Foundation Dalembert Curvature Gate Curvature Gate Dichotomy

The curvature gate

In the Recognition Science framework, the central object is a cost function, a measure of how expensive it is for reality to recognize one state from another. The framework's library, a machine-checked collection of formal theorems, studies this cost through a change of variables that turns it into a geometric metric. The curvature gate is the requirement that this metric have constant nonzero curvature, a condition that separates three possible geometries: flat, hyperbolic, and spherical.

The theorem curvature_gate_dichotomy proves that under two structural assumptions, normalization at zero and a calibration condition on the second derivative, the spherical case is impossible. The proof is direct: the spherical solution G(t) = 1 - cos(t) is nonpositive everywhere, violating the requirement that the cost be nonnegative. The theorem's conclusion is a disjunction: the metric satisfies either the flat equation G''(t) = 1 or the hyperbolic equation G''(t) = G(t) + 1. The flat case corresponds to G(t) = t²/2, where comparisons are independent, and the hyperbolic case to G(t) = cosh(t) - 1, where comparisons are entangled and nearby states diverge exponentially.

The framework's library proves the three candidate solutions satisfy their respective equations, and that the spherical candidate fails the nonnegativity condition at t = π. The dichotomy theorem then assembles these facts: given the structural axioms, the spherical branch is ruled out, leaving flat and hyperbolic as the only possibilities. This is a formal result, checked by a proof assistant, not a numerical approximation or a heuristic argument.

What the theorem does not claim is equally important. It does not prove that the hyperbolic case is the correct one; that requires an additional argument about why flat geometry is excluded, which the theorem does not supply. It does not establish that the cost function itself must have constant curvature; that is an assumption, formalized as the hypothesis of the theorem. And it does not derive the value of the golden ratio or any other physical constant; those results live elsewhere in the framework's library. The dichotomy narrows the space of possibilities, it does not select the unique answer.

/-- The curvature gate: spherical is ruled out by calibration, leaving only flat or hyperbolic. -/
theorem curvature_gate_dichotomy (G : ℝ → ℝ)
    (hNorm : G 0 = 0)
    (hCalib : deriv (deriv G) 0 = 1)
    (hConstCurv : SatisfiesFlatODE G ∨ SatisfiesHyperbolicODE G ∨ SatisfiesSphericalODE G) :
    SatisfiesFlatODE G ∨ SatisfiesHyperbolicODE G := by
  rcases hConstCurv with hFlat | hHyp | hSpher
  · left; exact hFlat
  · right; exact hHyp
  · exfalso
    have hcalib_spher := hSpher 0
    rw [hNorm] at hcalib_spher
    simp at hcalib_spher
    rw [hCalib] at hcalib_spher
    norm_num at hcalib_spher

/-- The spherical solution becomes negative (cos(t) - 1 ≤ 0, and < 0 for t ≠ 2πk). -/
theorem Gspher_nonpositive : ∀ t : ℝ, Gspher t ≤ 0 := by
  intro t
  simp only [Gspher]
  have : Real.cos t ≤ 1 := Real.cos_le_one t
  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

/-- Gquad satisfies the flat ODE: G''(t) = 1. -/
theorem Gquad_satisfies_flat : SatisfiesFlatODE Gquad := by
  intro t
  -- G(t) = t²/2, G'(t) = t, G''(t) = 1
  have h1 : deriv Gquad = fun t => t := by
    ext s
    unfold Gquad
    have hd : HasDerivAt (fun t => t ^ 2 / 2) s s := by
      have := hasDerivAt_pow 2 s
      simp only [Nat.cast_ofNat, pow_one] at this
      have h := this.div_const 2
      convert h using 1
      ring
    exact hd.deriv
  have h2 : deriv (deriv Gquad) t = 1 := by
    rw [h1]
    simp only [deriv_id'']
  exact h2

/-- 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

What this page does not claim

The theorem does not prove that the hyperbolic case is the correct one. The theorem does not establish that the cost function must have constant curvature; that is an assumption. The theorem does not derive the golden ratio or any other physical constant.

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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