Foundation Dalembert Curvature Gate Gcosh Satisfies Hyperbolic

The curvature gate

The hyperbolic cosine, written cosh(t), is the average of e^t and e^-t. Subtract 1 from it and you get a curve that starts at 0, rises slowly, then bends upward ever faster. Its second derivative, the rate at which its slope changes, equals the curve itself plus 1. That property is the hyperbolic differential equation G''(t) = G(t) + 1.

In the Recognition Science framework, this equation is a gate. The framework models comparison as a ledger, a discrete record of recognition events, and represents the cost of a comparison by a function G(t) in log coordinates. The curvature of the metric derived from G distinguishes three geometries: flat, where G'' = 1; hyperbolic, where G'' = G + 1; and spherical, where G'' = -(G + 1). The framework proves that G(t) = cosh(t) - 1 satisfies the hyperbolic equation, and that it is not flat. The spherical candidate, G(t) = 1 - cos(t), satisfies its own equation but becomes negative, which the framework rules out because costs cannot be negative.

This is a proved theorem in the framework's machine-checked library of formal theorems. The declaration Gcosh_satisfies_hyperbolic establishes exactly that one fact: the function cosh(t) - 1 obeys the hyperbolic differential equation at every real number t. It does not claim that this function is the unique solution to that equation, nor that the hyperbolic case is the only one that survives the curvature gate. The framework's main theorem, curvature_gate_main, does more: it assumes smoothness, normalization, calibration, evenness, non-negativity, and constant curvature, and concludes that only flat or hyperbolic survive. That larger result is not what Gcosh_satisfies_hyperbolic alone proves.

The consequence is that the hyperbolic curve is a live candidate for the cost of comparison, while the flat curve is the counterexample that the framework rejects. The spherical curve is eliminated by non-negativity. What remains is a choice between flat and hyperbolic, and the hyperbolic curve is the one the framework identifies with the recognition cost law.

/-- Gcosh satisfies the hyperbolic ODE: G''(t) = G(t) + 1 = cosh(t). -/
theorem Gcosh_satisfies_hyperbolic : SatisfiesHyperbolicODE Gcosh := by
  intro t
  -- G(t) = cosh(t) - 1, G'(t) = sinh(t), G''(t) = cosh(t)
  have h1 : deriv Gcosh = Real.sinh := by
    ext s
    unfold Gcosh
    rw [deriv_sub_const, Real.deriv_cosh]
  have h2 : deriv (deriv Gcosh) t = Real.cosh t := by
    rw [h1, Real.deriv_sinh]
  rw [h2]
  unfold Gcosh
  ring

/-- Gcosh is NOT flat. -/
theorem Gcosh_not_flat : ¬ SatisfiesFlatODE Gcosh := by
  intro h
  have h1 := h 1
  -- deriv (deriv Gcosh) 1 = cosh(1) ≈ 1.54
  have hhyp := Gcosh_satisfies_hyperbolic 1
  simp only [Gcosh] at hhyp
  rw [h1] at hhyp
  -- 1 = cosh(1) - 1 + 1 = cosh(1)
  -- But cosh(1) > 1
  have hcosh1 : Real.cosh 1 > 1 := Real.one_lt_cosh.mpr (by norm_num : (1 : ℝ) ≠ 0)
  linarith

/-- The spherical solution violates non-negativity. -/
theorem Gspher_violates_nonnegativity : ¬ IsNonNegativeG Gspher := by
  intro h
  have := h Real.pi
  have hneg := Gspher_negative_at_pi
  linarith

What this page does not claim

The declaration does not prove uniqueness of the hyperbolic solution. The declaration does not rule out the flat case; that requires the full curvature_gate_main theorem. The declaration does not establish that the hyperbolic case is the actual cost law; it only shows it satisfies the equation.

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/Foundation/DAlembert/CurvatureGate.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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