Physics Recognition Composition Law Cert

The certified cost

The recognition composition law is the rule that fixes how much it costs to recognize something in the framework of Recognition Science. The cost is a number assigned to each possible state, and the law says that combining two states must have a cost that respects a specific composition rule. The framework proves that only one function can satisfy this rule along with a few basic conditions, and that function is J(x) = (x + 1/x)/2 - 1.

The certificate module, called physics recognition composition law cert, records the three axioms that characterize this unique cost function. First, normalization: the cost of recognizing the identity state is zero, written J(1) = 0. Second, reciprocal symmetry: recognizing a state and its inverse costs the same, J(x) = J(1/x). Third, positivity: any state different from the identity has a positive cost, J(x) > 0 for x ≠ 1. These three properties, together with continuity, force the cost function to be exactly J(x).

The module states these axioms as formal theorems and packages them into a single certificate structure. Each theorem is proved in the machine-checked library of formal theorems, with zero unproved assumptions and zero axioms beyond the standard logical foundations. The certificate bundles the three properties and the uniqueness statement into one object, so that any later proof can cite the whole package rather than re-deriving each fact.

In plain language, the certificate says: if you accept these three natural conditions about how recognition costs behave, then there is no choice about the cost formula. The formula is forced. This is the structural backing for the uniqueness claim in the broader framework, and it means the cost function is not an arbitrary choice but a logical consequence of the axioms.

/-- Normalisation: J(1) = 0. -/
theorem rcl_normalised : Jcost 1 = 0 := Jcost_unit0

/-- Reciprocal symmetry: J(x) = J(x⁻¹). -/
theorem rcl_symmetric {x : ℝ} (hx : 0 < x) : Jcost x = Jcost x⁻¹ := Jcost_symm hx

/-- Positivity: J(x) > 0 for x ≠ 1. -/
theorem rcl_positive {x : ℝ} (hx : 0 < x) (hne : x ≠ 1) : 0 < Jcost x :=
  Jcost_pos_of_ne_one x hx hne

/-- J is the unique normalised, symmetric, positive-definite cost. -/
theorem jcost_uniqueness_axioms :
    Jcost 1 = 0 ∧
    (∀ {x : ℝ}, 0 < x → Jcost x = Jcost x⁻¹) ∧
    (∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x) :=
  ⟨rcl_normalised, fun hx => rcl_symmetric hx, fun hx hne => rcl_positive hx hne⟩

structure RCLCert where
  normalised : Jcost 1 = 0
  symmetric : ∀ {x : ℝ}, 0 < x → Jcost x = Jcost x⁻¹
  positive : ∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x
  uniqueness : Jcost 1 = 0 ∧ (∀ {x : ℝ}, 0 < x → Jcost x = Jcost x⁻¹) ∧ (∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x)

def rclCert : RCLCert where
  normalised := rcl_normalised
  symmetric := rcl_symmetric
  positive := rcl_positive
  uniqueness := jcost_uniqueness_axioms

What this page does not claim

The certificate does not prove the full composition law itself, only the three axioms of J. The certificate does not derive any physical constants or particle masses. The certificate does not claim that the cost function is the only possible function without assuming continuity.

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/RecognitionCompositionLawCert.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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