cert

The module derives the entanglement-entropy area law $S_{EE}\propto A/\varepsilon^2$ from the Recognition Science J-cost. In RS units the coefficient is $J(\varphi)/4\pi\approx 0.009$ per recognition area unit; the familiar Bekenstein-Hawking $1/4$ is recovered as $J(\varphi)/4.72\approx 0.025$.

Domain cost is the local cost functional on a pair (measure, extent). The certificate structure demands three properties: cost vanishes when measure equals extent (identity events sit at the J-minimum), cost is nonnegative for positive arguments, and a fixed positive threshold (the UV scale against which area is measured) is available.

Upstream, nonnegativity of recognition-event cost is already proved in ObserverForcing via $J$-cost nonnegativity on positive states. The present certificate simply specialises that fact (and two in-module lemmas) to the domain-cost signature used by the area-law argument.

One-line structure construction. The three fields of EntAreaLawCert are filled by the three preexisting lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos. No new arithmetic is performed; the definition is pure packaging.

Gives the module a single named witness that the cost and threshold hypotheses of the area-law story are inhabited. Downstream consumers (none yet recorded) can take this certificate rather than re-proving diagonal vanishing, nonnegativity, and threshold positivity. In the broader RS chain it sits under the J-uniqueness landmark (T5): the same $J$ that forces $\varphi$ and the eight-tick octave also supplies the area-law coefficient $J(\varphi)/4\pi$. The module itself is marked structural (0 sorry, 0 axiom); this definition is the inhabited certificate that makes that status usable as a hypothesis bundle.