Physics Combustion From Jcost

Combustion as recognition cost

Combustion is the rapid oxidation of a fuel that releases heat and light. In an engine or flame, the key quantity is the fuel-air equivalence ratio φ, the actual fuel-to-air ratio divided by the stoichiometric ratio. A flame burns stoichiometrically when φ equals 1, meaning exactly enough air is present to burn all the fuel. Rich mixtures (φ greater than 1) have excess fuel; lean mixtures (φ less than 1) have excess air. The adiabatic flame temperature, the hottest temperature a flame can reach with no heat lost, peaks at or near stoichiometric conditions.

The framework of Recognition Science models this balance with a cost function, a number that measures how far a system sits from equilibrium. The framework's central proved result is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. At x = 1, this cost is zero, which the framework identifies with stoichiometric combustion. Off-stoichiometric mixtures, whether rich or lean, carry a positive cost. The framework's machine-checked library of formal theorems proves both facts: J(1) = 0 and J(r) greater than 0 for any positive r not equal to 1.

In Recognition Science, the module also counts five canonical combustion regimes: lean deflagration, stoichiometric, rich deflagration, detonation, and distributed reaction zone. The framework proves this count equals 5 by a direct computation. The five regimes form a certificate structure that packages the count and the two cost facts into one object. The plain-language meaning is that the framework derives combustion's basic efficiency balance from its cost equation, rather than treating it as an independent assumption.

What this establishes in practice is a bridge from a purely mathematical cost function to a physical phenomenon. The stoichiometric point, where combustion is most efficient, is the same point where the recognition cost is zero. Rich and lean mixtures are exactly the states where the cost turns positive. This is a derivation, not a measurement: the framework shows that if the cost equation holds, combustion's central balance follows. It does not predict flame temperatures or reaction rates, and it does not replace the chemistry of oxidation.

/-- Stoichiometric combustion = recognition equilibrium (J = 0). -/
theorem stoichiometric_equilibrium : Jcost 1 = 0 := Jcost_unit0

/-- Off-stoichiometric combustion has positive J-cost. -/
theorem off_stoichiometric_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

theorem combustionRegimeCount : Fintype.card CombustionRegime = 5 := by decide

What this page does not claim

This module does not derive flame temperatures or reaction kinetics. The identification of stoichiometric with J = 0 is a framework modeling choice, not a measured law. The five regimes are enumerated, not derived from the cost equation.

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/Physics/CombustionFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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