Materials Superalloy From Phi Ladder

Superalloy precipitate sizes

Nickel-based superalloys, the workhorses of jet turbine blades, owe their high-temperature strength to precipitates: tiny particles of the gamma-prime phase, typically 100 to 500 nanometers across, that block the motion of dislocations. Metallurgists tune their size and spacing to maximize creep resistance. Recognition Science enters with a simple scaling guess: the optimal precipitate size equals phi raised to a power times the lattice parameter, where phi is the golden ratio, about 1.618. For a nickel lattice parameter near 0.36 nanometers, phi^8 through phi^11 gives roughly 47 to 71 nanometers, and slightly higher powers reach 300 nanometers, overlapping the observed range.

The framework's machine-checked library of formal theorems, however, proves nothing specific to superalloys. The module defines a cost function on a ratio of two real numbers, m over e, as J(m/e), where J is the framework's forced recognition cost. The three proved facts are general: the cost vanishes when m equals e, it is nonnegative for positive inputs, and the threshold phi minus 3/2 is positive. These statements hold for any positive m and e; they do not mention precipitates, nickel, or lattice parameters. The research note in the source openly records that the material application was the intended direction, not a result.

To turn the module into a theorem about superalloys, one would need a definition of m and e in metallurgical terms, for example m as a measured precipitate size and e as an optimal size derived from the lattice. Without that bridge, the formal content is a template shared verbatim with 2383 sibling modules. The honest summary: the framework provides a cost function with clean formal properties, and a phi-ladder estimate for precipitate size that is a hypothesis, not a proved consequence.

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

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]

What this page does not claim

The phi-ladder precipitate size is a proved theorem of Recognition Science. The module establishes any fact specific to nickel superalloys. The 47 to 300 nanometer range is a measured or certified optimal size.

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

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