Chemistry Thermal Runaway3 From Jcost Thermal Runaway3 Cert

A certificate with narrow scope

Thermal runaway in a lithium-ion battery is the self-accelerating rise in temperature that can end in fire. Engineers track it through an onset temperature, often near 130 degrees Celsius, where heat generation outpaces heat loss. The Recognition Science declaration ThermalRunaway3Cert is a machine-checked certificate: a packaged bundle of three proved facts about a mathematical cost function, not a measurement or a model of any physical battery.

The cost function, written Jcost, is a real-valued function of a ratio. The certificate's first fact says that when the two inputs are equal, the cost is zero. The second says that for positive inputs the cost is never negative. The third says that the golden-ratio constant phi minus 1.5 is greater than zero. These are general properties of the cost function, proved for any real inputs meeting the stated conditions. The certificate assembles them into a single structure, and the library shows the structure is inhabited, meaning the three facts hold together consistently.

The declaration itself is not a theorem about batteries. The module defines a domain cost as Jcost applied to a ratio m over e, but it never defines what m and e mean for a battery. The docstring records a research note, not a result: an idea that the thermal runaway threshold might scale with the golden ratio, with a rough estimate of 212 degrees Celsius that the note itself calls an overestimate. That paragraph is a plan, not a proof. The machine-checked part proves only the three general facts, which are shared verbatim across 2383 sibling modules, so citing this certificate as evidence about thermal runaway would be a category error.

What the certificate does establish is a template. It shows that a certain cost function has the three properties, and that a certificate structure can be formed from them. A theorem about thermal runaway would require defining m and e in the battery's own terms, such as a measured temperature ratio or an energy ratio, and then proving the relevant threshold property from that definition. Until that definition exists, the certificate remains a general mathematical object, not a chemistry result.

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 ThermalRunaway3Cert 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 any specific temperature threshold for thermal runaway. The certificate does not validate the research note's estimate of 212 degrees Celsius. The certificate does not claim that the cost function applies to batteries without a definition of m and e in battery terms.

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/Thermal_Runaway3_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.

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