The Decision-Theoretic Origin of the Admissibility Threshold

The Decision-Theoretic Origin of the Admissibility Threshold This note addresses the last order-one constant in the decoherence-cliff mechanism: the admissibility threshold Iadm. It argues that this threshold is not a free parameter, but the entropy required to decide between the two branches of a superposition. Purpose Article 20 derived a decoherence-cliff law whose location depends on the ratio between a total admissibility cost and a per-step information rate. The per-step rate was already identified with the mutual Fisher information. This note identifies the remaining total cost. Main Argument Reconstructive admissibility is operationalised as a binary decision between two branches of a superposition. The accumulated record is treated as a sequential log-likelihood process, and the closure occurs when this process first crosses the reconstructibility boundary. Rate versus Threshold The note separates two quantities that must not be confused: the per-step rate, fixed by Wald's identity as a Kullback–Leibler divergence, which reduces in the soft regime to the mutual Fisher information; the threshold, which is a total information and therefore must be an entropy, not another per-step divergence. Threshold Value The admissibility threshold is identified with the branch entropy: Iadm = H(p). For a balanced two-branch superposition, this gives: Iadm = ln 2. The resulting cliff location is therefore: Nc = H(p) τD / τmin, with no remaining free order-one constant. Physical Identification The key physical input is the standard entropy identity for decoherence: when a pure two-branch superposition decoheres into a classical mixture, the entropy increase is exactly the Shannon entropy of the branch weights. The note identifies this entropy increase with the relative-entropy recovery of the reconstructive closure. Status The decision-theoretic rate, the dimensional separation between rate and threshold, and the decoherence entropy identity are established under the framework of Articles 19 and 20. The exact equality Iadm = H(p) remains conditional on identifying Article 19's relative-entropy recovery with decoherence entropy, and on the initial branch being pure. Conclusion The note turns the admissibility threshold from an adjustable constant into a decision-theoretic and entropic quantity. In the clean balanced case, the decoherence cliff becomes parameter-free: the threshold is the entropy produced by making a branch definite.