Cpm Law Of Existence Energy Gap Ge Cmin Mul Defect

The energy gap floor

In the Recognition Science framework, a ledger (a discrete record of events) measures how far a candidate state is from being a perfect match. That distance is the defect: the mass of the components that do not fit. The framework also tracks an energy gap, the cost of moving from the current state to the next allowed one. The theorem energyGap_ge_cmin_mul_defect proves that the energy gap is always at least a fixed positive constant times the defect, whenever the three model constants are positive. In plain language: the worse the mismatch, the larger the minimum cost to escape it, and the constant is the same for every state in the model.

The constant, called cmin, is the reciprocal of the product of three model constants: Knet (a covering factor), Cproj (a projection bound), and Ceng (an energy control factor). For the framework's canonical cone model, those three constants are 1, 2, and 1, so cmin is 1/2. For the eight-tick model, Knet is refined to (9/7)^2, so cmin becomes 49/162. The theorem itself is abstract: it holds for any model that satisfies the framework's three defining inequalities, not for one specific choice of constants. The proof is a short algebraic chain from the projection-defect inequality and the energy-control inequality, and it is checked by the framework's machine-checked library of formal theorems.

The theorem does not claim that any particular physical system has a nonzero energy gap. It does not identify which states exist, what the constants are for a given material, or whether a gap is large enough to matter. It only states the conditional relationship: if the constants are positive and the model's inequalities hold, then the gap is bounded below by cmin times the defect. The positivity condition is essential; if any constant is zero or negative, the bound can fail. The theorem also does not say the bound is tight. The actual gap may be much larger than cmin times the defect; the theorem only guarantees it is not smaller.

What the result changes is the shape of the framework's reasoning. It turns a qualitative idea, that a bad match costs energy, into a quantitative floor that any concrete model must respect. When a later model plugs in its constants, this theorem gives a ready-made lower bound on the energy gap, which can then be compared against measurements or used to rule out configurations whose gap would fall below the floor. The bound is a tool for checking consistency, not a prediction of a specific gap value.

/-- Coercivity in the usual “energy gap ≥ c_min · defect” form.

Requires the product `K_net · C_proj · C_eng` to be strictly positive to
invert safely. -/
theorem energyGap_ge_cmin_mul_defect
  (M : Model β)
  (hpos : 0 < M.C.Knet ∧ 0 < M.C.Cproj ∧ 0 < M.C.Ceng)
  (a : β) :
  M.energyGap a ≥ cmin M.C * M.defectMass a := by
  have h := M.defect_le_constants_mul_energyGap a
  have hprodpos : 0 < M.C.Knet * M.C.Cproj * M.C.Ceng := by
    have := mul_pos (mul_pos hpos.1 hpos.2.1) hpos.2.2
    simpa [mul_assoc] using this
  -- From h: D ≤ (K·C·E)·gap, multiply both sides by (K·C·E)⁻¹
  -- Result: (K·C·E)⁻¹·D ≤ gap, i.e., c_min·D ≤ gap
  have hinv : (M.C.Knet * M.C.Cproj * M.C.Ceng)⁻¹ * (M.C.Knet * M.C.Cproj * M.C.Ceng) = 1 := by
    exact inv_mul_cancel₀ (ne_of_gt hprodpos)
  calc cmin M.C * M.defectMass a
      = (M.C.Knet * M.C.Cproj * M.C.Ceng)⁻¹ * M.defectMass a := by rfl
    _ ≤ (M.C.Knet * M.C.Cproj * M.C.Ceng)⁻¹ * ((M.C.Knet * M.C.Cproj * M.C.Ceng) * M.energyGap a) := by
        apply mul_le_mul_of_nonneg_left h
        exact le_of_lt (inv_pos.mpr hprodpos)
    _ = ((M.C.Knet * M.C.Cproj * M.C.Ceng)⁻¹ * (M.C.Knet * M.C.Cproj * M.C.Ceng)) * M.energyGap a := by ring
    _ = 1 * M.energyGap a := by rw [hinv]
    _ = M.energyGap 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

/-- The eight-tick coercivity constant is 49/162. -/
theorem c_value_eight_tick : cmin eightTickConstants = 49/162 := by
  simp [cmin, eightTickConstants]
  norm_num

/-- (AB) Coercivity link: `D ≤ (K_net·C_proj·C_eng) · (E−E_0)`.

This is the forward direction combining A + energy control.
We deliberately avoid dividing by the product, to keep sign issues out
of the core inequality. -/
theorem defect_le_constants_mul_energyGap
  (M : Model β) (a : β) :
  M.defectMass a ≤ (M.C.Knet * M.C.Cproj * M.C.Ceng) * M.energyGap a := by
  have hA : M.defectMass a ≤ M.C.Knet * M.C.Cproj * M.orthoMass a :=
    M.projection_defect a
  have hB : M.orthoMass a ≤ M.C.Ceng * M.energyGap a :=
    M.energy_control a
  calc M.defectMass a
      ≤ M.C.Knet * M.C.Cproj * M.orthoMass a := hA
    _ ≤ M.C.Knet * M.C.Cproj * (M.C.Ceng * M.energyGap a) := by
        apply mul_le_mul_of_nonneg_left hB
        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.Ceng) * M.energyGap a := by ring

What this page does not claim

The theorem does not claim that any particular physical system has a nonzero energy gap. The theorem does not claim that the bound is tight or that the gap equals cmin times the defect. The theorem does not claim that the constants Knet, Cproj, and Ceng are derived from first principles for any specific material.

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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