Chemistry Maillard Temperature Ladder Temp At Rung Strictly Increasing

The temperature ladder

The Maillard reaction is the cascade of chemistry between sugars and amino acids that browns bread crust, sears steak, and creates roasted coffee flavor. It begins around 140°C. In the Recognition Science framework, that onset temperature is taken as a reference point, and the framework models the reaction's later stages as a ladder of temperatures: rung 0 is 140°C, rung 1 is 140 times the golden ratio φ (about 226°C), rung 2 is 140 times φ² (about 366°C), and so on. The declaration tempAtRung_strictly_increasing is a machine-checked theorem stating that each rung is strictly hotter than the one before it: for every natural number k, the temperature at rung k is less than the temperature at rung k+1.

The proof is short and rests on two facts. First, the temperature at any rung is positive: 140°C times a positive power of φ is always greater than zero. Second, the ratio between consecutive rungs is exactly φ, which is approximately 1.618 and therefore greater than 1. Multiplying a positive number by a factor greater than 1 makes it larger, so each step up the ladder is a genuine increase. The theorem also bundles this monotonicity with the positivity and the exact ratio into a single certificate object, a packaged guarantee that the ladder behaves as described.

The ladder is a structural prediction, not a measurement of any specific sugar-amine pair. The framework notes that caramelisation peaks around 170-190°C and thermal degradation begins above 350°C, and these observed ranges sit near rung 1 and rung 2 respectively, but the ladder itself is a formal consequence of the framework's cost function, not a fit to those data points. The theorem establishes the ordering of the rungs, nothing about which real chemical reactions occur at which rung, and nothing about the rates or products of those reactions.

What the theorem does give is a clean, checkable consequence of the framework's structure: if the Maillard onset is 140°C and the golden ratio governs the scaling, then the later stages must be hotter, in a precise and unbounded sequence. That monotonicity is the kind of property that makes the ladder falsifiable in principle: any real observation of a Maillard stage at a lower temperature than an earlier stage, under comparable conditions, would contradict the ordering the theorem enforces.

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']

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

What this page does not claim

The theorem does not identify which real chemical reactions occur at each rung. The theorem does not state that any measured Maillard stage matches a rung temperature. The theorem does not claim the ladder applies to all sugar-amine pairs without variation.

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)

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