Chemistry Nmr T1rho From Phi Ladder Nmrt1rho Cert

The certificate

Nuclear magnetic resonance (NMR) measures how nuclear spins relax after being disturbed. In one variant, T1rho relaxation, the sample is held in a rotating magnetic field, the spin-lock field, and the relaxation time T1rho depends on the strength of that field. A research note in the Recognition Science library proposes that, at a certain scaling of molecular motions, the ratio T1rho(B_SL × φ)/T1rho(B_SL) should be approximately φ, the golden ratio, where B_SL is the spin-lock field strength.

The declaration NMRT1rhoCert is a formal structure in the framework's machine-checked library of formal theorems. It packages three proved facts about a cost function Jcost, a measure of recognition cost that the framework derives from first principles. The three facts are: the cost is zero when the two inputs are equal, the cost is nonnegative for positive inputs, and a certain threshold, φ − 3/2, is positive. These are general properties of the cost function, not results specific to NMR.

The certificate is built from a definition, domainCost, which sets the cost of a ratio m/e to Jcost(m/e). The three facts follow from general theorems about Jcost. The certificate itself is a structure that bundles these three facts, and the library proves the certificate is inhabited, meaning such a structure exists.

In Recognition Science, the framework models recognition events as a ledger, a discrete record of events whose cost is forced by a proved uniqueness theorem. The certificate here is a small piece of that larger framework. But the certificate does not claim anything about NMR T1rho itself. The connection to T1rho is only a research note, not a proved result. The library's own documentation says it proves nothing specific to this subject, because the cost is defined without reference to NMR. To make the certificate a theorem about T1rho, one would need to define m and e in terms of NMR quantities, such as the spin-lock field strength and some molecular motion rate.

What the certificate does establish is a clean, machine-checked example of the framework's cost function applied to a ratio. It shows that the cost is zero when the ratio is one, nonnegative for positive ratios, and that the golden-ratio threshold is positive. These are the kinds of facts that would be needed in a full derivation, but they are not the derivation.

structure NMRT1rhoCert 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)

structure NMRT1rhoCert 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

What this page does not claim

The certificate does not prove the T1rho scaling relation. The certificate does not define m or e in NMR terms. The certificate does not establish that the golden ratio appears in any measured NMR relaxation 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/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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