This paper provides explicit proofs of five theorems in the Inverted Hypersphere Cosmology (IHC) series and closes six foundational gaps identified by external critique. Three new lemmas establish the complete logical chain from axiom to RP4: Lemma C proves existence is forced by the spinodal instability of the trivial vacuum Psi=0 under the conformally-coupled quartic action, with Bunch-Davies quantum fluctuations providing the seed (Bunch-Davies 1978); Lemma D proves Euclidean metric signature is the unique signature consistent with the non-preferential axiom via the time-orientation argument and the Poincare-Hopf theorem applied to S4; Lemma E proves the antipodal map is the unique free Z2 action on S4 consistent with the axiom, using Milnor's (1957) classification of free involutions on spheres. Two existing lemmas (A and B) are retained and strengthened: Lemma A proves non-preference forces maximum symmetry (10 Killing vectors in 4D); Lemma B proves S4 is the unique compact simply-connected positively-curved 4-manifold. A declared framework remark establishes that the smooth, compact, boundaryless Riemannian category is not an additional assumption but the unique category consistent with the axiom. The complete T1 proof chain is A1 -> Lemma C (existence) -> Lemma D (signature) -> Lemma A (symmetry) -> Lemma B (S4) -> Lemma E (antipodal) -> RP4. Proofs of T3 (scheme independence), T4 (anti-periodic boundary conditions and xi=1/6), T5 (strong-CP dissolution), and T8 (N=33 from vacuum self-consistency) are unchanged.