realizedDefectCollapseScalar
plain-language theorem explainer
The realized defect collapse scalar for a nonzero-charge sensor and ring count N is the reciprocal of one plus the annular cost of the N-ring mesh from the canonical defect sampled family. Researchers working on the T1 exclusion argument in the unified RH architecture cite this scalar to track the forced approach of the collapse observable to zero under cost divergence. The definition is a direct one-line wrapper around annularCost applied to the sampled family mesh.
Claim. Let $s$ be a defect sensor with nonzero charge and let $N$ be a natural number. The realized collapse scalar is defined by $c_N = 1 / (1 + A_N)$, where $A_N$ is the total annular cost of the mesh produced by the canonical defect sampled family for sensor $s$ at level $N$.
background
The Unified RH module builds a T1-bounded realizability architecture. It replaces an earlier ledger asserting bounded total annular cost with three components: cost divergence forced by nonzero charge, Euler trace admissibility from instantiation data, and a physically realizable ledger whose scalar proxy stays bounded. The defect functional equals J(x) for positive x. Annular cost sums the individual ring costs over an N-ring mesh. The canonical defect sampled family supplies the mesh for each sensor and level N.
proof idea
One-line wrapper that applies annularCost to the mesh of the canonicalDefectSampledFamily sensor hm at N.
why it matters
This scalar supplies the concrete collapse observable that enters the direct T1 exclusion theorem, which proves its defect cannot remain uniformly bounded. It also populates the Euler ledger scalar state definition in the nonzero-charge sector and supports the obstruction structural asymmetry theorem. In the Recognition framework it realizes the boundary approach step in the T1 exclusion chain, where cost divergence forces the scalar toward zero while the realizability proxy remains bounded away from zero.
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