Operational Reconstruction of Jacobson's Equation of State by Overlapping Finite Observers

An earlier finite-observer extension of Jacobson thermodynamics introduced coherence-dependent temperature and gravitational-coupling corrections, an added record-entropy contribution, and stochastic geometric terms. Adversarial testing did not support those structures as derived gravitational effects. This paper supersedes that extension and asks a narrower operational question: can an embedded finite observer architecture instantiate Jacobson’s local thermodynamic argument without misidentifying detector, memory, transport, or overlap errors as spacetime curvature? The observer is modeled as a connected overlap atlas rather than an isolated point system. Local heat and area records are transported through matched causal channels, compared across shared null directions, retained in metastable memory, and tested against explicit horizon-focusing dynamics. In finite regulators, matched smearing preserves the Clausius relation, the resolved first variation reproduces Jacobson’s null equation of state, and a sufficiently complete null-direction bank reconstructs the corresponding tensor relation up to the usual cosmological-constant ambiguity. Individual observers are rank deficient, whereas the connected overlap atlas reconstructs the tensor relation and tolerates local corruption and substantial observer loss. Continuous microscopic record turnover leaves the gravitational coupling unchanged when the maintained coarse entropy density is locally stationary. Generic time-dependent entropy-density corrections fail tensor integrability, horizon-choice independence, or conservation and therefore do not define observer-dependent modified gravity. The numerical entropy density and Newton’s constant remain external physical inputs. The surviving result is a finite-regulator operational reconstruction of ordinary Jacobson gravity, together with explicit fault-tolerance conditions, falsification criteria, and boundaries on what the observer architecture does not derive.