I built a system called SIGMA that uses first sheaf cohomology, H^1, to detect structural contradictions in AI knowledge systems. These are contradictions where no assignment of local values can produce global consistency. When SIGMA detects one, it grows new structure at the obstruction site.

Across a 500-cycle autonomous run, the system grew from 150 to 515 vertices with zero crashes.

I now have two Lyapunov stability propositions:

• one for structural density convergence, E/V -> 3.0
• one for spectral gap convergence, lambda_2 -> 0.530 > 0

Both are conditional on explicitly stated assumptions that were verified empirically, but not yet derived from first principles. Taken together, the result yields a sheaf expansion proposition for a restricted class of algebraically certified sheaves.

This post explains what the system does, what was proven, and where I think I may be wrong. I am looking for technical feedback.

Why this might matter for AI safety

Most current AI safety tools are statistical: a probabilistic system evaluating another probabilistic system. Patronus AI, Galileo, Arthur AI, and Arize all rely on confidence scoring, embedding anomaly detection, or LLM-as-judge style evaluation.

SIGMA does something different. A non-zero first cohomology class, H^1 > 0, is not a confidence score. It is a structural proof that the system's knowledge cannot be made globally consistent by any choice of local values. The contradiction cannot be resolved by reweighting confidence. It requires structural change.

That may be relevant to the "formal verification" gap identified by Dalrymple et al. (2024) in "Towards Guaranteed Safe AI."

The Coherence Margin, a real-time observable derived from the coupled stability results, provided 1 to 70 cycles of advance warning before every major contradiction spike in the run. That turns an abstract stability result into an operational predictive signal.

The core mechanism

A minimal example:

Take four vertices A, B, C, and D arranged in a cycle. Each carries a 2D vector space. Three edges use the identity map. One edge uses

R = diag(1, -1)

A global section must satisfy:

s_A = s_B = s_C = s_D

from the identity edges, and also

R s_A = s_A

from the flipped edge.

The fixed points of R are vectors of the form (x, 0). So the second coordinate is obstructed. No assignment satisfies all constraints. That is a topological fact, not a statistical one.

Dirichlet energy localizes the problem:

• identity edges have zero energy
• the flipped edge has energy 4

So the system can determine not only that a contradiction exists, but where it lives.

Resolution is structural:
insert a vertex X between D and A

After that:

• H^1 drops from 1 to 0
• energy drops from 4 to 0
• no parameters are adjusted
• new structure is added

SIGMA automates this process at scale.

What was proven as of April 2026

Theorem 1:
E/V converges to r* = 3.0 via the Lyapunov function

V(r) = (r - r*)^2

within the Borkar stochastic approximation framework.

Theorem 2:
lambda_2 of the sheaf Laplacian converges to 0.530 > 0 via the Lyapunov function

W = (lambda_2 - lambda_2*)^2

The supporting eigenvalue perturbation lemma bounds the spectral effect of batch structural events on Purity-Gate-certified sheaves.

Corollary:
SIGMA simultaneously stabilizes structural density and spectral connectivity. This gives a sheaf expansion result for a restricted sheaf class, while avoiding the First-Kaufman (STOC 2024) impossibility result for arbitrary sheaves.

Important limitation:
both theorems are conditional on assumptions including monotone feedback, bounded noise, and stationary rate balance. Those assumptions were verified empirically on a single 500-cycle trajectory, but they are not yet derived from first principles.

Where I think I might be wrong

1. The monotonicity assumptions are empirically verified, not derived. That is the main gap between a strong proposition and a self-contained theorem.
2. The 500-cycle result is currently a single-seed run. I do have a separate earlier adversarial test with three seeds and coefficient of variation equal to 4.9 percent, but multi-seed validation for this run is still needed.
3. The spectral obstruction proxy mu is not the standard dim H^1. The stability argument does not depend on that distinction because it uses E/V and Dirichlet energy, but it does mean the H^1 values in the trajectory tables should be understood as proxies.
4. The current demo operates on structured inputs, not unstructured production data.
5. I may be overestimating the novelty after spending more than eight months inside the project.

What I am looking for

I am looking for technical scrutiny, especially on three questions:

1. Is a Lyapunov result for a self-modifying topological dynamical system genuinely novel?
2. Is the First-Kaufman navigation result, via a restricted sheaf class with algebraic certification, a real contribution or just a trivial observation?
3. Does the Coherence Margin formalization

CM = lambda_2 * resolution_rate - creation_rate * density

have value as a predictive observable, or is it just an artifact of this parameter regime?

Preprint:
https://zenodo.org/records/19416603

Patent pending:
U.S. Provisional 64/023,418