Physics Foundation

The foundation module

Physics in Recognition Science starts from a single idea: reality keeps a ledger, a discrete record of recognition events, and the cost of each recognition is forced by the mathematics itself. The foundation module is the formal seed of that idea. It defines a cost function, J, that measures the price of recognizing one quantity against another, and it proves three plain facts about that cost: it is zero when the two quantities are equal, it is never negative when both quantities are positive, and a certain threshold value built from the golden ratio is positive.

The cost function itself is J(x) = (x + 1/x)/2 - 1. This is not a guess. A separate theorem, proved in the machine-checked library of formal theorems, shows that any cost function satisfying five plain conditions must equal this exact form. The foundation module takes that general result and applies it to physics by defining the cost of a mass m against a reference mass e as J(m/e). The three proved facts then follow directly: the cost vanishes when m equals e, it stays nonnegative for positive masses, and the threshold phi - 3/2 is positive.

What the module does not do is just as important. It does not say what m and e are. The module defines the cost as J(m/e) without tying m or e to any particular particle, field, or interaction. The docstring is explicit about this: the module proves nothing specific to physics, because the definition does not reference a specific subject. The structure W_Boson_Mass_RS4 and its certificate cert merely package the three general facts into a single object. The name suggests a W boson mass application, but the content is universally quantified and shared verbatim with 2383 sibling modules.

In plain language, the foundation module is a template. It establishes the grammar of cost in physics: what a cost function must satisfy to be coherent. The actual physics, the identification of m and e with real particles, is left open. That is the honest state of the framework at this layer. The module proves the skeleton, and the flesh, the specific definitions of m and e, remains a target for future work.

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 module does not derive the W boson mass. The module does not identify m and e with any specific particle or field. The module does not prove that the cost function is the only possible one; that is a separate theorem.

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/Foundation.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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