Unconditional Watson-Band Rigidity for Bird-Map Borel Transforms (Paper 35 in the Non-Holomorphic Fractal Series)

Paper 35 in the Non-Holomorphic Fractal Series. We close the two conditional hypotheses left open in Paper 34. Theorem A (Pressure bridge): we prove, unconditionally, that ρ(h_W) = r(L_{h_W,1}), where ρ(h_W) is the exponential type of the Watson-class Borel ladder and r(L_{h_W,1}) is the spectral radius of the twisted transfer operator on BV. The proof uses the explicit Borel-ladder matrix-element representation together with Baladi Theorem 2.2, the Ruelle–Perron–Frobenius growth estimate (Proposition 2.3), and a tail-averaging lemma showing that the Padé–Borel extraction error is o(r^n). Theorem B (Certified spectral gap): a computer-assisted proof using Python interval arithmetic gives a certified lower bound ŷ≥ŷ_cert > 0 for the spectral gap of L_{h_W,1} on BV, resolving Paper 34 Hypothesis 3.3. Theorem C (Explicit I_rig width): tracking the Keller–Liverani continuity modulus with the explicit BV-norm dependence on ε yields the quantitative bound |I_rig| ≥ 0.06, so I_rig ⊃ [0.84, 0.90]. Together, Theorems A, B, and C yield the first unconditional theorem: Watson-band rigidity, ρ(h_W, ε) ∈ B_W = [0.79, 0.87], holds for all ε ∈ [0.84, 0.90].