# Section 27: Comparison and physical boundary

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<a id="section-27"></a>

## 27 Comparison and physical boundary

 The two theorems share a statistical and current-based proof strategy, but their constants cannot be interchanged. The periodic theorem allows its explicitly admitted inaccessible quantum reference and singular passive-coordinate laws. The radial theorem is stated for a normalized two-component qubit with no arbitrary inaccessible quantum-reference extension, and narrows the reference law to a full-cap class before adding its single joint auxiliary conditional. Its small entrance, preparation, finite radial cutoff and added writer require new proofs.

The periodic apparatus does not generate its prescribed potential from the retained elastic modes: they are decoupled spectators. The radial apparatus does not acquire a finite material source merely because a separate field candidate exists. Source preparation, actual-position loading or proxy agreement after that enlargement, finite propagation, clock/pointer support and whole-interval retention must be shown for one combined Hamiltonian. The unspent distance below one percent is only a possible allocation in such a future proof.