The Master p-Adic Deformed Arithmetic Diamond Pencil, Bivariate Cross-Washout Containment, and Noisy Split-Mark Channel Capacity

We establish three exact analytical theorems across algebraic number theory, evolutionary mathematical biology, and statistical psychometrics: The Master p-Adic Deformed Arithmetic Diamond Pencil (16+ OEIS Sequences): For any prime p ≥ 2, exponent r ≥ 1, and deformation parameter λ ∈ C, the p-adic valuation-coupled GCD matrix An,p⁽ʳ⁾(λ) = [gcd(i,j)^r (1 + λ δv_p(i), v_p(j))]1 ≤ i,j ≤ n has exact determinant det An,p⁽ʳ⁾(λ) = ∏1 ≤ m ≤ n, p ∤ m J_r(m)^{Vp,m+1} p^{r Vp,m(Vp,m-1)/2} Δ_{Vp,m}(p^r, λ), where Vp,m = ⌊log_p(n/m)⌋ and Δ_V(q,λ) = ((q+1)λ + q - 1)ΔV-1 - qλ^2 ΔV-2 (Δ_0 = 1+λ, Δ_1 = q(1+λ)^2 - 1) satisfies the universal Cassini identity Δ_V^2 - ΔV+1ΔV-1 = λ2V qV-1. Specializing (p,r,λ) unifies Pisot sequences T(2, 4q-1) (OEIS A020727, A022026, A234279), hollow Gaussian Lucas polynomials (-1)^V U_V(-2,q) (A009545, A088137, A088138, A045873, A088139, A098656), alternating sign-resonance sequences (A087170, A370191), and the Dual LCM Arithmetic Diamond Matrix L_n⁽ʳ⁾ = [i^r j^r / gcd(i^r+j^r, |i^r-j^r|)] at λ = -1/2. Bivariate Poisson-Bernoulli Cross-Washout & 86.12x Overlap Asymmetry: During an adaptive therapy switch from primary drug C_1(t) = C1,off e-γ_1 t (3RUK CYP17A1 inhibitor abiraterone, t1/2 = 15.0 h) to secondary collateral drug C_2(t) = C2,on(1 - e-γ_2 t) (2AM9 AR antagonist enzalutamide, t1/2 = 5.8 d = 139.2 h), the nonlinear crowding integral expands as an exact expectation over (M_1, M_2) ~ Poisson(Λ_1) × Poisson(-Λ_2), the normalized dual-exposure toxicity impulse obeys I2->1/I1->2 = (γ_1/γ_2)^2 = 86.1184, and 3-clone Volterra cycle integration yields the Double-Bind Extinction Wedge μ_1 < (1 - 1/α₂₁) + μ_2/α₂₁. Noisy Split-Mark Channel Capacity & Diachronic Clause-Bitrate Conservation: Under any stochastic DINA channel with slip s and guess g, antipodal states η = (1,0) and η′ = (0,1) have zero scalar-sum Kullback-Leibler divergence DKL(S_j) = 0 across all (s,g) ∈ [0,1]^2, whereas the ordered Split-Mark vector V_j = (Mj,L, Mj,R) achieves symmetric Jeffreys divergence J(V_j) = 2(1-s-g) log_2((1-s)(1-g)/(sg)) bits (15.29 bits at s=g=0.05). Across paired 1800-2026 English prose, per-word syntactic coding efficiency ηsyn surges by +53.14% (p = 8.0e-6) while sentence-level propositional load remains conserved (2.70 to 2.92, p = 0.4778).