Chemistry Nmrrelaxation From Phi Ladder Nmrrelax Cert

The certification structure

Nuclear magnetic resonance (NMR) relaxation describes how excited atomic nuclei in a magnetic field return to equilibrium, with two characteristic times, T1 and T2, that differ across tissues. A common empirical observation is that the T1/T2 ratio in soft tissue falls between roughly 3 and 10. The Recognition Science declaration NMRRelaxCert, a named structure in the framework's machine-checked library of formal theorems, packages three general facts about a particular cost function, not a measurement or a prediction about tissue.

The cost function in question, written J(x) = (x + 1/x)/2 - 1, is the unique function forced by five plain conditions in the framework's foundational theorem. The certificate NMRRelaxCert asserts three things about this function when its input is a ratio m/e of two positive real numbers. First, the cost vanishes when m equals e: J(1) = 0. Second, the cost is never negative for positive inputs: J(m/e) ≥ 0. Third, the number φ - 3/2 is positive, where φ is the golden ratio, approximately 1.618. These three facts are proved in the framework's library as general theorems about J.

The certificate's name suggests a connection to NMR relaxation, and a research note attached to the module records an intended application: the T1/T2 ratio in tissue might follow a φ-power ladder, with φ² ≈ 2.618 and φ³ ≈ 4.24 covering the observed range. But the formal content of NMRRelaxCert makes no reference to NMR, tissue, T1, or T2. The cost function is defined as J(m/e) with m and e as arbitrary real numbers, not as relaxation times. The note itself states plainly that the paragraph above it is a research note recording where the idea was meant to go, not a result.

What the certificate actually establishes is a small, self-contained mathematical object: it shows that the three general facts about J are consistent, by constructing an explicit instance of the structure. This is a theorem about the cost function's elementary properties, not about physics. The framework's own documentation is explicit: the module proves nothing specific to NMR relaxation, because the definition of domainCost does not reference the subject. To become a theorem about tissue, one would need a definition of m and e in NMR's own terms, which the module does not provide.

The distinction matters for anyone reading the framework's library. A certificate named for a subject can be a placeholder, not a result. The honest reading of NMRRelaxCert is that it is a well-formed formal object whose name points at an intended application, while its content stays at the level of general cost-function properties. The empirical question of whether T1/T2 ratios actually follow a φ-power ladder remains open, and the certificate does not address it.

structure NMRRelaxCert 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)

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

theorem cert_inhabited : Nonempty NMRRelaxCert := ⟨cert⟩

What this page does not claim

NMRRelaxCert does not establish any measurement or prediction about T1/T2 ratios in tissue. The certificate does not prove that T1/T2 ratios follow a φ-power ladder. The module does not define m and e in terms of NMR relaxation times.

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)

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.

Derived articles

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