Physics Chemistry

The cost of a ratio

Chemistry and physics both study how matter changes and moves. Recognition Science, the framework that derives structure from a ledger (a discrete record of events), treats both as the same kind of bookkeeping. The module in its library called Physics.Chemistry defines a single cost function that applies to any ratio of two positive real numbers. The cost measures how far that ratio is from 1, the point where the two quantities are equal. The function is J(m/e) = (m/e + e/m)/2 - 1, the same J that the framework proves is the only possible cost function under five plain conditions.

What the module actually proves is modest but exact. It shows three facts about that cost. First, when the two numbers are equal, the cost is zero: J(r/r) = 0. Second, for any two positive inputs, the cost is never negative: J(m/e) ≥ 0. Third, the number phi - 3/2, which the framework calls the canonical threshold, is greater than zero; phi is the golden ratio, about 1.618, so the threshold is about 0.118. These three facts are proved in the machine-checked library of formal theorems, and they hold for every positive pair of real numbers.

The module does not go further. It defines a structure called Avogadro_Number_RS4 that packages those three facts, and it proves that structure exists. But the name is a research note, not a result. The module never defines what m and e mean for chemistry specifically. It does not say that m is a mass and e is an energy, or that their ratio tracks Avogadro's number. The docstring says plainly: the content is shared verbatim with 2383 sibling modules, and what would make this a theorem about chemistry is a definition of m and e in chemistry's own terms.

So the honest summary is this. The framework's cost function J is universal in the sense that it applies to any ratio, and the module proves the basic analytic facts about that cost: zero at equality, nonnegative everywhere, and a positive threshold. What remains open is the bridge from that general cost to the specific quantities of chemistry. Until someone defines m and e as, say, a molar mass and a characteristic energy, the module stays a template. The general cost is proved; the chemical interpretation is a target, not a theorem.

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

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]

What this page does not claim

The module does not prove anything specific to chemistry or physics. The name Avogadro_Number_RS4 does not establish a connection to Avogadro's number. No claim is made that the cost function is derived from chemical principles; it is defined, not forced.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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