Chemistry Polarimetry3 From Jcost Polarimetry3 Cert

A certificate for optical rotation

Optical rotation is the angle by which a substance twists the plane of polarized light passing through it. Chemists measure it with a polarimeter, and the standard formula is [α] = α_obs / (c · l), where α_obs is the measured angle, c is concentration, and l is path length. The Recognition Science (RS) framework proposes a different formula, [α] = J(φ) · c · l · n, where n is a molar rotation contribution per rung and J is a cost function. The declaration Polarimetry3Cert is a formal object in the framework's machine-checked library that bundles three facts about that cost function.

The three facts are general properties of the cost function J(x) = (x + 1/x)/2 - 1, not facts about chemistry. First, the cost vanishes when its two inputs are equal: J(1) = 0. Second, the cost is never negative when both inputs are positive. Third, the golden-ratio constant φ exceeds 1.5, so φ - 3/2 is a positive number. The certificate packages these three statements into a single structure, and the library proves the certificate exists. The key equation is J(x) = (x + 1/x)/2 - 1, and the three facts follow directly from it.

The certificate does not establish anything about optical rotation. The cost function is defined as J(m/e), where m and e are real numbers, but nothing in the definition connects m to a measured angle or e to a concentration. The documentation is explicit: the code proves three general facts about J(m/e), and proves nothing specific to polarimetry, because domainCost is defined without reference to a chemical system. The optical rotation formula is a research note, a statement of where the idea was meant to go, not a result.

In Recognition Science, the framework models the cost of recognition as forced, and the function J is the unique cost function satisfying five plain conditions. The framework's library shows that J vanishes at unity, is nonnegative on positive inputs, and that φ - 3/2 is positive. These are the three facts the certificate bundles. What would make this a theorem about polarimetry is a definition of m and e in the subject's own terms, for example m as a molar quantity and e as a concentration. Until such a definition exists, Polarimetry3Cert remains a certificate about a cost function, not about light, molecules, or rotation.

structure Polarimetry3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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

Polarimetry3Cert does not prove the RS optical rotation formula [α] = J(φ) · c · l · n. The certificate does not connect its cost function to any measured optical rotation value. The three bundled facts are general properties of J, not results about polarimetry.

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/Polarimetry3FromJCost.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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