Chemistry Nmr T1rho From Phi Ladder

NMR T1rho and the phi ladder

Nuclear magnetic resonance (NMR) T1rho relaxation measures how quickly nuclear spins return to equilibrium while held by a radio-frequency locking field. The relaxation time T1rho depends on the strength of that locking field: a stronger lock field slows the relaxation, a phenomenon called T1rho dispersion. Researchers use this dispersion to study molecular motions, because the relaxation rate reflects how molecular tumbling matches the effective field frequency.

A proposed connection in Recognition Science links this dispersion to the golden ratio phi. The idea is that the ratio of T1rho at a lock field scaled by phi to T1rho at the original field approximates phi, when molecular motions follow the J-cost scaling of the framework. That sentence is a research note, not a proved result.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1 measures the cost of a recognition event. The module defines a domain cost as J(m/e), where m and e are real numbers. The machine-checked library of formal theorems proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the constant phi - 3/2 is positive. These facts hold for any positive real numbers m and e; they contain no chemistry-specific content.

The module does not define what m and e mean for NMR. Without such definitions, the theorems do not establish anything about T1rho dispersion. The research note records the intended direction, but the formal results are generic properties of the cost function, shared verbatim with 2383 sibling modules.

What the module does establish is a template: any future theory that defines m and e in NMR terms can inherit these cost properties. The positive threshold phi - 3/2 could serve as a marker for when the cost becomes significant, but that use remains a target, not a derivation.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove that T1rho dispersion follows the phi ratio. The module does not provide a physical definition of m and e in NMR terms. The module does not establish any empirical agreement with NMR measurements.

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_T1rhoFromPhiLadder.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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