Three-Pass Sparse Hypercubes, Fixed-Point Cayley-Dickson Zero-Divisor Gates, and Constructively Verified Interval Certificates for Supercritical Unforced 3D Navier-Stokes Flows

Computer-assisted regularity proofs and formal blow-up audits for the three-dimensional incompressible Navier–Stokes equations on T³ encounter two computational bottlenecks: (i) dense six-dimensional Cartesian wavevector triad convolutions (p + q = k on Z³ \ {0}) expend >99.9% of memory and arithmetic on algebraically inactive or symmetry-cancelled interactions, and (ii) conventional floating-point reductions obscure rounding drift while scalar energy netting conceals whether residual PDE bounds depend on external forcing terms (f := D[u,p]). Bridging functional R pipelines with multipass symbol-resolution techniques from 16-bit W65C816S assembler design (Asar/xkas/TRASM) and 64-byte Q8.8 fixed-point arithmetic, we introduce the BSA-R sparse hypercube compiler paired with an un-netted Eleven-Entry Audit Ledger. First, we evaluate dyadic shell cascades (k_n = 2^n) under scale-tailored external forcing (f*(x,t) ∝ k_n^1.65, Clay Option C/D) and verify in Base R that Ladyzhenskaya's (1967) shear-thickening modification ν_eff = ν_0 + ν_1 |∇u|² uniformly bounds shell vorticity below α/ν_1 = 3.0. Second, using outward-rounded IEEE-754 interval arithmetic, exact rational 3D Leray projection P_k (‖P_k‖ = 1), 64-byte Q8.8 16D Sedenion and 32D Pathion zero-divisor witnesses (7 octahedral Box-Kites, 42 assessors, 168 annihilation edges), and coupled infinite-dimensional Wiener algebra F⁰(T³) tail defect bounds, we establish computer-assisted global C^∞([0,∞) × T³) regularity certificates in the strictly unforced setting (f ≡ 0, Clay Option B) for supercritical benchmark initial data exceeding the Lei–Lin threshold (‖u_0‖F⁰ > ν = 0.50), eliminating 25,200,369 of 25,200,451 dense Cartesian triad candidates (99.99967% sparsity). Third, across six constructive Lean 4.35.0 modules (zero Classical.choice) cross-verified via Z3, we prove: (R1) the universal Moving Viscous Ceiling, Λ_BCC sublattice dilation (√3 ν), and Fourier-sparsity cool-down (N·E < ν² R²); (R2) infinite-dimensional perturbation Riccati balls and 3D Binet–Cauchy sin(θp,q) triad depletion; (R3) outward-rounded enclosure of the lattice Agmon–Wiener constant κT³ ∈ [6.183616, 6.469663], Foias–Temam Gevrey-1 ultraviolet decay, and the zero-step broadband global gate (17,076 modes at 4.50ν); (R4) centroid-shifted Laplacian spectral-variance depletion Υ_var = Υ·Θ² annihilating monochromatic shells at 10,000ν and reaching Tao's (2016) bimodal barrier; (R5) physical-space 3D trace-free strain (9× quartic sharpening 27/128 → 3/128) and Constantin–Fefferman Biot–Savart commutator bounds certifying scale-separated bimodal flows (m_2 > 4 m_1, up to 1,000ν); and (R6) singularity-free polar (Ω, ξ) palinstrophy splitting P = P_amp + P_dir (26.47×–28.56× ceiling widening), zero-gap four-channel Bony paraproduct UV barriers certifying supercritical 10-octave and 23-shell gapless cascades (mⱼ₊₁ ≤ 2 m_j), and C1/2 direction-cone D³ → D² Leray window-doubling closure. Explicit scope disclosures isolating unconditional ν → 0⁺ cone integrability on the single S² vorticity-direction PDE accompany our open-source verification artifacts.