The Price of a Fact: Information, Action, and Gravitation at the Level of One Event

Law. Reality advances by posting facts. We state one constitutive law for the elementary posting: the physical source covector reads one oriented occurrence as the action-dual of one native action quantum. In native units this says

S = φ⁵ = ℏ⁻¹, Sℏ = 1.

The statement is Primitive Occurrence-Action Flux Duality. Its tier is Law. It is an adopted physical postulate.

Theorem. The occurrence itself has already been isolated. One legal ledger tick is one owner-preserving channel insertion, and that insertion maps injectively to one oriented pair-kernel posting. The remaining normalization is exactly the law above. A machine-checked circularity test proves that the upstream package does not force the law.

Prediction (from Law). The law rejects the unit-source candidate. This rejection is a consequence of the postulate, not evidence for it.

Theorem. A discrete Green construction supplies the potential coefficient 1/(4π) independently of the law.

Theorem under Model (Bridge 2). The law supplies the per-event source scale. With the explicit four-phase continuum aggregate, a Model hypothesis, the resulting coupling gives

λ_rec/ℓ_P = √π.

For a general positive aggregate factor k, the ratio is 4π/k. Thus the √π signature occurs exactly at k = 4.

Measured. Nine derivation doors were mined under a frozen gate. Every door died at the same point: the current formal corpus has no physical interface that evaluates the source covector through the proposed stack. Thermal, information-theoretic, measure, capacity, geometric, modular, Hessian, and cascade routes were all premise-ungrounded. A cross-family adjudicator confirmed each verdict and the corpus-wide isolation fact. Two door refutations, for Clausius balance and Landauer calorimetry, are machine-checked theorems. This is corpus-relative exhaustion. It is not a universal impossibility proof. The result licenses a light-postulate architecture: state the primitive physical lock, derive its exact consequence, and keep the derivation problem open.