EndToEndChain
The end-to-end chain structure packages the discrete-to-continuum bridge certificate with the curvature axioms holding under explicit hypotheses on the Riemann tensor for the Minkowski metric. Researchers deriving general relativity from the Recognition Composition Law cite this to demarcate proved steps from external assumptions. The definition directly assembles the bridge field and a conditional statement on curvature axioms without additional proof.
claimLet $C$ denote the structure consisting of a discrete-to-continuum bridge certificate together with the assertion that, for every spacetime point $x$, whenever the lowered Riemann tensor coincides with its explicit form and the Riemann trace vanishes, the curvature axioms hold for the Minkowski metric at $x$.
background
The module connects the J-cost lattice theory of Recognition Science to continuum general relativity by showing how lattice defects source the metric and curvature. The discrete-to-continuum bridge certificate includes the flat chain from J-cost to the Minkowski metric, the coupling constant eight times phi to the fifth, and the four-dimensional spacetime, along with torsion-free Christoffel symbols and the existence and uniqueness of the Levi-Civita connection.
proof idea
As a definition of a Prop structure, this declaration introduces no proof steps. It simply declares the two fields: the discrete-to-continuum bridge and the function that maps a point and the two Riemann hypotheses to the curvature axioms holding.
why it matters in Recognition Science
This structure provides the certificate invoked by the end_to_end theorem to establish the complete path from the Recognition Composition Law through J-uniqueness, the phi fixed point, three spatial dimensions, and the eight-tick octave to the Einstein field equations. It highlights the five external hypotheses (Regge convergence, Schwarz theorem, Jacobi formula, Palatini boundary terms, and MP stationarity) as standard mathematics rather than Recognition Science assumptions. The declaration closes the discrete-to-continuum bridge in the module's architecture.
scope and limits
- Does not prove Regge convergence of the lattice to the Einstein-Hilbert action.
- Does not establish smoothness of the metric for application of the Schwarz theorem.
- Does not derive the explicit form of the Jacobi determinant.
- Does not show vanishing of Palatini divergence terms.
- Does not prove stationarity of the matter action via Euler-Lagrange.
Lean usage
theorem end_to_end (h_regge : ReggeConvergenceHypothesis) : EndToEndChain where bridge := bridge_certificate h_regge curvature_axioms := fun x h_eq h_trace => curvature_axioms_hold minkowski_tensor x h_eq h_traceformal statement (Lean)
198structure EndToEndChain : Prop where
199 bridge : DiscreteContinuumBridge
200 curvature_axioms : ∀ x,
201 (∀ ρ σ μ ν, riemann_lowered_eq_explicit_hypothesis minkowski_tensor x ρ σ μ ν) →
202 (∀ μ ν, riemann_trace_vanishes_hypothesis minkowski_tensor x μ ν) →
203 CurvatureAxiomsHold minkowski_tensor x
204