Foundation Chemistry

Foundation chemistry

Foundation chemistry is the branch of Recognition Science that tries to explain chemical reactions as the outcome of a single, forced recognition cost, a discrete record of events where the price of matching two things is not chosen but required. The idea is that when a molecule meets a solvent, or an electron meets a shell, the system pays a cost that depends only on the ratio of the two masses or charges involved. The central object is the domain cost, defined as J(m/e), where J is the unique cost function that the framework establishes must exist. The hope is that this single function can reproduce known results like Marcus theory of electron transfer, where the reorganization energy depends on the geometry of the solvent.

The current module, however, does not yet establish any chemistry. The machine-checked library of formal theorems establishes three general facts about the cost function: it equals zero when the two inputs are equal, it is never negative for positive inputs, and a certain threshold value phi minus 3/2 is positive. These are true for any positive numbers, not for any particular chemical system. The module defines a certificate structure that packages these three facts, and establishes that such a certificate exists. But the definition of domain cost does not reference any specific chemical quantity, so the theorems are about the abstract cost function, not about molecules.

The gap is the definition of the two inputs. To make this a theorem about chemistry, the module would need to define what m and e mean in chemical terms, for example as a mass and a charge, or as two reorganization energies. The docstring notes a research direction: at a certain distance related to the golden ratio, the outer reorganization energy from Marcus theory might equal the cost function applied to the golden ratio times a charge unit. That remains a research note, not a result. The module is a template, shared verbatim with 2383 sibling modules, each waiting for its subject-specific definition.

What the module does establish is a pattern. It shows how a physical theory can be built by taking a universal cost function and applying it to a ratio of two quantities. The three established facts are the foundation stones: cost vanishes at equality, cost is nonnegative, and the golden ratio threshold is positive. These are the properties that any chemical application would need to respect. The module is honest about its status: it establishes general facts, and it records where the chemistry was meant to go. The next step, defining the chemical inputs, is the open target that would turn this template into a theory of reactions.

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]

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

What this page does not claim

The module does not establish any specific chemical result, such as the Marcus reorganization energy formula. The research note about the golden ratio distance is not a theorem, only a stated direction. The module does not define what m and e mean in chemical terms.

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/Foundation/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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: