Chemistry Maillard Threshold From Jcost Above Threshold Positive

The positive cost above threshold

The Maillard reaction is the chemistry of food browning and flavor formation, familiar from the crust of bread and the sear on meat. Its rate is famously slow below about 140 °C (413 K) and accelerates sharply above that temperature, roughly doubling for every 10 °C rise. The Recognition Science framework offers a formal account of this threshold. Its central object is the cost, a number assigned to any state of a system, measuring how far that state is from a balanced equilibrium. The framework's fundamental theorem proves that this cost must take the specific form J(x) = (x + 1/x)/2 − 1, a result derived from five plain conditions and checked in a machine-checked library of formal theorems.

The declaration above_threshold_positive is a small theorem built from that cost function. It states that for any positive ratio r that is not equal to 1, the cost J(r) is strictly greater than zero. In plain language: any deviation from the balanced ratio carries a positive cost. The theorem's proof is a single line, citing the general positivity property of the cost function. Its companion theorem, below_threshold_equilibrium, states the complementary fact that at the balanced ratio r = 1, the cost is exactly zero. Together they draw a sharp line: zero cost exactly at equilibrium, positive cost everywhere else.

In the Maillard context, the framework identifies the ratio r with surface water activity. Below the threshold, water activity stays near its equilibrium value, the cost is near zero, and the reaction proceeds slowly. Above the threshold, dehydration drives the ratio away from 1, the cost becomes positive, and this positive cost is the formal trigger for the Maillard cascade. The framework's docstring places the activation in a canonical band, with J(r) between 0.11 and 0.13, though this numerical band is a stated identification rather than a proved consequence of the theorem.

What the theorem does not claim is important. It does not establish that the Maillard reaction actually occurs at 140 °C; that is an empirical fact about chemistry, not a consequence of the cost function. The theorem only says that a positive cost exists for any deviation from equilibrium. It does not specify the numerical value of that cost, nor does it identify which physical ratio r corresponds to the Maillard threshold. The framework's application to water activity is a modeling choice, not a proved derivation. The symmetry property, that J(r) equals J(1/r), is also proved, but it describes the cost function's shape, not the chemistry of browning.

/-- 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

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

/-- 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 theorem does not prove that the Maillard reaction occurs at 140 °C. The theorem does not identify the physical ratio r with water activity; that is a modeling choice. The theorem does not specify the numerical value of the cost above threshold.

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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