Chemistry Maillard Threshold From Jcost Maillard Symmetric

The symmetry of the threshold

The Maillard reaction is the chemistry of food browning and flavour formation: sugars and amino acids react when heated, producing the brown crust on bread and the savoury notes in roasted meat. Cooks know it has a sharp temperature threshold near 140 degrees Celsius, or 413 kelvin, above which the reaction accelerates dramatically. In the Recognition Science framework, this threshold is modelled as a crossing in the cost of recognition, a measure of how far a system's water-activity ratio has moved from its equilibrium value of one.

The declaration maillard_symmetric establishes a specific property of that cost function: it is symmetric under inversion of the water-activity ratio. If the ratio is r, then the cost at r equals the cost at its reciprocal, 1 divided by r. In plain terms, the cost of a system being twice as wet as equilibrium is exactly the same as the cost of it being half as wet. The theorem is proved in the machine-checked library of formal theorems, with no unproved assumptions, and it follows directly from the more general symmetry of the J-cost function that underpins the framework.

The symmetry has a concrete consequence for the Maillard threshold. Below the threshold, the water activity stays near one and the cost is zero, meaning the system is in recognition equilibrium. Above the threshold, dehydration pushes the ratio away from one and the cost becomes positive, triggering the cascade of browning reactions. Because the cost function treats r and its reciprocal identically, the framework's description of the threshold does not privilege wet over dry: the same magnitude of deviation from equilibrium carries the same cost in either direction, even though the chemistry itself is not symmetric in that way.

What the declaration does not claim is equally important. It establishes a formal property of a cost function, not a measurement of the Maillard reaction. The temperature value of 140 degrees Celsius and the phi-fold acceleration per 10 degrees are empirical facts that the framework's description references, but the symmetry theorem does not derive them from first principles. The declaration also does not claim that the chemistry of browning is symmetric: the reaction itself is not reversible, and the symmetry applies only to the abstract cost function, not to the molecular processes. The framework models the threshold with this symmetric cost; it does not assert that the underlying chemistry obeys the same symmetry.

/-- 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

/-- 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

def maillardThresholdCert : MaillardThresholdCert where
  equilibrium_below := below_threshold_equilibrium
  cascade_above := above_threshold_positive
  symmetric := maillard_symmetric

What this page does not claim

The declaration does not derive the 140 degree Celsius threshold value from first principles. The symmetry applies to the abstract cost function, not to the molecular chemistry of browning. The framework does not claim the Maillard reaction itself is reversible or symmetric.

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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