DiscreteContinuumBridge
DiscreteContinuumBridge packages the flat chain from J-cost lattices through Minkowski metric to vanishing Einstein tensor, the coupling 8 phi^5, dimension equality 1+3=4, torsion-free Christoffel symbols, Levi-Civita existence and uniqueness on Minkowski, plus the external Regge convergence hypothesis. Researchers deriving continuum GR from discrete Recognition Science models cite this certificate to justify the coarse-graining limit. The structure is assembled by direct field assignments from upstream proved lemmas in the same module and the
claimA structure asserting the flat chain from J-cost to Minkowski metric with vanishing Christoffel, Riemann, Ricci, and Einstein tensors; the inequality $8 phi^5 > 0$; the dimension equality $1 + 3 = 4$; the torsion-freeness of Christoffel symbols for the Minkowski metric tensor; the existence and uniqueness of the Levi-Civita connection for the Minkowski tensor; and the Regge convergence hypothesis that simplicial Regge actions converge to the Einstein-Hilbert action.
background
Recognition Science derives continuum geometry from discrete J-cost lattices. The J-cost satisfies the Recognition Composition Law and expands as J(1 + ε) = ε²/2 + O(ε⁴) in the flat limit, sourcing quadratic defects that induce the spatial metric δ_ij and the Minkowski tensor η = diag(-1, +1, +1, +1). The module connects this lattice to continuum GR via three tiers: Tier 1 proves the flat chain, coupling κ = 8φ⁵, and D = 3 from ContinuumManifoldEmergence; Tier 2 imports Christoffel, Riemann, Ricci, and Levi-Civita results from Curvature and LeviCivitaTheorem modules; Tier 3 packages the external Regge convergence hypothesis from Cheeger-Muller-Schrader 1984 as an explicit Prop.
proof idea
The structure is populated by direct field assignments: flat_chain from the proved flat_chain_holds lemma, coupling_positive from coupling_from_phi, dimension by norm_num, christoffel_torsion_free from levi_civita_torsion_free, levi_civita_exists and levi_civita_unique from the FundamentalTheorem results, and regge_convergence left as the external hypothesis. No additional tactics are required beyond these references.
why it matters in Recognition Science
This structure earns its place as the complete certificate bridging Recognition Science lattices to general relativity, feeding directly into the bridge_certificate theorem and the EndToEndChain that traces from the Recognition Composition Law through phi, the eight-tick octave, D = 3, to the Einstein field equations. It isolates the Regge convergence hypothesis in Tier 3, allowing the remainder of the chain to remain proved. The declaration touches the open question of whether nonlinear Regge calculus rigorously converges to Einstein-Hilbert without further assumptions.
scope and limits
- Does not prove the Regge convergence hypothesis, which remains external.
- Does not address curved backgrounds beyond the weak-field and flat limits.
- Does not incorporate quantum corrections or higher-order lattice effects.
- Does not specify the rate of convergence in the mesh limit.
Lean usage
theorem bridge_certificate (h_regge : ReggeConvergenceHypothesis) : DiscreteContinuumBridge where flat_chain := flat_chain_holds coupling_positive := coupling_from_phi dimension := by norm_num christoffel_torsion_free := levi_civita_torsion_free minkowski_tensorformal statement (Lean)
153structure DiscreteContinuumBridge : Prop where
154 -- Tier 1: PROVED (from ContinuumManifoldEmergence and this module)
155 flat_chain : FlatChain
156 coupling_positive : (8 : ℝ) * phi ^ (5 : ℝ) > 0
157 dimension : (1 : ℕ) + 3 = 4
158
159 -- Tier 2: PROVED (from Curvature.lean, LeviCivitaTheorem.lean)
160 christoffel_torsion_free : IsTorsionFree (christoffel minkowski_tensor)
161 levi_civita_exists : FundamentalTheoremExistence minkowski_tensor
162 levi_civita_unique : FundamentalTheoremUniqueness minkowski_tensor
163
164 -- Tier 3: HYPOTHESIS (external mathematics)
165 regge_convergence : ReggeConvergenceHypothesis
166
167/-- The bridge certificate is inhabited (modulo the Regge hypothesis). -/