The math behind the moat.

The core algorithm behind SIGMA. Shows how to maintain structural verification in constant time per edit, no matter how large the graph gets.

Classical computation of first sheaf cohomology H^1 on a cellular complex costs O(n^3) per edit through coboundary factorization, so a stream of edits costs O(m n^3). Under a bounded local geometry assumption (bounded cell size, stalk dimension, and nerve degree), each vertex insertion, edge insertion, or restriction map update touches only a bounded set of local coboundary blocks. The result is amortized edit processing that is constant in n, with deferred exact reconciliation at query time.

Applies SIGMA to AI agents that modify their own state. Proves that structural coherence can be maintained as agents evolve, with checks running in constant time per edit.

Update-time verification for self-evolving agents. Agent state is modeled as a growing graph equipped with a cellular sheaf, and coboundary checks, H^1 obstruction, Dirichlet energy, and Laplacian spectra detect structural contradictions before commit. Under explicit policy and regularity assumptions, the paper proves conditional Lyapunov stability for density and spectral-gap dynamics, and a cellular decomposition localizes recomputation to O(1) amortized per edit, sustaining maintenance at 5,000,000 vertices.

Proves that SIGMA's verification dynamics are stable: the structural properties the engine monitors converge to predictable equilibria rather than drifting or oscillating.

Conditional Lyapunov stability for a sheaf-governed graph whose topology grows under cohomological feedback. The edge-vertex ratio converges to a system equilibrium near 3.0, and the sheaf-Laplacian spectral gap converges to a positive equilibrium of 0.530. Validated on a 500-cycle run with ADF-confirmed stationarity, multi-seed (cross-seed CV 1.9 percent), and at scale via cellular decomposition: constant per-vertex eigensolve cost from 50,000 to 1,000,000 vertices, plus a streaming update path measured at 35 microseconds per edit.

Applies the same SIGMA verification engine to protein structures. Given a predicted structure and a native reference, it checks whether the prediction is structurally coherent and localizes failures to specific residues.

SATYA Protein builds a constant cellular sheaf over the residue constraint graph, where six-dimensional stalks encode per-residue spatial displacement and signed backbone dihedral deviation. The binary SAFE or UNSAFE verdict is gated on three independent checks: contact-network preservation (native contact fraction Q at least 0.90), localized damage, and chiral integrity. Failures are localized to specific residues, with no learned parameters in the verification loop.