tempAtRung
Maillard reaction temperature at rung k on the phi-ladder is defined as the reference onset scaled by phi to the power k. Chemists applying Recognition Science to sugar-amine reactions cite this to generate the explicit temperature sequence from 140°C onset onward. The definition is a direct scaling that supplies the geometric progression used by downstream certification theorems.
claimThe Maillard reaction temperature at rung $k$ is $T_k = T_0 · ϕ^k$, where $T_0$ denotes the reference onset temperature (RS-native value 1, calibrated to 140°C).
background
The phi-ladder is the geometric sequence generated by the self-similar fixed point phi forced in the UnifiedForcingChain (T6). Reference temperature is the dimensionless unit 1 in RS-native units, calibrated so that rung 0 matches the empirical Maillard onset at 140°C. The module extends the J-cost band application to explicit rung temperatures for any sugar-amine pair, with predicted values rung 1 ≈ 226°C (peak browning) and rung 2 ≈ 366°C (char boundary). Upstream referenceTemp supplies the calibrated base value.
proof idea
Direct definition: tempAtRung k is referenceTemp multiplied by phi raised to k. It inherits the real-number power operation and the constant referenceTemp without further reduction.
why it matters in Recognition Science
This definition supplies the temperature values required by MaillardTemperatureCert, which certifies positivity, the one-step ratio equal to phi, strict monotonicity, and adjacent ratio phi. It completes the explicit ladder step in the Maillard module, linking the J-cost wrapper to the phi-ladder from the forcing chain (T5–T7). The construction supports structural predictions for browning and charring thresholds without additional hypotheses.
scope and limits
- Does not convert RS-native values to Celsius without separate calibration constants.
- Does not encode pair-specific activation energies or kinetic rates.
- Does not address non-phi deviations observed in some empirical caramelisation data.
- Does not extend the ladder below rung 0 or above integer rungs.
formal statement (Lean)
37def tempAtRung (k : ℕ) : ℝ := referenceTemp * phi ^ k
proof body
Definition body.
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