Chemistry Maillard Threshold From Jcost Below Threshold Equilibrium

The equilibrium below the threshold

The Maillard reaction is the cascade of chemical steps that browns bread crust, darkens roasted coffee, and creates much of cooked food's flavor. It has a sharp temperature threshold, around 140 degrees Celsius (413 K), below which it proceeds slowly and above which its rate accelerates dramatically, roughly by the golden ratio per 10 degrees. In plain terms, below the threshold the reaction is in a balanced state; above it, the balance tips and the cascade runs.

Recognition Science models this balance with a cost, a number that measures how far a system's state is from a stable equilibrium, like a ledger that records the price of being out of balance. The framework's central cost function is J(x) = (x + 1/x)/2 - 1. The declaration below_threshold_equilibrium proves that when the water-activity ratio equals 1, the cost is exactly zero: J(1) = 0. That is the formal statement of the balanced state below the threshold, where normal hydration keeps the system in recognition equilibrium.

The same framework proves two companion facts. Above the threshold, where dehydration changes the water-activity ratio away from 1, the cost becomes strictly positive: J(r) > 0 for any r not equal to 1. And the cost is symmetric, J(r) = J(1/r), meaning the cascade treats a doubling of water activity the same as a halving. These three statements together form a certificate that the threshold behavior is built into the cost function itself, not added as a separate assumption.

The declaration does not claim that the Maillard reaction itself is derived from the cost function. It does not identify the exact temperature 140 degrees Celsius, nor does it prove that food browning occurs at that temperature. It establishes only the mathematical structure of the equilibrium below the threshold: at the balanced ratio, the cost is zero. The connection between this formal statement and the measured chemistry of food browning remains an empirical identification, not a theorem.

/-- Below threshold: normal hydration = recognition equilibrium. -/
theorem below_threshold_equilibrium : Jcost 1 = 0 := Jcost_unit0

/-- Above threshold: dehydration has positive recognition cost. -/
theorem above_threshold_positive {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- The Maillard cascade is symmetric in water-activity ratio. -/
theorem maillard_symmetric {r : ℝ} (hr : 0 < r) :
    Jcost r = Jcost r⁻¹ := Jcost_symm hr

What this page does not claim

The declaration does not prove that the Maillard reaction occurs at 140 degrees Celsius. The declaration does not derive the rate acceleration above the threshold. The declaration does not identify which chemical species set the water-activity ratio.

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

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