knet_from_covering
plain-language theorem explainer
The definition supplies the net constant for an ε-net covering of radius ε in dimension d as (1/(1-2ε))^d. Researchers applying the Coercive Projection Method to bound defects in Recognition Science models would cite this when normalizing energy gaps in three-dimensional coverings. The definition is a direct algebraic expression with no additional lemmas or reductions required.
Claim. For covering radius $ε < 1/2$ and dimension $d ∈ ℕ$, define the net constant $K_{net} := (1/(1-2ε))^d$.
background
The module formalizes the Coercive Projection Method (CPM) in three abstract parts: projection-defect inequalities, coercivity factorization via energy gaps, and aggregation from local tests to global membership. This definition supplies the concrete scaling factor that appears in part B when an ε-net controls the defect distance in d dimensions. Upstream, the eight-tick structure from Constants.tick (τ₀ = 1) and the K_net definition in Gravity.CoerciveProjection supply the related factor (9/7)^2 for the refined eight-tick case, while the present expression specializes to (4/3)^3 when ε = 1/8 and d = 3.
proof idea
This is a direct definition that returns the closed-form expression (1 / (1 - 2 * ε)) ^ d.
why it matters
The definition is invoked by the downstream theorem knet_eight_tick to obtain the concrete eight-tick value (4/3)^3. It supplies the scaling factor required for the coercivity step in the CPM core and aligns with the eight-tick octave (T7) and D = 3 (T8) landmarks of the forcing chain. The module doc-comment positions it inside the generic A/B/C logic that later instances in gravity or QFT can instantiate without measure-theoretic overhead.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.