Thermodynamics Heat Pump Cop Rs

The heat pump coefficient

A heat pump's coefficient of performance (COP) is the ratio of heat it delivers to the work you put in. A typical residential unit moves about three to four units of heat per unit of electricity, so its COP sits between 3 and 4. The number matters because it says how much heating you get for each watt you pay for; higher is better, and the theoretical ceiling comes from the temperatures involved, not from the machine's design.

In Recognition Science, the framework models the COP as a ratio of two quantities, recognition costs: the cost of recognizing a state m against a reference e, written as J(m/e). The framework's central cost function is J(x) = (x + 1/x)/2 - 1, which is proved to be the only function satisfying five plain conditions. That function vanishes when m equals e, and it is nonnegative for positive inputs, so the model's cost is zero when the two states match and grows as they diverge.

The golden ratio φ appears as the framework's natural operating point: φ² = 2.618, which lands in the typical COP range of 3 to 4. The framework also defines a threshold φ - 3/2, which is positive because φ exceeds 1.5. These are the framework's three proved facts: the cost is zero at equality, it is nonnegative for positive inputs, and the threshold is positive. The framework proves nothing specific to heat pumps, because its definition of m and e does not connect them to temperatures, pressures, or any physical heat-pump quantity.

What the framework establishes in plain language is a template, not a physical law. It shows that the cost function, applied to any ratio, has the minimal properties any sensible cost should have. The research note in the framework records the intended direction: that the COP at the golden-ratio operating point might match typical values. That note is a hypothesis, not a result; the framework itself does not define m and e in thermodynamic terms, so it cannot yet claim to derive a COP.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The framework derives the heat pump COP from thermodynamics. The golden-ratio COP value is a measured or proved physical result. The framework's cost function is specific to heat pumps.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Thermodynamics/Heat_Pump_COP_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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