Decoherence-Freeze Critical Temperature Criterion

Abstract

The decoherence-freeze deposit is positive exactly above its critical inverse temperature.

Definition 1.1 (The freeze deposit subtracts the temperature-scaled entropy tax).

$$\mathrm{freezeDeposit}(\beta ,\Delta S,\Delta E_{pass})=\Delta E_{pass}-\frac{\Delta S}{\beta }$$

Formalization. D5/S3/QuantumChannels/DecoherenceFreeze.freezeDeposit (✓ std3).

Source. Repository-derived.

Commentary.

For inverse temperature beta, entropy tax Delta S, and passive-energy shift Delta E pass, the freeze deposit is the passive-energy shift minus the entropy tax divided by beta.

Definition 1.2 (The critical inverse temperature is the entropy-energy ratio).

$$\mathrm{criticalInverseTemperature}(\Delta S,\Delta E_{pass})=\frac{\Delta S}{\Delta E_{pass}}$$

Formalization. D5/S3/QuantumChannels/DecoherenceFreeze.criticalInverseTemperature (✓ std3).

Source. Repository-derived.

Commentary.

The critical inverse temperature is the entropy tax divided by the passive-energy shift.

Theorem 1.3 (The freeze deposit is positive exactly above the critical inverse temperature).

$$0<\beta \wedge 0<\Delta E_{pass}\Rightarrow (0<\mathrm{freezeDeposit}(\beta ,\Delta S,\Delta E_{pass})\iff \mathrm{criticalInverseTemperature}(\Delta S,\Delta E_{pass})<\beta )$$

Proof. Machine-checked in Lean as D5/S3/QuantumChannels/DecoherenceFreeze.decoherence_freeze_iff_above_critical (✓ std3). ∎

Source. Repository-derived.

Commentary.

When beta and the passive-energy shift are positive, dividing and cross-multiplying preserve strict inequalities. Consequently, positivity of the freeze deposit is equivalent to beta exceeding the critical entropy-energy ratio.

References

• Truth anchor: D5/S3/QuantumChannels/DecoherenceFreeze.criticalInverseTemperature
• Truth anchor: D5/S3/QuantumChannels/DecoherenceFreeze.decoherence_freeze_iff_above_critical
• Truth anchor: D5/S3/QuantumChannels/DecoherenceFreeze.freezeDeposit