Foundation Electron Compton From Jcost Electron Compton3 Cert

The certificate

The electron Compton wavelength is the scale at which quantum effects become important for an electron: about 2.426 picometers, or 2426 femtometers. It is defined from the electron mass, the speed of light, and Planck's constant. In the Recognition Science framework, a machine-checked library of formal theorems contains a declaration named ElectronCompton3Cert, which is a certificate, a packaged collection of three proved facts about the framework's cost function.

The cost function, J, is the framework's measure of the cost of a recognition event, a discrete record of a comparison. The certificate establishes three general facts about this cost. First, when the two quantities being compared are equal, the cost is zero. Second, for positive quantities, the cost is never negative. Third, a certain threshold value, phi minus 3/2, is positive, where phi is the golden ratio. These are the three facts the certificate holds, and each is a proved theorem in the library.

The certificate does not establish the electron Compton wavelength itself. The declaration defines its cost using a ratio of two real numbers, m and e, but it does not define what m and e mean. The library's documentation notes the intended connection: the electron Compton wavelength divided by the Planck length is about 1.5 times 10 to the 20, and the logarithm of that ratio to base phi is about 97.5, so the wavelength is about phi to the 97.5 times the Planck length. But this is a research note, not a proved result.

In plain terms, the certificate is a formal guarantee about the cost function's properties, not about electrons. It is a template shared across many modules, stating facts that hold for any positive inputs. What would make it a theorem about the electron is a definition of m and e in terms of electron physics, which the declaration does not provide. The certificate is therefore a small, correct piece of the framework's logical structure, but it is not a derivation of the Compton wavelength.

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)

What this page does not claim

The certificate does not prove that the electron Compton wavelength equals phi to the 97.5 times the Planck length. The certificate does not define what m and e represent physically. The certificate does not derive the fine-structure constant or any other coupling constant.

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/Foundation/ElectronComptonFromJCost.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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