Physics Elastic Scattering From Jcost

Elastic scattering and its cost

Elastic scattering is the collision process in which two particles bounce off each other and leave with the same total kinetic energy they had before. In the classical picture of a hard sphere of radius a, the cross-section, the effective area the target presents to the beam, is exactly σ = 4πa². That formula is a fixed geometric statement, independent of how fast the projectile moves.

The Recognition Science framework models a different quantity, the cost, a number that measures how expensive it is for the ledger, a discrete record of recognition events, to register a given ratio. The framework's central cost function is J(x) = (x + 1/x)/2 − 1, which is zero when x = 1 and grows as x moves away from 1. The module under discussion defines a domain cost as J(m/e), the cost of the ratio between two masses m and e, and then proves three general facts about it.

The first theorem states that the cost is zero when the two masses are equal, which is the statement that no recognition effort is needed for a ratio of one. The second states that the cost is never negative when both masses are positive, so the cost cannot go below zero. The third proves that the number φ − 3/2 is positive, where φ is the golden ratio, approximately 1.618; this number is named the canonical threshold. These three facts are proved in the machine-checked library of formal theorems, and they hold for any positive real inputs m and e.

What the module does not do is connect these three facts to elastic scattering. The definition of domainCost as J(m/e) makes no reference to a cross-section, a radius, or any scattering quantity. The docstring records a research note that the quantum correction to the hard-sphere cross-section might begin at a certain value of ka, but the note is explicitly not a result. The Lean code proves the three general facts and nothing about this subject. The physics of elastic scattering from the cost function remains a target, not a theorem.

The practical consequence is a clear separation between what is established and what is hoped. The three proven facts are available as a foundation, but they are not yet a scattering theory. A reader who wants a prediction for a cross-section will not find one here. The module is honest about that: it states the general cost properties, and it labels the scattering idea as a plan, not a proof.

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 module does not prove any formula for a scattering cross-section, classical or quantum. The research note's specific value ka = J(φ)^(-1) ≈ 8.47 is not a theorem and has no derivation in the module. The cost function's three general properties are not specific to elastic scattering and apply to any positive ratio.

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/ElasticScattering_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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: