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+# ADR-CE-001: Sheaf Laplacian Defines Coherence Witness
+
+**Status**: Accepted
+**Date**: 2026-01-22
+**Parent**: ADR-014 Coherence Engine Architecture
+
+## Context
+
+Traditional AI systems use probabilistic confidence scores to gate decisions. These scores:
+- Can be confidently wrong (hallucination)
+- Don't provide structural guarantees
+- Are not provable or auditable
+
+## Decision
+
+**Sheaf Laplacian defines coherence witness, not probabilistic confidence.**
+
+The coherence energy E(S) = Σ w_e|r_e|² provides a mathematical measure of structural consistency where:
+- r_e = ρ_u(x_u) - ρ_v(x_v) is the edge residual
+- w_e is the edge weight
+- Zero energy means perfect global consistency
+
+## Consequences
+
+### Benefits
+- Mathematical proof of consistency, not statistical guess
+- Every decision has computable witness
+- Residuals pinpoint exact inconsistency locations
+
+### Risks
+- Restriction map design requires domain expertise
+- Initial setup more complex than confidence thresholds
+
+## References
+
+- Hansen & Ghrist (2019), "Toward a spectral theory of cellular sheaves"
+- ADR-014: Coherence Engine Architecture