latticeSpacing_tendsto_zero
plain-language theorem explainer
The theorem establishes that for fixed positive length L the quantity L/N tends to zero as the division count N grows without bound. A researcher constructing the continuum limit of a J-cost lattice in recognition-based gravity would cite this to pass from discrete events to the smooth manifold description. The proof is a direct epsilon-N construction that selects N0 larger than L/epsilon and verifies the strict inequality by rewriting the division and applying linear arithmetic on the cast naturals.
Claim. For every real number $L > 0$, the lattice spacing $a(N) = L/N$ satisfies $a(N) < 0$ as $N$ tends to infinity. Equivalently, for every $ε > 0$ there exists a natural number $N_0$ such that $N ≥ N_0$ implies $L/N < ε$.
background
In the Discrete-to-Continuum Bridge the lattice spacing is the ratio of a fixed length scale L to the number of lattice divisions N. This spacing enters the J-cost construction where recognition events sit at intervals a = L/N, and the quadratic defect approximation then yields the lattice Laplacian that converges to the continuum ∇² at order a². The module sets the local setting as the three-tier bridge from J-cost lattice through Minkowski flat limit to the Einstein tensor, with Tier 1 results already proved and Tier 3 packaged as the external Regge hypothesis. Upstream, the Chain structure supplies the minimal recognition sequence while the cost definitions from MultiplicativeRecognizerL4 and ObserverForcing supply the non-negative J-cost that underlies the lattice.
proof idea
The proof is a standard epsilon-N argument. It introduces ε, obtains N0 via exists_nat_gt (L/ε), refines the witness to N0+1, unfolds latticeSpacing, casts N to reals, rewrites the target inequality with div_lt_iff₀, and closes with nlinarith on the monotonicity of the division.
why it matters
The result supplies the vanishing-spacing step required for the flat spacetime chain inside the module: J-cost lattice to quadratic defect to vanishing Einstein tensor. It supports the Tier 1 proved items (Minkowski limit, Laplacian convergence at O(a²), coupling κ = 8φ⁵, D = 3) and thereby justifies passage to the continuum metric in the Recognition Science framework. The declaration sits immediately before the flat-limit chain comment and feeds the DiscreteContinuumBridge certificate that assembles the full discrete-to-continuum path.
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