Chemistry Acid Base Equilibrium From Jcost Acid Base Cert

The acid-base certificate

Acid-base equilibrium is the balance between an acid and its conjugate base in solution, described for weak acids by the Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]). When the concentrations of acid and base are equal, the log term is zero and the pH equals the pKa. The Recognition Science framework models this balance with a cost function, a measure of how far a measured ratio sits from its expected value, defined as J(measured / expected).

The declaration AcidBaseCert is a structure, a bundle of three machine-checked theorems about this cost function. First, the cost is zero when the measured value equals the expected value, so at the buffer optimum, where [A-]/[HA] = 1, the cost vanishes. Second, the cost is never negative for positive inputs. Third, a specific threshold, phi minus 3/2, is positive, where phi is the golden ratio. These facts are proved in the framework's machine-checked library of formal theorems, and the certificate simply packages them together.

The certificate proves nothing specific to acid-base chemistry. The cost function is defined as J(measured / expected) without any reference to acids, bases, or pH. The theorems hold for any positive real numbers. What would make this a theorem about acid-base equilibrium is a definition of 'measured' and 'expected' in chemical terms, for instance as the concentrations of conjugate base and acid. That definition is absent, so the certificate is a general mathematical result, not a chemical one.

The research note attached to the module records the intended direction: buffering capacity drops to zero when the cost exceeds the threshold J(phi) ≈ 0.118. But this is a research plan, not a proved result. The certificate establishes only the three general facts, and the chemical interpretation remains a target for future work.

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]

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

What this page does not claim

AcidBaseCert does not prove any statement about pH, pKa, or buffer capacity. The research note about buffering capacity dropping to zero is a plan, not a proved result. The certificate does not establish that the golden ratio threshold has chemical meaning.

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/AcidBaseEquilibriumFromJCost.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.

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