Chemistry Olefins From Phi Ladder

Olefin selectivity and the phi ladder

In chemistry, olefins are unsaturated hydrocarbons, molecules with at least one carbon-carbon double bond, while alkanes are saturated hydrocarbons with only single bonds. The ratio of olefins to alkanes produced in a reaction is a measure of selectivity, a key economic factor in industrial processes like fluid catalytic cracking. A common empirical observation is that the propylene-to-propane ratio in such cracking falls in a range of roughly 0.1 to 0.3 under standard conditions. The question is whether a deeper mathematical structure can explain where in that range a particular process lands.

The framework called Recognition Science starts from a different question: what is the forced cost of recognizing one thing as another? Its central result, proved in a machine-checked library of formal theorems, is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This J-cost has a unique self-similar scaling point at the golden ratio phi, approximately 1.618, which emerges from the mathematics rather than being chosen. The phi ladder is the framework's name for the sequence of powers of phi that appear throughout its derivations, from particle masses to this chemical selectivity question.

Within the framework, the module OlefinsFromPhiLadder defines a domain cost as J(m/e), where m and e are real numbers representing masses or energies of the alkane and alkene. The Lean code proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and phi minus 3/2 is positive. These are universal properties of the J-cost function itself, not results about any specific chemical system. The module also bundles these into a structure called OlefinSelCert, showing that such a certificate exists, but nothing in the code connects m and e to actual molecular masses or to any measured selectivity.

The research note attached to the module records where the idea was meant to go: FCC selectivity olefin/paraffin is approximately J(phi) times a catalyst acidity ratio, and the propylene/propane ratio of 0.1 to 0.3 brackets J(phi), which is about 0.118. This is a hypothesis, not a theorem. The note itself says the code proves nothing specific to this subject, because domainCost is defined without reference to actual chemistry. What would make this a theorem about olefins is a definition of m and e in the subject's own terms, which does not yet exist. The framework's library proves the cost function's shape; the chemical application remains a research target.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module proves any chemical selectivity result; it proves only universal properties of the J-cost function. The propylene-to-propane ratio of 0.1 to 0.3 is an empirical observation, not a derived consequence of the framework. The catalyst acidity ratio is mentioned in a research note but is not defined or derived anywhere in the module.

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

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