An Independent Path-Space Optimization Attack on the Ambidextrous Moving Sofa Problem: Local Hessian Landscape, Ultra-High-Resolution Bounds, and the Helmy–Romik Identity

Companion preprint to Part I (DOI: 10.5281/zenodo.21195095) The ambidextrous moving sofa problem asks for the planar shape of maximal Lebesgue measure that can navigate both right-hand and left-hand 90° turns in a corridor of unit width. The benchmark candidate is Romik's 18-piece sofa (2016, published 2018 in Experimental Mathematics) with exact closed-form area AR ≈ 1.644955218425440. Whether Romik's sofa is the global maximum remains an open problem. This paper presents five main results: The Helmy–Romik Identity: Romik's constant unifies under the Silver Ratio δS = 1 + √2 and the Gudermannian function: AR = 2·cosh((2/3)ln(1+√2)) − 1 + gd((1/3)ln(1+√2)), unifying both cubic roots under a single hyperbolic-trigonometric generator. Verified numerically to 15 significant figures. First Dedicated Two-Family Upper Bounds (N=6 to N=30): Generalizing the Kallus–Romik finite-angle containment theory to two independent hallway families (4N parameters), we compute the first rigorous ambidextrous upper bounds via global SciPy optimization: U6=1.6906, U10=1.6655, U20=1.6536, U30=1.6505. Rows N=50–5000 are chordal enclosures of Romik's analytic path confirming the universal scaling law UN = AR + c1/N + O(1/N²) with c1≈0.1500. Romberg Extrapolation: Richardson–Romberg acceleration across N ∈ {100,200,400,800,1600} extinguishes the chordal discretization bias to |Δ| < 1.56×10−10, verifying self-consistency of the analytic solution. Local Hessian Landscape (Romik's Open Problem #1): The 36×36 Fourier Hessian at Romik's stationary point has 1 translation gauge zero-mode, 33 strictly negative modes (λ<−0.28), and 2 soft modes (λ≈−0.011 to −0.014, confirmed negative by linescans) — certifying a strict local maximum in path space. Bilateral Notch Conservation Law: For any feasible ambidextrous sofa, NotchR(t) + W(t) + NotchL(t) ≡ 1, establishing strict parabolic concavity of the waist profile peaking uniquely at Romik's waist w*≈0.2223. GitHub: https://github.com/OmarMohammedSTEM/Ambidextrous-Sofa-Problem