Chemistry Maillard Temperature Ladder Temp At Rung Pos

The temperature ladder

The Maillard reaction is the browning that happens when sugars and amino acids meet heat, the chemistry behind crusts on bread, roasted coffee, and seared meat. Its onset is often quoted around 140°C. The Recognition Science framework models this onset as the base of a ladder: each step up multiplies the temperature by the golden ratio φ, about 1.618. So the predicted ladder runs 140°C at rung 0, roughly 226°C at rung 1, and roughly 366°C at rung 2.

The declaration tempAtRung_pos is a machine-checked theorem inside the framework's library of formal theorems. It states, in the framework's own units, that the temperature at every rung of the ladder is positive: for any natural number k, the temperature at rung k is greater than zero. This is a modest but necessary fact. It guarantees that the ladder never dips to zero or below, which would make the ratio between adjacent rungs undefined or meaningless.

The proof is short and relies on two facts already established in the library: the golden ratio is positive, and a positive number raised to any natural power stays positive. The theorem then follows by multiplying by the reference temperature, which is set to 1 in the framework's dimensionless units. The same file also proves that each rung is strictly hotter than the one before it, and that the ratio of adjacent rungs is exactly φ.

What the theorem does not claim is any empirical fact about real kitchens. It does not say that the Maillard reaction actually peaks at 226°C or that charring begins at 366°C. Those are predictions, checked against measurements like the caramelisation peak near 170-190°C and charring above 350°C, not consequences of the theorem. The theorem only certifies the internal consistency of the ladder: if the framework's model is right, then every rung names a positive temperature.

theorem tempAtRung_pos (k : ℕ) : 0 < tempAtRung k := by
  unfold tempAtRung referenceTemp
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

theorem tempAtRung_pos (k : ℕ) : 0 < tempAtRung k := by
  unfold tempAtRung referenceTemp
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

theorem tempAtRung_strictly_increasing (k : ℕ) :
    tempAtRung k < tempAtRung (k + 1) := by
  rw [tempAtRung_succ_ratio]
  have hk : 0 < tempAtRung k := tempAtRung_pos k
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have : tempAtRung k * 1 < tempAtRung k * phi :=
    mul_lt_mul_of_pos_left hphi_gt_one hk
  simpa using this

theorem temp_adjacent_ratio (k : ℕ) :
    tempAtRung (k + 1) / tempAtRung k = phi := by
  rw [tempAtRung_succ_ratio]
  field_simp [(tempAtRung_pos k).ne']

What this page does not claim

The theorem does not predict the actual onset, peak, or charring temperatures of the Maillard reaction in a real kitchen. The theorem does not prove that the Maillard reaction follows this ladder; it only certifies the internal consistency of the model. The framework's calibration of 140°C to rung 0 is a definitional choice, not a derived result.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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