VariationalFoundationCert
plain-language theorem explainer
A verification certificate bundling three status markers for the variational foundation: Einstein-field-equation grounding, Hamiltonian formalism, and energy conservation. Downstream verification theorems cite it to assert the variational bridge without importing sealed Relativity internals. The fields default to trivial true propositions, so the certificate is a typed status bundle rather than a deep proof.
Claim. A variational foundation certificate is a record of three propositions: (i) the Einstein field equations are grounded from the discrete variational structure, (ii) a Hamiltonian formalism is defined, and (iii) energy is conserved. The certificate is verified precisely when all three propositions hold.
background
This module sits in the Verification layer of Recognition Science. Its job is to keep the status of the variational bridge explicit as a first-class Lean object, without pulling sealed Relativity internals into the certificate surface.
The three markers name the expected pillars of a variational spacetime foundation. Upstream, Hamiltonian structure appears in two senses: as a bijective closed walk on the cube (T7 cycle realization: every vertex visited once) and as a bounded-below energy observable on an RS Hilbert space. Discrete Lichnerowicz work supplies operator-convergence status on the flat 3-torus for TT axis modes, with curved backgrounds still open. RS-native units fix $c=1$ and $\hbar=\varphi^{-5}$.
The certificate deliberately uses trivial default propositions so the type can exist and be checked while deeper grounding theorems live elsewhere.
proof idea
No proof obligation: this is a structure definition. Each field is a Prop with a trivially true default (0 \le 0, 1=1, 0+0=0). The companion verified predicate is the conjunction of the three fields, marked @[simp] so downstream theorems discharge it by simplification against a concrete instance.
why it matters
Gives the Verification layer a named, auditable handle on the variational foundation without coupling to sealed Relativity modules. Downstream, variational_foundation_verified builds a fully populated instance, and variational_foundation_is_verified proves that instance satisfies the conjunction. In the broader RS picture this sits beside T7 (eight-tick / Hamiltonian cycle on the cube) and the discrete-to-continuum gravity track (Lichnerowicz axis-sector convergence proved; curved and QNM spectra open). It is a status surface, not a substitute for EFE derivation or continuum energy theorems.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.