Chemistry Nmr Coupling From Phi Nmrjcoupling Cert

A certificate and its limits

Nuclear magnetic resonance (NMR) spectroscopy measures how atomic nuclei absorb and re-emit radio waves in a magnetic field. A key observable is the J-coupling constant, the splitting of a spectral line caused by neighboring nuclei influencing each other's spins. In organic chemistry, the Karplus equation relates this coupling to the dihedral angle between bonds, with vicinal hydrogen-hydrogen couplings typically falling between 6 and 10 hertz. The Recognition Science framework explores whether such constants can be derived from a single universal cost function, J(x) = (x + 1/x)/2 - 1, which the framework proves is the unique function satisfying five plain conditions.

Within this framework, the declaration NMRJCouplingCert is a formal structure, a machine-checked collection of three statements about the cost function applied to a ratio of two real numbers, m and e. The first states that when m equals e, the cost is zero. The second states that for positive inputs, the cost is never negative. The third states that the golden ratio minus 3/2 is positive. These are general facts about the cost function, proved in the framework's library of formal theorems, not facts specific to chemistry. The declaration itself is defined in a module that also contains a research note, a paragraph recording where the idea was meant to go, not a result.

In Recognition Science, the framework models the cost of recognition as forced, and this certificate is a small piece of that larger structure. The certificate does not define what m and e represent in NMR terms. It does not prove that J-coupling constants equal some function of the golden ratio. The research note mentions a possible connection, suggesting that a coupling of 10 hertz might relate to phi to the fifth power divided by 1000, but this is an unformalized idea, not a theorem. The certificate's three facts hold for any positive real numbers, which is precisely why they are too general to be a claim about NMR.

What the certificate does establish is a template. It shows that the cost function behaves sensibly at equality, stays nonnegative, and has a positive threshold related to the golden ratio. These are the kind of properties that a future, more specific definition of m and e would need to respect. The path from this certificate to a real NMR prediction requires a definition of m and e in the subject's own terms, a step the module itself does not take. Until then, the certificate remains a formal shell, a placeholder that records the intended shape of a result without filling in the chemistry.

structure NMRJCouplingCert 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]

What this page does not claim

This certificate does not prove any specific J-coupling constant value for any molecule. The research note's suggestion that 10 Hz relates to phi^5/1000 is not a proved theorem. The certificate does not define what m and e represent 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/Chemistry/NMR_Coupling_FromPhi.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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