H_HomogenizationLimit
plain-language theorem explainer
The declaration defines the homogenization hypothesis: simplicial recognition density converges to the square root of the metric determinant as maximum simplex volume tends to zero. Continuum-limit analysts in Recognition Science cite it to justify emergence of classical geometry from discrete ledgers. The statement is introduced directly as an epsilon-delta quantification with no lemmas or reductions.
Claim. Let $L$ be a simplicial ledger and $g$ a metric tensor. The homogenization hypothesis holds if for every $ε > 0$ there exists $ℓ_{0,max} > 0$ such that, whenever every simplex in $L$ has volume less than $ℓ_{0,max}$, then for all points $x$ the absolute difference between the simplicial density at $x$ and the square root of the absolute value of the metric determinant of $g$ (evaluated at the adjusted coordinate) is less than $ε$.
background
The module proves existence of the continuum limit for simplicial ledger transitions, with the objective that the macroscopic metric is the unique effective description of the underlying simplicial recognition density. MetricTensor supplies a local non-sealed interface whose determinant accessor is used in the limit expression. SimplicialLedger and SimplicialDensity are taken from the imported Foundation.SimplicialLedger module. Upstream, metric_det from the Hamiltonian module provides the placeholder determinant, while structures from NucleosynthesisTiers and LargeScaleStructureFromRS supply the discrete φ-tier densities and scaling that the continuum limit must recover.
proof idea
The definition directly encodes the hypothesis as a quantified statement: universal quantification over ε > 0, existential choice of a volume bound, an implication over simplices, and pointwise closeness of density to the square-root determinant. No lemmas are invoked; the body is the primitive statement itself.
why it matters
This hypothesis is the antecedent of the theorem homogenization_limit in the same module, which simply applies the definition to state the convergence. It supplies the empirical bridge required for the metric-from-density result in the Recognition Science framework. The statement connects discrete simplicial structures to the continuous volume form, consistent with the eight-tick octave and D = 3 spatial dimensions of the forcing chain, though formulated here in a 4D setting. It leaves open whether the limit holds for arbitrary ledgers or only those satisfying the Recognition Composition Law.
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