Chemistry Maillard Temperature Ladder Temp At Rung

The temperature ladder

The Maillard reaction is the cascade of chemical changes that browns bread crust, roasts coffee, and creates the flavor of grilled meat. Its onset temperature is commonly given as 140°C. The declaration tempAtRung takes that single number and defines an entire ladder: rung zero is 140°C, rung one is 140 times the golden ratio, approximately 226°C, and rung two is 140 times the golden ratio squared, approximately 366°C.

The golden ratio, usually written φ, is the number that solves r² = r + 1; its value is about 1.618. The ladder's rule is simple: each step multiplies the previous temperature by φ. The declaration proves three properties of this ladder. Every rung is a positive temperature. Each rung is strictly hotter than the one before it. And the ratio of any rung to its predecessor is exactly φ. These are formal theorems, checked by machine, with no hidden assumptions.

The framework's ledger, a discrete record of events, supplies the underlying cost function from which φ emerges as a forced scaling constant. The ladder is one application of that constant to chemistry. The reference point 140°C is a calibration choice, not a derivation. The framework models the Maillard onset as rung zero and then scales it by powers of φ.

Empirical cooking science offers a rough bench. Caramelization, a related browning process, peaks around 170 to 190°C, which sits below rung one. Thermal degradation and charring begin above 350°C, which is near rung two. The ladder predicts that for any sugar-amine pair, the peak browning should occur near 226°C and the char boundary near 366°C. Those are structural predictions, not measurements.

What the declaration does not claim is just as important. It does not prove that real Maillard reactions peak at exactly 226°C. It does not derive the 140°C onset from first principles. It defines a temperature scale and proves its internal arithmetic. Whether that scale matches reality is an empirical question, and the rough agreement with caramelization and charring is suggestive, not conclusive.

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 140°C onset is a calibration choice, not a derived constant. Real Maillard reactions are not proven to peak at exactly 226°C or char at exactly 366°C. The rough agreement with caramelization and charring is empirical observation, not a theorem.

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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