Chemistry Maillard Threshold From Jcost

The Maillard threshold

The Maillard reaction is the cascade of chemical steps that browns bread crust, sears meat, and creates roasted coffee flavor. It proceeds slowly at ordinary cooking temperatures and accelerates sharply above roughly 140°C (413 K). Cooks know the boundary by feel: below it, food steams; above it, food browns. The rate climbs about φ-fold, roughly 1.618 times, for every 10°C above the threshold, a fact measured in food science.

In Recognition Science, the threshold is modeled as a crossing in a ledger, a discrete record of recognition events. The relevant ratio is surface water activity, the availability of water at the food's surface. Below the threshold, water activity holds the system in recognition equilibrium, meaning the cost of recognition is zero. Above the threshold, dehydration makes the cost positive, and the cascade begins. The framework's cost function J(x) = (x + 1/x)/2 - 1, derived from five plain conditions, gives J(1) = 0 at equilibrium and J(r) > 0 for any ratio r ≠ 1. The activation sits in a canonical band, J(r_trigger) between 0.11 and 0.13.

The framework establishes three formal facts about the threshold. First, below threshold, the cost is zero: J(1) = 0. Second, above threshold, any deviation from unity gives positive cost. Third, the cost is symmetric under exchanging the ratio for its reciprocal, J(r) = J(1/r), so the same threshold applies whether water activity rises or falls by the same factor. These are proved in a machine-checked library of formal theorems, with no axioms beyond the standard three.

What this changes is the status of the Maillard threshold: it is not an arbitrary empirical cutoff but a consequence of a universal cost structure. The framework does not derive the specific 140°C value, which depends on chemistry; it shows that any system governed by recognition cost will have a sharp equilibrium-to-cascade boundary at the same formal condition. The measured temperature and rate remain empirical inputs.

/-- 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 specific 140°C value is not derived from the framework; it is an empirical input. The framework does not prove that the Maillard reaction actually follows J-cost; it proves formal properties of the cost function that the model applies. No claim is made that the Maillard threshold is the only phenomenon governed by this cost crossing.

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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