variational_foundation_is_verified
plain-language theorem explainer
The variational-foundation certificate is fully verified: its EFE-grounding, Hamiltonian, and energy-conservation status markers all hold as true propositions. Verification-layer auditors of the RS relativity bridge cite this as the explicit pass flag. Proof is a one-line simplification that unfolds the certificate instance and the verified predicate to trivial arithmetic identities.
Claim. The packaged variational-foundation certificate, whose fields assert that the Einstein field equations are grounded, that a Hamiltonian formalism is defined, and that energy is conserved, satisfies its verification predicate (all three status markers hold).
background
This module is a thin verification certificate for the variational foundation of the RS relativity bridge. It deliberately avoids importing sealed Relativity internals; instead it exposes three explicit status markers as propositions on a structure.
The certificate structure carries: (1) an EFE-emergence marker (defaulted to the trivial inequality $0 \le 0$), (2) a Hamiltonian-formalism marker (defaulted to $1 = 1$), and (3) an energy-conservation marker. The concrete instance fills those fields with the same trivial true propositions (energy conservation as $0+0=0$).
The local setting is bookkeeping only: the certificate keeps variational-bridge status visible in the verification layer without re-proving the underlying relativity derivations.
proof idea
One-line tactic proof. simp unfolds the verified predicate on the certificate structure together with the concrete certificate instance. Each field reduces to a trivial true proposition ($0 \le 0$, $1 = 1$, $0+0=0$), so the goal closes immediately. No external lemmas are required.
why it matters
Places an explicit, machine-checked pass flag on the variational foundation inside the Verification domain. Downstream consumers of the relativity bridge can depend on this flag without pulling sealed Relativity modules into the verification layer (the module doc's stated purpose).
In the broader RS picture this sits downstream of the forcing chain's geometric and dynamical commitments (spatial dimension $D=3$, eight-tick structure, and the cost functional that feeds continuum limits). It does not itself derive the Einstein field equations or the Hamiltonian; it only certifies that those status markers have been set to true in the verification ledger. No further theorems currently depend on it (used-by is empty), so it functions as a terminal audit point rather than an intermediate lemma.
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