Carnot efficiency η = 1 − T_c/T_h, recovered from ∮P dV and unbeaten by any cycle

Heat flows from hot to cold — what is the most work an engine can wring from that flow, and what sets the limit?

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Method

Numerically traverse a reversible ideal-gas Carnot cycle (γ=5/3) between reservoirs T_h, T_c — isotherm@T_h → adiabat → isotherm@T_c → adiabat — integrating the net work W=∮P dV and the hot-isotherm heat Q_h by trapezoid (4000 steps/leg), and recover η=W/Q_h without using the closed form. Repeat across five (T_h,T_c) pairs, sweep the expansion ratio V₂/V₁∈[1.5,12], and run an Otto cycle between the same temperature extremes as a second-law control.

The law it recovers

eta = 1 - T_cold/T_hot

Measurements, controls & cross-checks

Recovered abs error

3.8000e-15

Ratio invariance spread

4.9000e-15

Perturbation worst abs error

2.8000e-15

Control

Name

Otto cycle, r_v=8, 300-1500 K

Eta otto

0.75

Eta carnot same extremes

0.8

Margin

0.05

Module honesty

Status

honest-module: HUD deterministic, transcribed verbatim, pinned bit-exact, live read-back verbatim

Disclosure

The HUD's 'η = W_net/Q_h = 50.00% recovered by numerical ∮P dV' reads like discretization convergence; it is actually an EXACT grid-mapping cancellation. The adiabatic closing scaling s=(Th/Tc)^(1/(γ−1)) maps the cold-isotherm trapezoid grid point-by-point onto the hot one (Qc/Qh = Tc/Th at ANY step count; measured dev 4.4e-16) and the two adiabat grids onto each other (W2+W4 ≡ 0; measured |W2+W4|/Qh = 4.6e-16), so the displayed η is a structural identity of the ideal-gas trapezoid cycle — the same scaling that proves Carnot's theorem analytically — and can never disagree with 1−T_c/T_h at any n. Witness gate: at n=1 (one trapezoid per leg) Q_h is off analytic by 35.25% yet η is still exact to 2.2e-16. This holds equally for the oracle's Carnot gates A–C; the genuinely NUMERICAL gate is the Otto control D, whose heat input goes through the independent Cv·ΔT route (self-consistency 2.8e-7, a real O(n⁻²) residual). The legs themselves DO carry real discretization error (Q_h rel 1.27e-7 at n=2000, gated within [1e-8,1e-6]) — it cancels identically in the ratio. The module's Otto line (75.0% < 80.0%) is analytic formulas, pinned exactly.

Pins

η_numeric 0.5000000000000001, Q_h 6915.776462761074 J, W_net 3457.8882313805375 J, ratio-spread 5.440092820663267e-15, Otto 0.75/0.8, 4 corner V + 4 corner P + 6 sweep η — all strict ===; 7 HUD strings verbatim, matched live from a headless-Chrome read of ?world=carnot

Module vs oracle abs

3.6600e-15

What it reduces to

The Carnot efficiency 1 − T_c/T_h (Carnot 1824) — the founding result of thermodynamics and an equivalent statement of the second law. Validates a textbook bound: the recovered efficiency matches it to machine precision, is independent of the working substance and expansion ratio, and is never exceeded by the Otto control (Carnot's theorem). The recovery is non-circular (the traverse integrates P dV; η never enters the generator) but its machine precision is disclosed as structural, not numerical: the adiabatic scaling maps the trapezoid grids onto each other so discretization error cancels identically in η at any step count — the discrete cycle inherits the exact theorem (see result.module_honesty).

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- carnot (scripts/carnot-derisk.mjs)

Oracle

scripts/oracles/carnot.reference.json

Sources

S. Carnot, Réflexions sur la puissance motrice du feu (1824); Kelvin–Clausius second law.