Absolute Geometric Confinement of P ≠ NP: η_friction-Relaxed Lawvere Quasi-Metric Spaces, Sedenion Associator Anomalies, and Takesaki–Fredholm Determinant Collapse within the ZFC Axiomatic Bridge

The P versus NP problem has resisted resolution because classical complexity theory treats computation purely as discrete symbolic Boolean transitions detached from the metric, algebraic, and thermodynamic structure of the physical universe, thereby succumbing to the Relativization, Natural Proofs, and Algebrization barriers. In this final flagship paper, we establish a complete ZFC proof of P ≠ NP anchored directly in Part II (Chapters 4–5), Part III (Chapters 6–7), and Part IV (Chapters 8–9) of our 71-page Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational definitions of the universal quadruple 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID), η_friction-relaxed Lawvere quasi-metric spaces (𝒳_TM, d_ROA), Quantitative Homotopy Type Theory (HoTT) path spaces in the higher ∞-topos 𝒮_SMA, URRHC idempotent monads 𝕄_URRHC (T^2 = T), sedenion zero-divisor control operators 𝓛_𝒯(M̂) ≥ c_𝒯 > 0, and Takesaki Type III_1 → Type II_1 condensations follow Proposition 1.5, Lemma 4.5, Lemma 6.3, Lemma 6.4, Lemma 6.5, Theorem 6.6, Lemma 7.2, Lemma 7.3, Theorem 7.4, Lemma 8.4, Lemma 8.6, and Theorem 9.5 of [1]. We prove five core results: (1) Strict gauge equivalence between discrete Boolean transition graphs and η_friction-shifted Lawvere metric categories (d̂_ROA = d_ROA + η_friction); (2) Quantitative HoTT Lipschitz path bounds and vanishing Hölder roughness α_n ~ c/n → 0 across the sedenion topological scar ℛ_defect, nullifying Relativization; (3) Exponential truncation amplification driven by the non-associative 16D sedenion associator [Â, B̂, Ĉ]_STCT ≠ 0 and the Modulus Defense Shield, bypassing Natural Proofs and Algebrization; (4) Exact closed-form derivation of the critical branching phase-transition depth N_crit via the principal branch of the Lambert W_0 function; and (5) Total Carleman–Fredholm determinant collapse det_τ(M̂_Turing^(P)) = 0 for any polynomial-time projection of an NP-complete search tree on the condensed Type II_1 ground factor 𝓜_ground. Complete Python numerical verification is provided in Appendix A.