Cpm Law Of Existence Defect Le Constants Mul Tests

The defect bound

In Recognition Science, a ledger (a discrete record of events) tracks how well a system recognizes its own states. The declaration defect_le_constants_mul_tests is a theorem in the framework's machine-checked library of formal theorems. It states that for any model in the Coercive Projection Method, the defect mass (the cost of a failed recognition) is always at most a fixed constant multiplied by the number of tests performed. In symbols: D ≤ (K_net · C_proj · C_disp) · T, where D is defect mass, T is the test count, and the three constants come from the model's structure.

The theorem is a bound, not an equality. It says a system cannot hide its failures: if you run many tests, the total defect must stay within a predictable envelope. The three constants have plain meanings. K_net measures how much the system's network structure amplifies errors. C_proj is a projection constant, set to 2 in the standard Recognition Science normalization, which comes from the second derivative of the cost function at its minimum. C_disp is a dispersion constant, capturing how much the system's spread across states contributes. The product of these three, multiplied by the test count, gives the ceiling.

The theorem is proved in the Lean 4 proof assistant, meaning every step is checked by the kernel. It is axiom-clean, using only the three standard axioms of the underlying type theory. The proof is short: it chains three inequalities already in the model structure, namely that defect is bounded by projection, projection by energy gap, and projection by tests. The theorem is a composition of these, not a new physical law.

What the theorem does not claim is just as important. It does not say the bound is tight, nor that any particular model achieves it. It does not assign numerical values to the constants; those come from separate derivations, such as the cone model where K_net = 1, C_proj = 2, C_eng = 1, C_disp = 1, giving a cmin of 1/2. It does not say that tests are the only way to control defect; the energy gap provides an alternative bound. The theorem is a structural guarantee, not a recipe.

For a reader, the consequence is practical: in any Recognition Science model, the cost of errors is not arbitrary. It is controlled by a product of three constants and the number of tests. This makes the framework's claims about recognition costs checkable, and it is a step toward the larger result that the cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions.

/-- (AC) Aggregation: `D ≤ (K_net·C_proj·C_disp) · sup_W T[a]`.

Combines A + dispersion/interface without measure‑theoretic details. -/
theorem defect_le_constants_mul_tests
  (M : Model β) (a : β) :
  M.defectMass a ≤ (M.C.Knet * M.C.Cproj * M.C.Cdisp) * M.tests a := by
  have hA : M.defectMass a ≤ M.C.Knet * M.C.Cproj * M.orthoMass a :=
    M.projection_defect a
  have hD : M.orthoMass a ≤ M.C.Cdisp * M.tests a :=
    M.dispersion a
  calc M.defectMass a
      ≤ M.C.Knet * M.C.Cproj * M.orthoMass a := hA
    _ ≤ M.C.Knet * M.C.Cproj * (M.C.Cdisp * M.tests a) := by
        apply mul_le_mul_of_nonneg_left hD
        have h₁ : 0 ≤ M.C.Knet := M.C.Knet_nonneg
        have h₂ : 0 ≤ M.C.Cproj := M.C.Cproj_nonneg
        exact mul_nonneg h₁ h₂
    _ = (M.C.Knet * M.C.Cproj * M.C.Cdisp) * M.tests a := by ring

/-- (AC) Aggregation: `D ≤ (K_net·C_proj·C_disp) · sup_W T[a]`.

Combines A + dispersion/interface without measure‑theoretic details. -/
theorem defect_le_constants_mul_tests
  (M : Model β) (a : β) :
  M.defectMass a ≤ (M.C.Knet * M.C.Cproj * M.C.Cdisp) * M.tests a := by
  have hA : M.defectMass a ≤ M.C.Knet * M.C.Cproj * M.orthoMass a :=
    M.projection_defect a
  have hD : M.orthoMass a ≤ M.C.Cdisp * M.tests a :=
    M.dispersion a
  calc M.defectMass a
      ≤ M.C.Knet * M.C.Cproj * M.orthoMass a := hA
    _ ≤ M.C.Knet * M.C.Cproj * (M.C.Cdisp * M.tests a) := by
        apply mul_le_mul_of_nonneg_left hD
        have h₁ : 0 ≤ M.C.Knet := M.C.Knet_nonneg
        have h₂ : 0 ≤ M.C.Cproj := M.C.Cproj_nonneg
        exact mul_nonneg h₁ h₂
    _ = (M.C.Knet * M.C.Cproj * M.C.Cdisp) * M.tests a := by ring

/-- RS cone coercivity constant is 1/2. -/
theorem c_value_cone : cmin RS.coneConstants = 1/2 := by
  simp only [cmin, RS.cone_Knet_eq_one, RS.cone_Cproj_eq_two, RS.cone_Ceng_eq_one]
  norm_num

What this page does not claim

The theorem does not claim the bound is tight or achievable. The theorem does not claim tests are the only way to control defect. The theorem does not derive the numerical values of the constants.

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)

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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