H_HomogenizationLimit
H_HomogenizationLimit encodes the ε-δ condition under which the simplicial recognition density of ledger L converges pointwise to the square root of the absolute metric determinant of g as the maximum simplex volume tends to zero. Continuum-limit analysts and cosmologists deriving macroscopic geometry from discrete recognition structures would cite it. The definition is a direct universal quantification with no reduction steps or lemmas.
claimLet $L$ be a simplicial ledger and $g$ a metric tensor. The homogenization property holds when, for every $ε > 0$, there exists $ℓ_{0,max} > 0$ such that if every simplex volume in $L$ is less than $ℓ_{0,max}$, then for all coordinate points $x$, $|ρ_L(x) - √|det g(x)|| < ε$, where $ρ_L$ denotes the simplicial density extracted from $L$.
background
The module formalizes the continuum limit for simplicial ledger transitions, with the objective that the macroscopic metric $g_{μν}$ emerges as the unique effective description of the underlying simplicial recognition density. The referenced MetricTensor structure supplies a local determinant accessor det : (Fin 4 → ℝ) → ℝ. SimplicialDensity is the recognition density function on the ledger, drawn from the imported SimplicialLedger interface.
proof idea
Direct definition of the ε-δ convergence statement. No lemmas or tactics are invoked; the body is the primitive universal quantification over ε and the volume bound.
why it matters in Recognition Science
This supplies the hypothesis for the sibling homogenization_limit theorem, which simply applies the assumption to obtain the stated convergence. It occupies the meta-homogenization interface that links discrete simplicial structures (LedgerFactorization, PrimitiveDistinction) to the continuum metric, supporting the Recognition Science emergence of effective geometry from the phi-ladder and eight-tick scaling. It touches the open question of which ledgers obey the limit.
scope and limits
- Does not prove the limit holds for any concrete ledger.
- Does not specify the explicit form of SimplicialDensity.
- Does not derive the metric tensor from recognition axioms.
- Does not quantify convergence rates or error terms.
formal statement (Lean)
38def H_HomogenizationLimit (L : SimplicialLedger) (g : MetricTensor) : Prop :=
proof body
Definition body.
39 ∀ ε > 0, ∃ ell0_max > 0,
40 (∀ simplex ∈ L.simplices, simplex.volume < ell0_max) →
41 ∀ x, abs (SimplicialDensity L x - Real.sqrt (abs (metric_det g (fun i => if i.val = 0 then 0 else x i)))) < ε
42
43/-- **THEOREM: Metric from Density (Homogenization)**
44 As the simplicial scale $\ell_0 \to 0$, the simplicial recognition density
45 converges to the volume form of the macroscopic metric $g_{\mu\nu}$. -/