Physics Bremsstrahlung3 From Jcost

Bremsstrahlung and the cost function

Bremsstrahlung is the electromagnetic radiation emitted when a charged particle accelerates in the electric field of another charge, such as an electron slowing down near an atomic nucleus. The word comes from German for "braking radiation." In classical physics, the power radiated scales with the square of the charge and with the square of the velocity relative to light, a result known from Larmor's formula.

The module named physics-bremsstrahlung3-from-jcost in the Recognition Science framework does not prove any new bremsstrahlung physics. Its machine-checked content is three general facts about a cost function, a ledger (a discrete record of recognition events) that assigns a nonnegative price to any ratio of two positive numbers. The cost function J(x) = (x + 1/x)/2 - 1, which the framework proves is the unique function satisfying five plain conditions, appears here as J(m/e), where m and e stand for unspecified positive real numbers.

The three proved facts are: the cost vanishes when m equals e, the cost is never negative for positive inputs, and the number phi - 3/2 is positive, where phi is the golden ratio. These are collected into a structure called Brems3Cert, and the module shows such a certificate exists. The docstring explicitly states that the module proves nothing specific to bremsstrahlung, because the ratio m/e is defined without reference to any physical mass or charge.

A research note in the source records the intended direction: bremsstrahlung power proportional to J(phi)^2 times Z^2 times (v/c)^2, suggesting an emission efficiency around 1.4% for non-relativistic electrons at the golden-ratio cost. This is not a theorem, not a measured value, and not a derivation. It is a stated ambition, shared verbatim across 2383 sibling modules, waiting for a definition of m and e in the subject's own terms.

What the module does establish, in plain language, is a reusable template: any physical theory that can express its key ratio as m/e inherits three guaranteed properties of the cost function. The bremsstrahlung step, connecting that ratio to radiation power, remains open. The module is honest about this gap, and the framework's value here is the template, not the radiation 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]

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

What this page does not claim

No bremsstrahlung power formula is proved or derived here. No physical meaning is assigned to m or e in this module. The 1.4% efficiency figure is a research note, not a result.

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/Physics/Bremsstrahlung3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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