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 A_{n,p}^{(r)}(λ) = [gcd(i,j)^r (1 + λ δ_{v_p(i), v_p(j)})]_{1 ≤ i,j ≤ n} has exact determinant det A_{n,p}^{(r)}(λ) = ∏_{1 ≤ m ≤ n, p ∤ m} J_r(m)^{V_{p,m}+1} p^{r V_{p,m}(V_{p,m}-1)/2} Δ_{V_{p,m}}(p^r, λ), where V_{p,m} = ⌊log_p(n/m)⌋ and Δ_V(q,λ) = ((q+1)λ + q - 1)Δ_{V-1} - qλ^2 Δ_{V-2} (Δ_0 = 1+λ, Δ_1 = q(1+λ)^2 - 1) satisfies the universal Cassini identity Δ_V^2 - Δ_{V+1}Δ_{V-1} = λ^{2V} q^{V-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) = C_{1,off} e^{-γ_1 t} (3RUK CYP17A1 inhibitor abiraterone, t_{1/2} = 15.0 h) to secondary collateral drug C_2(t) = C_{2,on}(1 - e^{-γ_2 t}) (2AM9 AR antagonist enzalutamide, t_{1/2} = 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 I_{2->1}/I_{1->2} = (γ_1/γ_2)^2 = 86.1184, and 3-clone Volterra cycle integration yields the Double-Bind Extinction Wedge μ_1 < (1 - 1/α_{21}) + μ_2/α_{21}. 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 D_{KL}(S_j) = 0 across all (s,g) ∈ [0,1]^2, whereas the ordered Split-Mark vector V_j = (M_{j,L}, M_{j,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 η_{syn} surges by +53.14% (p = 8.0e-6) while sentence-level propositional load remains conserved (2.70 to 2.92, p = 0.4778).