Chemistry Nmrrelaxation From Phi Ladder

The phi ladder in NMR

Nuclear magnetic resonance (NMR) relaxation describes how excited atomic nuclei return to equilibrium after a radio pulse. Two time constants govern the return: T1, the spin-lattice relaxation, and T2, the spin-spin relaxation. In soft tissue, their ratio T1/T2 typically falls between 3 and 10. A research note in the Recognition Science framework observes that this range aligns with powers of the golden ratio φ ≈ 1.618: φ² ≈ 2.618 and φ³ ≈ 4.24, covering most soft tissues. Gray matter, with a ratio near 5-6, sits close to φ^3.4.

The module named NMRRelaxationFromPhiLadder does not prove this alignment. Its cost function, a measure of recognition effort in the framework's ledger, is defined as J(m/e), where m and e are real numbers. The Lean code proves three general facts: the cost is zero when m equals e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These are properties of the cost function itself, not of NMR. The module contains no definition of m or e in terms of tissue properties, so it establishes nothing specific to chemistry.

In plain language, the module is a template. It shows that a certain mathematical function behaves well under basic conditions, and it packages those results into a certificate. The research note records an intended application to NMR relaxation, but the formal content stops at the general cost properties. To make this a theorem about T1/T2 ratios, one would need to define m and e using measurable tissue parameters, a step the module does not take.

Within the framework, the golden ratio is not arbitrary; it emerges from the forcing chain as the unique self-similar scaling. But that derivation concerns the cost function's structure, not NMR physics. The empirical match between φ powers and tissue relaxation ratios is a hypothesis awaiting a mechanistic definition, not a proved result.

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 proves any specific fact about NMR relaxation. The golden ratio alignment with tissue T1/T2 ratios is a derived result. The framework's cost function is defined using NMR-specific parameters.

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

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