Paper 12: Thermal Elastocapillarity on the S3 Membrane: The Thermal Elastocapillary Length LT = G(TS)/c4, the BekensteinHawking Law as a Geometric Identity, and the ThermalMaxwell Duality

*Preface This paper is the twelfth in the Quantum Geometrodynamics (QGD) series. The programme derives gravity, electromagnetism, and thermodynamics fromthe geometry and mechanics of the S³ membrane under a unifiedelastocapillarity language. Paper~XII brings the thermal sector into this framework. Starting fromthe U (1) T gauge theory established in Paper~IV and theuniversal elastocapillarity formula of Paper~VII, the paper derives thethermal elastocapillary length\ LT = G (TS) c⁴, thermal analogue of the gravitational radius Rg = GE/c⁴, withrest energy E replaced by the thermodynamic variable TS. From this single structural substitution the following results followwithout additional postulates. enumerate Bekenstein--Hawking thermodynamics as a geometric identity. For any Schwarzschild black hole, \ LT^ (BH) = G\, TH S₁₇c⁴ = Rg2, by the 2 periodicity of the near-horizon Rindler geometry. Thermal Hooke's law. The thermal gauge four-potential acquires the Kerr--Schild correction\ _ = LTr\, k_, parallel to the gravitational and electromagnetic cases. Hagedorn threshold as a Plateau--Rayleigh instability. The ring condensate of radius aT = c/ (TS) isPlateau--Rayleigh stable if and only if LT aT, i. e. \TS EP. The extremal condition LT = aT is the Hagedornthreshold. Thermal Maxwell--elastocapillary duality. The thermal Maxwell field _ is the strain-rate tensorof the S³ membrane; the thermal Lagrangian-²/ (4ₜ) is its elastic energy density; and the thermalLorentz force is the restoring force against membrane deformation. Four condensed-matter correspondences. Thermophoresis, the pair-creation threshold, thermal censorship, andthe London penetration depth T 275\, nm in⁴He each acquire a mechanical interpretation within thesame framework. enumerate This version~ (v4) implements the sector-potential elastocapillarityprinciple in full, cleanly separates the two physically distinct thermallengths LT^ () = Kₜ S/T (within-sector) andLT^ (G) = G (TS) /c⁴ (cross-sector), and explicitly marks four openproblems (FIZ-1, MAT-1, MAT-2, FIZ-2) deferred to Paper~XIII. No new axioms are introduced. Every result follows from the establishedQGD postulates: the (H, ) vacuum, ISO (4) SO (4) symmetry breaking, Hopf quantisation c₁ = 1, null confinement|Ẋ|² = c², and the thermal gauge theory of Paper~IV.: Quantum Geometrodynamics S³ membrane thermal elastocapillarity Bekenstein--Hawking identity Hagedorn threshold thermal Maxwell duality elastocapillary length.