Two-Dimensional Numerical Proof of Principle for Geometry-Encoded Acoustic Direction Finding with a Second-Order Pseudohyperboloidal-Profile (PHB-2) Boundary

Conventional sound-source localization derives direction of arrival from inter-sensor delay, phase, or level differences, leaving nearly all directional discrimination to the sensor array and to post-sampling processing. This work realizes the complementary principle: the passive boundary itself performs part of the measurement before digitization, converting different arrival directions into different internal spectral and spatial states. A deliberately shaped profile thus acts as an analog encoder of direction — a performing part of the sensing chain, not a passive housing. We implement this geometry-first principle with the meridional profile of a second-order pseudohyperboloid (PHB-2). The enabling construction is exact, not heuristic. Within a restricted inherited meridional focal-conjugate ray family, successive focal transfers obey t₍ₙ₊₁₎ = κtₙ with κ = (√(1+β²)−1)/(√(1+β²)+1); for β = b/a = 2, κ = 0.381966 — a closed-form contraction factor with no free fitting parameter. The corresponding inter-branch segments converge, as sets, to the limiting chord L₀ = {(x,ρ): |x|≤a, ρ=R} with transverse error O(κⁿ). The geometry therefore marks ρ = R as a non-arbitrary, mathematically defined candidate readout location. Planar scalar-wave simulations (direct FDTD and a MEEP H_z analogue with rigid-wall Neumann conditions) show that this geometry-first boundary produces a reproducible, direction-dependent complex state whose defining signature is state organization rather than loudness. Exact PHB separates from matched controls primarily in complex amplitude–phase: normalized spectral shapes differ by ≈54.7°–56.1° (median 53.90° over 41 positions) at nearly equal norm, full-field state angles reach 25.2°–31.8° at essentially equal response proxies, and in the compact Regime-B test PHB retains larger directional-state separation at every position (mean margin +3.17°) despite a −0.17 dB response-proxy gain. A decoder frozen before held-out evaluation recovers held-out bearings within the tested ±20° sector, and the geometry-motivated readout chord places sensors by directional information content rather than local sensitivity. The central claim is deliberate and scoped: exact PHB is an exact analytical reference geometry inside a PHB-centered encoding family, while nearby near-PHB profiles prove to be the stronger aggregate performers (MAE 0.737° vs 0.975° for exact PHB and 0.991° for the equal-area smooth control). The contribution is therefore an analytically structured, reproducible framework in which geometry demonstrably performs part of the direction-finding chain — not a claim of universal accuracy superiority over conventional localization.