Cpm Law Of Existence Cproj Eq Two From J Normalization

The normalization theorem

The declaration cproj_eq_two_from_J_normalization is a theorem in the machine-checked library of formal theorems. It states that if the second derivative of a certain cost function, evaluated at a specific point, equals 1, then a constant called Cproj (the projection constant) is forced to be 2. The cost function in question is the recognition cost, which measures the price of a recognition event. The theorem is a formal statement about a specific normalization choice, not a claim about the physical world by itself.

The theorem is proved by a direct computation. The hypothesis is that the second derivative of the function t ↦ J(e^t) at t = 0 is 1. The conclusion is that the projection constant Cproj equals 2. The proof is a one-liner in the library, using the definition of the cone constants. The theorem does not derive the value 2 from first principles; it shows that if you accept the normalization condition, then the constant must be 2. This is a conditional statement, not an unconditional derivation.

The normalization condition itself is a definitional choice. The library also contains a companion theorem, cproj_from_J_second_deriv, which states both that Cproj = 2 and that the second derivative condition holds. This companion theorem is a conjunction of the normalization condition and the conclusion, so it is a stronger statement. The theorem cproj_eq_two_from_J_normalization is the one-way implication: the normalization forces the constant.

The significance of the constant 2 is that it appears in the Coercive Projection Method (CPM), a framework for analyzing how well a model can be approximated by a subspace. The constant Cproj appears in inequalities that bound the defect, or error, of a projection. The value 2 is a specific number that makes certain bounds tight. The theorem is a formal link between a normalization condition on a cost function and a constant that appears in a projection inequality.

What the theorem does not claim is important. It does not claim that the normalization condition is physically necessary. It does not claim that Cproj = 2 is a law of nature. It does not claim that the cost function J is the only possible cost function. The theorem is a formal implication: if you assume the normalization, then the constant is 2. The assumption is a choice, not a consequence of the theorem itself.

/-- Minimal justification: under the RS J-normalization, the Hermitian
rank-one projection constant exported by RS equals 2. (The detailed
Hermitian bound is proved in domain files; here we record the value and
the normalization that fixes it.) -/
theorem cproj_eq_two_from_J_normalization
  (_hJ : deriv (deriv (fun t : ℝ => IndisputableMonolith.Cost.Jcost (Real.exp t))) 0 = 1) :
  coneConstants.Cproj = 2 := by
  simp [cone_Cproj_eq_two]

/-- Minimal justification: under the RS J-normalization, the Hermitian
rank-one projection constant exported by RS equals 2. (The detailed
Hermitian bound is proved in domain files; here we record the value and
the normalization that fixes it.) -/
theorem cproj_eq_two_from_J_normalization
  (_hJ : deriv (deriv (fun t : ℝ => IndisputableMonolith.Cost.Jcost (Real.exp t))) 0 = 1) :
  coneConstants.Cproj = 2 := by
  simp [cone_Cproj_eq_two]

/-- C_proj = 2 follows from the J-cost second derivative normalization.

The Hermitian rank-one bound ‖Pψ‖² ≤ C_proj · ‖ψ‖² has optimal constant
C_proj = 2 when the projection is normalized so that J''(1) = 1 in
log-coordinates. This is the content of `RS.Jcost_log_second_deriv_normalized`. -/
theorem cproj_from_J_second_deriv :
    RS.coneConstants.Cproj = 2 ∧
    deriv (deriv (fun t : ℝ => IndisputableMonolith.Cost.Jcost (Real.exp t))) 0 = 1 :=
  ⟨rfl, RS.Jcost_log_second_deriv_normalized⟩

What this page does not claim

The theorem does not claim that the normalization condition is physically necessary. The theorem does not claim that Cproj = 2 is a law of nature. The theorem does not claim that the cost function J is the only possible cost function.

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/CPM/LawOfExistence.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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