coolingFraction_band
plain-language theorem explainer
The theorem establishes the strict bounds 0.11 < J(φ) < 0.13 on the cooling fraction for the identity-tick refrigerator. Engineers deriving performance certificates for phantom-cavity devices cite this interval to confirm per-cycle extraction lies near 0.118 of bath energy. The proof proceeds by unfolding the definition J(φ) = φ - 3/2 and applying linear arithmetic to the interval bounds on φ.
Claim. $0.11 < J(φ) < 0.13$ where $J(φ) := φ - 3/2$ and $φ$ is the golden ratio.
background
The Identity-Tick Refrigerator Spec module treats the phantom-cavity refrigerator as an engineering realization of Recognition Science track J5. Per-cycle cooling is given by Q_per_cycle = J(φ) · k_B · T_bath, with the J-cost coefficient defined as coolingFraction := phi - 3/2. Upstream lemmas supply the numerical bounds 1.61 < phi < 1.62 together with the algebraic structure of J from the PhiForcingDerived and DAlembert modules.
proof idea
Unfold coolingFraction to phi - 3/2. Introduce the two phi bound lemmas from Constants. Apply linarith to obtain both sides of the conjunction.
why it matters
The result supplies the fraction_band field in the IdentityTickRefrigeratorCert definition and appears in the refrigerator_one_statement theorem. It completes the engineering derivation for the RS_PAT_029 phantom-cavity refrigerator. The interval is consistent with the J-uniqueness property and the eight-tick octave structure in the foundational forcing chain.
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