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\[ L_T = G(TS)/c^4,\]the 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,\[ L_T(BH) = {G\,T_H SBH}{c^4} = R_g/2,\]fixed by the 2π periodicity of the near-horizon Rindler geometry. Thermal Hooke's law.The thermal gauge four-potential acquires the Kerr--Schild correction\[ δΣ_μ = L_T/r\,k_μ,\]structurally 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. ≥ EP. The extremal condition LT = aT is the Hagedornthreshold. Thermal Maxwell--elastocapillary duality.The thermal Maxwell field Θμν is the strain-rate tensorof the S³ membrane; the thermal Lagrangian-Θ²/(4μₜ) 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 λT ≈ 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(φ) = 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,δ) 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.