variational_foundation_verified
plain-language theorem explainer
A certificate instance marking the variational foundation as verified: EFE grounding, Hamiltonian formalism, and energy conservation each hold as status markers. Downstream verification theorems cite this witness. The construction fills each field with a trivial true proposition, keeping the bridge explicit without pulling sealed relativity internals into the verification layer.
Claim. There is a variational-foundation certificate whose three status markers all hold: the Einstein-field-equation grounding marker ($0 \le 0$), the Hamiltonian-formalism marker ($1 = 1$), and the energy-conservation marker ($0 + 0 = 0$).
background
The verification layer exposes a small certificate structure for the variational bridge to relativity, without importing sealed Relativity internals. The structure carries three propositional status markers: EFE emergence grounded, Hamiltonian formalism defined, and energy conserved. Each field defaults to a trivial true proposition, so the certificate is a deliberate status flag rather than a derivation of the field equations.
A companion predicate on the structure asks that all three markers hold simultaneously. This module therefore records bridge readiness as an explicit Lean object that other verification certificates can depend on, while the actual variational and relativity work stays sealed elsewhere.
proof idea
Definitional construction of a structure value. Each of the three fields is assigned a trivial true proposition: nonnegativity of zero for the EFE marker, reflexivity of one for the Hamiltonian marker, and $0+0=0$ for energy conservation. No lemmas are invoked; the terms typecheck as proofs of those Props by definitional equality and basic arithmetic.
why it matters
Gives the verification layer a named witness that the variational foundation certificate is populated. The immediate consumer is the theorem that this instance satisfies the structure's verified predicate (all three markers conjoined), proved by simp unfolding. In the broader Recognition stack this keeps the variational bridge status auditable beside other foundation certificates, without smuggling sealed relativity proofs into the verification surface. It does not itself advance the forcing chain (T0–T8) or the Recognition Composition Law; it only records that the variational side is marked ready.
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