If the oath is despised, then [O] = 0 in H₁(M). This creates a tear because the loop that once bounded a covenant now bounds a region of disobedience.

μ is left adjoint to the forgetful functor U, meaning that remembering the youth is the free construction on the torn covenant.

The shame function σ is proportional to the scalar curvature R of the covenantal metric. When R → ∞, shame → ∞. After atonement, R = 0.

When ∂M = ∅, the mouth cannot open. This is equivalent to χ(M) = 0 (Euler characteristic zero for a closed 2-manifold).

Attaching sisters increases the genus of M. The everlasting covenant has genus equal to the number of sisters attached, representing the inclusion of the nations.

Christianity

Appendix F: Lemmas and Corollaries

Lemma 1 (Oath Null-Homology)
If the oath is despised, then [O] = 0 in H₁(M). This creates a tear because the loop that once bounded a covenant now bounds a region of disobedience.

Lemma 2 (Memory is Left Adjoint)
μ is left adjoint to the forgetful functor U, meaning that remembering the youth is the free construction on the torn covenant.

Lemma 3 (Shame as Curvature)
The shame function σ is proportional to the scalar curvature R of the covenantal metric. When R → ∞, shame → ∞. After atonement, R = 0.

Corollary 1 (Silencing)
When ∂M = ∅, the mouth cannot open. This is equivalent to χ(M) = 0 (Euler characteristic zero for a closed 2-manifold).

Corollary 2 (Sisters as Handles)
Attaching sisters increases the genus of M. The everlasting covenant has genus equal to the number of sisters attached, representing the inclusion of the nations.