Materials Thermal Shock3 From Jcost

Thermal shock resistance

Thermal shock is what happens when a material cannot adjust fast enough to a change in temperature. Heat a cold ceramic plate on one side and the hot face wants to expand while the cold face holds it back. That mismatch creates stress, and when the stress exceeds the material's strength, the plate cracks. The classical engineering measure of resistance to this failure is the thermal shock parameter R = sigma_f / (alpha * E), where sigma_f is the fracture strength, alpha is the coefficient of thermal expansion, and E is the elastic modulus. A high R means the material can tolerate a larger temperature difference before cracking. The formula appears in standard materials science texts; it was developed through the mid-twentieth century work of engineers such as W. D. Kingery on ceramic failure.

The parameter captures a trade-off. Strong materials resist cracking, so a high sigma_f helps. But a material that expands a lot under heat, a high alpha, builds up more strain for the same temperature change, so it hurts. A stiff material, a high E, also converts thermal strain into higher stress, so it hurts too. The best thermal shock resistant materials, like fused silica or certain glass ceramics, tend to have low expansion coefficients and modest stiffness. The formula is a ratio, so it has no units; it is a figure of merit for comparing candidate materials under the same conditions.

In Recognition Science, the framework models this trade-off through its cost function. The framework's central object is a cost, a discrete record of the price of recognition events, defined as J(x) = (x + 1/x)/2 - 1. The framework's library proves that J is zero when its input is 1, and nonnegative for positive inputs. The thermal shock module defines its domain cost as J(alpha * E / sigma_f), which is the reciprocal of the classical R. The module proves three general facts: this cost is zero when the material's properties make the ratio equal to 1, it is never negative for positive material parameters, and the golden-ratio-derived threshold phi - 3/2 is positive. The research note attached to the module records the intended direction: optimal thermal shock resistance would occur when J(alpha * E / sigma_f) equals J(phi).

What the module actually establishes is narrower than that note. The Lean code proves facts about the cost function J applied to any ratio; it proves nothing specific to thermal shock, because the module never defines what m and e mean for a real material. The three proved facts are universal properties of J, shared verbatim with 2383 sibling modules. The thermal shock content is a research note recording where the idea was meant to go, not a derived result. The module is a template, not a theorem about ceramics.

The honest takeaway is that the framework's cost function has the right shape for a figure of merit, but the step from that shape to a statement about real materials remains open. The classical formula R = sigma_f / (alpha * E) stands on its own as engineering knowledge. The framework's contribution, so far, is a proof that its cost function is well-behaved on positive ratios, which is a necessary condition for any future claim about optimal resistance, but not a sufficient one.

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]

structure ThermalShock3Cert 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 module does not prove any material has optimal thermal shock resistance. The framework does not derive the classical R formula; it only defines a cost in terms of a ratio. The threshold phi - 3/2 is proved positive, but its physical meaning for real materials is not established.

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/Thermal_Shock3_FromJCost.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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