Physics Nuclear Magnetic Resonance From Phi

The NMR scale from φ

Nuclear magnetic resonance (NMR) is a technique that probes the magnetic environment of atomic nuclei. When a sample sits in a strong magnetic field, nuclei absorb radio waves at frequencies that depend on the surrounding electrons. Those electrons shield the nucleus slightly, so the same nucleus in different molecules resonates at slightly different frequencies. Chemists measure that shift on a scale of parts per million (ppm) relative to a reference compound. For protons, the standard scale runs from 0 to about 10 ppm.

The number 10 is not a law of nature; it is a convention chosen for convenience. In Recognition Science, the framework models this scale as a rung on a φ-ladder, where φ is the golden ratio, about 1.618. The fifth power of φ is about 11.09, which is close to the conventional 10 ppm upper end. The framework notes that the conventional scale corresponds to the first φ^5 rung of its recognition ladder in frequency units. That is a definitional choice, not a derived result.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1 measures the cost of a recognition event, where x is a ratio of two quantities. For NMR, the module defines a domain cost as J(m/e), where m and e are two real numbers, meant to stand for measured and expected frequencies. The machine-checked library proves three general facts about this cost: it vanishes when m equals e, it is nonnegative when both m and e are positive, and the threshold φ - 3/2 is positive. These are general properties of the cost function, not specific to NMR.

The module does not prove that NMR shifts follow the φ-ladder. The definition of domainCost uses J(m/e) without connecting m and e to actual NMR measurements. The docstring itself states that the module proves nothing specific to NMR, because the quantities m and e are not defined in NMR terms. What the library establishes is a template: if one can define m and e for a real NMR experiment, then the cost properties would apply. That step remains open.

For a reader, the takeaway is clear: the φ-ladder offers a suggestive numerical coincidence with the NMR proton scale, and the framework provides a cost structure that could host a derivation. But the bridge from the general cost function to actual chemical shifts is not built. The module is a placeholder, not a result about NMR.

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]

theorem cert_inhabited : Nonempty NMRChemShiftCert := ⟨cert⟩

What this page does not claim

The module does not prove that NMR chemical shifts follow a φ-ladder. The 10 ppm scale is not derived from first principles. No claim is made that the framework's cost function has been validated against NMR data.

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

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