cumulativeCooling
Cumulative cooling after n cycles is defined as n times the per-cycle J-cost fraction J(φ) = φ - 3/2. Engineers deriving the phantom-cavity refrigerator would cite this to scale total heat extraction linearly with cycle count. The definition is a direct real multiplication that inherits positivity and additivity from the upstream coolingFraction constant.
claimLet Q(n) be the cumulative cooling after n cycles in units of k_B T_bath. Then Q(n) = n · J(φ), where J(φ) = φ - 3/2 ≈ 0.118 is the J-cost coefficient.
background
The module derives the identity-tick refrigerator (RS_PAT_029) with per-cycle cooling Q_per_cycle = J(φ) · k_B · T_bath. The upstream definition coolingFraction sets J(φ) := φ - 3/2. Cumulative cooling is the linear extension to n cycles, as stated in the module doc: 'Cumulative cooling at cycle n is n · Q_per_cycle.'
proof idea
One-line definition that casts n to reals and multiplies by the constant coolingFraction.
why it matters in Recognition Science
This definition supplies the linear scaling required by the IdentityTickRefrigeratorCert structure and the refrigerator_one_statement theorem, which bundles positivity, the (0.11, 0.13) band, and strict monotonicity. It closes the engineering derivation step in the J5 track, connecting directly to J-uniqueness (T5) in the forcing chain.
scope and limits
- Does not derive or prove the numerical value of J(φ).
- Does not address physical implementation or cavity design.
- Does not incorporate temperature dependence or non-ideal losses.
formal statement (Lean)
46def cumulativeCooling (n : ℕ) : ℝ := (n : ℝ) * coolingFraction
proof body
Definition body.
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