Chemistry Oxidation Reduction Potential From Jcost

Redox potential and the cost ledger

An oxidation-reduction (redox) potential measures how strongly a chemical species pulls electrons. Standard tables list these potentials in volts, from lithium, which gives up electrons at -3.04 V, to fluorine, which grabs them at +2.87 V. The gap between those extremes is about 6 volts, a span that chemists use to predict which reactions will run spontaneously.

In Recognition Science, a ledger, a discrete record of events, keeps track of every recognition event, and the cost, the forced price of recording that event, obeys a fixed mathematical form. The framework's library, a machine-checked collection of formal theorems, defines the cost of a redox pair as J(measured / expected), where J(x) = (x + 1/x)/2 - 1. This function vanishes when measured equals expected, stays nonnegative for positive inputs, and has a positive threshold at phi - 3/2, where phi is the golden ratio.

The library proves three facts about this cost. First, at equilibrium, when the measured potential equals the expected one, the cost is exactly zero. Second, the cost is never negative when both potentials are positive. Third, the threshold phi - 3/2 is strictly positive. These are general properties of the cost function, not specific electrochemical results, because the module defines the ratio m/e without tying m and e to any particular chemical species.

The research note attached to the module observes that the 6-volt span of standard reduction potentials approximates phi^5 times a reference voltage, with phi^5 about 11.09 half-steps times 0.543 V. This observation is a hypothesis, not a theorem. What the module actually establishes is narrower: the cost function that governs recognition events also constrains any redox comparison, placing a formal floor under the chemistry.

The consequence for a reader is a way to see redox tables as instances of a single cost law. The framework does not derive the voltage of lithium or fluorine; it shows that the same function which forces the golden ratio also forces a nonnegative, equilibrium-vanishing cost on every measured-to-expected ratio in electrochemistry.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (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]

What this page does not claim

This module derives the standard reduction potential of any specific chemical species. The 6-volt span matching phi^5 times a reference is a proved theorem; it is a research note. The cost function is not claimed to be the only possible measure of redox favorability.

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

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