One Density In, One Coupling Out: The Price of Jacobson's Equation of State

Jacobson's letter does a real thing. It shows the Einstein equation is the equation of state that spacetime has to obey if entropy rides on horizon area and heat obeys Clausius. That is a structural result and this paper does not touch it. What this paper prices is the load path. The letter assumes the entropy density. It does not derive it. It imports the value from Bekenstein and Hawking, η = 1/(4L_P²), and it says so in its own words. Feed that density through the construction and Newton's constant comes out as G = 1/(4η). Vary the assumed density by any factor f and the implied G moves by exactly 1/f. The elasticity is exactly -1, by the form of the relation itself. The output coupling is the input density read backwards, and the number of independent digits of G the derivation produces is zero. The Clausius relation is demanded on every local Rindler horizon through every point. The number of those equilibrium conditions the letter derives is zero. The cosmological constant comes out the way it went in, as a free constant, and the letter's own verdict on it is "as enigmatic as ever." One density in, one coupling out, Λ untouched. Verdict: a reason was found, and the question remains open (not solved).