

\[\sigma_{F_2} = \frac{|\{e : w(e) \not\equiv 0 \pmod{2}\}|}{|E|} = 43.5\%\]

Complex Phase Statistics¶

\[\text{mean phase} = 52.6°, \quad \text{variance} = 2.71\]

p-adic Valuation¶

\[\nu_2(\text{weights}): \text{mean} = 0.73, \quad \max = 6\]

Relation Shape Spectrum¶

The 34 × 34 relation-to-relation adjacency matrix has eigenvalues:

\[\mu_1^{(2)} = 119.9, \quad \mu_2^{(2)} = 7.0, \quad \ldots, \quad \mu_{34}^{(2)} = -47.9\]

33 independent relationship shapes exist (dimension of the relation shape space).