Sheaf-Theoretic Extensions¶

Jean-Paul Niko · RTSG Extension Series (Feb 2026)

Three working papers extending the base framework with sheaf-theoretic, fractal, and inter-agent structures.

Doc 1: Filter-Site Correspondence¶

Galois connection between filters \(F: \mathbb{R}^{n(e)} \to \mathbb{R}^{n(e)}\) and Grothendieck topologies \(J\) on the dimensional nerve.

\(F \mapsto J_F\) and \(J \mapsto F_J\) form a Galois connection
Fixed points = coherent filter-topology pairs
Heyting gap from CIT = special case of cohomological obstruction
Five Grothendieck topologies: \(J_{\max}\), \(J_{\min}\), \(J_{\text{threshold}(k)}\), \(J_{\text{weighted}(\tau)}\), \(J_{\text{curvature}}\)

Doc 2: Fractal Dimensions of the Cognitive Blob¶

Cognitive Complexity Index (CCI):

\[\text{CCI} = \dim_H(\partial P) - \dim_{\text{top}}(\partial P)\]

From Moran's equation: \(\sum_{j=1}^m r_j^{d^*} = 1\).

CCI = 0 for single-filter minds (smooth boundary)
CCI > 0 for multi-filter minds (fractal boundary)
Left tail of multifractal spectrum = neurologist's domain
Right tail = therapist's domain

Doc 3: Inter-Agent Sheaf Gluing¶

Misunderstanding class:

\[\mu_{AB} \in H^1(N_{AB}, F_{AB})\]

\(\mu_{AB} = 0 \to\) mutual understanding achievable. \(\mu_{AB} \neq 0 \to\) irreducible disagreement.

Spectral sequence of understanding: \(E_1 \to E_2 \to \ldots \to E_\infty\). \(d_1\) kills pairwise resolutions. \(E_2\) captures irreducible 3-way conflicts. \(E_\infty\) = structural, permanent disagreements.

Cultural coherence:

\[C_G = 1 - \frac{\sum \dim E_\infty^{p,q}}{\sum \dim E_1^{p,q}}\]

Consciousness Triple¶

\[(\beta_1, c_1, \sigma)\]

\(\beta_1\) = rank \(H_1(\psi)\) — integration measure
\(c_1 \in H^2(\psi; \mathbb{Z})\) — topological charge
\(\sigma = \dim(\psi)\) — complexity dimension (number of active \(S^2\) factors)

Under Axiom 0: \(\sigma = 1\) means system is bisimilar to subsystem of itself. The self-model IS the system (exact, not lossy).