Cpm Law Of Existence C Value Derivation

The derived constant

The Coercive Projection Method (CPM) is a general framework for proving that a system's energy gap, the energy needed to escape a stable state, is bounded below by a positive multiple of its defect mass, the amount of "wrongness" or deviation from that state. The framework works with three abstract constants: K_net, C_proj, and C_eng. Their product, inverted, gives c_min, the largest universal multiplier that makes the energy gap bound hold. The theorem c_value_derivation computes this multiplier for a specific, concrete choice of constants.

That choice comes from the framework's eight-tick recognition cycle, a discrete sequence of eight steps the framework models as the basic unit of recognition. For this cycle, the framework sets K_net = (9/7)², a value derived from a covering argument in three dimensions, C_proj = 2, and C_eng = 1. Plugging these into the definition of c_min yields the theorem's conclusion: c_min = 49/162, or about 0.3025. The proof is a direct computation, verified by the machine-checked library of formal theorems.

What the declaration does not claim is just as important. It does not claim that these constants are the only possible ones, nor that they are forced by the framework's axioms. The framework's core theorems establish that a cost function must take a specific form, but the mapping from that cost to the concrete values of K_net, C_proj, and C_eng is a modeling choice, recorded in the eightTickConstants definition. The theorem's proof only verifies the arithmetic of that choice; it does not justify the choice itself.

The result earns its place because it turns an abstract inequality into a concrete number. With c_min = 49/162, any system that satisfies the CPM axioms with these constants has a guaranteed energy gap at least 0.3025 times its defect mass. This is a sharp, checkable prediction that the framework can carry forward into specific physical models, and it is the kind of exact output that separates a derived constant from a fitted parameter.

/-- Explicit computation: c_min = 1 / (K_net · C_proj · C_eng)
    = 1 / ((81/49) · 2 · 1) = 49 / 162. -/
theorem c_value_derivation :
    (1 : ℝ) / ((9/7)^2 * 2 * 1) = 49/162 := by
  norm_num

/-- CPM constants bundle for eight-tick geometry. -/
noncomputable def eightTickConstants : Constants := {
  Knet := (9/7)^2,
  Cproj := 2,
  Ceng := 1,
  Cdisp := 1,
  Knet_nonneg := by norm_num,
  Cproj_nonneg := by norm_num,
  Ceng_nonneg := by norm_num,
  Cdisp_nonneg := by norm_num
}

/-- 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

What this page does not claim

The eight-tick constants are not forced by the framework's axioms; they are a recorded modeling choice. The theorem does not establish that c_min = 49/162 is the value for any system outside the eight-tick model. The declaration does not derive the constants from the J cost function; it only uses their assigned values.

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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