obstruction_structural_asymmetry
plain-language theorem explainer
For any defect sensor with nonzero charge the realized collapse scalar can be driven below every positive threshold while the Euler-derived physically realizable ledger remains bounded away from the T1 boundary. Recognition Science workers on the unification bridge cite the result to separate the collapse observable from the realizability proxy. The proof is a direct conjunction that applies the nonzero-charge collapse lemma and the non-boundary property of the Euler ledger.
Claim. Let $S$ be a defect sensor with nonzero charge. Then the realized defect collapse scalar of $S$ satisfies: for every $ε>0$ there exists $N∈ℕ$ such that the scalar at step $N$ is less than $ε$. Moreover, the physically realizable ledger obtained from the Euler carrier for $S$ is not boundary-approaching.
background
The module replaces a single bounded-cost ledger with a three-component architecture. Cost Divergent records that nonzero charge forces annular cost to diverge. Euler Trace Admissible supplies a convergent Euler carrier whose logarithmic derivative remains bounded. Physically Realizable Ledger (defined at line 245) attaches to each sensor a scalar proxy state $x_N>0$ whose T1 defect, given by the defect functional $J(x)$, stays uniformly bounded: there exists $K$ such that defect$(x_N)≤K$ for all $N$ (see LawOfExistence.defect). BoundaryApproaching is the negation of this uniform bound away from zero. The local setting is the T1-bounded realizability chain that replaces the earlier OntologicalPrimeLedger.
proof idea
The proof is a one-line wrapper that forms the required conjunction. The left conjunct is supplied directly by the upstream lemma realizedDefectCollapseBoundaryApproaching_of_nonzero_charge. The right conjunct is obtained by instantiating the PhysicallyRealizableLedger class via euler_physically_realizable and then applying the lemma physicallyRealizableLedger_not_boundaryApproaching.
why it matters
The declaration packages the two incompatible scalar behaviors that make the bridge nontrivial: collapse approaches zero while the realizability proxy stays bounded away from zero. It therefore shows why the boundary-transport step in the module proof chain must remain an external hypothesis. The result sits inside the T1-bounded realizability architecture and directly supports the claim that no single-scalar identification can discharge the bridge. It touches the open question of whether an external mechanism can reconcile the two limits without violating T1 boundedness.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.